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WATKINS 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 Adam S. Watkins To my parents for their support, encouragement, and, most of all, love for a son who never calls nearly enough. To my brother for ahivx setting the bar so high that I have to stretch and for alv,i reminding me that, somn.Iiv, I need to graduate. To Jessica for making the last long years in Gainesville fun, exciting, and sometimes even more stressful than they should have been. To Allie for making sure that I get up every morning and put one foot in front of the other, if only to go for a short walk. ACKNOWLEDGMENTS I would like to thank my advisor Dr. Rick Lind for his guidance, criticisms, and motivation throughout my graduate school experience. I would like to express gratitude to my committee members for their patience and insight during my doctoral studies. I would also like to extend thanks to my lab fellow researchers in the Flight Controls Lab including AMi il !id Abdulrahim, Eric Branch, Ryan Causey, Daniel Grant, and Joe Kehoe. Thanks to M1,li I.!i Eric, and Ryan for dealing with my nearly constant disruptions of the workplace and generally sarcastic, downbeat attitude. Thanks go to Daniel "Tex" Grant for handling my verbal abuse so well and teaching me how to shoot a gun. Thanks to Joe Kehoe for his thousands of research related ideas, including the three that actually worked. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF TABLES ....................... ............. 8 LIST OF FIGURES .................................... 9 ABSTRACT . . . . . . . . . . 11 CHAPTER 1 INTRODUCTION ...................... .......... 13 1.1 Problem Statement ................... ........ 13 1.2 Motivation ...................... ........... 14 1.3 Historical Perspective .............................. 18 1.4 Contributions . . . . . . . . 22 2 SCENE RECONSTRUCTION FROM MONOCULAR VISION ........ 24 2.1 Introduction ...................... ........... 24 2.2 Structure from Motion Overview ................... ... 24 2.2.1 Pinhole Camera Model ................... ..... 25 2.2.2 FeaturePoint Tracking.... ......... ... ... .. .. 26 2.2.3 Epipolar Geometry .................. ........ .. 28 2.2.4 Euclidean Reconstruction ............... .. .. 30 2.3 Calculating Structure from Motion .............. . 31 2.3.1 EightPoint Algorithm ................ ... ... .. 31 2.3.2 Recursive Structure from Motion. .................... .. 32 2.4 Optical Flow State Estimation . ............ ... 34 2.5 Image Processing for Environment Mapping . . .. 35 3 ENVIRONMENT REPRESENTATION ........... .... . 37 3.1 Introduction ............... ............. .. 37 3.2 Environment Models ............... ........... ..39 3.2.1 Topological Maps .............. ....... ..39 3.2.2 Occupancy Maps ............... ......... ..39 3.2.3 Geometric Maps ............... .......... ..40 3.3 Dimensionality Reduction ............... ........ ..40 3.3.1 Principal Components Analysis ............. .. .. 41 3.3.2 Data Clustering ............... .......... ..43 3.3.3 Plane Fitting Algorithm .............. . .. 45 4 FILTERING FOR STATE ESTIMATION .......... 4.1 Introduction ...................... ........... 49 4.2 The Kalman Filter ............................. 49 4.2.1 The Extended Kalman Filter ................... ... .. 52 4.2.2 The Unscented Transform .................. 53 4.2.3 The Unscented Kalman Filter ............... . .. 54 4.2.4 The SquareRoot Unscented Kalman Filter . . 56 4.3 The Particle Filter ............... . . ... 58 4.4 The RaoBlackwellized Particle Filter ................... ... .. 60 5 SIMULTANEOUS LOCALIZATION AND MAP BUILDING . . ... 62 5.1 Introduction .................. ................ .. 62 5.2 Airborne SLAM Models .................. ......... .. 65 5.2.1 Vehicle M odel .................. ........... .. 66 5.2.2 Measurement Model. .................. ...... .. .. 67 5.3 Filtering for SLAM .............. . . . 68 5.3.1 RaoBlackwellized Particle Filter ............ .. .. 68 5.3.2 Unscented RaoBlackwellized Particle Filter . . 71 5.3.3 SquareRoot Unscented RaoBlackweillized Particle Filter . 73 5.4 Data Association ................ ............ .. 75 5.5 Map Management .................. ............ .. 76 6 TRAJECTORY PLANNING .............. . ..... 78 6.1 Introduction ............... . . .. 78 6.2 Direct Trajectory Planning Methods ................ .... .. 80 6.2.1 MixedInteger Linear Programming .... . . 80 6.2.2 Maneuver Automaton ............... .... .. 82 6.3 Decoupled Trajectory Planning .............. ...... 84 6.4 SamplingBased Path Planning .................. .... .. 85 6.4.1 ExpansiveSpaces Tree Planner ............. . 87 6.4.2 RapidlyExploring Random Tree Planner . . 88 6.5 The Dubins Paths .................. ............ .. 90 6.5.1 Extension to Three Dimensions ............. .. 93 6.5.2 Collision Detection ......... . . ... 94 6.5.2.1 Global path collision detection .. . . 94 6.5.2.2 Local path collision detection ..... . . 96 6.6 Environment Coverage Criteria ................ .. .. .. 97 6.7 Optimal Path Tracking ............... .......... .. 101 6.8 Trajectory Planning Algorithm .............. .. 102 7 DEMONSTRATIONS AND RESULTS ................ 105 7.1 Introduction . ............... ............ .. .. 105 7.2 Plane Fitting Results .................. ........... 105 7.2.1 Effects of Feature Point Noise ................ ... 106 7.2.2 Nonplanar Obstacles .................. .. ..... 108 7.3 SLAM Results ................... ... ... ..... 108 7.3.1 Simulation Environment .................. .. .. .. 109 7.3.2 Localization Result .................. ........ .. 111 7.3.3 State Estimation Error .................. ..... .. 113 7.3.4 Filter Consistency .................. ......... .. 113 7.3.5 Filter Efficiency .................. .......... .. 116 7.3.6 Example SLAM Result .................. ..... .. 117 7.4 Trajectory Planning Results .................. ..... .. 117 7.4.1 Environment Coverage .............. .... .. 117 7.4.2 Optimal Control Results .............. . .. .. 121 8 CONCLUSIONS .................. ............ .. .. 122 8.1 Conclusion ................. ............ .. .. 122 8.2 Future Directions ................ ............ 123 REFERENCES ... .. .. .. ... .. .. .. .. ... .. .. . . 124 BIOGRAPHICAL SKETCH .................. ............. 132 LIST OF TABLES 71 Efficiency comparison of the SLAM filters. .................. .... 117 Table page LIST OF FIGURES Figure 11 Map construction by an autonomous UAV. . . ..... 12 Examples of unmanned vehicles..... . . ..... 13 Examples of unmanned air vehicles.... . . ...... 14 Unmanned air vehicle mission profiles.. . . ..... 21 Pinhole camera model and perspective projection of a feature point onto the im age plane .. . . . . . .. . . .... 22 Featurepoint tracking result from an image sequence . . . 23 Epipolar geometry ....................... . . ..... 31 Environment representations....... . . ..... 32 2D Prinicipal Components Ain Ii; for a dataset with a linear trend . . 33 3D Principal Components Analysis for a planar dataset. Centroid splitting for iterative kmeans . .. Dimensionality reduction of feature point data. . Recursive state estimation . ......... The Unscented Transform . .......... Map building in the presence of noise . ... The SLAM estimation strategy . ....... Fixedwing aircraft model definitions . .... An overview of the trajectory planning process . An example Maneuver Automaton . ..... Probabilistic Roadmap planning in a twodimensional ExpansiveSpaces Tree planning . ...... RapidlyExploring Tree planning . ...... environment . . The coordinate system for calculating the paths in the Dubins set ED. . The Dubins set of paths for two configurations....... . ..... Examples of 3D Dubins paths ................ . ..... page 13 14 15 16 25 26 29 39 41 43 46 48 50 53 63 64 66 79 83 86 88 89 91 93 94 69 Collision detection for the global path. ............. ... 95 610 Interior to boundary test ............... ......... .. .. 96 611 Collision detection for the local path ................ ..... 98 612 Motion planning for coverage strategies ............... ... 99 613 Environmental coverage metric ............... ........ 100 71 Result of the plane fitting algorithm without image noise. . . .... 105 72 Angle of incidence constraint. ............... ..... .... 106 73 Results of the plane fitting algorithm in the presence of image noise. ...... .107 74 Result of the plane fitting algorithm when nonplanar obstacles are included. 109 75 The visionbased flight simulation environment for testing the SLAM algorithm. 110 76 Example localization result for the airborne SLAM simulation . ... 112 77 Average estimation error for the three SLAM filters and the nominal trajectory for 50 Monte Carlo runs. .................. .......... 114 78 Filter consistency results .................. ............ .. 116 79 An example result for the complete airborne SLAM solution using the URBPF 118 710 Example paths for environment coverage ................ ....... 119 711 Comparison of environment coverage results. ............... 120 712 Resultant trajectories from the path tracking optimization. . .... 121 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 VISIONBASED MAP BUILDING AND TRAJECTORY PLANNING TO ENABLE AUTONOMOUS FLIGHT THROUGH URBAN ENVIRONMENTS By Adam S. Watkins August 2007 ('!C in: Richard C. Lind, Jr. Major: Mechanical Engineering The desire to use Unmanned Air Vehicles (UAVs) in a variety of complex missions has motivated the need to increase the autonomous capabilities of these vehicles. This research presents autonomous visionbased mapping and trajectory planning strategies for a UAV navigating in an unknown urban environment. It is assumed that the vehicle's inertial position is unknown because GPS in unavailable due to environmental occlusions or jamming by hostile military assets. Therefore, the environment map is constructed from noisy sensor measurements taken at uncertain vehicle locations. Under these restrictions, map construction becomes a state estimation task known as the Simultaneous Localization and Mapping (SLAM) problem. Solutions to the SLAM problem endeavor to estimate the state of a vehicle relative to concurrently estimated environmental landmark locations. The presented work focuses specifically on SLAM for aircraft, denoted as airborne SLAM, where the vehicle is capable of six degree of freedom motion characterized by highly nonlinear equations of motion. The airborne SLAM problem is solved with a variety of filters based on the RaoBlackwellized particle filter. Additionally, the environment is represented as a set of geometric primitives that are fit to the threedimensional points reconstructed from gathered onboard imagery. The second half of this research builds on the mapping solution by addressing the problem of trajectory planning for optimal map construction. Optimality is defined in terms of maximizing environment coverage in minimum time. The planning process is decomposed into two phases of global navigation and local navigation. The global navigation strategy plans a coarse, collisionfree path through the environment to a goal location that will take the vehicle to previously unexplored or incompletely viewed territory. The local navigation strategy plans detailed, collisionfree paths within the currently sensed environment that maximize local coverage. The local path is converted to a trajectory by incorporating vehicle dynamics with an optimal control scheme which minimizes deviation from the path and final time. Simulation results are presented for the mapping and trajectory planning solutions. The SLAM solutions are investigated in terms of estimation performance, filter consistency, and computational efficiency. The trajectory planning method is shown to produce computationally efficient solutions that maximize environment coverage. CHAPTER 1 INTRODUCTION 1.1 Problem Statement The problem addressed in this research is the mapping of unknown, urban environments by an Unmanned Air Vehicle (UAV) equipped with a vision sensor. Map building is a vital task for autonomous vehicle navigation in that it provides information enabling vehicle localization, obstacle avoidance, and highlevel mission planning. The presented approach uses information gathered by an onboard camera to construct a threedimensional environment map while accounting for sensor noise and uncertainty in vehicle state estimates. Additionally, motion planning strategies are developed to generate trajectories which maximize environment coverage and avoid collisions with obstacles. The combined map building and motion planning algorithms yield a solution for autonomous flight through urban environments. The overall concept of this research is depicted in Figure 11 where a UAV navigating at low altitude uses sensor information to construct a threedimensional map and, subsequently, plan a trajectory. A B Figure 11. Map construction by an autonomous UAV. A) Missions envisioned for UAVs involve safe navigation in cluttered urban environments. B) Navigation in cluttered environments will require vehicles capable of autonomously building a map from sensor data and, subsequently, planning safe trajectories. 1.2 Motivation In recent years, autonomous vehicles have become increasingly ubiquitous. Unmanned vehicles are employ, 1 in a variety of situations where direct human involvement is infeasible. Such situations include tasks occurring in environments that are hazardous to humans, missions that humans are incapable of performing due to physical limitations, or assignments considered too mundane or tedious to be reliably performed by a human. Examples of autonomous vehicles currently in use, shown in Figure 12, include the Packbot EOD which assists in Explosive Ordnance Disposal (EOD) [1], the Mars Exploration Rovers (M\!lI ) Spirit and Opportunity used for planetary exploration [2], and the Roomba vacuum robot [3]. B Figure 12. Examples of unmanned vehicles. A) The JPL Mars Exploration Rover Spirit. B) The iRobot Packbot EOD. C) The iRobot Roomba. The unmanned vehicles depicted in Figure 12 exemplify the current level of autonomy reliably achieved by unmanned systems. The Packbot EOD requires constant guidance from a human operator and does not perform any truly autonomous tasks. The MER autonomously monitors its health in order to maintain safe operation and tracks vehicle attitude to aid in safe navigation over uneven terrain. Preloaded and communicated instructions, however, are executed by the MER after significant human analysis and interaction. The Roomba dip',1 a higher level of .ml.il..zii: by sensing its environment and updating the vehicle path accordingly. The floor cleaning task, however, is simplistic when compared to envisioned unmanned vehicle missions such as autonomous navigation of a car through traffic. The successful completion of complex missions requires technological advances that increase the level of autonomy currently di'.1 v, 1 by unmanned vehicles. A specific area of unmanned vehicles that have gained significant attention is UAVs. UAVs are currently used in a variety of reconnaissance, surveillance, and combat applications. Examples of UAVs, depicted in Figure 13, include the MQ1 Predator that has been used for reconnaissance missions and is capable of firing Hellfire missiles, the RQ5 Hunter designed for reconnaissance missions, and the Dragon Eye designated for reconnaissance, surveillance,and target acquisition [4]. The current UAV technology has a low level of .l, i I.o, .in,: requiring considerable human interaction to perform basic missions. Highly trained operators are necessary to remotely operate each individual unmanned aircraft. A B C Figure 13. Examples of unmanned air vehicles. A) The General Atomics Aeronautical Systems Inc. MQ1 Predator. B) The Northrop Grumman RQ5 Hunter. C) The AeroVironment Dragon Eye. Future missions envisioned for unmanned aircraft require a greater level of autonomy than is currently available in existing systems. Common autonomous behaviors exhibited by UAVs include waypoint navigation [5] and simple obstacle avoidance [6]. Such behaviors are wellsuited for relatively benign missions that occur at high altitudes and require little reaction to the surrounding environment as shown in Figure 14A. Advances in microprocessors, battery power, and sensor miniaturization, however, have produced smaller UAVs capable of flight through more complex environments. The ability of smaller UAVs to ., ..ressively maneuver through cluttered environments indicates that they are ideal platforms for completing lowaltitude missions, as depicted in Figure 14B, that occur outside of an operator's lineofsight. As vehicle missions require autonomous flight through increasingly elaborate, cluttered environments, a higher degree of environmental awareness is required. Therefore, two critical requirements for autonomous unmanned vehicles are the ability to construct a spatial representation of the sensed environment and, subsequently, plan safe trajectories through the vehicle's surroundings. A B Figure 14. Unmanned air vehicle mission profiles. A) Current missions occur at high altitude, therefore, not requiring rapid path planning and environmental awareness. B) Future missions will be low altitude assignments necessitating quick reaction to environmental obstacles. This dissertation concentrates on visionbased environmental sensing. Computer vision has been identified as an attractive environmental sensor for small UAVs due to the large amount of information that can be gathered from a single lightweight, lowpower camera. The sensor selection is a result of the decrease in vehicle size which severely constrains the available p liload both in terms of size and weight along with power consumption. Therefore, autonomous behaviours for UAVs must be designed within the framework of visionbased environmental sensing. In particular, two critical capabilities are investigated for creating a truly autonomous UAV: visionbased map building and computationallyefficient trajectory planning. The first capability is a visionbased map building task in which the vehicle gathers information concerning its surroundings with an onboard camera and constructs a spatial representation of the environment. For the past two decades, robotic mapping has been an actively researched topic because it is regarded as the first step toward absolute autonomy [7]. Robotic mapping allows the vehicle to localize itself with respect to the constructed map for navigating in GPSdenied environments. This capability is critical for military operations where GPS can be jammed or cluttered environments preclude the use of GPS due to occlusion. Additionally, robotic mapping is vitally important to safe, timeefficient navigation through unknown environments. Finding the shortest path to a goal location using purely reactive control methods, where the robot selects control actions based only on the current sensor measurements, is a trialanderror process in unknown environments with no provably optimal solution. By generating an obstacle map, a robot can retain knowledge of its surroundings and discriminate between obstructed and unobstructed paths. As knowledge of the environment grows, the task of traversing to a goal location can be accomplished in a timeefficient manner with the planning of complete, obstaclefree paths. The second capability developed in this dissertation is a computationally efficient trajectory planning routine which maximizes environment coverage to minimize the time to construct a complete environment map. In implementation, the method must be computationally efficient since the planning process occurs incrementally as new information is gathered. The rapidity of the planning process allows for path updates as knowledge of the environment increases which is critical for safe navigation in unknown environments. Additionally, vehicle dynamics are incorporated into the planning process in order to generate trajectories. This task is vital for aircraft which may not be capable of tracking every kinematically feasible path. Most ground vehicles are able to stop their forward motion and reverse direction which enables the tracking of any feasible path; however, aircraft motion is significantly more complex and dynamics must be considered to ensure safe navigation. This dissertation is motivated by the need to increase the level of autonomy present in UAV technology. To obtain this goal, a map building architecture is developed which constructs an environment representation from noisy sensor measurements and plans safe trajectories through the environment that ensure map completeness. 1.3 Historical Perspective The two main topics discussed in this dissertation, map building and trajectory planning for autonomous vehicles, are rich fields of study for which much research has been conducted. The presented work is an extension of previous research efforts which focuses on six degree of freedom vehicle motion, computational efficiency, and visionbased sensing. The robotic map building task described in this dissertation is an extension of solutions to the wellknown Simultaneous Localization and Mapping (SLAM) problem. The SLAM problem was first introduced in the seminal papers by Smith, Self, and C'!. i,, i [8, 9] which described a methodology for generating stochastic obstacle maps from a series of uncertain spatial relationships. In regards to unmanned vehicles, this work describes the SLAM problem: the construction of an environment map from noisy sensor measurements taken from uncertain vehicle positions while concurrently localizing the vehicle with respect to the learned map. The solution to the SLAM problem presented by Smith, et al. has since been extended to produce SLAM algorithms for numerous sensors, vehicles, and environment types. This dissertation focuses on SLAM for visionequipped aircraft navigating in highlystructured, urban environments. Visionbased SLAM refers to instances when vision is the only sensor available for measuring the environment. Historically, solutions to the SLAM problem have focused on vehicles equipped with sensors such as laser range finders and sonar [10, 11]. The dense depth information available from such sensors is not equivalent to the sparse depth information afforded a vehicle with a vision sensor. The use of vision in SLAM typically involves the identification and differentiation of landmarks, but relies on other sensors to measure the structure of the scene [12, 13]. In an example of purely visionbased SLAM, Davison uses vision to detect locally planar features which are incorporated as landmarks into the SLAM framework [1416]. The implementation emphasizes accurately estimating camera egomotion and is not concerned with generating an environment model. Consequently, the constructed map is a sparse approximation of the environment. The approach in this dissertation shifts the focus from accurate egomotion estimation to building a complete map of the environment for the purpose of efficiently planning collisionfree vehicle motions. Though the SLAM problem has been a highly researched topic in the ground vehicle community for the past two decades, very little research has been performed for aircraft. A few ground vehicle implementations have considered undulating terrain which requires the assumption of nonplanar motion [17]. Kim and Sukkarieh implemented SLAM on a UAV equipped with a vision sensor that locates artificial landmarks of known size [18, 19]. A bearingsonly SLAM approach for a Micro Air Vehicle (\!A V) equipped with a camera and an Inertial Measurement Unit (IMU) operating in a forest environment was introduced by Langelaan [2022]. Current approaches to airborne SLAM focus primarily on augmenting inertial navigation systems (INS) to reduce drift inherent in the calculated state estimates. These approaches do not address the difficulties associated with building a complete environment map as is addressed in this dissertation. Solutions to the SLAM problem require repeated observations of environmental landmarks in order to build a stochastic obstacle map. Therefore, information regarding each landmark must be stored and observations of a landmark must be correlated with past observations. The correlation task is known as data association and incorrect associations often lead to catastrophic failures of the SLAM estimation process. For these reasons, map representation is a vital aspect of any SLAM algorithm. Traditionally, SLAM algorithms define landmarks as threedimensional points of interest in the environment [10]. The practice of using point landmarks often yields environment models containing millions of features [23]. Such environment models of large size are infeasible due to the data storage requirements and computational power required to incorporate large numbers of landmarks into the SLAM calculation. Additionally, the data association problem becomes intractable because the relative measurements to landmarks cannot be correlated as a result of the large number of possible associations combined with uncertainty in vehicle location. Several SLAM formulations have constructed geometric environment maps for highly structured environment maps by fitting geometric primitives to the gathered pointwise data [24, 25]. The benefits of geometric primitives include the reduction of data required to store an environment map and landmarks that are more easily distinguished because they contain structural information. This dissertation employs a similar dimensionality reduction step in order to construct a complete environment map consisting of planar features. In order to autonomously and rapidly construct an environment map, this dissertation presents a trajectory planning algorithm. Ideally, the trajectory planning problem would be posed as an optimal control problem in order to find a globally optimal solution. An optimal solution, however, is computationally intractable because the problem is nonconvex and occurs in a highdimensional space. For this reason, several approximations to the full optimal control solution have been proposed which included Mixed Integer Linear Programming (\! IL.P) and Maneuver Automatons ( \ As). The MILP solution poses the full nonlinear optimization as a linear optimization [90, 91]. Therefore, the solution is only capable of handling a linearized dynamic aircraft model. The linearization restriction severely limits the efficacy of the MILP solution due to the reduction of available maneuvers. A linear model cannot properly characterize the  ressive maneuvers necessary to navigate in a cluttered environment. Additionally, the MILP solution cannot prove optimality because the problem is nonconvex and the optimization must be implemented in a Receding Horizon (RH) format to be computationally efficient. The MA is a method which employs a discrete representation of the vehicle dynamics in order to plan trajectories [3538]. The dynamics of the vehicle are discretized into a set of trim trajectories, or steadystate motions, and maneuvers that connect the trim states. The MA planning process is decomposed into a combinatorial problem which searches for the best maneuver sequence and an optimization that finds the optimal time durations for the trim trajectories. Unlike, MILP the discretized dynamics are based on a nonlinear model and can incorporate .I: riessive maneuvers. The MA solution, however, can become computationally intractable when solving for large trajectories requiring a long maneuver sequence. This issue is exacerbated if the discretization is not restricted to a small set of vehicle motions or if obstacles are considered. Therefore, the MA is combined with samplingbased planning methods to produce a solution in which optimality cannot be guaranteed. This dissertation employs a decoupled trajectoryplanning strategy [40, 41] in which a feasible collisionfree path is planned and optimal control is employ, 1 to track the path in minimal time. The path is subject to kinematic constraints and incorporates metrics for selecting paths that provide maximum coverage with respect to other paths generated during the planning process. The decoupled approach allows for the use of the complete nonlinear aircraft model; however, optimality in terms of coverage is not guaranteed since the coverage criteria is only included in the path planning task. The path to be tracked is generated with a samplingbased planning strategy [26]. Samplingbased algorithms were introduced by a methodology called Probabilistic Roadmaps (PRMs) [27]. PRMs construct a graph that captures the connectivity of the configuration space of a robot by randomly sampling configurations and connecting them to the existing map with kinematically feasible paths. Once a PRM is constructed for a given environment, it can be repeatedly used to find paths from a given initial configuration to a desired goal configuration. Navigation in unknown environments, however, does not require full exploration of the configuration space since new sensor information must be incrementally incorporated into the planning process. Therefore, extensions of PRMs such as the ExpansiveSpaces Trees (ESTs) [2830] and the RapidlyExploring Random Trees (RRTs) [31, 32] planners are emplo, l The EST and RRT planning strategies are more efficient since they are designed to find feasible paths between two specific configurations, rather than provide a general planning solution like the PRM. This dissertation employs a variant of the RRT to efficiently plan collisionfree paths through the environment. An abundance of research has been performed in the areas of SLAM and trajectory planning. This dissertation extends previous research efforts in both fields to provide a methodology for autonomous map building in highlystructured environments. 