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KEHOE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 ) 2007 Joseph J. Kehoe Dedicated to Sarah ACKNOWLEDGMENTS I would first like to acknowledge Dr. Rick Lind for providing steady guidance throughout my graduate education experience. His advice, support, and criticisms have ah,v kept my career interests in mind, and have prepared me to (finally) leave the safe confines of the education system for the i I! world." I thank the other members of my committee, Dr. Carl Crane, Dr. Warren Dixon, Dr. Peter Ifju, and Dr. Clint Slatton for taking the time to review my research and provide comments on the directions of my ideas. Additionally, thanks go out to the members of flight control lab for providing alhv  needed diversions from work, and for the countless indepth discussions related to nothing in particular. Sometimes steps in wrong direction are needed to make leaps in the right direction. I have learned things in the past four years that I never would have imagined learning in graduate school. Specifically, I thank M1uli !d Abdulrahim for greatly expand ing my knowledge regarding rally racing, rubber chickens, and unicycles. Adam Watkins deserves credit for teaching me the joy that can come from the right combination of cyn icism and sarcasm. I would also like to thank Daniel "Tex" Grant for teaching me that meatloaf made from venison tastes surprisingly just like regular meatloaf. I would also like to acknowledge Air Force Reasearch Labs, Munitions Directorate, specifically Johnny Evers, Neal Glassman, Sharon Heise, and Robert Sierakowski, for continued financial support. Thanks go to Dr. Jason Stack for providing additional funding as well as for his guidance during the two summers I spent as an intern at the N i,1 I Surface Warfare Center in Panama City, FL. Finally, I would like to thank my family and friends for their patience in dealing with me during these busy past few years. This work would not have been possible without the love and support of my parents, Joseph and Linda, and my sister, Kathleen. I would like to thank my Fatherinlaw, Bob Williams, for being .i.v li willing to talk shop, and my Motherinlaw, Pat Williams, for alv,v putting up with it. I owe my wife, Sarah, a great debt of gratitude for her patience, ( ,iii, and understanding throughout this process. On this note, her equine companion, Sundance, deserves my thanks for keeping her occupied during the times that I was unable. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. LIST O F TABLES . . . . . . . . . . LIST OF FIGURES . . . . . . . . . A B ST R A C T . . . . . . . . . . CHAPTER 1 INTRODUCTION .................................. Motivation ...... Problem Description Problem Statement Approach Overview Contributions .. 2 M OTION PLANNING ................................ 2.1 Introduction . . . . . . . . . 2.2 Path Constraints . . . . . . . . 2.2.1 Differential Constraints ......................... 2.2.2 Obstacle Constraints .......................... 2.3 O ptim al Control . . . . . . . . 2.3.1 Indirect Optimal Control .. .................... 2.3.2 Direct Optimal Control .. ..................... 2.4 Hybrid M otion M odels .. ........................ 2.4.1 2.4.2 2.4.3 2.4.4 Modeling with Motion Motion Planning with Kinematic Example Dynamic Example . Primitives Primitives. 3 RANDOMIZED SAMPLINGBASED MOTION PLANNING Introduction . . . . . Probabilistic Roadmap Methods (PRM) ... Random Dense Tree Methods (RDT)...... 3.3.1 RapidlyExploring Random Trees (RRT) 3.3.2 ExpansiveSpaces Trees (EST) ...... 3.3.3 Discussion . . . . . ..............................: 3.4 RDTBased Dynamic Planner for a PlanarMotion Vehicle . ... 68 3.4.1 Model ................... . . .. 69 3.4.2 Overview ................... . 73 3.4.3 Node Selection ............... ........... .. 74 3.4.4 Node Expansion ............... .......... .. 76 3.4.5 Solution C'!I .: ............................. .78 3.5 Exam ple . . . . . . . . .... 79 4 SENSING EFFECTIVENESS .................. ......... .. 85 4.1 Introduction .................. ................ .. 85 4.2 Remote Sensor Technologies ............... ..... .. 86 4.2.1 Computer Vision ............... ......... 87 4.2.2 Radar ................ ........... .. .. 87 4.2.3 Sonar . ............... ............ .. .. 88 4.2.4 Ladar ................... . . .. 89 4.3 Modeling the Sensing Task ............... ........ .. 90 4.3.1 Sensing Geometry ............... ......... .. 90 4.3.2 Visibility .. ... .. .. .. .. ... .. . . . 92 4.3.3 The Visibility Set ............... ......... .. 95 4.3.4 Proximity Effects .................. ......... .. 97 4.4 Effectiveness Metric ............... ........... .. 101 4.4.1 Formulation ............... ........... 102 4.4.2 The Quality Set ............... .......... 104 4.4.3 Sensing Mission Effectiveness .................. ... 104 4.4.4 Example: A Contrived Metric ............... .. 106 4.4.5 Example: Image Area .................. ..... 110 5 RANDOMIZED SENSOR PLANNING .................. .. .. 115 5.1 Introduction ................... . . 115 5.2 Environment Representation ................... . .. 117 5.3 A Randomized SensorPlanning Algorithm ..... . . ..... 120 5.3.1 Overview .................. .............. 120 5.3.2 Node Selection ... .. .. .. ... .. .. .. ... ... .. 121 5.3.3 Vantage Point Selection .................. ..... 123 5.3.4 Local Planning and Expansion .................. .. 125 5.3.5 Evaluation .................. ............. 126 5.4 Exam ples .. .. ... .. .. .. .. .. .. .. .. ... .. ... .. 127 5.4.1 M odel .. ... .. .. .. .. ... .. .. ... .. ..... 127 5.4.2 Effectiveness M etric .................. ........ 129 5.4.3 ObstacleFree Examples .................. .... 130 5.4.4 Examples with Obstacles .................. .. ... 136 6 OPTIMAL SENSOR PLANNING ................... ...... 143 6.1 Introduction .................. ................ .. 143 6.2 System .................. ................... .. 144 6.3 Problem Formulation .................. ........... 146 6.4 Variational Approach .................. ........... 149 6.4.1 Necessary Conditions .................. .. ..... 150 6.4.2 Boundary Value Problem .................. .. ... 153 6.5 Direct Transcription Approach .................. .... 155 6.5.1 Numerical Example .................. ........ 156 7 CONCLUSION .................. ................. .. 161 REFERENCES ........................... . . 165 BIOGRAPHICAL SKETCH ........... ........ . ... 175 LIST OF TABLES Table page 21 /AM components for each maneuver .................. ..... .. 57 41 Statistics for visibility parameter comparison ................ 101 LIST OF FIGURES Figure page 11 Prevalent examples of advancements in unmanned technology . .... 16 12 Comparison of sensing data for different mission scales . ..... 18 13 Touring a series of cells for a largescale sensing mission . 19 14 Touring a series of cells for a smallscale sensing mission . . 20 15 Motion planning to view a sequence of targets .............. .. 21 16 Sensing mission tasks .................... . 24 21 Obstacle boundary approximations ................ ...... 34 22 Vertexangle sum collision detection method ............. .. 35 23 Vertex edgevector collision detection method .................. 35 24 Obstacle expansion for pointwise safety .................. .. 36 25 Automaton model representation as a directed graph .............. ..45 26 Special reference frame, D for Dubins path solution ............... .51 27 Sample Dubins paths .................. ............ .. .. 54 28 Example state trajectory showing maneuver dynamics ............. ..56 29 Top view of environment for dynamic example .................. 58 210 Top view of solution for M {MSL, MLS, MSL} ..... . . 59 211 Heading and turn rate trajectories for M = {MSL, MLS, MSL} . ... 59 212 Top view of solution for M {MSL, MLS, MSL, MLS, MSL} . . ... 60 213 Heading and turn rate trajectories for M = {MSL, MLS, MSL, MLS, MSL} . 60 214 Top view of solution for M {MSL, MLS, MSR, MRS, MSR, MRs} ........ 61 215 Heading, turn rate trajectories for M = {MSL, MLS, MSR, MR, MSR, MRS} 61 31 The PRM algorithm ............... ............. 63 32 RRT algorithm .................. ................. .. 66 33 EST algorithm .................. ................. .. 67 34 Differences in exploration strategy for the RRT algorithm vs. the EST algorithm 68 35 Automaton structure for vehicle model used with current planning strategy 36 Distance function comparison . ........ 37 Distance function computation . . 38 Node expansion step . ............. 39 The new branch is subdivided to a set of nodes. . 310 Collisionfree solutions . ............ 311 Unique solution families for vehicle used in example . 312 Turnstraight solution sequences . ...... 313 314 315 316 41 42 43 44 Example planning environment . Incremental tree expansion . . Nodes in the final solution tree . Solution path refinement . . Sensing problem geometry . . Visibility parameters . ... Inverted visibility cone . . Construction of V . .... 45 Motion coupling effects for different problem scales 46 47 48 49 410 411 412 413 414 51 Range and incidence variation for different problem scales . Representative trajectories for sensor effectiveness metric . Quality parameter efficiency functions . ........ Simulated trajectory showing snapshots of effectiveness metric Maxvalue mission effectiveness . ............ Environment for image area sensing metric simulation . . Sensing effectiveness as a function of image area . .. Effectiveness trajectories . ............... Simulated mission effectiveness . ............ TSP problem comparison . ............... .. . 75 .. . 76 .. . 77 .. . 78 .. . 79 . 81 . 81 . 82 . 82 . 83 . 84 . 91 . 94 . 96 . 97 . 99 . 100 . 04 . 07 values . .. 109 . 110 . 112 . 113 . 114 . 114 . 115 52 53 54 55 56 57 58 59 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 61 62 63 Occlusion shadows . ................... 3D Occlusion shadow . ................. Discretization approaches for area coverage . ..... Sensing secondary targets . ............... Example node weighting function . ........... Vantage point reachability test . ... Quality parameter efficiency functions for examples . . Environments for obstaclefree sensor planning example . Trajectory refinement for obstaclefree reconnaissance . . Simulated effectiveness for obstaclefree reconnaissance . Trajectory refinement for obstaclefree surveillance . . Simulated effectiveness for obstaclefree surveillance . . Trajectory refinement for obstaclefree coverage . ... Simulated effectiveness for obstaclefree coverage . .. Environments for sensor planning example with obstacles . Initial tree for reconnaissance with obstacles . .... Trajectory refinement for reconnaissance with obstacles . Simulated effectiveness for reconnaissance with obstacles . Initial tree for surveillance with obstacles . ...... Trajectory refinement for surveillance with obstacles . . Simulated effectiveness for surveillance with obstacles . . Initial tree for coverage with obstacles . . Trajectory refinement for coverage with obstacles . .. Simulated effectiveness for coverage with obstacles . . Force coefficients . . . . . . . Environment for optimal sensing numerical example . . Solution position and velocity trajectories . . . 118 . 118 . 119 . 22 . 123 . 25 . 130 . . 131 . . 133 .. . 134 . 135 . 135 . 136 . 37 . . 138 . 138 . . 139 . . 139 . 40 . 141 . . 141 . 141 . 42 . 42 . 46 . 57 . 59 64 Solution a and o reference trajectories .................. ...... 159 65 Solution visibility parameter trajectories for each target . . ..... 160 66 Aircraft trajectory and sensor footprint over each target ............ .160 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRAJECTORY GENERATION FOR EFFECTIVE SENSING OF A CLOSE PROXIMITY ENVIRONMENT By Joseph J. Kehoe August 2007 ('C! i: Richard C. Lind, Jr. Major: Aerospace Engineering Unmanned systems stand to p1 iv a significant role in future sensing and information gathering applications. The scope of these scenarios is expanding to include those missions for which the sensor and carrier vehicle are forced to operate in close proximity to the surrounding environment. Several unique challenges are introduced for this class of sensing problems. First, the sensor projects a small footprint on environment surfaces relative to the scale of vehicle motion. Consequently, coupling exhibited between the motion of the vehicle and the direction of sensor pointing has a significant effect on gathered data. Second, view characteristics of sensed objects, such as resolution and aspect, can vary appreciably over the sensor fieldofview. These variations can have an adverse effect on data quality. Finally, vehicle maneuvering in close proximity to obstacles imposes significant risks to safe navigation. As such, consideration of vehicle dynamics within any motion planning strategy is critical to ensure that accurate and precise trajectories are generated. This dissertation addresses the problem of planning sensing trajectories through complex and cluttered environments. A randomized planning technique is developed which utilizes a hybrid modeling strategy to efficiently plan dynamicallyadmissible trajectories. Then, a generalized measure of sensing effectiveness is formulated to quantify the applicationspecific effects of sensor operation in close proximity to sensing targets. Finally, these elements are integrated into a randomized trajectory planner that ensures quality data collection regarding a specified set of targets while satisfying the system dynamics. The utility of the planner is demonstrated through several simulated examples. CHAPTER 1 INTRODUCTION 1.1 Motivation Unmanned systems technology has become an increasingly common presence in the world around us. In recent years, CNN has broadcast images and video that were captured by the Predator drone from behind enemy lines in Iraq and Afghanistan. An unexpectedlyrr__, 1 pair of robotic explorers have provided breathtaking views and unprecedented scientific data from the fourth planet from the Sun. A winner was declared in a race across 130 miles of desert in which no human drivers participated. These examples, shown in Figures 11A to 11C, represent a small crosssection of the diverse and growing field of unmanned vehicles. A B C Figure 11. Prevalent examples of advancements in unmanned technology. A) The RQ1 Predator. B) A mars rover. C) Stanford Racing's "Stanley" Of the expanding roles p1. i,, d by existing unmanned technologies, future vehicles stand to prove especially useful for applications that involve the gathering of information. Envisioned civiliandomain scenarios include border patrol, traffic monitoring, tactical law enforcement, maritime surveillance, and environmental sensing [1]. Likewise, intelligence, surveillance, and reconnaissance (ISR) missions are emerging as the dominant theme among projected military applications for future unmanned systems. The future combat paradigm is likely to include the use of cooperating teams of unmanned aerial vehicles (UAVs), unmanned ground vehicles (UGVs), and unmanned underwater vehicles (UUVs) equipped with various sensors that autonomc;ro;, navigate complex environments for the collection of ISR data [25]. Existing platforms typically p'1 ,i an ISR role from a standoff range. Such missions are characterized by sensing the environment from a significantly large distance relative to the dimensions of typical target surfaces. As such, a remote sensor with even a limited fieldofview (FOV) can cover a potentially large area from a single vantage point. For example, the Predator drone typically collects data from altitudes ranging from 15,000 20,000 ft. Figure 12A depicts the University of Florida campus as viewed from an altitude of approximately 15, 000 ft. The entire campus and surrounding area can be seen in a single image captured from a single vantage point. Advances in hardware and control technologies will soon enable mission descriptions on a much smaller scale in increasinglycomplex environments. For example, researchers foresee cooperative teams of UGVs and UAVs patrolling an urban peacekeeping environ ment [6]. N ,i i;, ii. capabilities for UAVs operating in such environments are already emerging through the development of miniature autopilot systems [7], visionbased control technologies [811], and agilityenhancing morphing strategies [1214]. Sensing missions in such a scenario are characterized by vehicle and sensor operation within close proximity to the environment. Figure 12B shows a simulated image of the University of Florida cam pus as viewed from a UAV operating at an altitude of approximately 50 ft. Typical data collected during such missions is obviously different in nature than that collected during standoff sensing missions. Smaller portions of the environment are covered from a partic ular viewpoint; however, the data is of much higher resolution and exhibits richer content. Moreover, additional sensor modalities, such as audio, gain relevance at close range. These key benefits expand possible mission scenarios to include tasks such as sensing under and inside structures. 1.2 Problem Description This dissertation deals with generating motion plans for smallscale unmanned sensing missions. Comparison of the images in Figure 12 shows that fundamental differences can be expected in the way the two depicted classes of sensing missions are approached. A B Figure 12. Comparison of sensing data for different mission scales. A) Images captured from standoff range. B) Images captured within close proximity to a cluttered environment. One aim of the work presented here is to identify and address some the unique challenges encountered for sensorplanning in closeproximity environments. Motion planning methods for standoff missions commonly make simplifying as sumptions that are consistent with the problem scale. For example, target touring is often considered sufficient to ensure sensing of a set of targets. Previous efforts have adopted this technique, which involves simply visiting each target location (or a location directly above, for the case of UAVs) [1518]. Similarly, area coverage in such cases is often achieved through decomposition of the area of interest into cells and then simply visiting each cell to meet sensing objectives [1921]. Figure 13 shows a representative example of a sensor motion plan that operates under such assumptions. In the depicted scenario, the sensor FOV is considered large enough such that cell visitation is sufficient to sense any targets contained within the cell. These approaches also typically assume that local motion constraints due to vehicle dynamics are negligible relative to the scale of the motion planning problem. For example, the vehicle turn radius in Figure 13 is small enough relative to the cell resolution and to the scale of maneuvering that such a constraint can be neglected in motion planning. Numerous technical challenges are intro duced by operation of the vehicle and sensor in close proximity to the environment. First, Figure 12B shows drastically disparate data in different areas of the image as compared to Figure 12A. Some objects are much closer to the sensor than others, and hence appear Figure 13. Touring a series of cells for a largescale sensing mission. in the foreground of the image with comparativelyhigh data resolution. Additionally, objects are seen at aspects that yield data with varying resolution on a single surface. The extent of these effects varies as the vehicle moves through the environment and the viewpoint changes. Data quality with respect to a particular sensed object is seen to vary as a function of both the vantage point and the location within the FOV. Further, sensor operation amongst obstacles results in the occlusion of large regions. As such, the view quality of a particular target could change drastically with small changes in vehicle and sensor position and orientation. Another issue is related to the comparatively small spatial area contained within the FOV for closeproximity cases. A small projected footprint relative to vehicle velocity causes target surfaces to pass through the FOV quickly as the vehicle moves through the environment. Resulting data losses are exacerbated by cluttered environments that require significant maneuvering for safe navigation. As such, a motion plan to view a particular target must be timed appropriately and tracked with enough precision to ensure that the sensor is pointing in the right direction at the right time. Several researchers have expanded the target touring approach to account for a particular sensor footprint shape and size; however, these efforts fail to account for motion coupling that results from maneuvering [17, 20, 2224]. Figure 14 shows a representative motion plan that dipli data quality and motion coupling effects related to proximity. In the figure, the sensor footprint is not significantly large relative to cell resolution. As such, sensor coverage is seen to cover some cells partially and nonuniformly due to the variable resolution issue discussed previously. Additionally, the spatial effects of vehicle dynamics are significant relative to the cell resolution and therefore must be considered in motion planning. The resulting motion coupling effects are seen in Figure 14 as sensor coverage that varies relative to the vehicle trajectory. Figure 14. Touring a series of cells for a smallscale sensing mission. Planning a trajectory to view a sequence of targets presents a difficult problem regardless of sensing proximity effects. Generally, an optimal solution to this problem requires that: 1. The optimal order of target visitation is determined 2. The optimal viewpoint for each target is determined 3. The optimal trajectory through the optimal viewpoints is computed These requirements are depicted in Figure 15. Efforts to address such problems typically involve the application of assumptions and the solution of suboptimal ap proximations. For example, the targettouring approach described previously essentially assumes that the optimal viewpoint for each target is the location of the target itself. When dynamics are neglected, this problem reverts to the wellknown traveling sales man problem (TSP) from graph theory, for which numerous solution and approximation techniques exist [25, 26]. Several efforts achieve an approximation to the optimal set of viewpoints to sense an area through random sampling, and then proceed to solve the en suing TSP [2729]. These efforts do not consider differentially constrained motion, which, while some approximation techniques do exist [30], can greatly complicate the solution of the TSP. Figure 15. Motion planning to view a sequence of targets. Finally, the close proximity sensing problem requires that planned trajectories accurately represent feasible vehicle motions such that trajectory tracking will ensure safe navigation and precision sensor placement. Trajectories that do not adequately represent feasible motions are subject to tracking error upon mission execution that could result in either failure to reach a preciselydefined viewpoint or in collision with an obstacle. Generally, this requirement implies that planned trajectories should be constrained by some representation of the vehicle dynamics. Inclusion of dynamicallyfeasible motions in a planned trajectory is typically treated in either a direct or a decoupled fashion [31]. Direct planning methods, such as optimal control, consider a representation of the vehicle dynamics in the formulation of the planning problem and directly solve for optimal system inputs. Alternatively, indirect methods use a simplified model of vehicle motion to plan a reference path and then "smooth" the path to satisfy dynamics using methods such as feedback control. Direct methods compute optimal trajectories but are often intractable for realistic problem descriptions. Indirect methods often exhibit tractable complexity properties that comes at the expense of optimality. Researchers have found clever vi to manipulate this tradeoff through a v ii I of techniques such that dynamics can be directly included in the planning process. For example, some researchers have recognized that systems which exhibit differential flatness properties admit solutions that can be represented parametrically in terms of a set of flat outputs and their derivatives [3234]. Others have applied mixedinteger linear programming (\!I .IP) to model dynamic constraints as a set of switching bounds on system velocities and accelerations [3537]. Frazzoli et al introduced a planning technique that utilizes a iiipl!'ddyin ,ii. model which employs a set of dynamicallyconsistent motion primitives [38, 39]. Additionally, recent advances in randomized planning allow the use of any of these techniques as local trajectory generation methods for growing a probabilistic tree of actions to explore the solution space [4042]. 1.3 Problem Statement This dissertation considers the problem of planning effective sensing trajectories for a vehiclecarried sensor operating in a closeproximity environment. Specifically, consider a class of remote sensors that rely on a clear lineofsight (LOS) to a target as a necessary condition for visibility. This class of sensors is assumed to exhibit variations in data quality resulting from geometric factors related to the relative position and orientation of the sensor with respect to the target. Consequently, these variations, and hence sensing effectiveness, can be quantified as a function of sensing geometry. In addition, consider a vehicle that is subject to differential motion constraints. This vehicle carries a sensor that belongs to the aforementioned class of LOS remote sensors. The focus of this dissertation is to develop a method for generating vehicle guidance trajectories that ensure effective sensing of prespecified locations embedded within a known operating environment. Further, the placement of these sensing locations is such that operation of the vehicle and the sensor are necessarily in close proximity to this environment. "Close proximity" is loosely defined such that the characteristic dimensions of the environment, characteristic dimensions of the vehicle dynamics, and typical sensor operating range are all of the same order of magnitude. Formally, this definition is intended to reflect several specific requirements that must be imposed upon computed motion plans: 1. The proximity of vehicle trajectories to obstacles in the environment requires that differential motion constraints are considered in motion planning. As such, trajectories must be generated that are consistent with the vehicle dynamics in order to ensure safe and accurate navigation and tracking. 2. The proximity of targets to sensor vantage points along planned trajectories results in significant variations in view quality with respect to changes in vantage character istics. As such, a metric must be formulated to quantify these variations and must be subsequently utilized by the motion planning algorithm to ensure effective sensing trajectories. 3. The proximity of the sensor to the surrounding environment results in a small projected footprint relative to the characteristic dimensions of the vehicle dynamics. As such, motion planning must consider effects related to the coupling of sensor pointing with vehicle maneuvering. Planned trajectories must therefore include constraints on vehicle/sensor orientation to ensure proper sensor pointing that accounts for FOV constraints. Three specific sensing task categories are selected based on their relevance to ISR applications. These tasks can be described as reconnaissance, surveillance, and area coverage. Each of these tasks focuses on slightly different mission criteria. The reconnaissance task assumes that a target or series of targets has been identified and located. The sensing system is required to visit each target and collect data prior to either returning home or continuing to some other specified goal. This mission task is depicted in Figure 16A. The purposes of this type of mission task include target identification, verification, classification, and other 'singlelook' reconnaissance objectives. The surveillance task also assumes that a target or series of targets has been recog nized and located; however, the goal has changed to persistent observation. The vehicle must patrol the series of targets and repeat the sequence until a prespecified termination condition is met. This mission task is depicted in Figure 16B. The purposes of this type of mission include target verification, classification, monitoring, change detection, and collection of modeling data. The coverage task implies no assumptions about target locations. The vehicle must pass the sensor footprint over the entire area of interest in such a way that some level of data quality is achieved over the entire task space. This mission task is depicted in Figure 16C. The purposes of this type of mission task include search, explore, target detection, change detection and wide area surveillance. A B Figure 16. Sensing mission tasks. A) Reconnaissance mission. B) Surveillance mission. C) Coverage mission. Finally, several assumptions are considered in formulating the solution approach. These main assumptions are as follows: * Obstacles in the environment are static * Obstacle and target locations are known a prior * A vehicle model is available that is capable of representing dynamics at various points in the operating envelope * Planned motions are designed as openloop guidance trajectories 1.4 Approach Overview The problem stated in Section 1.3 is addressed through three main steps. First, a method is developed for dynamic pointtopoint motion planning in environments that meet the definition of close proximity as discussed previously. Then, a framework for quantifying the effects of sensor operation in such environments is developed for use as a performance metric. Finally, a sensor planning method is developed which integrates this metric with the pointtopoint planning method into a randomized scheme designed to achieve the tasks specified in Section 1.3. Motion planning for differentially constrained vehicle systems is examined in Chap ter 2. A hybrid representation of the vehicle dynamics which utilizes the concept of motion primitives is modeled after previous efforts. This hybrid representation is used to formulate the trajectory planning problem as a hierarchical optimization problem. A special class of feasible, though suboptimal, explicit solutions are identified and utilized to approximate optimal solutions. The benefits and drawbacks of these approximations are demonstrated through simple examples. The concept of randomized samplingbased motion planning is introduced in ('!C ip ter 3 through discussion of several existing algorithms. A randomized planning algorithm is developed to compute approximate solutions to the optimization problem posed in C'! Ilpter 2. This method makes effective use of the explicit solution approximations intro duced in the previous chapter such that some of the 1n I 'r drawbacks are avoided. This planning method is demonstrated through a simulated example. ('!i ipter 4 discusses modeling of the sensing task. Specifically, a geometric sensing model is developed and used to formulate a definition of visibility. Then, the effects of sensor operation in close proximity to the environment are discussed. A general framework is introduced to quantify these effects as a measure of data quality. Examples demonstrate how this framework can be tailored to specific sensing applications. The sensorplanning task is addressed in ('!C ipter 5. First, a multiresolution environ ment representation is developed to represent the coverage task as a targetcentric task. Consequently, all three of the tasks listed in Section 1.3 can be treated using variations of one unified method. Then, the randomized planning and sensor effectiveness concepts from previous chapters are integrated to form a new randomized planning approach to sensing a series of targets in a closeproximity environment. The benefits and challenges associated with the application of optimal control principles to the effective sensing problem are discussed in ('! iplter 6. The necessary conditions for an optimal solution are derived for a simplified example. Optimal solutions to a simplified version of the sensing problem are computed using a numerical approach. Results are then demonstrated through a simulated example. Finally, C'!I ipter 7 summarizes the project and discusses the utility of the methods introduced throughout the dissertation. Additionally, recommendations for future work are provided. 1.5 Contributions This dissertation considers the problem of planning effective sensing trajectories for a vehiclecarried sensor operating in a closeproximity environment. Three main technologies are required to enable solutions to such problems: visibility and sensing effectiveness concepts for a point in the environment must be characterized, a pointtopoint motion planning method that considers vehicle dynamics must be developed, and a scheme that allows the application of this motion planning method to sensor planning problems must be contrived. Aspects of these subproblems have been researched independently in the past. The present effort pl1i the role of integrating technologies into a functional system. A variety of modifications are made to existing techniques in addition to the development of new concepts to enable this integration. As such, several contributions are made in the process. * An approximation method to solving both kinematic and dynamic maneuverbased motion planning problems is developed * A randomized planning algorithm utilizes this approximation to generate dynamic goalbased trajectories in complex environments * A metric of sensing effectiveness is developed to quantify the geometric effects of sensing at close range * Concepts are adopted from robotic manipulatorbased sensor planning to formulate the sensing problem as a goalbased planning problem where an effectivesensing goal set is defined in terms of vehicle configuration * A randomized approach is developed to sense a series of targets for which a number of issues related to operation within closeproximity environments are considered * A samplingbased multiresolution scheme allows an areabased sensing problem to be treated as a targetbased sensing problem * A novel application of a recently developed optimal control technique results in optimal sensing trajectories for a simplified case of the general problem CHAPTER 2 MOTION PLANNING 2.1 Introduction Motion planning describes the process of determining the transition of a system from an initial configuration to a terminal configuration. System configurations are defined as a vector, xY, on the configuration space, or Cspace. For robot and vehicle systems, the Cspace is spanned by the variables that describe the position and orientation of the body coordinate frame in addition to any other degrees of freedom the system may have. The terminal or goal configuration is typically specified as a point in the Cspace that achieves some objective such as to arrive at a base location or to point a sensor at a target. Motion planning determines the function that guides the system from the initial configuration to this goal. Motion planning problems for vehicle systems are typically classified according to three main categories [43]: pointtopoint motion, path following, and trajectory tracking. All three of these classes require that the vehicle move from an initial configuration to a goal configuration; however, the function describing this motion is constrained differently for each. The pointtopoint motion task places no restrictions or specifications on the intermediate motions occurring between two configurations as long as the goal is reached. The pathfollowing class of problems requires that the vehicle follow some specified continuous path in Cspace that satisfies any system differential constraints and which has the initial and goal configurations as endpoints. Similarly, a continuous path constraint is imposed for the trajectory tracking problem class; but, the additional requirement of an associated timing law is included as well. The result of these definitions is that pointto point motion plans are typically specified as a series of waypoints, path following motion plans are specified as a function defined on the Cspace, and trajectory plans are specified as either timeparameterized functions on the Cspace or a system input function, il, that is a function of time. A particular motion plan can be evaluated according to performance criteria in ad dition to the goal configuration endpoint constraint. Such criteria, such as path length, can be used to characterize a given motion plan as better or worse than another. Con sequently, the motion planning problem can be considered as a constrained optimization problem. An optimal motion plan minimizes (or maximizes) the performance criteria while satisfying the initial and terminal constraints in addition to any path constraints, such as those imposed by obstacles in the environment. This dissertation is concerned with generating guidance trajectories for a vehicle system given a model of the vehicle motion and a known environment. The environment is assumed to contain polygonal obstacles with known vertex locations. Further, all computed trajectories are understood to be openloop reference trajectories. Practical implementation of such methods would require some level of feedback control to effectively deal with disturbances and uncertainties; however, such issues are typically addressed using a separate control lr and are beyond the current scope. The remainder of this chapter discusses deterministic motion planning for a differentiallyconstrained system. First, typical path constraints that might be imposed by obstacles or through the system dynamics are defined and discussed in Section 2.2. Next, optimal control is examined in the context of benefits and drawbacks to practical planning problems involving realistic systems. Section 2.4 then introduces a modeling alternative that provides solutions through lowerdimensional optimization problems than those typ ically encountered when using direct optimal control methods. This method utilizes the concept of motion primitives and even admits closedform algebraic solutions to special cases of the planning problem formulation. 2.2 Path Constraints Path constraints impose restrictions on the values of the configuration variables, Yx E C, and/or their derivatives along the trajectory. These constraints can be written in general form as Equation 21. The constraint function, c zc, c, c,u,t) represents a vector of constraints such that the inequality is applied componentwise. c xcx, X ,, t) < 0 (21) Equation 21 can be used to require avoidance of obstacles, to enforce differential system behavior, to bound allowable input signals, and to restrict the operational envelope of the system. A path constraint can be applied to a particular point along a trajectory or to a continuum of points along a trajectory segment. The former case is required for finitedimensional trajectory optimization problems while either usage is acceptable for infinitedimensional functional optimization problems. 2.2.1 Differential Constraints The vehicle systems considered in this dissertation are described by systems of differential equations and are therefore subject to motion constraints imposed by the dynamic behavior of solutions to these equations. The system dynamics impose an important class of constraints that restrict allowable system velocities and allow kinematic and dynamic vehicle behavior to be considered in the motion plan. Integration of these differential behaviors into the planning process ensures enhanced accuracy of a given motion plan with respect to trajectories that are executable by the physical system. Differential motion constraints act on the time derivatives of the configuration variables, which can be described in terms of the statespace, X. The statespace is spanned by the minimum set of variables required to completely determine the motion of the system at a given point in time. The state vector, xs, therefore includes all of the configuration variables in addition to relevant velocities and accelerations. The Cspace is thus a subspace of X for a particular system, such that C C X, and where the dimension of X exhibits an increase of at least a twofold over the dimension of C. Differential constraints on system motion can be classified as kinematic or dynamic constraints. Typically, kinematics deals with descriptions of system motions independently from the forces and moments required to generate these motions. Physically speaking, dynamics relate the interaction of forces and moments with system inertial properties to generate accelerations; however, in general the term is often used in reference to any timevarying behavior. Further, the definition of a dynamic system model as compared to a kinematic system model is blurred by the direct relationship of the kinematic concept of acceleration to the dynamic concept of the forces that affect acceleration through Newton's second law. In other words, a "dyri .i... constraint regarding allowable system forces is directly related to a 1:.ii 11i iI ,:" constraint on system accelerations. These technicalities are the source of a great deal of confusion in motion planning lit erature, which lies at the intersection of robotics, control theory, and applied mathematics. The remainder of this dissertation considers firstorder differential constraints on the C space as kinematic constraints (Equation 22). Firstorder constraints on the statespace, which include first and secondorder differential constraints on the Cspace, are considered dynamic constraints (Equation 23). Motion planning under this definition of differential constraints is referred to in the robotics literature as i .:..,li, :. motion planning [44]. X = f (XY ), C (22) Xs = f(xs, ), s E X (23) This dissertation considers two broadlydefined approaches for generating a trajectory that satisfies system dynamics: 1. Directly plan a trajectory constrained by the system dynamics 2. Decouple the problem by first planning a trajectory with relaxed constraints, then applyingg the dynamics to the result The first approach is enabled using tools from optimal control theory and can yield optimal solutions, but is plagued by problem complexity for practical applications. The decoupled approach can be effective to a varying degree depending on the choice of planning method for the relaxedconstraint problem as well as the choice of method for application of the dynamics to the relaxedconstraint solution. For example, a simple pointtopoint planning approach can quickly generate a set of desired waypoints. Dynam ics can be applied through feedback control that steers the system toward the appropriate waypoint. Alternatively, solution of an optimal control problem that is constrained to pass through the waypoints would successfully introduce dynamics as well. Nontraditional methods exist as well, such as smoothing with primitives that satisfy the system dynamics, as emploi, 1 by Bottasso et al [45]. Each of these subproblems are easier than the global problem; however, some level of performance tradeoff is likely. As a matter of notation, x will be used interchangeably for the remainder of the dissertation to represent the state vector, Yx E X, and the configuration vector, Yx E C. Many of the described techniques operate in a similar fashion when a system is classified as either dynamic or kinematic. The definition will be stated explicitly when the correct usage is not apparent from the context. 2.2.2 Obstacle Constraints This dissertation considers polygonal obstacles for the case of a twodimensional environment and extruded p" ..lions for the case of a threedimensional environment. This definition allows threedimensional obstacles to be treated as planar polygonal obstacles at particular locations along the extrusion axis. As such, the subsequent discussion of collisionchecking considers the twodimensional case of testing a planar point for intersection with a polygonal obstacle. Without loss of generality, the extrusion axis is presently assumed parallel to the vertical direction such that planar position is described by the position vector, AD = [ p Py ]T. The planar positions of the n vertices of a particular obstacle are given by i,, = [ PT,,, Py,, ], i =1 .. n. These position vectors locate points relative to the origin of an inertial reference frame. Safe motion plans amongst obstacles require that planned trajectories never intersect any boundary of any obstacle, or equivalently, that the configuration variables describing position never reach a value on the boundary or inside any obstacle. This condition can be expressed in several different vv for the type of obstacles considered here. One method involves expressing the obstacle boundary as a radius function defined in cylindrical coordinates centered at an arbitrary point, c = [ pI,c Pv,c ]', that is located within the obstacle p" ..Il on. This obstacle coordinate frame is simply a translated <" iv of the inertial reference frame. The obstacle boundary is expressed as rebound( obs), where Oobs represents a bearing from the obstacle relative to the xob8axis, which is oriented parallel to the inertial xaxis. The path constraint restricting PD to values that lie outside the boundary of a particular obstacle can be expressed as Equation 24, where rp represents the range of AD from ic and 0obs,p represents the bearing of AD relative to ;c. These relations are given by Equations 25 and 26, respectively. rp rbound(Oobs,p)) < 0 (24) rp =  2D Pc  (25) 0obs,p = arctan Py ) (26) Computation of rbound(6obs) requires parameterization of the edges that represent the obstacle boundary with respect to Oobs. The inertial position of a point along the obstacle edge connecting the ith obstacle vertex to the jth obstacle vertex is expressed as Pe, = [ pxe Py,ej ]T The components of this vector can be parameterized as shown in Equations 27 and 28. The boundary of the obstacle is then computed using the piecewisecontinuous radius function shown in Equation 29, where Oobs,vi and Oobs,vj represent the bearings of the ith and jLh obstacle vertices relative to c, respectively. This radius function, rbound,act is depicted as a solid black line in Figure 21 for a square obstacle. The graphic shown in Figure 21A shows fe, relative to the reference frame centered at c = [ 0 0 ]T. The graphic shown in Figure 21B shows rbound(obs). S ( Py,vi + Py,c +pz,c +e Pvi tan obs Pxe[ (obs) Px,vi (Px,vj n+ + Px, +,) tanOobs_ (27) ( px,v, pz,Vi) tan Oobs Py,vj Py,vi) ( y,vi + Py,c + Px,c + Px,vi tan Oobs ( Py,e (Oobs) = Py,v (Py, Py, (p,vj p ) tan obs p ,v) (28) ( \Px,V Px,Vi) tan Bobos {Py,vj ~ Py,Vi) \/(Px,eI2 P,c)2 + (Py,e12 Py,c)2 Oobs,v1 < Oobs < 0obs,v2 00xe3 Px c)~ + (Pge03 Pgc)2 obs,vs no "ons K Hobsnvs rbound,act (Oobs) P < (29) \/(Px,en Pc)2 + (Py,e Py,c) 2 Oobs,v, < Oobs < Oobs,v1 Equation 29 accurately represents the boundary of a polygon as a radius function for use with constraints of the form of Equation 24. Unfortunately, the nonsmooth nature of this function can present problems for gradientbased optimization techniques. Alternatively, a smooth radius function that conservatively approximates the boundary can be computed. An example of such an approximation is shown in Equation 210. This function, rbound,circ, simply approximates the obstacle boundary as a circle with radius equal to the maximum value of rbound,act. This approximation is shown in Figure 21 as a dotted red line. The figure shows that the use of such a conservative approximation can restrict a significant portion of the obstaclefree space within the proximity of the obstacle being approximated. The number of feasible trajectories could be severely limited by restricting this space, especially for the case of closelyspaced obstacles. rbound,circ (Oobs) rbound,act oo (210) Another conservative approximation utilizes the concept of superellipses [46]. This family of curves represents a generalization of the ellipse that can be used to approximate rectangular shapes, among others. The obstacle boundary can be represented as a param eterized radius function given by a superellipse as shown in Equation 211, where as and b,, represent the maximum radius in the Xobs and yobs directions, respectively. The radius is thus given as a function of the relative bearing, Oobs. Equation 211 approximates a rect angle when the exponent, 7e, is a positive, even integer. A superelliptical approximation to the obstacle boundary is shown in Figure 21 as a dashed blue line. The curve shown in the figure is computed using as = 1.2, be, = 1.2, and 7,s = 8. The figure shows that this range function provides a more reasonable conservative estimate as compared to the radius function expressed in Equation 210. rboundse (Oobs) = os + ob, (211) a. se b() ( )i ) 1 2 2 1 1.5 , 0 :1 1 0.5 .  .,: 7 '" 2 0 2 0 45 90 135 180 Yobs 0obs (deg) A B Figure 21. Obstacle boundary approximations. Approximation functions are shown as follows: actual radius function (solid), circular radius function (dotted), and superelliptical radius function (dashed). A) Overhead view of obstacle and approximated boundary. B) Radius approximations parameterized on Oobs Alternative methods for collision detection avoid the need to explicitly represent the obstacle boundary. As such, these methods do not give a measure of the proximity of AD to a given obstacle. Rather, these methods simply provide a binary flag indicating the collision status of a test point with respect to each obstacle. These methods are therefore not suitable for use with gradientbased optimization methods. One such method requires that the set of vectors, ivpV, = v, AD, i = 1 n, is computed. These vectors represent the vectors from the test point to each obstacle vertex, and are shown in Figure 22. The interior angles between each .,li i,:ent pair of ffpvi,j , denoted aobs,ij can be computed using the dotproduct operation. A necessary and sufficient condition for a collisionfree point is given by the anglesum inequality shown in Equation 212. Saobs,ij < 2 i 1 n (212) i U obs, y Ul " U, Figure 22. Vertexangle sum collision detection method. Another method that requires computation of ipv,, is depicted in Figure 23. This method additionally requires computation of the set of edge vectors. Each of these vectors corresponds to a polygon edge and indicates edge direction in a clockwise sense. The edge vector based at the ith vertex points to the jth vertex and can be expressed as Zfe, = fj iv,. The unit normal to iec, is denoted nfj and is also based at the ith vertex. This normal vector can be computed via a rotation through r/2 rad, as shown in Equation 213. A necessary and sufficient condition for a collisionfree point is met if Equation 214 is satisfied for any (i, j) pair. This condition essentially states that if the test point is located in the same direction relative to each edge vector, the point lies inside the ]" Iv._on.  ( U, hAUi' Figure 23. Vertex edgevector collision detection method. 1 I cos(7r/2) sin (7r/2) (213) I 9 9I sin (7/2) cos (7/2) ) sign(nie, (4vJ)) < 0 As discussed previously, planning algorithms that rely upon a finitedimensional func tion optimization must enforce path constraints pointwise at a finite set of points placed along the trajectory. Enforcement of obstacle constraints therefore cannot guarantee a collisionfree trajectory between these test points. Example trajectories can be constructed that collide with an obstacle, yet technically satisfy all obstacle constraints, as shown in Figure 24A. Obstacle boundaries can be expanded as shown in Figure 24B to avoid such situations. The extent of expansion must be carefully selected to ensure that unsafe trajectories cannot feasibly satisfy obstacle constraints. A B Figure 24. Obstacle expansion for pointwise safety. A) Pointwise satisfaction of obstacle constraints. B) Pointwise satisfaction of expanded obstacle constraints. 2.3 Optimal Control Optimal control theory provides a rigorous approach to trajectory design for differentiallyconstrained systems. The objective of optimal control is to determine the vectorvalued input function, u(t), that drives the state trajectory, x(t), such that a scalarvalued performance index, J(x(t), u(t), t), is minimized within a time interval of interest. The state trajectory must satisfy differential constraints specified by the system dynamics in addition to any algebraic path constraints, such as obstacle regions or state and control bounds. Endpoint constraints are also imposed at the initial and terminal conditions of the trajectory. Mathematically, the optimal control problem can be posed as (214) an optimization problem as shown by Problem 215. (x(t)*, (t)*) arg min J((t), (t),t) (215) x(t),u(t) s.t. x(t) f(x(t),u(t)) = 0 S(xo,to,xf,tf) < 0 c(Q(t), u(t)) < 0 Time arguments are retained in Problem 215 to emphasize that the problem seeks to determine continuous functions, and therefore has infinite dimension. The differential path constraints are enforced in equality; therefore, they are written separately from the algebraic path constraints, c(x(t), u(t)). The function, '(xYo, to, Xf, tf), relates endpoint constraints on the trajectory. Finally, the functional performance index, J(7(t), i(t), t) is typically written in integral form as in Equation 216. J(M(t), (t)) = ((tf),tf) + t L(Y(t), (t))dt (216) 2.3.1 Indirect Optimal Control Indirect solution methods to optimal control problems seek the input function and corresponding state trajectory that satisfy the necessary conditions for an extremal of the performance index, subject to the constraints. These extremals occur at points where the performance index is considered stationary. Identification of stationary points requires the use of variational calculus concepts, as the performance index is described by a functional. Consider an instance of Problem 215 for which there are no algebraic path con straints and the endpoint constraint is applied in equality. The cost functional can be augmented with the differential and endpoint constraints through the introduction of a vector of Lagrange multipliers, 1i, and a vector of multiplier functions, X(t). Each ele ment of 7 is associated with an endpoint constraint while each element of X(t), denoted the costate, is associated with a differential constraint. The augmented cost functional is written as J, which is shown in Equation 217. In this expression, is denoted the Hamiltonian of the system and is shown as Equation 218. Function arguments have been omitted here to maintain clarity in the expressions. The constrained optimization problem in J is now reexpressed as an unconstrained optimization problem in J. J [ tf + A tjdt (217) = L +Tf(x, u) (218) A set of necessary conditions for a stationary point of J can be found by taking the first variation and setting it equal to zero, as 6J = 0. A full derivation of these conditions is beyond the scope of this dissertation but and is available in the literature [47]. The first set of conditions are given by the costate dynamics, as shown in Equation 219. These differential equations have boundary conditions given by the transversality conditions, as shown in Equations 220 and 221. A (2 19) XT(to) + T (220) aY(to) aO(to) (tf) (2 21) ax9tf) ax;tf) The optimal control is related by Pontryagin's minimum principle, which states that the optimal control minimizes the Hamiltonian [48]. The weak form of this condition is shown as Equation 222. S0 (222) Equation 222 relates the optimal control in terms of the state and costate. This control function can be recovered if the optimal state and costate trajectories are deter mined. As such, the optimal control problem can be solved indirectly through solution of the boundaryvalue problem (BVP) specified by the state dynamics, the costate dynamics, the state endpoint constraints, and the transversality conditions. When the state vector has a dimension of n, solution of this BVP involves solving a coupled differential system of 2n equations with splitend boundary conditions. Further, some of these boundary conditions may only be known as a function of the unknown Lagrange multipliers, i. Consequently, iterative numerical techniques often must be employ, ,1 to determine these boundary conditions. Further challenges are introduced when optimal control theory is applied to problems containing realistic constraints. Inclusion of algebraic path constraints, such as those imposed by obstacles or system operating restrictions, result in a differential algebraic system that introduces a number of additional complications [49]. Path or endpoint constraints that are enforced as inequalities can cause an increase in computational burden which results from the need to identify the active and inactive elements of these constraint functions. Further, the optimality condition does not alv, yield a closedform expression for il(t) in terms of the state and costate. Sometimes this condition does not even uniquely determine the input function. While the indirect approach to solving optimal control problems utilizes a rigorous mathematical framework, all but trivial problems are often rendered impractical. 2.3.2 Direct Optimal Control Various direct numerical methods for solving optimal control problems have emerged as an alternative to the burdensome indirect approach [4952]. Such methods involve transcription of the infinitedimensional functional optimization shown in Problem 215 to a finitedimensional function optimization. As such, derivation of necessary conditions and the use of variational calculus is unnecessary; instead, numerical optimization tech niques are used to solve the nonlinear program (NLP) which results from the problem transcription. The first step in the transcription process is to split the time interval, t E [to, tf], into a finite set of fixed subintervals, as shown by Equation 223. The N points at which the time interval is discretized are denoted nodes. to < ti < t2 < < tN1 tf (223) The continuous functions that represent the state and control trajectories are dis cretized, as well. These functions are replaced by finite sets of values comprised of each function evaluated at each of the N nodes. The combined set of these values make up the decision variables for the subsequent NLP formulation. The vector of these values is denoted X, as shown in Equation 224. S [ ((to0))T ((to0))T ((t1)) (UT(ti))7 ... ((tN 1)) ((tN 1)) (2 24) The cost functional, J, must be replaced by a finitedimensional counterpart, denoted F. The first term in a performance index of the form shown in Equation 216 remains unchanged through transcription if the terminal time, tf, coincides with one of the nodes in Equation 223. The integral term can be approximated as a finite sum using numerical integration techniques. Similarly, numerical integration is used to represent the differential constraints as a vector of defects, denoted as (. Each defect enforces the integration rule between the corresponding pair of nodes. The defect corresponding to the kt node is shown in Equation 225, where f represents the integration rule. Dependence of f is shown with respect to state and control values at the current and next time step to maintain generality; however, rn in: integration techniques only require information at the current time step. Examples of such techniques include Euler integration and the RungeKutta method. (k (t+1) k(t) f ((tk), 1(tk),i(tk+l), (tk+l)) (225) The endpoint constraints from Problem 215 can be applied directly within the transcribed formulation if to and tf are included as nodes. Additionally, algebraic path constraints are applied to the state and control pointwise at each node. These constraints can be combined with the vector of defect constraints into a single vector of constraint functions, as shown in Equation 226. S(xAo,to, xf,tf) C(X = c( (226) 4(X) Finally, the pieces can be assembled as the NLP shown in Problem 227. This prob lem contains at least N(n + m) variables for a system with n states and m controls, and thus is very large for even a coarse timediscretization. Fortunately, the relevant matrices involved in solution of the NLP take on a sparse form as a result of the transcription method [49]. This sparsity can be exploited to greatly reduce both storage and computa tion time. Several NLP solvers that exhibit this capability have been shown to perform quite well in handling problems of this nature [53]. X* argminF ) (227) s.t. c(s Direct transcription provides a practical alternative to indirect methods when consid ering realistic optimal control problems; however, the method does have some disadvan tages. First, constraints are only applied at the nodes. As such, behavior of the trajectory, including constraint satisfaction, cannot be guaranteed between nodes. Additionally, NLP solvers are typically susceptible to local minima and often exhibit dependence on the choice of initial condition. More importantly, Problem 227 is fundamentally a different problem than Problem 215. An interpolated function through the discrete solution values to the finitedimensional NLP may not coincide with the optimal solution to the infinite dimensional problem. Further, as the necessary conditions are not computed, there is no way to validate solutions to Problem 227. Progress has been made in this area recently through the development of costate mapping techniques that utilize pseudospectral col location methods [51, 52]. Such methods represent an active area of research in optimal control theory. 2.4 Hybrid Motion Models 2.4.1 Modeling with Motion Primitives A fundamental property that is common to many vehicle systems of interest is related to the concept of symmetry. Specifically, the trajectories of a certain subset of the state or configuration variables are seen to exhibit invariance with respect to certain classes of transformations. Let x represent either the state or configuration vector of a system whose dynamics are represented in the form of Equation 22 or 23, respectively. A system trajectory can be determined through integration of the dynamics with respect to time. Now consider a transformation function, H : C + C (or H : X + X). System trajectories are said to exhibit invariance with respect to the transformation, H, if Equation 228 holds true [39]. (t) =H( f (Fo, )dt) = f(H (fo) u)dt (228) Now, let two trajectories be considered equivalent if they can be exactly superimposed through time translation and application of H [39]. Satisfying Equation 228 thus implies that a trajectory generated from a particular initial condition, xo, through the application of an input function, u, is equivalent to all trajectories that result from the application of u and that are initialized at any point in the range, H(xo). Trajectories that satisfy this property are denoted motion primitives. Physically, H represents the class of transformations that do not affect the external forces acting on the vehicle. For example, consider a carlike vehicle operating on an expansive, flat, isotropic surface. Forces on the vehicle result mainly from gravity and friction. Relative displacements in position and heading that result from a particular input trajectory are invariant to absolute position and heading. Therefore, translations and rotations in the plane comprise the class of symmetric transformations, H. Alternatively, vehicle operation on a surface with variable material properties, such as transitions from concrete to gravel, exhibits variable friction forces as a function of position. Application of a particular input function results in different relative displacements depending on the position on the surface. For the former case, any trajectory on the surface can be reproduced at a different location and heading through application of H. Such a trajectory can be described as a motion primitive. Conversely, the latter case requires full integration of the dynamics for different initial conditions. Note that the invariance properties of some system variables depend on the operating environment. As such, the operating conditions must be considered when constructing motion primitives for a particular system. The concept of motion primitives leads to a useful framework for the simplification of complicated dynamic models. This framework involves combining sequences of compatible primitives to represent complicated trajectories. A set of compatibility conditions are detailed in the literature [39]. For the systems considered in this dissertation, the state displacement resulting from a finiteduration motion primitive can be represented by the transformation, GM, as shown in Equations 229 and 230. In Equation 230, the relative state displacement due to the motion primitive, ZA, is translated by xo and rotated by R(xo). A discrete set of reachable configurations can be achieved through successive application of compatible transformations of the form, GM. S= GM (o) (229) GM (o) = o + R(o)m (2 30) In addition to these finiteduration primitive solutions, the concept of trim motion can be used to generate continuouslyparameterized families of system trajectories. Trim motion is characterized by steadystate motions with fixed controls for the systems considered here [38]. A trim primitive then describes the timeevolution of noninvariant states resulting from constant values of the invariant states. Continuous state trajectories are represented by the timeparameterized transformation, GT, as shown in Equations 2 31 and 232. This transformation describes the state displacement from the initial condition, xo, along a trim trajectory after a time duration, r. As before, the relative displacement due to the motion primitive, Ar(T) is translated by 1o and is rotated by the transformation, R(xo). (r) = GT (xo,T) (231) GT (o, ') = Yo + R(xo) AT(r) (232) While executing a trim motion, the reachable set of the system lies along the curve described by Equation 232. This set can be greatly expanded by implementing a finite number of trim trajectories in a hybrid switching scheme that employs unsteady, finite duration primitives to transition from one steadystate trim motion to another. This model can be represented as a finitestate automaton, as depicted in Figure 25. Each state of the automaton, depicted as nodes of the directed graph in Figure 25, represents a trim primitive, Ti, and is defined by the steady trim velocities that characterize each particular trim. Motion associated with Ti is governed by a transformation of the form shown by Equation 231. The state transitions of the automaton are depicted as directed edges of the graph in Figure 25 and represent the finiteduration, unsteady transitions, if between each pair of trim states, Ti and Tj. These unsteady motions are denoted maneuvers and must originate and terminate in steadystate motion as a condition for inclusion in the automaton framework. Motion resulting from the execution of a maneuver is governed by a finite displacement of the form shown in Equation 229. The graph structure represents allowable switching behavior between primitives as required by compatibility conditions. Maneuvers can be realized as an instantaneous switch or a smooth dynamic motion, depending on the extent to which the system is differentiallyconstrained. Kinematic systems, as defined previously, allow instantaneous changes in configuration velocities and Figure 25. Automaton model representation as a directed graph can therefore transition between trims with no configuration changes. Dynamic systems exhibit bounded accelerations and, as such, smooth system configuration changes will accompany any transitions between trim states. The automaton modeling scheme depicted in Figure 25 can represent complicated system trajectories through specification of a sequence of maneuvers and the duration of the trim states between each consecutive maneuver. This model exhibits both continuous behavior and discrete switching behavior in the trim states and finiteduration maneuvers, respectively, and is therefore considered a hybrid system. Given a maneuver sequence of length n, M = {M1, M., }, and a corresponding length n + 1 sequence of trim durations, r {ri, 7 Tn+l }, system trajectories in the Cspace are computed via a series of transformations. Such trajectories alvi initiate and terminate in a trim configuration. Equations 233 and 234 show a general example of this process for maneuver and time sequences of M = {MI, M.}, and T {Tri, T2, 73}, respectively, where g represents the total state displacement resulting from the sequence. x= (x0o,M,r) (233) (fo,, T) GT3 (GM2 (GT2 (GM2 (GT, (fo, Ti)) ,72)) ,T3) (234) 2.4.2 Motion Planning with Primitives Planning a trajectory with motion primitives requires that a sequence of maneuvers, M, and the associated trim durations, r, are selected such that the system is driven from the initial configuration, xo, to a terminal configuration, xf, with favorable performance as evaluated by a performance function, J(M, r, xo). The trajectory planning problem for a system described by motion primitives can be described by the optimization problem 235. (M*,*) argmin J(M, r7,o) (235) M,T s.t. f Qo, M,) 0 (M,7,fYo) < 0 i I E M Vi ri > 0 Vi Problem 235 presents a difficult mixedinteger nonlinear program (\I NLP); however, the problem structure does lend itself to a hierarchical decomposition [39]. Given a maneuver sequence, M, Problem 235 reduces to a smooth NLP in the trim durations, 7, that is essentially analogous to an inverse kinematics problem. Such problems are a common and wellstudied class of problems in robot geometry [54]. Hence, a combinatorial search through the set of all possible M accompanied by the solution of a smooth NLP for each choice of M is generally required to solve Problem 235. The length, n*, of the optimal sequence, M*, is not known in general. As such, the set of all possible M is countably infinite; however, it can be shown that a finitelength optimal sequence exists and can be determined explicitly in some special cases. Additionally, the literature ,.. that pruning and branchandbound techniques can be applied to simplify and expedite the combinatorial search [39, 55]. Alternatively, consider a family of sequences where each member sequence consists of a fixed number of motion primitives. Each trim primitive in each member sequence differs in the magnitudes of the trim velocities; however,the general shape of the state trajectories remains unchanged. Maneuver primitives differ as needed between member sequences to accommodate the changes in trim states. As with the trim states, the general shape of these finiteduration trajectories should remain unchanged. A particular automaton model can be associated with a finite number of primitive sequences for a given family, as the model is comprised of finite number of primitives. Certain sequence families can be shown to admit unique solutions to the endpoint constraint of Problem 235. This constraint is repeated as Equation 236, where f represents the constrained terminal state and g(xo, M, r) represents the .. . regate state displacement of the primitive sequence characterized by M and r. At most one feasible solution exists to the continuous subproblem corresponding to each member sequence of families exhibiting this property. Further, Equation 236 can often be manipulated to yield a closedform expression for this family of solutions. As such, these potential solutions can be efficiently computed for each member sequence and evaluated for both performance and for satisfaction of path constraints. Note that differential path con straints are implicitly handled by the automaton representation of the dynamics. Feasible solutions can be enumerated for each sequence in a particular family and used to deter mine an approximate solution to Problem 235. Solutions obtained in this fashion can be shown to be optimal for some special cases, as will be demonstrated in Section 2.4.3. Xf g(X, M, 7) = 0 (236) This solution approximation method is imperfect, but efficient and effective for problems with few path constraints and with unique solution families consisting of short primitive sequences. Problems that contain more restrictive path constraints are less likely to yield feasible solutions via this method. Additionally, families that consist of many primitives admit a prohibitive number of sequence enumerations for all but trivial automaton models. The true utility of this method for the purposes of this dissertation comes as a local solution component within a randomized planning scheme, which is discussed in the next chapter. 2.4.3 Kinematic Example The Dubins car has served as a frequentlyused kinematic model to approximate solutions to a v ,ii I of nonholonomic motion planning problems [17, 18, 23, 30, 56, 57]. Further, the typical formulation of the model lends itself well to representation in terms of motion primitives. As such, this model serves as an appropriate and relevant model for use in a kinematic planning example. This example demonstrates the unique solution family approach to find approximate solutions to a problem of the form of Problem 235. Minimum time trajectories between two configurations are sought for the Dubins car model in an environment with no obstacles and, therefore, no path constraints. The Dubins car is a simple carlike vehicle model that operates in a Cspace spanned by two Euclidean position variables, p, and py, and an angle describing the heading, Q. The car moves with unit forward velocity and changes direction by assuming a unit turn rate, w, in either direction. As such, the motion of the Dubins car is described by the differential system shown in Equation 237, where the discrete set of values assumed by w is shown by Equation 238. 9p cos Q py sin (237) where: wE {1, 0, +1} /sec (238) A result of the limited set of turn rate commands is that the Dubins car moves with three basic motions: steady left turns, steady right turns, and steady forward motion. These three basic motions exhibit constant bodyfixed forward velocity, V 2 + p2, and bodyfixed turn rate, p, and therefore satisfy the definition of a trim primitive introduced in Section 2.4.1. These motions are invariant with respect to position and I. i1,i and therefore can be translated and rotated to originate from any point within the Cspace of the Dubins car. The transformations that describe evolution of the configuration variables along trim trajectories are described by Equations 239 to 2 41 for turning motions and Equations 242 to 244 for straight motion. Note that Equation 239 can represent motion along either a right or left turn based on the sign of w in Equation 241. Gtu (1xo, T) = xo + Rt,.(xfo) Aurn(r) (239) cos', sin',, 0 Rturn(io) = sin,',, cos ,',, 0 (240) 0 0 1 1 2(1 cos(Twr)cos () Atun(T) 1 2(1 cos(wr) sin () (241) aPT Gstraight(YO, 7') = + Rstraight(YO) straight(T) (242) cos ', sin 0 Rstraight(XY) = sin ',, cos ',, 0 (243) 0 0 1 T Astraight(T) = 0 0 (244) These three trim primitives can be integrated into a simple threestate automaton switching structure like that described in Section 2.4.1. The trim states are denoted L , S and R which correspond to the left, straight, and right trim motions, respectively. The six maneuvers that interconnect the trim states consist of instantaneous switches cor responding to an instantaneous change of turn rate, w. Recall that such an instantaneous switch is permissible for a kinematic model. In general, solution of a particular motion planning problem would require the solution of the MINLP expressed in Problem 235. A potentially infinite set of automaton sequences could be enumerated and the resulting constrained NLP solved for each. Alternatively, two families of solution sequences exist that uniquely satisfy Equation 2 36 when turning motion is restricted to heading changes of less than 27 rad. These lengththree sequence families consist of the set of all turnstraightturn sequences and the set of all turnturnturn sequences. The member sequences of these families can be enumerated for the current automaton model in terms of trim motions as D1 = {LSL, RSR, RSL, LSR} and D2 = {RLR, LRL}. The solution to Problem 235 can then be approximated by computing the solutions corresponding to each of the six sequences contained in D = (D1 U D2). Specifically, Equation 236 can be solved to obtain feasible trim durations for each sequence. The resulting set of six trajectories are then evaluated for time performance, where the minimum time trajectory is kept as the solution. Dubins showed in 1957 that the optimal minimum time trajectory for the Dubins car consists of motions described by one of the sequences in the set, ED [56]. Consequently, the approximate solution computed using the described method corresponds to the actual optimal solution for this special case. T ii:y sequences that are not contained in iD also satisfy the endpoint constraint; however, these solution sequences generally do not yield a unique solution. Feasible sequences of lengthfour or greater overdetermine Equation 236 in the trim durations and therefore generally admit an infinite number of solutions. While an optimal solution with respect to a particular performance criterion can be found through the solution of an NLP, these suboptimization problems do not necessarily guarantee desirable convexity and conditioning properties, ie., local and poorly conditioned solution cases may exist. Actual computation of the trim durations that satisfy the configuration endpoint constraint for sequences contained in ED can be achieved using several different tech niques from the literature [5863]. Here, a particular method which utilizes algebraic solutions is adopted [61]. This method requires that the initial and goal configurations are transformed to a special canonical coordinate frame, D, as shown in Figure 26. This coordinate frame has (px,oPyo) located at the origin and the goal configuration located a distance, d, along the XDaxis. Initial and final headings are measured from the line con necting the initial and final positions, as shown in Figure 26. Therefore the transformed initial and final configurations are given as Equations 245 and 246, respectively. YD 1 2 IF  W _ _   _ .... __ (0,0) XD (d, 0) Figure 26. Special reference frame, D for Dubins path solution. X,D = 0 0 (245) T Xf,D = d 0 1)2 (246) The transformations describing the left turn, right turn, and straight ahead motions are rewritten in an alternate form as Equations 247, 248, and 249, respectively. The resulting motions are unchanged as compared to the primitives defined previously. L(px,py, r) = (px + sin(p + r) sin py cos(Q + r) + cos + + r) (247) R(ppy,, r) = (px sin( r) + sin py + cos( r) cos b r) (248) S(px, py, ) = (px + cos y + Trsin Q) (249) For each sequence in D, the transformations in Equations 247, 248, and 249 can be applied to the initial configuration consecutively in the proper order through the composition operation. The result in each case is then equated to final configuration. A series of algebraic and trigonometric substitutions are then used to achieve closedform expressions for the trim durations, Ti, T2, T3. Infeasible solutions return complex values for at least one trim duration. The resulting closedform expressions for each sequence are [61]: 1. L(S(L(po,Ppyo,', 71), T2), T3)) (d, 0, 2) Ti 61 + arctan CS sinCOS {mod 2w} (250a) Sd + sin "1 sinS2 2 72 /2 + d2 2 cos(i 2) + 2d(sin sin,' .) (250b) ( COS' COS~ 7 73 2 w/ arctan CS" {mod 271} (250c) d + sin 1 sin s2 2. R (S(R(p., o, '71), 72), 73)) (d, 0, 2) 1 = "1 arctan S {od 2} (2 51a) d sini + sin2 mod 2 T2 /2 + d2 2cos(2i "b2) + 2d(sin '. sin 1) (251b) ( COSI COS d r r73 w 2 + arctan  sinC S I {rmod 27} (251c) 3 d sin 6 + sin' _. { 3. R(S(L(p,Ppyo,',, 71), 72),73)) (d,0, 2) ( cos i cos P2 2 T = ' + arctan sin CsOS' 2 arctan {mod 27} (252a) d + sin + sin 2 72 72 \2 + d2+ 2 cos(Q1i b2) + 2d(sin' i + sin 2) (252b) ( cosin cos '(r. (2 25 73 '+ arctan OS 6 cCSI arctan mod 2}) (252c) ( d + sin 6 + sin , 72 4. L(S(R(pz, 'o,, 71), 72),73)) (d,0, 2) ( cos, +cos, (2) Ti i arctan sin ' sin'2 + arctan ( ) {mod 27} (253a) ( d sin sin b2 )72 72 2 + d2 + 2 cos( 2) 2d(sin + sin 2) (253b) ( cos ~1i +cos ~2 2('2 73 arctan d in1 + COsin + arctan ( 2 {mod 27} (253c) d sin6 sin,'. 72 5. R(L(R(p ,py,',, 71,),T72),73)) (d,0, 2) ( cos 1 cos 2 72 T arctan sIin + ) {mod 27} (254a) d sin Q1 + sin ]2 2/ 72 arccos ((1/8) (6d2 + 2 cos( 2) + 2d (sin sin 2))) (2 54b) 73 2 71+ 72 (254c) 6. L(R(L(peo po',, T1),,T2),,T3)) (d, 0, 2) Ti 6 + arctan i( si + COS ) + ( {)mod 24} (255a) Sd + sin sin 22 2/ T2 = arTCOS ((1/8) (6d2 + 2 cos (Qi 2) + 2d ( sin i + sin b2))) (255b) 73 + 2 71 72 (255c) Figures 27A to 27F show sample solutions to the configuration endpoint constraint that have been generated using the explicit expressions for the Dubins car trim durations. Figures 27A to 27D show solutions from the turnstraightturn family of sequences while Figures 27E and 27F show solutions from the turnturnturn family sequences. The examples for each of these two families are generated using different goal configurations. Configurations were selected to show interesting trajectories for each sequence class. Generally, the optimal trajectory between two configurations is found by enumerating the six possibilities for a given configuration and then comparing the feasible results to find the optimum. Infeasible solutions could occur for a particular sequence and goal configuration, particularly if the goal is located very close to the initial condition. Additionally, infeasible solutions could occur for paths that intersect an obstacle region. As such, use of this technique as the primary approximation method to solve the full version of Problem 2 35 with many obstacle path constraints is not likely to yield a feasible trajectory. This technique demonstrates the most utility for obstaclefree problems or, alternatively, as a local method integrated into a randomized scheme, such as the methods introduced in C'!i ,l>ter 3. 3 2 1 0 1 2 01 0 1 2 1 2 2 0 2 4 6 0 2 4 6 XD 0 2 4 6 2 0 2 2 1 0 1 XD XD XD D E F Figure 27. Sample Dubins paths. A) Sample LSL trajectory. B) Sample RSR trajectory. C) Sample LSR trajectory. D) Sample RSL trajectory. E) Sample RLR trajectory. F) Sample LRL trajectory. 2.4.4 Dynamic Example This example considers the dynamic system described by Equation 256, which is an extension of the simple Dubins car model examined in Section 2.4.3. The vehicle described by Equation 256 moves with a constant forward velocity, V, and is controlled via the commanded turn rate, wc. Dynamics are introduced into the system in the form of a heading acceleration constraint. This acceleration constraint results in simple firstorder lag behavior of turn rate with respect to commanded turn rate. The lag filter exhibits a time constant of 7, = 0.25 sec. As such, the turnrate response to a step command achieves steadystate in approximately 11.25 sec, or 45 time constants following the command. V cos b V sin a)c (256) The system described by Equation 256 admits trim trajectories that behave accord ing to the kinematic conditions shown in Equations 257 to 260. A motion primitive model is formed through quantization of the dynamics according to a discrete set of turn rates. Specifically, this set is chosen such that the resulting trim trajectories are character ized by turn rates given by cE {Qa, 0, +max where max =30 deg/sec represents the maximum achievable turn rate. V = const. (257) = const. (258) = 0 (259) c = (260) The resulting hybrid system trim trajectories corresponding to this discrete set of motions consist of a maximumrate left turn, TL, straightahead motion, Ts, and a maximumrate right turn, TR. State evolution along the turning trims, TL and TR, is characterized by the transformation given in Equations 261 to 263. Similarly, state evolution along the straightahead trim trajectory, Ts, is characterized by Equations 2 64 to 266. Gturn (o, 7) = xo + Rturn(xo) aturn(r) (261) cos ',, sin ',, 0 0 sin, ,, cos ',, 0 0 Rturn(xo)= (262) 0 0 1 0 0 0 0 0 /2(1 cos(wr)cos(T) S V 2(1 cos(wr) sin ) () turn(T) =2 (263) aPT 0 Gstraight('o, 7) = F + Rstraight(ro) Astraight(T) (264) cos ,, sin',, 0 0 sin,',, cos ',, 0 0 Rstraight(Xo) = s (265) 0 0 1 0 0 0 0 0 T Astraight(T) = V 0 0 0 (266) A set of six maneuvers are computed such that a feasible transition exists between each pair of trim trajectories. Each of these maneuvers consists of the systems's dynamic response to a step command in wc. The initial and final values of this step command are determined by the initial and final trim trajectories connected by the maneuver. An example maneuver is shown in Figure 28, which shows the time history of each state for evolution along the maneuver connecting Ts and TR. The step input is shown as the dashed red line in the u plot. 0.5 1 1.5 2 0.5 1 1.5 2 t (sec) t (sec) 0.5 1 1.5 2 0.5 1 1.5 2 t (sec) t (sec) Figure 28. Example state trajectory showing maneuver dynamics. The time history for each state is carried out to t = 5 ,, such that w has reached 99.;:'. of its final value. The statedisplacement over each finiteduration maneuver is given by the transformation shown in Equations 267 to 269. The timeinvariant components of the displacement vector, AM, take on different values for each maneuver. The values for these components are dip, i,' .1 for all maneuvers in Table 21, where the maneuver connecting Ti to Tj is designated as .1 ;. GM(Yo) = o + RM(YO)AM (267) cos',, sin', 0 0 sin '1,, cos ',, 0 0 RM(xo)= (268) 0 0 1 0 0 0 0 1 AM [A, A A^ A ,] (269) Table 21. AM components for each maneuver. MSL MSR MLS MI R MLR MRL Az (ft) +54.1487 +54.1487 +55.9118 +55.9118 +55.3357 +55.3357 Ap, (ft) 12.2331 +12.2331 5.8870 +5.8870 +6.5181 6.5181 A, (deg) 30.0505 +30.0505 7.4495 +7.4495 +22.6011 22.6011 A, (deg/sec) 30.0000 +30.0000 +30.000 30.0000 +60.0000 60.0000 The effect of inserting maneuvers between each pair of trim trajectories is to cause a slight state displacement at the end of the first trim trajectory before the point at which evolution along the second trim trajectory begins. Consequently, closedform solutions derived in a similar manner to that described for the Dubins car in Section 2.4.3 are the oretically possible. However, solutions computed in this manner do not explicitly consider path constraints beyond merely checking for constraint satisfaction after generating the trajectory, as discussed previously. As an alternative, this example demonstrates a partial solution to the full MINLP shown in Problem 235 for which obstacle constraints are considered. The subsequent solution is denoted a Ip itial" solution because only a select few sequences are enumerated for the combinatorial sequence selection step. Enumerating the full set of all feasible sequences is impractical, as this set is countablyinfinite. Solving this problem therefore involves enumerating some of these sequences and comparing the local solutions that result from solving the smooth NLP associated with each. As such, any solution of this problem could be considered a Ip itial" solution. The example considers the environment shown in Figure 29. This environment contains two polygonal obstacles with a maximum dimension of 200 ft. The initial and goal vehicle states are given by Equations 270 and 271, respectively, where (*) indicates that the value is unconstrained. Both of these states are assumed to be along a trim trajectory. The trajectory optimization requires that a sequence of maneuvers, M, and a corresponding set of trim durations, 7, are found that take the vehicle from xo to Yx in minimum time while satisfying the differential path constraints given by Equation 2 56 and obstacle path constraints of the form shown in Equation 24. The differential constraints are encoded as part of the endpoint constraints through G(xo, M, r). The obstacle path constraints utilize the superellipse formulation of the obstacle boundary, as described in Section 2.2.2. Also note that each maneuver sequence, M, must have either MSL or MSR as its first element because the initial state lies along the trim trajectory, Ts. xo = 0 ft Oft 90deg 0 deg/sec (270) Y9 = 100 ft 650 ft 180 deg (*) deg/sec (271) 400 200 0 200 0 200 400 600 800 1000 Y (ft) Figure 29. Top view of environment for dynamic example. Trajectory generation proceeds by selecting a primitive sequence and solving the resulting smooth NLP subject to the endpoint constraints and path constraints. Recall that for the finitedimensional optimization considered here, the obstacle path constraints are enforced at a finite number of points along the trajectory. As such, the state must be computed using g(xo, M, r) at various intermediate points along each trim and maneuver for the NLP associated with each maneuver sequence, M. Optimization is performed using the TOMLAB/SNOPT optimization software package [53, 64]. The first sequence examined is M = {MSL, MLS, MSLt}, which requires solution of four trim durations. An overhead view of the solution is shown in Figure 210. Intermediate points at which obstacle constraints are enforced are depicted as circles along the solution trajectory. Blue circles indicate a trim segment while red circles indicate a maneuver segment. The corresponding heading angle and turn rate trajectories are shown in Figures 211A and 211B, respectively. In each of these figures, trim segments are depicted as solid blue lines and dynamic maneuver segments are depicted as dashed red lines. The solution trajectory requires a total of 29.7140 sec to reach x, from xo. 400 200  0  200 0 200 400 600 800 1000 y(ft) Figure 210. Top view of solution for M = {MSL, MLS, MSL}. 5 10 15 20 25 30 5 10 15 20 25 30 t (sec) t (sec) A B Figure 211. Heading and turn rate trajectories for M {MSL, MLS, MSL}. A) Heading angle trajectory. B) Turn rate trajectory. The next sequence examined is M = {MSL, MLS, MSL, MLS, MSL}, which requires solution of a set of six trim durations. An overhead view of the solution is shown in Figure 212. As before, intermediate points at which obstacle constraints are enforced are depicted as blue and red circles along the solution trajectory. The corresponding heading angle and turn rate trajectories are shown in Figures 213A and 213B, respectively. The solution trajectory requires a total of 29.3593 sec to reach xY from xo. 400 20 200 0 200 400 600 800 1000 Y(ft) Figure 212. Top view of solution for M = {MSL, MLS, MSL, MLS, MSL} 5 10 15 20 25 30 5 10 15 20 25 30 t(sec) t (sec) A B Figure 213. Heading and turn rate trajectories for M {MMSL, MLS, MSL, MLS, MSL} A) Heading angle trajectory. B) Turn rate trajectory. The final sequence examined is M = {MSL, MLS, MSR, MRS, MSR, MRs}, which requires solution of a set of seven trim durations. An overhead view of the solution is shown in Figure 214. As before, intermediate points at which obstacle constraints are enforced are depicted as blue and red circles along the solution trajectory. The corresponding heading angle and turn rate trajectories are shown in Figures 215A and 2 15B, respectively. The solution trajectory requires a total of 21.7796 sec to reach Xg from xo. 400 200 0 200 0 200 400 600 800 1000 y(ft) Figure 214. Top view of solution for M = {MSL, MLS, MSR, MRS, MSR, MRS}. 5 10 15 20 25 5 10 15 20 25 t (sec) t (sec) A B Figure 215. H. ilii: turn rate trajectories for M = {MSL, MLS, MSR, MRs, MSR, MRS}. A) Heading angle trajectory. B) Turn rate trajectory. Several relevant observations and conclusions can be drawn upon examination and comparison of the presented solution sequences. First, the longest sequence is seen to yield the minimumtime solution. As such, it is apparent that short sequences do not necessarily correlate to low cost trajectories. Second, the first two sequences yield similar solutions while the third sequence follows a markedly different trajectory through the environment. After enumeration of the first two, there is no indication that a better solution exists. The third solution sequence demonstrates a significant improvement in performance over the previous two solutions; however, there is no way to determine if this solution is optimal or if more sequences should be enumerated. Further enumeration of additional sequences can introduce combinatorial issues, as there are potentially an infinite number of sequences that yield a feasible solution. An automaton containing m maneuvers can be shown to yield m (m 1)"1 different sequences of length n. As such, termination criteria must be carefully selected based on the specific application. CHAPTER 3 RANDOMIZED SAMPLINGBASED MOTION PLANNING 3.1 Introduction Randomized motion planning has recently emerged as a powerful technique for solving complex problems. These algorithms proceed through a probabilistic search of the solution space, and are characterized by subdividing a difficult global problem into a set of easier local planning problems. Essentially, a large number of points are sampled from the C space and are connected to neighboring samples through a planning method that is locally valid. These connections form a i ... hi 'Ip that provides access to sampled areas of the Cspace and connectivity to other similarly sampled areas. Initial and goal configurations can be connected to the roadmap network if the space has been sampled effectively. This chapter discusses Probabilistic Roadmap Methods (PRM) and Rapidlyexploring Dense Tree (RDT) methods. PRM planners form a roadmap network to map out the configuration space for a particular system and can be used to solve planning problems in multitude. The approach adopted by PRM planners can become impractical for differentiallyconstrained systems, however. RDT methods generate a tree of feasible trajectories that is rooted at the initial condition of the system and expand in a depthfirst fashion such that the reachable set of the system is explored rapidly. Such planners have been effectively demonstrated for systems that are subject to differential constraints. Finally, Section 3.4 describes a specific implementation of a randomized planning approach that uses motion primitives to plan trajectories for a differentiallyconstrained system. 3.2 Probabilistic Roadmap Methods (PRM) The PRM planning algorithm was introduced by Kavraki as a method to find feasible solutions to problems with a complicated Cspace [41]. PRM planning algorithms probabilistically construct a roadmap of the Cspace of a system through sampling and connecting configurations. The PRM planner has been shown to be complete in a probabilistic sense [65, 66]. A planning algorithm is said to be complete if it is guaranteed to find a solution, provided one exists, and to terminate if one does not. Planning with PRMbased algorithms typically proceeds in four main steps: 1. Sampling: A large but finite number of configurations are sampled randomly from the Cspace. The distribution from which these points are sampled is a design choice and can drastically affect the performance of the motion planner [31, 44, 67]. Complex systems that operate in complex environments often require the use of heuristicbased sampling methods that bias sampling near obstacles, inside narrow passages, or in other areas of the Cspace that are deemed important for the particular system, environment, or task. 2. Nearest Neighbor: Each sample is grouped with a set of samples that are deter mined to be within a specified neighborhood. This neighborhood is determined using a distance function that is a valid metric or at least an approximation to a valid metric on the Cspace. 3. Local Planning: A path or trajectory is planned locally from each sample to each sample within its neighborhood. These interconnections form a network of local solutions. 4. Query: The initial configuration and goal configuration are added to the existing roadmap network by planning a path or trajectory between each of these configu rations and existing nodes located within a specified neighborhood. Once added, welldeveloped graph search techniques such as Dijksta's algorithm or the A* algorithm can be applied to search the roadmap network for feasible and/or favorable solutions [25, 68]. Figure 31 depicts an example of a roadmap network that has been formed in a simple twodimensional Cspace that contains three obstacles. Each node in the graph depicted in Figure 31 represents a system configuration and each edge of the graph represents a feasible trajectory between the two configurations at its endpoints. .0 A B C Figure 31. The PRM algorithm. A) Sampling step. B) Local planning step. C) Query and solution steps. PRM algorithms can simplify a planning problem on a complex Cspace to a simple graph search problem; however, computational issues arise for differentiallyconstrained systems and for highdimensional Cspaces. Sampling of the Cspace is not sufficient for systems with differential constraints, as local planning solutions between samples must consider configuration velocities as well. As such, the statespace must be sampled in lieu of the Cspace, thus doubling the dimension. Further, edges added through local planning will be necessarily unidirectional in many cases due to nonholonomic constraints present in many vehicle systems. As a result, twice as many local plans must be generated to achieve the same network connectivity as a system without these constraints. 3.3 Random Dense Tree Methods (RDT) RDTbased planners provide an alternative to the basic PRM paradigm that enables efficient solutions to differentiallyconstrained problems. While PRM generates a roadmap that describes the connectivity of many configurations to many other configurations, RDT methods generate a tree that is rooted at a specific initial condition and which describes connectivity of this initial condition to as many reachable configurations as possible. Algorithmic details ensure efficient and rapid exploration of the space. A drawback of RDT methods is that they are designed to solve a single planning problem at a time. This limitation is in contrast to PRM planning algorithms, which establish a network that spans the Cspace and can be used many times for many different planning tasks. A major benefit in this tradeoff is that RDT methods can often handle problems involving dynamic systems. In general, RDT methods incrementally build a search tree from an initial node in three main steps: 1. Node Selection: A node from the existing tree is selected as a location to add a branch. Selection of a particular node is usually based on probabilistic criteria that may require use of a valid distance metric. 2. Node Expansion: A local planning method is used to extend a feasible trajectory from the selected node. The local goal for this trajectory branch is determined probabilistically. 3. Evaluation: The new branch is evaluated according to performance criteria and often for connection to the goal configuration. Additionally, the new branch may be subdivided into multiple segments, thus adding several new nodes to the existing tree. A variety of RDTbased planners have been developed with numerous variations on the main steps listed previously, often to optimize performance for a specific application or to address a pathological case [6974]. Two algorithms, the Rapidlyexploring Random Tree (RRT) algorithm and the Expansive Spaces Tree (EST) algorithms, demonstrate different core exploration philosophies through the manner in which nodes are selected and expanded. These algorithms also serve as a basis for many of the existing variations on the general method, and hence prove useful as demonstrative examples. 3.3.1 RapidlyExploring Random Trees (RRT) The RRT algorithm was developed by Lavalle and Kuffner specifically to handle problems that involve dynamics and differential constraints [42, 71]. The algorithm biases tree growth toward unexplored areas of the space and hence focuses on rapid exploration. The node selection step is initiated with a sampled configuration that is chosen from a uniform distribution of the Cspace. A distance metric is then used to determine the closest point in the existing tree. During the expansion step, the selected node is extended incrementally "tcv ,iil the sampled configuration using a local planning method. This incremental extension is performed to varying degree in different versions of the algorithm and is ultimately a design parameter. Some versions use a fixed step size, others use a step size proportional to the distance from the sample, while others attempt to completely connect the sampled configuration to the existing tree. Figures 32A and 32B depict the RRT expansion process. Both images show a tree grown from the root node, No, in a twodimensional Cspace that contains obstacles. Figure 32A depicts the sampling step, in which a random configuration, Nrand, is selected and the nearest node in the existing tree, Nne,,, is determined. Figure 32B shows the expansion step, where a branch is incrementally extended from Nnear toward Nr"nd along the trajectory connecting the two configurations. A new node, Nnew, is added at the end of the new branch. The algorithm proceeds in this fashion until a branch of the tree reaches the goal within some specified tolerance. Nrand Nrand 0 ne0 A B Figure 32. RRT algorithm. A) Sampling step. B) Expansion step. 3.3.2 ExpansiveSpaces Trees (EST) The EST algorithm was developed by Hsu et al as a planning method to address problems involving highdimensional Cspaces and was later adopted to handle kinody namic planning problems [40, 75]. The EST algorithm explores space in a fundamentally different way than the RRT algorithm. Specifically, node selection occurs through the random selection of an existing node according to a probability distribution that is left as a design choice. This node is expanded within a local neighborhood that is defined by a valid distance metric. A configuration is sampled randomly from within this neighbor hood and a local planning method is used to connect the selected node to the sampled configuration. Figures 33A and 33B depict the EST expansion process. Both images show a tree grown from the root node, No, in a twodimensional Cspace that contains obstacles. Figure 33A depicts the node selection step, in which the expansion node, Ngxp, is selected from the existing nodes. The neighborhood of Nexp is defined here using a Euclidean distance metric and is shown as the area within the dashed circle in Figure 33A. Figure 3 3B shows the expansion step, where a random configuration, Nrand is selected from the neighborhood of Nexp and then a trajectory is planned from Nexp to Nrand. The algorithm proceeds in this fashion until a branch of the tree reaches the goal within some specified tolerance. o 0 ' 'Nrand Nrand A B Figure 33. EST algorithm. A) Node selection. B) Sampling and expansion. 