1.4 Contributions The main contribution of this work is a complete solution for autonomous map building by a UAV in an unknown, urban environment. The solution incrementally constructs a map from noisy, relative sensor measurements without knowledge of the vehicle's location. The map is used as feedback to generate collisionfree trajectories that maximize environment coverage. In the process of developing the main contribution of this work, the following contributions were made: * A plane fitting algorithm is developed to reduce the dimensionality of gathered threedimensional, pointwise sensor data. * A method for incorporating planar features into the SLAM algorithm is developed. * A new SLAM filter based on the RaoBlackwellized particle filter and the Square Root Unscented Kalman filter is developed. * Three SLAM filters are investigated in relation to the airborne SLAM problem. * A method for for rapid path planning in an unknown environment based on local and global navigation tasks is developed. * An environment coverage criteria is developed which does not require discretization of the environment. * An optimization problem is posed for tracking the planned path in minimal time. CHAPTER 2 SCENE RECONSTRUCTION FROM MONOCULAR VISION 2.1 Introduction Computer vision is an attractive sensor option for autonomous vehicle navigation tasks due to the wealth of information that can be inferred from digital imagery [42]. In addition, the lowcost, small p loadd requirements, and passive nature of vision has made it the primary sensor for many small Unmanned Air Vehicles (UAVs) [4]. This study assumes that computer vision is the only available onboard sensor capable of measuring the surrounding environment. Mapping an unknown environment with a moving, monocular camera requires that threedimensional information be inferred from a sequence of twodimensional images. This process of threedimensional reconstruction from imagery is an actively researched area in the image processing community commonly referred to as Structure from Motion (SFM) [43, 44]. This chapter presents an overview of SFM in Section 2.2. Specific solutions to the SFM problem are then discussed in Sections 2.3.1 and 2.3.2. Section 2.4 describes a vehicle state estimation task similar to SFM which calculates vehicle velocity states. Subsequently, C!i plters 3 and 5 describe how the visionbased state estimates and environmental structure information is employ, 1 in the robotic mapping mission. 2.2 Structure from Motion Overview The class of algorithms known as SFM endeavor to reconstruct the pose of a moving camera as well as the threedimensional structure of its environment. Calculating the motion of the camera typically begins with an operation known as featurepoint tracking, as described in Section 2.2.2, which calculates correspondences between points of interest in successive images. Section 2.2.3 demonstrates that by exploiting epipolar geometry between correspondence pairs the relative rotation and translation between the two camera views can be estimated. The calculated motion of the camera can then be used to triangulate the threedimensional position of the tracked feature points as shown in Section 2.2.4. 2.2.1 Pinhole Camera Model This chapter derives image processing algorithms which assume a pinhole camera model. The pinhole camera is an idealized model of camera function in which all light converges on an infinitesimally small hole that represents a camera lens with zero aperture. In the frontal pinhole imaging model, as shown in Figure 21, the twodimensional projection f = [p, v]T of a threedimensional point p = [q 92, 913]T is the intersection of the ray emanating from the camera optical center ending at point p and the image plane. The image plane is located at a distance f, called the focal length. The camera coordinate frame C is located at the camera optical center with coordinates o1 and 82 oriented parallel to the image plane in the negative vertical and horizontal directions, respectively. The coordinate 83 is oriented perpendicular to the image plane and denotes the camera axis. C11 3 9f Pf3 7 r/2 ... 173 p Figure 21. Pinhole camera model and perspective projection of a feature point onto the image plane. For the pinhole camera model, perspective projection of a threedimensional point can be written in terms of the camera focal length [P f 7 (21) V T3 T2 Equation 21 shows that only the depth of the threedimensional feature point is required to resolve the location of the point from the image projection. 2.2.2 FeaturePoint Tracking Featurepoint tracking is a wellestablished image processing operation and is commonly a prerequisite for solving the SFM problem. Essentially, a featurepoint tracking algorithm tracks neighborhoods of pixels, referred to as feature points, through a sequence of images. Feature points are defined as any distinguishable image region that can be reliably located within an image which are commonly corners and edges of objects. An example of tracked feature points are shown in Figure 22. A B Figure 22. Featurepoint tracking result from an image sequence. A) Frame 1. B) Frame 10. The LucasKanade featurepoint tracker [45] is a standard algorithm for small baseline camera motion. The algorithm solves for the displacement of all feature points in an image where the displacement of a feature point is measured in the horizontal and vertical image coordinates denoted by p and v, respectively. For a series of two images, the displacement of a feature point, d = [d,, d,]T, can be described as the translation of a group of image pixels as follows I1(f)w = I2(f + d)lw (22) where Ii and 12 are the image intensity values for two images taken from infinitesimally close vantage points, W is a window that defines the neighborhood of pixels that comprise the feature point, and f = [p, v]T is the center of the feature window. The solution to Equation 22 can be formulated as a leastsquares minimization of the change in image intensity values between feature point windows in two consecutive images as given by d =argmin [i(f) 12(f +d) (23) d w The firstorder Taylor series expansion of 12(f + d) about f is 2(f + d) I2(f) + 12 612 (24) The firstorder approximation of the Taylor series can be substituted into Equation 23 to produce d arglin I(f) 12(f) 6I2 I2 (25) dw r i 6 By differentiating the sum in Equation 25 with respect to feature point displacement the solution to the linear leastsquares estimate of feature point displacement can be written as follows 2 T2 [ 1(f) I2(f) 61 ]] 0 (26) w d, which can be rewritten in the form d G'b (27) where G is a matrix encoding the twodimensional gradient information of the second image in the feature point window Y ( 6I2 Y2612 s612 G= W w (28) YW2 WI12 y (62 z6P SV 6v WW and b is a matrix representing the difference between the two images E ~ (Ii(f) 12(f)) b 6 w (29) E (Ii(f) 12(f)) W This form of the LucasKanade feature tracker suffers from two weakness. The first is its inability to track features for large camera motions. This is a critical problem for image sequences taken from vehicles with fast dynamics in comparison to the camera frame rate. The second weakness is that each calculated feature location resides at discrete pixel values. The discretization of the feature locations makes the tracking operation very susceptible to noise. Two alterations to the above algorithm can be implemented to overcome the aforementioned issues. First, a multiresolution tracking step is added to aid in the tracking of larger feature point displacements. Second, an iterative step is added to calculate the location of the feature points to subpixel accuracy. These modifications significantly increase the performance of the algorithm for images streams indicative of vehicle navigation [46]. 2.2.3 Epipolar Geometry The most basic form of SFM calculates camera motion between two sequential image frames and concatenates the solution with previous motion estimates. This calculation is commonly accomplished by exploiting the epipolar constraint, depicted in Figure 23, which is a geometric constraint relating camera motion to the image projections of a tracked feature point. Epipolar geometry dictates that there is a plane which is defined by a three dimensional feature point p and two camera coordinate frames C1 and C2 from which the point is visible. The epipolar plane intersects both image planes at the epipolar lines 11 and 12 which contain the image projections fi and f2 of the feature point p and the epipoles el and e2. cl R, T) c Figure 23. Epipolar geometry. The epipolar constraint describes the geometric relationship between camera motion and the projection of a feature point onto the image plane. The epipolar constraint relates the image projections, fl and f2, of the threedimensional point, p, to the relative position, T, and orientation, R, of the two camera locations. Since these three vectors are coplanar their scalar triple product is zero as follows f2(T x Rf) 0 (210) The relative position and orientation of the camera are typically encoded as a single entity known as the essential matrix, EE IR33 ETxR (211) The core calculation of the SFM algorithm is estimation of the essential matrix using a set of tracked feature points and the epipolar constraint given by Equation 210. A method for esimating the essential matrix is described in Section 2.3.1. Once the essential matrix is estimated, it can be decomposed into the relative rotation and translation of the camera. The relative translation, however, can only be estimated to a scalefactor since Equation 210 is homogeneous with respect to the essential matrix (Equation 211) and is not modified by multiplying the essential matrix by a nonzero constant. Estimation to a scalefactor is an inherent ambiguity in the SFM problem stemming from the inability to distinguish between the motion through one environment and twice the motion through a scene twice as large, but two times further away. 2.2.4 Euclidean Reconstruction Once camera motion is known, the final step in the SFM algorithm involves estimating the threedimensional location of the tracked twodimensional feature points. The calculation of pointwise scene structure is accomplished by triangulating the correlated feature points. Recall that the threedimensional location of a feature point is denoted p = [1, /92, 93]T. Therefore, the triangulation step can be formulated by relating the unknown depths of the ith feature point, Im and qi,, measured with respect to two different camera poses, Cm and Cn, yielding T,,f = 9,mRfR + 7T (212) where 7 is the unknown translational scalefactor and f, denotes the ith feature point projected onto the image plane of the mth camera pose. Since ],m and ]3, represent the same depth measured with respect to different camera frames Equation 212 can be rewritten in terms of a single unknown depth as follows Tmf x Rf; + f xT 0 (213) Equation 213 can be solved for j correspondence pairs simultaneously by expressing the problem in the form Mr= 0 (214) where y is a vector of the unknown feature point depths and unknown translational scale factor S[Tru1 m/ 2 .. (215) and M is the matrix f xRf 0 0 0 f xT 0 f x Rf 0 0 f xT (216) M (216) 0 0 "*. 0 0 0 0 f x Rf' f xT The solution of Equation 214 provides the depth of the tracked feature points. Therefore, this section has outlined the solution to the SFM problem from twodimensional image data to estimated camera motion and threedimensional Euclidean scene structure. 2.3 Calculating Structure from Motion This section presents two methods for calculating SFM. The first method is the eightpoint algorithm which requires only two camera images and a minimum of eight tracked feature points. The second method is a recursive algorithm which employs a Kalman filter in order to incorporate vehicle dynamics into the motion estimation process. 2.3.1 EightPoint Algorithm The eightpoint algorithm introduced by LonguetHT.ii:i, [47] calculates the rotation and translation between two images given a minimum set of eight tracked feature points. Larger numbers of tracked feature points can be used in the calculation by solving a linear leastsquares minimization problem. The inclusion of additional tracked points has been shown to improve the results of the algorithm. The eightpoint algorithm exploits the epipolar constraint by solving a linear system of equations in the form Ae = 0 (217) where e E R9x1 is a stacked version of the essential matrix E and A E RIx9 is constructed from the twodimensional correspondence pairs. Each row of A is constructed by calculating the Kronecker product of the ith feature point correspondence. The ith feature point correspondence is two feature points correlated between two images denoted by f, = [p, vi, 1]T and fA = [p, v, 1]T. The Kronecker product of the correspondence pair is equal to the ith row of A as follows A(i,:) [ tu tv, P, n4P', V4j, V4, P' 1] (2 18) The essential matrix can be recovered from A by calculating its singular value decomposition (SVD) A UA AV" (219) The ninth column of VA can be unstacked to recover the essential matrix. Once the essential matrix is found, camera rotation and translation can be extracted as seen by Equation 211. It should be noted that this presentation of the eightpoint algorithm is highly sensitive to noise and requires a normalizing step as presented by Har'!, [48]. 2.3.2 Recursive Structure from Motion The estimates of camera egomotion produced by twoframe SFM algorithms, such as the eightpoint algorithm, are subject to systematic, correlated errors since the current calculation is appended to past reconstructions. Several batch processing algorithms have been presented to eliminate these additive errors [49]. These algorithms, however, are not suited for trajectory generation because they incur a delay from the data gathering process and cannot be implemented in realtime. For causal, realtime SFM a Kalman filter can be employ, to recursively calculate structure and motion estimates [50]. The standard recursive SFM algorithms gradually refine SFM state estimates by incorporating new data [51]. Alternatively, batch recursive algorithms create a series of intermediate reconstructions which are fused into a final reconstruction [52]. Both recursive techniques show improved results over nonrecursive techniques, however, no single algorithm works well in all situations [44]. The problems specific to SFM reconstruction for a UAV are identified by Prazenica et al. [46]. These issues include the loss of feature tracking due to .,..ressive vehicle motion, unreliable reconstructions due to small baselines, and the wellknown scalefactor ambiguity associated with SFM. The SFM approach proposed by Prazenica mitigates errors inherent in SFM by incorporating vehicle dynamics into the Kalman filter. This approach adds robustness to the SFM calculation by producing state estimates even when the SFM calculation fails. The scalefactor can be resolved by a gradual refinement of estimates made by incorporating information from the physical model of the vehicle [53, 54]. The Kalman filtering approach defines both a state transition model g(.) and a measurement model h(.) written as Xk = (Xk, Uk1,K) +qk1, qk1 ~A (0, q) (220) Yk = (xk, K)+rk, rk ~ '(0, Er) where x denotes the complete state vector including vehicle states and threedimensional feature point locations. The vector y denotes the projection of the feature points onto the image plane, u represents the control inputs to the vehicle equations of motion, K is the camera pose relative to the bodyfixed coordinate system, and the additive noise parameters q and r are zeromean Gaussian random vectors with covariances Eq and Er, respectively. The notation a ~ p(b) denotes that random variable a is drawn from a probability distribution defined by p(b). Therefore, f(.) is a nonlinear function that updates the vehicle states and estimated feature point locations according to the previous system state and vehicle control input. The nonlinear function h(.) predicts the projection of the threedimensional feature points onto the image plane given the estimated inertial feature point locations and the camera pose. Consequently, the measurement model is the epipolar constraint (Section 2.2.3). The filtering approach solves the SFM problem by estimating the state of the model given by Equation 220. The estimation process is computed with an Extended Kalman Filter (EKF). The EKF first predicts the state and covariance of the system according to X/]k1 f= (xkllkl, Uk_I, m) (221) Pklk1 = FkPkllklFT + Eq where xklk1 denotes a state estimate at time k given measurements up to time k 1, Pkk1 is the covariance matrix of the state without a measurement update, and Fk is the linearized form of the state transition function g(.). Next the optimal Kalman gain K is calculated according to Sk = HkPklk1iH + r (223) Kk Pklk1HSk 1 (224) where Hk is the linearized form of the measurement model h(.). The final step in the filtering process is to update the state and covariance of the system by incorporating the sensor measurement as follows Xkk Xkk1 + Kk(yk hxk1, k )) (225) Fk I KkHk (226) Pkk rkPklk lr + KkErK[ (227) The updated state estimate xkk gives both structure and motion for the current time step. 2.4 Optical Flow State Estimation Several research efforts have pursed optical flow state estimation for UAVs [5557]. Optical flow is similar to feature tracking with the distinction that displacement is calculated at a fixed location as opposed to tracking a specific image feature through an image stream. Additionally, optical flow refers to feature point velocity as opposed to displacement, but for a given fixed frame rate the difference in calculation is trivial. For optical flow state estimation, the focal plane velocity of the feature point, approximated by the optical flow, can be obtained by differentiating Equation 21. The result is an equation for the focal plane velocity of a feature point in terms of its (222) threedimensional location and relative velocity with respect to the camera as follows S f 13 0 i (22 2 (228) S T3 0 r/3 r/2 The optical flow of the feature points is a known quantity determined in the feature tracking algorithm that is an integral part of the SFM calculation. The equations of motion of an aircraft can be coupled with Equation 228 in order to express optical flow in terms of aircraft states, as given by v S [ 0 1 v (1 +/2) w = f q3 q3 (229) S/ 1 0 / uV (1+ V2) p q where {u, v, w} are aircraft linear velocities, {p, q, r} are aircraft angular velocities, and 73 is the unknown depth of the feature point. Various solutions to Equation 229 have been proposed using optimization routines. The resulting solutions have been shown to produce accurate estimates of aircraft linear and angular velocities though the total velocity of the aircraft must be known to account for the scale ambiguity that also affects SFM. 2.5 Image Processing for Environment Mapping This chapter has presented several key image processing results suitable for aiding UAV navigation. The described SFM algorithms are suitable algorithms for providing information regarding the threedimensional structure of the environment. This information will be used in Chapter 3 to generate a geometric representation of the environment. Furthermore, both SFM and the optical flow approach provide vehicle state information that is subject to drift over time. In C'!i pter 5 this information will be fused with onboard state estimates and the external environment measurements in a Simultaneous Localization and Mapping (SLAM) algorithm to reduce drift in vehicle state estimates and concurrently decrease map errors. CHAPTER 3 ENVIRONMENT REPRESENTATION 3.1 Introduction The proposed goal of autonomously generating an environment map requires a vehicle to sense its surroundings and infer meaningful information concerning scene structure from sensor measurements. Conventional sensors commonly used for measuring a vehicle's surroundings include laser range finders, sonar, and cameras. The three dimensional environmental information provided by such sensors is pointwise, relative range information. An obstacle map defined by a cloud of threedimensional feature points is not amenable to motion planning algorithms [26] and is a poor map in terms of data storage, landmark definition, and enabling highlevel mission planning. Motion planning algorithms typically used for robotic navigation presuppose a known obstacle map [58]. The obstacle map enables the identification of safe routes through a vehicle's surroundings by limiting the set of admissible positions a vehicle can achieve. A pointwise representation of obstacle geometry is ambiguous in terms of defining safe regions within the environment. A more concise representation of the environment, such as geometric objects, in which complete obstacles can be accurately described is better suited to the process of motion planning. Mapping largescale environments requires that many sensor measurements be taken to accurately define scene structure. Storing individual sensor measurements during a mapping operation can quickly lead to a computationally intractable problem [59]. Lowerdimensional representations of sensor data can greatly reduce the number of parameters needed to define an obstacle map. For example, a single plane can reasonably represent thousands of threedimensional points using four parameters including three to define a normal direction and one to define the plane's distance from the origin along that normal direction. For highly structured environments, using geometric primitives such as planes to model the sensor data can greatly reduce the amount of data required to generate a spatial model. In addition, dimensionality reduction can filter the effects of sensor noise by creating a smooth representation of noisy data. Robotic navigation can be divided into different methodologies depending on the level of knowledge afforded the vehicle concerning its surroundings and relative position within the environment. The most general case requires that the vehicle estimate both a map of the environment and its location within that map simultaneously [10, 60, 61]. This case is known as the Simultaneous Localization and Mapping (SLAM) problem as discussed in detail in C'! lpter 5. The SLAM framework requires that distinguishable and reobservable landmarks be present in the environment. Landmarks allow the vehicle to ascertain its relative location to the environment through sensing operations. The only discernible characteristic of a 3D feature point is its location in the environment which makes 3D points a poor choice for landmarks. A geometric landmark, however, can provide additional information in terms of orientation and size. Additionally, feature points that are visible from a given vantage point are not necessarily visible from another location due to occlusion and lighting variations. Geometric maps define landmarks that are robustly observable with respect to camera pose. M 11 : missions envisioned for autonomous vehicles require human interaction in terms of highlevel mission planning. Surveilling targets in an environment such as a specific building or roadway depends upon the ability to properly designate the known objective. Similarly, reconnaissance of an area of interest in the environment requires that the region be clearly specified. Such highlevel decisions will be made by a human operator who must designate the targets or regions of interest in some manner. In this capacity, a spatial model is advantageous as it provides an intuitive interface for target selections as opposed to a pointwise model which is an ambiguous representation that requires further interpretation of the data. The need to enable motion pl1 iii i construct compact map representations, define distinguishable landmarks, and assist highlevel mission planning motivates the generation of reduceddimensionality geometric models of the environment. The following chapter describes the chosen map representation and the method for dimensionality reduction. 3.2 Environment Models Mapping operations are typically categorized by how the environment is modeled. Maps commonly used for robotic navigation can be classified as either topological, occupancy grid, or geometric representations as depicted in Figure 31. A B C Figure 31. Environment representations. A) Topographic maps depict spatial connectivity by storing locations of interest and connecting paths. B) Occupancy maps record the probability that a discrete location contains an obstacle. C) Geometric maps define obstacles as geometric primitives. 3.2.1 Topological Maps Topological maps model the environment as a series of nodes and connecting arcs [6264] as shown in Figure 31A. Typically, the nodes consist of places of interest such as the intersection of corridors and arcs provide connectivity information about the nodes, to i 1 navigation paths between nodes. Topological models are not necessarily spatial models as the nodes can represent various states of the vehicle. Such maps, however, rely on spatial models for their construction since the arcs need to define maneuvers that avoid obstacles. This fact blurs the distinction between topological and geometric maps since topological maps rely on geometric information. 3.2.2 Occupancy Maps Occupancy grid representations discretize the environment and each generated cell is assigned a value that corresponds to the likelihood that the cell is occupied [65, 66]. Though occupancy grid approaches are easy to construct they suffer from the amount of space required to handle an environment that is discretized finely enough to capture the necessary amount of detail. For large, threedimensional environments this lack of compactness causes this representation to be intractable in terms of the data storage requirements for the UAV mapping problem. 3.2.3 Geometric Maps Geometric representations attempt to fit geometric primitives to sensor data. In two dimensions this fit is accomplished by replacing point data with line segments [25, 67]. Threedimensional maps generated from dense depth information typically use triangle meshes to represent the environment [60, 68]. Examples exist for representing structured, threedimensional environments by a less general set of geometric primitives such as distinct planar surfaces [24, 69, 70]. This dissertation employs a geometric environment map because the properties are beneficial for both the SLAM (C'!i plter 5) and trajectory planning (C'!i plter 6) tasks. The geometric map provides a compact representation that reduces the computational burden for both processes. Additionally, a geometric description of the landmarks makes them more observable than point landmarks or visually defined landmarks. Lastly, the geometric landmarks are amenable to collision checking algorithms required for planning safe trajectories and are an intuitive representation for aiding in higherlevel mission planning concerns. The geometric map is constructed from the pointwise observations of the environment provided by SFM. This task is accomplished via a dimensionality reduction process. 3.3 Dimensionality Reduction Dimensionality reduction refers to the process of representing a dataset with a smaller set of parameters while preserving the information conveyed by the dataset. For example, a set of m twodimensional points that are approximately linear can be represented by four parameters for the endpoints of the linear fit. For large values of m, the dimensionally reduced representation vastly decreases the storage requirements for a given dataset. The dimensionality reduction step in this dissertation is achieved by fitting planes to the threedimensional points generated by the SFM algorithm. 3.3.1 Principal Components Analysis The plane fitting task is accomplished using Principal Components Analysis (PCA) [71], a statistical linear transformation used for reducing the dimensionality of a dataset. PCA finds the directions in a dataset with the most variation, as shown in Figure 32 in order to capture the variability of the data with fewer parameters. These directions are also the eigenvectors that correspond to the largest eigenvalues of the data's covariance matrix. Therefore, PCA can be employ, ,1 to transform a dataset to a new basis where the first coordinate, called the first principal component, defines the direction of maximum variance. All subsequent directions defined by the basis are orthogonal to all previous directions and define the remaining direction of greatest variance. 14 1 Data points 12" S. 2c covariance 10 . P  2 P 8 4 2 .. .. . * 2 dd, ' 4 1i 10 5 0 5 10 15 Figure 32. 2D Prinicipal Components Analysis for a dataset with a linear trend. The principal components are labeled p,, for the nth principal component and the 2o covariance of the dataset is shown. The principal components can be calculated by finding the eigenvectors of the data's covariance matrix or, more succinctly, by performing a Singular Value Decomposition (SVD). The calculation begins by centering the dataset X by subtracting its mean t,. Because the goal of the calculation is to find directions of greatest variance, a nonzero mean will produce erroneous results. The SVD of the meansubtracted dataset X is given by Xs UEVT (31) where U and V define matrices of orthonormal vectors and E defines a diagonal matrix of singular values assumed to be ordered from largest to smallest. The columns of V define the principal components of X with the column corresponding to largest singular value designating the first principal component. The result is typically ordered so that the first principal component is the first entry of E with subsequent components listed in a descending order. The dataset can be projected on to the basis formed by the principal components according to Y XV (32) where Y is the transformed dataset. The SVD computation is related to the eigen decomposition method because the SVD of the rectangular matrix X calculates the eigenvectors of XTX and XXT which are then the matrices U and V in Equation 31, respectively. Additionally, the singular values E are the square roots of the eigenvalues of XTX and XXT. Since the dataset has zero mean, the covariance matrix of X is XTX and the eigen decomposition of the covariance gives eigenvectors identical to V and produces eigenvalues equal to the square of the singular values. Therefore, SVD and eigen decomposition can be used interchangeably for PCA. For threedimensional information being mapped to a planar representation, shown in Figure 33, the first two principal components define two orthogonal inplane directions while the third component defines the plane's normal direction. Therefore, three principal components and a centroid will define each planar surface in the environment regardless of how many feature points comprise the surface. 7 Data points m 2o covariance 20 , P p/ ' P, 10  10. 15 20 10 0 10 20 10 10 20 20 Figure 33. 3D Principal Components Analysis for a planar dataset. The principal components are labeled p, for the nth principal component and the 2a covariance of the dataset is shown. 3.3.2 Data Clustering The aforementioned PCA algorithm cannot be naively implemented to generate planar landmarks for a given sensing operation. The threedimensional data points returned by the computer vision algorithm must be clustered into candidate planar regions before PCA can return the plane parameters. A difficulty arises because no knowledge regarding the number of visible planes is available. Therefore, the number of planar clusters must be estimated for each sensing operation. Planar clustering is accomplished with an iterative algorithm based on kmeans. Kmeans is an unsupervised learning algorithm that defines a set of clusters for a dataset by defining cluster centers, or centroids, that minimize the distance between all data points and their associated cluster centroid [72]. Effectively, the kmeans algorithm minimizes the objective function k EJ Yx 11Xi 2 (33) j=1 xiESj where xi is a data point contained in the data vector X, Sj is the set of all data points associated with the jth centroid, and cj is the centroid location of the jth cluster. The minimization of the objective function in Equation 33 is carried out according to Algorithm 1 which iterates until all centroid locations converge to locally optimal solution. The algorithm begins by guessing the number of centroids k and randomly initializing the centroid locations {cI, c2,..., Ck}. For each iteration, the data points are associated to the nearest centroid according to Euclidean distance. The centroid locations are then updated to the mean of the newly associated data points. Convergence occurs when the maximum centroid displacement is less than a given tolerance tolk. Algorithm 1 The kmeans algorithm 1: select the number of clusters k 2: randomly initialize cluster centroids {ci, c2,... Ck} 3: repeat 4: for each data point xi E X do 5: for j 1 to k do 6: compute the squared 2norm dj = x ci 2 7: end for 8: classify x, E Sj where j argmin(d3) 9: end for 10: for j = 1 to k do 11: compute new centroid location cj = mean(xi E Sj) 12: calculate centroid movement Dj = Icj cj2 13: update centroid location cj = cj 14: end for 15: until max(Dj) < tolk The kmeans algorithm, however, requires that the number of clusters k is known a prior an the number of planar clusters in unknown for each sensing operation. An iterative extension of kmeans, Gmeans, locates an unspecified number of clusters based on a statistical test that determines if a cluster fits a Gaussian distribution [73]. Similarly, the employ, .1 clustering algorithm iteratively searches for clusters that satisfy a planarity test. 3.3.3 Plane Fitting Algorithm The clustering algorithm, presented in Algorithm 2, iteratively searches for planar clusters by starting with a single cluster (k = 1) and increasing the number of clusters to best fit the dataset. Each iteration begins with the kmeans algorithm which produces a set of clustered feature points. Each cluster is checked for planarity with PCA by inspecting the singular value associated with the third principle component p3. The singular values produced by PCA denote the amount of the data's variance captured by the associated principal component. Therefore, the magnitude of the third singular value, a3, denotes how planar is the data cluster with zero magnitude describing a perfectly planar data set. Planarity is determined by imposing a threshold at on a3. A threshold value of 0.05 < at < 0.1 produces good results for realistically noisy datasets. C('lI, found to be planar are removed from further consideration in the clustering procedure. All feature points, including points currently attributed to different clusters, are projected onto the planar basis according to Equation 32. The feature points are tested for inclusion into the current plane definition according to connectivity and distance along the normal direction tests. The connectivity test searches for all points within a specified distance from any currently included point beginning with the point closest to the cluster's centroid. The normal direction test removes points that are a specified distance from an inplane position. The remaining points are classified as a single planar object and removed from the dataset. The convex hull of the points H, the plane's normal direction, and the perpendicular distance from the sensor position to the plane are saved as an observation z. All observations are used in the SLAM algorithm described in C'!h pter 5. The centroids of nonplanar clusters are split, as shown in Figure 34, in order to choose new centroids that properly capture the variability of the dataset. The centroids + ..+........ + ... ... ' ,,, ,,,, ., ... +.. " +" + ++ + + I  I I  A centroid ..... 2o covariance . P2 + data points + data points + + + ++ + t t+, + + + x x x x x . X V '* '' centroid 1 E centroid 2 "..... 2 covariance  P1  P2 + data points 1 x data points 1 Figure 34. Centroid splitting for iterative kmeans. A) The initial result of kmeans which does not fit the dataset. B) Centroid splitting results in two new centroids that more aptly characterize the dataset. C) The final kmeans result that properly clusters the data. * centroid 1 o centroid 2 + data points 0 4 K are split in the direction of the first principal component pl as described by vector r whose magnitude is a function of the components singular value ar as given by r = p 1 (34) The expression in Equation 34 is equal to the splitting distance employ, ,1 in the Gmeans algorithm [73]. After the centroids have been split or removed kmeans is rerun on the remaining data set. This process is performed repeatedly until the number of remaining data points falls below a threshold tolf. This tolerance is typically set as 10' of the total number of feature points. Algorithm 2 Plane fitting algorithm 1: initialize a single cluster centroid (k = 1) 2: initialize observation counter j = 0 3: repeat 4: perform kmeans on the dataset returning centroid locations {cI, c2,..., Ck} 5: for i = 1 to k clusters do 6: perform PCA on cluster ci 7: extract third singular value r3 8: if 03 < at then 9: increment observation counter j j + 1 10: project all feature points onto PCA basis Y = XV 11: find all feature points that fit the planar definition 12: compute convex hull H of associated feature points 13: store H and PCA result as sensor observation zj 14: remove centroid ci and associated feature points from data set 15: else 16: split centroid c, along first principal component pi 17: end if 18: end for 19: until number of fpts < tolf The result of the plane fitting algorithm is shown in Figure 35. The algorithm transforms the pointwise data provided by SFM into a set of geometric primitives. Concatenating these observations will create an incorrect geometric map due to drift in vehicle state estimates. C!i Ilter 5 will present an algorithm that fuses the vehicle state estimates and environment observations in order to create a spatially consistent map. 400 S300 M 200 100 0 1500 o feature points vision frustum planes B Figure 35. Dimensionality reduction of feature point data. A) Pointwise data from the SFM algorithm. B) The dimensionallyreduced dataset consists of geometric primitives in the form of planar features. CHAPTER 4 FILTERING FOR STATE ESTIMATION 4.1 Introduction ('!i ipter 3 described a method for extracting planar features from a highly structured environment. These features will be used in ('! Ilpter 5 to incrementally construct an environment map with a filtering technique known as Simultaneous Localization and Mapping (SLAM). The SLAM task is a state estimation approach in which the state of the system is a combination of the vehicle state, defined in terms of position and orientation, and the state of the map, designated by the position and orientation of environmental landmarks. State estimation is commonly addressed as a filtering problem in which estimates are calculated recursively by incrementally incorporating sensor information [79]. This chapter describes several filters that follow a similar predictorcorrector structure depicted in Figure 41. The filtering process begins with the prediction of a noisy estimate of the system state according a state transition model. Measurements of the system are predicted as if the true system state is equal to the state prediction. The measurement prediction is compared to an actual, noisy system measurement and the state estimate is updated according to a weighted measurement residual. The weight of the measurement residual represents the confidence of the measurement with respect to the modelbased state estimate. Two filters that form the foundation of the i, Ii ii ly of SLAM solutions are the Kalman filter [80] and particle filter [82]. This chapter describes the general form of these filters and several variants that will be used in the SLAM filtering approaches described in ('!C ,pter 5. 4.2 The Kalman Filter The Kalman filter is an estimator for linear systems that yields minimum mean square error (il\ IlE) estimates. Kalman filtering estimates the state x E R" of a dynamic r *predict system state *predict measurement I Update Ose.rvatona *update state prediction *measure system state according to measurement *compare to predicted residual measurement Figure 41. Recursive state estimation. The recursive process involves three steps: prediction, observation, and update. system described by the linear stochastic difference equation Xk = Axk1 + Buk + qk1 (41) where AE '. is the state matrix, B E is the input matrix, u E RI is the input vector, and qk1 E R' is a vector of random variables representing additive process noise. The noise variable qk1 is a zeromean Gaussian variable with covariance Eq. The state estimation process incorporates measurements z e R' of the system that are modeled as Zk Hxk + rk (42) where H E Rmxm is an output matrix mapping states to observations and r E R" is a zeromean Gaussian random variable vector with covariance E, representing measurement noise. For a given state estimate x, the error e and error covariance P are defined as e = x x (43) P E [eeT] (44) where E[.] denotes the expectation or mean value of a random variable. The Kalman filtering procedure begins with a prediction step that updates the state estimate and error covariance as follows Xkk1 Axkik1 + Buk1 (45) Pklk1 = APk1Ik1AT + Eq (46) where the subscript klk 1 denotes a value at time k given measurement information, or evidence, up to time k 1 also known as an a priori estimate. The state and covariance at k k 1 define the prior probability distribution of the state estimate. The notation klk is referred to as an a posteriori estimate, with the mean and covariance denoting the posterior probability distribution, since such estimates include measurements taken up to the current time step. The optimal Kalman gain, which weights the measurement residual, for MMSE estimation is defined as Kk Pi k1HT (HPklk1HT + r)1 (47) The final step of the Kalman filter is to update the predictions given in Equations 45 and 46 after a measurement zk is taken. The measurement update equations are Xkk Xkk1 + Kk (Zk Hxklk) (4 8) Pkk (I KkH) Pkk1 (49) The resulting estimates :kjk and Pk k are used as previous time estimates in Equations 45 and 46 for the next prediction as the recursive process continues. The Kalman filter is designed for linear systems with linear measurement models. The SLAM problem, however, deals with nonlinear systems in the form of vehicle dynamic models and sensors with nonlinear measurement models. Therefore, the nonlinear extensions of the Kalman filter are employ, l for the estimation task. The nonlinear Kalman filters include the Extended Kalman Filter (EKF) [81], the Unscented Kalman Filter (UKF) [83], and the SquareRoot Unscented Kalman Filter (SQUKF) [84]. 4.2.1 The Extended Kalman Filter The EKF allows for a nonlinear process model f(.) and a nonlinear measurement model h(.) Xk = (Xk1, Uk1, qk1) (410) Zk h(xk,rk) (4 11) It should be noted that the process noise and measurement noise are no longer restricted to be additive uncertainties. The EKF follows the Kalman filter approach by linearizing the models with respect to the current state estimate of the system as follows af Ak a l xk 1,,uK1 (412) Hk Oh 1, (4 13) Hk x Additionally, the models are linearized with respect to the noise values af Qk x 1/c 1,Uk 1 (414) aq ah Rk O ,' (415) which means the EKF filter equations, analogous to Equations 45 through 49, can be written as Xkk1 =f (xk1k1, Ukl, 0) Pkck1 = AIkPkIk1AT + QkI qQ Kk PkIk1HU (HkPklk1H + Rk kCR)1 (4 16) Xkk = XkIc1 + Kk (zk h (xklkc1, 0)) Pkck c (I KkHk) Pkk1 which defines the complete EKF. It should be noted that the state a measurement predictions use the full nonlinear equations while the linearized models are required to estimate the distribution's covariance. This linearization is the primary source of error in the EKF state estimation process. 4.2.2 The Unscented Transform The EKF does not retain the optimality of the Kalman filter because the linearization step does not properly handle the statistics of the problem. This inaccuracy is because the random variables no longer have normal distributions after undergoing a nonlinear transformation. The UKF employs the Unscented Transform (UT) to more accurately estimate the statistics of a random variable that is passed through a nonlinear transformation. The UT calculates the mean and covariance of a variable undergoing a nonlinear transformation by propagating a set of points (called sigma points) through the nonlinear function and estimating the new statistics of the distribution as shown in Figure 42. The sigma points are chosen according to a deterministic algorithm so that they accurately approximate the true mean and covariance of the untransformed distribution.  ~ \ Nonlinear STransformation  Figure 42. The Unscented Transform. The UT propagates a set of sigma points through a nonlinear function to calculate the posterior probability distribution. In general, the UT defines a set of 2L + 1 sigma points X and associated weights for an Ldimensional random variable with mean g and covariance P as follows Xo = k1 WA + ( l a2+) XO Ak ^vL' 0 LW=A Xi / k1 + (V(L + A)Pk1~1 i WI 'i( 2(L+A) (417) Xi+= Ak1 (V(L + A)Pk1k1)i Ws Wc A = a2(L + ) L where the subscript of the square root denotes the ith column of the matrix square root and a, 3, and K are tuning parameters for controlling the placement and weighting of the sigma points. The sigma points are transformed by the nonlinear function i = f(Xj), V i ... ,2L + 1 (418) and the new mean and covariance are the weighted average and weighted outer product of the transformed points 2L I'k WYWSi (419) i=0 2L P = Wf [Y k] [Yi tk] (420) i=0 4.2.3 The Unscented Kalman Filter The unscented transform is applied to the EKF to replace the prediction and update equations of Equation 416 used to create the UKF. This derivation is accomplished with an augmented state vector x' e R"+', where q is the dimension of the noise vector q, defined as X x' (421) q The process model, Equation 410, is rewritten to accept the augmented state vector as input, Xk f (Xk1, Uk_1, qk1) (422) so that the sigma points in Equation 417 are drawn from a probability distribution with a mean value x k Xk1k(423) Oqxl and covariance SPxv,kllk1 Eq where Pxv is the cross correlation matrix between the state errors and the process noise. The measurement model, Equation 411, is rewritten so that measurement noise is restricted to be additive as was assumed in the linear Kalman filter Zk h' (Xk1) + rk1 (425) The resulting UKF filter equations begin with the prediction step where sigma points X', as given in Equation 417, are drawn from the distribution defined by Equations 423 and 424. These sigma points are transformed by the nonlinear process model with zero noise Xi,klk f (,klk1, Ukl, 0) V i 1,..., 2L + 1 (426) The mean and covariance of the sigma points are calculated from Equations 419 and 420 as 2L ^Xklk1 WXklk1 (427) i=0 Pkk1 Wi[Xiklk1 Xklk1 [Xi,k k1 Xkk1 ] (428) i=0 Next, observations are predicted by transforming the updated sigma points with the measurement model Zik h' (Xikk1, Uk, 0) Vi 1,...,2L+ 1 (4 29) where the mean of the probability distribution of the current observation is 2L Zk i ( (430) i=0 with corresponding covariance 2L Pzz1[ZW[Zk Zk][,k ZkT (4 31) i=t The Kalman gain for the UKF is calculated according to Pz,kk1 ZL0 iWX i,klk1 1Xkkl1 [Zi,k Zkf( (432) Kk Pxz,klk1 (Pzz,k)1 Finally, the state estimate and covariance matrix can be updated to generate the current state estimate Xkk = Xkk1 + Kk (zk Zk) (433) (433) Pklk P Pkk1 KkPzz,kK[ Therefore, the state estimate of the system has been computed without requiring linearization of the process and measurement models. 4.2.4 The SquareRoot Unscented Kalman Filter The UKF approach to state estimation has been shown to produce superior results to the EKF for nonlinear systems [83]. The UKF, however, is a computationally expensive algorithm in comparison to the EKF. This expense is because the sigma point calculation, Equation 417, requires 2L + 1 evaluations of the nonlinear process model and matrix square roots of the state covariance. The SquareRoot Unscented Kalman Filter (SRUKF) is an efficient variant of the UKF in which the matrix square root required to generate the sigma points is eliminated by propagating the square root of the covariance matrix in lieu of the full covariance [84]. Therefore, the sigma point calculation, replacing Equation 417, is X1 X[ 1,xL ( L+ A)S ) (4 34) X "kllk_1 Xk~l P k I + A)Siklk1)l where S is the matrix square root of the covariance matrix initialized by a C'!. .,! l:y factorization of Po. The C'!I i!. l:y factor S is updated in the prediction step using a QR decomposition to to incorporate the weighted sigma points excluding the centroid as follows Si qr ( i:2L,k k1 Xk) q]) (435) where qr(A) represents a QR decomposition A = QR where Q is a square unitary matrix and R is an upper triangular matrix returned by the qr operator. Since the weight corresponding to the centroidal sigma point may be negative it is incorporated separately according to Sk 1 = chollupdate (Sik, Xo,kk ), W) (436) where cholupdate(A, X, s) represents a C'! .!. l:y update returning the C'!I!., l:y factor of A XX* with s dictating the sign on the update. Equations 435 and 436 replace the UKF state covariance update Equation 428. The measurement covariance update equations, replacing the analogous UKF Equation 431, follow the same formulation as the state prediction methodology as shown in the following equations S,k q ([ W (Z :2Lk (4 37) ,k cholupdate (Sk, Zo, z), ) (4 38) U K S, S ,, cholupdate (si ,, (4 39) where the Kalman gain is calculated as follows K = (P /(S) )/S, (440) k Xkzkl S'z,k) Therefore, the SRUKF calculates state estimates without linearizing the system models as in the EKF and without the computationally expensive matrix square root in the UKF approach. 4.3 The Particle Filter The Kalman filter and its nonlinear variants (EKF,UKF,SRUKF) operate under the assumption that the probability distribution of the state estimate is Gaussian, or able to be parameterized by a mean and covariance. The particle filter is a Monte Carlo ( MC) method for state estimation which represents the probability distribution as a set of weighted random samples [82]. This formulation allows the estimation process to handle nonGaussian and multimodal probability density functions. Like the Kalman filtering approaches, the particle filter operates in a recursive fashion. The state is predicted according to a process model and, subsequently, updated by comparing sensor measurements to measurement predictions calculated with a measurement model. However, while the Kalman filter propagates a normal distribution through the nonlinear models, the particle filter propagates a set of weighted samples called particles. Particle filters employ a set of N particles defined as S {xw} Vi ,... ,N (4 41) where x4 are random samples of the state's probability density function p(Xk Xk_1, Uk) and w' are associated importance weights calculated according to environmental observations. The particles differ from the sigma points employ, ,1 by the UKF in that they are randomly selected and not deterministically calculated. Additionally, the particles are weighted in order to maintain a general probability distribution while the sigma points are used to reconstitute a Gaussian distribution. The particle filtering algorithm, outlined in Algorithm 3, begins by drawing state samples from a probability density defined as Xk ~ p(XkcXk k1, Uk) = f(xkl, Uk, qk) (442) which is the same nonlinear process model definition used in the EKF (Equation 410). The notation a ~ b denotes that the random variable a has the probability distribution b and the notation p(cld) represents a probability distribution for variable c given the evidence d. Each particle drawn from the process model distribution is assigned a weight w oc w p(zk X) (4 43) where p(zk lx) is defined as the nonlinear measurement model Zkj ~ p(zk Xk) h(xk, rk) (444) and the proportionality is eliminated by normalizing the weights at each step. Therefore, each particle is assigned a weight which is a function of its previous weight and the probability that the sample would produce the current sensor measurement. The weights can be seen as the importance of the particles with respect to maintaining an accurate representation of the probability distribution. The final step of the particle filter is the resampling of the particles with replacement according to their weights. The resampling allows for particles with a low weight, that no longer represent the actual probability distribution, to be removed from further consideration. Since the particle set is sampled with replacement, particles with a large weight are often selected multiple times. The sampling step, however, can lead to particle deficiency if sampling occurs too frequently. Particle deficiency refers to a lack of particle diversity in which the distribution is approximated by N particles, but few are unique. .ii;, sampling heuristics have been developed to determine when best to perform the resampling step. The method used in this dissertation is a resampling procedure based on the number of effective particles N~ff [85]. The number of effective particles Neff,k N= (445) EI(Wfc)2 estimates how effectively the current particle set approximates the true posterior distribution. Resampling occurs when Neff falls below a threshold NT. Algorithm 3 Particle filter 1: loop 2: for i = 1 to N samples do 3: draw sample xk p(xk Xk 1, Uk) 4: assign a weight to the particle w, w_ p(zk x) 5: end for 6: calculate the total weight w, = Z w, 7: for i 1 to N samples do wk 8: normalize the importance weights wk =  9: end for 10: calculate Neff 11: if Neff < NT then 12: resample with replacement 13: end if 14: end loop 4.4 The RaoBlackwellized Particle Filter Particle filtering can be extremely inefficient in highdimensional spaces [86]. This inefficiency is vitally important with respect to the SLAM problem where the state space includes the vehicle states and the map landmark states. Therefore, particle filtering has been applied to the SLAM problem with the RaoBlackwellized Particle Filter (RBPF). The RaoBlackwellization is a process in which the number of states that need to be sampled is reduced by partitioning the joint state. Therefore, for a joint distribution p(x, X2) the state is partitioned using the product rule p(xi,X2) p(x2 xl)p(xi) (446) If the joint state is partitioned in such a way that the conditional probability p(x2 xl) can be determined analytically then only the probability distribution of the partial state vector p(xl) needs to be sampled as x ~ p(xl). Therefore, the new particle set becomes S {x p(x2 Xi), W} Vi 1, .. ., (447) which means that each particle contains the partial state estimate x' and a probability density p(x2 xi) that is conditioned on the particle's specific estimate of x,. The sampling of a reduced state space increases the computational efficiency of the RBPF by decreasing the dimension of the problem. This approach is applied to SLAM by partitioning the state space so that the vehicle pose is sampled and the map's landmark states are determine analytically. This application is described further in C'! lpter 5 where the RBPF is combined with the aforementioned EKF, UKF, and SRUKF to produce three filters for solving the SLAM problem. CHAPTER 5 SIMULTANEOUS LOCALIZATION AND MAP BUILDING 5.1 Introduction Trajectory planning for a vehicle navigating an unknown environment motivates the need to incrementally construct a spatial map of the vehicle's surroundings. Motion planning strategies that react only to current environmental sensor measurements produce inefficient results since no knowledge outside the current sensor field of view is incorporated into the planning process [74]. Timedependent missions, such as target acquisition and tracking, require a map to produce time efficient solutions. Additionally, the ini i ily of motion planning strategies presuppose a known environment map [26]. Therefore, map construction becomes a vitally important task for efficient motion planning. This research is concerned with navigation in GPS denied environments meaning knowledge of the vehicle's inertial location is not known. Therefore, navigation must rely on relative vehicle position estimates which are subject to drift due to model uncertainties and external disturbances. Additionally, sensor measurements of the external environment are susceptible to noise. The combination of errors incurred from uncertain vehicle location and noisy senor measurements produces spatially inconsistent maps as shown in Figure 51. The issues incurred by placing a vehicle at an unknown location in an unknown environment and requiring it to incrementally build a spatially consistent map while concurrently computing its location relative to the map are known as the Simultaneous Localization and Mapping (SLAM) problem [75]. Research into the SLAM problem has produced a variety of solutions based on probabilistic recursive algorithms with implementations on a variety of platforms including ground vehicles, aircraft, and underwater vehicles [19, 76, 77]. All successful SLAM solutions follow a similar overall estimation strategy as depicted in Figure 52.   