3.3.3 Discussion It is important to note the fundamental difference between the vv in which the RRT and EST explore the space. Samples from empty space have a tendency to "pull" branches off of the tree built in the RRT algorithm. Thus, the space is rapidly spanned with coarse resolution. Continued sampling has the effect of improving the resolution of this exploration without appreciably changing the form of the solution. This concept is depicted in Figure 34A. Conversely, the EST selects a node randomly and tends to "push" branches from the selected node toward empty space as shown in Figure 34B. A benefit to this "piI!hi;, tendency is that the shape of the tree is continually evolving such that expansion is guided by the node sampling distribution. A wise choice of this distribution can favorably affect solution performance qualities; however, care must be taken to avoid biasing exploration toward previously explored areas. The performance qualities of the resulting trajectories planned using these algorithms can be affected by altering the various degrees of freedom for each planner. The explo ration behavior of the RRT is clearly dependent upon the distribution from which random configurations are sampled. For example, a nonuniform distribution can be chosen to bias exploration toward the goal; however, care must be taken to avoid pitfalls such as local minima. Additionally, RRT performance has been shown to exhibit sensitivity to the choice of a distance metric [42, 76]. C('! i.., of the exact distance metric, which is the op timal costtogo, may be impractical for many planning problems. Hence, an appropriate 1000 1000 1000 800 800 800 600 / 600 600 > 400 >< 400 >< 400 200 200 200 0 0 0 0 500 1000 0 500 1000 0 500 1000 Y (ft) Y (ft) Y (ft) A 1000 1000 1000 800 800 800 600 600 00 400 A 400 < 400 200 200 200 0 0 0 0 500 1000 0 500 1000 0 500 1000 Y (ft) Y (ft) Y (ft) B Figure 34. Differences in exploration strategy for the RRT algorithm vs. the EST algorithm. A) RRT expansion. B) EST expansion. approximation to this optimal distance metric must be chosen. Selection and computation of such an approximation can be nontrivial for complicated systems. As discussed previously, the EST provides a particularly useful design choice in the freedom to select the distribution from which expansion nodes are determined. A modified node selection scheme, denoted the Guided EST (GEST) assigns each node a probabilistic weight that reflects proximity to other nodes spatially and temporally and, more importantly, reflects a measure of performance [73, 77]. This explicit inclusion of performance cost in the sampling process enables the GEST to consider solution quality while exploring space. Moreover, the "push" tendency described previously allows potential solutions to continually improve. This improvement tendency results from the increased likelihood that as branches exhibiting desirable performance will be selected for expansion under the GEST framework. 3.4 RDTBased Dynamic Planner for a PlanarMotion Vehicle This section considers an RDTbased planner to generate approximate minimumtime trajectories for a differentiallyconstrained system, such as an aircraft. The planning problem considers a cluttered environment that forces the vehicle to operate within close proximity to obstacles. Consequently, differential motion constraints must be considered to ensure safe tracking of trajectories. An RDT planner is used as a practical alternative to optimal solutions offered by optimal control or solutions to the MINLP posed in Problem 235. The method presented here utilizes algebraic solutions for planning with motionprimitive models as described in Section 2.4.2. These algebraic solutions are used to generate local, obstaclefree trajectories as part of the tree expansion process. 3.4.1 Model This planner considers a hybrid motion primitive model that behaves according to the dynamics described by Equation 31. The single input to the differential system is the turnrate, w. The constant translational velocity, V, is constrained differentially to act in the direction of the vehicle heading. The system described by Equation 31 admits trim trajectories that belong to two families: constant rate turns and straight forward motion. S V cos Q p4 V sin Q (31) Here, trim trajectories behave according to the kinematic conditions shown in Equations 32 and 33. A motion primitive model can be formed by selecting a set of trim primitives that behave according to these conditions. V = const. (32) = = const. (33) A set of 2n + 1 trim primitives are selected which consist of constantrate turns at n different turn rates in each direction along with a straightahead primitive that corresponds to w = 0. This set is shown as Equation 34, where Qmax =30 deg/sec. The velocity, V, is held fixed over the set of all primitives. S{E fO, w1L, 2 ,.n} wil < Qmax,i = I, (34) The allowable switching behavior between these primitives is characterized by the automaton structure shown as a directed graph in Figure 35. Each trim trajectory is depicted in the figure as a state of the automaton and is designated by the turn rate corresponding to that trim state. 1 1 Figure 35. Automaton structure for vehicle model used with current planning strategy. State evolution along each turning trim characterized by wi is described by the time parameterized transformation given in Equations 35 to 37. Similarly, state evolution along the straightahead trim trajectory characterized by w = 0 is characterized by Equations 38 to 310. Gturn, i(fo, ) = Fo+ R (iurnYo) tAurn,&ir) (35) cos' ,, sin ',, 0 0 sin,',, cos ',, 0 0 Return (xo) = (36) 0 0 1 0 0 0 0 0 vV 2(1 cos(wrT) cos (j) S2(1 cos(w,) sin () 0PT 0 Gstraight(Yo, 7) Yo + Rstracight(o) A straight() cos ',, sin',, 0 sin,',, cos ',, 0 0 Rstraight (Xo) = c (39) 0 0 1 0 0 0 0 0 T Astraight(T) = VT 0 0 0 (310) As presented, the model is classified as kinematic, ie. uw can change value instanta neously. Therefore, maneuvers are considered as instantaneous switches between trim trajectories, and the corresponding transformation which describes the configuration transition for maneuvers is simply an identity matrix. As described for the Dubins car model, the hybrid model presented here admits unique solutions between any two configurations. These solutions exist as members of up to two families of sequences: a turn followed by a straight segment followed by another turn, or three consecutive turns. If the model consists of three trim primitives, a straight forward motion at constant velocity and steady, maximum rate turns in each direction, the set of unique solutions is equivalent to the Dubins set, D, which has six members. When 2n turning trim primitives are considered which consist of opposite direction turns at rates of n different magnitudes, the set of turnstraightturn sequences is expanded to size 4n2 and the set of turnturnturn sequences is expanded to size 2n3. The trim durations corresponding to each of these sequences can be expressed in closed form, where infeasible solutions yield complexvalued trim durations. Thus, computation of an approximation to the optimal trajectory requires that 4n2 + 2n3 expressions are evaluated and are then compared for cost performance. Only turnstraightturn sequences are considered with variable turn rate to reduce computational burden, and only turnturnturn sequences at the nominal (maximum) turn (38) rate are considered. Thus, consideration of variable turn rate Dubins sequences results in 4n2 + 2 unique solutions for the trajectory between any two configurations. The equations that determine the trim durations are similar to those presented in Section 2.4.3; however, the added consideration of variable turn rates makes explicit presentation somewhat cumbersome. While the actual expressions are not shown here, the solution procedure is identical and is easily performed using symbolic algebra software. In addition to the unique solution families between two points in the obstaclefree Cspace, unique solution families also exist on a useful subspace. This subspace consists of the position variables, px and py, and is denoted, P. Motion plans ending at a point on P are not subject to constrained heading, ), at the endpoint. These solution families consist of turnstraight member sequences. Hence, for a model containing 2n turning trim primitives comprised of n rates in opposite directions, there are 2n turnstraight members in the unique solution set. Solutions for the trim durations are shown for the leftstraight family in Equations 31 a and 31 b and for the rightstraight family in Equations 312a and 312b. V aVd cos + sin b + 7,i = + arctan ) {mod 2} (311a) 72 Sin +d) +( COS 2Vd r2i sin + d2 (311 b) Wi 7V2 Vcosi (()2si l Vd 7r1i = i+ arctan si n I {mrnod 2w} (312a) 72 Sin +d) cos 1 2Vd 2,i = Sin 1i + d2 (312b) LWi While the model considered here exhibits kinematicallyconstrained motion, the addition of simple dynamics in the form of monotonic transitions between trim states does not appreciably change the behavior of the model. Actually, the procedure for determining unique solutions for trim durations remains identical: a sequence of transformations are equated to the desired terminal configuration and the expressions are solved in terms of the trim durations. The dynamic motions appear in the string of transformations as a finite displacement in each configuration variable accompanied by a finite shift in time. Thus, generality is not affected in the subsequent development. 3.4.2 Overview The algorithm presented here proceeds with the "pull" expansion ]i1i .. .l hi, exhib ited by the RRT. A point from the Cspace is selected and a new branch is "pulled" from an existing branch that is determined to be nearby. While the EST "puil! approach demonstrates a number of benefits, especially in achieving improved performance of resulting trajectories, the RRT approach is selected for its efficiency and robustness in finding feasible solutions. Experience with the EST approach found significant sensitivity of convergence properties with respect to the various degrees of freedom allowed by the algorithm. The main steps of the algorithm are detailed in the next three sections. These three main steps can be summarized as follows: 1. Select a Node: A point is selected from the subspace of the Cspace, P, which is spanned by the position variables. An approximate, obstaclefree distance metric is used to determine the nearest node in the existing tree. 2. Extend a Branch: The set of 2n unique solutions on the position subspace are enumerated, evaluated, and pruned. Selection criteria is used to choose a branch from the set for addition to the solution tree. 3. Check for Solutions: The new branch is split into an intermediate set of nodes. Each node is tested for obstaclefree connection to the goal configuration on the full Cspace. If appropriate, new solutions are added to a solution list and the current upper bound is updated. Finally, a significant difference between the nominal RRT algorithm discussed in Section 3.3.1 and the algorithm presented here is related to algorithm termination. The nominal RRT algorithm seeks a single feasible solution. Here, the first feasible solution is adopted as a performance upper bound. The algorithm continues to generate new solutions and update this upper bound in an attempt to arrive at a better solution. Thus, an independent termination condition must be set that reflects either a measure of convergence or a measure of computational resource usage. 3.4.3 Node Selection The first step in each iteration is to select a node, Nexp, from the existing tree for expansion. This expansion node is selected as the closest node to a randomly sampled point, psamp E P, where P represents the twodimensional position subspace. The reduction in search dimension that is achieved through sampling on P is allowable because there are no attitudebased constraints on the trajectory besides those enforced at the initial and goal configurations. Planning on this subspace is sufficient to generate trajectories that navigate the obstacle field. Moreover, there is a substantial complexity benefit to the upcoming node expansion step in that the number of explicit solution sequences that must be evaluated is reduced by an order of magnitude from 4n2 + 2 to 2n, as described in Section 3.4.1. The distribution on P from which psamp is chosen pl i, a role in planner perfor mance.A uniform distribution over P maximizes space exploration such that trajectories might 'wander' about the space before achieving a feasible solution. Conversely, a distribu tion that heavily emphasizes the goal configuration di pl'v' greedy goalseeking behavior that is subject to local minima [31]. Ideally a balance should be achieved between these two behaviors. As such, samples are selected from one of two distributions, each of which is assigned a probability that it will be used to generate the current sample. The two distributions consist of a uniform distribution on P and a Gaussian distribution centered at the goal position on P. Thus, sampling proceeds by first selecting a distribution and then sampling a position from the selected distribution. Once a point is selected, nearby nodes in the existing tree must be determined. An intuitive distance metric such as the Euclidean 2norm does not capture the dependence of path length on differentiallyconstrained motions. Specifically, the vehicle is generally required to change direction from the configuration at Nexp before proceeding towards Psamp. This behavior results in .ivmmetric distance measures, eg. a point immediately in front of the vehicle is closer than a point immediately to the side of the vehicle, even if the two points are the same Euclidean distance from the current position. The ideal distance metric is the optimal costtogo; however, computation of this value is equally as difficult as the original planning problem in many cases [42]. Therefore, the minimum obstaclefree turnstraight trajectory from a configuration to a point on P is considered as an approximate distance metric. This minimum occurs when the turning segment is executed at maximum turn rate. A plot of distances measured using this metric is shown in Figure 36A. The vehicle has a configuration of (px,Py, 0) (0,0,0) in Figure 36A, such that distance is measured from the center of the plot where the vehicle is initially pointing along the p1axis. For comparison, a plot of the Euclidean 2norm is shown in Figure 36B for the same vehicle configuration. Inspection of the figures clearly shows discrepancies between the two metrics. O!t ..600ft 2Ea0t ( ) oEast (f)f A B 1. I.I.... 400 ft East (ft) East (ft) A B Figure 36. Distance function comparison. A) Approximation of obstaclefree costtogo. B) Euclidean distance function. Actual computation of the distance from each node to Psamp using the metric depicted in Figure 36A is accomplished by computing the minimum turnstraight trim durations to reach Psamp from each node in the tree. The minimum for each of these primitives individually occurs at the maximum turn rate. As such, Equations 311a, 311b, 312a, and 312b with wi = max are used to compute the candidate sequences. The global minimum for the turnstraight family is then found through direct comparison of the minimum leftstraight and minimum rightstraight sequences. This process is shown in Figure 37. 400 300 300 200 200 100 100 10 0 100 100 200 200 0 200 400 200 0 200 400 Y (ft) Y (ft) A B Figure 37. Distance Function Computation. A) Sampled point. B) Turnstraight paths from each node. The Node Selection step exposes a troubling practical reality associated with imple mentation of RRT algorithms: that a distance computation must be performed at every node at each iteration. While this operation is often computationally cheap, as is the case here, new nodes are added to the tree at each step. In the limit, an infinite number of nodes are involved in a distance measurement operation. Fortunately, in practice, the algorithm tends to converge before this issue becomes problematic; however, such an observation is difficult in theory to prove nor guarantee. 3.4.4 Node Expansion After Nexp is selected, a branch is extended from the configuration at Nexp to the sampled position, psamp. Unique solution sequences for the trajectory from a configuration to a point on P are members of the family of 2n turnstraight sequences, as discussed in Section 3.4.1. Trim duration times are computed using Equations 311a, 311b, 312a, and 312b. Some values of psamp yield infeasible solutions to certain sequences. These infeasibilities are reflected as complexvalued trim durations. Thus, enumerating the 2n members of the turnstraight family of sequences in fact yields a maximum of 2n candidate branches. Each candidate branch must be evaluated for both cost performance and safety. The minimumtime cost performance is trivially evaluated by summing the trim durations and switching maneuver time for each candidate.Maneuver time evaluates to zero for the instantaneous switch case considered here. Safety is evaluated by checking points along each candidate trajectory for collisions. Points are generated at intermediate time values and each is checked using a collision detection method as described in Section 2.2.2. Care must be taken in selection of a time step in order to ensure that the points are spaced closely enough to adequately represent the continuous trajectory without imposing undue computational burden. Upon detection of a collision, a candidate branch is pruned such that the end of the branch lies a distance of one minimum turn radius short of the first collision point along the original candidate branch. This step ensures that the branch can be safely extended beyond the current endpoint without a collision. This process is depicted in Figure 38. After pruning, a candidate is selected according to Euclidean distance from psamp and minimum time properties. 400 400 200 ._ 200 0 0 200 0 200 400 600 200 0 200 400 600 Y (ft) Y (ft) A B Figure 38. Node expansion step. A) The new branch is checked for pointwise collisions. B) The new branch is pruned beyond the first collision. 3.4.5 Solution Check After the addition of a new branch to the tree, new nodes, N,,, must be added. The new branch is subdivided by a fixed time step as shown in Figure 39. The time step for subdivision is left as a design choice. More nodes per branch generate a greater v ,i, I, i of starting points for future node expansion, Ngxp, but induce greater computational burden in terms of number of operations required for the nearest node operation as well as with respect to memory capacity required to store tree data. Information regarding total cost, automaton sequence, trim durations, and node precedence is computed and stored at each node as it is added to the tree. 600 400 X 200 0 200 0 200 400 600 Y (ft) Figure 39. The new branch is subdivided to a set of nodes After computing and adding nodes, each node is tested for connection to the goal configuration. This connection test requires that trim durations for the 4n2 + 2 variable rate sequences are computed for each new node, N,,w,i, as shown in Figure 310. The lengththree sequences must be used to satisfy the heading constraint at the goal config uration. The solution for each of these sequences is expressed in closed form and as such, complexity does not exceed O(n2) operations. The cost for feasible sequences originating at Ne,,i is added to the total cost for Ne,,i. This value is compared to the current upper bound for the minimum safe and feasible solution. Each feasible sequence that represents an improvement to the current upper bound is checked for safety with respect to obstacle collisions. Sequences that improve the upper bound while maintaining feasibility and safety then replace the current best trajectory. Other sequences are disregarded. 600 400 400 200200 0 0 200 0 200 400 600 200 0 200 400 600 Y (ft) Y (ft) A B Figure 310. Collisionfree solutions. A) Rightstraightright family. B) Leftstraightright family. After each new node has been tested for feasible and safe reachability of the goal configuration, the termination condition is checked. As discussed previously, this termi nation condition can vary according to the specific restrictions on current operation of the planner. Termination can depend on runtime, memory usage, or even after a certain number of valid solutions have been achieved. Upon termination, the planner returns a precedence list of nodes that lie along the solution trajectory from the initial configuration to the goal configuration. If the termination condition has not been reached at the end of a particular iteration, execution returns to the Node Selection step. 3.5 Example A sample planning problem is solved in this section to demonstrate the utility of the RDTbased planning algorithm introduced in Section 3.4. This problem considers a vehicle whose motion is governed by the continuous differential system shown previously as Equation 31. As such, the vehicle moves with forward velocity, V, and is subject to a bounded turn rate, i\ < 30 deg/sec. The single system input is the commanded turn rate, Cu. As previously, the model is characterized as kinematic. The motion of this vehicle is modeled using a hybrid motion model with seven trim states. Formulation of this model requires the selection of trim and maneuver primitives. Maneuvers between trim states occur instantaneously due to the kinematic description of vehicle motion. The trim states are characterized by constant system velocities as follows: To =(Vo,wo) = (45 ft/sec, 0 deg/sec) T, = (Via, wa) = (45 ft/sec, + 30 deg/sec) T1b = (V1b, Ulb) = (45 ft/sec, 30 deg/sec) T2a = (V2a, 2a) = (45 ft/sec, + 20 deg/sec) T2b = (V2b, 2b) = (45 ft/sec, 20 deg/sec) T3a = (V3a, 13a) = (45 ft/sec, + 15 deg/sec) T3b = (V3b, W3b) = (45 ft/sec, 15 deg/sec) The problem of planning a trajectory between two configurations admits two families of unique solution sequences for the obstaclefree case. These families consist of turn straightturn sequences and turnturnturn sequences, respectively. Sequences exhibiting variable turn rate are considered for the turnstraightturn family. Here, n = 3 giving a total of 4n2 = 36 sequences in the turnstraightturn family. Example solutions are plotted in Figure 311A, which shows trajectories that drive the system from an initial condition of (px,, Py,, '',,) = (0, 0, 0) to a terminal configuration of (Pf,Pyf, f) = (0, 800, 0). Only maximum rate turns are considered for the turnturnturn family of sequences, as discussed in Section 3.4.1. Examples of the two resulting solution trajectories are shown in Figure 311B, which shows trajectories that drive the system from an initial condition of (Px,o,Py,o, ',') = (0,0,0) to a terminal configuration of (p,f,Pyf, f) = (0,100,0). In addition to the solution sequence families for planning between two configurations, a solution family exist for planning from a configuration to a point on the subspace, 2, which consists of the position variables, p, and py. Variable rate turns are considered for this solution family, which is characterized by turnstraight sequences. There are 500 100 0 0 100 500 200 0 500 1000 200 0 200 Y (ft) Y (ft) A B Figure 311. Unique solution families for vehicle used in example. A) Turnstraightturn solution sequences. B) Turnturnturn solution sequences. a total of 2n = 6 members in this solution family. Example solutions are plotted in Figure 312, which shows trajectories that drive the system from an initial condition of (px,o,Py,o,',) = (0,0,0) to a terminal point on P located at (Px,f,Py,f) = (200,500). 200 200 200 0 200 400 Y (ft) Figure 312. Turnstraight solution sequences. The planning environment considered for the example problem is shown in Figure 3 13. For this example, the inertial reference frame is oriented such that the xaxis points in the North direction and the yaxis points in the East direction. In the figure, the bounds on the extent of the environment are shown as the blue square that contains 0 < x < 1000 ft and 0 < y < 1000 ft. All sampling steps draw samples from within these bounds. Also, p" Iv._aonal obstacles are randomly placed within this region. 200 The example problem considers eight square obstacles with sides of length 150 ft, as shown in Figure 313. Sampling distributions are not defined within obstacle regions. Finally, the initial and goal configurations are specified as (px,o,py,o ,,) (0, 0, 0) and (px,fPf, P f) = (1000, 1000, 0), respectively. 1000 S500 7 U 0 0 500 1000 East (ft) Figure 313. Example planning environment. The algorithm described in Section 3.4 is executed for sixty iterations. Figure 314 shows the expansion of the search tree at various intermediate stages progressing from left to right. Samples on P are drawn from either a uniform distribution defined over the obstaclefree portions of the planning environment or a Gaussian distribution centered at the goal location with a standard deviation of 250 ft that is also only valid over the obstaclefree portions of the environment. The uniform distribution is chosen with a probability of 0.75 and the Gaussian distribution is chosen with a probability of 0.25. 1000 1000 1000 500 500 500 o o , 0 0 0 0 500 1000 0 500 1000 0 500 1000 East (ft) East (ft) East (ft) A B C Figure 314. Incremental tree expansion. A) After 1 iteration. B) After 25 iterations. C) After 60 iterations. The image at the left of Figure 314 shows the turnstraight branch extended from the initial configuration to the first sample. The center image dipl ', the tree after twenty five iterations. Note that branches are not necessarily added during every iteration, as certain obstacle collisions can prune an entire branch. The image at the right of Figure 3 14 shows tree expansion after sixty iterations. Nodes are added at intermediate time steps along each branch as it is appended to the tree, as discussed in Section 3.4.5. Nodes in the final tree after sixty iterations are shown in Figure 315. These nodes serve as starting points for additional branches as well as for solution trajectories. 1000 500 0 0 500 1000 East (ft) Figure 315. Nodes in the final solution tree. Each node is tested for feasible connection to the goal configuration as it is generated. Successful connections that improve the currentbest solution replace this solution for future iterations. Figure 316 shows the refined solution for the present example as tree expansion proceeds. In each of the plots shown in Figure 316, the precedent path from the initial configuration along the tree branches to the tested node is depicted as a solid blue line. The solution trajectory from the tested node to the goal configuration is depicted as a solid red line. 0 500 1000 0 500 1000 0 500 1000 East (ft) East (ft) East (ft) A B C Figure 316. Solution path refinement. A) After 7 iterations. B) After 14 iterations. C) After 37 iterations. The image at the left of Figure 316 depicts the first valid solution found by the algorithm. This solution was computed during the seventh iteration. Hence, this example demonstrates the ability of the planner to a find a feasible, but suboptimal, solution very quickly. The solution shown in the center image of Figure 316 is computed during the fourteenth iteration. Finally, the minimumtime solution found for the example run is shown at the right of Figure 316. This solution was computed during the thirtyseventh iteration. As such, the final twentythree iterations did not contribute to the solution and could be considered a waste of computational resources. Also note that the improvement of each solution over previous solutions is fairly benign, due to the nature of the RRT expansion procedure. Unfortunately, there is no way to determine if a particular solution is potentiallyimprovable or how long such potential improvement will take. Therefore, the termination condition remains a problemspecific parameter. 1000 1000 1000 CHAPTER 4 SENSING EFFECTIVENESS 4.1 Introduction Sensing tasks are identified in C'! ipter 1 as one of the primary application domains for unmanned systems. Guidance trajectory design methods that ensure desirable sensor placement represent a required technology to enable such missions. Before trajectory design methods can be discussed, a metric to define desirable viewpoints must be estab lished. The formulation of such a metric is seen to be especially relevant for the socalled i i!!11 .., sensing missions discussed in this dissertation. This dissertation considers the class of LOS remote sensors. These sensors collect data in the form of reflected energy from remote surfaces that exhibit clear LOS from the sensor and which are contained within the sensor FOV. Data collection can be either active or passive in nature, depending on the specific sensor. In either case, the sensing operation can be characterized geometrically. The geometric relationships that relate energy reflections seen by the sensor are dependent mainly on the relative position and orientation of the reflecting surface with respect to the sensor. Sensor performance is affected by variations in the energy that is reflected from each particular target surface. Additionally, variations in the relative orientation of the target surface with respect to the sensor can result in data distortions and degraded resolution. The perspective from which targets are seen continually varies while the sensor moves through the environment. As such, sensing performance with respect to a particular target surface can vary appreciably as the sensor is moved relative to that target's position. Furthermore, .idi i:ent objects are seen from a slightly different perspective even for a stationary sensor. Hence, sensing performance can show significant variation over the sensor FOV. This latter effect is especially pronounced for sensor operation in close proximity to the environment. Such a scenario involves target surfaces appearing at various relative ranges and aspects with respect to the sensor. Close proximity operation also increases the severity of effects resulting from timevarying behavior of these relative geometric parameters. These effects indicate that the data product can vary significantly with respect to small variations in guidance trajectory planning and tracking. As such, these variations should be quantified for integration into the motion planning process to ensure that planned trajectories yield the desired data product. AT !ii: current research efforts that involve vehiclebased sensor planning treat visibility in a binary sense and therefore fail to model these variations effectively [17, 23, 7880]. This chapter discusses a geometric approach to quantify the effectiveness of a partic ular sensortarget configuration. First, a brief discussion of some of the relevant remote sensor technologies to the applications considered provides context to the subsequent pre sentation. Then, a geometric model of LOS sensor operation is introduced in Section 4.3. A parameterization of the relative geometry is used to define the concept of visibility, which is discussed in the context of sensing tasks in closeproximity environments. Finally, a generalized sensing effectiveness metric is formulated that quantifies the quality with which a particular point in the environment has been sensed. Several specific examples demonstrate how this concept can be used to evaluate sensing mission performance. 4.2 Remote Sensor Technologies This section introduces some of the general characteristics of the sensors relevant to this dissertation. These sensors fall into the class of vehiclecarried remote sensors which gather information about the environment using reflected electromagnetic energy. Hence, contact is not required for data collection and sensing tasks can be performed from a distance. The sensors discussed in this section include video cameras, radar, sonar, and laser sensors. The quality of data collected by each of these sensors is affected by generallysimilar geometric factors. Sensorspecific quality factors will not be considered in order to maintain generality. Hence, the detailed physical processes that enable sensor functionality are considered as beyond the scope of this dissertation. 4.2.1 Computer Vision Computer vision is a passive, informationrich sensor technology. Sequences of images collected using a video camera provide a twodimensional representation of the three dimensional environment. The mapping of the environment to a twodimensional image is described by projective geometry. A point in the environment is projected onto the focal plane along the LOS. As such, the depth dimension is lost in the mapping [81]. Despite this loss of depth information, a great deal of information can be gathered from two dimensional images using vision processing techniques. Points of special signif icance denoted 'feature points' can be identified and extracted from a twodimensional image. Overlapping views allow feature points in sequential images to be correlated and tracked. The motion of these tracked features within the image plane can be used with structure from motion (SFM) techniques to reconstruct the depth dimension in addition to estimating camera motion [82]. Such techniques have received increasing attention in recent years for the purposes of vehicle navigation and control [8387]. Full reconstruction of the threedimensional geometry is not necessarily required to render twodimensional images useful. A variety of applications in both the civil and military domains employ visiblelight imagery. Color and shape information related to objects within the camera FOV can be used for identification and tracking tasks. 4.2.2 Radar Radar is an active sensor that uses electromagnetic energy in the microwave band. Pulses of electromagnetic energy are emitted from the transmitting antenna and are scattered by objects that lie within the antenna footprint. Some of the scattered energy is reflected back to a receiving antenna. The nature of this scattering is affected by the material properties of the targets and by problem geometry. The time lapse between these events is used to compute a range measurement. Resolution of radar data can present issues for mapping and imaging applications. Essentially, the range resolution is dictated by the pulse wavelength and incidence angle while the bearing resolution is dictated by the physical size of the aperture, or an tenna [88]. The range resolution has been addressed in recent years through the use of ul tra wideband (UWB) radar systems. These systems use a wide range of energy frequencies to produce better range resolution than standard radar. The bearing resolution depen dency presents a difficult tradeoff in that highresolution data requires unrealisticallylarge antennae. Syntheticaperture radar (SAR) processing is a technique that uses the vehicle motion to enhance the b. ,iii. or azimuth, resolution. The concept behind SAR processing is that successive measurements from a small, moving radar antenna can be synthesized to a much larger beamwidth. Thus, when a particular point in the environment is illuminated by multiple radar pulses, the returns can be processed as if the point is illuminated by a single pulse from a large synthetic aperture. Advances in UWB radar systems and SAR processing techniques have begun to allow new capabilities and applications for radar. The information content of radar data is dependent on the frequency band in which the system operates. Pulsed UWB systems utilize a shortduration pulse with support at frequencies that are both low enough to penetrate foliage and soil and that are high enough to resolve small objects [8992]. Use of SAR processing techniques enables high spatial resolution. Predicted advances in computational and processing techniques i'. 1 that these radar systems will eventually exhibit imaging capabilities at frame rates comparable to visiblelight imaging sensors [92, 93]. Hence, UWB radar poses great benefit to a variety of applications dealing with the detection of concealed objects such as buried landmines or enemy forces in deep cover [89, 92, 94, 95]. 4.2.3 Sonar Sonar operates on principles similar to radar; however, acoustic energy is transmitted into the environment and reflected back to the sensor as opposed to electromagnetic energy. The return is rangegated based on the timelapse since the pulse was transmitted. Sonar arrays can be implemented to yield bearing measurements in addition to the range measurements. Sonar range sensors are used in a variety of applications. Acoustic energy travels more efficiently through water than it does through the lessdense atmosphere. Therefore, sonar is often the remote sensor of choice for underwater applications. Sonar sensors are also frequently used for shortdistance ranging in mobile robotics applications [9699]. The resulting data can be used for navigation, target detection, and even for mapping and imaging applications. The mapping and imaging applications require the use of syntheticaperture sonar (SAS) processing, which is analogous to SAR processing of radar data [100, 101]. 4.2.4 Ladar Scannerless imaging ladar (LAser Detection And Ranging) is an emerging sensor technology that exhibits threedimensional imaging capability. Ladar systems transmit pulses of coherent light in the visible, nearinfrared (NIR), and infrared (IR) energy bands. These energy pulses are reflected back to the sensor by objects in the environment. The time lapse between transmission and reflection is used along with knowledge of the pulse physical properties to determine a range measurement to the reflecting object. Complex mechanical scanner systems are typically used to cover a wide swath of the environment with laser pulses. Accurate range and bearing measurements are possible using such a system; however, extensive processing is often required to resolve the threedimensional measurements. Recent efforts have applied standard optics to project the laser reflections to a focal plane array (FPA) [102, 103]. Essentially, a single transmitted laser pulse can result in a twodimensional image that contains both range and reflection intensity information for each pixel. Ladar systems are currently under development with the expectation that these range images will be produced at framerates comparable to standard video cameras. These research efforts have resulted in several prototype sensors that exhibit much smaller size, lighter weight, and lower power requirements than typical laser scanning systems [102104]. The potential applications for a sensor with threedimensional imaging capabilities are numerous and varied. The reduction in size, complexity, and power compared to scanning systems allows a much broader class of vehicle to consider imaging ladar as a sensor sys tem. Scannerless imaging ladar technology has been investigated for use with autonomous navigation systems and could be extended to other applications, such as remote detection of concealed objects or targets [104]. The NATO Research and Technology Organization predicts that ii. i, pixel arrays making realtime multidimensional measurements at 10 km with frame rates of 30 Hz or f , I will be a reality within a decade [2]. Addi tionally, the energy properties of these future advanced ladar systems could be used to identify chemicals, pollutants, and organisms along with characteristic vibration properties of living or manmade objects [2]. 4.3 Modeling the Sensing Task 4.3.1 Sensing Geometry The geometry for the general case of a vehicle carrying a single LOS sensor is described by the vector diagram of Figure 41. The vehicle body basis, B, is fixed at the vehicle center of gravity and is located relative to the inertial frame, E, by the time varying vector iB. The orientation of B is described relative to E through the typical Euler angle parameterization, which is expressed as KB as in Equation 41. The Euler angles are all timevarying as well. The timeevolution of both JB and KB is governed by the vehicle dynamics. B B OB OB QB (41) The timevarying transformation from E to B is denoted TEB and is a function of IB. This transformation consists of the standard 321 sequence of singleaxis rotations shown in Equation 42. PB At Figure 41. Sensing problem geometry. 1 0 0 cos OB 0 sin OB cos 5B sin fB 0 TEB B) 0 cos B sin fB 0 1 0 sin aB COs B 0 0 sin (B cos B sin OB 0 cos 0B 0 0 1 (42) The sensor reference frame is denoted S. The basis vector, 83, is denoted the sensor axis and is oriented such that it points along the LOS at the center of the sensor FOV. Without loss of generality, the sensor is assumed to be fixed to the vehicle and is located relative to B by the vector p1. The inertial position of S relative to E can then be com puted as given by Equation 43. The sensor is assumed to have a known orientation with respect to B, which is given by the constant transformation, TBS. The transformation from E to S is then given by Equation 44. The orientation of S with respect to E can be expressed using an Euler angle parameterization, as well. The timevarying vector of sensor Euler angles is shown in Equation 45. The transformation, TES, can be expressed relative to these parameters as in Equation 46. The sensor Euler angles, 4I, can then be determined in terms of KB by equating the right hand sides of Equations 44 and 46 then solving the resulting system. PA Ps P= B + (TEB) l (4 3) TES (B) TBS TEB B) (4 4) s = s IT s (45) 1 0 0 cos O 0 sin Os cos ~ sin ~ 0 TES (s) 0 cos, sing 0 1 0 sing cos&, 0 (46) 0 sin ( cos sin 0O 0 cos Os 0 0 1 The ith pointfeature, or target, in the environment is located relative to E by ;t,i. This point is considered to lie on a surface with an associated normal direction at pt,i designated by the unit vector, hi. The components of these vectors are expressed relative to E. Here, ~,i and hi are considered constant in time, ie. the environment is static relative to E. The relative position of the ith target with respect to S can be expressed as the vector sum shown in Equation 47. Relative orientation is described by the orientation of ii relative to the sensor axis, S3. This relationship is characterized through expression of ii relative to the orientation of S. The required transformation is shown in Equation 4 8, where is,i represents ii in terms of the basis vectors of S. P(sti) Pt,i Ps (4 7) s,i = TES ( i (4 8) 4.3.2 Visibility The characterization of visibility has been extensively studied in the field of robotics for computer vision tasks related to industrial automation and manipulation [27, 105, 106]. A viewpoint is typically defined on a ',l 'liiy space." This parameter space is generally spanned by a set of variables that affect visibility and view quality. These parameters often include optical specifications of the camera and other sensor specific factors in addition to the geometric definitions discussed in Section 4.3.1. Various definitions of visibility can then be constructed through the application of sensor and task specific constraints on this parameter space. Here, the relative position, ips(ti), sensor orientation, tI, and target orientation, hi, are projected onto a limited geometric parameter space to maintain generality with respect to sensor type and environment characteristics. In addition, the specific parameterization allows a simple and intuitive definition of visibility through bound constraints on the parameters. The threedimensional parameterization with respect to the ith target is defined by range, ri, incidence angle, a7, and FOV angle, Ofi. The range is simply the norm of the relative position vector, as described by Equation 49. Variations in range are associated with the intensity and resolution of data. The incidence angle measures the degree to which the sensor LOS i. the target surface. This effect is characterized by the angle between the LOS along As(ti) and the target surface normal hi, as shown in Equation 410. High values for ai can result in a distorted data product. Finally, the FOV angle measures the angular discrepancy between the LOS along t(s.ti) and the LOS along s3, which is the sensorframe basis vector that indicates the center of the FOV. This relationship is described by Equation 411. Variations in Oft are associated with angular position within the FOV. Figure 42 depicts a cross section of the FOV that shows the visibility parameters for a particular target location. r P = A(s*) (49) aJ = arccos ()) (410) Of7i = arccos ((TEs ) (411) Once expressed in terms of the range, incidence, and FOV angle, visibility can be defined through the application of simple bounds. These bounds rely on several basic assumptions concerning the limitations of the sensor. First, the sensor is assumed to Figure 42. Visibility parameters. have a finite FOV, ie., the sensor is not omnidirectional. Second, the existence of a range beyond which objects within the FOV cannot be resolved is assumed. These conditions result in the visibility constraints given by Equations 412 and 413. ri < rmax (412) Of,i < Of,max (413) In physical space, the simultaneous application of these constraints are realized as the intersection of a sphere that is centered at the origin of S and and infinite circular cone that originates from this same origin. The sphere has radius rmax. The axis of the cone is collinear with the sensor axis, s3, and has a halfangle of Of,max. This intersection thus forms a spherical cone that is aligned with the sensor axis. A necessary condition for visibility of the ith target is that i(s.ti) must locate the target within this spherical visibility cone for the current sensor orientation, s,. Another visibility constraint results from the relative orientation of the target surface with respect to the location of the sensor. This condition constrains the incidence angle, ai, as shown in Equation 414. This constraint reflects the notion that a point on a surface cannot be seen when the view is parallel to the surface. A parallel view occurs when ai = 7/2, or when the LOS is orthogonal to the target surface normal, hii. Equation 414 also restricts incidence angles greater than r/2, as this range of ai corresponds to viewpoints behind the target surface. A more conservative upper bound of ,max can be considered as well to account for sensorspecific incidence requirements. a < rmax < (414) 2 Physically, the incidence constraint shown in Equation 414 is realized as an infinite cone originating at the target location. The axis of this cone lies along the surface normal, hi, and the halfangle is related by ,max. In the limiting case where cmax = 7/2, this cone approaches an open halfspace that is bounded by a parallel plane to the target surface. A necessary condition for visibility is that f(s.ti) must locate the target within this infinite cone. A final condition requires that a viewpoint must not be occluded for visibility. This condition requires that the LOS from the sensor to the ith target does not intersect any obstacles or surfaces between the sensor and the target. As such, this constraint depends on the specific environment in which the target and sensor are located and must therefore be applied in a taskspecific fashion, as needed. 