groundtruth  estimate I / ' Figure 51. Map building in the presence of noise. Noise in the map building process produces a spatially inconsistent map. The SLAM estimation strategy begins with a prediction step (Figure 52A) in which the vehicle position and orientation are estimated according to a motion model. Additionally, sensor measurements are predicted using a measurement model that estimates sensor readings based on the uncertain state of the vehicle and the current map of the environment. Essentially, the prediction step generates a guess of what the vehicle should observe given the current state estimates. Next, actual sensor measurements of the environment are taken by the vehicle's onboard sensor suite as depicted in Figure 52B. Finally, the actual observations are compared with the predicted observations and the map and vehicle states are updated accordingly (Figure 52C). There are several open problems in the SLAM community including the reduction of computation complexity to enable realtime implementation, correct association of observed landmarks with mapped landmarks, and environment representation [78]. This dissertation addresses these problems by employing a dimensionally reduced map representation as described in Ch'! pter 3. The map representation is a set of planar landmarks that fit threedimensional feature points gathered by an onboard camera. The geometric map alleviates the computational burden of the SLAM solution by reducing the ,, I I I I I I ri  7 '"" Figure 52.  groundtruth estimate   sensor prediction ......... error ellipse  groundtruth  estimate sensor measurement  groundtruth  estimate .......... error ellipse  updated error The SLAM estimation strategy. A) In the prediction step the pose of the vehicle and sensor measurements are predicted according to motion and measurement models, respectively. The error ellipses denote uncertainty in vehicle state and map estimates. B) The observation step involves acquiring relative measurements from onboard sensors to environmental landmarks. C) In the update step the vehicle state and map estimates are updated by comparing the predicted sensor measurements with the actual sensor measurements. ,,,~ number of landmarks that must be stored and incorporated into the estimation process. Additionally, the geometric landmarks are easily distinguishable which aids the association of sensor observations with map features. This chapter builds on the overview of recursive filters that are important to the SLAM estimation process presented in C'!i lpter 4. Section 5.2 introduces the vehicle and measurement models developed for the presented SLAM formulation. Section 5.3 describes three specific filters that were investigated in relation to the airborne SLAM problem. Lastly, Sections 5.4 and 5.5 describe the data association process and map management procedures employ, 1 to incrementally incorporate sensed geometric primitives into a global map. 5.2 Airborne SLAM Models The goal of the airborne SLAM algorithm is to concurrently estimate the state of an aircraft, x E R6, defined by position and attitude, and a relative map consisting of j static environmental landmarks m E R6j. The landmarks in this formulation are distinct planar features defined by a centroidal location and a unit normal direction m = [p,, f~] . As described in ('!i lpter 4, state estimation requires the definition of process and measurement models that describe the system of interest. The general process and measurement model definitions for the SLAM problem are probabilistically defined as the following distributions Xk ~ p(xk Xk 1, Uk) f (xk1, Uk1, qk1) (51) Zkj ~ p(zk,j Xk, my) =h (xk, my, rk) (52) where f() is a nonlinear state transition model that calculates the current state of the system xk based on the previous state xk1, control input uk1, and process noise qk1 with covariance Cq. The nonlinear measurement model h() generates a sensor measurement zk,j for each landmark my based on the vehicle state xk and measurement noise rk with covariance E,. 5.2.1 Vehicle Model The addressed airborne SLAM problem assumes the vehicle is a fixedwing aircraft as depicted in Figure 53. Additionally, it is assumed that vehicle linear velocity [u, v, w] and angular velocity [p, q, r] information is provided via an IMU or visionbased state estimation algorithm (Sections 2.3.2 and 2.4). Therefore, the input to the vehicle model is selected to be u [u, v, w,p, q, r]T (ND it [,r ,y, ]T 0, q Ob, r Yb Figure 53. Fixedwing aircraft model definitions. The aircraft linear and angular rates are defined with respect to a bodyfixed basis B. The inertial position and orientation of the vehicle is measured with respect to a North East Down (NED) inertial frame E. The vehicle model f(.) accepts noisy estimates of vehicle velocities as inputs. Position and orientation state information is generated according to the aircraft kinematic equations S= u cos 0 cos Q + v( cos Q sin Q + sin Q sin 0 cos Q) +w(sin Q sin i + cos Q sin 0 cos b) y = cos 0 sin Q + v(cos Q cos Q + sin Q sin 0 sin Q) +w( sin cos + cos sinsin ) ( ) i u sin v sin cos 0 w cos 0 cos 0 Sp + tan (q sin + rcos ) S= q cos Q r sin ~ (qsin + rcos 0)/cos 0 where [x, y, z]T denotes the inertial position measure in a NED frame and [, 0, Ob]T are the vehicle Euler angles defining roll, pitch, and yaw. The errors in linear and angular velocity qk are assumed to be Gaussian with a covariance 2 0 0 0 0 0 0 a2 0 0 0 0 0 0 a 2 0 0 0 v, (54) 0 0 0 2 0 0 0 0 0 0 a2 0 0 0 0 0 0 ao 5.2.2 Measurement Model The measurement model h(.) maps planes defined in the inertial coordinate system to a bodyfixed definition so that predicted measurements z can be compared with actual sensor readings z. The definition of a landmark mj consists of an inertially defined centroid and normal direction mj = [,j, fj] The sensor observations consist of planes defined by a normal direction fi and a perpendicular distance d from the plane to the sensor. Therefore, the observation model h(.) is defined as kj ,j Rbj (55) dkj nikj R B(Pi Pk) where Rb is the rotation matrix from the inertial frame to the bodyfixed frame at time k and pk is the current vehicle position. The sensor induced errors for the perceived normal direction i = [ni, n2, n3]T and perpendicular distance are characterized by S 0 0 0 0 a2 0 0 Er n2 (56) 0 0 ~2 0 0 0 0 0 0 0 ad 5.3 Filtering for SLAM The airborne SLAM problem addressed in this dissertation is subject to difficulties exceeding ground vehicle SLAM with respect to computational complexity and high nonlinearity in both vehicle dynamics and environmental measurements [19]. The airborne SLAM problem has been investigated with the Extended Kalman Fitler (EKF) [18] and the Unscented Kalman Filter (UKF) [20]. The EKF approach has been shown to be somewhat successful in implementation for a very limited number of landmarks. Additionally, UKF approach has proven to be superior to the EKF in terms of estimation accuracy, but suffers from increased computational complexity. This dissertation investigates three additional SLAM filters based on the RaoBlackwellized Particle Filter (RBPF). The goal is to find a solution to airborne SLAM that is more computationally efficient than both the EKF and UKF approaches and handles the nonlinearities as well as the UKF. The complete formulation of these filters, including the equations necessary for implementation, is described in this section. The SLAM estimation task is framed as a probabilistic process with the goal of calculating the joint posterior distribution of the vehicle and landmark states p(xk, m ZO:k, UO:k, XO) (57) where ZO:k is the history of all landmark observations, Uo:k is the history of all control inputs, and xo is the known initial vehicle state. The following three filters estimate this distribution solving the airborne SLAM problem. 5.3.1 RaoBlackwellized Particle Filter The RBPF was introduced as an alternative to EKF SLAM in an approach referred to in literature as FastSLAM [23, 87]. The RBPF is a samplingbased approach which factors the joint SLAM posterior distribution according to the observation that landmark positions can be estimated independently if the vehicle path is known. The resulting RaoBlackwellized form contains a sampled term representing the probability of vehicle paths and M terms that estimate the landmark locations M p(XO:k, ,m U:k, Zo:k, xo) = p(Xk:o ZO:k, Uo:k, Xo) j p(mi Xk:o, Zo:k) (58) j 1 where XO:k denotes a vehicle state history. According to this formulation the posterior distribution is represented as a set of N particles {wk, X:k, p(m X:k, ZO:k)} where wi is a weight assigned to each particle and p(mXO:k, Zo:k) is an individual map associated with each particle. Therefore, one benefit of the RBPF approach is that each particle contains a separate map which allows for multiple data association hypothesis to be assigned at each observation update. This allowance means that the RBPF approach is more robust, in terms of filter convergence, than EKF SLAM with respect data to association. The FastSLAM approach, which is based on the RBPF with EKF i updates, proceeds as follows: 1. Generate N vehicle pose samples ^i and associated observations z 1ki ~ PXkfX_1, Uk) = f(X'_1, Ukqk1) (59) Pxkk VfxPjx,k1klVfx + Vf q VfT (510) ki i i i( z ~ p(z xk, m) h(i, mji,k1,) (511) where Pxx is the covariance of the vehicle pose, Vfx is the Jacobian of the vehicle model with respect to the vehicle state, and Vfu is the Jacobian of the vehicle model with respect to the vehicle control inputs. 2. Calculate importance weights wk and resample according to weights to promote particle diversity: Wk w _1C (512) where the constant C is the likelihood of all observations zk (the subscript j has been dropped to show that the following is performed for all landmarks in a batch fashion) given the estimated position of the vehicle. The likelihood is calculated as follows C (2) pi eI ((z k P 1(z (513) where PW VhxEqVhx + VhmP.,klVhm + ,E (5 14) and Vhx is the Jacobian of the observation model with respect to vehicle pose evaluated at ^5kjk1 and mk1, Vhm is the Jacobian of the observation model with respect to landmark location evaluated at ^ of the map states. 3. Calculate the proposal distribution and draw N samples xk defining the updated vehicle state where the distribution is Gaussian with parameters: T ^i i i ii)5) XkIk k lk1 + Pk kh z(B k) (515) P',k = (h (BR) hx + (q,)1)1 (516) where Bk is defined as B" = ,E + VhmP",k_1Vh (517) 4. Update definitions of observed landmarks according to the EKF measurement update: Kk PmrkVh (B')1 (518) m m_1 + K,(zk (519) P, = (I KVhm)P,1 (520) If a new landmark is observed it's covariance is initialized to Pn,k =VhmErVhm (521) From the above FastSLAM algorithm it can be seen that another advantage of the RBPF is that only observed landmarks are updated. Therefore, the RBPF is more efficient than a naive implementation of an EKF where all landmark and vehicle states are concatenated into a single matrix and updated simultaneously. Current research approaches use submaps to increase in the efficiency of KFbased solutions. 5.3.2 Unscented RaoBlackwellized Particle Filter The Unscented RaoBlackwellized Particle Filter (URBPF) is formulated by applying the Unscented Transform (UT) (Section 4.2.2) to the RBPF SLAM formulation [88]. The URBPF is an improvement over the RBPF in terms of estimation performance similar to the comparison between the EKF and UKF discussed in Section 4.2.2. Additionally, the URBPF is easier to implement because it does not require linearization of the state transition and measurement models. 1. Generate N vehicle pose samples x^i and associated observations z, according to the UKF prediction rule: 1kk1 L [ ,x_ 1, (L + A)Pzzkl (522) Xk fkl,t f(X lkl,t Uk) X lk1 EO WiXkl (523) 2L Px,klk1 > w[kXk1,t Xkk l [X kl,t XAkk (5 24) t=0 ,tj = h( _kl2,t, mJ,k1) 2L t Z (525) Zj h(Xk WtZ k,t,j 2. Calculate importance weights wk and resample according to weights to promote particle diversity: k = w _1C (526) where the constant C is the likelihood of all observations zk (the subscript j has been dropped to show that the following is performed for all landmarks in a batch fashion) given the estimated position of the vehicle. The likelihood is calculated as follows C = (27)m jp (ze 2 k)1( ( (527) where P, (Pf )T P f,k + PT,,k + (528) where 2L P,k z W [t ] [z ],t (5 29) t=0 2L P7,k E>\ W i1 [Z^ yt (5 30) xx,k tc kkl,t Xkk1 [,t ] (530) t=0 3. Calculate the proposal distribution and draw N samples to define the new vehicle state xk where the distribution is Gaussian with parameters: ^k k + K (Zk ) (531) Px,kk P?,klk KPzzk(K)T (532) and the Kalman gain is K PX',k(PI ,k)1 (5 33) 4. Calculate sigma points for the observed landmarks and transform according to the observation model: S= ml, m_1 ((L A)P (534) 2L0 (535) 2L y7,  Y i ~yy,k E [c [ l,t zk] k k, z] (536) t=0 5. Update definitions of observed landmarks according to the UKF update rule: Kr, k Pn,k lklPyy,k((Pyy,k) Pn,kklPyy,k + r)1 (5 37) m m_ + K Zk k ) (538) P,k, k (I Kk(P y ) (539) Knkk r,k(P~y,k) )P'rn,klk1 5 9 5.3.3 SquareRoot Unscented RaoBlackweillized Particle Filter This dissertation introduces a new option for efficiently solving the SLAM problem based on the SquareRoot Unscented Kalman Filter (SRUKF) (Section 4.2.4). The SRUKF is an efficient variant of the UKF in which the matrix square root required to generate the sigma points is eliminated by propagating the square root of the covariance matrix in lieu of the full covariance. Similarly, the SquareRoot Unscented RaoBlackwellized Particle Filter (SRURBPF) is formulated by replacing the UKF 1. calculations in the URBPF with the SRUKF approach. 1. Generate N vehicle pose samples x^i and associated observations zi according to SRUKF prediction rule: k1 klx Xk1 ( L + A)Sxn,klk1) (540) ^ l2L (541) XkXkI,t 1=kikl,t, UkX) Xki W kX (541) Sxz,klki (v (I:2Lkk1k Xkk i ) q]) (5 42) Six,kk cholupdate (S,k k1 ,kk Xkk1)i, W&) (543) xo klW) (5l 43) Zk,, j h(X4 kl,t, mj,k1) Z = Et0 WsZ (5 44) Szz,k qr ([w i:2L,k Z)i r)(545) S = cholupdate (Sz,k ,k Zk, W) (546) 2. Calculate importance weights wk and resample to promote particle diversity: Wk = W 1C (547) where the constant C is the likelihood of all observations zk (the subscript j has been dropped to show that the following is performed for all landmarks in a batch fashion) given the estimated position of the vehicle. The likelihood is calculated as C" (2)T pl i (ze (P) 1(ze (548) where pi pi )Tr p pi +Rk (549) PW (P kzQkP k + Pz + Rk (549) 2L Pi _i S7 tc [ziL 2] [zt 2] (5 50) t=0 2L Pi [xi ^i 1 T aXkz Ltc k1,t Xl] L~k ,t J ] (5 51) t=0 3. Calculate the proposal distribution and draw N samples to define the new vehicle state xk where the distribution is Gaussian with parameters: Xk Xkk + K Z(Zk 2) (552) U K Szz,kk S ,kk cholupdate (sk1, U (553) and the Kalman gain is K = (PAz,k/(Szz,klk_l))/Szz,klk1 (554) 4. Calculate sigma points for the observed landmarks and transform according to the observation model: M1, = mi, (L (555) mkl, mkl I (V/L+ A)Siml 55) ,yi = h(x, . A^ zyi2L (5 56) ,t,j h( Z 0 ts(5 6 S i,klk1 ([ [ (Y~:2L,k Zk) (557) S'klk 1 cholupdate (S l,k y1, Zki, W&) (558) Syy,k qr ([I( : ( 2Lk Zk (559) S' cholupdate (SY z), W) (5 60) Syy,k =y;k C O a yyk ,k Zki) (560) 5. Update definitions of observed landmarks according to the SRUKF measurement update rule: K ,k (Pklk/(Syy,kckl))/Syy,kck1 (561) m = m_1 + Kk(zk z4) (562) Uk KkSy,k S ,klk cholupdate (Sk,_ 1 (563) The SRURBPF procedure is more computationally efficient than the URBPF approach. A comparison of all three filters is given in C'!i pter 7. 5.4 Data Association Several important implementation issues were not addressed in the previous filter development section. The first to be discussed is data association, the process of relating existing map landmarks to the current environmental observations. Data association is a critical step in the SLAM problem with incorrect associations often leading to a catastrophic failure of the state estimation process. The RBPF mitigates the data association problem by allowing for multiple association hypotheses because each particle contains a separate estimate of the environment map. Additionally, defining landmarks as geometric primitives further alleviates the problem by greatly reducing the number of landmarks and making the landmarks more distinguishable by incorporating orientation information. The vast in i ii i iy of SLAM implementations deal with a large number of landmarks defined by inertial location. These situations lend themselves to batch gating which exploits the geometric relationships between landmarks by considering multiple landmark associations simultaneously [78]. The observed landmark information provided by the dimensionality reduction step, however, is sparse and distinguishable. Furthermore, texture variations on building facades can lead to multiple observations of the same landmark during a single sensing operation. Therefore, a nearest neighbor approach is used to allow for multiple associations to the same target. It should be noted that individual gating techniques such as the nearest neighbor method fail in environments that are not sparsely populated or structured. This failure is a result of the individual gating method's investigation of a single landmark and sensor measurement pair at a time. Therefore, the overall structure of the environment is not considered and many incorrect associations can be made when the measurements are noisy. The nearest neighbor approach uses Mahalanobis distance as a statistical measure of similarity for an observed landmark and a predicted landmark. The Mahalanobis distance is a distance metric that is scaled by the statistical variation of the data points. For the case of data association, the squared Mahalanobis distance is calculated as follows Mij = (z, 2j)T(Pij) (z 2j) (564) where Pij is the covariance of the ith observation zi with respect to the observation estimate 2j of the jth landmark. In the Gaussian case the Mahalanobis distance forms a X2 distribution dependent on the dimension of the innovation vector and the desired percentage of accepted associations. For a 4dimensional innovation vector and an acceptance percentage of 95'. the Mahalanobis gate value for which an association will be accepted is 9.5. An additional step is added to the nearest neighbor search which checks the the distance between the estimated and observed plane boundary and does not accept associations with boundaries that are too distant from each other. The data association step is used in the preceding SLAM filters whenever a measurement residual is calculated. The residual is a function of the actual measurement and the predicted measurement, but the mapping between which actual measurement relates to which prediction is unknown. Therefore, the true value and predicted value are associated as the minimum arguments to Equation 564 for all possible pairs of true and predicted measurements. All associated pairs are used to update the SLAM filter. 5.5 Map Management The observations in the described SLAM formulation consist of partially perceived planes. New observations must be incrementally fused into the global map as the environment is continuously surv i' 1 Incremental map building is accomplished by mapping the boundary of an observation, described by a convex hull H, to the inertial frame according to Hk,e = RHk,b + Pk (565) The points that comprise the boundary H are updated to match the current landmark after the landmark update step of the SLAM filter by moving the points along the updated normal direction as follows Hk,e = (in p n Hk,)n + Hfk, (566) The combined observed and original threedimensional boundary points Ht are projected onto updated landmark basis V according to Yk = (Hk,t g)V (567) and the plane boundary is updated by computing the convex hull of the projected set y and transforming the points back to the inertial frame. The centroid of the updated landmark is computed as the mean of the updated hull. This chapter has presented three complete filters for solving the airborne SLAM problem. Additionally, methods for handling planar landmarks within the SLAM framework were presented. Simulation results and comparison between the filters are presented in Chapter 7. CHAPTER 6 TRAJECTORY PLANNING 6.1 Introduction Trajectory planning refers to any process which determines the history of control inputs, u(t), for a system that yield a time parameterized curve, q(t), called a trajectory [26]. The trajectory for system can be derived to avoid obstacles, regulate the system to a desired state, optimize an objective function, or a combination of these tasks. Trajectory planning is distinct from path planning because it accounts for system dynamics by planning motions with physically realizable velocities and accelerations. This distinction means that the curve q(t) is twice differentiable with respect to time. Path 1p1 i11Hi:7. however, is limited to constructing a kinematically feasible curve, q(s), in the system's configuration space. The goal of trajectory planning addressed in this research is to generate collisionfree trajectories for an aircraft navigating in an unknown environment. The trajectories are designed to build a complete environment map in minimal time. The employ, 1 mapping procedure which incrementally constructs a map of planar features was discussed in C'! lpter 5. This map is used as feedback for obstacle avoidance in the trajectory planning process. Optimal trajectory planning for aircraft in the presence of obstacles is a difficult problem because the optimization is nonconvex [89]. This nonconvexity leads to multiple locally optimal solutions which makes most standard optimization routines computationally intractable. Several approaches for directly solving the trajectory optimization problem for aircraft have been proposed. Two prominent methods based on MixedInteger Linear Programming (l\ILI.P) [90, 91] and the Maneuver Automaton [3537] are described in Section 6.2. These methods do not, in general, guarantee optimality, are sensitive to initial conditions, and can become computationally intractable in realistic situations. This research employs a decoupled trajectory planning strategy which converts a path planning result to a trajectory using optimal control theory. This approach is similar to offering an accurate initial condition to the direct trajectory planning methods, but limits the optimization to a minimumtime path following calculation. The path planning task is segmented into a two phase problem as shown in Figure 61. First, a coarse path is constructed to a selected goal location. The portion of the global path that occurs within the current sensor field of view is refined to provide a detailed local path. As a final step, the detailed local path is converted to a trajectory with optimal control. u global path  local path I S trajectory I I I ........... planning horizon I I Figure 6t. An overview of the trajectory planning process. A coarse global path is planned to a prescribed goal lolocatioation defined to explore the environment. A detailed local path is planned to provide local sensor coverage. Lastly, the The emplo I path planning methodology is based on well established samplingbased criteria which selects the best path for environment coverage are introduce in Section 6.6. The optimal control formulation employ, 1 for converting the kinematically feasible path to a dynamically feasible trajectory is described in Section 6.7. The final section, Section 6.8, describes the full algorithm for trajectory planning for optimal map building. I I l J L.I Figure 61. An overview of the trajectory planning process. A coarse global path is planned to a prescribed goal location defined to explore the environment. A detailed local path is planned to provide local sensor coverage. Lastly, the local path is converted to a trajectory using optimal control theory. The emploiv ,t path planning methodology is based on well established samplingbased strategies [27, 32] described in Section 6.4. Section 6.5 describes a curvature constrained kinematic model used to generate kinematically feasible paths. The environment coverage criteria which selects the best path for environment coverage are introduce in Section 6.6. The optimal control formulation emploiv ,t for converting the kinematically feasible path to a dynamically feasible trajectory is described in Section 6.7. The final section, Section 6.8, describes the full algorithm for trajectory planning for optimal map building. 6.2 Direct Trajectory Planning Methods Direct trajectory planning methods generate motion plans directly in the state space of the system. Several methods for direct trajectory planning include artificial potential functions [92], optimal control [89], and samplingbased methods [93]. The latter two methods have been directly applied to the aircraft motion planning problem and are described below. 6.2.1 MixedInteger Linear Programming MILP is the minimization of a linear function subject to linear constraints in which some of the variables are constrained to have integer values. The integer valued variables are key because they allow the incorporation of obstacle avoidance into the optimization. MILP has been selected to solve the trajectory optimization problem because of its ability to handle obstacle constraints and due to recent improvements in the computational efficiency of commercially available software for solving MILP problems. The general form of a mixed integer program for the variable vector x E R" with m constraints can be written as x* = argmin J(c,x) = (cTx) x s.t. Cx < b (61) l where J is the objective function to be minimized, c E R" and b E R" are vectors of known coefficients, C E is a constraint matrix, u is the upper bound of the solution vector, and 1 is the corresponding lower bound. The following formulation of the MILP problem for vehicle navigation is a simplified version for a twodimensional vehicle [91], but can be extended to three dimensions [90]. The goal of the trajectory planning problem is to compute a control history {uk+i E R2 : i= 0,1,..., L 1} that defines a trajectory {xk+i E R2 : i= 1,...,L}. The optimization seeks to find a minimum time trajectory that terminates at a specified goal location x,. This minimum time, terminallyconstrained problem can be formulated as the following MILP u* argmin J(bg(u),t) L E bi (u1) s.t. K( b) < x < bg) (62) L b 1 i= 1 "g 1 where bg e BL is a set of binary decision variables, B = 0, 1}, that define the point where the goal location is reached and K is large positive number. When the goal is reached at point Xk+i the value of b, is set to 1, otherwise, a value of 0 makes the first constraint inactive. The second constraint ensures that the goal location is reached only once. The objective function J guarantees that the goal is reached in minimum time. Vehicle dynamics, written in terms of position x and velocity x, are included as a constraint on the optimization in the linearized form Xkc+l Xkc k+l A +Buk (63) Xk+l Xk where A is the vehicle state transition matrix, B is the vehicle input matrix, and u is the vehicle control input. The vehicle dynamic model is subject to limits on velocity and control input (64) Uk < Umax Lastly, collision avoidance is included in the optimization as a constraint stating that trajectory points must not reside within an obstacle. The following form of the constraint is written for rectangular obstacles High K[bin,1, bin,2]T < Xk+i < Olow + K[bin,3, bin,4]T (65) Z: binj < 3 ib where K is a large positive value, bi EC B4 is a binary vector that must have at least one active constraint (bj = 0) to be outside a given obstacle, and the obstacle upper right corner and lower left corner are given by Ohigh and orow, respectively. The extension of this constraint to three dimensions requires geometrically regular p" I.v.onal obstacles. The above formulation for minimum time travel to a goal location is computationally intractable for largescale trajectories and highdimensional systems with more complex dynamic models and a greater number of obstacles. Therefore, the MILP trajectory planning problem is implemented in a Receding Horizon (RH) fashion to force a computationally efficient solution which, by construct, is not provably optimal. This is true for all RH implementations because the problem is solved incrementally and does not account for the full scope of the problem at once. The MILP solution, like most optimizationbased approaches, is sensitive to initial conditions and there is no guarantee that the solution is globally optimal as opposed to locally optimal. Additionally, a vast number of objective functions and constraints cannot be expressed linearly and thus fall outside the umbrella of admissible MILP problems. 6.2.2 Maneuver Automaton The MA motion planning strategy is a direct motion planning strategy which realizes a computationally efficient solution by discretizing the system dynamics and controls into a maneuver library. The maneuver library is a finite set of motion primitives that can be concatenated to create a complete trajectory. The motion primitives consist of a set of trim trajectories and maneuvers. The trim trajectories are timeparameterized vehicle motions that are characterized by constant control input and bodyaxis velocities. The maneuvers are finite time transitions that connect trim trajectories. The MA can be depicted as a directed graph as shown in Figure 62. The MA is a directed graph, as seen in Figure 62, consisting of a set of nodes and connecting arcs. The nodes represent the trim trajectories A that define equilibrium states for the vehicle. For example, the three trim trajectory model shown could represent an aircraft operating at constant altitude with one trim trajectory representing steady level flight and the two other trim trajectories representing left and right turns. The maneuvers Figure 62. An example Maneuver Automaton. A Maneuver Automaton defined by a maneuver library of three trim trajectories and nine maneuvers. I1,, defines the initial state transition by which the aircraft enters the directed graph. M define transitions between trim trajectory operating conditions with I1,, defining the initial state transition by which the aircraft enters the directed graph. The MA defines a set of dynamically admissible motion primitives for a given vehicle which can be concatenated to form a trajectory. To generate a motion plan, an optimization is performed which searches for the optimal sequence of maneuvers and the associated time duration for each connecting primitive. Therefore, the optimization searches for a trajectory defined by the sequence (w, ) A, (Ti)MwAI2(2) ...... ( ), A > 0, Vi {1,N+ 1} (66) where w is a vector determining the maneuver sequence and is a vector of coasting times that define how long the trim primitives are executed. The trim primitives are uniquely defined by the maneuver sequence and the constraint on 7 ensures that all coasting times are positive. The trim trajectories and maneuvers can be multiplied in this fashion because they represent transformations of the vehicle state written in a homogeneous representation. Therefore, general rigidbody motion denoted by a rotation R and translation T, such as a maneuver, is written in the form R T g = (67) 0 1 Similarly, timeparameterized rigid body motions, such as trim primitives, can be written in the form g(t ) t T(t) (68) 0 1 which can be written in the exponential form g(t) = exp(rt) where r] is called the twist. Optimal trajectories are found by solving the Nonlinear Program (NLP) (w*, r*) arg min J(w, 7r) ( MW + yA,1ri) + A7wA T (w,r)1 s.t. : gw n i'exp(qTi) go1 9 go 0 (69) w e L(MA) r > 0, Vi c {1, N+ 1} where the cost function J is defined by a summation of all maneuver costs Fm. and primitive costs 77T, which are scaled linearly by their duration of execution r7,. The first constraint ensures the system is driven to a goal state and second constraint assures that the solution maneuver exists in the set of all admissible maneuver sequences L(MA) defined by the MA. The posed optimization becomes computationally intractable as the length of the maneuver sequence and the number of available motion primitives increases. Therefore, the problem is decomposed into a search for the optimal maneuver sequence w* and an optimization to determine the optimal coasting times r*. This formulation, however, does not include constraints for obstacle avoidance. For obstacle avoidance, the MA motion planning strategy is coupled with samplingbased methods [94] (Section 6.4). 6.3 Decoupled Trajectory Planning The aforementioned direct trajectory planning methods employ strategies that plan directly in the state space of the system. Decoupled, or indirect methods, plan paths in the vehicle's configuration space and convert the resulting collisionfree path q(s) into a trajectory. Therefore, this method can be used to find the timeoptimal trajectory that follows a specified path in the configuration space [40, 41]. This dissertation employs a decoupled trajectory planning strategy in order to incorporate a more complicated cost function into the planning process than can be employ, ,1 efficiently into a direct method. Additionally, the full vehicle dynamics are used. The main drawback of this method is that the coverage cost is not included in the optimization. However, since the environment is unknown the planning strategy must be implemented in a RH fashion so optimality can never be guaranteed. 6.4 SamplingBased Path Planning Decoupled trajectory planning requires that a path be planned in the vehicle's configuration space. A popular methodology for efficiently solving the path planning problem in high dimensional spaces is known as samplingbased planners. Samplingbased planners randomly select collisionfree configurations from the vehicle's configuration space and connect the samples with kinematically feasible paths to form a solution. It should be noted that the MA can be combined with samplebased planning to directly produce dynamically feasible trajectories. The first samplingbased path planner to be developed was the Probabilistic Roadmap (PRM) [27]. The PRM algorithm constructs a roadmap in the workspace consisting of nodes and connecting arcs as shown in Figure 63. The nodes are vehicle configurations sampled from the vehicle's free configuration space Qfree. The connecting arcs are collisionfree paths generated by a local planner A that connects two sampled configurations and satisfies the kinematic constraints of the vehicle. Therefore, A contains a collision detection step alerting the planner if a collisionfree path cannot be constructed. The roadmap is intended to capture the connectivity of the vehicle's configuration space in order to efficiently plan feasible paths without an exhaustive search. The PRM process involves a learning phase in which the roadmap is constructed and a query phase in which multiple path planning problems are solved with the constructed roadmap. Therefore, the roadmap only needs to be constructed once. The roadmap is constructed incrementally in the learning phase by sampling a configuration from Qfree and connecting it to its k closest neighbors according to A. The distance metric is user defined and not necessarily Euclidean distance. This process is repeated until a predetermined number of nodes has been added as shown in Figure 63A. Once the roadmap has been constructed path planning queries, in the form of an initial configuration qinit and goal configuration qgoal, are answered. As shown in Figure 63B, qinit and qgoal are connected to the to their nearest neighbor in the roadmap q' and q", respectively. Lastly, a graph search is performed to find the shortest path connecting the initial and goal locations. i q A B Figure 63. Probabilistic Roadmap planning in a twodimensional environment. A) A constructed roadmap where the circles are nodes designating vehicle configurations and the straight lines are arcs connecting the nodes with kinematically feasible paths. B) The query phase where the shortest path is found through the roadmap given an initial and goal configuration. The PRM was designed as a computationally efficient strategy for capturing the connectivity of Qfree in order to answer multiple path planning queries in regards to the same environment. M\,f z planning strategies, however, require the answer to a single query and do not require exploration of the entire Qfree. Two examples of sampling based single query planners are the ExpansiveSpaces Tree (EST) planner and RapidlyExploring Random Tree (RRT) planner. The difference between the two approaches is the manner in which the roadmap, or tree, is constructed. 6.4.1 ExpansiveSpaces Tree Planner The EST planner was developed to efficiently cover the space between qinit and qgoal without needlessly exploring the entire Qfree. Instead of building a complete roadmap, like the PRM approach, the EST incrementally constructs a tree T that is guided toward the goal location. The tree construction process is outlined in Algorithm 4. Algorithm 4 ExpansiveSpaces Tree Construction 1: initialize tree T as a set of nodes V = qo and edges E = 0 2: for i = 1 to n number of tree expansions do 3: randomly select q from T with probability 7rT(q) 4: select a random collisionfree configuration qrand near q 5: if A finds a collisionfree connection between q and qrand then 6: V VU qrand 7: E EU {q, qrand} 8: end if 9: end for 10: return T The standard form of the EST constructs a tree Tinit rooted at qinit by incrementally adding branches to the tree in the form of nodes and connecting arcs. The tree construction process begins by randomly selecting a node q from the tree. A random sample qrand is then selected from the neighborhood of q and the local planner A attempts to connect the two configurations. If the connection is successful then qrand and the connecting arc is added to the tree. This process, depicted in Figure 64 continues until a termination criteria such as the maximum number of nodes in the tree or nodal proximity to the goal is satisfied. The most important aspect of the EST is the ability to direct the tree growth according to a prescribed metric. Guided sampling of Qfree is achieved by assigning a probability density function 7rT to the nodes of the tree. The probability density function biases the selection of tree nodes from which to extend the tree. This bias enables tree growth toward specified regions such as areas that are sparsely populated by nodes or the goal location. Figure 64. ExpansiveSpaces Tree planning. The EST involves constructing a tree T by extending the tree from randomly sampled configuration in the tree q. A random sample qrand local to q is connected to the tree by a local planner. Both a successful qrand and unsuccessful connection q"and are shown. The last step of the EST planning process is to connect the tree to the goal location Pgoal. This connecting is accomplished by checking for connections with the local planner A. If no connection is made, further expansion of the tree is required. Another variation involves simultaneously growing a tree Tgoal rooted at pgoal and proceeding with a connection process between q E Tinit and q E Tgoal to merge the trees. 6.4.2 RapidlyExploring Random Tree Planner The RRT planner, like the EST, was developed to efficiently cover the space between qinit and qgoal. The RRT differs from the EST with respect to the tree expansion process as described in Algorithm 5. Instead of extending the tree from randomly selected points in the tree T, as in the EST approach, the RRT extends the tree towards randomly selected points qrand in the free configuration space Qfree. Tree construction begins by selecting qrand and finding the nearest configuration near in T to the sample. The tree is extend from near a distance 6 toward qrand. The extension is performed by a local operator A terminating at qnew. If the constructed path is collisionfree, it is added to T. Tree construction, depicted in Figure V Algorithm 5 RapidlyExploring Tree Construction 1: initialize tree T as a set of nodes V = qo and edges E = 0 2: for i = 1 to n number of tree expansions do 3: randomly select qrand from Qfree with probability distribution 7Q 4: assign near to be the closest neighbor q in T to qrand 5: find path P between near and qrand with A 6: extract subpath p along P terminating at qnew a distance 6 from near 7: if p is collisionfree then 8: V V U qnew 9: E EU neara, qnew} 10: end if 11: end for 12: return T 65, continues until a termination criteria such as the maximum number of nodes in the tree or nodal proximity to the goal is satisfied. init qnear qinit q Y ,new qrand Figure 65. RapidlyExploring Tree planning. The RRT involves constructing a tree T by extending the tree toward a randomly sampled configuration qrand in Qfree. The tree is extended a distance 6 toward qrand terminating at qnew. Guided sampling can be incorporated into the RRT in a similar fashion to the EST. To guide sampling, qrand is sampled according to a probability distribution 7Q. Therefore, the main difference between the EST and RRT planners is the mechanism for guiding the sampling. The EST guides the sampling by selecting attractive nodes of the tree to extend while the RRT expands by selecting regions in the environment to move towards. Though the expansion methods are similar, this difference can produce very different results and computational performance. Like the EST, the RRT planning process culminates in an attempt to connect the current tree to the goal location pgo,,. If the local planner A cannot construct a collisionfree path to the goal from any configuration in the tree further tree expansion is required. This section has describe two strategies for efficient path planning in cluttered environments. The next section describes the local planning function A which connects the nodes in the trees of both the EST and RRT. 6.5 The Dubins Paths The samplingbased planners described in Section 6.4 rely on a local planner A to connect vehicle configurations. The local planners are defined by vehicle kinematic constraints so that the constructed path is kinematically feasible. For nonholonomic vehicle motion pl1 "'ii:ir a popular local planning strategy is the Dubins optimal path [34, 95, 96]. The Dubins optimal path is a shortest path solution for connecting two vehicle configurations subject to a minimum curvature constraint. The solution pertains to a carlike vehicle operating in a threedimensional configuration space defining location and orientation qi = [xi, yi,' ] E R3. The curvature constraint relates to the minimum turning radius of the vehicle pmin. The seminal work by Dubins [33] proved that the optimal curvature constrained path consists of three path segments of sequence CCC or CSC where C is a circular arc of radius pmin and S is a straight line segment. Therefore, by enumerating all possibilities the Dubins set is defined by six paths D = {LSL, RSR, RSL, LSR, RLR, LRL} where L denotes a left turn and R denotes a right turn. The optimal Dubins path q*(s) is the shortest path q(s) E D. For a mobile robot, the optimal Dubins path is the timeoptimal path for nonaccelerating motion where pmin is unique to the given constant velocity. The Dubins set can be calculated analytically for any pair of configurations qinit and qgoal. These general configurations are first transformed to the coordinate system pictured in Figure 66. The transformed configurations have the form qinit = [0, 0, ]T and goal = [d, 0, igT where qinit is located at origin and qgoal is located on the yaxis. X Figure 66. I_ I Ii Y (0,0) / (d,0) goal The coordinate system for calculating the paths in the Dubins set D. The initial configuration is qinit is translated to the origin and rotated so that qgoal is located on yaxis. The Dubins paths are the concatenation of three canonical transformations scaled so that pmin = 1 and the vehicle moves with unit speed. This formulation allows the general solution to be scaled by the vehicle's pmin because the maximum heading rate is equal to . The transformations are written in terms the distance of motion v along the path Pmin segments for an arbitrary configuration q = [x, y, ]T as follows L, (x, y, Q) R,(x, y, Q) S,(x, y, Q) [x + sin(Q + U) sin Q, y cos(Q + v) + cos Q, Q + V]T [x sin(Q vU) + sin Q, y + cos( v v) cos Q, V]T [x + v cos Q, y + v sin b, Q]T (610) The total length of a Dubins path is the sum of the individual segment lengths vi, v2, v3 that transform the initial configuration [,0,0,' ] to the final configuration [d, 0, 'T, L = VI + V2 + V3 (6 11) The Dubins paths for 1. Dubins path LSL V V2 V3 2. Dubins path RSR V1 3. Dubins path 4. Dubins path 5. Dubins path V2 V3 LSR V1 V2 V3 RSL vi V2 V3 RLR the entire Dubins set are: +tan' cos gcos \ I' +tan1 (C O d+sin ybsin g) /2+ d2 2 cos(' '. + 2d(sin sin ,g) ta.n1 cosg cos S, tan1 (d+sinCsing tan 1 cos Ybcos ,g = dsin tbi +sin bg = 2+ d2 2 cos(' ') + 2d(sin g sin,' )  + tan'1 COS iCOS lbg g' ( t d sin YHsin Yg + tan C1 ( cos is g ( d+sin 9bi+sin 9g V/2 + d2 + 2 cos( ,) + +tan' cos cos g \ d+sin ii+sin tb tan1(2) 2d(sin + sin ,) tan1 (2 v2^ Stan cos1 +cosg) + an1 (2 dsin isin g ) V2 2 +d2 + 2cos(,' g)+ 2d(sin +sin,',) tan1 ( cosicos#g )+ an1 (2) dsinisinpg V2a S Ota I ( COS 9bCOSbg \ V2 V1 I tan dsin sin bg, ) 2 V2 COS1 ((6 d2 2 cos(,'  V3 = g V +V2 6. Dubins path LRL +',) +2d(sin,' sin ',))) , + tanl ( cosgcosb )+ Sd+sin isinbg ) cos ( (6 d2 2 cos('  + ', V I +V2 .,') +2d(sin,', sin' )) (612) (613) (614) (615) (616) (617) It should be noted that all segment lengths in these Dubins set calculations that represent angular distance are constrained to be on the interval [0, 27). An example of the Dubins set is shown in Figure 67. 2 1.5 ... ..x. 1 4 '" ' qini 1 q*() LSLR 1.5  O R SR : 2 x LSR 7V SL 2.5 RLR ** ** 3 I I I I I I I I I 2 1 0 1 2 3 4 5 6 7 xaxis Figure 67. The Dubins set of paths for two configurations. Notice that the LRL path does not give a feasible solution for this set. The timeoptimal is denoted as q* (s). 6.5.1 Extension to Three Dimensions The Dubins set determines the shortest curvatureconstrained path between two configurations in a planar environment. Aircraft are not constrained to inplane motion so altitude variations must be included to properly characterize the vehicle kinematics. A recent application of the Dubins set to three dimensions assumes a linear variation of altitude limited by the maximum climb rate of the vehicle [97]. A method similar to the MA (Section 6.2.2) plans an initial path which places the vehicle at a position and orientation that forms a plane with the final position and orientation [98]. A 2D Dubins path is then planned between the two inplane configurations. Because the dynamics of the vehicle will be incorporated after the path planning task, the 3D Dubins solution employ ,1 in this dissertation assumes that changes in flight path angle 7 are instantaneous. This solution follows the assumption for 2D Dubins paths that assume changes in heading rate are instantaneous. The flight path angle for each individual segment of the Dubins path can vary so that the maximum flight path angle 7max for curved and straight segments can differ. A smoothness constraint limits the maximum change of fight path angle 6 ,max between segments. Therefore, the altitude variation for the 3D Dubins path is piecewise linear. A few example 3D Dubins paths are presented in Figure 68. 250 120 0 502250  5105 250 2001 2D0 O 50 15200  100 200 150 20 xaxis 20 I 1, 250 250 20 150 100 50 50 100 150 200 250 yaxis xaxis A B Figure 68. Examples of 3D Dubins paths. The paths originate from for the same start configuration and terminate at randomly generated goal configurations. A) Perspective view. B) Topdown view. 6.5.2 Collision Detection 6.5.2.1 Global path collision detection The 3D Dubins path defines the local planner A to be used for connecting aircraft configurations. Each path must be checked for collisions before it can be added to the tree T according to the samplingbased planning strategies (Section 6.4). The result of A is a path composed of piecewise linear segments. The sensed environment is represented by a set of bounded planes. Therefore, the collision detection algorithm must search for collisions along the entire path by detecting intersections between linear path segments and planes as depicted in Figure 69. Figure 69. Collision detection for the global path. The collision detection algorithm searches for any intersections between the piecewise linear paths of a tree planner and the planar environment obstacles. The first step of the collision detection algorithm is to search for intersections between each piecewise linear segment of the path and all infinite planes in the current map. The intersection point pint is calculated by a pi.f+D vfl (618) Pint = Pi av where pi is the first point of the path segment, v is the vector of the path segment originating at pi, fi is the normal direction of the plane, and D is the intercept of the plane from the plane equation Ax + By + Cz + D = 0. If the scalar value a is on the interval [0, 1] then an intersection exists along the finite path segment. Alternatively, if the denominator v in is zero the line segment is parallel to the plane. Therefore, the path segment is parallel to the plane and may be contained within the plane. These cases can be differentiated by checking if one of the end points satisfies the plane equation Ax + By + Cz + D 0. Once the intersections between the path segments and the infinite planar obstacles have been calculated a search for collisions can be completed. A collision exists only if the intersection point for each segmentplane pair is contained within the specified Figure 610. Interior to boundary test. A point that intersects with the infinite plane is within the boundary defined by the convex hull H if the sum of the interior angles is equal to 2r. plane boundary. C'!h pter 3 defined each planar boundary by a convex hull H {hi, hi,..., h.} e c of n boundary points. Therefore, the test for determining if a collision has occurred becomes determining if the intersection point pint is interior to the threedimensional boundary H. This determination is accomplished by inspecting the two dimensional projections onto the planar landmark of pint and H denoted by pint E IR2 and H . As shown in Figure 610, pint can be connected with every point in H. If the sum of the angles between the connecting vectors is equal to 27 then the point is interior to the boundary. This constraint can be written for n boundary points as Scos1 (h pint) (h+l pint) 2 (69) i h, pint (hi+ Pint) where the value h +1 is equal to hi. 6.5.2.2 Local path collision detection The above formulation for collision checking is wellsuited for the global planning task. It is a computationally efficient solution that can quickly search a large number of lengthy paths for collision with a large environment map. The method assumes that the vehicle is a point and that there are no uncertainties in vehicle and obstacle location. The local planning strategy, however, requires a more detailed approach to account for the various uncertainties in the problem. The current map and relative vehicle position are subject to uncertainty due to sensor noise, modeling errors, and external disturbances. Additionally, the constructed kinematically feasible path is, in general, not dynamically feasible due to instantaneous changes in heading rate and climb rate. Therefore, the trajectory will not exactly follow the planned path. To account for these uncertainties, the local path collision checking algorithm ensures safe navigation by increasing the required distance between the vehicle and obstacles. The local collision checking algorithm represents the path as a cylinder of radius R. The radius of the cylinder is a function of the various uncertainties in the problem. The radius must be as large as the greatest variance in the vehicle's and obstacles' inertial location. Additionally, the radius must account for deviations of the trajectory from the prescribed path which can be determined from simulation of the optimal control problem presented in Section 6.7. Therefore, the local collision checking algorithm searches for intersections between the piecewise linear path segments represented by finite length cylinders and obstacles represented by bounded planes (see Figure 6.5.2.2). This search is accomplished by first finding the intersection point pint between the central axis of the cylinder and the obstacle using Equation 618. The intersection between the cylinder and plane is an ellipse with center pint. As shown in Figure 6.5.2.2, the i i, i axis of the ellipse will be R where p is cos the angle between the plane's normal vector and the cylinder's axis. The minor axis of the ellipse has a value R. The orientation of the ellipse is determined by the in ii' axis which is oriented in the plane defined by the plane's normal vector and the cylinder's axis. The discretized ellipse can be tested for being interior to the plane boundary with Equation 619. 