4.3.3 The Visibility Set The visibility conditions discussed in Section 4.3.2 provide a set of necessary condi tions for visibility of a particular point from a particular sensor location and orientation. The range and FOV angle conditions describe a set of points relative to a specific sensor configuration that may be visible if other conditions are met. Conversely, the incidence angle condition describes a set of sensor configurations relative to a specific Inl, / con figuration from which this target may be visible if other conditions are met. A unified representation of these constraints in terms of sensor configurations is desirable from a sensor planning standpoint. Such a representation would allow the definition of a target set of sensor configurations from which a particular target is visible. A set of sensor configurations is constructed from which the ith target meets visibility conditions for any configuration within the set. This set is denoted the iv l'!ly set" with respect to i, or Vi. Construction of this set requires inversion of the constraints that define the visibility cone such that these conditions describe a set of sensor configurations relative to a particular i instead of a set of target locations relative to a particular sensor configuration. The visibility set is then defined as the intersection of the inverted visibility cone with the infinite cone describing the incidence angle constraint. Inversion of the visibility cone is achieved by recognizing that the sensor must be visible from a target for the target to be visible by the sensor. Consider a scenario which has an identical sensor to that carried by the vehicle located at the ith target. This sensor is described by the reference frame S', which is basically a copy of S that is translated to the target location. The sensor axis corresponding to the sensor at S' points in the direction of s'. This direction is therefore .1 li, exactly opposite to the direction in which the sensor at S points. The sensor at S' now has a visibility cone like that depicted in Figure 43. This cone describes a set of sensor positions, ps, that are visible from S'. Therefore the ith target is also visible from sensor positions within the S' visibility cone. ni, S 83 Figure 43. Inverted visibility cone. As discussed previously, Vi now consists of the intersection of this inverted spherical visibility cone with the infinite cone that relates the incidence constraint for the ith target. This set describes the sensor positions and orientations that satisfy the geometric visibility conditions for the ith target, as described by Equation 415. Valid sensor positions are defined by inclusion within the intersection of these cones while valid sensor orientations are described implicitly by the orientation of the inverted visibility cone. As such, the boundaries of this set vary with sensor orientation. A visualization of Vi for a particular target for a particular sensor orientation at some instant in time is depicted in Figure 44. The visualization shows the full set intersection resulting from the bounds applied to ri, oi, and Ofi for the inverted visibility cone. In the figure, Vi is shown as the shaded region. Vi { Q(p ,ts) ri < rmx, o7 < 7/2, Oft < Oimax (415) Figure 44. Construction of Vi. The visibility set (shaded) is constructed through application of set intersections applied to an inverted visibility cone 4.3.4 Proximity Effects Operating a sensor in close proximity to the environment introduces several issues that are not present for the case of sensing from standoff range. Standoff data collection is characterized by a large footprint projected by a sensor that is significantly removed from sensed surfaces. Conversely, close proximity data collection involves a small projected footprint from a sensor that must navigate amongst and potentially avoid sensed surfaces. These fundamental differences in mission scale can fundamentally change the nature of sensor data. One issue deals with occluded views, as introduced in Section 4.3.2. When a surface intersects a particular LOS within the sensor FOV, nothing beyond the point of inter section is visible along this LOS. As such, obstacles that fall within the visibility cone effectively cast a shadow of nonvisible points. For the case where obstacles are close to the origin of S, and therefore close to the apex of the visibility cone associated with S, this shadow can prevent visibility of a large portion of the environment.Cluttered environments may severely limit Vi for certain targets such that sensor placement must be performed with extreme care. Another significant issue is related to the coupling of sensor position and orientation to vehicle motions. C '! .i;, in vehicle configuration directly affect the location and pointing direction of the visibility cone regardless of problem scale; however, the effect on collected data is significantly different for standoff sensing as compared to close proximity sensing. For the standoff case, a large footprint is projected onto the sensed environment even for a narrow FOV sensor. This effect results from the angular nature of the FOV constraint combined with the large distance from sensed surfaces. For this case, even large translations of the sensor maintain significant overlap in projected footprint between the start and end of the translation, as shown in Figure 45A. Consequently, angular motions that occur along such translations do not have a great effect on the . .r'egate data collected as long as the visibility cone points in roughly the same direction at the end of the motion as it does at the beginning. An example of such a case is a downward pointing sensor on an aircraft that must change its course. The aircraft must bank to turn and may temporarily point the sensor away from sensing objectives; however, if a large portion of the sensor footprint upon completion of the maneuver overlaps with the footprint preceding the maneuver, there is no significant loss of data. Alternatively, sensing at close proximity results in a much smaller projected footprint, even for a fairly wide FOV sensor. As such, there is a high turnover of covered area as the sensor translates with the vehicle. The sensor footprint following a translation may be projected onto a completely different set of surfaces than were visible prior to the translation, as shown in Figure 45B. The example of a turning aircraft with a downward pointing sensor now exhibits significant data loss as a result of the previously benign maneuver. Moreover, navigation through close proximity environments often requires frequent and .. ressive maneuvering to avoid obstacles, thus causing motion coupling effects to be even more pronounced. l \ A B Figure 45. Motion coupling effects for different problem scales. A) Translation at standoff range. B) Translation at close range. Finally, significant relative variations in range and incidence are seen over the FOV as a result of the close proximity of the sensor to the environment. When the sensor is located a large distance from target surfaces, the range of all visible points varies very little relative to the standoff distance. Objects appear within the FOV at disparate distances and aspects when sensing at close range. The incidence angle can vary significantly over a surface that would appear at a nearly constant incidence from standoff range. Figure 46 shows this effect as compared to the variations seen in typical standoff sensing missions. The two images on the left of Figure 46 show range and incidence angle data over a simulated urban environment for a sensor located at an altitude 10,000 ft. This altitude is typical for current airborne sensor platforms such as the RQ1 Predator drone [3]. The images on the right of Figure 46 show range and incidence data when the sensor is located at an altitude of 200 ft along the same LOS as the left images. This altitude is typical of projected urban operations for small and micro UAVs. I90 4000 I0 I_ 6 / I 4Woo iDO / ao 410 0 400 La0 1 000 180ooo ,00 East (fi loo' 'L 00 East (t) North (R) North () C D from the two cases depicted in Figure 46 are di it in Table 41. The table shows that the two cases exhibit a similar absolute spread of range data, as the values for Fthe standard dev and iniiation similar magnitude for each case. However, a standoff sensing. D) Incidence for close proximity sensing The statistics describing the variations in range and incidence for all visible points from the two cases depicted in Figure 46 are di 1,v ,1 in Table 41. The table shows that the two cases exhibit a similar absolute spread of range data, as the values for the standard deviation of this data is of similar magnitude for each case. However, a discrepancy of a full order in magnitude is seen when these values are normalized on the mean range. This discrepancy is reflected by the color variations in the top right image of Figure 46 as compared to those seen in the top left image. Table 41. Statistics for visibility parameter comparison. Mean ri ri Std. Normalized ri Avg. ri Max ri Deviation Std. Deviation Std. Deviation Std. Deviation Standoff 20,023 ft 205.74 ft 0.0103 0.1471 deg 0.3673 deg Close Proximity 543.87 ft 181.84 ft 0.3343 4.2563 deg 12.0783 deg Incidence angle must be examined on a surfacetosurface basis, as target surfaces with different normal directions would artificially skew the statistics. The values in the last two columns of Table 41 dipl wi average values for each visible surface over all visible points. Very little absolute variation in incidence angle is seen for the standoff case as compared to the close proximity case, as indicated by the average ri standard deviation over all surfaces. Further, an increase in standard deviation by two orders of magnitude is seen by at least one sensed surface. These results are reflected by the significant color variations on each surface of the lower right image of Figure 46 as compared to those seen on surfaces in the lower left image. Frequent occlusions, motion coupling, and large relative variations in the visibility parameters are demonstrated for sensing missions in close proximity to the sensed environ ment. These effects can adversely affect data quality if not properly considered in mission planning. Proper consideration requires that the effects of these issues be quantified for inclusion within sensor planning optimization algorithms. The next section describes a sensor effectiveness metric that is designed to address this problem. 4.4 Effectiveness Metric The developments of Section 4.3.2 and 4.3.3 allow a geometric definition of visibility for a generic LOS remote sensor. The ith target is visible for sensor positions and orien tations that lie within the bounds of Vi, while for all other configurations the target is not visible. This relationship can be characterized by a simple binary flag as expressed in Equation 4 16. This definition of visibility is limiting in terms of describing sensor performance, or equivalently, quality of the resulting data product. Many cases exist for which this definition is not sufficient to describe the utility of a particular set of collected data. Realistically, data quality can be expected to vary substantially over Vi. Moreover, regions within Vi may exist that correspond to unusable or poor quality data due to in sufficient resolution or distortions. While the characterization of these effects is unique to specific sensor modalities, a framework can be established that allows their consideration in various sensing problems. f {j {.; s 4.4.1 Formulation Consider a scalar function, qj, that is associated with the ith target in the environ ment. This function takes as arguments the relative position of the sensor with respect to the target, pst,, sensor orientation, ts, and the target surface normal, ii. These arguments are mapped to a unit interval on the real number line, as expressed in Equa tion 417. This function is considered as a metric of the effectiveness with which target i has been sensed. qi : ( ) v [0,1] (417) The function q, necessarily evaluates to zero for all isti and s, that fall outside Vi. As such, expression of qj in terms of visibility parameters is desirable for convenient inclusion of this constraint. Additionally, these parameters are also often directly measur able from the sensor data itself. Expression of qi in terms of these parameters then allows direct computation of sensing effectiveness from raw sensor data without requiring access to the vehicle and sensor configurations. This parameterized form of qi is expressed in Equation 418, where the visibility parameters take on functional dependencies as shown in Equations 419 to 421. Note that the explicit form of qi is specific to both the sensor and the application. qi : (ri, ,o, O,) ) [0, 1] (418) r ,: (,ti) R+ (419) a : (sti ht,i) [0, 7/2] (420) Of,i" (: s, e, ht+) [0, Ofi]x (421) In addition to the dependencies expressed in Equations 418 to 421, qi has an implicit dependence on time, such that qi = q(t). This implicit dependence results from the timevarying nature of the relative position and sensor orientation, ,sti and ts, and consequently the visibility parameters, ri, a, and Of. As the sensor frame, S, is fixed relative to the vehicle bodybasis, B, these timevarying quantities are affected by the vehicle dynamics. As such, qi can be considered as an output function of the vehicle dynamics for all i. A representative trajectory of q, might appear as shown in Figure 47A. Notice that the function exhibits apparent discontinuities along the trajectory shown. These discontinuities result from target i leaving the sensor FOV during these time segments. As such, the visibility criterion is not met and the effectiveness metric evaluates to zero during these time segments. S!,il:/ remote sensor modalities actually operate in discretetime where data is cap tured at an instant in time at a finite rate. The effectiveness metric should be considered a function of the timediscretized dynamics for these cases, and should be expressed as a discretetime function such that qi = qi[k]. Likewise, a representative trajectory of qi for these cases might appear as in Figure 47B. While necessary to maximize realism, this discretetime formulation could introduce a number of problems in gradientbased trajectory optimization schemes. Thus, the continuoustime form of qi is primarily used throughout the remainder of this dissertation, unless otherwise indicated. This assumption requires that the sampling rate of the sensor is fast relative to the dynamics of the vehicle. This condition is met for the 1i i .ri y of the sensors considered here. 1 1 0.8 0.8 30.6 0.6 S0.4 0.4 0.2 0.2 o c 0 10 20 30 10 20 30 t (sec) t[k] (sec) A B Figure 47. Representative trajectories for sensor effectiveness metric. A) qi(t). B) qi(k). 4.4.2 The Quality Set The sensor effectiveness metric formulated in Section 4.4.1 can be used to define a subset of Vi. This subset, Qi C Vi, represents the sensor positions and orientations that view target i with an effectiveness of at least qi,desired, as shown by Equation 422. The actual makeup of Qi depends on the specific form of qi, and is not necessarily convex with respect to the visibility parameters, ri, ai, and Ofi. i ( st, s) I qi > 9i,desired} (422) The benefit to constructing a set of viewpoints that achieve a desired value of qi is similar to the purpose of constructing V. Such a set provides a target set for motion planning problems that can be used to formulate constraints and boundary conditions on allowable sensor configurations. 4.4.3 Sensing Mission Effectiveness The metric formulated in the Section 4.4.1 provides an instantaneous measure of the quality with which a remote sensor is gathering data with respect to a particular point in space. Realistic use of this metric for mission planning purposes would likely require evaluation of qi at many different points in addition to a method for tracking accumulated effectiveness. Perhaps viewing a target multiple times adds utility to the data, even if sensed with somewhat low values for qi. Alternatively, perhaps data can become "outdated" if a certain period of time passes between views. Here, the function Qi is introduced to reflect the ..':regate sensing effectiveness over a specified time segment. This function is related in general by Equation 423. Qi : (i,te[o,t]) [0, 1] (423) As with qi, explicit representation of this function is sensor and application specific; however, several forms commonly appear for realistic sensing applications. Two useful examples that are considered here relate the i: .::value" effectiveness and a binary probabilitybased effectiveness. The maxvalue version of the mission effectiveness metric is shown in Equation 424. This function keeps the maximum value of the instantaneous quality metric, qi, over the time interval, t E [0, t]. The value of Qmax,i indicates that target i has been sensed with that value at least once since the beginning of the mission segment. This version of mission effectiveness can be useful for applications involving imaging sensors that collect data subject to human analysis. One good view of a sensing objective can often provide a great deal of information to a human analyst. Qmax,i = max (i,te[o,t]) (424) Another form of the mission effectiveness metric can be expressed using basic prob ability theory. Many active sensors, such as radar and sonar, pl.l a role of detecting and classifying objects. Mine detection is an example of such an application. The probability of detecting or classifying such objects can often be characterized in terms of visibility parameters such as range and aspect. Equation 425 shows the case for which qi rep resents the probability of an event, such as detection, occurring at the sensing instance corresponding to qi. Then the probability of that event not occurring can be expressed as the complement of qi, as shown in Equation 426. The probability of the event not occurring for ,:,;1 of k sensing instances can be expressed as the product in Equation 427. Finally, the probability that the event occurs at least once is given by the complement of the probability that the event does not occur at all, as shown by Equation 428. P (event) = q (425) Pi( event) = 1 q, (426) Pi,tot( event) P( event)k (427) k Pi,once (event) 1 Pi,ta( event) (428) The probability that the event occurs at least once can then be implemented in recursion to express sensing mission effectiveness for detection probability, QpD,i. This recursion is shown as in Equation 429. The use of a recursive formula requires that the discretetime form of the sensing effectiveness metric is used. Qp,, [k] 1 (1 Qpi[k 1])(1 q [k]) (429) 4.4.4 Example: A Contrived Metric This example examines a specific formulation of qi that is contrived to reflect the sensing performance of a generic airborne imaging sensor. This formulation uses the concept of "efficiency functions," f,ei, for each of the visibility parameters that affect sensor performance and data quality. The efficiency function corresponding to a particular visibility parameter tracks the loss in data quality as a function of that parameter. Sensing effectiveness at the ith target is then determined by multiplying these functions together as in Equation 430. qc,i= ffe,i (430) Here, efficiency functions are formulated for the range, incidence angle, and FOV angle, as shown in Equations 431 to 433. These functions necessarily evaluate to zero outside the sensor FOV. Within the FOV, each efficiency function is shaped to reflect data quality losses within appropriate ranges for the associated visibility parameter. The effectiveness metric for this generic sensor is then given by Equation 434. 1+(r ..)4 ) 0.7exp (400 1, ri < rmax f m)i,ax /(431) 0, else 2 ni',, ) J1 i 4 0, else f (1+(0.750, f l,maz)4O) f, < Of,max fefi =i/Of~max ) (433) 0, else q,i = fr". f, f0e, (434) The efficiency functions, f,i fe1, and foe,i, are plotted in Figure 48. The efficiency function corresponding to range, fri, is shown in the left image. This function is seen to exhibit losses at close range and long range. This shaping reflects poor spatial coverage at close range and poor resolution at long range. 0.8 0.8 0.8 0.6 \ 0.6 \ 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 500 1000 1500 2000 2500 0 50 100 20 40 60 rft o (deg) f (deg) A B C Figure 48. Quality parameter efficiency functions. A) Range efficiency. B) Incidence efficiency. C) FOV angle efficiency. The function corresponding to incidence angle, fi, is shown in the middle image of Figure 48. This function exhibits losses as the view direction approaches tangency with the target surface. Increasing i has the effect of distorting surface data and limiting visibility. The figure shows that data quality is not considered to be substantially affected for ai < 50 deg. The function corresponding to FOV angle, foe,i, is shown in the right image of Figure 48. This function is seen to exhibit minimal losses over all FOV angles, with the greatest effects appearing near Of,max. The maximum FOV angle corresponds to the FOV boundary. As such, the losses are incorporated here to reflect the radial distortion effects common to camera sensors [82]. Use of the metric described by Equation 434 is demonstrated through a simple simulation. The simulation consists of a kinematic aircraft model flying at a constant altitude of 200 ft above ground level over an urban environment. This environment consists of three buildings arranged such that they partially enclose an area. Each building has a maximum height of 50 ft. A sample mission scenario might require data collection along the rooftop edges that face the partiallyenclosed area to check for snipers prior to a troop movement. The aircraft carries an imaging sensor that is located at the origin of the aircraft bodybasis such that fx = 6. The sensor points forward and down at an angle of 60 deg. As such, the transformation from B to S is given by Equation 435. The visibility bounds for this sensor are given by Equations 436 to 438. These bounds are used with Equations 431 to 434 to formulate the quality metric. 0.866 0 0.5 TBS 0 1 0 (435) 0.5 0 0.866 ra = 2500 ft (436) mrax = 90 deg (437) Of,ma = 35 deg (438) The simulated trajectory is depicted by the series of images in Figure 49. For display purposes, targets are generated by gridding the environment surfaces at fixed resolution of 25 ft. This selection of targets allows the metric value to be shown at all points in the projected footprint with the specified resolution. The images in Figure 49 show snapshots of instantaneous metric values within the sensor footprint at several points along the trajectory. Notice the change in shape and effectiveness makeup of the footprint as the aircraft banks to turn in the top right and bottom left images. Also, notice the stark differences in metric value on different surfaces that are perpendicular to each other as the sensor views building edges. 300 300 000 S0\1000, 500 500 1000\ 1500 1000 1500 1000 1 000 500 500 East (f)150 0 North (f) East (f) 1500 North (n ) 300, i30o 20\ .  _ S00 oo 10 0 0 o 500 \ / 500 , 1000 1500 1000 1500  1000 \  1000 ) 500 0 5o0 500 East () 1500 1 North (ft) East (ft) 1500 0North (ft) Figure 49. Simulated trajectory showing snapshots of effectiveness metric values. The maxvalue mission effectiveness metric is used to track .,. egate sensing ef fectiveness with respect to each target for the present example. Recall that this metric maintains the maximum effectiveness value over the entire trajectory, as given by Equa tion 424. The final values for mission effectiveness are shown in Figure 410 for the trajectory depicted in Figure 49. 1000 Z 500 O 500 1000 1500 East (ft) Figure 410. Maxvalue mission effectiveness. 4.4.5 Example: Image Area This example constructs a version of qi for use with a video camera sensor. This function, qA,i, is related to the image area occupied by a reference object. A metric formulated in this fashion could be used to ensure that a target object appears large enough in the image to extract desired information. For example, a certain number of pixels might be required to resolve the license plate number on a car. The spatial dimensions of the ith reference object must be transformed to pixel coordinates in order to quantify image area. This projective transformation is achieved through normalizing lengths using the distance along the sensor axis, d, as described by the standard pinhole camera model [82]. Simple geometric relationships yield an expression for di in terms of the visibility parameters, ri and Ofi. This expression is given as Equation 439. di = ri cos Of,i (439) The physical area of a rectangular reference object is given by the product of the lengths of its two sides, Wrf and Href, as shown in Equation 440. The corresponding area in image coordinates is given by Equation 441, where N represents the number of pixels in a single line of the image. The length of each side of reference object is transformed to canonical image coordinates through a pinhole projection and is then scaled to pixel units. The dimensions of the reference object in pixel units are given by Wref and Hrf. Similarly, the image area in pixels2 is denoted A. A = WrefHef (440) A = Wre fHtef (441) ( (W,,f N H,,f N (((rcos Of) (2 tanOf,,,max) )(( (r cosf) (2 tan Of,,,x N When the image area is constrained to fall between two values, Amin and Amax, allowable values for sensor range, ri, are restricted. The range must fall between two values that are determined as a function of FOV angle, f7,i, and characteristics specific to the sensor, N and Of,max. These bounds result from algebraic manipulation of Equation 4 41 and are shown in Equation 442. (WfH f < r The effectiveness metric is formulated as a function of image area as shown in Equation 443. This function exponentially approaches qA,i 1 as A Amax. The numerical parameters are chosen here such that a rapid rise occurs for A that are slightly larger that Amin, followed by a large interval for which there is little gain. Such behavior is chosen to reflect the notion that increasing image area of the reference object does not yield additional relevant information beyond a certain size. For the aforementioned example of a license plate, little is to be gained by increasing A once the numbers are clearly legible. Conversely, a significant amount of information is gained closer to Amin as slight increases in resolution help to differentiate the characters from each other. The formulation in Equation 443 also includes a scaling efficiency function, f,, to account for distortions resulting from grazing incidence angles. The same formulation of f, that was emploiv, in the previous example is used here as well (Equation 432). qA,i f exp 50 0.99(A A o) +0.)) (443) The image area formulation of the sensing metric is demonstrated through a simple simulation. As before, the simulation consists of a kinematic aircraft model flying through an urban environment. This environment consists of one building .i.1i i:ent to five cars fac ing the North direction located along a roadway, as shown by Figure 411. The simulated aircraft trajectory is designed to perform a diving turn from its initial configuration near the building and then follow along the roadway. A target is designated on the rear surface of each car near the likely location of the license plate. As such, target labels are given by i 1, ,5. 0 ) I 5"  n 0 20(0 , 150 North (ft) 0 0 50 100 50 East (ft) Figure 411. Environment for image area sensing metric simulation. The aircraft carries a video camera that is located at the origin of the aircraft body basis such that fi = 6. The camera points forward and down at an angle of 30 deg. As such, the transformation from B to S is given by Equation 444. The camera used here has N = 1280 lines of resolution and a FOV of 30 deg, such that Of,max = 15 deg. 0.5 0 0.866 TBS= 0 1 0 (444) 0.866 0 0.5 The reference object for this example is a 1 ft by 0.5 ft license plate. The minimum image area to resolve the numbers is chosen as Ai = 100 pixels2. The maximum is chosen as the area that would fill the FOV, A,,x = N2 pixels2. The metric formulated in Equation 443 is plotted in Figure 412 after substitution of these numbers. The xaxis of the plot in Figure 412 is shown on a logarithmic scale. 1 0.8 5 0.6 0.4 0.2 0 5 10 10 Image Area (pixels)2 Figure 412. Sensing effectiveness as a function of image area. Execution of the aircraft trajectory results in the camera FOV passing over four of the five target surfaces. The second target is not viewed as a result of motion coupling. The sensing effectiveness trajectories for each of the remaining four targets are shown in Figure 413. These trajectories are zero for all time segments during which the target is outside the camera FOV. The maxvalue mission effectiveness metric is used here to track .I.:regate sensing effectiveness for each target. Recall that this metric assumes the maximum effectiveness value over the entire trajectory, as given by Equation 424. The final values for mission effectiveness are shown in Figure 414. Mission Effectiveness values are shown for all visible surfaces in addition to the five specified targets. u0 1 2 t (sec) 1 2 t (sec) 3 4 3 4 0t 1 2 t (sec) J0 1 2 t (sec) Figure 413. Effectiveness trajectories. North (ft) 100 East (ii) Figure 414. Simulated mission effectiveness. 3 4 3 4 0.8 II *0.6 0.2 0 CHAPTER 5 RANDOMIZED SENSOR PLANNING 5.1 Introduction Generation of minimum time, dynamicallyconstrained trajectories that view multiple targets poses a challenging problem. Most motion planning methods, such as those described in C'! lpters 2 and 3, are generally tailored to drive the system from one specific configuration to another. The problem of finding minimum time trajectories that visit multiple points represents an instance of the wellknown traveling salesman problem (TSP). While this problem is wellstudied in the literature [25, 26], solutions and approximations typically rely on the assumption that optimal trajectory consists of optimal segments, as shown in Figure 51A. Optimal solutions do not generally take this form for dynamic systems. Figures 51B and 51C each show a threepoint tour for a curvatureconstrained system starting from the same initial condition. Each pointto point segment shown in Figure 51B is locally optimal. Figure 51C shows the optimal trajectory for the entire tour, which includes pointtopoint segments that are locally suboptimal. Generally, both the optimal sequence of points and the optimal trajectory must be determined; however, these solution elements are usually coupled for systems with nontrivial dynamics. 700 700 700 600 600 600 S500 = 500 500 400 400 400 300200 300 400 500 300200 300 400 500 300200 300 400 500 East (ft) East (ft) East (ft) A B C Figure 51. TSP problem comparison. A) Standard TSP. B) Curvatureconstrained TSP with locallyoptimal segments. C) Optimal curvatureconstrained TSP The curvature constrained TSP has been addressed to some extent by the research community [15, 17, 18, 30, 57]. These efforts provide a good start to the multitarget sensing problem, but they all equate target visitation with target sensing. Close proximity sensing problems realistically need to consider the full sensor FOV and the coupling of vehicle motion to sensor pointing, as discussed in C'! lpter 4. In actuality, targets can be viewed from a set of configurations. Consideration of this property adds another lv r of complexity to the planning problem. The sensing problem now requires that the target sequence is determined, the viewpoints within the visibility set, V, or the quality set, Q, are determined for each target in the sequence, and the optimal trajectory through these viewpoint configurations is determined. This dynamicallyconstrained, setvisitation TSP has been addressed to a far lesser extent in the literature. Researchers in the field of manipulator robotics have made efforts to define ;ood" viewpoints, but typically have not been concerned with generating opti mal trajectories. Those that have investigated optimal tours do not treat nonholonomic dynamics, as manipulator arms are usually fullyactuated systems [2729]. These efforts decouple the problem by generating a set of acceptable viewpoints and subsequently solving a standard TSP through these viewpoints. References [23] and [107] consider FOV dimensions in their study of a nonholonomic vehicle with a downwardpointing sensor. Nearoptimal trajectories are generated to view a series of targets using a simple three element discrete motion model and an heuristicbased enumerative search. The method performs well for the presented examples, but the enumerative nature of the search would introduce severe computational issues for problems that involve more than a few isolated targets or even a slightly more complicated motion model. Additionally, the restriction that the sensor ah,ii points downward neglects motion coupling effects. This chapter develops a randomized motion planning approach to generate good, though suboptimal, trajectories that sense a specified set of targets with a specified measure of effectiveness in minimum time. Concepts from previous chapters are integrated to generate a search tree of sensing trajectories that implicitly determines a target sequence and an associated set of viewpoints. Specifically, branches are added by planning local trajectories to viewpoint configurations that are sampled from the quality set, Q, of individual targets. These local motion plans must negotiate a typical closeproximity environment that is cluttered with obstacles. The nature of this environment and that of the embedded targets are discussed in Section 5.2. The actual algorithm is then detailed in Section 5.3. Finally, Section 5.4 demonstrates application of the planner to the core ISR missions described in ('! plter 1. Recall that these missions consisted of a multitarget, single view reconnaissance task, a multitarget, multiview surveillance task, and an area coverage task. 5.2 Environment Representation The planning environment is bounded to a finite, three dimensional operating region in the vicinity of the sensing objectives. This region contains extruded p" .Iv.onal obstacles which are each defined by a set of vertex coordinates, as described in Chapter 2. Specific locations in the operating region are designated as targets, which essentially serve as output locations for measurement of sensing performance throughout the mission. In other words, these targets represent the environment locations for which sensing performance is critical, and are therefore the only locations at which sensing performance is measured. Each target is fixed in the environment and is described by a threedimensional position and unit normal vector. Visibility and sensing performance at these target locations are characterized by geometric relationships between the sensor and the target. Specifically, the relative position and orientation are projected onto a parameter space spanned by the range, r, the incidence angle, o, and the FOV angle, Of, as described in ('!i lpter 4. As before, a set of configurations, V, can be defined for each target from which that target is visible. Similarly, the set, Q C V, is defined for configurations that sense the target with at least some specified value of the effectiveness criterion formulated in C'!h ,pter 4. Additionally, occlusion regions can be associated with each target. Occluded view points are described in C'! lpter 4 as as points that do not have a clear LOS to the target, such as when an obstacle lies between the sensor and the target. For the purposes of this dissertation, stationary targets are embedded in a stationary environment. Hence, regions of occlusion are timeinvariant and are defined for each targetobstacle pair. These regions can each be conservatively approximated as a convex polygon that corresponds to the maximum !i ,n[.v cast by a point source located at the target, as depicted in Figure 52. A more exact approximation can be achieved by parameterizing the vertex locations of these p" v.. ons on the vertical position, as shown in Figure 53. 1000 " 500 7 1000 U UI 500 0 500 1000 East (ft) A 0 500 1000 East (ft) B Figure 52. Occlusion shadows. A) A target in a threeobstacle environment. B) Conservative occlusion ! I iI... in the target plane. Figure 53. 3D Occlusion shadow. The occlusion shadow is constructed through parameterization of occlusion polygon vertices on vertical position Finally, measurement of sensing effectiveness must be addressed for the special case of area coverage missions. C'! lpter 4 formulates an effectiveness metric as a function of the geometric relationship between the sensor and a single, discrete point in space. Evaluation of sensing performance over an area generates a continuum of values and therefore complicates some of the definitions that prove useful for motion p11 .iiiiili such as the concept of the visibility set, V, and the quality set, Q. Alternatively, the continuum of points in the coverage area can be discretized into a set of "virtual t i1, I that represent the area with a desired resolution. Such a dis cretization could be achieved by simply dividing the entire environment into a grid, as depicted in the left image of Figure 54A; however, this discretization scheme introduces severe computational inefficiencies. Realistically, a multiresolution scheme is required to emphasize the coverage area without devoting unnecessary resources to noncritical areas of the environment. An example of such a scheme is depicted by the octtreelike approach shown in the right image of Figure 54A. This approach involves discretizing the environ ment with a coarse resolution and then subdividing cells in the coverage area to increase emphasis. Alternatively, virtual targets can be sampled from an importanceweighted distribution that emphasizes the coverage area. The resulting resolution from choosing virtual targets in this fashion is depicted as a vornoi plot in Figure 54B. The voronoi plot consists of polygons with edges that are equidistant from .,.i ,i:ent virtual targets, and hence reflect resolution of the discretization. 08 08 06 : 06 04 04 02 02 0 0 0 05 1 0 05 1 Y Y A B Figure 54. Discretization approaches for area coverage. A) Octreelike approach. B) Virtual targets approach. 5.3 A Randomized SensorPlanning Algorithm This section develops a randomized motion planning approach to achieve the ob jective of effectively sensing a specified set of targets in minimum time. This planner generates trajectories for a dynamicallyconstrained vehicle which carries a remote sensor having fixed position and orientation relative to the body frame of the vehicle. The motion plan is also subject to endpoint constraints, ie. a specified initial and terminal configura tion. Additionally, path constraints imposed by obstacles must also be satisfied. Sensing criteria for the motion plan is reflected by a set of endpoint inequality constraints imposed on the sensing mission effectiveness metric, which is defined with respect to the ith target as Qi, as described in ('!i lpter 4. These constraints are expressed as in Equation 51, where Qd, represents the desired sensing effectiveness for the ith target. Qd, Qi(tf) < 0 (51) 5.3.1 Overview As described in ('!i lpter 3, random dense tree methods generate a tree of feasible trajectories that is designed to rapidly explore the reachable set of the system. Here, this expansion is biased toward achieving sensing objectives. Specifically, each branch added the tree is designed to incrementally satisfy the mission sensing constraints. As such, each branch satisfies at least one sensing constraint in addition to those satisfied at the expansion node. Nodes that have satisfied all sensing objectives are expanded to the mission terminal configuration. An upper bound on the timeperformance of feasible solutions is maintained while additional solutions are generated. Feasible solutions are generated until an independent termination condition is met, such as a limit on computational resources or on maximum allowable planning time. The tree structure is typically initialized as a single node at the initial condition. An alternate option involves using a standard randomized planning technique, such as the RRT algorithm, to generate an exploratory initial tree. Use of this option provides a greater variety of starting locations throughout the environment prior to considering any sensing constraints. Following tree initialization, each iteration of the algorithm proceeds in four main steps which are detailed in the following sections. These steps are summarized here as follows: 1. Node Selection: A node from the existing tree is selected from a weighted dis tribution. Node weights can be chosen to reflect path performance such that tree expansion is biased towards betterperforming solutions. 2. Vantage Point Selection: A set of vantage points are sampled from the set, Ui Qi, where each Qi represents the quality set associated with the ith target for which sensing objectives have not yet been achieved. 3. Branch Extension: Candidate branches are generated by computing local trajecto ries from the selected node to each of the sampled vantage points. A branch is then selected from among the candidates for addition to the tree. 4. Solution Check: The new branch is split into nodes and evaluated for possible solutions. If a segment of the new branch has achieved all sensing objectives, a trajectory is planned to the terminal configuration and the solution upper bound is updated, if necessary. A key concept that contributes to the efficiency of this algorithm is that the inherentlyclose target spacing in considered environments often results in the satisfac tion of multiple sensing constraints along a given trajectory. This concept is depicted in Figure 55. The left image depicts a scenario where six targets along a road must be sensed. Generally, there are 6! = 720 possible sequences in which these targets can be sensed; however, planning trajectories to view these targets individually yields a trajectory that senses all six, as shown in the right image of Figure 55. A sufficient condition for optimality of this implicitlydetermined sequence is optimality of the trajectory to view the individual target. 5.3.2 Node Selection The node selection step determines which node is to be expanded for the current iteration. The choice of a node selection strategy can be an important factor in biasing AND. Figure 55. Sensing secondary targets. Planning a trajectory to sense a single target often results in secondary sensing of other targets. tree expansion in directions that favor betterperforming solutions. Previous research ef forts have achieved such a bias by randomly sampling an expansion node from a weighted probability distribution [73, 77]. Each node is assigned a weight that is dependent on performance characteristics. A weighting scheme that reflects the tradeoff between sensing performance and cost performance is shown in Equation 52, where #vis represents the number of targets that have been successfully sensed and L represents the path length in seconds between the root node and N ,,,. Additionally, #tgts represents the total number of targets, Lmax is the maximum path length over all nodes in the tree, and 6 is a shift factor included to vary the emphasis on unsensed targets. A representative plot of this weighting function is shown in Figure 56. The figure shows that nodes which have achieved sensing objectives in a short period of time are favored. The formulation of this weighting function allows for nodes that have sensed more targets to exhibit longer paths yet still maintain a high weight value. High weights are also seen for branches that have not seen any targets and have short length. WN = (#vis)2 + (#tgts) ( ) )2 (52) V \ \l max / / ~~ 3 jr 0U Lmax Paill Length Figure 56. Example node weighting function. 5.3.3 Vantage Point Selection At least one vantage point is selected that will incrementally step the current node, N,,,,, towards achieving sensing objectives. As such, this vantage point is selected such that sensing requirements are met for a target that has not yet been sensed along the path between the root and N,,,. These requirements are typically a function of the visibility parameters associated with the ith unsensed target, ri, ai, and Of,, as described in ('!i lpter 4. Specifically, a vantage point characterized by the sensor position and attitude must lie in the quality set, Qi, that is associated with the ith unsensed target. Additionally, the view from the selected vantage point must not be occluded and must be reachable from Nr,,,. The actual process which determines how a particular vantage point is selected de pends on how the vehicle is modeled and how the sensor is mounted relative to the vehicle. For example, a vehicle modeled using motion primitives, as discussed in ('! Ilpter 2, has a continuum of allowable positions and heading angles but a finite set of bodyaxis ori entations. These quantized elements of the allowable vehicle motions must be considered in viewpoint selection for this case. Generally, viewpoints are selected pseudorandomly through sampling from the allowable vehicle configurations that meet the sensing criteria for a selected target. Sampling is pseudorandom in that heuristics can be incorporated to yield favorable viewpoints. For example, a position can be sampled randomly from a projection of Qi and a favorable heading can be determined based on the relative position of the sample to the target along with the orientation of the sensor relative to the vehicle. Prior to acceptance, a sampled vantage point must be checked for occlusions and reachability. As discussed in Section 5.2, each target has an associated set of p]" Iv.onal regions for which the LOS is obstructed. Therefore, a sampled vantage point can be tested for occlusion by verifying that it does not lie in any of the occlusion p]" Iv.ons associated with the selected target. This condition can be checked using one of the methods described in Section 2.2.2. Verification of reachability poses a slightly more difficult task. This task involves the concept of forward and backwards reachable sets. The forward reachable set consists of all points reachable from the sampled vantage configuration while the backwards reachable set consists of all points from which the sampled vantage configuration can be reached. For the present development, a vantage point is considered reachable if the goal location is contained in the forward reachable set and the current node is contained in the backwards reachable set. Unfortunately, characterization of these sets is highly nontrivial. A vantage point sampled inside an obstacle region can be easily identified and rejected; however, selection of vantage points that cannot be reached without an obstacle collision or that will result in an imminent collision in the next expansion are difficult to identify without actually planning a trajectory to or from the sample. An approximation is obtained by computing and collisionchecking several short trajectory segments near the boundaries of the forward and backwards reachable sets, as depicted in Figure 57. The figure shows short turning and straight segments into and out from a sampled position and heading. When these representative trajectories are collision free, a degree of shorttime reachability is indicated. While this approach cannot guarantee full reachability, it can serve as a quick test to rule out the most obvious cases. 1000 S500 U 0 0 500 1000 East (ft) Figure 57. Vantage point reachability test. Vantage points for several targets are tested for reachability using representative trajectories from the forward (green) and backwards (blue) reachable sets. 5.3.4 Local Planning and Expansion After vantage point selection, a set of candidate branches are generated by planning trajectories from the current node, N,,,, to each of the sampled vantage configurations. Each of these candidate branches must reach their respective local goals to ensure that at least one additional sensing objective is satisfied beyond N,,r,. Consequently, a local planning method that is capable of effectively dealing with path constraints related to both obstacles and vehicle dynamics is required. Recall that the randomized method developed in C'! plter 3 exhibits the capability to efficiently generate feasible trajectories that meet these criteria. As such, this method is appropriate for use as the local planning method here; however, use of a randomized planner as the local planning method for use with another randomized planner requires that some caution is taken in tree and node data representation. Specifically, the tree structure used in the local planner should be consistent with the main planning tree to facilitate integration of local solution nodes into the main solution. Further, a termination condition for the local planner should be carefully selected. Recall that the planner described in ('!i plter 3 is capable of generating a feasible solution quickly, but can offer better approximations to the optimal solution given additional iterations. A reasonable tradeoff between performance and efficiency must be made based on the application. Note also that while a randomized approach is chosen here, other local planning options, such as optimal control, are allowable. Different choices of a local planning method may yield better solutions at the cost of computational resources. Use of an approximation technique, such as the randomized approach which is selected here, may require a high number of computationally cheap iterations to achieve a desirable solution. Specific scenarios that favor one method over another can be contrived. One 1i i'. i benefit to using a randomized local planning method is that the nodes along the solution branch can be directly incorporated into the main solution tree. These intermediate nodes provide a greater degree of versatility to the main solution tree in that these nodes may be selected for expansion in the next iteration. 5.3.5 Evaluation Candidate branches must be evaluated prior to tree addition, as only a single branch is added at each iteration. Branches are evaluated based on sensing performance as well as path length. Intuitively, a branch is considered desirable based on two main criteria: 1) the number of sensing constraints that are newlysatisfied along the branch, and 2) the time duration of the trajectory along the branch. A variety of methods are suitable for evaluating the candidate branches based on these competing criteria. For example, a function such as that described by Equation 52 provides a measure of the tradeoff seen in these path properties. Alternatively, these criteria could be applied independently and consecutively. The number of newly satisfied sensing constraints could be used to reduce the set of candidates followed by a selection of the minimum time candidate from those that remain. Regardless of the exact criteria formulation, this approach to evaluating candidate branches rewards trajectories that satisfy the sensing constraints in "chunks" while using as little time as possible. Selection of such branches essentially amounts to gradient fol lowing, which is not ahv, the most effective optimization approach for highly nonlinear and nonconvex problems. Gradient following approaches are notoriously attracted to local minima. Scenarios can be contrived for the present application where the optimal solution contains a segment of long duration along which few targets are sensed. Therefore, it is critical that the vantage point selection step retains a random element to ensure that the planning solution can escape these local minima with nonzero probability. Finally, any newly added nodes that have satisfied all of the sensing constraints for a reconnaissance or coverage mission are expanded toward the mission goal. Surveillance missions require that all targets are viewed multiple times, and, as such, the mission goal is considered as the configuration from which the first viewpoint in the sequence is sensed. The local planner is used for this purpose, as there are no additional sensing requirements imposed on the trajectory. Once the planner reaches the goal, the path length is compared to the current best upper bound and is added to a list of completed paths. The upper bound is replaced, if necessary, and the algorithm continues on to the next iteration. 5.4 Examples The examples in this section demonstrate an implementation of the algorithm de scribed in Section 5.3. These examples employ a simple vehicle model that is constrained to operate in two dimensions yet exhibits threedimensional sensing capabilities. Mission scenarios are constructed such that this vehicle is required to collect data regarding targets in a closeproximity environment. As such, a sensing metric is selected that appropri ately characterizes the effects of the environment proximity on sensing capabilities. The mission scenarios reflect the three sensing tasks identified in Chapter 1: reconnaissance, surveillance, and coverage. These tasks are first addressed in an obstaclefree environment such that occlusions and trajectory safety are not considered. This simplified example is followed by an example that includes polygonal obstacles. 