6.6 Environment Coverage Criteria Sections 6.4 and 6.5.2 described a method for constructing collisionfree paths in a cluttered environment. This section defines a method for selecting paths with the goal of constructing a complete environment map in minimum time. For this situation, optimality 2R? Cos c\ SCOS~ A B Figure 611. Collision detection for the local path. The local collision detection algorithm incorporates a safety region which encompasses the path segments radially. A) The path segment represented by a cylinder with radius R intersects an obstacle described as a plane with normal vector fi. B) A twodimensional view rotated orthogonal to the plane containing the in i, r axis of the intersecting ellipse. cannot be proven because initially the environment is entirely unknown. Therefore, an antagonistic example can be constructed for any planner to provide a nonoptimal result. The robotic coverage problem refers to optimal coverage of an environment with known obstacles [99]. Solutions to the robotic coverage problem generally involve decomposing the free space of the environment into connected regions. Each region of the environment is sensed with a back and forth lawnmower, or boustrophedon, pattern as shown in Figure 6.6. The boustrophedon path is optimal for a twodimensional holonomic vehicle under the assumption that passing through a small region of the environment is equivalent to sensing the region. The space decomposition process reduces the planning task to finding the optimal route for which to connect the individual paths. When the environment is unknown and the planning task must be directed by gathered sensor information, the problem is called sensorbased coverage or sensorbased exploration. Sensorbased coverage tasks typically operate under several strict assumptions including a twodimensional environment, an omnidirectional range sensor, and a A B Figure 612. Motion planning for coverage strategies. Traditional motion planning strategies for environmental coverage are not well suited for vehicles with dynamic constraints and directional sensors. A) Robotic coverage tasks assume the environment is known and decompose the free space into manageable search spaces. B) Sensorbased coverage strategies often assume omnidirectional sensors. robot capable of nonholonomic motion [100, 101] (see Figure 6.6). Therefore, current sensorbased coverage algorithms do address the problems inherent in mapping a threedimensional environment with a dynamically constrained vehicle equipped with a directional senor. The coverage methodology used in this dissertation affects both the global and local planning strategies. The strategy is based on tracking all locations to which the vehicle has traveled and the attitudes from which individual locations have been sensed. The goal of the planner is maximize the variance of these individual quantities. Therefore, the discrete vehicle locations and the vectors defining location and orientation of sensed regions are tracked over time as shown in Figure 613. The vectors defining sensed region are constructed from the attitude of the vehicle during the sensing operation and the center of the field of view of the sensor. The total variance of the vehicle position is calculated as the trace of the covariance of the vehicle positions qpos as given by as = tr(EpoS) (620) p I I I I I I I I ]l~ii I I ,., Figure 613. Environmental coverage metric. Environment coverage is quantified by the variance of the position of the vehicle and sensing locations over the entire path. Sensing locations are defined by the vector originating from the end of the sensor field of view oriented back towards the sensor. where Epos is the sample covariance of the all vehicle position vectors. The same equation can compute the variance of the the sensor pose vectors a21n given the covariance of the sensor pose vectors sen. The variance of the vehicle position apos is used to determine the global location by selecting the point in the environment that, when added to the current vehicle location history, maximizes apos. This calculation is computed on a coarse grid of the vehicle environment for the purpose of computational efficiency. The global path planner constructs a path to this location with a greedy search and a coarse discretization of the path. The local path is determined by maximizing the variance of the sensor pose vectors gen over all candidate paths. As opposed to the quick, greedy search used for global path 1p1 iiiiiiii the local samplingbased strategy generates a tree that more completely explores the environment. When a specified number of candidate paths are found the path that maximizes aen is selected. 6.7 Optimal Path Tracking The presented path planning strategy computes short detailed paths within the sensor field of view. The local collision detection algorithm (Section 6.5.2.2) ensures that the planned paths traverse the environment inside of a safe corridor. The optimal path tracking strategy seeks to find a time optimal trajectory that follows the planned path within a distance r < R subject to the vehicle dynamics. The constraint r is less than the radius of the safety corridor because additional uncertainties incurred in the map building process must be accounted for. The dynamic vehicle model used for optimization is the navigationlevel aircraft model V Tcos(a) CDV2 sin7 S (sin(a) + CLV) cos Q C` = ( sin(Ta) + CL V) s(in2 cos 7 (6 21) S= V sin 7 S = V cos 7 cos Q S = V cos 7 sin Q where V is total vehicle velocity, a is angle of attack, 7 is flight path angle, 0 is bank angle, Q is heading, T is thrust, CD is the drag coefficient, and CL is the lift coefficient. The thrust and lift and drag coefficients are approximated by polynomial fits parameterized by V and a T =Ao+AIV+A2V2 CD Bo + Ba + B2a2 (622) CL =C Co + C1 + C2(a 1i)2 where the constant values are given by Bryson [39]. The model was selected because aircraft lift and drag coefficients are incorporated into the formulation which provides a realistic aircraft response. Additionally, lift and drag curves from UAV wind tunnel test could be included into the trajectory tracking optimization. The model was augmented so that the control inputs, a and Q, have second order and first order dynamics, respectively, as given by S(623) where cj,~ is the the natural frequency for the a response, (Q is the damping ratio corresponding to a, Tr is the time constant for the Q response, and the new control inputs to the system are us and u6. The optimization problem is formulated as the following nonlinear program (NLP) (u*) argmin J(u) t + Zt i 1(Xk,q(s)) s.t. Xk+ f(xk, uk)dt (624) (624) XN qf Tk where T is a function that calculates distance between the discrete trajectory points Xk and the path q(s). The objective function J minimizes final time tf and the sum of the distances between the path and the trajectory by varying the control input u = [ua, u]1. The first constraint is an Euler integration step between two successive trajectory points ensuring the trajectory is dynamically feasible. The second constraint forces the endpoints of the trajectory and the path to be equal. The final constraint ensures each calculated values of the error function T are less than the safety radius r. The NLP was solved with the nonlinear optimization routine SNOPT as a part of the Matlab TOMLAB toolbox [102]. 6.8 Trajectory Planning Algorithm An overview of the trajectory planning approach is presented in Algorithm 6. The presented formulation is executed until the change in sensor variance a2e falls below a threshold tolsen. Therefore, the planning process stops when additional vehicle motions do not provide additional environment coverage. Supplementary metrics such as a maximum final time can easily be employ, ,1 in the same framework. Algorithm 6 Trajectory Planner for Environment Coverage 1: repeat 2: calculate covariance Epos of all vehicle locations in the trajectory 3: select the global goal location qG that maximizes apos 4: plan greedy RRT to qG using Algorithm 5 5: select the shortest distance path qc(s) from the global RRT 6: select local goal location qL on qc(s) inside sensor field of view 7: repeat 8: extend exploratory RRT towards qL using Algorithm 5 9: until N desired candidate paths q(s) have been generated 10: calculate sensor covariance Esen along each q(s) 11: select q(s) that maximizes aen 12: convert local path to trajectory using Equation 624 13: execute path to gather sensor data and build environment map 14: until aent + 1 Int < tolsen The planning process begins with the determination of the vehicle's global goal location qc. The global goal is selected so that it's addition to the vehicle position history maximizes the variance of vehicle position Rpos. This process greedily drives the vehicle to locations in the environment that have not been explored. For example, under this strategy a vehicle starting in one corner of a rectangular environment will select a goal location in the opposite corner. Therefore, the vehicle will cross the environment diagonally performing a greedy search. An RRT is planned from the initial vehicle configuration qint to the goal location qc. The global RRT is extended in a greedy fashion in order to rapidly generate a path solution to the goal location. The tree expansion is made greedy by sampling qG with a high probability. The fact that other randomly generated configurations are sampled for connection to the tree differentiates the greedy RRT from an artificial potential function planner. This differentiation is important because it prevents the planning strategy from becoming mired in a local minimum. The solution path is denoted qc(s). Next, the local goal configuration qL is selected. The configuration qL is found by selecting the first point of intersection between the sensor field of view and qG(s). A specified distance from the intersection point along qg(s) towards the interior of the field of view is selected as qL. A local RRT is planned to this goal from the initial configuration qint. Unlike the global RRT, the local RRT is performed in a less greedy fashion until N number of candidate paths have been generated. The path which maximizes the senor variance an2 is selected as the local path qL(s). The local RRT branching strategy is biased towards regions that maximize variance. Therefore, the space within the sensor field of view is efficiently searched for informative paths. The paths are restricted to the field of view so that the detailed planning occurs within the region of lowest uncertainty. Lastly, the local path path is converted to a trajectory by solving the NLP given in Equation 624. The flexibility of the planning framework allows for the optimal control formulation to be substituted with a MA implementation as described in 6.2.2. Both methodologies would incorporate dynamics into the final solution. The final planned trajectory is implemented and the planning process is repeated when the vehicle arrives at qL. This chapter has presented a framework that plans aircraft trajectories for complete sensor coverage of an unknown, uncertain environment. Results of the planning strategy are presented in C'!i Iter 7. CHAPTER 7 DEMONSTRATIONS AND RESULTS 7.1 Introduction This chapter presents results and analysis of the algorithms presented throughout this research. Section 7.2 gives results of the plane fitting algorithm introduced in ('! Ilpter 3. The plane fitting algorithm is employ, 1 in the SLAM algorithm (C'!i lpter 5) to produce the results in Section 7.3. Finally, results of the trajectory planning algorithm (C'!i lpter 6) are presented in Section 7.4. 7.2 Plane Fitting Results The plane fitting algorithm introduced in ('!, lpter 3 generates a low dimensional environment representation of the pointwise sensor data provided by SFM (Chapter 2). The quality of the fit is investigated with respect to feature point noise in Section 7.2.1 and the inclusion of nonplanar obstacles in Section 7.2.2. The simulation environment employ, 1 for analyzing the plane fitting algorithm is shown in Figure 71. feature points  vision frustum planes 600, 480. 360, 240. 120, 1000 1000 Figure 71. Result of the plane fitting algorithm without image noise. The simulation environment in Figure 71 consists of polyhedral obstacles viewed by a camera at a given vantage point denoted by a vision frustum. The obstacles represent an urban environment and the frustum indicates the field of view for a camera attached to a UAV navigating the environment. Feature points are added to the obstacle facades and checked for visibility according to a pinhole camera model. The visibility constraints account for environmental occlusion, sensor field of view, and angle of incidence. The angle of incidence constraint for a feature point is depicted in Figure 72 where feature points with an angle of incidence, 0 are deemed visible if 15 < i < 75. Lastly, all visible feature points are input into the plane fitting algorithm to generate the planar fit of the pointwise data. C2 C1 Figure 72. Angle of incidence constraint. The angle of incidence, 0, for feature point f is the angle between the feature point's position,f, measured in the camera coordinate frame and the normal direction, fi, of the plane on which the feature point is situated. 7.2.1 Effects of Feature Point Noise Image noise is simulated by adding noise to the image plane positions of the tracked feature points. This is accomplished by calculated the twodimensional projections of the feature points at two distinct vantage points. Noise is added to the twodimensional feature point locations in both images and noisy three dimensional points are calculated by triangulation (Equation 214). Results of the plane fitting algorithm in the presence of noise are shown in Figure 73. 480 360 240 \ 120 800 \\\ 600\ North ift) 400 240 S20 / 800 North (ft) 400 200 Feature points viSl on frustum ; , , , 400 00 400 East (ft) Feature points vision frustum 600 p planes 480 360 120 /: 800 600 1000 \\ North (ft) 400 \0 800 20 4 600 200 \ / 400 S200 East (ft) 200 ai S 600 400 East (ft) Figure 73. Results of the plane fitting algorithm in the presence of image noise. A) Plane fitting result with a = 0.5 pixel image noise and a camera displacement of 25 ft. B) Plane fitting result with a = 1.0 pixel image noise and a camera displacement of 25 ft. C) Plane fitting result with a = 0.5 pixel image noise and a camera displacement of 15 ft. D) Plane fitting result with a = 1.0 pixel image noise and a camera displacement of 15 ft. The plane fitting results, shown in Figure 73, were generated with Gaussian noise of 0.5 and a = 1.0 pixels applied to the images. Additionally, the  feature points vision frustum 600 North (ft) 600 200 400 200 East (ft) 200 standard deviation a distance between the two vantage points was varied to values of 25 and 15 feet simulating different values for camera translational motion. The results show that noise does not affect the plane fitting algorithm for nearby obstacles. This is a result of large apparent motion of the feature points associated with nearby obstacles. The large displacement in the image plane of such feature points produces more accurate results because the image noise is small in comparison to the actual feature point displacement. The plane fitting algorithm, however, produces poorer results for obstacles at a large distance, especially for shorter motions between image vantage points. The effect of camera displacement can be seen by comparing Figures 73B and 73D. These two results show that increasing the baseline between camera vantage points produces more accurate triangulation results leading to a more complete planar representation. 7.2.2 Nonplanar Obstacles The plane fitting algorithm assumes that the environment is highly structured. This is not entirely accurate for urban environments which often include nonstructured obstacles such as foliage. Figure 74 presents plane fitting results when nonstructured obstacles are included in the form of several treelike objects. The additional feature points generate spurious planar features. Such features are inaccurate representations of the physical world and provide poor landmarks for the SLAM algorithm. Specifically, the spurious planes are not necessarily observable from multiple vantage points. In order to reduce the number of spurious planar obstacles in the map, the SLAM algorithm only incorporates obstacles that have been reobserved a minimum number of times. A minimum value of 3 reobservations was found to provide successful map results in the presence of spurious planar features. 7.3 SLAM Results Simulations of the SLAM formulation described in C'!i pter 5 are performed for the three filters presented in this dissertation: the RBPF, URBPF, and SRURBPF. Feature points vision frustum 600 planes 480 360, 240 120 0 1000 800 600 1000 North (ft) 400 800 200 \ 4008 East (ft) 200 0 0 Figure 74. Result of the plane fitting algorithm when nonplanar obstacles are included. The filters were compared by performing Monte Carlo (' \C) simulations in which the vehicle state measurement noise and sensor noise were randomly generated within the confines of the covariance matrices Eq and Er, respectively. This process tests the average performance of the filters over many estimation processes. 7.3.1 Simulation Environment The simulation environment, depicted in Figure 75A, consists of geometrically regular obstacles indicative of an urban environment. The aircraft navigates along a precomputed trajectory generated by a highfidelity nonlinear aircraft model. SFM, as discussed in C'! lpter 2, is simulated by distributing threedimensional feature points on the building facades and ground. The feature points provide pointwise structure information to the plane fitting algorithm. The plane fitting algorithm, introduced in C'! lpter 3, generates observations as the vehicle navigates as shown in Figure 75B. The vehicle state prediction occurs at 30 Hz. Though an IMU could provide a faster stateestimation rate, it is assumed that the IMU is fused with image processing 780 520 East ( East (ft) Figure 75. The visionbased flight simulation environment for testing the SLAM algorithm. A) The urban simulation environment is composed of geometrically regular obstacles representing buildings. A precomputed trajectory generated from a nonlinear aircraft model is shown circling the environment. B) As the aircraft navigates along the precomputed trajectory sensor information is gathered. The gathered sensor information is restricted to a senor cone defined by a maximum depth of sensing and horizontal and vertical fields of view. Threedimensional feature points are distributed on the building faces and planes are fit to the pointwise data. 500, 400 300 200 100 0 1500 1200 North (II. 1040 information which is limited to 30 Hz. The observation updates in the SLAM process occur at 3 Hz to allow for sufficient parallax for robust triangulation in creating threedimensional feature points. The threedimensional feature points are generated from the triangulation of twodimensional image points, therefore, noise can be reduced by allowing for a larger camera translation between calculations. If the translation is too larger, however, twodimensional feature points will leave the image plane and no threedimensional information will be calculated. This problem can be alleviated by testing SFM algorithms on the actual flight data to determine the optimal rate for triangulation or by instituting an adaptive step where triangulation occurs at the maximum camera motion that retains the largest number of tracked feature points. 7.3.2 Localization Result A localization result of the SLAM algorithm is shown in Figure 76. The result, obtained with the URBPF approach, simulates a known environment by initializing the landmarks to their actual locations. This is equivalent to using the SLAM framework strictly for localizing the vehicle in the presence of noisy state measurements and noisy sensor measurements. The noise values, defined in Equations 53 and 54, were given the following standard deviations au =5 ft/s ac = 5 ft/s O, 0.1 ft 5 ft/s ao,, 0.1 ft (71) a = 0.05 rad/s a,3 0.1 ft ao = 0.05 rad/s ad = 5 ft a, = 0.05 rad/s The localization result(Figure 76) shows the filter is capable of producing very good localization results in the presence of considerable noise. The estimated trajectory given by the SLAM algorithm is defined by the particle with the largest weight and the 500, 500, 400 400 01000 S00 1300 S measured trajectory 1040 A B North (ft) 52 4 13001300 20 .13 S8 780 200 1 0 0 500004 300,780El780l 30 / 520 East (ft 100 1200 900 000 300 0 0 North (ft) C D Figure 76. Example localization result for the airborne SLAM simulation. A) The complete localization result with constructed map and estimated trajectory. B) The constructed map after a single loop through the environment. C) The localization result without the constructed map. D) A topdown view of the actual vehicle trajectory, trajectory predicted from measured vehicle states only, and the trajectory estimated by the SLAM algorithm. measured trajectory is calculated from the noisy state measurements without updates from the vision sensor information. The estimated trajectory quickly converges to the groundtruth trajectory as is considerably more accurate than the model based prediction over the entire trajectory. This shows the versatility of the SLAM algorithm in that it can be employ, ,1 if the structure of the environment is known to provide accurate estimates of vehicle location despite very noisy sensor measurements. 7.3.3 State Estimation Error MC simulations were performed to analyze and compare the performance of the three SLAM filters. Fifty MC runs were generated for each filter using the same environment with randomly generated noise added to sensor and vehicle rate measurements. The noise was generated according to the following standard deviations a, = 0.5 ft/s a, = 0.5 ft/s a, 0.1 ft ,, = 0.5 ft/s a,2 0.1 ft (72) a, = 0.005 rad/s a,,3 0.1 ft aq = 0.005 rad/s ad = 10 ft a, = 0.005 rad/s Figure 77 shows the average position and orientation errors for the three SLAM filters along with the average error of the measured trajectory. The measured trajectory is the trajectory calculated purely from the aircraft kinematic model and velocity measurements without updates from the environmental sensing. These graphs indicate the accuracy of the position and attitude estimation. The results show that the SLAM algorithms provide a significant decrease in estimation errors for aircraft position, roll angle, and pitch angle. The heading angle estimates, however, begin to diverge halfway through the simulation. The apparent sharp decreases in error for the measured trajectory can be attributed the erratic behavior of the measured trajectory. The presented error plots show the absolute value of the errors, therefore, when the error changes from positive to negative it crosses the zero error axis and appears as a decrease in error. This sharp decrease should not be mistaken for acceptable estimation performance. 7.3.4 Filter Consistency Filter inconsistency refers to a the production of optimistic or conservative estimates of estimation error. In other words, the actual variance of the estimates is significantly larger or smaller than that predicted by the model. Filters that are inconsistent produce RBPF URBPF SRURBPF SMeasured 10 20 30 40 50 60 70 80 90 time (s) RBPF URBPF SRURBPF Measured 10 20 30 40 50 60 70 80 90 time (s) RBPF URBPF SR URBPF Measured 4t 2 12 10 20 30 40 50 60 70 80 90 time (s) 2 18 16 14 112 '08 06 RBPF URBPF SRURBPF Measured 10 20 30 40 50 60 70 80 90 time (s) RBPF URBPF SRURBPF Measured 0 0 10 20 30 40 50 60 70 80 90 time (s) RBPF URBPF S SRURBPF Measured S i ^i,,fl 10 20 30 40 50 60 70 80 90 time (s) Figure 77. Average estimation error for the three SLAM filters and the nominal trajectory for 50 Monte Carlo runs. A) Estimation error in the North direction. B) Estimation error in the East direction. C) Estimation error in altitude. D) Estimation error in roll angle. E) Estimation error in pitch angle. F) Estimation error in heading angle. 114 poor state estimates without any internal knowledge that the estimates are erroneous. A common method used for determining the consistency of a SLAM filter is called the Normalized Estimation Error Squared (NEES)[79] calculated according to lk = xk Xkk)(Pklk)(Xk Xkk) (7 3) where xk is true vehicle state and ^klk is the vehicle state estimate. This metric relates the actual variance of the estimation to the variance predicted by the filter and should have a value equal to the dimension of the state vector. Therefore, the NEES is a measure of how well the filter is predicting it own performance. If the filter has small covariance values for the the states, but the actual errors are large, then the NEES value will be large and the filter is optimistic. Conversely, if the state values are assigned a large covariance by the filter, but the actual estimation errors are small, the NEES value will be small and the filter is conservative. An optimistic filter is sensitive to noise and may diverge easily while a conservative filter will produce poor estimates relative to the performance of a consistent filter. The NEES is calculated as an average over 50 MC runs in order to properly characterized the SLAM filters. For a system with 50 Monte Carlo runs and a state vector of dimension 6, the twosided probability region for the NEES is [5.08, 7.00]. This means that a conservative filter will have an NEES below 5.08 and an optimistic filter will have NEES values above 7.00. Ground vehicle simulations have shown that EKF SLAM and RBPF SLAM, though shown to work in practice, are theoretically inconsistent filters [103, 104]. The consistency of the filters, however, is an acceptable metric for assessing the robustness of the filter because the length of time that a filter remains consistent relates to how well the filter performs over long time periods. The results presented in Figure 78 show that the URBPF provides the best results in terms of filter consistency. This is likely due to the fact that the URBPF does not require linearization as does the RBPF. Also, it is believed that the numerical approach to calculating the squareroot of the covariances in the SRURBPF compromises the filter's consistency. Additionally, is should be noted that increasing the number of particles improves the consistency of the filters. 200 600 S RBPF 100 particles 180 URBPF 1000 particles 160 SRURBPF 5 loo0 300 z 80 Z 60 200 , 40 100 120 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (s) tme A B 80 602 200 1 2100 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (s) time A B Figure 78. Filter consistency results. A) Average NEES for all three SLAM filters over 50 Monte Carlo runs. B) Average NEES for the URBPF filter for 50 Monte Carlo runs with 100 and 1000 particles. 7.3.5 Filter Efficiency The efficiency of the investigated filters is shown in Table 71. Since the filters are implemented with nonoptimized MATLAB code, the efficiency results have been presented as computation time normalized to the RBPF result for N particles. The comparison includes results for N, 2N, and 4N particles to compare the effects of increased sampling on filter efficiency. The efficiency results show that the RBPF filter is considerably faster than the other two filters and that the filters scale linearly with the number of particles. The inefficiency of the unscented filters is due to the sigma point propagation step in which 2L + 1 sigma points must be transformed according to the system dynamics and sensor model. The URBPF, however, is a more consistent filter meaning that it is more robust with respect the length of the estimation time. Therefore, fewer particles could be used in the URBPF to produce results comparable to the RBPF over the same time period. This decrease in the number of particles would decrease the computational efficiency difference between the URBPR and RBPF. Table 71. Efficiency comparison of the SLAM filters. Number of particles RBPF URBPF SRURBPF N 1.00 2.06 1.60 2N 2.07 3.97 2.85 4N 3.98 7.63 5.64 7.3.6 Example SLAM Result A sample result of the complete SLAM algorithm is shown in Figure 79 where the estimated trajectory given by the SLAM algorithm is defined by the particle with the largest weight at each time step. The error covariances for this solution are given in Equation 72. 7.4 Trajectory Planning Results This section presents results for the the trajectory planning algorithm introduced in C'i lpter 6. The trajectories were planned within the simulation environment presented in Section 7.3.1. The constructed maps used in the planning process are exact, representing a SLAM solution with no error. 7.4.1 Environment Coverage The trajectory planning process is decomposed into two phases denoted the global and local planners. The global planner moves the vehicle to previously unexplored regions of the environment and the local planner selects paths within the currently visible environment that increase sensor coverage. In the complete planning process both strategies are employ, 1 in tandem so that the global planner produces a global goal location that performs a greedy search of the environment, and the local planner generates detailed obstaclepaths that locally maximize sensor coverage. The global planning process represents traditional planning strategies in that the physics of the sensor are not considered. The global planner selects unexplored regions of the environment in order to perform a greedy search under the assumption that once 500 55k00 S300 00 200 1200 0 0 '", L 1500 15000 " \ 1 S1200 0 < 13000 9 O 90oC" ' th (" 1040 N.Ah (ft) 60 1040 Nodh (fh 600 078 Northft) 0 <"o \ 260 0/ 260 A B 1300r " .nan/  measured trajectory  actua trajectory I i S... .~ 'measured trajectory S* estimated trajectory 5000 4 00 S00  S200 < 01200 900 600 300 N t () 2Noth (78) 0 M" 3 50260 20 100 1200 900 600 300 0 N North of7) C D Figure 79. An example result for the complete airborne SLAM solution using the URBPF. A) The complete SLAM result with constructed map and estimated trajectory. B) The constructed map after a single loop through the environment. C) The SLAM result without the constructed map. D) A top down view of the actual vehicle trajectory, trajectory predicted from measured vehicle states only, and the trajectory estimated by the SLAM algorithm. a region has been reached it has been sensed. This, however, is not correct for a directed sensor such as an onboard camera. In this case, direct paths through the environment will not provide the needed information to build a complete map. The local planning strategy incorporates the sensor physics by tracking vehicle poses and selecting paths which maximize the variance in sensing vantages. Therefore, paths are selected according to the amount of new information that will be provided by traversing the path. The local planning process occurs within the vision cone defined by the onboard camera in order to ensure that the planner incorporates the most recently sensed environmental information. This ensures that the local path is collision free. 260 East (ft) 1500,, Njfl L Nv. (* L / / 0 260 520 780 1040 1300 East (ft) 260 520 780 1040 1300 East (ft) Figure 710. Example paths for environment coverage. A) Example result for the global planning strategy. B) Example result for the complete planning strategy. C) Topdown view of the global planning strategy result. D) Topdown view of the complete planning strategy result. In order to quantify the benefit of the overall planning strategy, Figure 710 presents trajectory results for cases when the planning process is restricted to the global planner and when the complete planning strategy is employ, l The first case, where just the * 300 1500 1200\ North (ft) S/ 1040 300 7 BO 520 260 East ift) 30150 1300 780 1200 900 90 North (ft) 600 300 520 global planner is employ, 1 represents traditional planning strategies that do not account for a directed sensor. The second case is the complete planning strategy which accounts for a directed sensor by tracking vehicle poses and selecting paths which maximize the variance in sensing vantages. For the example result, each case was run for an equal amount of time. Since the global planner selects more direct paths through the map, its resultant path traverses a larger portion of the environment. The path that results from including the local planning strategy follows the same trend as the global path, but is more circuitous due to the goal of maximizing the variance in sensor vantage. Both paths follow the same trend because the first global goal location, the northeast corner of the environment, is identical in both cases. The benefit of this methodology can be seen in Figure 711 which compares both methods in terms of total obstacle surface area viewed. The result shows that adding the local planning strategy increases environment coverage despite reducing the overall distance traversed to the first global goal location. x 10/ 7 6a C5 40 C 3 global strategy ., complete strategy 2 , 1 0 5 10 15 20 25 times) Figure 711. Comparison of environment coverage results. The amount of obstacle surface area viewed during the transversal of paths given in Figure 710. 7.4.2 Optimal Control Results The NLP presented in C'! ipter 6 was used to convert Dubins paths to aircraft trajectories. The objective function of the NLP minimizes the distance between the trajectory and the given paths. Therefore, the NLP defines a control strategy for optimal path following. Results of the optimal control problem are presented in Figure 712. 80kinematic path optimal trajectory kinematic path 60 traje optimal trajectory 40 20 120 / S100 20 \ Z 200 40 50\ 150 60 0 100 80 0 North (ft) East (ft) 50' 50 1000 50 100 150 200 250 100 0 North (ft) A B Figure 712. Resultant trajectories from the path tracking optimization. A) Results of the optimal control strategy used to convert Dubins paths to aircraft trajectories. B) Topdown view of the optimal path following result. The optimal control results are generated by supplying the Dubins path as the initial condition to the optimization. The Dubins paths specify initial conditions for the states V, 7, y, x, y, h, and 0. The angle of attack, c, was calculated for each segment of the Dubins path by calculating the trim angle of attack according to the vehicle's equations of motion (Equation 621). The optimal control results show that the aircraft is capable of tracking the Dubins paths in the lateral direction. The longitudinal directional tracking, however, produces poorer results. It was found that the optimization did not alvb converge, generating poor results for .I. ressive maneuvers. Additionally, the optimizations required computation time on the order of 20 to 60 seconds. For these reasons, it is believed that an MA strategy would provide a better solution to incorporating dynamics into the planning strategy. CHAPTER 8 CONCLUSIONS 8.1 Conclusion This research has presented a complete framework for solving the problem of autonomous airborne mapping of unknown environments. The problem was decomposed into two parts: map building in the presence of noisy measurements and uncertain vehicle location and the planning of trajectories for minimumtime environment coverage. The focus of the SLAM formulation is to generate consistent maps of the environment in order to enable higherlevel control decisions. Three filters were investigated in terms of estimation performance, consistency, and efficiency. The results show that the URBPF is superior to RBPF and SRURBPF with respect to estimation performance and consistency. The URBPF suffers due to its inefficiency in comparison to the other filters. This problem can be alleviated by reducing the number of particles and reducing the rate at which SLAM algorithm updates the aircraft's INS estimates. Regardless, it has been shown that it is possible to build an environment map from noisy sensor data gathered by a visionbased UAV. The trajectory planning portion of this dissertation showed that it is possible to efficiently plan paths for sensor coverage of an unknown environment. The map construction process was used as feedback for obstacle avoidance and the paths were planned incrementally as knowledge of the environment increased. By definition, however, optimality cannot be proven for this process since the structure of the environment is unknown prior to the planning task. Presented results show that the trajectory planning process planned local paths that increased the gained knowledge of the environment with respect to a purely greedy exploration strategy. Additionally, it was found that an optimal path following approach to incorporating dynamics is feasible, but can be computationally inefficient. However, a MA approach could also be easily incorporated into the framework to possibly produce more computationally efficient results. 8.2 Future Directions The presented framework for map building and trajectory planning for environment coverage shows that there are still many open problems to be solved before a UAV can autonomously navigate within an urban environment. Sir, .,Il future directions include investigations of the proposed architecture with respect to real aircraft state data and gathered onboard imagery. Such investigations would involve gathering vision data and groundtruth state information from a UAV navigating an urban environment. From a real dataset the plane fitting and SLAM algorithms could be tested in a context more accurately representing the actual operating conditions. 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[104] Bailey, T., Neito J., and Nebot, E., "Consistency of the EKFSLAM Algorithm," IEEE/RSJ International Conference on Intelligent Robots and S;,il/. ii 2006. BIOGRAPHICAL SKETCH Adam Scott Watkins was born in Orlando, Florida on November 11, 1980. After graduating from Lake Mary High School, Adam attended the University of Florida receiving a Bachelor of Science in Mechanical Engineering in 2003. Adam continued his education at the University of Florida receiving a Master of Science in Mechanical Engineering in 2005. Upon obtaining his MS, Adam pursued a doctoral degree under the advisement of Dr. Richard Lind. Adam's research involves the control of unmanned systems and he is prepared to graduate with his Ph.D. in the summer of 2007. Upon graduation Adam will begin work at the Johns Hopkins University Applied Physics Lab continuing his research in the field of unmanned systems. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IwouldliketothankmyadvisorDr.RickLindforhisguidance,criticisms,andmotivationthroughoutmygraduateschoolexperience.Iwouldliketoexpressgratitudetomycommitteemembersfortheirpatienceandinsightduringmydoctoralstudies.IwouldalsoliketoextendthankstomylabfellowresearchersintheFlightControlsLabincludingMujahidAbdulrahim,EricBranch,RyanCausey,DanielGrant,andJoeKehoe.ThankstoMujahid,Eric,andRyanfordealingwithmynearlyconstantdisruptionsoftheworkplaceandgenerallysarcastic,downbeatattitude.ThanksgotoDaniel\Tex"Grantforhandlingmyverbalabusesowellandteachingmehowtoshootagun.ThankstoJoeKehoeforhisthousandsofresearchrelatedideas,includingthethreethatactuallyworked. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1ProblemStatement ............................... 13 1.2Motivation .................................... 14 1.3HistoricalPerspective .............................. 18 1.4Contributions .................................. 22 2SCENERECONSTRUCTIONFROMMONOCULARVISION ......... 24 2.1Introduction ................................... 24 2.2StructurefromMotionOverview ....................... 24 2.2.1PinholeCameraModel ......................... 25 2.2.2FeaturePointTracking ......................... 26 2.2.3EpipolarGeometry ........................... 28 2.2.4EuclideanReconstruction ........................ 30 2.3CalculatingStructurefromMotion ...................... 31 2.3.1EightPointAlgorithm ......................... 31 2.3.2RecursiveStructurefromMotion .................... 32 2.4OpticalFlowStateEstimation ......................... 34 2.5ImageProcessingforEnvironmentMapping ................. 35 3ENVIRONMENTREPRESENTATION ...................... 37 3.1Introduction ................................... 37 3.2EnvironmentModels .............................. 39 3.2.1TopologicalMaps ............................ 39 3.2.2OccupancyMaps ............................ 39 3.2.3GeometricMaps ............................. 40 3.3DimensionalityReduction ........................... 40 3.3.1PrincipalComponentsAnalysis .................... 41 3.3.2DataClustering ............................. 43 3.3.3PlaneFittingAlgorithm ........................ 45 5 PAGE 6 ..................... 49 4.1Introduction ................................... 49 4.2TheKalmanFilter ............................... 49 4.2.1TheExtendedKalmanFilter ...................... 52 4.2.2TheUnscentedTransform ....................... 53 4.2.3TheUnscentedKalmanFilter ..................... 54 4.2.4TheSquareRootUnscentedKalmanFilter .............. 56 4.3TheParticleFilter ............................... 58 4.4TheRaoBlackwellizedParticleFilter ..................... 60 5SIMULTANEOUSLOCALIZATIONANDMAPBUILDING ........... 62 5.1Introduction ................................... 62 5.2AirborneSLAMModels ............................ 65 5.2.1VehicleModel .............................. 66 5.2.2MeasurementModel ........................... 67 5.3FilteringforSLAM ............................... 68 5.3.1RaoBlackwellizedParticleFilter .................... 68 5.3.2UnscentedRaoBlackwellizedParticleFilter ............. 71 5.3.3SquareRootUnscentedRaoBlackweillizedParticleFilter ...... 73 5.4DataAssociation ................................ 75 5.5MapManagement ................................ 76 6TRAJECTORYPLANNING ............................ 78 6.1Introduction ................................... 78 6.2DirectTrajectoryPlanningMethods ..................... 80 6.2.1MixedIntegerLinearProgramming .................. 80 6.2.2ManeuverAutomaton .......................... 82 6.3DecoupledTrajectoryPlanning ........................ 84 6.4SamplingBasedPathPlanning ........................ 85 6.4.1ExpansiveSpacesTreePlanner ..................... 87 6.4.2RapidlyExploringRandomTreePlanner ............... 88 6.5TheDubinsPaths ................................ 90 6.5.1ExtensiontoThreeDimensions .................... 93 6.5.2CollisionDetection ........................... 94 6.5.2.1Globalpathcollisiondetection ................ 94 6.5.2.2Localpathcollisiondetection ................ 96 6.6EnvironmentCoverageCriteria ........................ 97 6.7OptimalPathTracking ............................. 101 6.8TrajectoryPlanningAlgorithm ........................ 102 7DEMONSTRATIONSANDRESULTS ....................... 105 7.1Introduction ................................... 105 6 PAGE 7 .............................. 105 7.2.1EectsofFeaturePointNoise ..................... 106 7.2.2NonplanarObstacles .......................... 108 7.3SLAMResults .................................. 108 7.3.1SimulationEnvironment ........................ 109 7.3.2LocalizationResult ........................... 111 7.3.3StateEstimationError ......................... 113 7.3.4FilterConsistency ............................ 113 7.3.5FilterEciency ............................. 116 7.3.6ExampleSLAMResult ......................... 117 7.4TrajectoryPlanningResults .......................... 117 7.4.1EnvironmentCoverage ......................... 117 7.4.2OptimalControlResults ........................ 121 8CONCLUSIONS ................................... 122 8.1Conclusion .................................... 122 8.2FutureDirections ................................ 123 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 132 7 PAGE 8 Table page 71EciencycomparisonoftheSLAMlters. ..................... 117 8 PAGE 9 Figure page 11MapconstructionbyanautonomousUAV ..................... 13 12Examplesofunmannedvehicles ........................... 14 13Examplesofunmannedairvehicles ......................... 15 14Unmannedairvehiclemissionproles ........................ 16 21Pinholecameramodelandperspectiveprojectionofafeaturepointontotheimageplane. ...................................... 25 22Featurepointtrackingresultfromanimagesequence ............... 26 23Epipolargeometry .................................. 29 31Environmentrepresentations ............................. 39 322DPrinicipalComponentsAnalysisforadatasetwithalineartrend. ...... 41 333DPrincipalComponentsAnalysisforaplanardataset. ............. 43 34Centroidsplittingforiterativekmeans ....................... 46 35Dimensionalityreductionoffeaturepointdata. .................. 48 41Recursivestateestimation .............................. 50 42TheUnscentedTransform .............................. 53 51Mapbuildinginthepresenceofnoise ........................ 63 52TheSLAMestimationstrategy ........................... 64 53Fixedwingaircraftmodeldenitions ........................ 66 61Anoverviewofthetrajectoryplanningprocess ................... 79 62AnexampleManeuverAutomaton ......................... 83 63ProbabilisticRoadmapplanninginatwodimensionalenvironment ....... 86 64ExpansiveSpacesTreeplanning ........................... 88 65RapidlyExploringTreeplanning .......................... 89 66ThecoordinatesystemforcalculatingthepathsintheDubinssetD. ...... 91 67TheDubinssetofpathsfortwocongurations. .................. 93 68Examplesof3DDubinspaths ............................ 94 9 PAGE 10 ....................... 95 610Interiortoboundarytest ............................... 96 611Collisiondetectionforthelocalpath ........................ 98 612Motionplanningforcoveragestrategies ....................... 99 613Environmentalcoveragemetric ........................... 100 71Resultoftheplanettingalgorithmwithoutimagenoise. ............. 105 72Angleofincidenceconstraint. ............................ 106 73Resultsoftheplanettingalgorithminthepresenceofimagenoise. ....... 107 74Resultoftheplanettingalgorithmwhennonplanarobstaclesareincluded. .. 109 75ThevisionbasedightsimulationenvironmentfortestingtheSLAMalgorithm. 110 76ExamplelocalizationresultfortheairborneSLAMsimulation .......... 112 77AverageestimationerrorforthethreeSLAMltersandthenominaltrajectoryfor50MonteCarloruns. ............................... 114 78Filterconsistencyresults ............................... 116 79AnexampleresultforthecompleteairborneSLAMsolutionusingtheURBPF 118 710Examplepathsforenvironmentcoverage ...................... 119 711Comparisonofenvironmentcoverageresults. .................... 120 712Resultanttrajectoriesfromthepathtrackingoptimization. ............ 121 10 PAGE 11 11 PAGE 12 12 PAGE 13 11 whereaUAVnavigatingatlowaltitudeusessensorinformationtoconstructathreedimensionalmapand,subsequently,planatrajectory. B MapconstructionbyanautonomousUAV.A)MissionsenvisionedforUAVsinvolvesafenavigationinclutteredurbanenvironments.B)Navigationinclutteredenvironmentswillrequirevehiclescapableofautonomouslybuildingamapfromsensordataand,subsequently,planningsafetrajectories. 13 PAGE 14 12 ,includethePackbotEODwhichassistsinExplosiveOrdnanceDisposal(EOD)[ 1 ],theMarsExplorationRovers(MER)SpiritandOpportunityusedforplanetaryexploration[ 2 ],andtheRoombavacuumrobot[ 3 ]. Figure12. Examplesofunmannedvehicles.A)TheJPLMarsExplorationRoverSpirit.B)TheiRobotPackbotEOD.C)TheiRobotRoomba. TheunmannedvehiclesdepictedinFigure 12 exemplifythecurrentlevelofautonomyreliablyachievedbyunmannedsystems.ThePackbotEODrequiresconstantguidancefromahumanoperatoranddoesnotperformanytrulyautonomoustasks.TheMERautonomouslymonitorsitshealthinordertomaintainsafeoperationandtracksvehicleattitudetoaidinsafenavigationoveruneventerrain.Preloadedandcommunicatedinstructions,however,areexecutedbytheMERaftersignicanthumananalysisandinteraction.TheRoombadisplaysahigherlevelofautonomybysensing 14 PAGE 15 13 ,includetheMQ1PredatorthathasbeenusedforreconnaissancemissionsandiscapableofringHellremissiles,theRQ5Hunterdesignedforreconnaissancemissions,andtheDragonEyedesignatedforreconnaissance,surveillance,andtargetacquisition[ 4 ].ThecurrentUAVtechnologyhasalowlevelofautonomyrequiringconsiderablehumaninteractiontoperformbasicmissions.Highlytrainedoperatorsarenecessarytoremotelyoperateeachindividualunmannedaircraft. B C Examplesofunmannedairvehicles.A)TheGeneralAtomicsAeronauticalSystemsInc.MQ1Predator.B)TheNorthropGrummanRQ5Hunter.C)TheAeroVironmentDragonEye. Futuremissionsenvisionedforunmannedaircraftrequireagreaterlevelofautonomythaniscurrentlyavailableinexistingsystems.CommonautonomousbehaviorsexhibitedbyUAVsincludewaypointnavigation[ 5 ]andsimpleobstacleavoidance[ 6 ].SuchbehaviorsarewellsuitedforrelativelybenignmissionsthatoccurathighaltitudesandrequirelittlereactiontothesurroundingenvironmentasshowninFigure 14A .Advances 15 PAGE 16 14B ,thatoccuroutsideofanoperator'slineofsight.Asvehiclemissionsrequireautonomousightthroughincreasinglyelaborate,clutteredenvironments,ahigherdegreeofenvironmentalawarenessisrequired.Therefore,twocriticalrequirementsforautonomousunmannedvehiclesaretheabilitytoconstructaspatialrepresentationofthesensedenvironmentand,subsequently,plansafetrajectoriesthroughthevehicle'ssurroundings. B Unmannedairvehiclemissionproles.A)Currentmissionsoccurathighaltitude,therefore,notrequiringrapidpathplanningandenvironmentalawareness.B)Futuremissionswillbelowaltitudeassignmentsnecessitatingquickreactiontoenvironmentalobstacles. Thisdissertationconcentratesonvisionbasedenvironmentalsensing.ComputervisionhasbeenidentiedasanattractiveenvironmentalsensorforsmallUAVsduetothelargeamountofinformationthatcanbegatheredfromasinglelightweight,lowpowercamera.Thesensorselectionisaresultofthedecreaseinvehiclesizewhichseverelyconstrainstheavailablepayloadbothintermsofsizeandweightalongwithpowerconsumption.Therefore,autonomousbehavioursforUAVsmustbedesignedwithintheframeworkofvisionbasedenvironmentalsensing.Inparticular,twocriticalcapabilities 16 PAGE 17 7 ].RoboticmappingallowsthevehicletolocalizeitselfwithrespecttotheconstructedmapfornavigatinginGPSdeniedenvironments.ThiscapabilityiscriticalformilitaryoperationswhereGPScanbejammedorclutteredenvironmentsprecludetheuseofGPSduetoocclusion.Additionally,roboticmappingisvitallyimportanttosafe,timeecientnavigationthroughunknownenvironments.Findingtheshortestpathtoagoallocationusingpurelyreactivecontrolmethods,wheretherobotselectscontrolactionsbasedonlyonthecurrentsensormeasurements,isatrialanderrorprocessinunknownenvironmentswithnoprovablyoptimalsolution.Bygeneratinganobstaclemap,arobotcanretainknowledgeofitssurroundingsanddiscriminatebetweenobstructedandunobstructedpaths.Asknowledgeoftheenvironmentgrows,thetaskoftraversingtoagoallocationcanbeaccomplishedinatimeecientmannerwiththeplanningofcomplete,obstaclefreepaths.Thesecondcapabilitydevelopedinthisdissertationisacomputationallyecienttrajectoryplanningroutinewhichmaximizesenvironmentcoveragetominimizethetimetoconstructacompleteenvironmentmap.Inimplementation,themethodmustbecomputationallyecientsincetheplanningprocessoccursincrementallyasnewinformationisgathered.Therapidityoftheplanningprocessallowsforpathupdatesasknowledgeoftheenvironmentincreaseswhichiscriticalforsafenavigationinunknownenvironments.Additionally,vehicledynamicsareincorporatedintotheplanningprocessinordertogeneratetrajectories.Thistaskisvitalforaircraftwhichmaynotbecapableoftrackingeverykinematicallyfeasiblepath.Mostgroundvehiclesareabletostoptheir 17 PAGE 18 8 9 ]whichdescribedamethodologyforgeneratingstochasticobstaclemapsfromaseriesofuncertainspatialrelationships.Inregardstounmannedvehicles,thisworkdescribestheSLAMproblem:theconstructionofanenvironmentmapfromnoisysensormeasurementstakenfromuncertainvehiclepositionswhileconcurrentlylocalizingthevehiclewithrespecttothelearnedmap.ThesolutiontotheSLAMproblempresentedbySmith,etal.hassincebeenextendedtoproduceSLAMalgorithmsfornumeroussensors,vehicles,andenvironmenttypes.ThisdissertationfocusesonSLAMforvisionequippedaircraftnavigatinginhighlystructured,urbanenvironments.VisionbasedSLAMreferstoinstanceswhenvisionistheonlysensoravailableformeasuringtheenvironment.Historically,solutionstotheSLAMproblemhavefocusedonvehiclesequippedwithsensorssuchaslaserrangendersandsonar[ 10 11 ].The 18 PAGE 19 12 13 ].InanexampleofpurelyvisionbasedSLAM,DavisonusesvisiontodetectlocallyplanarfeatureswhichareincorporatedaslandmarksintotheSLAMframework[ 14 { 16 ].Theimplementationemphasizesaccuratelyestimatingcameraegomotionandisnotconcernedwithgeneratinganenvironmentmodel.Consequently,theconstructedmapisasparseapproximationoftheenvironment.Theapproachinthisdissertationshiftsthefocusfromaccurateegomotionestimationtobuildingacompletemapoftheenvironmentforthepurposeofecientlyplanningcollisionfreevehiclemotions.ThoughtheSLAMproblemhasbeenahighlyresearchedtopicinthegroundvehiclecommunityforthepasttwodecades,verylittleresearchhasbeenperformedforaircraft.Afewgroundvehicleimplementationshaveconsideredundulatingterrainwhichrequirestheassumptionofnonplanarmotion[ 17 ].KimandSukkariehimplementedSLAMonaUAVequippedwithavisionsensorthatlocatesarticiallandmarksofknownsize[ 18 19 ].AbearingsonlySLAMapproachforaMicroAirVehicle(MAV)equippedwithacameraandanInertialMeasurementUnit(IMU)operatinginaforestenvironmentwasintroducedbyLangelaan[ 20 { 22 ].CurrentapproachestoairborneSLAMfocusprimarilyonaugmentinginertialnavigationsystems(INS)toreducedriftinherentinthecalculatedstateestimates.Theseapproachesdonotaddressthedicultiesassociatedwithbuildingacompleteenvironmentmapasisaddressedinthisdissertation.SolutionstotheSLAMproblemrequirerepeatedobservationsofenvironmentallandmarksinordertobuildastochasticobstaclemap.Therefore,informationregardingeachlandmarkmustbestoredandobservationsofalandmarkmustbecorrelatedwithpastobservations.ThecorrelationtaskisknownasdataassociationandincorrectassociationsoftenleadtocatastrophicfailuresoftheSLAMestimationprocess.Forthese 19 PAGE 20 10 ].Thepracticeofusingpointlandmarksoftenyieldsenvironmentmodelscontainingmillionsoffeatures[ 23 ].SuchenvironmentmodelsoflargesizeareinfeasibleduetothedatastoragerequirementsandcomputationalpowerrequiredtoincorporatelargenumbersoflandmarksintotheSLAMcalculation.Additionally,thedataassociationproblembecomesintractablebecausetherelativemeasurementstolandmarkscannotbecorrelatedasaresultofthelargenumberofpossibleassociationscombinedwithuncertaintyinvehiclelocation.SeveralSLAMformulationshaveconstructedgeometricenvironmentmapsforhighlystructuredenvironmentmapsbyttinggeometricprimitivestothegatheredpointwisedata[ 24 25 ].Thebenetsofgeometricprimitivesincludethereductionofdatarequiredtostoreanenvironmentmapandlandmarksthataremoreeasilydistinguishedbecausetheycontainstructuralinformation.Thisdissertationemploysasimilardimensionalityreductionstepinordertoconstructacompleteenvironmentmapconsistingofplanarfeatures.Inordertoautonomouslyandrapidlyconstructanenvironmentmap,thisdissertationpresentsatrajectoryplanningalgorithm.Ideally,thetrajectoryplanningproblemwouldbeposedasanoptimalcontrolprobleminordertondagloballyoptimalsolution.Anoptimalsolution,however,iscomputationallyintractablebecausetheproblemisnonconvexandoccursinahighdimensionalspace.Forthisreason,severalapproximationstothefulloptimalcontrolsolutionhavebeenproposedwhichincludedMixedIntegerLinearProgramming(MILP)andManeuverAutomatons(MAs).TheMILPsolutionposesthefullnonlinearoptimizationasalinearoptimization[ 90 91 ].Therefore,thesolutionisonlycapableofhandlingalinearizeddynamicaircraftmodel.ThelinearizationrestrictionseverelylimitstheecacyoftheMILPsolutionduetothereductionofavailablemaneuvers.Alinearmodelcannotproperlycharacterizetheaggressivemaneuversnecessarytonavigateinaclutteredenvironment.Additionally, 20 PAGE 21 35 { 38 ].Thedynamicsofthevehiclearediscretizedintoasetoftrimtrajectories,orsteadystatemotions,andmaneuversthatconnectthetrimstates.TheMAplanningprocessisdecomposedintoacombinatorialproblemwhichsearchesforthebestmaneuversequenceandanoptimizationthatndstheoptimaltimedurationsforthetrimtrajectories.Unlike,MILPthediscretizeddynamicsarebasedonanonlinearmodelandcanincorporateaggressivemaneuvers.TheMAsolution,however,canbecomecomputationallyintractablewhensolvingforlargetrajectoriesrequiringalongmaneuversequence.Thisissueisexacerbatedifthediscretizationisnotrestrictedtoasmallsetofvehiclemotionsorifobstaclesareconsidered.Therefore,theMAiscombinedwithsamplingbasedplanningmethodstoproduceasolutioninwhichoptimalitycannotbeguaranteed.Thisdissertationemploysadecoupledtrajectoryplanningstrategy[ 40 41 ]inwhichafeasiblecollisionfreepathisplannedandoptimalcontrolisemployedtotrackthepathinminimaltime.Thepathissubjecttokinematicconstraintsandincorporatesmetricsforselectingpathsthatprovidemaximumcoveragewithrespecttootherpathsgeneratedduringtheplanningprocess.Thedecoupledapproachallowsfortheuseofthecompletenonlinearaircraftmodel;however,optimalityintermsofcoverageisnotguaranteedsincethecoveragecriteriaisonlyincludedinthepathplanningtask.Thepathtobetrackedisgeneratedwithasamplingbasedplanningstrategy[ 26 ].SamplingbasedalgorithmswereintroducedbyamethodologycalledProbabilisticRoadmaps(PRMs)[ 27 ].PRMsconstructagraphthatcapturestheconnectivityofthecongurationspaceofarobotbyrandomlysamplingcongurationsandconnectingthemtotheexistingmapwithkinematicallyfeasiblepaths.OnceaPRMisconstructed 21 PAGE 22 28 { 30 ]andtheRapidlyExploringRandomTrees(RRTs)[ 31 32 ]plannersareemployed.TheESTandRRTplanningstrategiesaremoreecientsincetheyaredesignedtondfeasiblepathsbetweentwospeciccongurations,ratherthanprovideageneralplanningsolutionlikethePRM.ThisdissertationemploysavariantoftheRRTtoecientlyplancollisionfreepathsthroughtheenvironment.AnabundanceofresearchhasbeenperformedintheareasofSLAMandtrajectoryplanning.Thisdissertationextendspreviousresearcheortsinbotheldstoprovideamethodologyforautonomousmapbuildinginhighlystructuredenvironments. 