5.4.1 Model The examples presented here consider a simple model of a small, fixedwing aircraft that flies at a constant altitude of 200 ft. The motion of this vehicle can be described by the same model used for the example in C'!i pter 3. As such, the vehicle moves with forward velocity, V, and is subject to a bounded turn rate, I < 30 deg/sec. The single system input is the commanded turn rate, w. An additional requirement imposed for the present example is that the vehicle must bank to turn. The assumption is made here that the bank angle, o, is coupled directly to the turn rate, Q. A hybrid motionprimitive model is formulated with seven trim states, as in C'! ,p ter 2. As previously, these trim states consist of steady turns in each direction at three different turn rates and a steady, straightahead motion. All trim primitives maintain constant forward velocity. The model is characterized as kinematic such that transition maneuvers between trim states occur instantaneously. The trim states are characterized by constant system velocities and bankangles as follows: To = (Vo, wo, o) = (40 ft/sec, 0 deg/sec, 0 deg) Ta = (Via, Wia, ~ia) = (40 ft/sec, + 30 deg/sec, + 60 deg) Tb = (VIb, blb, ylb) = (40 ft/sec, 30 deg/sec, 60 deg) T2a (Via, W2a, 2a)o) (40 ft/sec, + 20 deg/sec, + 45 deg) T2b = (V2b, 2b, 2b) (40 ft/sec, 20 deg/sec, 45 deg) T3a = (V3a, 3a, 3a) = (40 ft/sec, + 15 deg/sec, + 30 deg) T3b = (V3b, L3b, 3b) = (40 ft/sec, 15 deg/sec, 30 deg) The aircraft carries a downwardpointing video camera. The sensing capabilities of this camera are characterized by a maximum FOV halfangle of Of,max = 35 deg. Maximum range for a video camera is generally dependent on the desired image resolution. A maximum range of rma = 750 ft is chosen arbitrarily as an appropriate value for the present example. Also, note that the orientation of the camera is fixed with respect to the aircraft body reference frame. As such, motion coupling effects are nonnegligible and must be considered in the motion plan. These effects are seen for the present example as sensorpointing direction varying with respect to bank angle. 5.4.2 Effectiveness Metric Imperfect tracking performance can be expected in realistic situations, even with accurate models and welldesigned controllers. Deviations from the planned trajectory in either position or orientation can result in diminished sensing performance. Specifically, targets that are viewed from the planned reference trajectory may not be visible from the perturbed actual trajectory. Therefore, the effectiveness metric used for the example is formulated as a "robust visibility" parameter that essentially builds an allowable margin of error into the visibility criteria considered by the planner. The contrived metric example discussed in Section 4.4.4 essentially quantifies this no tion of robust visibility. Specifically, a sensing effectiveness metric is constructed using the product of a series of functions that exhibit dependence on the visibility parameters, as shown by Equation 53. The functions, fri, fci, and fo,,,, take values that reflect the detri mental effects on data quality resulting from variations in the visibility parameters, r, a, and Of, relative to the ith target. These functions take the same form as in Section 4.4.4, with the exception of f,,i, which is related by the binary condition shown in Equation 54. The parameter values that specify the exact expressions are given by ra = 750 ft, rmax = 70 deg, Of,max = 35 deg. qrv,i = fr fo, fof,, (53) 1, Ji ` Jmax fU = <(54) 0, else The examples presented here require that the maxvalue effectiveness criteria cor responding to each target achieve a value of at least Qi = 0.75. This requirement is incorporated into the motionplanning problem through a series of inequality constraints of the form of Equation 5 1, where Qd = 0.75. The maxvalue criteria require that each target is viewed with an effectiveness metric value of qr = Qd at least once along the trajectory. Satisfaction of these constraints is guaranteed through a conservative defi nition of the quality set, Qi C Vi. Specifically, a valid veiewpoint must achieve a value of Vd = 0.866 for each of the two continuouslydefined efficiency functions, f,, and fe,o. This value is shown oval i, d1 as a dashed red line on plots of the three efficiency functions in Figure 58. The plots show that simple bounds on the visibility parameters for a particular target ensure that sensing criteria are met. These bounds are seen as: 49 < ri < 385 ft, Ofi < 29 deg, and, from Equation 54, ai < 70 deg. These conservative requirements allow a nonvarying definition of Qi with respect to the visibility parameters, as shown in Equation 55. S     ^          0.8 0.8 0.6 e 0.6 0.4 0.4 0.2 \0.2 0 0 20 40 60 0 200 400 60 0 0 (deg) rft Of(deg) A B Figure 58. Quality parameter efficiency functions for examples. A) Range efficiency function. B) FOV angle efficiency function. Qi,r { s (psti, s) 49 < r, < 385 ft, ci < 70 deg, Of,i < 29 deg} (55) 5.4.3 ObstacleFree Examples Motion plans for effective sensing are generated in an obstaclefree environment. An inertial referenceframe is embedded within the environment such that the xaxis points in the North direction and the yaxis points in the East direction. This environment is bounded such that all targets are located within an area defined by the bounds, 0 < x < 1000 ft and 0 < y < 1000 ft. All sampling steps of the algorithm draw from locations that satisfy these bounds. Additionally, targets are placed throughout this environment. For the present example, the normal vector associated with each target is orthogonal to the ground plane, and can be described by Equation 56. i = 0 0 Vi (56) This example actually considers two sets of specified targets. The first set of targets is shown in Figure 59A. The figure shows the environment boundary as a solid blue line and twelve randomlyselected target locations as red asterisks. These targets are used for the reconnaissance and surveillance mission descriptions. Figure 59B shows the second set of targets, which is used for the coverage mission description. The figure shows a series of roads passing through an open area. An area to the right of the plot in Figure 59B is enclosed by the roads and is designated for sensor coverage. This sensing requirement is represented by a set of eighty virtual targets which are randomlyselected from the coverage region. These targets are depicted as red asterisks in Figure 59B. 1000 1000 S500 500 t" o 5 00* " *O O L .*** 4 0 0 0 500 1000 0 500 1000 East (ft) East (ft) A B Figure 59. Environments for obstaclefree sensor planning example. A) Environment for reconnaissance and surveillance missions. B) Environment for coverage mission. Several degrees of freedom in the algorithm described in Section 5.3 are set specif ically for the present example. First, each iteration of the algorithm begins with the selection of an expansion node from a weighted distribution, where node weights are given by the weighting function described as previously by Equation 52. Viewpoint selection proceeds by selecting an unsensed target randomly from a uniform distribution, then sampling an inertial point that satisfies the range condition associated with Q for the selected target. A heading is computed that places the target directly broadside the vehicle. This choice of heading increases the likelihood that the downwardpointing sensor will view the target while in a banked turn. Finally, each bank angle corresponding to the finite set of primitives is evaluated for sensing effectiveness. Those primitives that satisfy the sensing constraint for the selected target are retained as allowable terminal configurations for the local planning method. Next, candidate branches are generated from the current node to each of the ten viewpoint configurations. The goal configuration consists of a position, a heading angle, and a finite set of allowable terminal trim states. For the present example, the randomized planning method described in C'! lpter 3 is used to generate the candidate branches; however, other approaches such as optimal control are certainly valid. Finally, each candidate branch is evaluated for sensing and time performance. Branches that have newlysatisfied the most sensing constraints are set aside. The minimumtime branch from this subset is selected for addition to the trajectory tree. If all sensing constraints have been satisfied, a trajectory is planned to the goal corresponding to the sensing task. The initial condition serves as a goal for the reconnaissance and coverage tasks. The first point along the trajectory for which a sensing constraint is satisfied is used as a goal location for the surveillance task. The first mission description examined is the reconnaissance task. Recall that this task requires that each target is viewed at least once with a desired value for the sensing metric. The trajectory tree is initialized by a single node at the origin of the inertial frame. The initial automaton state at this node is represented by the trim state, To. The initial configuration is given by (px, p, ) = (0, 0, 0). For the twelve targets shown in Figure 59A, there are 12! a 479 million possible sequences in which to visit targets. Here, the algorithm described in Section 5.3 is run for 100 iterations. As the algorithm proceeds, a minimum upper bound is maintained for computed solutions. Trajectories corresponding to several of these refinements are shown in Figure 5 10. In the figure, solution trajectories are depicted as a red solid line. Additionally, the projection of the restricted sensor footprint on the environment surface is shown for the point along the trajectory which first satisfies the sensing constraint associated with each target. These projections restrict the sensor FOV to the values that satisfy the definition of Q, and are shown in Figure 510 as dashed contours. Solutions are shown after, 1, 10, and 64 iterations and correspond to trajectories of 87.91 sec, 63.30 sec, and 56.95 sec, respectively. 1000 1000 1000 ...... S500 .. 500 v 500 _} .... ...... 0 0 0 0 500 1000 0 500 1000 0 500 1000 East (ft) East (ft) East (ft) Figure 510. Trajectory refinement for obstaclefree reconnaissance. A simulation is run to plot the maxvalue effectiveness metric over the entire environ ment for the minimumtime sensing trajectory, which corresponds to the solution found in the 64'' iteration. The result of this simulation is shown in Figure 511. The solution trajectory is shown as a solid black line and the targets are shown as black circles. Notice that the value of the effectiveness metric is at least Q = Qd = 0.75 at all target locations. Also, note that the planner makes use of the motion coupling properties exhibited in turning flight to sense targets from a distance. As such, the solution trajectory, while suboptimal, is shorter than any trajectory that passes directly over all twelve targets. Application of the planner to the surveillance task proceeds in similar fashion. The same set of targets in the same environment is used for this demonstration. As previously, the tree is initialized with a single node at (px, py, Q) = (0, 0, 0) in the trim state, To. S300( S200,  100. 0 1000 500 North (ft) 4 1000 East (fR) 1 0.9 0.8 07 0.6 05 04 0.3 0.2 0.1 0 Figure 511. Simulated effectiveness for obstaclefree reconnaissance. The algorithm is run for 100 iterations. As branches are added, a list of viewpoints is maintained that tracks the first target sensed along a trajectory. Viewpoint configurations from this list are used as goal locations when a child branch has satisfied all of the sensing constraints. Use of these configurations as goal locations ensures that the surveillance loop is closed once all targets have been sensed once. This loop is then traversed repeatedly to view each target multiple times. A minimum upper bound on time performance is maintained as surveillance loops are closed. The upper bound is revised as better solutions are computed. Several of these improving solution trajectories are shown in Figure 512. As before, solution trajectories are depicted as a red solid line and the "firstv, '.v projected sensor footprints are depicted as dashed black lines. Solutions are shown after, 1, 7, and 10 iterations and correspond to trajectories of 80.44 sec, 59.66 sec, and 54.80 sec, respectively. These values reflect the time required to the reach the end of the first loop from the initial condition. Figure 513 shows the result of a simulation that plots the maxvalue effectiveness metric for the minimumtime surveillance loop. As previously, the solution trajectory is shown as a solid black line and the targets are shown as black circles. Notice that the sensing criteria are achieved for all target locations.  IUUU  IUUU .. IUUU c.K *? ,, ,."* .., *.' 4, .'....*.. '"C:i ... ... ... ... ..... 500 t. 500 500 o .... .. .. 0 0 0 0 500 1000 0 500 1000 0 500 1000 East (ft) East (ft) East (ft) Figure 512. Trajectory refinement for obstaclefree surveillance. 0.9 0.8 0 .7 g 300 0.5 500 0.2 100.1 North (ft) East (ft) Figure 513. Simulated effectiveness for obstaclefree surveillance. The algorithm is applied to the coverage task in the environment depicted by Fig ure 59B. The coverage area is represented by a set of eighty virtual targets which are sampled from a uniform distribution. The planner and tree are initialized to the same val ues as used for the reconnaissance and surveillance tasks. The algorithm proceeds through 100 iterations in the same fashion as the other sensing tasks. Despite the larger set of tar gets, the order of computational complexity is the same as for the other mission tasks due to selection of the same number of viewpoints at each iteration. The algorithm benefits significantly from the fact that many targets are often seen from a single viewpoint given the targetspacing that results from the sampling approach. A rise in storage complexity does result from the need to store effectiveness metric values for each target at each node in the trajectory tree. A r r t\ A r r t\ A r r t\ Upon satisfaction of all sensing constraints, a trajectory is planned from a branch node to the initial condition. The best upper bound is maintained as in the previous cases. Several solutions that refine this upper bound are depicted in Figure 514. Solution trajectories are shown as a black solid line in the figure. Projected sensor footprints are not shown to avoid clutter resulting from the large number of virtual targets. Solutions are shown after, 1, 15, and 84 iterations and correspond to trajectories of 93.19 sec, 78.89 sec, and 59.02 sec, respectively. 1000 1000 1000 500 '. 500 500 . 0 0 0 0 500 1000 0 500 1000 0 500 1000 East (ft) East (ft) East (ft) Figure 514. Trajectory refinement for obstaclefree coverage. Figure 515 shows a simulation of the final trajectory that plots the effectiveness metric over the entire environment. As before, the trajectory is depicted as a solid black line and the virtual targets are depicted as black circles. Notice from the figure that the final trajectory does not actually result in perfect coverage. This deficiency demonstrates the suboptimality of the planner, and silr. I that a larger set of virtual targets are needed to adequately represent the considered coverage area. 5.4.4 Examples with Obstacles The present example considers an environment with identical bounds as those considered for the examples in Section 5.4.3. Additionally, four polygonal obstacles are placed randomly within these bounds. These obstacles might represent buildings in an urban setting. As previously, two different sets of targets are embedded within this environment: one to demonstrate reconnaissance and surveillance mission tasks 1 0.9 0.8 300, 200 .0 7 )0  0 4 500 1o *0.2 1000 0 500 .1 North (ft) 0 0 East (f) Figure 515. Simulated effectiveness for obstaclefree coverage. and another to demonstrate area coverage. These targets have the same surface normal properties, as expressed in Equation 56. Unlike the previous case which had targets randomly scattered throughout the area, the present case considers target locations that are strategically placed close to the obstacles. Figure 516A shows the environment used for reconnaissance and surveillance missions. Nine targets are seen in locations that could represent entrances to buildings. Information gathered from such locations could provide valuable intelligence regarding people or vehicles that enter and leave each building. Figure 516B shows the environment used for coverage missions. Onehundred virtual targets are sampled from the region surrounding one of the buildings. An example application for coverage of such an area is to identify the presence of any unauthorized persons or even an explosive device within the vicinity of the building. The majority of the algorithm parameters are set identically to those described for the examples in Section 5.4.3. Nodes are selected from a weighted distribution, a set of ten viewpoints are sampled from Ui Qi, and candidate branches are planned and evaluated to these viewpoints. Contrary to the previous example, the tree is initialized here using an exploratory run of an RRT algorithm. Execution of this algorithm is concerned primarily with spanning the environment and does not consider sensing objectives. Use of this S U. 0 500 1000 East (ft) A 1000 500 0 500 1000 East (ft) B Figure 516. Environments for sensor planning example with obstacles. A) Environment for reconnaissance and surveillance missions. B) Environment for coverage mission. technique to initialize the tree generates configuration nodes that might not have been computed using the sensor planning approach. The first mission description examined is the reconnaissance task. Each target must be viewed at least once with a desired value for the sensing metric before a trajectory is planned back to the initial condition. The trajectory tree is initialized with an exploratory run of an RRT algorithm resulting in 50 initial nodes. This tree is rooted at the vehicle initial configuration, (px,p,, ) = (0, 0, 0), and is shown in Figure 517. As previously, the initial automaton state at the root node is represented by the trim state, To. The algorithm is run for 100 iterations. 1000 500 0 500 East (ft) 1000 Figure 517. Initial tree for reconnaissance with obstacles. 1000  500 o . As the algorithm proceeds, a minimum upper bound is maintained for computed solutions. Trajectories corresponding to several of these refinements are shown in Figure 5 18. In the figure, solution trajectories are depicted as a red solid line. As in the previous set of examples, the projection of the restricted sensor footprint on the environment surface is shown for a point along the trajectory which satisfies the sensing constraint associated with each target. These projections are shown as dashed contours. Solutions are shown after, 1, 20, and 98 iterations and correspond to trajectories of 68.56 sec, 67.06 sec, and 62.97 sec, respectively. 1000 S500 0 0 1000 500 0 1= 1000 500 0 10 0 500 1000 East (ft) 0 500 1000 East (ft) 0 500 1000 East (ft) Figure 518. Trajectory refinement for reconnaissance with obstacles. The results of a simulation showing the sensing effectiveness over the entire environ ment is shown in Figure 519. The simulated trajectory in the figure corresponds to the solution shown in Figure 518C. Two views are shown to demonstrate that all targets have been sensed with desired effectiveness. 200 1000 500 North (ft) 500 East (ft) 1200 z 1000 200 0 200 500 East (ft) Figure 519. Simulated effectiveness for reconnaissance with obstacles. Next, the surveillance task requires that a closed orbit is generated such that all targets can be viewed multiple times. The trajectory tree is again initialized with an RRT containing fifty nodes and which is rooted at (py, Py, ) = (0, 0, 0) in a trim state corresponding to To. This initial tree is shown in Figure 520. During the 100 iterations for which the algorithm is run, the first point along each solution path to satisfy a sensing constraint must be stored. As before, this point serves as a goal location for child branches that have achieved all sensing objectives. 1000 500 0 0 500 1000 East (ft) Figure 520. Initial tree for surveillance with obstacles. As surveillance loops are closed, a minimum upper bound on trajectory time per formance is maintained. The upper bound is revised as better solutions are computed. Several of these improving solution trajectories are shown in Figure 521. As before, solution trajectories are depicted as a red solid line and projected sensor footprints are depicted as dashed black lines. Solutions are shown after 4, 33, and 83 iterations and correspond to trajectories of 71.40 sec, 66.01 sec, and 61.43 sec, respectively. These values reflect the time required to the reach the end of the first loop from the initial condition. Figure 522 shows the result of a simulation that plots the maxvalue effectiveness metric for the minimumtime surveillance loop shown by Figure 521C. As previously, the solution trajectory is shown as a solid black line and the targets are shown as black circles. Notice that the sensing criteria are achieved for all target locations. 1000  500 0 500 0 0 0 500 1000 East (ft) 0 500 1000 East (ft) 0 500 1000 East (ft) Figure 521. Trajectory refinement for surveillance with obstacles. 1200 .. A 6011 6400 o 200 0 North (ft) 500 East (ft) 500 1000 East (ft) Figure 522. Simulated effectiveness for surveillance with obstacles. The algorithm is then applied to the coverage task in the environment depicted by Figure 516B. The coverage area is represented by a set of onehundred virtual targets which are sampled from a uniform distribution. The trajectory tree is again initialized with an RRT containing fifty nodes and which is rooted at (px,py, ) =(0,0,0) in a trim state corresponding to To. This initial tree is shown in Figure 523. 1000 500 0 0 500 1000 East (ft) Figure 523. Initial tree for coverage with obstacles. S500 0 1000 1000 : ::..~j~~7 The algorithm proceeds through 100 iterations in the same fashion as the other sensing tasks. Upon satisfaction of all sensing constraints, a trajectory is planned to the initial condition from the most recently added node. The best upper bound is maintained as in the previous cases. Several solutions that refine this upper bound are depicted in Figure 524. Solution trajectories are shown as a black solid line in the figure. Projected sensor footprints are not shown to avoid clutter resulting from the large number of virtual targets. Solutions are shown after, 1 and 5 iterations that correspond to trajectories of 106.38 sec and 87.30 sec, respectively. Notice that the best solution out of all 100 itera tions was found very early in the process. 1000  1000 500 500 0 0 0 500 1000 0 500 1000 East (ft) East (ft) Figure 524. Trajectory refinement for coverage with obstacles. Finally, Figure 525 shows the result of a simulation that plots the maxvalue effec tiveness metric for the minimumtime coverage trajectory shown by Figure 524B. As previously, the solution trajectory is shown as a solid black line and the targets are shown as black circles. Notice that the sensing criteria are achieved for all target locations. 0 00 0 08 a 600 06 East (f) 00 100 East (f) East (ft) Figure 525. Simulated effectiveness for coverage with obstacles. CHAPTER 6 OPTIMAL SENSOR PLANNING 6.1 Introduction Optimal control theory was presented in C'!i pter 2 as an approach for the design of optimal trajectories for systems constrained by dynamics. While tools from optimal control can provide solutions to many problems of interest, consideration of realistic constraints and mission descriptions often introduces complications which render optimal control solutions intractable. Several such complications are seen for the sensorplanning problem considered in this dissertation. In particular, the discontinuous nature of visibility and sensing effectiveness conditions violate smoothness requirements. These conditions can be treated as endpoint constraints in a series of subproblems; however, such an approach requires knowledge of the optimal sequence in which to view targets. Further, obtacles and occlusion regions add path constraints that can increase computational burden and introduce local minima. This chapter examines the application of optimal control methods to a simplified version of the closeproximity effective sensing problem. First, the problem of sensing a single target in an obstaclefree environment is considered. This scenario allows the formulation of sensing requirements as endpoint constraints and avoids the necessity to consider path constraints resulting from obstacles and occlusions. The necessary conditions for an optimal solution are derived to demonstrate some of the difficulties associated with the use of indirect methods to solve realistic problems. Next, numerically based direct optimal control methods are used to transcribe the problem to a finite dimensional NLP. The problem is extended to consider the sensing of a small set of targets in an obstaclefree environment. This scenario allows enumeration of all possible target sequences such that each can be formulated as a multiphase optimal control problem. The optimal solution for each sequence can be computed and compared to solutions which correspond to all other sequences to determine the optimum. Each of these problems utilizes concepts from C'! lpter 4 to define a goal set of vehicle configurations that ensure effective sensing. Specifically, sensing a particular target does not require visitation. Conversely, the visibilityset concept is employ' 1 to use the entire sensor FOV and thus account for motion coupling effects that result from vehicle dynamics. The specific dynamic model used with these examples is described in Section 6.2. The problem of determining a minimumtime trajectory that terminates with a desired view of a specified target is then formulated in Section 6.3. The indirect, variational approach to solving this problem is discussed in Section 6.4 and is followed by the application of a direct numerical method to the problem in Section 6.5. 6.2 System The sensing problems considered here utilize a dynamic aircraft model represented in statespace form as = f(x, i). The state and control vectors are given by Equations 6 1 and 62, respectively. The states that comprise Y are: total velocity, flight path angle, heading angle, altitude, North position, East position, angleofattack, angleofattack rate, and roll angle. The controls are commanded angleofattack and commanded roll angle. These control variables represent nontraditional system inputs and can be considered as generators of reference commands for lowerlevel control systems. Design of innerloop a and o controllers is typical in the field of aircraft control [108]. x= V 7 Q ph px p, a l y (61) u= U 1 (62) The equations of motion for this ninestate system are given by Equations 63 to 6 11, where c = 2 (7/180) and the force coefficients, T, CL, and CD, are taken from curvefitted data [109]. Each of these equations represents an element in the vectorvalued function, x= f(x, i). This model is based on a sixstate guidance level model from the literature [109]. The original model used a and o as controls; however, these variables are related to the aircraft/sensor attitude and are needed as states to formulate boundary conditions for the visibility problem. Therefore, secondorder angleofattack dynamics and firstorder roll dynamics are simulated through the addition of the three states, a, Q, and o, along with the two new controls, u, and u,. The parameters that specify the transient response of these added dynamics were adopted from the literature [108]. V = T cos (a + c) CDV2 sin 7 (63) 7 = ((T sin (a + c) + CLV2) cos O cos 7) (64) 1 S (T sin (a + e) + CV2) sin o (65) V cos 7 Ph =V sin7 (66) x = V cos 7 cos Q (67) y = V cos 7 sin Q (68) & = Q (69) 2 ,a 2Qwn,a + w ,L (6 10) 1 S1 (u 0) (611) The polynomial curves that are used to approximate the aerodynamic force coeffi cients are shown by Equations 612 to 614, where a1 = (127/180). The data corresponds to a model of a Boeing 727 [109]. Such an aircraft would not typically be involved in a closeproximity sensing mission; consequently, a scaledup version of the problem must be envisioned. The force coefficient curves are are plotted in Figure 61. T(V) = 0.2476 0.04312V + 0.008392V2 (612) CD(a) = 0.07351 0.086170 + 1.996c 2 (613) 0.1667 + 6.231c, a < ca CL(a) (614) 0.1667+ 6.231a 21.65 (a a)2, else 025 07, 2 06 15 024 05 1 022 0 30 02 0 021 0 1 1 0 100 200 300 400 500 15 10 0 5 10 5 20 25 30 '1 10 5 0 5 10 15 20 25 30 V (ft/sec) a (deg) a (deg) A B C Figure 61. Force coefficients. A) Thrust. B) Drag. C) Lift. In addition to scaling the problem to match the maneuvering capabilities of the aircraft, the model is scaled to facilitate numerical optimization performance [109]. Velocity is expressed in units of /g lc, where g is the acceleration due to gravity and 1c is a characteristic length. This length is given by Equation 615, where W is the aircraft weight and S is the planform area of the wing. All measures of distance are expressed in units of lc and time is scaled by vg lc/lc for optimization purposes. 2W Ic (615) 6.3 Problem Formulation Consider an aircraft whose motion is described by Equations 63 to 611 and which carries a LOS remote sensor. Motion is described relative to an inertial frame that is oriented such that the xaxis points in the North direction, the yaxis points in the East direction, and the zaxis points downward. The geometry of this vehiclesensor system is described as in ('! Ilpter 4. Here, the sensor reference frame, S, is placed to coincide with the vehicle body reference frame, B. As such, the sensor position and orientation are equivalently described by the vehicle position and orientation, as shown by Equations 6 16 and 617, where 0 represents the aircraft pitch angle. Additionally, this sensor has a FOV characterized by a maximum range of rm, = 10, 000 ft and a FOV halfangle of Of,max = 7/4. An incidence angle of a < r/2 is also required for visibility. 1T Ps =PB = Py Ph (616) Cs = = 0 (617) A target is located relative to an inertial coordinate frame by jt with an associated unit normal direction indicated by n, as shown by Equations 618 and 619, respectively. This target has an associated visibility set, V, which is defined in terms of the visibility parameters, r, o, and Of as described in Chapter 4. t= Pt,x Py Pt,h (618) T T = 0 0 o1 (619) Explicit forms of r and Of are expressed in Equations 620 and 621, respectively. The transformation term in Equation 621 can be written as Equation 622 for the sensor configuration considered here. Notice that this expression requires knowledge of the aircraft pitch angle, 0, which is not explicitly modeled in the equations of motion presented in Section 6.2. The pitch angle can be recovered through the kinematic rate ofclimb constraint, which is expressed in Equation 623 under the assumption that the sideslip angle, 3 is regulated to zero [108]. r = (px Pt,x)2 + (Py Pt,)2 + (Ph Pt,h)2 (620) P(st (TBES Of = arccos ) (T 3) (621) sin p sin u + cos p sin 0 cos u TBES3 sin O cos ) + cos cp sin 0 sin ) (622) cos p cos 0 sin 7 = cos a sin 0 cos 9p sin a cos 0 (623) The objective of the problem considered in this chapter is to plan a minimumtime trajectory from a specified initial condition at t = to to a terminal condition at t = tf which is characterized by (ps(tf), ((tf)) V. The minimumtime performance objective, J, is given by Equation 624. (624) The system is subjected to state equality and inequality constraints at the trajectory endpoints. These constraints are expressed as Teq = 0 and Tineq < 0, where the constraint vectors are given by Equations 625 and 626, respectively. Each state is constrained at t = to to take the value of a specified initial condition. The flight path angle, 7, is constrained for level flight at t = tf and the final altitude is constrained to match the initial altitude. Note that the inequality constraint vector is comprised of the conditions required for visibility, as shown in Equation 626. The incidence condition is not included here because the terminal constraint on altitude ensures a < 7r/2. S'(7(to), to, x(tf), tf)eq I(x(to), to, x(tf), tf) )ineq V(to) Vo/lc 7(to) 7o p(to) ',o/ ph(to) Ph,o/lc Px (to) Pol/lc p,(to) Py,0o/I a(to) ao (to) Qo p(to) o 7(tf) Ph(tf) Ph,o O(X(tf)) Omax Of (2?(^)) Of"... (625) (626) J= (x(tf),tf) = tf The resulting optimal control problem is described by Equations 627 to 632. The operating conditions have been restricted by placing bounds on allowable values for the elements of the control vector, u, and u,. These control limits are represented as Equations 631 and 632. min J (627) x f(, i) = 0 (628) '(I'(to),to, (tf),tf)eq 0 (629) (x(to), to, (tf),tf)ineq < 0 (630) Ua,min U < Ua,max (631) Up,min < Up < Uyp,max (632) 6.4 Variational Approach This section applies the variational calculus approach to determining the optimal state and control trajectories for the problem described by Equations 627 to 632. Such an approach is often called an indirect method because the solution is obtained indirectly through derivation of the firstorder necessary conditions for an optimal solution. State and control trajectories that satisfy these conditions are determined through the solution of boundaryvalue problem that results from the necessary conditions. Unfortunately, solution of such problems have numerous associated difficulties, including small radii of convergence, a good initial guess for the costate, and prior knowledge of trajectory segments for which path constraints are active [51, 52]. As a result of these difficulties, the current presentation formulates, but does not solve, the boundaryvalue problem. The primary purpose of this section is to demonstrate some of the practical issues that arise in applying optimal control theory to even a simplified version of the sensing problem. As such, the subsequent development actually considers a reducedform of the problem formulated in the previous section. Specifically, the constraints restricting allowable control values, given by Equations 631 and 632, are neglected. This assumption removes the need for prior knowledge of the constrained trajectory segments. Additionally, the endpoint inequality constraints represented by '(x0(to), to, x(tf), tf)ineq are treated as active qualities for derivation purposes. 6.4.1 Necessary Conditions The firstorder necessary conditions for a stationary point of the augmented cost functional, J, are derived using the first variation, 6J. The general form of each of these conditions was introduced in C'i plter 2. Here, these conditions are derived for the problem described by Equations 627 to 632. The Hamiltonian of the system must be computed to in order to form J prior to derivation of the necessary conditions. Recall that the Hamiltonian is given by = L + ATf(x, ui), where A is the vector of costate variables. The costates are timedependent and match the state vector in dimension, as shown by Equation 633. Here, H is expressed as in Equation 634. A Av A, A APh Apx py Aa AQ A T (633) H Av (T cos (a + c) CDV2 sin) + +A7 ((T sin (a + )+CLV2)cos cos)) + S(T sin (a + ) + CV2 sin +APh (V sin) + (634) S v cos 7 + Ap (V cos 7 cos ) + Ap, (V cos 7 sin )) + A () + + AQ (LL^0a w(aL2n,aQ + wua~a) + A, (1 ( ) The first condition considered specifies the differential behavior of the costate. Specifically, the costate dynamics are given by A = For the present system, the elements of this vector are expressed as shown in Equations 636 to 644. These expressions require partial derivatives of the forcecoefficient curve fits, Tv, CL., and CD. These derivatives are shown in Equations 645 to 647. Av ((Tvcos (a + ) 2CDV)AV + (635) +((sin (a + e)(VTv T) + CL) cos V + cos 7) +(sin ( + (VTv T) A, L sin i +(sin (a + e) CL) + sin ph + V2 cos y + cos 7 cos bApX + cos 7 sin bAp,) sin 7A. sin 7 A ( cos Xv + + s (T sin(a + ) +CLV2)sin p + (636) V V cos2 7 +V cos 7Ah V sin 7 cos bApX V sin 7 sin b ) A (Vcos7sin mAp + Vcos7cos ,Ay) (637) Aph 0 (638) AP, 0 (639) Ay = 0 (640) A ((T sin (a + ) CD V2)Av + (641) +(T cos (a + c) + CL V2) ( + ) A) (642) An = (, 2(,a,,An) (643) A, ((Tsin(a + )+CLV2)( co A.sin ) (644) Vcos 7 V 7r1 Tv(V) = 0.04312 + 2(0.008392)V (645) CD, (a) 0.08617 + 2(1.996)a (646) 6.231, a < ca CL, (a) (647) 6.231 2(21.65) (a ac) else The boundary conditions on the costate are set by the transversality conditions. These conditions were described in C'!I pter 2 as Equations 220 and 221, and require the introduction of a vector of Lagrange multipliers, given here as Equation 648. T V Vv Vo V,0 VPh0 VP,0 Vp V0 V00 0 Vf Vhf Vf VOf 1 (648) Evaluation of the transversality conditions also requires that partial derivatives are taken of the endpoint term in the cost functional, f(x(tf), tf), and of the endpoint T constraints, T = q Fineq j with respect to the state evaluated at the initial and terminal conditions. The cost functional exhibits no explicit dependence on the state such that a a= = 0. The partial derivative of the endpoint constraint ax(to) ax(t1) vector with respect to the initial state yields the matrix shown in Equation 649, where I represents an identity matrix and 0 represents a zero matrix. The partial derivative of the endpoint constraint vector with respect to the terminal state yields the matrix shown in Equation 650. The last row of this matrix contains several complicated derivatives resulting from the appearance of TBE in the expression for Of. The actual expressions for these derivatives are not shown here to maintain clarity. The costate boundary conditions are then formed by combining these expressions as specified by Equations 220 and 221. To 19x9 (649) aY(to) [49j 04x9 09x9 a( ) 0 1 0 0 0 0 0 0 0 (650) 0 0 0 1 0 0 0 0 0 0 0 0 ar or or 0 0 0 5h 5x 9y 0 ae, aso, aso, aso, aso, aso 0 a a a9i5 aph px Py ap 9 The consideration of open final time requires the introduction of a condition not cov ered in C'! lpter 2. This condition is given by Equation 651 and is denoted a Hamiltonian condition [47]. The result of this expression for the present derivation is to fix 'H at = 1 for all time, as here H is not an explicit function of time. f t f F (tf) 0 (651) Finally, the condition for the optimal control is given by the Minimum Principle, which states that the optimal control will minimize the Hamiltonian [48]. Recall from C'! lpter 2 that this condition can be expressed as H = 0, which identifies a stationary point of the Hamiltonian. For the present example, the optimality condition is expressed as in Equations 652 and 653. Ideally, these expressions could be used to solve for the optimal control vector in terms of the states, x, and costates, A. The Hamiltonian for the present example is linear in the control variables. Consequently, Equations 652 and 653 do not directly yield any relevant information regarding the control. 9H,0 0 A= A (6 52) Hn, = 0 = A, (653) 6.4.2 Boundary Value Problem The previous section has derived the firstorder necessary conditions for an optimal solution to a simplified version of the problem stated in Equations 627 to 632. Obtain ing the optimal state and control trajectories from these conditions is not necessarily an intuitive and straightforward process. When the Minimum Principle yields an expression for the optimal control in terms of the state and costate, the result can be substituted back into the state and costate dynamics. The augmented system is then described by the differential equations shown in Equations 654 and 655. .a= (654) A= ( (655) The state endpoint constraints and the transversality conditions provide boundary conditions for the augmented system of Equations 654 and 655; however, these con ditions are split between the endpoints and are not all known values. For example, the costate boundary conditions given by the transversality conditions derived in the previous section are seen as: VVo 0 90f 7T h0 V hfi + row a r n i r 0Pho + Phf a9 ( V r os "f Knowledge of these values would allow forward integration or the augmented dynamics to an optimal control trajectory. Unfortunately, determination of the set of initial costate s t ss t o os ipos a t r r a The case presented here cannot even proceed in this fashion, as Equations 6 52 90f Vaco VOf O Vo 0 9O f L V 0o Josf ao These expressions are dependent on the vector of unknown Lagrange multipliers, i . Knowledge of these values would allow forward integration of the augmented dynamics to determine the optimal state and costate trajectories. The Minimum Principle then yields an optimal control trajectory. Unfortunately, determination of the set of initial costate values that satisfy the conditions imposed at t = t typically requires an intensive and iterative process commonly referred to as "shooting." The case presented here cannot even proceed in this fashion, as Equations 652 and 653 do not provide an expression for u in terms of x and A. Methods for treating such problems are at least as complex as solution of the boundary value problem. The problem becomes even more complex when the inequality constraints are considered and path constraints are introduced. Consequently, an alternative approach to variational methods is desired. 6.5 Direct Transcription Approach Transcription of Equations 627 to 632 from a functional optmization problem to a finitedimensional parameter optimization problem presents a practical alternative to the variational approach presented in the previous section. The general process for state and control parameterization was described in C'! lpter 2. Basically, the time interval is split into a finite set of subintervals and a vector of unknown parameters is formed using the state and control values at the subinterval nodes. Dynamic constraints are applied in a piecewise fashion through the use of numerical methods to ensure that the state and control values satisfy the dynamics at these nodes. Path constraints can also be applied at these nodes, as well. Finally, a cost function is formed in terms of the parameterized state and control and solutions are determined through the use of wellestablished NLP solvers. This section applies a pseudospectral transcription method to the effective sensing problem. Pseudospectral methods parameterize the state and control using a basis of global orthogonal polynomials as opposed to the typical piecewise approximations [51, 52, 110, 111]. The specific method utilized here is denoted the Gauss pseudospectral method. This method collocates the dynamics at LegendreGauss points using a basis of Lagrange polynomials to interpolate the state and control trajectories[51]. The key benefits to pseudospectral methods are related to their efficiency, accuracy, and ease of implementation [110, 111]. The multitarget effective sensing problem is treated through subdivision of the problem into a series of phases. Each phase represents a nonoverlapping trajectory segment that involves different problem specifications such as cost, endpoint constraints, or even dynamics. These segments are linked by a set of conditions that constrain relevant parameters across the boundaries of sequential phases. Here, the trajectory segments associated with sensing each target individually are treated as separate phases. Each of these phases has an initial condition associated with the endpoint of the previous phase (or the global initial condition) and terminal constraints associated with sensing a particular target. For a given target sequence, an NLP can be formulated with a global state and control parameterization that is constrained individually within each phase as well as across phase boundaries. 6.5.1 Numerical Example A numerical example was performed using direct transcription with the Gauss Pseudospectral method. This example considers an aircraft described by the dynamics given in Section 6.2. This aircraft carries a downwardpointing sensor as described in Section 6.3. The objective of the example is to sense two targets in a specified sequence and then to return to the initial position. Simple visibility is considered sufficient to meet sensing effectiveness requirements. This example considers the environment depicted in Figure 62, which has an embedded inertial referenceframe that is oriented with the xaxis pointing in the North direction and the yaxis pointing in the East direction. The two targets are located relative to the inertial frame by jt, and 2,, as shown by Equations 6 56 and 657. The targets at these locations are oriented such that each has a unit normal vector described by hi, shown by Equation 658. These targets are shown in Figure 62 as red circles. t = 0, 000 5,000 0 (656) aP2 10,000 20,000 0 (657) S= 0 0 1 i 1,2 (658) The state initial conditions at to = 0 are given by Equations 659 to 667. This initial condition represents trimmed steady, straight and level flight at an altitude of 5, 000 ft heading due North. Note that the states are nondimensionalized for scaling purposes, as 5000 0 5000 10000 East 15000 20000 Figure 62. Environment for optimal sensing numerical example. described previously. V(O) 7(0) p(0) ph (0) P.(0) Py(0) a (0) Q (0) 535/lc 0 5000/lc 0 0 1.7882 (7/180) 0 (667) The problem is set up in three sequential phases. The first phase contains the trajectory segment along which the aircraft must move from the initial condition to a point where the target located by jp is visible. The second phase contains the trajectory segment from the endpoint of the first phase to a point where the target located by it2 is visible. Finally, the third phase contains the trajectory segment from the endpoint of the second phase back to the initial position. Each phase is discretized using 25 nodes at which the dynamics are collocated and path constraints are enforced. Terminal constraints are enforced at the end of each phase. These constraints are consistent with Equations 629 and 630; however, the inequality constraints that reflect target visibility are only enforced for the first and second phases. Further, these visibility constraints are defined relative to the appropriate target for each phase. An additional path constraint is imposed that restricts altitude to remain constant at ph(0) 5000/lc. This constraint is added to simplify the computation of the visibility parameter, Of. Generally, computation of Of requires access to the pitch angle, 0, as described by Equations 621 and 622. Recall that the pitch angle is recovered from the kinematic constraint given as Equation 623. This expression requires a numerical solution procedure for nonzero 7. The constant altitude constraint results in constant 7 = 0 over the entire trajectory. As such, 0 can be computed analytically as Equation 668. 0 = arctan (cos p tan a) (668) The NLP is set up and solved using GPOCS optimization software, which utilizes TOMLAB/SNOPT [53, 64, 112]. The cost function for the NLP is specified for the total time elapsed at the end of the third phase. An optimal solution was found that achieved all objectives with a trajectory of 62.0356 sec in duration. The resulting trajectories of the six states which describe position and velocity are plotted in Figure 63. The position T relative to the inertial frame is described by JB P= p pi Ph while the total velocity is described by V. The direction of total velocity is specified by the flight path angle, 7, and the heading angle y. The trajectories corresponding to the angle of attack, a, and the roll angle, p, are shown in Figure 64. These trajectories would serve as the input reference trajectories to a lowerlevel controller for a vehicle tasked with executing the solution trajectory. 10000 5000 0 ^^^ 0 10 20 30 40 50 60 x104 0 10 20 30 40 50 60 0 10 20 30 40 50 60 C,6000 4000 2000 0 10 20 30 40 50 60 Figure 63. Solution position and velocity tra 15 5T 10 5 4500 0 10 20 30 40 50 60 0 10 20 30 300 a 200 100 0 jectories. 40 50 60 10 20 30 40 50 60 0 10 20 30 40 50 60 40 20 e t'' 0 10 20 30 40 50 60 Figure 64. Solution a and o reference trajectories. The trajectories for the visibility parameters associated with the first and second targets are shown in Figures 65A and 65B, respectively. The maximum range and FOV angle are indicated by dashed lines in each plot. Notice that these constraints are not necessarily all active at the phase boundaries. The plots show that the visibility bounds are satisfied for the first target at the end of the first phase and are satisfied for the second target at the end of the second phase. Finally, Figure 66 shows an overhead view of the total trajectory with projected sensor footprints shown by dashed contours at the end of each of the first two phases. The figure shows that the full sensor FOV and motion coupling were considered and utilized in the optimal solution. M 2 x 10,  1 05 0 0 10 20 30 40 50 60 S100 1 50 0 10 20 30 40 50 60 2x 10 2 05 0 0 10 20 30 40 50 60 150 100 0 50 0 10 20 30 40 50 60 Figure 65. Solution visibility parameter trajectories for each target. 0 5000 10000 East 15000 20000 Figure 66. Aircraft trajectory and sensor footprint over each target. 16000 14000 12000 10000 8000 6000 4000 2000 0 2000 4000 5000 CHAPTER 7 CONCLUSION The work presented in this dissertation has addressed the problem of mission planning for autonomous sensing tasks. More specifically, mission scenarios have been considered which require operation of a vehiclecarried remote sensor in close proximity to both obstacles and sensing objectives. Methods were developed to address some of the unique challenges introduced for sensing missions fitting this description. A pointtopoint motion planning method that accounts for vehicle dynamics was developed to ensure safe navigation amongst obstacles, where there is little margin for error. A framework was introduced to quantify and evaluate the general quality of a particular view with respect to geometric disparities exhibited by various surfaces within the sensor FOV. Finally, a randomized motion planning method was developed to generate trajectories that utilize the entire sensor FOV and account for coupling between sensor pointing and vehicle maneuvering. Motion planning for differentially constrained vehicle systems was examined in C'i ip ter 2. The concept of motion primitives were used to represent vehicle dynamics in a hybrid modeling framework. The pointtopoint motion planning problem was then formu lated as a hierarchical optimization problem that contained a combinatorial element and a continuous NLP element. The combinatorial element was shown to greatly complicate the solution procedure for realistic scenarios. As such, a special class of feasible, though sub optimal, explicit solutions were identified as a means by which approximate solutions can be computed efficiently. Utility of these solutions was demonstrated for the obstaclefree case; however, global solutions were seen to be limited when obstacles are considered. As such, these solutions are tailored more for local planning tasks, such as that required for branch extension in randomized methods. Several areas for future work related to the methods described in C'!i pter 2 can be identified. The examples discussed in C'! lpter 2 utilized very simple dynamic models to maintain clarity of presentation; however, the concept of modeling with motion primitives is general enough to handle more complicated and realistic systems. The introduction of complicated dynamics is likely to be accompanied by new issues not covered in this presentation. For example, the identification of families of unique solution sequences was straightforward for the models considered in C'!i lpter 2, but identification of these sequences has not been treated here for the general case. Additionally, consideration of more complicated solution sequences could adversely affect the efficiency of these solutions. Generalization of this approach provides an interesting direction for future research. The concept of randomized samplingbased motion planning was introduced in ('! i p ter 3. A randomized planning algorithm was developed that utilized the solution methods introduced in C'!i lpter 2. This planner generated feasible solutions by growing a tree of subsolutions that search the planning space in a probabilistic fashion. A simulated example demonstrated the efficiency and effectiveness of the approach; however, several drawbacks were apparent as well. First, solutions are suboptimal and not necessarily repeatable due to the randomized nature of the algorithm. These issues result from the tradeoff between optimality of oversimplified problems versus feasible, but suboptimal, so lutions to complicated problems. Such a tradeoff is often acceptable for many applications. Future work might include an investigation into the extent to which performance can be improved by a postprocessing trajectory refinement. A main criticism of randomized approaches is that actual implementation of such algorithms requires a large number of design parameters to be set. While these planners have been shown to perform well in practice, this performance is often dependent on the proper selection of problemspecific parameters and heuristics. The process of selecting the most effective combination can be lengthy and involves a significant degree of trial and error. ('!i lpter 4 discussed a geometric approach to modeling sensor visibility. The effects of sensor operation in close proximity to the environment were discussed and a general framework was introduced to quantify these effects as a measure of data quality. The work presented in C'!i ipter 4 represents an initial treatment of a topic that should pi. i a significant role in future missions for autonomous vehicles. Numerous potential extensions to the concept of sensing effectiveness can be identified. Future work might include research into the integration of a FOVbased effectiveness metric with existing data quality metrics. For example, the characterization of views that provide highquality data related to certain features could aid in target recognition tasks. Additionally, a logical next step involves the inclusion of temporal effects in the formulation of the effectiveness metric. Such a formulation could account for effects related to the velocity with which an object moves through the FOV, such as motion blur. The sensorplanning task was addressed with a randomized planning approach in C'! Ilpter 5. The randomized planning and sensor effectiveness concepts from previous chapters are integrated to form a new randomized planning approach to sensing a series of targets in a close proximity environment. This method generates efficient solutions to a difficult problem that exhibits combinatorial and dynamic elements that are highly coupled. Such solutions are enabled by the property that a singletarget sensing trajectory often senses multiple targets as a result of the mission and environment scale. The randomized approach suffers from similar drawbacks to those discussed for the planner developed in C'!i lpter 2; however, the simulated results are quite reasonable considering the difficulty of the problem. The efficient but imperfect randomized approach was contrasted by a rigorous but cumbersome application of optimal control to the sensing problem in Chapter 6. This chapter derived necessary conditions for an optimal solution to a simplified version of the problem with the purpose of demonstrating the challenges associated with adopting this solution method. Then, a direct method is implemented to compute a numerical solution to the simplified problem. While this implementation neglects some of the issues related to sensing in a close proximity environment, the results are promising. There is significant potential in combining concepts from randomized planning methods with direct optimal control methods. 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[108] Stevens, B., and Lewis, F., Aircraft Control and Simulation, John Wiley & Sons, Inc., New York, NY, 1992. [109] Bryson, A.E., D im:. Optimization, AddisonWesley, Menlo Park, CA, 1999. [110] Rao, A.V., "Extension of a Pseudospectral Legendre Method to NonSequential MultiplePhase Optimal Control Problems," Proceedings of the 2003 AIAA Guid ance, Navigation, and Control Conference, Austin, TX, August 2003. [111] Elnagar, G., Kazemi, M.A., and Razzaghi, M., "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," IEEE Transactions on Auto matic Control, Vol. 40, No. 10, October 1995, pp. 18931796. [112] Rao, A.V., "User's Manual for GPOCS Version 1.0/: A MATLAB Package to Solve NonSequential MultuplePhase Optimal Control Problems," Gainesville, FL, 2007. BIOGRAPHICAL SKETCH Joseph John Kehoe was born in Cooperstown, NY, on July 16, 1980. He grew up in Oneonta, NY, and graduated from Oneonta High School in June of 1998. He then attended Virginia Polytechnic Institute and State University, where he received a Bachelor of Science degree in aerospace engineering in May of 2002. He has been a graduate student at the University of Florida in the Department of Mechanical and Aerospace Engineering since August of 2003. His research in the Flight Control Laboratory has involved the development of .L,, .i ii i.:venabling guidance and control technologies for small unmanned aerial vehicles (UAVs). Joseph received a Master of Science degree in aerospace engineer ing from the University of Florida in December, 2004. Upon finishing his doctoral degree, he will remain at UF as a postdoctoral researcher for the fall semester, after which he plans to pursue a career researching UAV control technologies. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IwouldrstliketoacknowledgeDr.RickLindforprovidingsteadyguidancethroughoutmygraduateeducationexperience.Hisadvice,support,andcriticismshavealwayskeptmycareerinterestsinmind,andhavepreparedmeto(nally)leavethesafeconnesoftheeducationsystemforthe\realworld."Ithanktheothermembersofmycommittee,Dr.CarlCrane,Dr.WarrenDixon,Dr.PeterIfju,andDr.ClintSlattonfortakingthetimetoreviewmyresearchandprovidecommentsonthedirectionsofmyideas.Additionally,thanksgoouttothemembersofightcontrollabforprovidingalwaysneededdiversionsfromwork,andforthecountlessindepthdiscussionsrelatedtonothinginparticular.Sometimesstepsinwrongdirectionareneededtomakeleapsintherightdirection.IhavelearnedthingsinthepastfouryearsthatIneverwouldhaveimaginedlearningingraduateschool.Specically,IthankMujahidAbdulrahimforgreatlyexpandingmyknowledgeregardingrallyracing,rubberchickens,andunicycles.AdamWatkinsdeservescreditforteachingmethejoythatcancomefromtherightcombinationofcynicismandsarcasm.IwouldalsoliketothankDaniel\Tex"Grantforteachingmethatmeatloafmadefromvenisontastessurprisinglyjustlikeregularmeatloaf.IwouldalsoliketoacknowledgeAirForceReasearchLabs,MunitionsDirectorate,specicallyJohnnyEvers,NealGlassman,SharonHeise,andRobertSierakowski,forcontinuednancialsupport.ThanksgotoDr.JasonStackforprovidingadditionalfundingaswellasforhisguidanceduringthetwosummersIspentasaninternattheNavalSurfaceWarfareCenterinPanamaCity,FL.Finally,Iwouldliketothankmyfamilyandfriendsfortheirpatienceindealingwithmeduringthesebusypastfewyears.Thisworkwouldnothavebeenpossiblewithouttheloveandsupportofmyparents,JosephandLinda,andmysister,Kathleen.IwouldliketothankmyFatherinlaw,BobWilliams,forbeingalwayswillingtotalkshop,andmyMotherinlaw,PatWilliams,foralwaysputtingupwithit.Iowemywife,Sarah,agreat 4 PAGE 5 5 PAGE 6 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 16 1.1Motivation .................................... 16 1.2ProblemDescription .............................. 17 1.3ProblemStatement ............................... 22 1.4ApproachOverview ............................... 24 1.5Contributions .................................. 26 2MOTIONPLANNING ................................ 27 2.1Introduction ................................... 27 2.2PathConstraints ................................ 28 2.2.1DierentialConstraints ......................... 29 2.2.2ObstacleConstraints .......................... 31 2.3OptimalControl ................................ 36 2.3.1IndirectOptimalControl ........................ 37 2.3.2DirectOptimalControl ......................... 39 2.4HybridMotionModels ............................. 42 2.4.1ModelingwithMotionPrimitives ................... 42 2.4.2MotionPlanningwithPrimitives .................... 45 2.4.3KinematicExample ........................... 48 2.4.4DynamicExample ............................ 54 3RANDOMIZEDSAMPLINGBASEDMOTIONPLANNING .......... 62 3.1Introduction ................................... 62 3.2ProbabilisticRoadmapMethods(PRM) ................... 62 3.3RandomDenseTreeMethods(RDT) ..................... 64 3.3.1RapidlyExploringRandomTrees(RRT) ............... 65 3.3.2ExpansiveSpacesTrees(EST) ..................... 66 3.3.3Discussion ................................ 67 6 PAGE 7 ......... 68 3.4.1Model .................................. 69 3.4.2Overview ................................. 73 3.4.3NodeSelection .............................. 74 3.4.4NodeExpansion ............................. 76 3.4.5SolutionCheck ............................. 78 3.5Example ..................................... 79 4SENSINGEFFECTIVENESS ............................ 85 4.1Introduction ................................... 85 4.2RemoteSensorTechnologies .......................... 86 4.2.1ComputerVision ............................ 87 4.2.2Radar .................................. 87 4.2.3Sonar ................................... 88 4.2.4Ladar ................................... 89 4.3ModelingtheSensingTask ........................... 90 4.3.1SensingGeometry ............................ 90 4.3.2Visibility ................................. 92 4.3.3TheVisibilitySet ............................ 95 4.3.4ProximityEects ............................ 97 4.4EectivenessMetric ............................... 101 4.4.1Formulation ............................... 102 4.4.2TheQualitySet ............................. 104 4.4.3SensingMissionEectiveness ...................... 104 4.4.4Example:AContrivedMetric ..................... 106 4.4.5Example:ImageArea .......................... 110 5RANDOMIZEDSENSORPLANNING ....................... 115 5.1Introduction ................................... 115 5.2EnvironmentRepresentation .......................... 117 5.3ARandomizedSensorPlanningAlgorithm .................. 120 5.3.1Overview ................................. 120 5.3.2NodeSelection .............................. 121 5.3.3VantagePointSelection ......................... 123 5.3.4LocalPlanningandExpansion ..................... 125 5.3.5Evaluation ................................ 126 5.4Examples .................................... 127 5.4.1Model .................................. 127 5.4.2EectivenessMetric ........................... 129 5.4.3ObstacleFreeExamples ......................... 130 5.4.4ExampleswithObstacles ........................ 136 7 PAGE 8 .......................... 143 6.1Introduction ................................... 143 6.2System ...................................... 144 6.3ProblemFormulation .............................. 146 6.4VariationalApproach .............................. 149 6.4.1NecessaryConditions .......................... 150 6.4.2BoundaryValueProblem ........................ 153 6.5DirectTranscriptionApproach ......................... 155 6.5.1NumericalExample ........................... 156 7CONCLUSION .................................... 161 REFERENCES ....................................... 165 BIOGRAPHICALSKETCH ................................ 175 8 PAGE 9 Table page 21~Mcomponentsforeachmaneuver ......................... 57 41Statisticsforvisibilityparametercomparison .................... 101 9 PAGE 10 Figure page 11Prevalentexamplesofadvancementsinunmannedtechnology .......... 16 12Comparisonofsensingdatafordierentmissionscales .............. 18 13Touringaseriesofcellsforalargescalesensingmission .............. 19 14Touringaseriesofcellsforasmallscalesensingmission ............. 20 15Motionplanningtoviewasequenceoftargets ................... 21 16Sensingmissiontasks ................................. 24 21Obstacleboundaryapproximations ......................... 34 22Vertexanglesumcollisiondetectionmethod .................... 35 23Vertexedgevectorcollisiondetectionmethod ................... 35 24Obstacleexpansionforpointwisesafety ....................... 36 25Automatonmodelrepresentationasadirectedgraph ............... 45 26Specialreferenceframe,DforDubinspathsolution ................ 51 27SampleDubinspaths ................................. 54 28Examplestatetrajectoryshowingmaneuverdynamics .............. 56 29Topviewofenvironmentfordynamicexample ................... 58 210TopviewofsolutionforM=fMSL;MLS;MSLg 59 211HeadingandturnratetrajectoriesforM=fMSL;MLS;MSLg 59 212TopviewofsolutionforM=fMSL;MLS;MSL;MLS;MSLg 60 213HeadingandturnratetrajectoriesforM=fMSL;MLS;MSL;MLS;MSLg 60 214TopviewofsolutionforM=fMSL;MLS;MSR;MRS;MSR;MRSg 61 215Heading,turnratetrajectoriesforM=fMSL;MLS;MSR;MRS;MSR;MRSg 61 31ThePRMalgorithm ................................. 63 32RRTalgorithm .................................... 66 33ESTalgorithm .................................... 67 34DierencesinexplorationstrategyfortheRRTalgorithmvs.theESTalgorithm 68 10 PAGE 11 ... 70 36Distancefunctioncomparison ............................ 75 37Distancefunctioncomputation ........................... 76 38Nodeexpansionstep ................................. 77 39Thenewbranchissubdividedtoasetofnodes. .................. 78 310Collisionfreesolutions ................................ 79 311Uniquesolutionfamiliesforvehicleusedinexample ................ 81 312Turnstraightsolutionsequences ........................... 81 313Exampleplanningenvironment ........................... 82 314Incrementaltreeexpansion .............................. 82 315Nodesinthenalsolutiontree ........................... 83 316Solutionpathrenement ............................... 84 41Sensingproblemgeometry .............................. 91 42Visibilityparameters ................................. 94 43Invertedvisibilitycone ................................ 96 44ConstructionofVi 97 45Motioncouplingeectsfordierentproblemscales ................ 99 46Rangeandincidencevariationfordierentproblemscales ............ 100 47Representativetrajectoriesforsensoreectivenessmetric ............. 104 48Qualityparametereciencyfunctions. ....................... 107 49Simulatedtrajectoryshowingsnapshotsofeectivenessmetricvalues ...... 109 410Maxvaluemissioneectiveness ........................... 110 411Environmentforimageareasensingmetricsimulation ............... 112 412Sensingeectivenessasafunctionofimagearea .................. 113 413Eectivenesstrajectories ............................... 114 414Simulatedmissioneectiveness ........................... 114 51TSPoroblemcomparison ............................... 115 11 PAGE 12 .................................. 118 533DOcclusionshadow ................................. 118 54Discretizationapproachesforareacoverage ..................... 119 55Sensingsecondarytargets .............................. 122 56Examplenodeweightingfunction .......................... 123 57Vantagepointreachabilitytest ........................... 125 58Qualityparametereciencyfunctionsforexamples ................ 130 59Environmentsforobstaclefreesensorplanningexample .............. 131 510Trajectoryrenementforobstaclefreereconnaissance ............... 133 511Simulatedeectivenessforobstaclefreereconnaissance .............. 134 512Trajectoryrenementforobstaclefreesurveillance ................ 135 513Simulatedeectivenessforobstaclefreesurveillance ................ 135 514Trajectoryrenementforobstaclefreecoverage .................. 136 515Simulatedeectivenessforobstaclefreecoverage .................. 137 516Environmentsforsensorplanningexamplewithobstacles ............. 138 517Initialtreeforreconnaissancewithobstacles .................... 138 518Trajectoryrenementforreconnaissancewithobstacles .............. 139 519Simulatedeectivenessforreconnaissancewithobstacles ............. 139 520Initialtreeforsurveillancewithobstacles ...................... 140 521Trajectoryrenementforsurveillancewithobstacles ............... 141 522Simulatedeectivenessforsurveillancewithobstacles ............... 141 523Initialtreeforcoveragewithobstacles ....................... 141 524Trajectoryrenementforcoveragewithobstacles ................. 142 525Simulatedeectivenessforcoveragewithobstacles ................. 142 61Forcecoecients ................................... 146 62Environmentforoptimalsensingnumericalexample ................ 157 63Solutionpositionandvelocitytrajectories ..................... 159 12 PAGE 13 ....................... 159 65Solutionvisibilityparametertrajectoriesforeachtarget .............. 160 66Aircrafttrajectoryandsensorfootprintovereachtarget ............. 160 13 PAGE 14 14 PAGE 15 15 PAGE 16 11A to 11C ,representasmallcrosssectionofthediverseandgrowingeldofunmannedvehicles. B CFigure11. Prevalentexamplesofadvancementsinunmannedtechnology.A)TheRQ1Predator.B)Amarsrover.C)StanfordRacing's\Stanley" Oftheexpandingrolesplayedbyexistingunmannedtechnologies,futurevehiclesstandtoproveespeciallyusefulforapplicationsthatinvolvethegatheringofinformation.Envisionedciviliandomainscenariosincludeborderpatrol,tracmonitoring,tacticallawenforcement,maritimesurveillance,andenvironmentalsensing[ 1 ].Likewise,intelligence,surveillance,andreconnaissance(ISR)missionsareemergingasthedominantthemeamongprojectedmilitaryapplicationsforfutureunmannedsystems.Thefuturecombatparadigmislikelytoincludetheuseofcooperatingteamsofunmannedaerialvehicles(UAVs),unmannedgroundvehicles(UGVs),andunmannedunderwatervehicles(UUVs)equippedwithvarioussensorsthatautonomouslynavigatecomplexenvironmentsforthecollectionofISRdata[ 2 { 5 ]. 16 PAGE 17 12A depictstheUniversityofFloridacampusasviewedfromanaltitudeofapproximately15;000ft.Theentirecampusandsurroundingareacanbeseeninasingleimagecapturedfromasinglevantagepoint.Advancesinhardwareandcontroltechnologieswillsoonenablemissiondescriptionsonamuchsmallerscaleinincreasinglycomplexenvironments.Forexample,researchersforeseecooperativeteamsofUGVsandUAVspatrollinganurbanpeacekeepingenvironment[ 6 ].NavigationcapabilitiesforUAVsoperatinginsuchenvironmentsarealreadyemergingthroughthedevelopmentofminiatureautopilotsystems[ 7 ],visionbasedcontroltechnologies[ 8 { 11 ],andagilityenhancingmorphingstrategies[ 12 { 14 ].Sensingmissionsinsuchascenarioarecharacterizedbyvehicleandsensoroperationwithincloseproximitytotheenvironment.Figure 12B showsasimulatedimageoftheUniversityofFloridacampusasviewedfromaUAVoperatingatanaltitudeofapproximately50ft.Typicaldatacollectedduringsuchmissionsisobviouslydierentinnaturethanthatcollectedduringstandosensingmissions.Smallerportionsoftheenvironmentarecoveredfromaparticularviewpoint;however,thedataisofmuchhigherresolutionandexhibitsrichercontent.Moreover,additionalsensormodalities,suchasaudio,gainrelevanceatcloserange.Thesekeybenetsexpandpossiblemissionscenariostoincludetaskssuchassensingunderandinsidestructures. 12 showsthatfundamentaldierencescanbeexpectedinthewaythetwodepictedclassesofsensingmissionsareapproached. 17 PAGE 18 BFigure12. Comparisonofsensingdatafordierentmissionscales.A)Imagescapturedfromstandorange.B)Imagescapturedwithincloseproximitytoaclutteredenvironment. Oneaimoftheworkpresentedhereistoidentifyandaddresssometheuniquechallengesencounteredforsensorplanningincloseproximityenvironments.Motionplanningmethodsforstandomissionscommonlymakesimplifyingassumptionsthatareconsistentwiththeproblemscale.Forexample,targettouringisoftenconsideredsucienttoensuresensingofasetoftargets.Previouseortshaveadoptedthistechnique,whichinvolvessimplyvisitingeachtargetlocation(oralocationdirectlyabove,forthecaseofUAVs)[ 15 { 18 ].Similarly,areacoverageinsuchcasesisoftenachievedthroughdecompositionoftheareaofinterestintocellsandthensimplyvisitingeachcelltomeetsensingobjectives[ 19 { 21 ].Figure 13 showsarepresentativeexampleofasensormotionplanthatoperatesundersuchassumptions.Inthedepictedscenario,thesensorFOVisconsideredlargeenoughsuchthatcellvisitationissucienttosenseanytargetscontainedwithinthecell.Theseapproachesalsotypicallyassumethatlocalmotionconstraintsduetovehicledynamicsarenegligiblerelativetothescaleofthemotionplanningproblem.Forexample,thevehicleturnradiusinFigure 13 issmallenoughrelativetothecellresolutionandtothescaleofmaneuveringthatsuchaconstraintcanbeneglectedinmotionplanning.Numeroustechnicalchallengesareintroducedbyoperationofthevehicleandsensorincloseproximitytotheenvironment.First,Figure 12B showsdrasticallydisparatedataindierentareasoftheimageascomparedtoFigure 12A .Someobjectsaremuchclosertothesensorthanothers,andhenceappear 18 PAGE 19 Touringaseriesofcellsforalargescalesensingmission. intheforegroundoftheimagewithcomparativelyhighdataresolution.Additionally,objectsareseenataspectsthatyielddatawithvaryingresolutiononasinglesurface.Theextentoftheseeectsvariesasthevehiclemovesthroughtheenvironmentandtheviewpointchanges.DataqualitywithrespecttoaparticularsensedobjectisseentovaryasafunctionofboththevantagepointandthelocationwithintheFOV.Further,sensoroperationamongstobstaclesresultsintheocclusionoflargeregions.Assuch,theviewqualityofaparticulartargetcouldchangedrasticallywithsmallchangesinvehicleandsensorpositionandorientation.AnotherissueisrelatedtothecomparativelysmallspatialareacontainedwithintheFOVforcloseproximitycases.AsmallprojectedfootprintrelativetovehiclevelocitycausestargetsurfacestopassthroughtheFOVquicklyasthevehiclemovesthroughtheenvironment.Resultingdatalossesareexacerbatedbyclutteredenvironmentsthatrequiresignicantmaneuveringforsafenavigation.Assuch,amotionplantoviewaparticulartargetmustbetimedappropriatelyandtrackedwithenoughprecisiontoensurethatthesensorispointingintherightdirectionattherighttime.Severalresearchershaveexpandedthetargettouringapproachtoaccountforaparticularsensorfootprintshapeandsize;however,theseeortsfailtoaccountformotioncouplingthatresultsfrommaneuvering[ 17 20 22 { 24 ].Figure 14 showsarepresentativemotionplanthatdisplaysdataqualityandmotioncouplingeectsrelatedtoproximity.Inthegure,thesensorfootprintisnotsignicantlylargerelativetocellresolution.Assuch,sensorcoverageisseentocoversomecellspartiallyandnonuniformlyduetothevariableresolutionissuediscussedpreviously. 19 PAGE 20 14 assensorcoveragethatvariesrelativetothevehicletrajectory. Figure14. Touringaseriesofcellsforasmallscalesensingmission. Planningatrajectorytoviewasequenceoftargetspresentsadicultproblemregardlessofsensingproximityeects.Generally,anoptimalsolutiontothisproblemrequiresthat:1.Theoptimalorderoftargetvisitationisdetermined2.Theoptimalviewpointforeachtargetisdetermined3.TheoptimaltrajectorythroughtheoptimalviewpointsiscomputedTheserequirementsaredepictedinFigure 15 .Eortstoaddresssuchproblemstypicallyinvolvetheapplicationofassumptionsandthesolutionofsuboptimalapproximations.Forexample,thetargettouringapproachdescribedpreviouslyessentiallyassumesthattheoptimalviewpointforeachtargetisthelocationofthetargetitself.Whendynamicsareneglected,thisproblemrevertstothewellknowntravelingsalesmanproblem(TSP)fromgraphtheory,forwhichnumeroussolutionandapproximationtechniquesexist[ 25 26 ].Severaleortsachieveanapproximationtotheoptimalsetofviewpointstosenseanareathroughrandomsampling,andthenproceedtosolvetheensuingTSP[ 27 { 29 ].Theseeortsdonotconsiderdierentiallyconstrainedmotion,which,whilesomeapproximationtechniquesdoexist[ 30 ],cangreatlycomplicatethesolutionoftheTSP. 20 PAGE 21 Motionplanningtoviewasequenceoftargets. Finally,thecloseproximitysensingproblemrequiresthatplannedtrajectoriesaccuratelyrepresentfeasiblevehiclemotionssuchthattrajectorytrackingwillensuresafenavigationandprecisionsensorplacement.Trajectoriesthatdonotadequatelyrepresentfeasiblemotionsaresubjecttotrackingerroruponmissionexecutionthatcouldresultineitherfailuretoreachapreciselydenedviewpointorincollisionwithanobstacle.Generally,thisrequirementimpliesthatplannedtrajectoriesshouldbeconstrainedbysomerepresentationofthevehicledynamics.Inclusionofdynamicallyfeasiblemotionsinaplannedtrajectoryistypicallytreatedineitheradirectoradecoupledfashion[ 31 ].Directplanningmethods,suchasoptimalcontrol,considerarepresentationofthevehicledynamicsintheformulationoftheplanningproblemanddirectlysolveforoptimalsysteminputs.Alternatively,indirectmethodsuseasimpliedmodelofvehiclemotiontoplanareferencepathandthen\smooth"thepathtosatisfydynamicsusingmethodssuchasfeedbackcontrol.Directmethodscomputeoptimaltrajectoriesbutareoftenintractableforrealisticproblemdescriptions.Indirectmethodsoftenexhibittractablecomplexitypropertiesthatcomesattheexpenseofoptimality.Researchershavefoundcleverwaystomanipulatethistradeothroughavarietyoftechniquessuchthatdynamicscanbedirectlyincludedintheplanningprocess.Forexample,someresearchershaverecognizedthatsystemswhichexhibitdierentialatnesspropertiesadmitsolutionsthatcanberepresentedparametricallyintermsofasetofatoutputsandtheirderivatives[ 32 { 34 ].Othershaveappliedmixedintegerlinearprogramming(MILP)tomodeldynamicconstraintsasasetofswitchingboundson 21 PAGE 22 35 { 37 ].Frazzolietalintroducedaplanningtechniquethatutilizesa\sampleddynamics"modelwhichemploysasetofdynamicallyconsistentmotionprimitives[ 38 39 ].Additionally,recentadvancesinrandomizedplanningallowtheuseofanyofthesetechniquesaslocaltrajectorygenerationmethodsforgrowingaprobabilistictreeofactionstoexplorethesolutionspace[ 40 { 42 ]. 22 PAGE 23 16A .Thepurposesofthistypeofmissiontaskincludetargetidentication,verication,classication,andother`singlelook'reconnaissanceobjectives.Thesurveillancetaskalsoassumesthatatargetorseriesoftargetshasbeenrecognizedandlocated;however,thegoalhaschangedtopersistentobservation.Thevehiclemustpatroltheseriesoftargetsandrepeatthesequenceuntilaprespeciedterminationconditionismet.ThismissiontaskisdepictedinFigure 16B .Thepurposesofthistypeofmissionincludetargetverication,classication,monitoring,changedetection,andcollectionofmodelingdata.Thecoveragetaskimpliesnoassumptionsabouttargetlocations.Thevehiclemustpassthesensorfootprintovertheentireareaofinterestinsuchawaythatsomelevelofdataqualityisachievedovertheentiretaskspace.ThismissiontaskisdepictedinFigure 16C .Thepurposesofthistypeofmissiontaskincludesearch,explore,targetdetection,changedetectionandwideareasurveillance. 23 PAGE 24 B CFigure16. Sensingmissiontasks.A)Reconnaissancemission.B)Surveillancemission.C)Coveragemission. Finally,severalassumptionsareconsideredinformulatingthesolutionapproach.Thesemainassumptionsareasfollows:ObstaclesintheenvironmentarestaticObstacleandtargetlocationsareknownaprioriAvehiclemodelisavailablethatiscapableofrepresentingdynamicsatvariouspointsintheoperatingenvelopePlannedmotionsaredesignedasopenloopguidancetrajectories 1.3 isaddressedthroughthreemainsteps.First,amethodisdevelopedfordynamicpointtopointmotionplanninginenvironmentsthatmeetthedenitionofcloseproximityasdiscussedpreviously.Then,aframeworkforquantifyingtheeectsofsensoroperationinsuchenvironmentsisdevelopedforuseasaperformancemetric.Finally,asensorplanningmethodisdevelopedwhichintegratesthismetricwiththepointtopointplanningmethodintoarandomizedschemedesignedtoachievethetasksspeciedinSection 1.3 .MotionplanningfordierentiallyconstrainedvehiclesystemsisexaminedinChapter 2 .Ahybridrepresentationofthevehicledynamicswhichutilizestheconceptofmotionprimitivesismodeledafterpreviouseorts.Thishybridrepresentationisusedtoformulatethetrajectoryplanningproblemasahierarchicaloptimizationproblem.Aspecialclassoffeasible,thoughsuboptimal,explicitsolutionsareidentiedandutilizedto 24 PAGE 25 3 throughdiscussionofseveralexistingalgorithms.ArandomizedplanningalgorithmisdevelopedtocomputeapproximatesolutionstotheoptimizationproblemposedinChapter 2 .Thismethodmakeseectiveuseoftheexplicitsolutionapproximationsintroducedinthepreviouschaptersuchthatsomeofthemajordrawbacksareavoided.Thisplanningmethodisdemonstratedthroughasimulatedexample.Chapter 4 discussesmodelingofthesensingtask.Specically,ageometricsensingmodelisdevelopedandusedtoformulateadenitionofvisibility.Then,theeectsofsensoroperationincloseproximitytotheenvironmentarediscussed.Ageneralframeworkisintroducedtoquantifytheseeectsasameasureofdataquality.Examplesdemonstratehowthisframeworkcanbetailoredtospecicsensingapplications.ThesensorplanningtaskisaddressedinChapter 5 .First,amultiresolutionenvironmentrepresentationisdevelopedtorepresentthecoveragetaskasatargetcentrictask.Consequently,allthreeofthetaskslistedinSection 1.3 canbetreatedusingvariationsofoneuniedmethod.Then,therandomizedplanningandsensoreectivenessconceptsfrompreviouschaptersareintegratedtoformanewrandomizedplanningapproachtosensingaseriesoftargetsinacloseproximityenvironment.ThebenetsandchallengesassociatedwiththeapplicationofoptimalcontrolprinciplestotheeectivesensingproblemarediscussedinChapter 6 .Thenecessaryconditionsforanoptimalsolutionarederivedforasimpliedexample.Optimalsolutionstoasimpliedversionofthesensingproblemarecomputedusinganumericalapproach.Resultsarethendemonstratedthroughasimulatedexample.Finally,Chapter 7 summarizestheprojectanddiscussestheutilityofthemethodsintroducedthroughoutthedissertation.Additionally,recommendationsforfutureworkareprovided. 25 PAGE 26 26 PAGE 27 43 ]:pointtopointmotion,pathfollowing,andtrajectorytracking.Allthreeoftheseclassesrequirethatthevehiclemovefromaninitialcongurationtoagoalconguration;however,thefunctiondescribingthismotionisconstraineddierentlyforeach.Thepointtopointmotiontaskplacesnorestrictionsorspecicationsontheintermediatemotionsoccurringbetweentwocongurationsaslongasthegoalisreached.ThepathfollowingclassofproblemsrequiresthatthevehiclefollowsomespeciedcontinuouspathinCspacethatsatisesanysystemdierentialconstraintsandwhichhastheinitialandgoalcongurationsasendpoints.Similarly,acontinuouspathconstraintisimposedforthetrajectorytrackingproblemclass;but,theadditionalrequirementofanassociatedtiminglawisincludedaswell.Theresultofthesedenitionsisthatpointtopointmotionplansaretypicallyspeciedasaseriesofwaypoints,pathfollowingmotionplansarespeciedasafunctiondenedontheCspace,andtrajectoryplansarespeciedaseithertimeparameterizedfunctionsontheCspaceorasysteminputfunction,~u,thatisafunctionoftime. 27 PAGE 28 2.2 .Next,optimalcontrolisexaminedinthecontextofbenetsanddrawbackstopracticalplanningproblemsinvolvingrealisticsystems.Section 2.4 thenintroducesamodelingalternativethatprovidessolutionsthroughlowerdimensionaloptimizationproblemsthanthosetypicallyencounteredwhenusingdirectoptimalcontrolmethods.Thismethodutilizestheconceptofmotionprimitivesandevenadmitsclosedformalgebraicsolutionstospecialcasesoftheplanningproblemformulation. 2{1 .Theconstraintfunction,c~xc;_~xc;~xc;~u;trepresentsa 28 PAGE 29 2{1 canbeusedtorequireavoidanceofobstacles,toenforcedierentialsystembehavior,toboundallowableinputsignals,andtorestricttheoperationalenvelopeofthesystem.Apathconstraintcanbeappliedtoaparticularpointalongatrajectoryortoacontinuumofpointsalongatrajectorysegment.Theformercaseisrequiredfornitedimensionaltrajectoryoptimizationproblemswhileeitherusageisacceptableforinnitedimensionalfunctionaloptimizationproblems. 29 PAGE 30 2{2 ).Firstorderconstraintsonthestatespace,whichincluderstandsecondorderdierentialconstraintsontheCspace,areconsidereddynamicconstraints(Equation 2{3 ).Motionplanningunderthisdenitionofdierentialconstraintsisreferredtointheroboticsliteratureaskinodynamicmotionplanning[ 44 ]._~xc=f(~xc;~u);~xc2C 30 PAGE 31 45 ].Eachofthesesubproblemsareeasierthantheglobalproblem;however,somelevelofperformancetradeoislikely.Asamatterofnotation,~xwillbeusedinterchangeablyfortheremainderofthedissertationtorepresentthestatevector,~xs2X,andthecongurationvector,~xc2C.Manyofthedescribedtechniquesoperateinasimilarfashionwhenasystemisclassiedaseitherdynamicorkinematic.Thedenitionwillbestatedexplicitlywhenthecorrectusageisnotapparentfromthecontext. 31 PAGE 32 2{4 ,whererprepresentstherangeof~p2Dfrom~pcandobs;prepresentsthebearingof~p2Drelativeto~pc.TheserelationsaregivenbyEquations 2{5 and 2{6 ,respectively. (2{4) Computationofrbound(obs)requiresparameterizationoftheedgesthatrepresenttheobstacleboundarywithrespecttoobs.Theinertialpositionofapointalongtheobstacleedgeconnectingtheithobstaclevertextothejthobstaclevertexisexpressedas~peij=[px;eijpy;eij]T.ThecomponentsofthisvectorcanbeparameterizedasshowninEquations 2{7 and 2{8 .TheboundaryoftheobstacleisthencomputedusingthepiecewisecontinuousradiusfunctionshowninEquation 2{9 ,whereobs;viandobs;vjrepresentthebearingsoftheithandjthobstacleverticesrelativeto~pc,respectively.Thisradiusfunction,rbound;act,isdepictedasasolidblacklineinFigure 21 forasquareobstacle.ThegraphicshowninFigure 21A shows~peijrelativetothereferenceframecenteredat~pc=[00]T.ThegraphicshowninFigure 21B showsrbound(obs).px;eij(obs)=px;vi(px;vjpx;vi)py;vi+py;c+px;c+px;vitanobs 32 PAGE 33 2{9 accuratelyrepresentstheboundaryofapolygonasaradiusfunctionforusewithconstraintsoftheformofEquation 2{4 .Unfortunately,thenonsmoothnatureofthisfunctioncanpresentproblemsforgradientbasedoptimizationtechniques.Alternatively,asmoothradiusfunctionthatconservativelyapproximatestheboundarycanbecomputed.AnexampleofsuchanapproximationisshowninEquation 2{10 .Thisfunction,rbound;circ,simplyapproximatestheobstacleboundaryasacirclewithradiusequaltothemaximumvalueofrbound;act.ThisapproximationisshowninFigure 21 asadottedredline.Thegureshowsthattheuseofsuchaconservativeapproximationcanrestrictasignicantportionoftheobstaclefreespacewithintheproximityoftheobstaclebeingapproximated.Thenumberoffeasibletrajectoriescouldbeseverelylimitedbyrestrictingthisspace,especiallyforthecaseofcloselyspacedobstacles. 46 ].Thisfamilyofcurvesrepresentsageneralizationoftheellipsethatcanbeusedtoapproximaterectangularshapes,amongothers.TheobstacleboundarycanberepresentedasaparameterizedradiusfunctiongivenbyasuperellipseasshowninEquation 2{11 ,whereaseandbserepresentthemaximumradiusinthexobsandyobsdirections,respectively.Theradiusisthusgivenasafunctionoftherelativebearing,obs.Equation 2{11 approximatesarectanglewhentheexponent,se,isapositive,eveninteger.AsuperellipticalapproximationtotheobstacleboundaryisshowninFigure 21 asadashedblueline.Thecurveshowninthegureiscomputedusingase=1:2,bse=1:2,andse=8.Thegureshowsthatthis 33 PAGE 34 2{10 BFigure21. Obstacleboundaryapproximations.Approximationfunctionsareshownasfollows:actualradiusfunction(solid),circularradiusfunction(dotted),andsuperellipticalradiusfunction(dashed).A)Overheadviewofobstacleandapproximatedboundary.B)Radiusapproximationsparameterizedonobs 22 .Theinterioranglesbetweeneachadjacentpairof~vp!vi;j,denotedobs;ij,canbecomputedusingthedotproductoperation.AnecessaryandsucientconditionforacollisionfreepointisgivenbytheanglesuminequalityshowninEquation 2{12 34 PAGE 35 Vertexanglesumcollisiondetectionmethod. Anothermethodthatrequirescomputationof~vp!viisdepictedinFigure 23 .Thismethodadditionallyrequirescomputationofthesetofedgevectors.Eachofthesevectorscorrespondstoapolygonedgeandindicatesedgedirectioninaclockwisesense.Theedgevectorbasedattheithvertexpointstothejthvertexandcanbeexpressedas~veij=~pvj~pvi.Theunitnormalto~veijisdenoted^neijandisalsobasedattheithvertex.Thisnormalvectorcanbecomputedviaarotationthrough=2rad,asshowninEquation 2{13 .AnecessaryandsucientconditionforacollisionfreepointismetifEquation 2{14 issatisedforany(i;j)pair.Thisconditionessentiallystatesthatifthetestpointislocatedinthesamedirectionrelativetoeachedgevector,thepointliesinsidethepolygon. Figure23. Vertexedgevectorcollisiondetectionmethod. ^neij=1 35 PAGE 36 24A .ObstacleboundariescanbeexpandedasshowninFigure 24B toavoidsuchsituations.Theextentofexpansionmustbecarefullyselectedtoensurethatunsafetrajectoriescannotfeasiblysatisfyobstacleconstraints. BFigure24. Obstacleexpansionforpointwisesafety.A)Pointwisesatisfactionofobstacleconstraints.B)Pointwisesatisfactionofexpandedobstacleconstraints. 36 PAGE 37 2{15 (~x(t);~u(t))=argmin~x(t);~u(t)J(~x(t);~u(t);t) (2{15) 2{15 toemphasizethattheproblemseekstodeterminecontinuousfunctions,andthereforehasinnitedimension.Thedierentialpathconstraintsareenforcedinequality;therefore,theyarewrittenseparatelyfromthealgebraicpathconstraints,c(~x(t);~u(t)).Thefunction,(~x0;t0;~xf;tf),relatesendpointconstraintsonthetrajectory.Finally,thefunctionalperformanceindex,J(~x(t);~u(t);t)istypicallywritteninintegralformasinEquation 2{16 2{15 forwhichtherearenoalgebraicpathconstraintsandtheendpointconstraintisappliedinequality.ThecostfunctionalcanbeaugmentedwiththedierentialandendpointconstraintsthroughtheintroductionofavectorofLagrangemultipliers,~,andavectorofmultiplierfunctions,~(t).Eachelementof~isassociatedwithanendpointconstraintwhileeachelementof~(t),denoted 37 PAGE 38 2{17 .Inthisexpression,HisdenotedtheHamiltonianofthesystemandisshownasEquation 2{18 .Functionargumentshavebeenomittedheretomaintainclarityintheexpressions.TheconstrainedoptimizationprobleminJisnowreexpressedasanunconstrainedoptimizationprobleminJ. J=~Tjtf+Ztft0hH~T_~xidt (2{18) AsetofnecessaryconditionsforastationarypointofJcanbefoundbytakingtherstvariationandsettingitequaltozero,asJ=0.Afullderivationoftheseconditionsisbeyondthescopeofthisdissertationbutandisavailableintheliterature[ 47 ].Therstsetofconditionsaregivenbythecostatedynamics,asshowninEquation 2{19 .Thesedierentialequationshaveboundaryconditionsgivenbythetransversalityconditions,asshowninEquations 2{20 and 2{21 _~=@H @~x(t0)+~T@ (2{20) @~x(tf)~T@ (2{21) TheoptimalcontrolisrelatedbyPontryagin'sminimumprinciple,whichstatesthattheoptimalcontrolminimizestheHamiltonian[ 48 ].TheweakformofthisconditionisshownasEquation 2{22 2{22 relatestheoptimalcontrolintermsofthestateandcostate.Thiscontrolfunctioncanberecoverediftheoptimalstateandcostatetrajectoriesaredetermined.Assuch,theoptimalcontrolproblemcanbesolvedindirectlythroughsolutionoftheboundaryvalueproblem(BVP)speciedbythestatedynamics,thecostatedynamics, 38 PAGE 39 49 ].Pathorendpointconstraintsthatareenforcedasinequalitiescancauseanincreaseincomputationalburdenwhichresultsfromtheneedtoidentifytheactiveandinactiveelementsoftheseconstraintfunctions.Further,theoptimalityconditiondoesnotalwaysyieldaclosedformexpressionfor~u(t)intermsofthestateandcostate.Sometimesthisconditiondoesnotevenuniquelydeterminetheinputfunction.Whiletheindirectapproachtosolvingoptimalcontrolproblemsutilizesarigorousmathematicalframework,allbuttrivialproblemsareoftenrenderedimpractical. 49 { 52 ].SuchmethodsinvolvetranscriptionoftheinnitedimensionalfunctionaloptimizationshowninProblem 2{15 toanitedimensionalfunctionoptimization.Assuch,derivationofnecessaryconditionsandtheuseofvariationalcalculusisunnecessary;instead,numericaloptimizationtechniquesareusedtosolvethenonlinearprogram(NLP)whichresultsfromtheproblemtranscription.Therststepinthetranscriptionprocessistosplitthetimeinterval,t2[t0;tf],intoanitesetofxedsubintervals,asshownbyEquation 2{23 .TheNpointsatwhichthe 39 PAGE 40 2{24 2{16 remainsunchangedthroughtranscriptioniftheterminaltime,tf,coincideswithoneofthenodesinEquation 2{23 .Theintegraltermcanbeapproximatedasanitesumusingnumericalintegrationtechniques.Similarly,numericalintegrationisusedtorepresentthedierentialconstraintsasavectorofdefects,denotedas.Eachdefectenforcestheintegrationrulebetweenthecorrespondingpairofnodes.ThedefectcorrespondingtothekthnodeisshowninEquation 2{25 ,wherefrepresentstheintegrationrule.Dependenceoffisshownwithrespecttostateandcontrolvaluesatthecurrentandnexttimesteptomaintaingenerality;however,manyintegrationtechniquesonlyrequireinformationatthecurrenttimestep.ExamplesofsuchtechniquesincludeEulerintegrationandtheRungeKuttamethod. 2{15 canbeapplieddirectlywithinthetranscribedformulationift0andtfareincludedasnodes.Additionally,algebraicpathconstraintsareappliedtothestateandcontrolpointwiseateachnode.Theseconstraints 40 PAGE 41 2{26 2{27 .ThisproblemcontainsatleastN(n+m)variablesforasystemwithnstatesandmcontrols,andthusisverylargeforevenacoarsetimediscretization.Fortunately,therelevantmatricesinvolvedinsolutionoftheNLPtakeonasparseformasaresultofthetranscriptionmethod[ 49 ].Thissparsitycanbeexploitedtogreatlyreducebothstorageandcomputationtime.SeveralNLPsolversthatexhibitthiscapabilityhavebeenshowntoperformquitewellinhandlingproblemsofthisnature[ 53 ]. 2{27 isfundamentallyadierentproblemthanProblem 2{15 .AninterpolatedfunctionthroughthediscretesolutionvaluestothenitedimensionalNLPmaynotcoincidewiththeoptimalsolutiontotheinnitedimensionalproblem.Further,asthenecessaryconditionsarenotcomputed,thereisnowaytovalidatesolutionstoProblem 2{27 .Progresshasbeenmadeinthisarearecently 41 PAGE 42 51 52 ].Suchmethodsrepresentanactiveareaofresearchinoptimalcontroltheory. 2.4.1ModelingwithMotionPrimitivesAfundamentalpropertythatiscommontomanyvehiclesystemsofinterestisrelatedtotheconceptofsymmetry.Specically,thetrajectoriesofacertainsubsetofthestateorcongurationvariablesareseentoexhibitinvariancewithrespecttocertainclassesoftransformations.Let~xrepresenteitherthestateorcongurationvectorofasystemwhosedynamicsarerepresentedintheformofEquation 2{2 or 2{3 ,respectively.Asystemtrajectorycanbedeterminedthroughintegrationofthedynamicswithrespecttotime.Nowconsideratransformationfunction,H:C7!C(orH:X7!X).Systemtrajectoriesaresaidtoexhibitinvariancewithrespecttothetransformation,H,ifEquation 2{28 holdstrue[ 39 ]. 39 ].SatisfyingEquation 2{28 thusimpliesthatatrajectorygeneratedfromaparticularinitialcondition,~x0,throughtheapplicationofaninputfunction,~u,isequivalenttoalltrajectoriesthatresultfromtheapplicationof~uandthatareinitializedatanypointintherange,H(~x0).Trajectoriesthatsatisfythispropertyaredenotedmotionprimitives.Physically,Hrepresentstheclassoftransformationsthatdonotaecttheexternalforcesactingonthevehicle.Forexample,consideracarlikevehicleoperatingonanexpansive,at,isotropicsurface.Forcesonthevehicleresultmainlyfromgravityandfriction.Relativedisplacementsinpositionandheadingthatresultfromaparticularinputtrajectoryareinvarianttoabsolutepositionandheading.Therefore,translationsand 42 PAGE 43 39 ].Forthesystemsconsideredinthisdissertation,thestatedisplacementresultingfromanitedurationmotionprimitivecanberepresentedbythetransformation,GM,asshowninEquations 2{29 and 2{30 .InEquation 2{30 ,therelativestatedisplacementduetothemotionprimitive,~m,istranslatedby~x0androtatedbyR(~x0).Adiscretesetofreachablecongurationscanbeachievedthroughsuccessiveapplicationofcompatibletransformationsoftheform,GM. (2{29) Inadditiontothesenitedurationprimitivesolutions,theconceptoftrimmotioncanbeusedtogeneratecontinuouslyparameterizedfamiliesofsystemtrajectories.Trimmotionischaracterizedbysteadystatemotionswithxedcontrolsforthesystemsconsideredhere[ 38 ].Atrimprimitivethendescribesthetimeevolutionofnoninvariantstatesresultingfromconstantvaluesoftheinvariantstates.Continuousstatetrajectories 43 PAGE 44 2{31 and 2{32 .Thistransformationdescribesthestatedisplacementfromtheinitialcondition,~x0,alongatrimtrajectoryafteratimeduration,.Asbefore,therelativedisplacementduetothemotionprimitive,~T(),istranslatedby~x0andisrotatedbythetransformation,R(~x0). (2{31) (2{32) Whileexecutingatrimmotion,thereachablesetofthesystemliesalongthecurvedescribedbyEquation 2{32 .Thissetcanbegreatlyexpandedbyimplementinganitenumberoftrimtrajectoriesinahybridswitchingschemethatemploysunsteady,nitedurationprimitivestotransitionfromonesteadystatetrimmotiontoanother.Thismodelcanberepresentedasanitestateautomaton,asdepictedinFigure 25 .Eachstateoftheautomaton,depictedasnodesofthedirectedgraphinFigure 25 ,representsatrimprimitive,Ti,andisdenedbythesteadytrimvelocitiesthatcharacterizeeachparticulartrim.MotionassociatedwithTiisgovernedbyatransformationoftheformshownbyEquation 2{31 .ThestatetransitionsoftheautomatonaredepictedasdirectededgesofthegraphinFigure 25 andrepresenttheniteduration,unsteadytransitions,Mij,betweeneachpairoftrimstates,TiandTj.Theseunsteadymotionsaredenotedmaneuversandmustoriginateandterminateinsteadystatemotionasaconditionforinclusionintheautomatonframework.MotionresultingfromtheexecutionofamaneuverisgovernedbyanitedisplacementoftheformshowninEquation 2{29 .Thegraphstructurerepresentsallowableswitchingbehaviorbetweenprimitivesasrequiredbycompatibilityconditions.Maneuverscanberealizedasaninstantaneousswitchorasmoothdynamicmotion,dependingontheextenttowhichthesystemisdierentiallyconstrained.Kinematicsystems,asdenedpreviously,allowinstantaneouschangesincongurationvelocitiesand 44 PAGE 45 Automatonmodelrepresentationasadirectedgraph canthereforetransitionbetweentrimswithnocongurationchanges.Dynamicsystemsexhibitboundedaccelerationsand,assuch,smoothsystemcongurationchangeswillaccompanyanytransitionsbetweentrimstates.TheautomatonmodelingschemedepictedinFigure 25 canrepresentcomplicatedsystemtrajectoriesthroughspecicationofasequenceofmaneuversandthedurationofthetrimstatesbetweeneachconsecutivemaneuver.Thismodelexhibitsbothcontinuousbehavioranddiscreteswitchingbehaviorinthetrimstatesandnitedurationmaneuvers,respectively,andisthereforeconsideredahybridsystem.Givenamaneuversequenceoflengthn,M=fM1;M2;;Mng,andacorrespondinglengthn+1sequenceoftrimdurations,=f1;2;;n+1g,systemtrajectoriesintheCspacearecomputedviaaseriesoftransformations.Suchtrajectoriesalwaysinitiateandterminateinatrimconguration.Equations 2{33 and 2{34 showageneralexampleofthisprocessformaneuverandtimesequencesofM=fM1;M2g,and=f1;2;3g,respectively,whereGrepresentsthetotalstatedisplacementresultingfromthesequence. 45 PAGE 46 2{35 (M;)=argminM;J(M;;~x0) (2{35) 2{35 presentsadicultmixedintegernonlinearprogram(MINLP);however,theproblemstructuredoeslenditselftoahierarchicaldecomposition[ 39 ].Givenamaneuversequence,M,Problem 2{35 reducestoasmoothNLPinthetrimdurations,,thatisessentiallyanalogoustoaninversekinematicsproblem.Suchproblemsareacommonandwellstudiedclassofproblemsinrobotgeometry[ 54 ].Hence,acombinatorialsearchthroughthesetofallpossibleMaccompaniedbythesolutionofasmoothNLPforeachchoiceofMisgenerallyrequiredtosolveProblem 2{35 .Thelength,n,oftheoptimalsequence,M,isnotknowningeneral.Assuch,thesetofallpossibleMiscountablyinnite;however,itcanbeshownthatanitelengthoptimalsequenceexistsandcanbedeterminedexplicitlyinsomespecialcases.Additionally,theliteraturesuggeststhatpruningandbranchandboundtechniquescanbeappliedtosimplifyandexpeditethecombinatorialsearch[ 39 55 ].Alternatively,considerafamilyofsequenceswhereeachmembersequenceconsistsofaxednumberofmotionprimitives.Eachtrimprimitiveineachmembersequencediersinthemagnitudesofthetrimvelocities;however,thegeneralshapeofthestatetrajectoriesremainsunchanged.Maneuverprimitivesdierasneededbetweenmembersequencesto 46 PAGE 47 2{35 .ThisconstraintisrepeatedasEquation 2{36 ,where~xfrepresentstheconstrainedterminalstateandG(~x0;M;)representstheaggregatestatedisplacementoftheprimitivesequencecharacterizedbyMand.Atmostonefeasiblesolutionexiststothecontinuoussubproblemcorrespondingtoeachmembersequenceoffamiliesexhibitingthisproperty.Further,Equation 2{36 canoftenbemanipulatedtoyieldaclosedformexpressionforthisfamilyofsolutions.Assuch,thesepotentialsolutionscanbeecientlycomputedforeachmembersequenceandevaluatedforbothperformanceandforsatisfactionofpathconstraints.Notethatdierentialpathconstraintsareimplicitlyhandledbytheautomatonrepresentationofthedynamics.FeasiblesolutionscanbeenumeratedforeachsequenceinaparticularfamilyandusedtodetermineanapproximatesolutiontoProblem 2{35 .Solutionsobtainedinthisfasioncanbeshowntobeoptimalforsomespecialcases,aswillbedemonstratedinSection 2.4.3 47 PAGE 48 17 18 23 30 56 57 ].Further,thetypicalformulationofthemodellendsitselfwelltorepresentationintermsofmotionprimitives.Assuch,thismodelservesasanappropriateandrelevantmodelforuseinakinematicplanningexample.ThisexampledemonstratestheuniquesolutionfamilyapproachtondapproximatesolutionstoaproblemoftheformofProblem 2{35 .MinimumtimetrajectoriesbetweentwocongurationsaresoughtfortheDubinscarmodelinanenvironmentwithnoobstaclesand,therefore,nopathconstraints.TheDubinscarisasimplecarlikevehiclemodelthatoperatesinaCspacespannedbytwoEuclideanpositionvariables,pxandpy,andanangledescribingtheheading,.Thecarmoveswithunitforwardvelocityandchangesdirectionbyassumingaunitturnrate,!,ineitherdirection.Assuch,themotionoftheDubinscarisdescribedbythedierentialsystemshowninEquation 2{37 ,wherethediscretesetofvaluesassumedby!isshownbyEquation 2{38 where: 2.4.1 .Thesemotionsareinvariantwithrespecttopositionandheading,andthereforecanbetranslatedandrotatedtooriginatefromanypoint 48 PAGE 49 2{39 to 2{41 forturningmotionsandEquations 2{42 to 2{44 forstraightmotion.NotethatEquation 2{39 canrepresentmotionalongeitherarightorleftturnbasedonthesignof!inEquation 2{41 (2{39) (2{42) ThesethreetrimprimitivescanbeintegratedintoasimplethreestateautomatonswitchingstructurelikethatdescribedinSection 2.4.1 .ThetrimstatesaredenotedL,S,andR,whichcorrespondtotheleft,straight,andrighttrimmotions,respectively.Thesixmaneuversthatinterconnectthetrimstatesconsistofinstantaneousswitchescorrespondingtoaninstantaneouschangeofturnrate,!.Recallthatsuchaninstantaneousswitchispermissibleforakinematicmodel. 49 PAGE 50 2{35 .ApotentiallyinnitesetofautomatonsequencescouldbeenumeratedandtheresultingconstrainedNLPsolvedforeach.Alternatively,twofamiliesofsolutionsequencesexistthatuniquelysatisfyEquation 2{36 whenturningmotionisrestrictedtoheadingchangesoflessthan2rad.Theselengththreesequencefamiliesconsistofthesetofallturnstraightturnsequencesandthesetofallturnturnturnsequences.ThemembersequencesofthesefamiliescanbeenumeratedforthecurrentautomatonmodelintermsoftrimmotionsasD1=fLSL;RSR;RSL;LSRgandD2=fRLR;LRLg.ThesolutiontoProblem 2{35 canthenbeapproximatedbycomputingthesolutionscorrespondingtoeachofthesixsequencescontainedinD=(D1[D2).Specically,Equation 2{36 canbesolvedtoobtainfeasibletrimdurationsforeachsequence.Theresultingsetofsixtrajectoriesarethenevaluatedfortimeperformance,wheretheminimumtimetrajectoryiskeptasthesolution.Dubinsshowedin1957thattheoptimalminimumtimetrajectoryfortheDubinscarconsistsofmotionsdescribedbyoneofthesequencesintheset,D[ 56 ].Consequently,theapproximatesolutioncomputedusingthedescribedmethodcorrespondstotheactualoptimalsolutionforthisspecialcase.ManysequencesthatarenotcontainedinDalsosatisfytheendpointconstraint;however,thesesolutionsequencesgenerallydonotyieldauniquesolution.FeasiblesequencesoflengthfourorgreateroverdetermineEquation 2{36 inthetrimdurationsandthereforegenerallyadmitaninnitenumberofsolutions.WhileanoptimalsolutionwithrespecttoaparticularperformancecriterioncanbefoundthroughthesolutionofanNLP,thesesuboptimizationproblemsdonotnecessarilyguaranteedesirableconvexityandconditioningproperties,ie.,localandpoorlyconditionedsolutioncasesmayexist.ActualcomputationofthetrimdurationsthatsatisfythecongurationendpointconstraintforsequencescontainedinDcanbeachievedusingseveraldierenttechniquesfromtheliterature[ 58 { 63 ].Here,aparticularmethodwhichutilizesalgebraic 50 PAGE 51 61 ].Thismethodrequiresthattheinitialandgoalcongurationsaretransformedtoaspecialcanonicalcoordinateframe,D,asshowninFigure 26 .Thiscoordinateframehas(px0;py0)locatedattheoriginandthegoalcongurationlocatedadistance,d,alongtheXDaxis.Initialandnalheadingsaremeasuredfromthelineconnectingtheinitialandnalpositions,asshowninFigure 26 .ThereforethetransformedinitialandnalcongurationsaregivenasEquations 2{45 and 2{46 ,respectively. Figure26. Specialreferenceframe,DforDubinspathsolution. Thetransformationsdescribingtheleftturn,rightturn,andstraightaheadmotionsarerewritteninanalternateformasEquations 2{47 2{48 ,and 2{49 ,respectively.Theresultingmotionsareunchangedascomparedtotheprimitivesdenedpreviously.L(px;py;;)=(px+sin(+)sin;pycos(+)+cos;+) (2{47)R(px;py;;)=(pxsin()+sin;py+cos()cos;) (2{48)S(px;py;;)=(px+cos;y+sin;) (2{49)ForeachsequenceinD,thetransformationsinEquations 2{47 2{48 ,and 2{49 canbeappliedtotheinitialcongurationconsecutivelyintheproperorderthroughthecompositionoperation.Theresultineachcaseisthenequatedtonalconguration.A 51 PAGE 52 61 ]:1.L(S(L(px0;py0;0;1);2);3))=(d;0;2) (2{50b) 2.R(S(R(px0;py0;0;1);2);3))=(d;0;2) (2{51b) 3.R(S(L(px0;py0;0;1);2);3))=(d;0;2) (2{52b) 4.L(S(R(px0;py0;0;1);2);3))=(d;0;2) (2{53b) 52 PAGE 53 6.L(R(L(px0;py0;0;1);2);3))=(d;0;2) Figures 27A to 27F showsamplesolutionstothecongurationendpointconstraintthathavebeengeneratedusingtheexplicitexpressionsfortheDubinscartrimdurations.Figures 27A to 27D showsolutionsfromtheturnstraightturnfamilyofsequenceswhileFigures 27E and 27F showsolutionsfromtheturnturnturnfamilysequences.Theexamplesforeachofthesetwofamiliesaregeneratedusingdierentgoalcongurations.Congurationswereselectedtoshowinterestingtrajectoriesforeachsequenceclass.Generally,theoptimaltrajectorybetweentwocongurationsisfoundbyenumeratingthesixpossibilitiesforagivencongurationandthencomparingthefeasibleresultstondtheoptimum.Infeasiblesolutionscouldoccurforaparticularsequenceandgoalconguration,particularlyifthegoalislocatedveryclosetotheinitialcondition.Additionally,infeasiblesolutionscouldoccurforpathsthatintersectanobstacleregion.Assuch,useofthistechniqueastheprimaryapproximationmethodtosolvethefullversionofProblem 2{35 withmanyobstaclepathconstraintsisnotlikelytoyieldafeasibletrajectory.Thistechniquedemonstratesthemostutilityforobstaclefreeproblemsor,alternatively,asalocalmethodintegratedintoarandomizedscheme,suchasthemethodsintroducedinChapter 3 53 PAGE 54 B C D E FFigure27. SampleDubinspaths.A)SampleLSLtrajectory.B)SampleRSRtrajectory.C)SampleLSRtrajectory.D)SampleRSLtrajectory.E)SampleRLRtrajectory.F)SampleLRLtrajectory. 2{56 ,whichisanextensionofthesimpleDubinscarmodelexaminedinSection 2.4.3 .ThevehicledescribedbyEquation 2{56 moveswithaconstantforwardvelocity,V,andiscontrolledviathecommandedturnrate,!c.Dynamicsareintroducedintothesystemintheformofaheadingaccelerationconstraint.Thisaccelerationconstraintresultsinsimplerstorderlagbehaviorofturnratewithrespecttocommandedturnrate.Thelaglterexhibitsatimeconstantof!=0:25sec.Assuch,theturnrateresponsetoastepcommandachievessteadystateinapproximately11.25sec,or45timeconstantsfollowingthecommand. 54 PAGE 55 2{56 admitstrimtrajectoriesthatbehaveaccordingtothekinematicconditionsshowninEquations 2{57 to 2{60 .Amotionprimitivemodelisformedthroughquantizationofthedynamicsaccordingtoadiscretesetofturnrates.Specically,thissetischosensuchthattheresultingtrimtrajectoriesarecharacterizedbyturnratesgivenby_2n_max;0;+_maxo,where_max=30deg/secrepresentsthemaximumachievableturnrate.V=const. (2{57)_=const. (2{58)_!=0 (2{59)!c=_ 2{61 to 2{63 .Similarly,stateevolutionalongthestraightaheadtrimtrajectory,TS,ischaracterizedbyEquations 2{64 to 2{66 .Gturn(~x0;)=~x0+Rturn(~x0)~turn() (2{61)Rturn(~x0)=266666664cos0sin000sin0cos00000100000377777775 !p !p 55 PAGE 56 (2{64)Rstraight(~x0)=266666664cos0sin000sin0cos00000100000377777775 28 ,whichshowsthetimehistoryofeachstateforevolutionalongthemaneuverconnectingTSandTR.Thestepinputisshownasthedashedredlineinthe!plot. Figure28. Examplestatetrajectoryshowingmaneuverdynamics. Thetimehistoryforeachstateiscarriedouttot=5!,suchthat!hasreached99.3%ofitsnalvalue.ThestatedisplacementovereachnitedurationmaneuverisgivenbythetransformationshowninEquations 2{67 to 2{69 .Thetimeinvariant 56 PAGE 57 21 ,wherethemaneuverconnectingTitoTjisdesignatedasMij.GM(~x0)=~x0+RM(~x0)~M Table21. +54.1487 +54.1487 +55.9118 +55.9118 +55.3357 +55.3357py(ft) 12.2331 +12.2331 5.8870 +5.8870 +6.5181 6.5181(deg) 30.0505 +30.0505 7.4495 +7.4495 +22.6011 22.6011!(deg/sec) 30.0000 +30.0000 +30.000 30.0000 +60.0000 60.0000 Theeectofinsertingmaneuversbetweeneachpairoftrimtrajectoriesistocauseaslightstatedisplacementattheendofthersttrimtrajectorybeforethepointatwhichevolutionalongthesecondtrimtrajectorybegins.Consequently,closedformsolutionsderivedinasimilarmannertothatdescribedfortheDubinscarinSection 2.4.3 aretheoreticallypossible.However,solutionscomputedinthismannerdonotexplicitlyconsiderpathconstraintsbeyondmerelycheckingforconstraintsatisfactionaftergeneratingthetrajectory,asdiscussedpreviously.Asanalternative,thisexampledemonstratesapartialsolutiontothefullMINLPshowninProblem 2{35 forwhichobstacleconstraintsareconsidered.Thesubsequentsolutionisdenoteda\partial"solutionbecauseonlyaselectfewsequencesareenumeratedforthecombinatorialsequenceselectionstep.Enumeratingthefullsetofallfeasiblesequencesisimpractical,asthissetiscountablyinnite.Solvingthis 57 PAGE 58 29 .Thisenvironmentcontainstwopolygonalobstacleswithamaximumdimensionof200ft.TheinitialandgoalvehiclestatesaregivenbyEquations 2{70 and 2{71 ,respectively,where(?)indicatesthatthevalueisunconstrained.Bothofthesestatesareassumedtobealongatrimtrajectory.Thetrajectoryoptimizationrequiresthatasequenceofmaneuvers,M,andacorrespondingsetoftrimdurations,,arefoundthattakethevehiclefrom~x0to~xginminimumtimewhilesatisfyingthedierentialpathconstraintsgivenbyEquation 2{56 andobstaclepathconstraintsoftheformshowninEquation 2{4 .ThedierentialconstraintsareencodedaspartoftheendpointconstraintsthroughG(~x0;M;).Theobstaclepathconstraintsutilizethesuperellipseformulationoftheobstacleboundary,asdescribedinSection 2.2.2 .Alsonotethateachmaneuversequence,M,musthaveeitherMSLorMSRasitsrstelementbecausetheinitialstateliesalongthetrimtrajectory,TS. Figure29. Topviewofenvironmentfordynamicexample. TrajectorygenerationproceedsbyselectingaprimitivesequenceandsolvingtheresultingsmoothNLPsubjecttotheendpointconstraintsandpathconstraints.Recallthatforthenitedimensionaloptimizationconsideredhere,theobstaclepathconstraints 58 PAGE 59 53 64 ].TherstsequenceexaminedisM=fMSL;MLS;MSLg,whichrequiressolutionoffourtrimdurations.AnoverheadviewofthesolutionisshowninFigure 210 .Intermediatepointsatwhichobstacleconstraintsareenforcedaredepictedascirclesalongthesolutiontrajectory.Bluecirclesindicateatrimsegmentwhileredcirclesindicateamaneuversegment.ThecorrespondingheadingangleandturnratetrajectoriesareshowninFigures 211A and 211B ,respectively.Ineachofthesegures,trimsegmentsaredepictedassolidbluelinesanddynamicmaneuversegmentsaredepictedasdashedredlines.Thesolutiontrajectoryrequiresatotalof29.7140sectoreach~xgfrom~x0. Figure210. TopviewofsolutionforM=fMSL;MLS;MSLg. BFigure211. HeadingandturnratetrajectoriesforM=fMSL;MLS;MSLg.A)Headingangletrajectory.B)Turnratetrajectory. 59 PAGE 60 212 .Asbefore,intermediatepointsatwhichobstacleconstraintsareenforcedaredepictedasblueandredcirclesalongthesolutiontrajectory.ThecorrespondingheadingangleandturnratetrajectoriesareshowninFigures 213A and 213B ,respectively.Thesolutiontrajectoryrequiresatotalof29.3593sectoreach~xgfrom~x0. Figure212. TopviewofsolutionforM=fMSL;MLS;MSL;MLS;MSLg. BFigure213. HeadingandturnratetrajectoriesforM=fMSL;MLS;MSL;MLS;MSLg.A)Headingangletrajectory.B)Turnratetrajectory. ThenalsequenceexaminedisM=fMSL;MLS;MSR;MRS;MSR;MRSg,whichrequiressolutionofasetofseventrimdurations.AnoverheadviewofthesolutionisshowninFigure 214 .Asbefore,intermediatepointsatwhichobstacleconstraintsareenforcedaredepictedasblueandredcirclesalongthesolutiontrajectory.ThecorrespondingheadingangleandturnratetrajectoriesareshowninFigures 215A and 215B ,respectively.Thesolutiontrajectoryrequiresatotalof21.7796sectoreach~xgfrom~x0. 60 PAGE 61 TopviewofsolutionforM=fMSL;MLS;MSR;MRS;MSR;MRSg. BFigure215. Heading,turnratetrajectoriesforM=fMSL;MLS;MSR;MRS;MSR;MRSg.A)Headingangletrajectory.B)Turnratetrajectory. Severalrelevantobservationsandconclusionscanbedrawnuponexaminationandcomparisonofthepresentedsolutionsequences.First,thelongestsequenceisseentoyieldtheminimumtimesolution.Assuch,itisapparentthatshortsequencesdonotnecessarilycorrelatetolowcosttrajectories.Second,thersttwosequencesyieldsimilarsolutionswhilethethirdsequencefollowsamarkedlydierenttrajectorythroughtheenvironment.Afterenumerationofthersttwo,thereisnoindicationthatabettersolutionexists.Thethirdsolutionsequencedemonstratesasignicantimprovementinperformanceovertheprevioustwosolutions;however,thereisnowaytodetermineifthissolutionisoptimalorifmoresequencesshouldbeenumerated.Furtherenumerationofadditionalsequencescanintroducecombinatorialissues,astherearepotentiallyaninnitenumberofsequencesthatyieldafeasiblesolution.Anautomatoncontainingmmaneuverscanbeshowntoyieldm(m1)n1dierentsequencesoflengthn.Assuch,terminationcriteriamustbecarefullyselectedbasedonthespecicapplication. 61 PAGE 62 3.4 describesaspecicimplementationofarandomizedplanningapproachthatusesmotionprimitivestoplantrajectoriesforadierentiallyconstrainedsystem. 41 ].PRMplanningalgorithmsprobabilisticallyconstructaroadmapoftheCspaceofasystemthroughsamplingandconnectingcongurations.ThePRMplannerhasbeenshowntobecompleteinaprobabilisticsense[ 65 66 ].Aplanningalgorithmissaidtobecompleteifitisguaranteed 62 PAGE 63 31 44 67 ].Complexsystemsthatoperateincomplexenvironmentsoftenrequiretheuseofheuristicbasedsamplingmethodsthatbiassamplingnearobstacles,insidenarrowpassages,orinotherareasoftheCspacethataredeemedimportantfortheparticularsystem,environment,ortask.2.NearestNeighbor:Eachsampleisgroupedwithasetofsamplesthataredeterminedtobewithinaspeciedneighborhood.ThisneighborhoodisdeterminedusingadistancefunctionthatisavalidmetricoratleastanapproximationtoavalidmetricontheCspace.3.LocalPlanning:Apathortrajectoryisplannedlocallyfromeachsampletoeachsamplewithinitsneighborhood.Theseinterconnectionsformanetworkoflocalsolutions.4.Query:Theinitialcongurationandgoalcongurationareaddedtotheexistingroadmapnetworkbyplanningapathortrajectorybetweeneachofthesecongurationsandexistingnodeslocatedwithinaspeciedneighborhood.Onceadded,welldevelopedgraphsearchtechniquessuchasDijksta'salgorithmortheA*algorithmcanbeappliedtosearchtheroadmapnetworkforfeasibleand/orfavorablesolutions[ 25 68 ].Figure 31 depictsanexampleofaroadmapnetworkthathasbeenformedinasimpletwodimensionalCspacethatcontainsthreeobstacles.EachnodeinthegraphdepictedinFigure 31 representsasystemcongurationandeachedgeofthegraphrepresentsafeasibletrajectorybetweenthetwocongurationsatitsendpoints. B CFigure31. ThePRMalgorithm.A)Samplingstep.B)Localplanningstep.C)Queryandsolutionsteps. 63 PAGE 64 64 PAGE 65 69 { 74 ].Twoalgorithms,theRapidlyexploringRandomTree(RRT)algorithmandtheExpansiveSpacesTree(EST)algorithms,demonstratedierentcoreexplorationphilosophiesthroughthemannerinwhichnodesareselectedandexpanded.Thesealgorithmsalsoserveasabasisformanyoftheexistingvariationsonthegeneralmethod,andhenceproveusefulasdemonstrativeexamples. 42 71 ].Thealgorithmbiasestreegrowthtowardunexploredareasofthespaceandhencefocusesonrapidexploration.ThenodeselectionstepisinitiatedwithasampledcongurationthatischosenfromauniformdistributionoftheCspace.Adistancemetricisthenusedtodeterminetheclosestpointintheexistingtree.Duringtheexpansionstep,theselectednodeisextendedincrementally\towards"thesampledcongurationusingalocalplanningmethod.Thisincrementalextensionisperformedtovaryingdegreeindierentversionsofthealgorithmandisultimatelyadesignparameter.Someversionsuseaxedstepsize,othersuseastepsizeproportionaltothedistancefromthesample,whileothersattempttocompletelyconnectthesampledcongurationtotheexistingtree.Figures 32A and 32B depicttheRRTexpansionprocess.Bothimagesshowatreegrownfromtherootnode,N0,inatwodimensionalCspacethatcontainsobstacles.Figure 32A depictsthesamplingstep,inwhicharandomconguration,Nrand,isselectedandthenearestnodeintheexistingtree,Nnear,isdetermined.Figure 32B showstheexpansionstep,whereabranchisincrementallyextendedfromNneartowardNrandalongthetrajectoryconnectingthetwocongurations.Anewnode,Nnew,isaddedatthe 65 PAGE 66 BFigure32. RRTalgorithm.A)Samplingstep.B)Expansionstep. 40 75 ].TheESTalgorithmexploresspaceinafundamentallydierentwaythantheRRTalgorithm.Specically,nodeselectionoccursthroughtherandomselectionofanexistingnodeaccordingtoaprobabilitydistributionthatisleftasadesignchoice.Thisnodeisexpandedwithinalocalneighborhoodthatisdenedbyavaliddistancemetric.Acongurationissampledrandomlyfromwithinthisneighborhoodandalocalplanningmethodisusedtoconnecttheselectednodetothesampledconguration.Figures 33A and 33B depicttheESTexpansionprocess.Bothimagesshowatreegrownfromtherootnode,N0,inatwodimensionalCspacethatcontainsobstacles.Figure 33A depictsthenodeselectionstep,inwhichtheexpansionnode,Nexp,isselectedfromtheexistingnodes.TheneighborhoodofNexpisdenedhereusingaEuclideandistancemetricandisshownastheareawithinthedashedcircleinFigure 33A .Figure 33B showstheexpansionstep,wherearandomconguration,NrandisselectedfromtheneighborhoodofNexpandthenatrajectoryisplannedfromNexptoNrand.Thealgorithmproceedsinthisfashionuntilabranchofthetreereachesthegoalwithinsomespeciedtolerance. 66 PAGE 67 BFigure33. ESTalgorithm.A)Nodeselection.B)Samplingandexpansion. 34A .Conversely,theESTselectsanoderandomlyandtendsto\push"branchesfromtheselectednodetowardemptyspaceasshowninFigure 34B .Abenettothis\pushing"tendencyisthattheshapeofthetreeiscontinuallyevolvingsuchthatexpansionisguidedbythenodesamplingdistribution.Awisechoiceofthisdistributioncanfavorablyaectsolutionperformancequalities;however,caremustbetakentoavoidbiasingexplorationtowardpreviouslyexploredareas.Theperformancequalitiesoftheresultingtrajectoriesplannedusingthesealgorithmscanbeaectedbyalteringthevariousdegreesoffreedomforeachplanner.TheexplorationbehavioroftheRRTisclearlydependentuponthedistributionfromwhichrandomcongurationsaresampled.Forexample,anonuniformdistributioncanbechosentobiasexplorationtowardthegoal;however,caremustbetakentoavoidpitfallssuchaslocalminima.Additionally,RRTperformancehasbeenshowntoexhibitsensitivitytothechoiceofadistancemetric[ 42 76 ].Choiceoftheexactdistancemetric,whichistheoptimalcosttogo,maybeimpracticalformanyplanningproblems.Hence,anappropriate 67 PAGE 68 BFigure34. DierencesinexplorationstrategyfortheRRTalgorithmvs.theESTalgorithm.A)RRTexpansion.B)ESTexpansion. approximationtothisoptimaldistancemetricmustbechosen.Selectionandcomputationofsuchanapproximationcanbenontrivialforcomplicatedsystems.Asdiscussedpreviously,theESTprovidesaparticularlyusefuldesignchoiceinthefreedomtoselectthedistributionfromwhichexpansionnodesaredetermined.Amodiednodeselectionscheme,denotedtheGuidedEST(GEST)assignseachnodeaprobabilisticweightthatreectsproximitytoothernodesspatiallyandtemporallyand,moreimportantly,reectsameasureofperformance[ 73 77 ].ThisexplicitinclusionofperformancecostinthesamplingprocessenablestheGESTtoconsidersolutionqualitywhileexploringspace.Moreover,the\push"tendencydescribedpreviouslyallowspotentialsolutionstocontinuallyimprove.ThisimprovementtendencyresultsfromtheincreasedlikelihoodthatasbranchesexhibitingdesirableperformancewillbeselectedforexpansionundertheGESTframework. 68 PAGE 69 2{35 .ThemethodpresentedhereutilizesalgebraicsolutionsforplanningwithmotionprimitivemodelsasdescribedinSection 2.4.2 .Thesealgebraicsolutionsareusedtogeneratelocal,obstaclefreetrajectoriesaspartofthetreeexpansionprocess. 3{1 .Thesingleinputtothedierentialsystemistheturnrate,!.Theconstanttranslationalvelocity,V,isconstraineddierentiallytoactinthedirectionofthevehicleheading.ThesystemdescribedbyEquation 3{1 admitstrimtrajectoriesthatbelongtotwofamilies:constantrateturnsandstraightforwardmotion. 3{2 and 3{3 .Amotionprimitivemodelcanbeformedbyselectingasetoftrimprimitivesthatbehaveaccordingtotheseconditions. (3{2) _=!=const. (3{3) Asetof2n+1trimprimitivesareselectedwhichconsistofconstantrateturnsatndierentturnratesineachdirectionalongwithastraightaheadprimitivethatcorrespondsto!=0.ThissetisshownasEquation 3{4 ,where_max=30deg/sec.Thevelocity,V,isheldxedoverthesetofallprimitives. 69 PAGE 70 35 .Eachtrimtrajectoryisdepictedinthegureasastateoftheautomatonandisdesignatedbytheturnratecorrespondingtothattrimstate. Figure35. Automatonstructureforvehiclemodelusedwithcurrentplanningstrategy. Stateevolutionalongeachturningtrimcharacterizedby!iisdescribedbythetimeparameterizedtransformationgiveninEquations 3{5 to 3{7 .Similarly,stateevolutionalongthestraightaheadtrimtrajectorycharacterizedby!=0ischaracterizedbyEquations 3{8 to 3{10 (3{5) !ip !ip 70 PAGE 71 (3{8) Aspresented,themodelisclassiedaskinematic,ie.!canchangevalueinstantaneously.Therefore,maneuversareconsideredasinstantaneousswitchesbetweentrimtrajectories,andthecorrespondingtransformationwhichdescribesthecongurationtransitionformaneuversissimplyanidentitymatrix.AsdescribedfortheDubinscarmodel,thehybridmodelpresentedhereadmitsuniquesolutionsbetweenanytwocongurations.Thesesolutionsexistasmembersofuptotwofamiliesofsequences:aturnfollowedbyastraightsegmentfollowedbyanotherturn,orthreeconsecutiveturns.Ifthemodelconsistsofthreetrimprimitives,astraightforwardmotionatconstantvelocityandsteady,maximumrateturnsineachdirection,thesetofuniquesolutionsisequivalenttotheDubinsset,D,whichhassixmembers.When2nturningtrimprimitivesareconsideredwhichconsistofoppositedirectionturnsatratesofndierentmagnitudes,thesetofturnstraightturnsequencesisexpandedtosize4n2andthesetofturnturnturnsequencesisexpandedtosize2n3.Thetrimdurationscorrespondingtoeachofthesesequencescanbeexpressedinclosedform,whereinfeasiblesolutionsyieldcomplexvaluedtrimdurations.Thus,computationofanapproximationtotheoptimaltrajectoryrequiresthat4n2+2n3expressionsareevaluatedandarethencomparedforcostperformance.Onlyturnstraightturnsequencesareconsideredwithvariableturnratetoreducecomputationalburden,andonlyturnturnturnsequencesatthenominal(maximum)turn 71 PAGE 72 2.4.3 ;however,theaddedconsiderationofvariableturnratesmakesexplicitpresentationsomewhatcumbersome.Whiletheactualexpressionsarenotshownhere,thesolutionprocedureisidenticalandiseasilyperformedusingsymbolicalgebrasoftware.InadditiontotheuniquesolutionfamiliesbetweentwopointsintheobstaclefreeCspace,uniquesolutionfamiliesalsoexistonausefulsubspace.Thissubspaceconsistsofthepositionvariables,pxandpy,andisdenoted,P.MotionplansendingatapointonParenotsubjecttoconstrainedheading,,attheendpoint.Thesesolutionfamiliesconsistofturnstraightmembersequences.Hence,foramodelcontaining2nturningtrimprimitivescomprisedofnratesinoppositedirections,thereare2nturnstraightmembersintheuniquesolutionset.SolutionsforthetrimdurationsareshownfortheleftstraightfamilyinEquations 3{11a and 3{11b andfortherightstraightfamilyinEquations 3{12a and 3{12b !i2sin1+Vd !i !isin1+d+V !i2cos11CAfmod2g !isin1+d2 !i2sin1+Vd !i !isin1+d+V !i2cos11CAfmod2g !isin1+d2 Whilethemodelconsideredhereexhibitskinematicallyconstrainedmotion,theadditionofsimpledynamicsintheformofmonotonictransitionsbetweentrimstatesdoesnotappreciablychangethebehaviorofthemodel.Actually,theprocedurefordetermining 72 PAGE 73 3.3.1 andthealgorithmpresentedhereisrelatedtoalgorithmtermination.ThenominalRRTalgorithmseeksasinglefeasiblesolution.Here,therstfeasiblesolution 73 PAGE 74 3.4.1 .ThedistributiononPfromwhichpsampischosenplaysaroleinplannerperformance.AuniformdistributionoverPmaximizesspaceexplorationsuchthattrajectoriesmight`wander'aboutthespacebeforeachievingafeasiblesolution.Conversely,adistributionthatheavilyemphasizesthegoalcongurationdisplaysgreedygoalseekingbehaviorthatissubjecttolocalminima[ 31 ].Ideallyabalanceshouldbeachievedbetweenthesetwobehaviors.Assuch,samplesareselectedfromoneoftwodistributions,eachofwhichisassignedaprobabilitythatitwillbeusedtogeneratethecurrentsample.ThetwodistributionsconsistofauniformdistributiononPandaGaussiandistributioncenteredatthegoalpositiononP.Thus,samplingproceedsbyrstselectingadistributionandthensamplingapositionfromtheselecteddistribution.Onceapointisselected,nearbynodesintheexistingtreemustbedetermined.AnintuitivedistancemetricsuchastheEuclidean2normdoesnotcapturethedependence 74 PAGE 75 42 ].Therefore,theminimumobstaclefreeturnstraighttrajectoryfromacongurationtoapointonPisconsideredasanapproximatedistancemetric.Thisminimumoccurswhentheturningsegmentisexecutedatmaximumturnrate.AplotofdistancesmeasuredusingthismetricisshowninFigure 36A .Thevehiclehasacongurationof(px;py;)=(0;0;0)inFigure 36A ,suchthatdistanceismeasuredfromthecenteroftheplotwherethevehicleisinitiallypointingalongthepxaxis.Forcomparison,aplotoftheEuclidean2normisshowninFigure 36B forthesamevehicleconguration.Inspectionoftheguresclearlyshowsdiscrepanciesbetweenthetwometrics. BFigure36. Distancefunctioncomparison.A)Approximationofobstaclefreecosttogo.B)Euclideandistancefunction. ActualcomputationofthedistancefromeachnodetopsampusingthemetricdepictedinFigure 36A isaccomplishedbycomputingtheminimumturnstraighttrimdurationstoreachpsampfromeachnodeinthetree.Theminimumforeachoftheseprimitives 75 PAGE 76 3{11a 3{11b 3{12a ,and 3{12b with!i=_maxareusedtocomputethecandidatesequences.Theglobalminimumfortheturnstraightfamilyisthenfoundthroughdirectcomparisonoftheminimumleftstraightandminimumrightstraightsequences.ThisprocessisshowninFigure 37 BFigure37. DistanceFunctionComputation.A)Sampledpoint.B)Turnstraightpathsfromeachnode. TheNodeSelectionstepexposesatroublingpracticalrealityassociatedwithimplementationofRRTalgorithms:thatadistancecomputationmustbeperformedateverynodeateachiteration.Whilethisoperationisoftencomputationallycheap,asisthecasehere,newnodesareaddedtothetreeateachstep.Inthelimit,aninnitenumberofnodesareinvolvedinadistancemeasurementoperation.Fortunately,inpractice,thealgorithmtendstoconvergebeforethisissuebecomesproblematic;however,suchanobservationisdicultintheorytoprovenorguarantee. 3.4.1 .TrimdurationtimesarecomputedusingEquations 3{11a 3{11b 3{12a ,and 3{12b .Somevaluesofpsampyieldinfeasiblesolutionstocertainsequences.These 76 PAGE 77 2.2.2 .Caremustbetakeninselectionofatimestepinordertoensurethatthepointsarespacedcloselyenoughtoadequatelyrepresentthecontinuoustrajectorywithoutimposingunduecomputationalburden.Upondetectionofacollision,acandidatebranchisprunedsuchthattheendofthebranchliesadistanceofoneminimumturnradiusshortoftherstcollisionpointalongtheoriginalcandidatebranch.Thisstepensuresthatthebranchcanbesafelyextendedbeyondthecurrentendpointwithoutacollision.ThisprocessisdepictedinFigure 38 .Afterpruning,acandidateisselectedaccordingtoEuclideandistancefrompsampandminimumtimeproperties. BFigure38. Nodeexpansionstep.A)Thenewbranchischeckedforpointwisecollisions.B)Thenewbranchisprunedbeyondtherstcollision. 77 PAGE 78 39 .Thetimestepforsubdivisionisleftasadesignchoice.Morenodesperbranchgenerateagreatervarietyofstartingpointsforfuturenodeexpansion,Nexp,butinducegreatercomputationalburdenintermsofnumberofoperationsrequiredforthenearestnodeoperationaswellaswithrespecttomemorycapacityrequiredtostoretreedata.Informationregardingtotalcost,automatonsequence,trimdurations,andnodeprecedenceiscomputedandstoredateachnodeasitisaddedtothetree. Figure39. Thenewbranchissubdividedtoasetofnodes Aftercomputingandaddingnodes,eachnodeistestedforconnectiontothegoalconguration.Thisconnectiontestrequiresthattrimdurationsforthe4n2+2variableratesequencesarecomputedforeachnewnode,Nnew;i,asshowninFigure 310 .Thelengththreesequencesmustbeusedtosatisfytheheadingconstraintatthegoalconguration.Thesolutionforeachofthesesequencesisexpressedinclosedformandassuch,complexitydoesnotexceedO(n2)operations.ThecostforfeasiblesequencesoriginatingatNnew;iisaddedtothetotalcostforNnew;i.Thisvalueiscomparedtothecurrentupperboundfortheminimumsafeandfeasiblesolution.Eachfeasiblesequencethatrepresentsanimprovementtothecurrentupperboundischeckedforsafetywithrespecttoobstacle 78 PAGE 79 BFigure310. Collisionfreesolutions.A)Rightstraightrightfamily.B)Leftstraightrightfamily. Aftereachnewnodehasbeentestedforfeasibleandsafereachabilityofthegoalconguration,theterminationconditionischecked.Asdiscussedpreviously,thisterminationconditioncanvaryaccordingtothespecicrestrictionsoncurrentoperationoftheplanner.Terminationcandependonruntime,memoryusage,orevenafteracertainnumberofvalidsolutionshavebeenachieved.Upontermination,theplannerreturnsaprecedencelistofnodesthatliealongthesolutiontrajectoryfromtheinitialcongurationtothegoalconguration.Iftheterminationconditionhasnotbeenreachedattheendofaparticulariteration,executionreturnstotheNodeSelectionstep. 3.4 .ThisproblemconsidersavehiclewhosemotionisgovernedbythecontinuousdierentialsystemshownpreviouslyasEquation 3{1 .Assuch,thevehiclemoveswithforwardvelocity,V,andissubjecttoaboundedturnrate,j_j30deg=sec.Thesinglesysteminputisthecommandedturnrate,!.Aspreviously,themodelischaracterizedaskinematic.Themotionofthisvehicleismodeledusingahybridmotionmodelwithseventrimstates.Formulationofthismodel 79 PAGE 80 311A ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalcongurationof(px;f;py;f;f)=(0;800;0).Onlymaximumrateturnsareconsideredfortheturnturnturnfamilyofsequences,asdiscussedinSection 3.4.1 .ExamplesofthetworesultingsolutiontrajectoriesareshowninFigure 311B ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalcongurationof(px;f;py;f;f)=(0;100;0).Inadditiontothesolutionsequencefamiliesforplanningbetweentwocongurations,asolutionfamilyexistforplanningfromacongurationtoapointonthesubspace,P,whichconsistsofthepositionvariables,pxandpy.Variablerateturnsareconsideredforthissolutionfamily,whichischaracterizedbyturnstraightsequences.Thereare 80 PAGE 81 BFigure311. Uniquesolutionfamiliesforvehicleusedinexample.A)Turnstraightturnsolutionsequences.B)Turnturnturnsolutionsequences. atotalof2n=6membersinthissolutionfamily.ExamplesolutionsareplottedinFigure 312 ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalpointonPlocatedat(px;f;py;f)=(200;500). Figure312. Turnstraightsolutionsequences. TheplanningenvironmentconsideredfortheexampleproblemisshowninFigure 313 .Forthisexample,theinertialreferenceframeisorientedsuchthatthexaxispointsintheNorthdirectionandtheyaxispointsintheEastdirection.Inthegure,theboundsontheextentoftheenvironmentareshownasthebluesquarethatcontains0x1000ftand0y1000ft.Allsamplingstepsdrawsamplesfromwithinthesebounds.Also,polygonalobstaclesarerandomlyplacedwithinthisregion. 81 PAGE 82 313 .Samplingdistributionsarenotdenedwithinobstacleregions.Finally,theinitialandgoalcongurationsarespeciedas(px;0;py;0;0)=(0;0;0)and(px;f;py;f;f)=(1000;1000;0),respectively. Figure313. Exampleplanningenvironment. ThealgorithmdescribedinSection 3.4 isexecutedforsixtyiterations.Figure 314 showstheexpansionofthesearchtreeatvariousintermediatestagesprogressingfromlefttoright.SamplesonParedrawnfromeitherauniformdistributiondenedovertheobstaclefreeportionsoftheplanningenvironmentoraGaussiandistributioncenteredatthegoallocationwithastandarddeviationof250ftthatisalsoonlyvalidovertheobstaclefreeportionsoftheenvironment.Theuniformdistributionischosenwithaprobabilityof0.75andtheGaussiandistributionischosenwithaprobabilityof0.25. B CFigure314. Incrementaltreeexpansion.A)After1iteration.B)After25iterations.C)After60iterations. 82 PAGE 83 314 showstheturnstraightbranchextendedfromtheinitialcongurationtotherstsample.Thecenterimagedisplaysthetreeaftertwentyveiterations.Notethatbranchesarenotnecessarilyaddedduringeveryiteration,ascertainobstaclecollisionscanpruneanentirebranch.TheimageattherightofFigure 314 showstreeexpansionaftersixtyiterations.Nodesareaddedatintermediatetimestepsalongeachbranchasitisappendedtothetree,asdiscussedinSection 3.4.5 .NodesinthenaltreeaftersixtyiterationsareshowninFigure 315 .Thesenodesserveasstartingpointsforadditionalbranchesaswellasforsolutiontrajectories. Figure315. Nodesinthenalsolutiontree. Eachnodeistestedforfeasibleconnectiontothegoalcongurationasitisgenerated.Successfulconnectionsthatimprovethecurrentbestsolutionreplacethissolutionforfutureiterations.Figure 316 showstherenedsolutionforthepresentexampleastreeexpansionproceeds.IneachoftheplotsshowninFigure 316 ,theprecedentpathfromtheinitialcongurationalongthetreebranchestothetestednodeisdepictedasasolidblueline.Thesolutiontrajectoryfromthetestednodetothegoalcongurationisdepictedasasolidredline. 83 PAGE 84 B CFigure316. Solutionpathrenement.A)After7iterations.B)After14iterations.C)After37iterations. TheimageattheleftofFigure 316 depictstherstvalidsolutionfoundbythealgorithm.Thissolutionwascomputedduringtheseventhiteration.Hence,thisexampledemonstratestheabilityoftheplannertoandafeasible,butsuboptimal,solutionveryquickly.ThesolutionshowninthecenterimageofFigure 316 iscomputedduringthefourteenthiteration.Finally,theminimumtimesolutionfoundfortheexamplerunisshownattherightofFigure 316 .Thissolutionwascomputedduringthethirtyseventhiteration.Assuch,thenaltwentythreeiterationsdidnotcontributetothesolutionandcouldbeconsideredawasteofcomputationalresources.Alsonotethattheimprovementofeachsolutionoverprevioussolutionsisfairlybenign,duetothenatureoftheRRTexpansionprocedure.Unfortunately,thereisnowaytodetermineifaparticularsolutionispotentiallyimprovableorhowlongsuchpotentialimprovementwilltake.Therefore,theterminationconditionremainsaproblemspecicparameter. 84 PAGE 85 1 asoneoftheprimaryapplicationdomainsforunmannedsystems.Guidancetrajectorydesignmethodsthatensuredesirablesensorplacementrepresentarequiredtechnologytoenablesuchmissions.Beforetrajectorydesignmethodscanbediscussed,ametrictodenedesirableviewpointsmustbeestablished.Theformulationofsuchametricisseentobeespeciallyrelevantforthesocalled\smallscale"sensingmissionsdiscussedinthisdissertation.ThisdissertationconsiderstheclassofLOSremotesensors.ThesesensorscollectdataintheformofreectedenergyfromremotesurfacesthatexhibitclearLOSfromthesensorandwhicharecontainedwithinthesensorFOV.Datacollectioncanbeeitheractiveorpassiveinnature,dependingonthespecicsensor.Ineithercase,thesensingoperationcanbecharacterizedgeometrically.Thegeometricrelationshipsthatrelateenergyreectionsseenbythesensoraredependentmainlyontherelativepositionandorientationofthereectingsurfacewithrespecttothesensor.Sensorperformanceisaectedbyvariationsintheenergythatisreectedfromeachparticulartargetsurface.Additionally,variationsintherelativeorientationofthetargetsurfacewithrespecttothesensorcanresultindatadistortionsanddegradedresolution.Theperspectivefromwhichtargetsareseencontinuallyvarieswhilethesensormovesthroughtheenvironment.Assuch,sensingperformancewithrespecttoaparticulartargetsurfacecanvaryappreciablyasthesensorismovedrelativetothattarget'sposition.Furthermore,adjacentobjectsareseenfromaslightlydierentperspectiveevenforastationarysensor.Hence,sensingperformancecanshowsignicantvariationoverthesensorFOV.Thislattereectisespeciallypronouncedforsensoroperationincloseproximitytotheenvironment.Suchascenarioinvolvestargetsurfacesappearingatvariousrelative 85 PAGE 86 17 23 78 { 80 ].Thischapterdiscussesageometricapproachtoquantifytheeectivenessofaparticularsensortargetconguration.First,abriefdiscussionofsomeoftherelevantremotesensortechnologiestotheapplicationsconsideredprovidescontexttothesubsequentpresentation.Then,ageometricmodelofLOSsensoroperationisintroducedinSection 4.3 .Aparameterizationoftherelativegeometryisusedtodenetheconceptofvisibility,whichisdiscussedinthecontextofsensingtasksincloseproximityenvironments.Finally,ageneralizedsensingeectivenessmetricisformulatedthatquantiesthequalitywithwhichaparticularpointintheenvironmenthasbeensensed.Severalspecicexamplesdemonstratehowthisconceptcanbeusedtoevaluatesensingmissionperformance. 86 PAGE 87 81 ].Despitethislossofdepthinformation,agreatdealofinformationcanbegatheredfromtwodimensionalimagesusingvisionprocessingtechniques.Pointsofspecialsignificancedenoted`featurepoints'canbeidentiedandextractedfromatwodimensionalimage.Overlappingviewsallowfeaturepointsinsequentialimagestobecorrelatedandtracked.Themotionofthesetrackedfeatureswithintheimageplanecanbeusedwithstructurefrommotion(SFM)techniquestoreconstructthedepthdimensioninadditiontoestimatingcameramotion[ 82 ].Suchtechniqueshavereceivedincreasingattentioninrecentyearsforthepurposesofvehiclenavigationandcontrol[ 83 { 87 ].Fullreconstructionofthethreedimensionalgeometryisnotnecessarilyrequiredtorendertwodimensionalimagesuseful.Avarietyofapplicationsinboththecivilandmilitarydomainsemployvisiblelightimagery.ColorandshapeinformationrelatedtoobjectswithinthecameraFOVcanbeusedforidenticationandtrackingtasks. 87 PAGE 88 88 ].Therangeresolutionhasbeenaddressedinrecentyearsthroughtheuseofultrawideband(UWB)radarsystems.Thesesystemsuseawiderangeofenergyfrequenciestoproducebetterrangeresolutionthanstandardradar.Thebearingresolutiondependencypresentsadiculttradeointhathighresolutiondatarequiresunrealisticallylargeantennae.Syntheticapertureradar(SAR)processingisatechniquethatusesthevehiclemotiontoenhancethebearing,orazimuth,resolution.TheconceptbehindSARprocessingisthatsuccessivemeasurementsfromasmall,movingradarantennacanbesynthesizedtoamuchlargerbeamwidth.Thus,whenaparticularpointintheenvironmentisilluminatedbymultipleradarpulses,thereturnscanbeprocessedasifthepointisilluminatedbyasinglepulsefromalargesyntheticaperture.AdvancesinUWBradarsystemsandSARprocessingtechniqueshavebeguntoallownewcapabilitiesandapplicationsforradar.Theinformationcontentofradardataisdependentonthefrequencybandinwhichthesystemoperates.PulsedUWBsystemsutilizeashortdurationpulsewithsupportatfrequenciesthatarebothlowenoughtopenetratefoliageandsoilandthatarehighenoughtoresolvesmallobjects[ 89 { 92 ].UseofSARprocessingtechniquesenableshighspatialresolution.Predictedadvancesincomputationalandprocessingtechniquessuggestthattheseradarsystemswilleventuallyexhibitimagingcapabilitiesatframeratescomparabletovisiblelightimagingsensors[ 92 93 ].Hence,UWBradarposesgreatbenettoavarietyofapplicationsdealingwiththedetectionofconcealedobjectssuchasburiedlandminesorenemyforcesindeepcover[ 89 92 94 95 ]. 88 PAGE 89 96 { 99 ].Theresultingdatacanbeusedfornavigation,targetdetection,andevenformappingandimagingapplications.Themappingandimagingapplicationsrequiretheuseofsyntheticaperturesonar(SAS)processing,whichisanalogoustoSARprocessingofradardata[ 100 101 ]. 102 103 ].Essentially,asingletransmittedlaserpulsecanresultinatwodimensionalimagethatcontainsbothrangeandreectionintensityinformationforeachpixel.Ladarsystemsarecurrentlyunderdevelopmentwiththeexpectationthattheserangeimageswillbeproducedatframeratescomparabletostandardvideocameras.Theseresearcheortshaveresultedinseveralprototypesensorsthatexhibitmuch 89 PAGE 90 102 { 104 ].Thepotentialapplicationsforasensorwiththreedimensionalimagingcapabilitiesarenumerousandvaried.Thereductioninsize,complexity,andpowercomparedtoscanningsystemsallowsamuchbroaderclassofvehicletoconsiderimagingladarasasensorsystem.Scannerlessimagingladartechnologyhasbeeninvestigatedforusewithautonomousnavigationsystemsandcouldbeextendedtootherapplications,suchasremotedetectionofconcealedobjectsortargets[ 104 ].TheNATOResearchandTechnologyOrganizationpredictsthat\megapixelarraysmakingrealtimemultidimensionalmeasurementsat10kmwithframeratesof30Hzorfaster"willbearealitywithinadecade[ 2 ].Additionally,theenergypropertiesofthesefutureadvancedladarsystemscouldbeusedtoidentifychemicals,pollutants,andorganismsalongwithcharacteristicvibrationpropertiesoflivingormanmadeobjects[ 2 ]. 4.3.1SensingGeometryThegeometryforthegeneralcaseofavehiclecarryingasingleLOSsensorisdescribedbythevectordiagramofFigure 41 .Thevehiclebodybasis,B,isxedatthevehiclecenterofgravityandislocatedrelativetotheinertialframe,E,bythetimevaryingvector~pB.TheorientationofBisdescribedrelativetoEthroughthetypicalEulerangleparameterization,whichisexpressedas~BasinEquation 4{1 .TheEuleranglesarealltimevaryingaswell.Thetimeevolutionofboth~pBand~Bisgovernedbythevehicledynamics. 4{2 90 PAGE 91 Sensingproblemgeometry. 4{3 .ThesensorisassumedtohaveaknownorientationwithrespecttoB,whichisgivenbytheconstanttransformation,TBS.ThetransformationfromEtoSisthengivenbyEquation 4{4 .TheorientationofSwithrespecttoEcanbeexpressedusinganEulerangleparameterization,aswell.ThetimevaryingvectorofsensorEuleranglesisshowninEquation 4{5 .Thetransformation,TES,canbeexpressedrelativetotheseparametersasinEquation 4{6 .ThesensorEulerangles,~s,canthenbedeterminedintermsof~BbyequatingtherighthandsidesofEquations 4{4 and 4{6 thensolvingtheresultingsystem. 91 PAGE 92 4{7 .Relativeorientationisdescribedbytheorientationof^nirelativetothesensoraxis,^s3.Thisrelationshipischaracterizedthroughexpressionof^nirelativetotheorientationofS.TherequiredtransformationisshowninEquation 4{8 ,where^ns;irepresents^niintermsofthebasisvectorsofS.~p(s!ti)=~pt;i~ps 27 105 106 ].Aviewpointistypicallydenedona\visibilityspace."Thisparameterspaceisgenerallyspannedbyasetofvariablesthataectvisibilityandviewquality.TheseparametersoftenincludeopticalspecicationsofthecameraandothersensorspecicfactorsinadditiontothegeometricdenitionsdiscussedinSection 4.3.1 .Variousdenitionsof 92 PAGE 93 4{9 .Variationsinrangeareassociatedwiththeintensityandresolutionofdata.TheincidenceanglemeasuresthedegreetowhichthesensorLOS\grazes"thetargetsurface.ThiseectischaracterizedbytheanglebetweentheLOSalong~p(s!ti)andthetargetsurfacenormal^ni,asshowninEquation 4{10 .Highvaluesforicanresultinadistorteddataproduct.Finally,theFOVanglemeasurestheangulardiscrepancybetweentheLOSalong~p(s!ti)andtheLOSalong^s3,whichisthesensorframebasisvectorthatindicatesthecenteroftheFOV.ThisrelationshipisdescribedbyEquation 4{11 .Variationsinf;iareassociatedwithangularpositionwithintheFOV.Figure 42 depictsacrosssectionoftheFOVthatshowsthevisibilityparametersforaparticulartargetlocation. Onceexpressedintermsoftherange,incidence,andFOVangle,visibilitycanbedenedthroughtheapplicationofsimplebounds.Theseboundsrelyonseveralbasicassumptionsconcerningthelimitationsofthesensor.First,thesensorisassumedto 93 PAGE 94 Visibilityparameters. haveaniteFOV,ie.,thesensorisnotomnidirectional.Second,theexistenceofarangebeyondwhichobjectswithintheFOVcannotberesolvedisassumed.TheseconditionsresultinthevisibilityconstraintsgivenbyEquations 4{12 and 4{13 Inphysicalspace,thesimultaneousapplicationoftheseconstraintsarerealizedastheintersectionofaspherethatiscenteredattheoriginofSandandinnitecircularconethatoriginatesfromthissameorigin.Thespherehasradiusrmax.Theaxisoftheconeiscollinearwiththesensoraxis,^s3,andhasahalfangleoff;max.Thisintersectionthusformsasphericalconethatisalignedwiththesensoraxis.Anecessaryconditionforvisibilityoftheithtargetisthat~p(s!ti)mustlocatethetargetwithinthissphericalvisibilityconeforthecurrentsensororientation,~s.Anothervisibilityconstraintresultsfromtherelativeorientationofthetargetsurfacewithrespecttothelocationofthesensor.Thisconditionconstrainstheincidenceangle,i,asshowninEquation 4{14 .Thisconstraintreectsthenotionthatapointonasurfacecannotbeseenwhentheviewisparalleltothesurface.Aparallelviewoccurswheni==2,orwhentheLOSisorthogonaltothetargetsurfacenormal,^ni.Equation 4{14 alsorestrictsincidenceanglesgreaterthan=2,asthisrangeofi PAGE 95 4{14 isrealizedasaninniteconeoriginatingatthetargetlocation.Theaxisofthisconeliesalongthesurfacenormal,^ni,andthehalfangleisrelatedbymax.Inthelimitingcasewheremax==2,thisconeapproachesanopenhalfspacethatisboundedbyaparallelplanetothetargetsurface.Anecessaryconditionforvisibilityisthat~p(s!ti)mustlocatethetargetwithinthisinnitecone.Analconditionrequiresthataviewpointmustnotbeoccludedforvisibility.ThisconditionrequiresthattheLOSfromthesensortotheithtargetdoesnotintersectanyobstaclesorsurfacesbetweenthesensorandthetarget.Assuch,thisconstraintdependsonthespecicenvironmentinwhichthetargetandsensorarelocatedandmustthereforebeappliedinataskspecicfashion,asneeded. 4.3.2 provideasetofnecessaryconditionsforvisibilityofaparticularpointfromaparticularsensorlocationandorientation.TherangeandFOVangleconditionsdescribeasetofpointsrelativetoaspecicsensorcongurationthatmaybevisibleifotherconditionsaremet.Conversely,theincidenceangleconditiondescribesasetofsensorcongurationsrelativetoaspecictargetcongurationfromwhichthistargetmaybevisibleifotherconditionsaremet.Auniedrepresentationoftheseconstraintsintermsofsensorcongurationsisdesirablefromasensorplanningstandpoint.Sucharepresentationwouldallowthedenitionofatargetsetofsensorcongurationsfromwhichaparticulartargetisvisible.Asetofsensorcongurationsisconstructedfromwhichtheithtargetmeetsvisibilityconditionsforanycongurationwithintheset.Thissetisdenotedthe\visibilityset" 95 PAGE 96 43 .Thisconedescribesasetofsensorpositions,~ps,thatarevisiblefromS0.ThereforetheithtargetisalsovisiblefromsensorpositionswithintheS0visibilitycone. Figure43. Invertedvisibilitycone. Asdiscussedpreviously,Vinowconsistsoftheintersectionofthisinvertedsphericalvisibilityconewiththeinniteconethatrelatestheincidenceconstraintfortheithtarget.Thissetdescribesthesensorpositionsandorientationsthatsatisfythegeometricvisibilityconditionsfortheithtarget,asdescribedbyEquation 4{15 .Validsensorpositionsare 96 PAGE 97 44 .Thevisualizationshowsthefullsetintersectionresultingfromtheboundsappliedtori,i,andf;ifortheinvertedvisibilitycone.Inthegure,Viisshownastheshadedregion. Figure44. ConstructionofVi.Thevisibilityset(shaded)isconstructedthroughapplicationofsetintersectionsappliedtoaninvertedvisibilitycone 97 PAGE 98 4.3.2 .WhenasurfaceintersectsaparticularLOSwithinthesensorFOV,nothingbeyondthepointofintersectionisvisiblealongthisLOS.Assuch,obstaclesthatfallwithinthevisibilityconeeectivelycastashadowofnonvisiblepoints.ForthecasewhereobstaclesareclosetotheoriginofS,andthereforeclosetotheapexofthevisibilityconeassociatedwithS,thisshadowcanpreventvisibilityofalargeportionoftheenvironment.ClutteredenvironmentsmayseverelylimitViforcertaintargetssuchthatsensorplacementmustbeperformedwithextremecare.Anothersignicantissueisrelatedtothecouplingofsensorpositionandorientationtovehiclemotions.Changesinvehiclecongurationdirectlyaectthelocationandpointingdirectionofthevisibilityconeregardlessofproblemscale;however,theeectoncollecteddataissignicantlydierentforstandosensingascomparedtocloseproximitysensing.Forthestandocase,alargefootprintisprojectedontothesensedenvironmentevenforanarrowFOVsensor.ThiseectresultsfromtheangularnatureoftheFOVconstraintcombinedwiththelargedistancefromsensedsurfaces.Forthiscase,evenlargetranslationsofthesensormaintainsignicantoverlapinprojectedfootprintbetweenthestartandendofthetranslation,asshowninFigure 45A .Consequently,angularmotionsthatoccuralongsuchtranslationsdonothaveagreateectontheaggregatedatacollectedaslongasthevisibilityconepointsinroughlythesamedirectionattheendofthemotionasitdoesatthebeginning.Anexampleofsuchacaseisadownwardpointingsensoronanaircraftthatmustchangeitscourse.Theaircraftmustbanktoturnandmaytemporarilypointthesensorawayfromsensingobjectives;however,ifalargeportionofthesensorfootprintuponcompletionofthemaneuveroverlapswiththefootprintprecedingthemaneuver,thereisnosignicantlossofdata.Alternatively,sensingatcloseproximityresultsinamuchsmallerprojectedfootprint,evenforafairlywideFOVsensor.Assuch,thereisahighturnoverofcoveredareaas 98 PAGE 99 45B .Theexampleofaturningaircraftwithadownwardpointingsensornowexhibitssignicantdatalossasaresultofthepreviouslybenignmaneuver.Moreover,navigationthroughcloseproximityenvironmentsoftenrequiresfrequentandaggressivemaneuveringtoavoidobstacles,thuscausingmotioncouplingeectstobeevenmorepronounced. BFigure45. Motioncouplingeectsfordierentproblemscales.A)Translationatstandorange.B)Translationatcloserange. Finally,signicantrelativevariationsinrangeandincidenceareseenovertheFOVasaresultofthecloseproximityofthesensortotheenvironment.Whenthesensorislocatedalargedistancefromtargetsurfaces,therangeofallvisiblepointsvariesverylittlerelativetothestandodistance.ObjectsappearwithintheFOVatdisparatedistancesandaspectswhensensingatcloserange.Theincidenceanglecanvarysignicantlyoverasurfacethatwouldappearatanearlyconstantincidencefromstandorange.Figure 46 showsthiseectascomparedtothevariationsseenintypicalstandosensingmissions.ThetwoimagesontheleftofFigure 46 showrangeandincidenceangledataoverasimulatedurbanenvironmentforasensorlocatedatanaltitude10,000ft.ThisaltitudeistypicalforcurrentairbornesensorplatformssuchastheRQ1Predatordrone[ 3 ].TheimagesontherightofFigure 46 showrangeandincidencedatawhen 99 PAGE 100 B C DFigure46. Rangeandincidencevariationfordierentproblemscales.A)Rangeforstandosensing.B)Rangeforcloseproximitysensing.C)Incidenceforstandosensing.D)Incidenceforcloseproximitysensing ThestatisticsdescribingthevariationsinrangeandincidenceforallvisiblepointsfromthetwocasesdepictedinFigure 46 aredisplayedinTable 41 .Thetableshowsthatthetwocasesexhibitasimilarabsolutespreadofrangedata,asthevaluesforthestandarddeviationofthisdataisofsimilarmagnitudeforeachcase.However,adiscrepancyofafullorderinmagnitudeisseenwhenthesevaluesarenormalizedonthemeanrange.ThisdiscrepancyisreectedbythecolorvariationsinthetoprightimageofFigure 46 ascomparedtothoseseeninthetopleftimage. 100 PAGE 101 Statisticsforvisibilityparametercomparison. Meanri Normalizedri Std.Deviation Std.Deviation Std.Deviation Stando 20,023ft 0.1471deg 543.87ft 4.2563deg 41 displayaveragevaluesforeachvisiblesurfaceoverallvisiblepoints.Verylittleabsolutevariationinincidenceangleisseenforthestandocaseascomparedtothecloseproximitycase,asindicatedbytheaverageistandarddeviationoverallsurfaces.Further,anincreaseinstandarddeviationbytwoordersofmagnitudeisseenbyatleastonesensedsurface.TheseresultsarereectedbythesignicantcolorvariationsoneachsurfaceofthelowerrightimageofFigure 46 ascomparedtothoseseenonsurfacesinthelowerleftimage.Frequentocclusions,motioncoupling,andlargerelativevariationsinthevisibilityparametersaredemonstratedforsensingmissionsincloseproximitytothesensedenvironment.Theseeectscanadverselyaectdataqualityifnotproperlyconsideredinmissionplanning.Properconsiderationrequiresthattheeectsoftheseissuesbequantiedforinclusionwithinsensorplanningoptimizationalgorithms.Thenextsectiondescribesasensoreectivenessmetricthatisdesignedtoaddressthisproblem. 4.3.2 and 4.3.3 allowageometricdenitionofvisibilityforagenericLOSremotesensor.TheithtargetisvisibleforsensorpositionsandorientationsthatliewithintheboundsofVi,whileforallothercongurationsthetargetisnotvisible.ThisrelationshipcanbecharacterizedbyasimplebinaryagasexpressedinEquation 4{16 .Thisdenitionofvisibilityislimitingintermsofdescribingsensorperformance,orequivalently,qualityoftheresultingdataproduct.Manycasesexistforwhichthisdenitionisnotsucienttodescribetheutilityofaparticularsetofcollected 101 PAGE 102 agvis;i=8>><>>:1;~ps!ti;~s2Vi0;~ps!ti;~s=2Vi(4{16) 4{17 .Thisfunctionisconsideredasametricoftheeectivenesswithwhichtargetihasbeensensed. 4{18 ,wherethevisibilityparameterstakeonfunctionaldependenciesasshowninEquations 4{19 to 4{21 .Notethattheexplicitformofqiisspecictoboththesensorandtheapplication. 102 PAGE 103 (4{20) (4{21) InadditiontothedependenciesexpressedinEquations 4{18 to 4{21 ,qihasanimplicitdependenceontime,suchthatqi=qi(t).Thisimplicitdependenceresultsfromthetimevaryingnatureoftherelativepositionandsensororientation,~ps!tiand~s,andconsequentlythevisibilityparameters,ri,,andf.Asthesensorframe,S,isxedrelativetothevehiclebodybasis,B,thesetimevaryingquantitiesareaectedbythevehicledynamics.Assuch,qicanbeconsideredasanoutputfunctionofthevehicledynamicsforalli.ArepresentativetrajectoryofqimightappearasshowninFigure 47A .Noticethatthefunctionexhibitsapparentdiscontinuitiesalongthetrajectoryshown.ThesediscontinuitiesresultfromtargetileavingthesensorFOVduringthesetimesegments.Assuch,thevisibilitycriterionisnotmetandtheeectivenessmetricevaluatestozeroduringthesetimesegments.Manyremotesensormodalitiesactuallyoperateindiscretetimewheredataiscapturedataninstantintimeataniterate.Theeectivenessmetricshouldbeconsideredafunctionofthetimediscretizeddynamicsforthesecases,andshouldbeexpressedasadiscretetimefunctionsuchthatqi=qi[k].Likewise,arepresentativetrajectoryofqiforthesecasesmightappearasinFigure 47B .Whilenecessarytomaximizerealism,thisdiscretetimeformulationcouldintroduceanumberofproblemsingradientbasedtrajectoryoptimizationschemes.Thus,thecontinuoustimeformofqiisprimarilyusedthroughouttheremainderofthisdissertation,unlessotherwiseindicated.Thisassumptionrequiresthatthesamplingrateofthesensorisfastrelativetothedynamicsofthevehicle.Thisconditionismetforthemajorityofthesensorsconsideredhere. 103 PAGE 104 BFigure47. Representativetrajectoriesforsensoreectivenessmetric.A)qi(t).B)qi(k). 4.4.1 canbeusedtodeneasubsetofVi.Thissubset,QiVi,representsthesensorpositionsandorientationsthatviewtargetiwithaneectivenessofatleastqi;desired,asshownbyEquation 4{22 .TheactualmakeupofQidependsonthespecicformofqi,andisnotnecessarilyconvexwithrespecttothevisibilityparameters,ri,i,andf;i. 4.4.1 providesaninstantaneousmeasureofthequalitywithwhicharemotesensorisgatheringdatawithrespecttoaparticularpointinspace.Realisticuseofthismetricformissionplanningpurposeswouldlikelyrequireevaluationofqiatmanydierentpointsinadditiontoamethodfortrackingaccumulatedeectiveness.Perhapsviewingatargetmultipletimesaddsutilitytothe 104 PAGE 105 4{23 4{24 .Thisfunctionkeepsthemaximumvalueoftheinstantaneousqualitymetric,qi,overthetimeinterval,t2[0;t].ThevalueofQmax;iindicatesthattargetihasbeensensedwiththatvalueatleastoncesincethebeginningofthemissionsegment.Thisversionofmissioneectivenesscanbeusefulforapplicationsinvolvingimagingsensorsthatcollectdatasubjecttohumananalysis.Onegoodviewofasensingobjectivecanoftenprovideagreatdealofinformationtoahumananalyst. 4{25 showsthecaseforwhichqirepresentstheprobabilityofanevent,suchasdetection,occurringatthesensinginstancecorrespondingtoqi.Thentheprobabilityofthateventnotoccurringcanbeexpressedasthecomplementofqi,asshowninEquation 4{26 .Theprobabilityoftheeventnot 105 PAGE 106 4{27 .Finally,theprobabilitythattheeventoccursatleastonceisgivenbythecomplementoftheprobabilitythattheeventdoesnotoccuratall,asshownbyEquation 4{28 (4{28) Theprobabilitythattheeventoccursatleastoncecanthenbeimplementedinrecursiontoexpresssensingmissioneectivenessfordetectionprobability,QPD;i.ThisrecursionisshownasinEquation 4{29 .Theuseofarecursiveformularequiresthatthediscretetimeformofthesensingeectivenessmetricisused. 4{30 4{31 to 4{33 .ThesefunctionsnecessarilyevaluatetozerooutsidethesensorFOV.WithintheFOV,eacheciencyfunctionisshapedtoreect 106 PAGE 107 4{34 1+(ri=rmax)40:7exp400r2i 1+(i=max)41;imax0;else 1+(0:75f;i=f;max)4;f;if;max0;else 48 .Theeciencyfunctioncorrespondingtorange,fri,isshownintheleftimage.Thisfunctionisseentoexhibitlossesatcloserangeandlongrange.Thisshapingreectspoorspatialcoverageatcloserangeandpoorresolutionatlongrange. B CFigure48. Qualityparametereciencyfunctions.A)Rangeeciency.B)Incidenceeciency.C)FOVangleeciency. Thefunctioncorrespondingtoincidenceangle,fi,isshowninthemiddleimageofFigure 48 .Thisfunctionexhibitslossesastheviewdirectionapproachestangencywiththetargetsurface.Increasingihastheeectofdistortingsurfacedataandlimiting 107 PAGE 108 48 .ThisfunctionisseentoexhibitminimallossesoverallFOVangles,withthegreatesteectsappearingnearf;max.ThemaximumFOVanglecorrespondstotheFOVboundary.Assuch,thelossesareincorporatedheretoreecttheradialdistortioneectscommontocamerasensors[ 82 ].UseofthemetricdescribedbyEquation 4{34 isdemonstratedthroughasimplesimulation.Thesimulationconsistsofakinematicaircraftmodelyingataconstantaltitudeof200ftabovegroundleveloveranurbanenvironment.Thisenvironmentconsistsofthreebuildingsarrangedsuchthattheypartiallyencloseanarea.Eachbuildinghasamaximumheightof50ft.Asamplemissionscenariomightrequiredatacollectionalongtherooftopedgesthatfacethepartiallyenclosedareatocheckforsniperspriortoatroopmovement.Theaircraftcarriesanimagingsensorthatislocatedattheoriginoftheaircraftbodybasissuchthat~p=~0.Thesensorpointsforwardanddownatanangleof60deg.Assuch,thetransformationfromBtoSisgivenbyEquation 4{35 .ThevisibilityboundsforthissensoraregivenbyEquations 4{36 to 4{38 .TheseboundsareusedwithEquations 4{31 to 4{34 toformulatethequalitymetric. 108 PAGE 109 49 .Fordisplaypurposes,targetsaregeneratedbygriddingtheenvironmentsurfacesatxedresolutionof25ft.Thisselectionoftargetsallowsthemetricvaluetobeshownatallpointsintheprojectedfootprintwiththespeciedresolution.TheimagesinFigure 49 showsnapshotsofinstantaneousmetricvalueswithinthesensorfootprintatseveralpointsalongthetrajectory.Noticethechangeinshapeandeectivenessmakeupofthefootprintastheaircraftbankstoturninthetoprightandbottomleftimages.Also,noticethestarkdierencesinmetricvalueondierentsurfacesthatareperpendiculartoeachotherasthesensorviewsbuildingedges. Figure49. Simulatedtrajectoryshowingsnapshotsofeectivenessmetricvalues. Themaxvaluemissioneectivenessmetricisusedtotrackaggregatesensingeffectivenesswithrespecttoeachtargetforthepresentexample.Recallthatthismetricmaintainsthemaximumeectivenessvalueovertheentiretrajectory,asgivenbyEquation 4{24 .ThenalvaluesformissioneectivenessareshowninFigure 410 forthetrajectorydepictedinFigure 49 109 PAGE 110 Maxvaluemissioneectiveness. 82 ].Simplegeometricrelationshipsyieldanexpressionfordiintermsofthevisibilityparameters,riandf;i.ThisexpressionisgivenasEquation 4{39 4{40 .ThecorrespondingareainimagecoordinatesisgivenbyEquation 4{41 ,whereNrepresentsthenumberofpixelsinasinglelineoftheimage.Thelengthofeachsideofreferenceobjectis 110 PAGE 111 ~A=~Wref~Href =Wref 4{41 andareshowninEquation 4{42 4{43 .ThisfunctionexponentiallyapproachesqA;i=1as~A!~Amax.Thenumericalparametersarechosenheresuchthatarapidriseoccursfor~Athatareslightlylargerthat~Amin,followedbyalargeintervalforwhichthereislittlegain.Suchbehaviorischosentoreectthenotionthatincreasingimageareaofthereferenceobjectdoesnotyieldadditionalrelevantinformationbeyondacertainsize.Fortheaforementionedexampleofalicenseplate,littleistobegainedbyincreasing~Aoncethenumbersareclearlylegible.Conversely,asignicantamountofinformationisgainedcloserto~Aminasslightincreasesinresolutionhelptodierentiatethecharactersfromeachother.TheformulationinEquation 4{43 alsoincludesascalingeciencyfunction,f,toaccountfordistortionsresultingfromgrazingincidenceangles.Thesameformulationoffthatwasemployedinthepreviousexampleisusedhereaswell(Equation 4{32 ). 111 PAGE 112 ~Amax~Amin+0:01!!!(4{43)Theimageareaformulationofthesensingmetricisdemonstratedthroughasimplesimulation.Asbefore,thesimulationconsistsofakinematicaircraftmodelyingthroughanurbanenvironment.ThisenvironmentconsistsofonebuildingadjacenttovecarsfacingtheNorthdirectionlocatedalongaroadway,asshownbyFigure 411 .Thesimulatedaircrafttrajectoryisdesignedtoperformadivingturnfromitsinitialcongurationnearthebuildingandthenfollowalongtheroadway.Atargetisdesignatedontherearsurfaceofeachcarnearthelikelylocationofthelicenseplate.Assuch,targetlabelsaregivenbyi=1;;5. Figure411. Environmentforimageareasensingmetricsimulation. Theaircraftcarriesavideocamerathatislocatedattheoriginoftheaircraftbodybasissuchthat~p=~0.Thecamerapointsforwardanddownatanangleof30deg.Assuch,thetransformationfromBtoSisgivenbyEquation 4{44 .ThecamerausedherehasN=1280linesofresolutionandaFOVof30deg,suchthatf;max=15deg. 112 PAGE 113 4{43 isplottedinFigure 412 aftersubstitutionofthesenumbers.ThexaxisoftheplotinFigure 412 isshownonalogarithmicscale. Figure412. Sensingeectivenessasafunctionofimagearea. ExecutionoftheaircrafttrajectoryresultsinthecameraFOVpassingoverfourofthevetargetsurfaces.Thesecondtargetisnotviewedasaresultofmotioncoupling.ThesensingeectivenesstrajectoriesforeachoftheremainingfourtargetsareshowninFigure 413 .ThesetrajectoriesarezeroforalltimesegmentsduringwhichthetargetisoutsidethecameraFOV.Themaxvaluemissioneectivenessmetricisusedheretotrackaggregatesensingeectivenessforeachtarget.Recallthatthismetricassumesthemaximumeectivenessvalueovertheentiretrajectory,asgivenbyEquation 4{24 .ThenalvaluesformissioneectivenessareshowninFigure 414 .MissionEectivenessvaluesareshownforallvisiblesurfacesinadditiontothevespeciedtargets. 113 PAGE 114 Eectivenesstrajectories. Figure414. Simulatedmissioneectiveness. 114 PAGE 115 2 and 3 ,aregenerallytailoredtodrivethesystemfromonespeciccongurationtoanother.Theproblemofndingminimumtimetrajectoriesthatvisitmultiplepointsrepresentsaninstanceofthewellknowntravelingsalesmanproblem(TSP).Whilethisproblemiswellstudiedintheliterature[ 25 26 ],solutionsandapproximationstypicallyrelyontheassumptionthatoptimaltrajectoryconsistsofoptimalsegments,asshowninFigure 51A .Optimalsolutionsdonotgenerallytakethisformfordynamicsystems.Figures 51B and 51C eachshowathreepointtourforacurvatureconstrainedsystemstartingfromthesameinitialcondition.EachpointtopointsegmentshowninFigure 51B islocallyoptimal.Figure 51C showstheoptimaltrajectoryfortheentiretour,whichincludespointtopointsegmentsthatarelocallysuboptimal.Generally,boththeoptimalsequenceofpointsandtheoptimaltrajectorymustbedetermined;however,thesesolutionelementsareusuallycoupledforsystemswithnontrivialdynamics. B CFigure51. TSPproblemcomparison.A)StandardTSP.B)CurvatureconstrainedTSPwithlocallyoptimalsegments.C)OptimalcurvatureconstrainedTSP 115 PAGE 116 15 17 18 30 57 ].Theseeortsprovideagoodstarttothemultitargetsensingproblem,buttheyallequatetargetvisitationwithtargetsensing.CloseproximitysensingproblemsrealisticallyneedtoconsiderthefullsensorFOVandthecouplingofvehiclemotiontosensorpointing,asdiscussedinChapter 4 .Inactuality,targetscanbeviewedfromasetofcongurations.Considerationofthispropertyaddsanotherlayerofcomplexitytotheplanningproblem.Thesensingproblemnowrequiresthatthetargetsequenceisdetermined,theviewpointswithinthevisibilityset,V,orthequalityset,Q,aredeterminedforeachtargetinthesequence,andtheoptimaltrajectorythroughtheseviewpointcongurationsisdetermined.Thisdynamicallyconstrained,setvisitationTSPhasbeenaddressedtoafarlesserextentintheliterature.Researchersintheeldofmanipulatorroboticshavemadeeortstodene\good"viewpoints,buttypicallyhavenotbeenconcernedwithgeneratingoptimaltrajectories.Thosethathaveinvestigatedoptimaltoursdonottreatnonholonomicdynamics,asmanipulatorarmsareusuallyfullyactuatedsystems[ 27 { 29 ].TheseeortsdecoupletheproblembygeneratingasetofacceptableviewpointsandsubsequentlysolvingastandardTSPthroughtheseviewpoints.References[ 23 ]and[ 107 ]considerFOVdimensionsintheirstudyofanonholonomicvehiclewithadownwardpointingsensor.Nearoptimaltrajectoriesaregeneratedtoviewaseriesoftargetsusingasimplethreeelementdiscretemotionmodelandanheuristicbasedenumerativesearch.Themethodperformswellforthepresentedexamples,buttheenumerativenatureofthesearchwouldintroduceseverecomputationalissuesforproblemsthatinvolvemorethanafewisolatedtargetsorevenaslightlymorecomplicatedmotionmodel.Additionally,therestrictionthatthesensoralwayspointsdownwardneglectsmotioncouplingeects.Thischapterdevelopsarandomizedmotionplanningapproachtogenerategood,thoughsuboptimal,trajectoriesthatsenseaspeciedsetoftargetswithaspeciedmeasureofeectivenessinminimumtime.Conceptsfrompreviouschaptersareintegrated 116 PAGE 117 5.2 .TheactualalgorithmisthendetailedinSection 5.3 .Finally,Section 5.4 demonstratesapplicationoftheplannertothecoreISRmissionsdescribedinChapter 1 .Recallthatthesemissionsconsistedofamultitarget,singleviewreconnaissancetask,amultitarget,multiviewsurveillancetask,andanareacoveragetask. 2 .Speciclocationsintheoperatingregionaredesignatedastargets,whichessentiallyserveasoutputlocationsformeasurementofsensingperformancethroughoutthemission.Inotherwords,thesetargetsrepresenttheenvironmentlocationsforwhichsensingperformanceiscritical,andarethereforetheonlylocationsatwhichsensingperformanceismeasured.Eachtargetisxedintheenvironmentandisdescribedbyathreedimensionalpositionandunitnormalvector.Visibilityandsensingperformanceatthesetargetlocationsarecharacterizedbygeometricrelationshipsbetweenthesensorandthetarget.Specically,therelativepositionandorientationareprojectedontoaparameterspacespannedbytherange,r,theincidenceangle,,andtheFOVangle,f,asdescribedinChapter 4 .Asbefore,asetofcongurations,V,canbedenedforeachtargetfromwhichthattargetisvisible.Similarly,theset,QV,isdenedforcongurationsthatsensethetargetwithatleastsomespeciedvalueoftheeectivenesscriterionformulatedinChapter 4 117 PAGE 118 4 asaspointsthatdonothaveaclearLOStothetarget,suchaswhenanobstacleliesbetweenthesensorandthetarget.Forthepurposesofthisdissertation,stationarytargetsareembeddedinastationaryenvironment.Hence,regionsofocclusionaretimeinvariantandaredenedforeachtargetobstaclepair.Theseregionscaneachbeconservativelyapproximatedasaconvexpolygonthatcorrespondstothemaximum\shadow"castbyapointsourcelocatedatthetarget,asdepictedinFigure 52 .Amoreexactapproximationcanbeachievedbyparameterizingthevertexlocationsofthesepolygonsontheverticalposition,asshowninFigure 53 BFigure52. Occlusionshadows.A)Atargetinathreeobstacleenvironment.B)Conservativeocclusion\shadows"inthetargetplane. Figure53. 3DOcclusionshadow.Theocclusionshadowisconstructedthroughparameterizationofocclusionpolygonverticesonverticalposition 118 PAGE 119 4 formulatesaneectivenessmetricasafunctionofthegeometricrelationshipbetweenthesensorandasingle,discretepointinspace.Evaluationofsensingperformanceoveranareageneratesacontinuumofvaluesandthereforecomplicatessomeofthedenitionsthatproveusefulformotionplanning,suchastheconceptofthevisibilityset,V,andthequalityset,Q.Alternatively,thecontinuumofpointsinthecoverageareacanbediscretizedintoasetof\virtualtargets"thatrepresenttheareawithadesiredresolution.Suchadiscretizationcouldbeachievedbysimplydividingtheentireenvironmentintoagrid,asdepictedintheleftimageofFigure 54A ;however,thisdiscretizationschemeintroducesseverecomputationalineciencies.Realistically,amultiresolutionschemeisrequiredtoemphasizethecoverageareawithoutdevotingunnecessaryresourcestononcriticalareasoftheenvironment.AnexampleofsuchaschemeisdepictedbytheocttreelikeapproachshownintherightimageofFigure 54A .Thisapproachinvolvesdiscretizingtheenvironmentwithacoarseresolutionandthensubdividingcellsinthecoverageareatoincreaseemphasis.Alternatively,virtualtargetscanbesampledfromanimportanceweighteddistributionthatemphasizesthecoveragearea.TheresultingresolutionfromchoosingvirtualtargetsinthisfashionisdepictedasavornoiplotinFigure 54B .Thevoronoiplotconsistsofpolygonswithedgesthatareequidistantfromadjacentvirtualtargets,andhencereectresolutionofthediscretization. BFigure54. Discretizationapproachesforareacoverage.A)Octreelikeapproach.B)Virtualtargetsapproach. 119 PAGE 120 4 .TheseconstraintsareexpressedasinEquation 5{1 ,whereQdirepresentsthedesiredsensingeectivenessfortheithtarget. 3 ,randomdensetreemethodsgenerateatreeoffeasibletrajectoriesthatisdesignedtorapidlyexplorethereachablesetofthesystem.Here,thisexpansionisbiasedtowardachievingsensingobjectives.Specically,eachbranchaddedthetreeisdesignedtoincrementallysatisfythemissionsensingconstraints.Assuch,eachbranchsatisesatleastonesensingconstraintinadditiontothosesatisedattheexpansionnode.Nodesthathavesatisedallsensingobjectivesareexpandedtothemissionterminalconguration.Anupperboundonthetimeperformanceoffeasiblesolutionsismaintainedwhileadditionalsolutionsaregenerated.Feasiblesolutionsaregenerateduntilanindependentterminationconditionismet,suchasalimitoncomputationalresourcesoronmaximumallowableplanningtime.Thetreestructureistypicallyinitializedasasinglenodeattheinitialcondition.Analternateoptioninvolvesusingastandardrandomizedplanningtechnique,suchastheRRTalgorithm,togenerateanexploratoryinitialtree.Useofthisoptionprovidesa 120 PAGE 121 55 .Theleftimagedepictsascenariowheresixtargetsalongaroadmustbesensed.Generally,thereare6!=720possiblesequencesinwhichthesetargetscanbesensed;however,planningtrajectoriestoviewthesetargetsindividuallyyieldsatrajectorythatsensesallsix,asshownintherightimageofFigure 55 .Asucientconditionforoptimalityofthisimplicitlydeterminedsequenceisoptimalityofthetrajectorytoviewtheindividualtarget. 121 PAGE 122 Sensingsecondarytargets.Planningatrajctorytosenseasingletargetoftenresultsinsecondarysensingofothertargets. treeexpansionindirectionsthatfavorbetterperformingsolutions.Previousresearcheffortshaveachievedsuchabiasbyrandomlysamplinganexpansionnodefromaweightedprobabilitydistribution[ 73 77 ].Eachnodeisassignedaweightthatisdependentonperformancecharacteristics.AweightingschemethatreectsthetradeobetweensensingperformanceandcostperformanceisshowninEquation 5{2 ,where#visrepresentsthenumberoftargetsthathavebeensuccessfullysensedandLrepresentsthepathlengthinsecondsbetweentherootnodeandNcurr.Additionally,#tgtsrepresentsthetotalnumberoftargets,Lmaxisthemaximumpathlengthoverallnodesinthetree,andisashiftfactorincludedtovarytheemphasisonunsensedtargets.ArepresentativeplotofthisweightingfunctionisshowninFigure 56 .Thegureshowsthatnodeswhichhaveachievedsensingobjectivesinashortperiodoftimearefavored.Theformulationofthisweightingfunctionallowsfornodesthathavesensedmoretargetstoexhibitlongerpathsyetstillmaintainahighweightvalue.Highweightsarealsoseenforbranchesthathavenotseenanytargetsandhaveshortlength. Lmax12(5{2) 122 PAGE 123 Examplenodeweightingfunction. 4 .Specically,avantagepointcharacterizedbythesensorpositionandattitudemustlieinthequalityset,Qi,thatisassociatedwiththeithunsensedtarget.Additionally,theviewfromtheselectedvantagepointmustnotbeoccludedandmustbereachablefromNcurr.Theactualprocesswhichdetermineshowaparticularvantagepointisselecteddependsonhowthevehicleismodeledandhowthesensorismountedrelativetothevehicle.Forexample,avehiclemodeledusingmotionprimitives,asdiscussedinChapter 2 ,hasacontinuumofallowablepositionsandheadinganglesbutanitesetofbodyaxisorientations.Thesequantizedelementsoftheallowablevehiclemotionsmustbeconsideredinviewpointselectionforthiscase.Generally,viewpointsareselectedpseudorandomlythroughsamplingfromtheallowablevehiclecongurationsthatmeetthesensingcriteriaforaselectedtarget.Samplingispseudorandominthatheuristicscanbeincorporated 123 PAGE 124 5.2 ,eachtargethasanassociatedsetofpolygonalregionsforwhichtheLOSisobstructed.Therefore,asampledvantagepointcanbetestedforocclusionbyverifyingthatitdoesnotlieinanyoftheocclusionpolygonsassociatedwiththeselectedtarget.ThisconditioncanbecheckedusingoneofthemethodsdescribedinSection 2.2.2 .Vericationofreachabilityposesaslightlymorediculttask.Thistaskinvolvestheconceptofforwardandbackwardsreachablesets.Theforwardreachablesetconsistsofallpointsreachablefromthesampledvantagecongurationwhilethebackwardsreachablesetconsistsofallpointsfromwhichthesampledvantagecongurationcanbereached.Forthepresentdevelopment,avantagepointisconsideredreachableifthegoallocationiscontainedintheforwardreachablesetandthecurrentnodeiscontainedinthebackwardsreachableset.Unfortunately,characterizationofthesesetsishighlynontrivial.Avantagepointsampledinsideanobstacleregioncanbeeasilyidentiedandrejected;however,selectionofvantagepointsthatcannotbereachedwithoutanobstaclecollisionorthatwillresultinanimminentcollisioninthenextexpansionarediculttoidentifywithoutactuallyplanningatrajectorytoorfromthesample.Anapproximationisobtainedbycomputingandcollisioncheckingseveralshorttrajectorysegmentsneartheboundariesoftheforwardandbackwardsreachablesets,asdepictedinFigure 57 .Thegureshowsshortturningandstraightsegmentsintoandoutfromasampledpositionandheading.Whentheserepresentativetrajectoriesarecollisionfree,adegreeofshorttimereachabilityisindicated.Whilethisapproachcannotguaranteefullreachability,itcanserveasaquicktesttoruleoutthemostobviouscases. 124 PAGE 125 Vantagepointreachabilitytest.Vantagepointsforseveraltargetsaretestedforreachabilityusingrepresentativetrajectoriesfromtheforward(green)andbackwards(blue)reachablesets. 3 exhibitsthecapabilitytoecientlygeneratefeasibletrajectoriesthatmeetthesecriteria.Assuch,thismethodisappropriateforuseasthelocalplanningmethodhere;however,useofarandomizedplannerasthelocalplanningmethodforusewithanotherrandomizedplannerrequiresthatsomecautionistakenintreeandnodedatarepresentation.Specically,thetreestructureusedinthelocalplannershouldbeconsistentwiththemainplanningtreetofacilitateintegrationoflocalsolutionnodesintothemainsolution.Further,aterminationconditionforthelocalplannershouldbecarefullyselected.RecallthattheplannerdescribedinChapter 3 iscapableofgeneratingafeasiblesolutionquickly,butcanoerbetterapproximationstotheoptimalsolutiongivenadditionaliterations.Areasonabletradeobetweenperformanceandeciencymustbemadebasedontheapplication. 125 PAGE 126 5{2 providesameasureofthetradeoseeninthesepathproperties.Alternatively,thesecriteriacouldbeappliedindependentlyandconsecutively.Thenumberofnewlysatisedsensingconstraintscouldbeusedtoreducethesetofcandidatesfollowedbyaselectionoftheminimumtimecandidatefromthosethatremain.Regardlessoftheexactcriteriaformulation,thisapproachtoevaluatingcandidatebranchesrewardstrajectoriesthatsatisfythesensingconstraintsin\chunks"whileusingaslittletimeaspossible.Selectionofsuchbranchesessentiallyamountstogradientfollowing,whichisnotalwaysthemosteectiveoptimizationapproachforhighlynonlinearandnonconvexproblems.Gradientfollowingapproachesarenotoriouslyattractedtolocal 126 PAGE 127 5.3 .Theseexamplesemployasimplevehiclemodelthatisconstrainedtooperateintwodimensionsyetexhibitsthreedimensionalsensingcapabilities.Missionscenariosareconstructedsuchthatthisvehicleisrequiredtocollectdataregardingtargetsinacloseproximityenvironment.Assuch,asensingmetricisselectedthatappropriatelycharacterizestheeectsoftheenvironmentproximityonsensingcapabilities.ThemissionscenariosreectthethreesensingtasksidentiedinChapter 1 :reconnaissance,surveillance,andcoverage.Thesetasksarerstaddressedinanobstaclefreeenvironmentsuchthatocclusionsandtrajectorysafetyarenotconsidered.Thissimpliedexampleisfollowedbyanexamplethatincludespolygonalobstacles. 3 .Assuch,thevehiclemoveswith 127 PAGE 128 2 .Aspreviously,thesetrimstatesconsistofsteadyturnsineachdirectionatthreedierentturnratesandasteady,straightaheadmotion.Alltrimprimitivesmaintainconstantforwardvelocity.Themodelischaracterizedaskinematicsuchthattransitionmaneuversbetweentrimstatesoccurinstantaneously.Thetrimstatesarecharacterizedbyconstantsystemvelocitiesandbankanglesasfollows: 128 PAGE 129 4.4.4 essentiallyquantiesthisnotionofrobustvisibility.Specically,asensingeectivenessmetricisconstructedusingtheproductofaseriesoffunctionsthatexhibitdependenceonthevisibilityparameters,asshownbyEquation 5{3 .Thefunctions,fri,fi,andff;i,takevaluesthatreectthedetrimentaleectsondataqualityresultingfromvariationsinthevisibilityparameters,r,,andf,relativetotheithtarget.ThesefunctionstakethesameformasinSection 4.4.4 ,withtheexceptionoff;i,whichisrelatedbythebinaryconditionshowninEquation 5{4 .Theparametervaluesthatspecifytheexactexpressionsaregivenbyrmax=750ft,max=70deg,f;max=35deg. 5{1 ,whereQd=0:75.Themaxvaluecriteriarequirethateachtargetisviewedwithaneectivenessmetricvalueofqrv=Qdatleastoncealongthe 129 PAGE 130 58 .Theplotsshowthatsimpleboundsonthevisibilityparametersforaparticulartargetensurethatsensingcriteriaaremet.Theseboundsareseenas:49ri385ft,f;i29deg,and,fromEquation 5{4 ,i<70deg.TheseconservativerequirementsallowanonvaryingdenitionofQiwithrespecttothevisibilityparameters,asshowninEquation 5{5 BFigure58. Qualityparametereciencyfunctionsforexamples.A)Rangeeciencyfunction.B)FOVangleeciencyfunction. 130 PAGE 131 5{6 ^ni=001T8i(5{6)Thisexampleactuallyconsiderstwosetsofspeciedtargets.TherstsetoftargetsisshowninFigure 59A .Thegureshowstheenvironmentboundaryasasolidbluelineandtwelverandomlyselectedtargetlocationsasredasterisks.Thesetargetsareusedforthereconnaissanceandsurveillancemissiondescriptions.Figure 59B showsthesecondsetoftargets,whichisusedforthecoveragemissiondescription.Thegureshowsaseriesofroadspassingthroughanopenarea.AnareatotherightoftheplotinFigure 59B isenclosedbytheroadsandisdesignatedforsensorcoverage.Thissensingrequirementisrepresentedbyasetofeightyvirtualtargetswhicharerandomlyselectedfromthecoverageregion.ThesetargetsaredepictedasredasterisksinFigure 59B BFigure59. Environmentsforobstaclefreesensorplanningexample.A)Environmentforreconnaissanceandsurveillancemissions.B)Environmentforcoveragemission. SeveraldegreesoffreedominthealgorithmdescribedinSection 5.3 aresetspecificallyforthepresentexample.First,eachiterationofthealgorithmbeginswiththeselectionofanexpansionnodefromaweighteddistribution,wherenodeweightsaregivenbytheweightingfunctiondescribedaspreviouslybyEquation 5{2 131 PAGE 132 3 isusedtogeneratethecandidatebranches;however,otherapproachessuchasoptimalcontrolarecertainlyvalid.Finally,eachcandidatebranchisevaluatedforsensingandtimeperformance.Branchesthathavenewlysatisedthemostsensingconstraintsaresetaside.Theminimumtimebranchfromthissubsetisselectedforadditiontothetrajectorytree.Ifallsensingconstraintshavebeensatised,atrajectoryisplannedtothegoalcorrespondingtothesensingtask.Theinitialconditionservesasagoalforthereconnaissanceandcoveragetasks.Therstpointalongthetrajectoryforwhichasensingconstraintissatisedisusedasagoallocationforthesurveillancetask.Therstmissiondescriptionexaminedisthereconnaissancetask.Recallthatthistaskrequiresthateachtargetisviewedatleastoncewithadesiredvalueforthesensingmetric.Thetrajectorytreeisinitializedbyasinglenodeattheoriginoftheinertialframe.Theinitialautomatonstateatthisnodeisrepresentedbythetrimstate,T0.Theinitialcongurationisgivenby(px;py;)=(0;0;0).ForthetwelvetargetsshowninFigure 59A ,thereare12!479millionpossiblesequencesinwhichtovisittargets.Here,thealgorithmdescribedinSection 5.3 isrunfor100iterations. 132 PAGE 133 510 .Inthegure,solutiontrajectoriesaredepictedasaredsolidline.Additionally,theprojectionoftherestrictedsensorfootprintontheenvironmentsurfaceisshownforthepointalongthetrajectorywhichrstsatisesthesensingconstraintassociatedwitheachtarget.TheseprojectionsrestrictthesensorFOVtothevaluesthatsatisfythedenitionofQ,andareshowninFigure 510 asdashedcontours.Solutionsareshownafter,1,10,and64iterationsandcorrespondtotrajectoriesof87:91sec,63:30sec,and56:95sec,respectively. Figure510. Trajectoryrenementforobstaclefreereconnaissance. Asimulationisruntoplotthemaxvalueeectivenessmetricovertheentireenvironmentfortheminimumtimesensingtrajectory,whichcorrespondstothesolutionfoundinthe64thiteration.TheresultofthissimulationisshowninFigure 511 .Thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.NoticethatthevalueoftheeectivenessmetricisatleastQ=Qd=0:75atalltargetlocations.Also,notethattheplannermakesuseofthemotioncouplingpropertiesexhibitedinturningighttosensetargetsfromadistance.Assuch,thesolutiontrajectory,whilesuboptimal,isshorterthananytrajectorythatpassesdirectlyoveralltwelvetargets.Applicationoftheplannertothesurveillancetaskproceedsinsimilarfashion.Thesamesetoftargetsinthesameenvironmentisusedforthisdemonstration.Aspeviously,thetreeisinitializedwithasinglenodeat(px;py;)=(0;0;0)inthetrimstate,T0. 133 PAGE 134 Simulatedeectivenessforobstaclefreereconnaissance. Thealgorithmisrunfor100iterations.Asbranchesareadded,alistofviewpointsismaintainedthattracksthersttargetsensedalongatrajectory.Viewpointcongurationsfromthislistareusedasgoallocationswhenachildbranchhassatisedallofthesensingconstraints.Useofthesecongurationsasgoallocationsensuresthatthesurveillanceloopisclosedoncealltargetshavebeensensedonce.Thisloopisthentraversedrepeatedlytovieweachtargetmultipletimes.Aminimumupperboundontimeperformanceismaintainedassurveillanceloopsareclosed.Theupperboundisrevisedasbettersolutionsarecomputed.SeveraloftheseimprovingsolutiontrajectoriesareshowninFigure 512 .Asbefore,solutiontrajectoriesaredepictedasaredsolidlineandthe\rstview"projectedsensorfootprintsaredepictedasdashedblacklines.Solutionsareshownafter,1,7,and10iterationsandcorrespondtotrajectoriesof80:44sec,59:66sec,and54:80sec,respectively.Thesevaluesreectthetimerequiredtothereachtheendoftherstloopfromtheinitialcondition.Figure 513 showstheresultofasimulationthatplotsthemaxvalueeectivenessmetricfortheminimumtimesurveillanceloop.Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. 134 PAGE 135 Trajectoryrenementforobstaclefreesurveillance. Figure513. Simulatedeectivenessforobstaclefreesurveillance. ThealgorithmisappliedtothecoveragetaskintheenvironmentdepictedbyFigure 59B .Thecoverageareaisrepresentedbyasetofeightyvirtualtargetswhicharesampledfromauniformdistribution.Theplannerandtreeareinitializedtothesamevaluesasusedforthereconnaissanceandsurveillancetasks.Thealgorithmproceedsthrough100iterationsinthesamefashionastheothersensingtasks.Despitethelargersetoftargets,theorderofcomputationalcomplexityisthesameasfortheothermissiontasksduetoselectionofthesamenumberofviewpointsateachiteration.Thealgorithmbenetssignicantlyfromthefactthatmanytargetsareoftenseenfromasingleviewpointgiventhetargetspacingthatresultsfromthesamplingapproach.Ariseinstoragecomplexitydoesresultfromtheneedtostoreeectivenessmetricvaluesforeachtargetateachnodeinthetrajectorytree. 135 PAGE 136 514 .Solutiontrajectoriesareshownasablacksolidlineinthegure.Projectedsensorfootprintsarenotshowntoavoidclutterresultingfromthelargenumberofvirtualtargets.Solutionsareshownafter,1,15,and84iterationsandcorrespondtotrajectoriesof93:19sec,78:89sec,and59:02sec,respectively. Figure514. Trajectoryrenementforobstaclefreecoverage. Figure 515 showsasimulationofthenaltrajectorythatplotstheeectivenessmetricovertheentireenvironment.Asbefore,thetrajectoryisdepictedasasolidblacklineandthevirtualtargetsaredepictedasblackcircles.Noticefromthegurethatthenaltrajectorydoesnotactuallyresultinperfectcoverage.Thisdeciencydemonstratesthesuboptimalityoftheplanner,andsuggeststhatalargersetofvirtualtargetsareneededtoadequatelyrepresenttheconsideredcoveragearea. 5.4.3 .Additionally,fourpolygonalobstaclesareplacedrandomlywithinthesebounds.Theseobstaclesmightrepresentbuildingsinanurbansetting.Aspreviously,twodierentsetsoftargetsareembeddedwithinthisenvironment:onetodemonstratereconnaissanceandsurveillancemissiontasks 136 PAGE 137 Simulatedeectivenessforobstaclefreecoverage. andanothertodemonstrateareacoverage.Thesetargetshavethesamesurfacenormalproperties,asexpressedinEquation 5{6 .Unlikethepreviouscasewhichhadtargetsrandomlyscatteredthroughoutthearea,thepresentcaseconsiderstargetlocationsthatarestrategicallyplacedclosetotheobstacles.Figure 516A showstheenvironmentusedforreconnaissanceandsurveillancemissions.Ninetargetsareseeninlocationsthatcouldrepresententrancestobuildings.Informationgatheredfromsuchlocationscouldprovidevaluableintelligenceregardingpeopleorvehiclesthatenterandleaveeachbuilding.Figure 516B showstheenvironmentusedforcoveragemissions.Onehundredvirtualtargetsaresampledfromtheregionsurroundingoneofthebuildings.Anexampleapplicationforcoverageofsuchanareaistoidentifythepresenceofanyunauthorizedpersonsorevenanexplosivedevicewithinthevicinityofthebuilding.ThemajorityofthealgorithmparametersaresetidenticallytothosedescribedfortheexamplesinSection 5.4.3 .Nodesareselectedfromaweighteddistribution,asetoftenviewpointsaresampledfromSiQi,andcandidatebranchesareplannedandevaluatedtotheseviewpoints.Contrarytothepreviousexample,thetreeisinitializedhereusinganexploratoryrunofanRRTalgorithm.Executionofthisalgorithmisconcernedprimarilywithspanningtheenvironmentanddoesnotconsidersensingobjectives.Useofthis 137 PAGE 138 BFigure516. Environmentsforsensorplanningexamplewithobstacles.A)Environmentforreconnaissanceandsurveillancemissions.B)Environmentforcoveragemission. techniquetoinitializethetreegeneratescongurationnodesthatmightnothavebeencomputedusingthesensorplanningapproach.Therstmissiondescriptionexaminedisthereconnaissancetask.Eachtargetmustbeviewedatleastoncewithadesiredvalueforthesensingmetricbeforeatrajectoryisplannedbacktotheinitialcondition.ThetrajectorytreeisinitializedwithanexploratoryrunofanRRTalgorithmresultingin50initialnodes.Thistreeisrootedatthevehicleinitialconguration,(px;py;)=(0;0;0),andisshowninFigure 517 .Aspreviously,theinitialautomatonstateattherootnodeisrepresentedbythetrimstate,T0.Thealgorithmisrunfor100iterations. Figure517. Initialtreeforreconaissancewithobstacles. 138 PAGE 139 518 .Inthegure,solutiontrajectoriesaredepictedasaredsolidline.Asintheprevioussetofexamples,theprojectionoftherestrictedsensorfootprintontheenvironmentsurfaceisshownforapointalongthetrajectorywhichsatisesthesensingconstraintassociatedwitheachtarget.Theseprojectionsareshownasdashedcontours.Solutionsareshownafter,1,20,and98iterationsandcorrespondtotrajectoriesof68:56sec,67:06sec,and62:97sec,respectively. Figure518. Trajectoryrenementforreconnaissancewithobstacles. TheresultsofasimulationshowingthesensingeectivenessovertheentireenvironmentisshowninFigure 519 .ThesimulatedtrajectoryinthegurecorrespondstothesolutionshowninFigure 518C .Twoviewsareshowntodemonstratethatalltargetshavebeensensedwithdesiredeectiveness. Figure519. Simulatedeectivenessforreconnaissancewithobstacles. 139 PAGE 140 520 .Duringthe100iterationsforwhichthealgorithmisrun,therstpointalongeachsolutionpathtosatisfyasensingconstraintmustbestored.Asbefore,thispointservesasagoallocationforchildbranchesthathaveachievedallsensingobjectives. Figure520. Initialtreeforsurveillancewithobstacles. Assurveillanceloopsareclosed,aminimumupperboundontrajectorytimeperformanceismaintained.Theupperboundisrevisedasbettersolutionsarecomputed.SeveraloftheseimprovingsolutiontrajectoriesareshowninFigure 521 .Asbefore,solutiontrajectoriesaredepictedasaredsolidlineandprojectedsensorfootprintsaredepictedasdashedblacklines.Solutionsareshownafter4,33,and83iterationsandcorrespondtotrajectoriesof71:40sec,66:01sec,and61:43sec,respectively.Thesevaluesreectthetimerequiredtothereachtheendoftherstloopfromtheinitialcondition.Figure 522 showstheresultofasimulationthatplotsthemaxvalueeectivenessmetricfortheminimumtimesurveillanceloopshownbyFigure 521C .Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. 140 PAGE 141 Trajectoryrenementforsurveillancewithobstacles. Figure522. Simulatedeectivenessforsurveillancewithobstacles. ThealgorithmisthenappliedtothecoveragetaskintheenvironmentdepictedbyFigure 516B .Thecoverageareaisrepresentedbyasetofonehundredvirtualtargetswhicharesampledfromauniformdistribution.ThetrajectorytreeisagaininitializedwithanRRTcontainingftynodesandwhichisrootedat(px;py;)=(0;0;0)inatrimstatecorrespondingtoT0.ThisinitialtreeisshowninFigure 523 Figure523. Initialtreeforcoveragewithobstacles. 141 PAGE 142 524 .Solutiontrajectoriesareshownasablacksolidlineinthegure.Projectedsensorfootprintsarenotshowntoavoidclutterresultingfromthelargenumberofvirtualtargets.Solutionsareshownafter,1and5iterationsthatcorrespondtotrajectoriesof106:38secand87:30sec,respectively.Noticethatthebestsolutionoutofall100iterationswasfoundveryearlyintheprocess. Figure524. Trajectoryrenementforcoveragewithobstacles. Finally,Figure 525 showstheresultofasimulationthatplotsthemaxvalueeectivenessmetricfortheminimumtimecoveragetrajectoryshownbyFigure 524B .Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. Figure525. Simulatedeectivenessforcoveragewithobstacles. 142 PAGE 143 2 asanapproachforthedesignofoptimaltrajectoriesforsystemsconstrainedbydynamics.Whiletoolsfromoptimalcontrolcanprovidesolutionstomanyproblemsofinterest,considerationofrealisticconstraintsandmissiondescriptionsoftenintroducescomplicationswhichrenderoptimalcontrolsolutionsintractable.Severalsuchcomplicationsareseenforthesensorplanningproblemconsideredinthisdissertation.Inparticular,thediscontinuousnatureofvisibilityandsensingeectivenessconditionsviolatesmoothnessrequirements.Theseconditionscanbetreatedasendpointconstraintsinaseriesofsubproblems;however,suchanapproachrequiresknowledgeoftheoptimalsequenceinwhichtoviewtargets.Further,obtaclesandocclusionregionsaddpathconstraintsthatcanincreasecomputationalburdenandintroducelocalminima.Thischapterexaminestheapplicationofoptimalcontrolmethodstoasimpliedversionofthecloseproximityeectivesensingproblem.First,theproblemofsensingasingletargetinanobstaclefreeenvironmentisconsidered.Thisscenarioallowstheformulationofsensingrequirementsasendpointconstraintsandavoidsthenecessitytoconsiderpathconstraintsresultingfromobstaclesandocclusions.Thenecessaryconditionsforanoptimalsolutionarederivedtodemonstratesomeofthedicultiesassociatedwiththeuseofindirectmethodstosolverealisticproblems.Next,numericallybaseddirectoptimalcontrolmethodsareusedtotranscribetheproblemtoanitedimensionalNLP.Theproblemisextendedtoconsiderthesensingofasmallsetoftargetsinanobstaclefreeenvironment.Thisscenarioallowsenumerationofallpossibletargetsequencessuchthateachcanbeformulatedasamultiphaseoptimalcontrolproblem.Theoptimalsolutionforeachsequencecanbecomputedandcomparedtosolutionswhichcorrespondtoallothersequencestodeterminetheoptimum. 143 PAGE 144 4 todeneagoalsetofvehiclecongurationsthatensureeectivesensing.Specically,sensingaparticulartargetdoesnotrequirevisitation.Conversely,thevisibilitysetconceptisemployedtousetheentiresensorFOVandthusaccountformotioncouplingeectsthatresultfromvehicledynamics.ThespecicdynamicmodelusedwiththeseexamplesisdescribedinSection 6.2 .TheproblemofdeterminingaminimumtimetrajectorythatterminateswithadesiredviewofaspeciedtargetisthenformulatedinSection 6.3 .Theindirect,variationalapproachtosolvingthisproblemisdiscussedinSection 6.4 andisfollowedbytheapplicationofadirectnumericalmethodtotheprobleminSection 6.5 6{1 and 6{2 ,respectively.Thestatesthatcomprise~xare:totalvelocity,ightpathangle,headingangle,altitude,Northposition,Eastposition,angleofattack,angleofattackrate,androllangle.Thecontrolsarecommandedangleofattackandcommandedrollangle.Thesecontrolvariablesrepresentnontraditionalsysteminputsandcanbeconsideredasgeneratorsofreferencecommandsforlowerlevelcontrolsystems.Designofinnerloopand'controllersistypicalintheeldofaircraftcontrol[ 108 ].~x=Vphpxpy'T 6{3 to 6{11 ,where=2(=180)andtheforcecoecients,T,CL,andCD,aretakenfromcurvetteddata[ 109 ].Eachoftheseequationsrepresentsanelementinthevectorvaluedfunction,_~x=f(~x;~u).Thismodelisbasedonasixstateguidancelevelmodelfromtheliterature[ 109 ].Theoriginalmodelusedand'ascontrols;however,thesevariables 144 PAGE 145 108 ]._V=Tcos(+)CDV2sin (6{9)_=!2n;2!n;+!2n;u (6{11)ThepolynomialcurvesthatareusedtoapproximatetheaerodynamicforcecoecientsareshownbyEquations 6{12 to 6{14 ,where1=(12=180).ThedatacorrespondstoamodelofaBoeing727[ 109 ].Suchanaircraftwouldnottypicallybeinvolvedinacloseproximitysensingmission;consequently,ascaledupversionoftheproblemmustbeenvisioned.TheforcecoecientcurvesareareplottedinFigure 61 .T(V)=0:24760:04312V+0:008392V2 145 PAGE 146 B CFigure61. Forcecoecients.A)Thrust.B)Drag.C)Lift. Inadditiontoscalingtheproblemtomatchthemaneuveringcapabilitiesoftheaircraft,themodelisscaledtofacilitatenumericaloptimizationperformance[ 109 ].Velocityisexpressedinunitsofp 6{15 ,whereWistheaircraftweightandSistheplanformareaofthewing.AllmeasuresofdistanceareexpressedinunitsoflCandtimeisscaledbyp gS(6{15) 6{3 to 6{11 andwhichcarriesaLOSremotesensor.MotionisdescribedrelativetoaninertialframethatisorientedsuchthatthexaxispointsintheNorthdirection,theyaxispointsintheEastdirection,andthezaxispointsdownward.ThegeometryofthisvehiclesensorsystemisdescribedasinChapter 4 .Here,thesensorreferenceframe,S,isplacedtocoincidewiththevehiclebodyreferenceframe,B.Assuch,thesensorpositionandorientationareequivalentlydescribedbythevehiclepositionandorientation,asshownbyEquations 6{16 and 6{17 ,whererepresentstheaircraftpitchangle.Additionally,thissensorhasaFOVcharacterizedbyamaximumrangeofrmax=10;000ftandaFOVhalfangleoff;max==4.Anincidenceangleof<=2isalsorequiredforvisibility. 146 PAGE 147 Atargetislocatedrelativetoaninertialcoordinateframeby~ptwithanassociatedunitnormaldirectionindicatedby^n,asshownbyEquations 6{18 and 6{19 ,respectively.Thistargethasanassociatedvisibilityset,V,whichisdenedintermsofthevisibilityparameters,r,,andfasdescribedinChapter 4 .~pt=pt;xpt;ypt;hT 6{20 and 6{21 ,respectively.ThetransformationterminEquation 6{21 canbewrittenasEquation 6{22 forthesensorcongurationconsideredhere.Noticethatthisexpressionrequiresknowledgeoftheaircraftpitchangle,,whichisnotexplicitlymodeledintheequationsofmotionpresentedinSection 6.2 .Thepitchanglecanberecoveredthroughthekinematicrateofclimbconstraint,whichisexpressedinEquation 6{23 undertheassumptionthatthesideslipangle,isregulatedtozero[ 108 ].r=q 147 PAGE 148 6{24 6{25 and 6{26 ,respectively.Eachstateisconstrainedatt=t0totakethevalueofaspeciedinitialcondition.Theightpathangle,,isconstrainedforlevelightatt=tfandthenalaltitudeisconstrainedtomatchtheinitialaltitude.Notethattheinequalityconstraintvectoriscomprisedoftheconditionsrequiredforvisibility,asshowninEquation 6{26 .Theincidenceconditionisnotincludedherebecausetheterminalconstraintonaltitudeensures<=2.(~x(t0);t0;~x(tf);tf)eq=266666666666666666666666666666664V(t0)V0=lC(t0)0(t0)0ph(t0)ph;0=lCpx(t0)px;0=lCpy(t0)py;0=lC(t0)0(t0)0'(t0)'0(tf)ph(tf)ph;0377777777777777777777777777777775 148 PAGE 149 6{27 to 6{32 .Theoperatingconditionshavebeenrestrictedbyplacingboundsonallowablevaluesfortheelementsofthecontrolvector,uandu'.ThesecontrollimitsarerepresentedasEquations 6{31 and 6{32 minuJ _~xf(~x;~u)=0 (6{28) (~x(t0);t0;~x(tf);tf)eq=0 (6{29) (~x(t0);t0;~x(tf);tf)ineq0 (6{30) 6{27 to 6{32 .Suchanapproachisoftencalledanindirectmethodbecausethesolutionisobtainedindirectlythroughderivationoftherstordernecessaryconditionsforanoptimalsolution.Stateandcontroltrajectoriesthatsatisfytheseconditionsaredeterminedthroughthesolutionofboundaryvalueproblemthatresultsfromthenecessaryconditions.Unfortunately,solutionofsuchproblemshavenumerousassociateddiculties,includingsmallradiiofconvergence,agoodinitialguessforthecostate,andpriorknowledgeoftrajectorysegmentsforwhichpathconstraintsareactive[ 51 52 ].Asaresultofthesediculties,thecurrentpresentationformulates,butdoesnotsolve,theboundaryvalueproblem.Theprimarypurposeofthissectionistodemonstratesomeofthepracticalissuesthatariseinapplyingoptimalcontroltheorytoevenasimpliedversionofthesensingproblem.Assuch,thesubsequentdevelopmentactuallyconsidersareducedformoftheproblemformulatedintheprevioussection.Specically, 149 PAGE 150 6{31 and 6{32 ,areneglected.Thisassumptionremovestheneedforpriorknowledgeoftheconstrainedtrajectorysegments.Additionally,theendpointinequalityconstraintsrepresentedby(~x(t0);t0;~x(tf);tf)ineqaretreatedasactiveequalitiesforderivationpurposes. 2 .Here,theseconditionsarederivedfortheproblemdescribedbyEquations 6{27 to 6{32 .TheHamiltonianofthesystemmustbecomputedtoinordertoformJpriortoderivationofthenecessaryconditions.RecallthattheHamiltonianisgivenbyH=L+Tf(~x;~u),whereisthevectorofcostatevariables.Thecostatesaretimedependentandmatchthestatevectorindimension,asshownbyEquation 6{33 .Here,HisexpressedasinEquation 6{34 @~x.Forthepresentsystem,theelementsofthisvectorareexpressedasshowninEquations 6{36 to 6{44 .Theseexpressionsrequirepartialderivativesoftheforcecoecientcurvets,TV,CL,andCD.ThesederivativesareshowninEquations 6{45 to 6{47 150 PAGE 151 (6{35) +((sin(+)(VTVT)+CL)cos'+cos) Vcos2(Tsin(+)+CLV2)sin'+ (6{36) +VcosphVsincospxVsinsinpy)_=(Vcossinpx+Vcoscospy) (6{37) _ph=0 (6{38) _px=0 (6{39) _py=0 (6{40) _=((Tsin(+)CDV2)V+ (6{41) +(Tcos(+)+CLV2)cos' V+sin' Vcos!2n;) (6{42) _=(2!n;) (6{43) _'=((Tsin(+)+CLV2)cos' Vcossin' V+1 (6{44) 2 asEquations 2{20 and 2{21 ,andrequiretheintroductionofavectorofLagrangemultipliers,givenhereasEquation 6{48 151 PAGE 152 @~x(t0)=@ @~x(tf)=0.ThepartialderivativeoftheendpointconstraintvectorwithrespecttotheinitialstateyieldsthematrixshowninEquation 6{49 ,whereIrepresentsanidentitymatrixand;representsazeromatrix.ThepartialderivativeoftheendpointconstraintvectorwithrespecttotheterminalstateyieldsthematrixshowninEquation 6{50 .ThelastrowofthismatrixcontainsseveralcomplicatedderivativesresultingfromtheappearanceofTBEintheexpressionforf.Theactualexpressionsforthesederivativesarenotshownheretomaintainclarity.ThecostateboundaryconditionsarethenformedbycombiningtheseexpressionsasspeciedbyEquations 2{20 and 2{21 .@ @h@r @x@r @y0000@f 2 .ThisconditionisgivenbyEquation 6{51 andisdenotedaHamiltonian 152 PAGE 153 47 ].TheresultofthisexpressionforthepresentderivationistoxHatH=1foralltime,ashereHisnotanexplicitfunctionoftime. @tf~T@ 48 ].RecallfromChapter 2 thatthisconditioncanbeexpressedas@H @u=0,whichidentiesastationarypointoftheHamiltonian.Forthepresentexample,theoptimalityconditionisexpressedasinEquations 6{52 and 6{53 .Ideally,theseexpressionscouldbeusedtosolvefortheoptimalcontrolvectorintermsofthestates,~x,andcostates,.TheHamiltonianforthepresentexampleislinearinthecontrolvariables.Consequently,Equations 6{52 and 6{53 donotdirectlyyieldanyrelevantinformationregardingthecontrol.Hu=0=!2n; 6{27 to 6{32 .Obtainingtheoptimalstateandcontroltrajectoriesfromtheseconditionsisnotnecessarilyanintuitiveandstraightforwardprocess.WhentheMinimumPrincipleyieldsanexpressionfortheoptimalcontrolintermsofthestateandcostate,theresultcanbesubstitutedbackintothestateandcostatedynamics.TheaugmentedsystemisthendescribedbythedierentialequationsshowninEquations 6{54 and 6{55 ._~x=@H(~x;~) 153 PAGE 154 6{54 and 6{55 ;however,theseconditionsaresplitbetweentheendpointsandarenotallknownvalues.Forexample,thecostateboundaryconditionsgivenbythetransversalityconditionsderivedintheprevioussectionareseenas: @ph+f@f @px+f@f @py+f@f 6{52 and 6{53 donotprovideanexpressionfor~uintermsof~xand~.Methodsfortreatingsuchproblemsareatleastascomplexassolutionoftheboundaryvalueproblem.Theproblembecomesevenmorecomplexwhentheinequalityconstraintsareconsideredandpathconstraintsareintroduced.Consequently,analternativeapproachtovariationalmethodsisdesired. 154 PAGE 155 6{27 to 6{32 fromafunctionaloptmizationproblemtoanitedimensionalparameteroptimizationproblempresentsapracticalalternativetothevariationalapproachpresentedintheprevioussection.ThegeneralprocessforstateandcontrolparameterizationwasdescribedinChapter 2 .Basically,thetimeintervalissplitintoanitesetofsubintervalsandavectorofunknownparametersisformedusingthestateandcontrolvaluesatthesubintervalnodes.Dynamicconstraintsareappliedinapiecewisefashionthroughtheuseofnumericalmethodstoensurethatthestateandcontrolvaluessatisfythedynamicsatthesenodes.Pathconstraintscanalsobeappliedatthesenodes,aswell.Finally,acostfunctionisformedintermsoftheparameterizedstateandcontrolandsolutionsaredeterminedthroughtheuseofwellestablishedNLPsolvers.Thissectionappliesapseudospectraltranscriptionmethodtotheeectivesensingproblem.Pseudospectralmethodsparameterizethestateandcontrolusingabasisofglobalorthogonalpolynomialsasopposedtothetypicalpiecewiseapproximations[ 51 52 110 111 ].ThespecicmethodutilizedhereisdenotedtheGausspseudospectralmethod.ThismethodcollocatesthedynamicsatLegendreGausspointsusingabasisofLagrangepolynomialstointerpolatethestateandcontroltrajectories[ 51 ].Thekeybenetstopseudospectralmethodsarerelatedtotheireciency,accuracy,andeaseofimplementation[ 110 111 ].Themultitargeteectivesensingproblemistreatedthroughsubdivisionoftheproblemintoaseriesofphases.Eachphaserepresentsanonoverlappingtrajectorysegmentthatinvolvesdierentproblemspecicationssuchascost,endpointconstraints,orevendynamics.Thesesegmentsarelinkedbyasetofconditionsthatconstrainrelevantparametersacrosstheboundariesofsequentialphases.Here,thetrajectorysegmentsassociatedwithsensingeachtargetindividuallyaretreatedasseparatephases.Eachofthesephaseshasaninitialconditionassociatedwiththeendpointofthepreviousphase(ortheglobalinitialcondition)andterminalconstraintsassociatedwithsensinga 155 PAGE 156 6.2 .ThisaircraftcarriesadownwardpointingsensorasdescribedinSection 6.3 .Theobjectiveoftheexampleistosensetwotargetsinaspeciedsequenceandthentoreturntotheinitialposition.Simplevisibilityisconsideredsucienttomeetsensingeectivenessrequirements.ThisexampleconsiderstheenvironmentdepictedinFigure 62 ,whichhasanembeddedinertialreferenceframethatisorientedwiththexaxispointingintheNorthdirectionandtheyaxispointingintheEastdirection.Thetwotargetsarelocatedrelativetotheinertialframeby~pt1and~pt2,asshownbyEquations 6{56 and 6{57 .Thetargetsattheselocationsareorientedsuchthateachhasaunitnormalvectordescribedby^ni,shownbyEquation 6{58 .ThesetargetsareshowninFigure 62 asredcircles.~pt1=10;0005;0000T (6{58)Thestateinitialconditionsatt0=0aregivenbyEquations 6{59 to 6{67 .Thisinitialconditionrepresentstrimmedsteady,straightandlevelightatanaltitudeof5;000ftheadingdueNorth.Notethatthestatesarenondimensionalizedforscalingpurposes,as 156 PAGE 157 Environmentforoptimalsensingnumericalexample. describedpreviously. (6{60) (6{61) (6{63) (6{64) (6{65) (0)=0 (6{66) (6{67) Theproblemissetupinthreesequentialphases.Therstphasecontainsthetrajectorysegmentalongwhichtheaircraftmustmovefromtheinitialconditiontoapointwherethetargetlocatedby~pt1isvisible.Thesecondphasecontainsthetrajectorysegmentfromtheendpointoftherstphasetoapointwherethetargetlocatedby~pt2isvisible.Finally,thethirdphasecontainsthetrajectorysegmentfromtheendpointof 157 PAGE 158 6{29 and 6{30 ;however,theinequalityconstraintsthatreecttargetvisibilityareonlyenforcedfortherstandsecondphases.Further,thesevisibilityconstraintsaredenedrelativetotheappropriatetargetforeachphase.Anadditionalpathconstraintisimposedthatrestrictsaltitudetoremainconstantatph(0)=5000=lC.Thisconstraintisaddedtosimplifythecomputationofthevisibilityparameter,f.Generally,computationoffrequiresaccesstothepitchangle,,asdescribedbyEquations 6{21 and 6{22 .RecallthatthepitchangleisrecoveredfromthekinematicconstraintgivenasEquation 6{23 .Thisexpressionrequiresanumericalsolutionprocedurefornonzero.Theconstantaltitudeconstraintresultsinconstant=0overtheentiretrajectory.Assuch,canbecomputedanalyticallyasEquation 6{68 53 64 112 ].ThecostfunctionfortheNLPisspeciedforthetotaltimeelapsedattheendofthethirdphase.Anoptimalsolutionwasfoundthatachievedallobjectiveswithatrajectoryof62:0356secinduration.TheresultingtrajectoriesofthesixstateswhichdescribepositionandvelocityareplottedinFigure 63 .Thepositionrelativetotheinertialframeisdescribedby~pB=pxpyphTwhilethetotalvelocityisdescribedbyV.Thedirectionoftotalvelocityisspeciedbytheightpathangle,,andtheheadingangle.Thetrajectoriescorrespondingtotheangleofattack,,andtherollangle,',areshowninFigure 64 .Thesetrajectorieswouldserveastheinputreferencetrajectoriestoalowerlevelcontrollerforavehicletaskedwithexecutingthesolutiontrajectory. 158 PAGE 159 Solutionpositionandvelocitytrajectories. Figure64. Solutionand'referencetrajectories. ThetrajectoriesforthevisibilityparametersassociatedwiththerstandsecondtargetsareshowninFigures 65A and 65B ,respectively.ThemaximumrangeandFOVangleareindicatedbydashedlinesineachplot.Noticethattheseconstraintsarenotnecessarilyallactiveatthephaseboundaries.Theplotsshowthatthevisibilityboundsaresatisedforthersttargetattheendoftherstphaseandaresatisedforthesecondtargetattheendofthesecondphase.Finally,Figure 66 showsanoverheadviewofthetotaltrajectorywithprojectedsensorfootprintsshownbydashedcontoursattheendofeachofthersttwophases.ThegureshowsthatthefullsensorFOVandmotioncouplingwereconsideredandutilizedintheoptimalsolution. 159 PAGE 160 BFigure65. Solutionvisibilityparametertrajectoriesforeachtarget. Figure66. Aircrafttrajectoryandsensorfootprintovereachtarget. 160 PAGE 161 2 .Theconceptofmotionprimitiveswereusedtorepresentvehicledynamicsinahybridmodelingframework.ThepointtopointmotionplanningproblemwasthenformulatedasahierarchicaloptimizationproblemthatcontainedacombinatorialelementandacontinuousNLPelement.Thecombinatorialelementwasshowntogreatlycomplicatethesolutionprocedureforrealisticscenarios.Assuch,aspecialclassoffeasible,thoughsuboptimal,explicitsolutionswereidentiedasameansbywhichapproximatesolutionscanbecomputedeciently.Utilityofthesesolutionswasdemonstratedfortheobstaclefreecase;however,globalsolutionswereseentobelimitedwhenobstaclesareconsidered.Assuch,thesesolutionsaretailoredmoreforlocalplanningtasks,suchasthatrequiredforbranchextensioninrandomizedmethods.SeveralareasforfutureworkrelatedtothemethodsdescribedinChapter 2 canbeidentied.TheexamplesdiscussedinChapter 2 utilizedverysimpledynamicmodelsto 161 PAGE 162 2 ,butidenticationofthesesequenceshasnotbeentreatedhereforthegeneralcase.Additionally,considerationofmorecomplicatedsolutionsequencescouldadverselyaecttheeciencyofthesesolutions.Generalizationofthisapproachprovidesaninterestingdirectionforfutureresearch.TheconceptofrandomizedsamplingbasedmotionplanningwasintroducedinChapter 3 .ArandomizedplanningalgorithmwasdevelopedthatutilizedthesolutionmethodsintroducedinChapter 2 .Thisplannergeneratedfeasiblesolutionsbygrowingatreeofsubsolutionsthatsearchtheplanningspaceinaprobabilisticfashion.Asimulatedexampledemonstratedtheeciencyandeectivenessoftheapproach;however,severaldrawbackswereapparentaswell.First,solutionsaresuboptimalandnotnecessarilyrepeatableduetotherandomizednatureofthealgorithm.Theseissuesresultfromthetradeobetweenoptimalityofoversimpliedproblemsversusfeasible,butsuboptimal,solutionstocomplicatedproblems.Suchatradeoisoftenacceptableformanyapplications.Futureworkmightincludeaninvestigationintotheextenttowhichperformancecanbeimprovedbyapostprocessingtrajectoryrenement.Amaincriticismofrandomizedapproachesisthatactualimplementationofsuchalgorithmsrequiresalargenumberofdesignparameterstobeset.Whiletheseplannershavebeenshowntoperformwellinpractice,thisperformanceisoftendependentontheproperselectionofproblemspecicparametersandheuristics.Theprocessofselectingthemosteectivecombinationcanbelengthyandinvolvesasignicantdegreeoftrialanderror.Chapter 4 discussedageometricapproachtomodelingsensorvisibility.Theeectsofsensoroperationincloseproximitytotheenvironmentwerediscussedandageneralframeworkwasintroducedtoquantifytheseeectsasameasureofdataquality.The 162 PAGE 163 4 representsaninitialtreatmentofatopicthatshouldplayasignicantroleinfuturemissionsforautonomousvehicles.Numerouspotentialextensionstotheconceptofsensingeectivenesscanbeidentied.FutureworkmightincluderesearchintotheintegrationofaFOVbasedeectivenessmetricwithexistingdataqualitymetrics.Forexample,thecharacterizationofviewsthatprovidehighqualitydatarelatedtocertainfeaturescouldaidintargetrecognitiontasks.Additionally,alogicalnextstepinvolvestheinclusionoftemporaleectsintheformulationoftheeectivenessmetric.SuchaformulationcouldaccountforeectsrelatedtothevelocitywithwhichanobjectmovesthroughtheFOV,suchasmotionblur.ThesensorplanningtaskwasaddressedwitharandomizedplanningapproachinChapter 5 .Therandomizedplanningandsensoreectivenessconceptsfrompreviouschaptersareintegratedtoformanewrandomizedplanningapproachtosensingaseriesoftargetsinacloseproximityenvironment.Thismethodgeneratesecientsolutionstoadicultproblemthatexhibitscombinatorialanddynamicelementsthatarehighlycoupled.Suchsolutionsareenabledbythepropertythatasingletargetsensingtrajectoryoftensensesmultipletargetsasaresultofthemissionandenvironmentscale.TherandomizedapproachsuersfromsimilardrawbackstothosediscussedfortheplannerdevelopedinChapter 2 ;however,thesimulatedresultsarequitereasonableconsideringthedicultyoftheproblem.TheecientbutimperfectrandomizedapproachwascontrastedbyarigorousbutcumbersomeapplicationofoptimalcontroltothesensingprobleminChapter 6 .Thischapterderivednecessaryconditionsforanoptimalsolutiontoasimpliedversionoftheproblemwiththepurposeofdemonstratingthechallengesassociatedwithadoptingthissolutionmethod.Then,adirectmethodisimplementedtocomputeanumericalsolutiontothesimpliedproblem.Whilethisimplementationneglectssomeoftheissuesrelatedtosensinginacloseproximityenvironment,theresultsarepromising.Thereissignicantpotentialincombiningconceptsfromrandomizedplanningmethodswithdirectoptimal 163 PAGE 164 164 PAGE 165 [1] DeGarmo,M.,andNelson,G.,\ProspectiveUnmannedAerialVehicleOperationsintheFutureNationalAirspaceSystem,"4thAIAAAviationTechnology,Integration,andOperationsForum,Chicago,IL,September2004. 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