22 PAGE 23 23 PAGE 24 42 ].Inaddition,thelowcost,smallpayloadrequirements,andpassivenatureofvisionhasmadeittheprimarysensorformanysmallUnmannedAirVehicles(UAVs)[ 4 ].Thisstudyassumesthatcomputervisionistheonlyavailableonboardsensorcapableofmeasuringthesurroundingenvironment.Mappinganunknownenvironmentwithamoving,monocularcamerarequiresthatthreedimensionalinformationbeinferredfromasequenceoftwodimensionalimages.ThisprocessofthreedimensionalreconstructionfromimageryisanactivelyresearchedareaintheimageprocessingcommunitycommonlyreferredtoasStructurefromMotion(SFM)[ 43 44 ].ThischapterpresentsanoverviewofSFMinSection 2.2 .SpecicsolutionstotheSFMproblemarethendiscussedinSections 2.3.1 and 2.3.2 .Section 2.4 describesavehiclestateestimationtasksimilartoSFMwhichcalculatesvehiclevelocitystates.Subsequently,Chapters 3 and 5 describehowthevisionbasedstateestimatesandenvironmentalstructureinformationisemployedintheroboticmappingmission. 2.2.2 ,whichcalculatescorrespondencesbetweenpointsofinterestinsuccessiveimages.Section 2.2.3 demonstratesthatbyexploitingepipolargeometrybetweencorrespondencepairstherelativerotationandtranslationbetweenthetwocameraviewscanbeestimated.Thecalculatedmotionofthecameracanthenbeused 24 PAGE 25 2.2.4 21 ,thetwodimensionalprojectionf=[;]Tofathreedimensionalpoint=[1;2;3]Tistheintersectionoftherayemanatingfromthecameraopticalcenterendingatpointandtheimageplane.Theimageplaneislocatedatadistancef,calledthefocallength.ThecameracoordinateframeCislocatedatthecameraopticalcenterwithcoordinates^c1and^c2orientedparalleltotheimageplaneinthenegativeverticalandhorizontaldirections,respectively.Thecoordinate^c3isorientedperpendiculartotheimageplaneanddenotesthecameraaxis. Figure21. Pinholecameramodelandperspectiveprojectionofafeaturepointontotheimageplane. Forthepinholecameramodel,perspectiveprojectionofathreedimensionalpointcanbewrittenintermsofthecamerafocallength 326412375(2{1) 25 PAGE 26 2{1 showsthatonlythedepthofthethreedimensionalfeaturepointisrequiredtoresolvethelocationofthepointfromtheimageprojection. 22 B Featurepointtrackingresultfromanimagesequence.A)Frame1.B)Frame10. TheLucasKanadefeaturepointtracker[ 45 ]isastandardalgorithmforsmallbaselinecameramotion.Thealgorithmsolvesforthedisplacementofallfeaturepointsinanimagewherethedisplacementofafeaturepointismeasuredinthehorizontalandverticalimagecoordinatesdenotedbyand,respectively.Foraseriesoftwoimages,thedisplacementofafeaturepoint,d=[d;d]T,canbedescribedasthetranslationofagroupofimagepixelsasfollows 26 PAGE 27 2{2 canbeformulatedasaleastsquaresminimizationofthechangeinimageintensityvaluesbetweenfeaturepointwindowsintwoconsecutiveimagesasgivenby 2{3 toproduce 2{5 withrespecttofeaturepointdisplacementthesolutiontothelinearleastsquaresestimateoffeaturepointdisplacementcanbewrittenasfollows 27 PAGE 28 46 ]. 23 ,whichisageometricconstraintrelatingcameramotiontotheimageprojectionsofatrackedfeaturepoint.EpipolargeometrydictatesthatthereisaplanewhichisdenedbyathreedimensionalfeaturepointandtwocameracoordinateframesC1andC2fromwhichthepointisvisible.Theepipolarplaneintersectsbothimageplanesattheepipolarlinesl1andl2whichcontaintheimageprojectionsf1andf2ofthefeaturepointandtheepipolese1ande2. 28 PAGE 29 Epipolargeometry.Theepipolarconstraintdescribesthegeometricrelationshipbetweencameramotionandtheprojectionofafeaturepointontotheimageplane. Theepipolarconstraintrelatestheimageprojections,f1andf2,ofthethreedimensionalpoint,,totherelativeposition,T,andorientation,R,ofthetwocameralocations.Sincethesethreevectorsarecoplanartheirscalartripleproductiszeroasfollows 2{10 .AmethodforesimatingtheessentialmatrixisdescribedinSection 2.3.1 .Oncetheessentialmatrixisestimated,itcanbedecomposedintotherelativerotationandtranslationofthecamera.Therelativetranslation,however,canonlybeestimatedtoascalefactorsinceEquation 2{10 ishomogeneouswithrespecttotheessentialmatrix(Equation 2{11 )andis 29 PAGE 30 2{12 canberewrittenintermsofasingleunknowndepthasfollows 2{13 canbesolvedforjcorrespondencepairssimultaneouslybyexpressingtheproblemintheform 30 PAGE 31 2{14 providesthedepthofthetrackedfeaturepoints.Therefore,thissectionhasoutlinedthesolutiontotheSFMproblemfromtwodimensionalimagedatatoestimatedcameramotionandthreedimensionalEuclideanscenestructure. 47 ]calculatestherotationandtranslationbetweentwoimagesgivenaminimumsetofeighttrackedfeaturepoints.Largernumbersoftrackedfeaturepointscanbeusedinthecalculationbysolvingalinearleastsquaresminimizationproblem.Theinclusionofadditionaltrackedpointshasbeenshowntoimprovetheresultsofthealgorithm.Theeightpointalgorithmexploitstheepipolarconstraintbysolvingalinearsystemofequationsintheform 31 PAGE 32 2{11 .ItshouldbenotedthatthispresentationoftheeightpointalgorithmishighlysensitivetonoiseandrequiresanormalizingstepaspresentedbyHartley[ 48 ]. 49 ].Thesealgorithms,however,arenotsuitedfortrajectorygenerationbecausetheyincuradelayfromthedatagatheringprocessandcannotbeimplementedinrealtime.Forcausal,realtimeSFMaKalmanltercanbeemployedtorecursivelycalculatestructureandmotionestimates[ 50 ].ThestandardrecursiveSFMalgorithmsgraduallyreneSFMstateestimatesbyincorporatingnewdata[ 51 ].Alternatively,batchrecursivealgorithmscreateaseriesofintermediatereconstructionswhicharefusedintoanalreconstruction[ 52 ].Bothrecursivetechniquesshowimprovedresultsovernonrecursivetechniques,however,nosinglealgorithmworkswellinallsituations[ 44 ].TheproblemsspecictoSFMreconstructionforaUAVareidentiedbyPrazenicaetal.[ 46 ].Theseissuesincludethelossoffeaturetrackingduetoaggressivevehiclemotion,unreliablereconstructionsduetosmallbaselines,andthewellknownscalefactor 32 PAGE 33 53 54 ].TheKalmanlteringapproachdenesbothastatetransitionmodelg()andameasurementmodelh()writtenas 2.2.3 ).ThelteringapproachsolvestheSFMproblembyestimatingthestateofthemodelgivenbyEquation 2{20 .TheestimationprocessiscomputedwithanExtendedKalmanFilter(EKF).TheEKFrstpredictsthestateandcovarianceofthesystemaccordingto ^xkjk1=f(^xk1jk1;uk1;)(2{21) 33 PAGE 34 ^xkjk=^xkjk1+Kk(ykh(^xkjk1;))(2{25) 55 { 57 ].Opticalowissimilartofeaturetrackingwiththedistinctionthatdisplacementiscalculatedataxedlocationasopposedtotrackingaspecicimagefeaturethroughanimagestream.Additionally,opticalowreferstofeaturepointvelocityasopposedtodisplacement,butforagivenxedframeratethedierenceincalculationistrivial.Foropticalowstateestimation,thefocalplanevelocityofthefeaturepoint,approximatedbytheopticalow,canbeobtainedbydierentiatingEquation 2{1 .Theresultisanequationforthefocalplanevelocityofafeaturepointintermsofits 34 PAGE 35 23264301032375266664_1_2_3377775(2{28)TheopticalowofthefeaturepointsisaknownquantitydeterminedinthefeaturetrackingalgorithmthatisanintegralpartoftheSFMcalculation.TheequationsofmotionofanaircraftcanbecoupledwithEquation 2{28 inordertoexpressopticalowintermsofaircraftstates,asgivenby 301 31 2{29 havebeenproposedusingoptimizationroutines.TheresultingsolutionshavebeenshowntoproduceaccurateestimatesofaircraftlinearandangularvelocitiesthoughthetotalvelocityoftheaircraftmustbeknowntoaccountforthescaleambiguitythatalsoaectsSFM. 3 togenerateageometricrepresentationoftheenvironment.Furthermore,bothSFMandtheopticalowapproachprovidevehicle 35 PAGE 36 5 thisinformationwillbefusedwithonboardstateestimatesandtheexternalenvironmentmeasurementsinaSimultaneousLocalizationandMapping(SLAM)algorithmtoreducedriftinvehiclestateestimatesandconcurrentlydecreasemaperrors. 36 PAGE 37 26 ]andisapoormapintermsofdatastorage,landmarkdenition,andenablinghighlevelmissionplanning.Motionplanningalgorithmstypicallyusedforroboticnavigationpresupposeaknownobstaclemap[ 58 ].Theobstaclemapenablestheidenticationofsaferoutesthroughavehicle'ssurroundingsbylimitingthesetofadmissiblepositionsavehiclecanachieve.Apointwiserepresentationofobstaclegeometryisambiguousintermsofdeningsaferegionswithintheenvironment.Amoreconciserepresentationoftheenvironment,suchasgeometricobjects,inwhichcompleteobstaclescanbeaccuratelydescribedisbettersuitedtotheprocessofmotionplanning.Mappinglargescaleenvironmentsrequiresthatmanysensormeasurementsbetakentoaccuratelydenescenestructure.Storingindividualsensormeasurementsduringamappingoperationcanquicklyleadtoacomputationallyintractableproblem[ 59 ].Lowerdimensionalrepresentationsofsensordatacangreatlyreducethenumberofparametersneededtodeneanobstaclemap.Forexample,asingleplanecanreasonablyrepresentthousandsofthreedimensionalpointsusingfourparametersincludingthreetodeneanormaldirectionandonetodenetheplane'sdistancefromtheoriginalongthatnormaldirection.Forhighlystructuredenvironments,usinggeometricprimitivessuchasplanestomodelthesensordatacangreatlyreducetheamountofdatarequired 37 PAGE 38 10 60 61 ].ThiscaseisknownastheSimultaneousLocalizationandMapping(SLAM)problemasdiscussedindetailinChapter 5 .TheSLAMframeworkrequiresthatdistinguishableandreobservablelandmarksbepresentintheenvironment.Landmarksallowthevehicletoascertainitsrelativelocationtotheenvironmentthroughsensingoperations.Theonlydiscerniblecharacteristicofa3Dfeaturepointisitslocationintheenvironmentwhichmakes3Dpointsapoorchoiceforlandmarks.Ageometriclandmark,however,canprovideadditionalinformationintermsoforientationandsize.Additionally,featurepointsthatarevisiblefromagivenvantagepointarenotnecessarilyvisiblefromanotherlocationduetoocclusionandlightingvariations.Geometricmapsdenelandmarksthatarerobustlyobservablewithrespecttocamerapose.Manymissionsenvisionedforautonomousvehiclesrequirehumaninteractionintermsofhighlevelmissionplanning.Surveillingtargetsinanenvironmentsuchasaspecicbuildingorroadwaydependsupontheabilitytoproperlydesignatetheknownobjective.Similarly,reconnaissanceofanareaofinterestintheenvironmentrequiresthattheregionbeclearlyspecied.Suchhighleveldecisionswillbemadebyahumanoperatorwhomustdesignatethetargetsorregionsofinterestinsomemanner.Inthiscapacity,aspatialmodelisadvantageousasitprovidesanintuitiveinterfacefortargetselectionsasopposedtoapointwisemodelwhichisanambiguousrepresentationthatrequiresfurtherinterpretationofthedata.Theneedtoenablemotionplanning,constructcompactmaprepresentations,denedistinguishablelandmarks,andassisthighlevelmissionplanningmotivatesthegeneration 38 PAGE 39 31 B C Environmentrepresentations.A)Topographicmapsdepictspatialconnectivitybystoringlocationsofinterestandconnectingpaths.B)Occupancymapsrecordtheprobabilitythatadiscretelocationcontainsanobstacle.C)Geometricmapsdeneobstaclesasgeometricprimitives. 62 { 64 ]asshowninFigure 31A .Typically,thenodesconsistofplacesofinterestsuchastheintersectionofcorridorsandarcsprovideconnectivityinformationaboutthenodes,tosuggestnavigationpathsbetweennodes.Topologicalmodelsarenotnecessarilyspatialmodelsasthenodescanrepresentvariousstatesofthevehicle.Suchmaps,however,relyonspatialmodelsfortheirconstructionsincethearcsneedtodenemaneuversthatavoidobstacles.Thisfactblursthedistinctionbetweentopologicalandgeometricmapssincetopologicalmapsrelyongeometricinformation. 65 66 ]. 39 PAGE 40 25 67 ].Threedimensionalmapsgeneratedfromdensedepthinformationtypicallyusetrianglemeshestorepresenttheenvironment[ 60 68 ].Examplesexistforrepresentingstructured,threedimensionalenvironmentsbyalessgeneralsetofgeometricprimitivessuchasdistinctplanarsurfaces[ 24 69 70 ].ThisdissertationemploysageometricenvironmentmapbecausethepropertiesarebenecialforboththeSLAM(Chapter 5 )andtrajectoryplanning(Chapter 6 )tasks.Thegeometricmapprovidesacompactrepresentationthatreducesthecomputationalburdenforbothprocesses.Additionally,ageometricdescriptionofthelandmarksmakesthemmoreobservablethanpointlandmarksorvisuallydenedlandmarks.Lastly,thegeometriclandmarksareamenabletocollisioncheckingalgorithmsrequiredforplanningsafetrajectoriesandareanintuitiverepresentationforaidinginhigherlevelmissionplanningconcerns.ThegeometricmapisconstructedfromthepointwiseobservationsoftheenvironmentprovidedbySFM.Thistaskisaccomplishedviaadimensionalityreductionprocess. 40 PAGE 41 71 ],astatisticallineartransformationusedforreducingthedimensionalityofadataset.PCAndsthedirectionsinadatasetwiththemostvariation,asshowninFigure 32 inordertocapturethevariabilityofthedatawithfewerparameters.Thesedirectionsarealsotheeigenvectorsthatcorrespondtothelargesteigenvaluesofthedata'scovariancematrix.Therefore,PCAcanbeemployedtotransformadatasettoanewbasiswheretherstcoordinate,calledtherstprincipalcomponent,denesthedirectionofmaximumvariance.Allsubsequentdirectionsdenedbythebasisareorthogonaltoallpreviousdirectionsanddenetheremainingdirectionofgreatestvariance. Figure32. 2DPrinicipalComponentsAnalysisforadatasetwithalineartrend.Theprincipalcomponentsarelabeledpnforthenthprincipalcomponentandthe2covarianceofthedatasetisshown. 41 PAGE 42 ^XSVD=UVT(3{1)whereUandVdenematricesoforthonormalvectorsanddenesadiagonalmatrixofsingularvaluesassumedtobeorderedfromlargesttosmallest.ThecolumnsofVdenetheprincipalcomponentsofXwiththecolumncorrespondingtolargestsingularvaluedesignatingtherstprincipalcomponent.Theresultistypicallyorderedsothattherstprincipalcomponentistherstentryofwithsubsequentcomponentslistedinadescendingorder.Thedatasetcanbeprojectedontothebasisformedbytheprincipalcomponentsaccordingto 3{1 ,respectively.Additionally,thesingularvaluesarethesquarerootsoftheeigenvaluesof^XT^Xand^X^XT.Sincethedatasethaszeromean,thecovariancematrixof^Xis^XT^XandtheeigendecompositionofthecovariancegiveseigenvectorsidenticaltoVandproduceseigenvaluesequaltothesquareofthesingularvalues.Therefore,SVDandeigendecompositioncanbeusedinterchangeablyforPCA.Forthreedimensionalinformationbeingmappedtoaplanarrepresentation,showninFigure 33 ,thersttwoprincipalcomponentsdenetwoorthogonalinplanedirectionswhilethethirdcomponentdenestheplane'snormaldirection.Therefore,threeprincipal 42 PAGE 43 Figure33. 3DPrincipalComponentsAnalysisforaplanardataset.Theprincipalcomponentsarelabeledpnforthenthprincipalcomponentandthe2covarianceofthedatasetisshown. 72 ].Eectively,thekmeansalgorithm 43 PAGE 44 3{3 iscarriedoutaccordingtoAlgorithm 1 whichiteratesuntilallcentroidlocationsconvergetolocallyoptimalsolution.Thealgorithmbeginsbyguessingthenumberofcentroidskandrandomlyinitializingthecentroidlocationsfc1;c2;:::;ckg.Foreachiteration,thedatapointsareassociatedtothenearestcentroidaccordingtoEuclideandistance.Thecentroidlocationsarethenupdatedtothemeanofthenewlyassociateddatapoints.Convergenceoccurswhenthemaximumcentroiddisplacementislessthanagiventolerancetolk. 73 ].Similarly, 44 PAGE 45 2 ,iterativelysearchesforplanarclustersbystartingwithasinglecluster(k=1)andincreasingthenumberofclusterstobesttthedataset.Eachiterationbeginswiththekmeansalgorithmwhichproducesasetofclusteredfeaturepoints.EachclusterischeckedforplanaritywithPCAbyinspectingthesingularvalueassociatedwiththethirdprinciplecomponentp3.ThesingularvaluesproducedbyPCAdenotetheamountofthedata'svariancecapturedbytheassociatedprincipalcomponent.Therefore,themagnitudeofthethirdsingularvalue,3,denoteshowplanaristhedataclusterwithzeromagnitudedescribingaperfectlyplanardataset.Planarityisdeterminedbyimposingathresholdton3.Athresholdvalueof0:05t0:1producesgoodresultsforrealisticallynoisydatasets.Clustersfoundtobeplanarareremovedfromfurtherconsiderationintheclusteringprocedure.Allfeaturepoints,includingpointscurrentlyattributedtodierentclusters,areprojectedontotheplanarbasisaccordingtoEquation 3{2 .Thefeaturepointsaretestedforinclusionintothecurrentplanedenitionaccordingtoconnectivityanddistancealongthenormaldirectiontests.Theconnectivitytestsearchesforallpointswithinaspecieddistancefromanycurrentlyincludedpointbeginningwiththepointclosesttothecluster'scentroid.Thenormaldirectiontestremovespointsthatareaspecieddistancefromaninplaneposition.Theremainingpointsareclassiedasasingleplanarobjectandremovedfromthedataset.TheconvexhullofthepointsH,theplane'snormaldirection,andtheperpendiculardistancefromthesensorpositiontotheplanearesavedasanobservationz.AllobservationsareusedintheSLAMalgorithmdescribedinChapter 5 .Thecentroidsofnonplanarclustersaresplit,asshowninFigure 34 ,inordertochoosenewcentroidsthatproperlycapturethevariabilityofthedataset.Thecentroids 45 PAGE 46 B C Centroidsplittingforiterativekmeans.A)Theinitialresultofkmeanswhichdoesnottthedataset.B)Centroidsplittingresultsintwonewcentroidsthatmoreaptlycharacterizethedataset.C)Thenalkmeansresultthatproperlyclustersthedata. 46 PAGE 47 3{4 isequaltothesplittingdistanceemployedintheGmeansalgorithm[ 73 ].Afterthecentroidshavebeensplitorremovedkmeansisrerunontheremainingdataset.Thisprocessisperformedrepeatedlyuntilthenumberofremainingdatapointsfallsbelowathresholdtolf.Thistoleranceistypicallysetas10%ofthetotalnumberoffeaturepoints. 35 .ThealgorithmtransformsthepointwisedataprovidedbySFMintoasetofgeometricprimitives.Concatenatingtheseobservationswillcreateanincorrectgeometricmapduetodriftinvehiclestateestimates.Chapter 5 willpresentanalgorithmthatfusesthevehiclestateestimatesandenvironmentobservationsinordertocreateaspatiallyconsistentmap. 47 PAGE 48 B Dimensionalityreductionoffeaturepointdata.A)PointwisedatafromtheSFMalgorithm.B)Thedimensionallyreduceddatasetconsistsofgeometricprimitivesintheformofplanarfeatures. 48 PAGE 49 3 describedamethodforextractingplanarfeaturesfromahighlystructuredenvironment.ThesefeatureswillbeusedinChapter 5 toincrementallyconstructanenvironmentmapwithalteringtechniqueknownasSimultaneousLocalizationandMapping(SLAM).TheSLAMtaskisastateestimationapproachinwhichthestateofthesystemisacombinationofthevehiclestate,denedintermsofpositionandorientation,andthestateofthemap,designatedbythepositionandorientationofenvironmentallandmarks.Stateestimationiscommonlyaddressedasalteringprobleminwhichestimatesarecalculatedrecursivelybyincrementallyincorporatingsensorinformation[ 79 ].ThischapterdescribesseveralltersthatfollowasimilarpredictorcorrectorstructuredepictedinFigure 41 .Thelteringprocessbeginswiththepredictionofanoisyestimateofthesystemstateaccordingastatetransitionmodel.Measurementsofthesystemarepredictedasifthetruesystemstateisequaltothestateprediction.Themeasurementpredictioniscomparedtoanactual,noisysystemmeasurementandthestateestimateisupdatedaccordingtoaweightedmeasurementresidual.Theweightofthemeasurementresidualrepresentsthecondenceofthemeasurementwithrespecttothemodelbasedstateestimate.TwoltersthatformthefoundationofthemajorityofSLAMsolutionsaretheKalmanlter[ 80 ]andparticlelter[ 82 ].ThischapterdescribesthegeneralformoftheseltersandseveralvariantsthatwillbeusedintheSLAMlteringapproachesdescribedinChapter 5 49 PAGE 50 Recursivestateestimation.Therecursiveprocessinvolvesthreesteps:prediction,observation,andupdate. systemdescribedbythelinearstochasticdierenceequation 50 PAGE 51 ^xkjk1=A^xk1jk1+Buk1(4{5) 4{5 and 4{6 afterameasurementzkistaken.Themeasurementupdateequationsare ^xkjk=^xkjk1+KkzkH^xkjk1(4{8) 4{5 and 4{6 forthenextpredictionastherecursiveprocesscontinues.TheKalmanlterisdesignedforlinearsystemswithlinearmeasurementmodels.TheSLAMproblem,however,dealswithnonlinearsystemsintheformofvehicledynamicmodelsandsensorswithnonlinearmeasurementmodels.Therefore,thenonlinearextensionsoftheKalmanlterareemployedfortheestimationtask.Thenonlinear 51 PAGE 52 81 ],theUnscentedKalmanFilter(UKF)[ 83 ],andtheSquareRootUnscentedKalmanFilter(SQUKF)[ 84 ]. @x^xk1jk1;uk1(4{12) @x^xkjk1;(4{13)Additionally,themodelsarelinearizedwithrespecttothenoisevalues @q^xk1jk1;uk1(4{14) @r^xkjk1;(4{15)whichmeanstheEKFlterequations,analogoustoEquations 4{5 through 4{9 ,canbewrittenas ^xkjk1=f^xk1jk1;uk1;0Pkjk1=AkPk1jk1ATk+QkqQTkKk=Pkjk1HTkHkPkjk1HTk+RkrRTk1^xkjk=^xkjk1+Kkzkhxkjk1;0Pkjk=(IKkHk)Pkjk1(4{16)whichdenesthecompleteEKF.Itshouldbenotedthatthestateameasurementpredictionsusethefullnonlinearequationswhilethelinearizedmodelsarerequiredto 52 PAGE 53 42 .Thesigmapointsarechosenaccordingtoadeterministicalgorithmsothattheyaccuratelyapproximatethetruemeanandcovarianceoftheuntransformeddistribution. Figure42. TheUnscentedTransform.TheUTpropagatesasetofsigmapointsthroughanonlinearfunctiontocalculatetheposteriorprobabilitydistribution. 53 PAGE 54 L+;Wc0= L++(12+)Xi=k1+p 2(L+)Xi+L=k1p 2(L+)=2(L+)L(4{17)wherethesubscriptofthesquarerootdenotestheithcolumnofthematrixsquarerootand,,andaretuningparametersforcontrollingtheplacementandweightingofthesigmapoints.Thesigmapointsaretransformedbythenonlinearfunction 4{16 usedtocreatetheUKF.Thisderivationisaccomplishedwithanaugmentedstatevectorx02Rn+q,whereqisthedimensionofthenoisevectorq,denedas 4{10 ,isrewrittentoaccepttheaugmentedstatevectorasinput, 54 PAGE 55 4{17 aredrawnfromaprobabilitydistributionwithameanvalue ^x0k1jk1=264^xk1jk10q1375(4{23)andcovariance 4{11 ,isrewrittensothatmeasurementnoiseisrestrictedtobeadditiveaswasassumedinthelinearKalmanlter 4{17 ,aredrawnfromthedistributiondenedbyEquations 4{23 and 4{24 .Thesesigmapointsaretransformedbythenonlinearprocessmodelwithzeronoise 4{19 and 4{20 as ^xkjk1=2LXi=0WsiX0i;kjk1(4{27) 55 PAGE 56 ^zk=2LXi=0WsiZ0i;k(4{30)withcorrespondingcovariance ^xkjk=^xkjk1+Kk(zk^zk)Pkjk=Pkjk1KkPzz;kKTk(4{33)Therefore,thestateestimateofthesystemhasbeencomputedwithoutrequiringlinearizationoftheprocessandmeasurementmodels. 83 ].TheUKF,however,isacomputationallyexpensivealgorithmincomparisontotheEKF.Thisexpenseisbecausethesigmapointcalculation,Equation 4{17 ,requires2L+1evaluationsofthenonlinearprocessmodelandmatrixsquarerootsofthestatecovariance.TheSquareRootUnscentedKalmanFilter(SRUKF)isanecientvariantoftheUKFinwhichthematrixsquarerootrequiredtogeneratethesigmapointsiseliminatedbypropagatingthesquarerootofthecovariancematrixinlieuofthefullcovariance[ 84 ].Therefore,thesigmapointcalculation,replacingEquation 4{17 ,is 56 PAGE 57 4{35 and 4{36 replacetheUKFstatecovarianceupdateEquation 4{28 .Themeasurementcovarianceupdateequations,replacingtheanalogousUKFEquation 4{31 ,followthesameformulationasthestatepredictionmethodologyasshowninthefollowingequations 57 PAGE 58 82 ].ThisformulationallowstheestimationprocesstohandlenonGaussianandmultimodalprobabilitydensityfunctions.LiketheKalmanlteringapproaches,theparticlelteroperatesinarecursivefashion.Thestateispredictedaccordingtoaprocessmodeland,subsequently,updatedbycomparingsensormeasurementstomeasurementpredictionscalculatedwithameasurementmodel.However,whiletheKalmanlterpropagatesanormaldistributionthroughthenonlinearmodels,theparticlelterpropagatesasetofweightedsamplescalledparticles.ParticleltersemployasetofNparticlesdenedas 3 ,beginsbydrawingstatesamplesfromaprobabilitydensitydenedas xkp(xkjxk1;uk)=f(xk1;uk;qk)(4{42)whichisthesamenonlinearprocessmodeldenitionusedintheEKF(Equation 4{10 ).Thenotationabdenotesthattherandomvariableahastheprobabilitydistribution 58 PAGE 59 85 ].Thenumberofeectiveparticles 59 PAGE 60 4.4TheRaoBlackwellizedParticleFilterParticlelteringcanbeextremelyinecientinhighdimensionalspaces[ 86 ].ThisineciencyisvitallyimportantwithrespecttotheSLAMproblemwherethestatespaceincludesthevehiclestatesandthemaplandmarkstates.Therefore,particlelteringhasbeenappliedtotheSLAMproblemwiththeRaoBlackwellizedParticleFilter(RBPF).TheRaoBlackwellizationisaprocessinwhichthenumberofstatesthatneedtobesampledisreducedbypartitioningthejointstate.Therefore,forajointdistributionp(x1;x2)thestateispartitionedusingtheproductrule 60 PAGE 61 5 wheretheRBPFiscombinedwiththeaforementionedEKF,UKF,andSRUKFtoproducethreeltersforsolvingtheSLAMproblem. 61 PAGE 62 74 ].Timedependentmissions,suchastargetacquisitionandtracking,requireamaptoproducetimeecientsolutions.Additionally,themajorityofmotionplanningstrategiespresupposeaknownenvironmentmap[ 26 ].Therefore,mapconstructionbecomesavitallyimportanttaskforecientmotionplanning.ThisresearchisconcernedwithnavigationinGPSdeniedenvironmentsmeaningknowledgeofthevehicle'sinertiallocationisnotknown.Therefore,navigationmustrelyonrelativevehiclepositionestimateswhicharesubjecttodriftduetomodeluncertaintiesandexternaldisturbances.Additionally,sensormeasurementsoftheexternalenvironmentaresusceptibletonoise.ThecombinationoferrorsincurredfromuncertainvehiclelocationandnoisysenormeasurementsproducesspatiallyinconsistentmapsasshowninFigure 51 .TheissuesincurredbyplacingavehicleatanunknownlocationinanunknownenvironmentandrequiringittoincrementallybuildaspatiallyconsistentmapwhileconcurrentlycomputingitslocationrelativetothemapareknownastheSimultaneousLocalizationandMapping(SLAM)problem[ 75 ].ResearchintotheSLAMproblemhasproducedavarietyofsolutionsbasedonprobabilisticrecursivealgorithmswithimplementationsonavarietyofplatformsincludinggroundvehicles,aircraft,andunderwatervehicles[ 19 76 77 ].AllsuccessfulSLAMsolutionsfollowasimilaroverallestimationstrategyasdepictedinFigure 52 62 PAGE 63 Mapbuildinginthepresenceofnoise.Noiseinthemapbuildingprocessproducesaspatiallyinconsistentmap. TheSLAMestimationstrategybeginswithapredictionstep(Figure 52A )inwhichthevehiclepositionandorientationareestimatedaccordingtoamotionmodel.Additionally,sensormeasurementsarepredictedusingameasurementmodelthatestimatessensorreadingsbasedontheuncertainstateofthevehicleandthecurrentmapoftheenvironment.Essentially,thepredictionstepgeneratesaguessofwhatthevehicleshouldobservegiventhecurrentstateestimates.Next,actualsensormeasurementsoftheenvironmentaretakenbythevehicle'sonboardsensorsuiteasdepictedinFigure 52B .Finally,theactualobservationsarecomparedwiththepredictedobservationsandthemapandvehiclestatesareupdatedaccordingly(Figure 52C ).ThereareseveralopenproblemsintheSLAMcommunityincludingthereductionofcomputationcomplexitytoenablerealtimeimplementation,correctassociationofobservedlandmarkswithmappedlandmarks,andenvironmentrepresentation[ 78 ].ThisdissertationaddressestheseproblemsbyemployingadimensionallyreducedmaprepresentationasdescribedinChapter 3 .Themaprepresentationisasetofplanarlandmarksthattthreedimensionalfeaturepointsgatheredbyanonboardcamera.ThegeometricmapalleviatesthecomputationalburdenoftheSLAMsolutionbyreducingthe 63 PAGE 64 B C TheSLAMestimationstrategy.A)Inthepredictionsteptheposeofthevehicleandsensormeasurementsarepredictedaccordingtomotionandmeasurementmodels,respectively.Theerrorellipsesdenoteuncertaintyinvehiclestateandmapestimates.B)Theobservationstepinvolvesacquiringrelativemeasurementsfromonboardsensorstoenvironmentallandmarks.C)Intheupdatestepthevehiclestateandmapestimatesareupdatedbycomparingthepredictedsensormeasurementswiththeactualsensormeasurements. 64 PAGE 65 4 .Section 5.2 introducesthevehicleandmeasurementmodelsdevelopedforthepresentedSLAMformulation.Section 5.3 describesthreespecicltersthatwereinvestigatedinrelationtotheairborneSLAMproblem.Lastly,Sections 5.4 and 5.5 describethedataassociationprocessandmapmanagementproceduresemployedtoincrementallyincorporatesensedgeometricprimitivesintoaglobalmap. 4 ,stateestimationrequiresthedenitionofprocessandmeasurementmodelsthatdescribethesystemofinterest.ThegeneralprocessandmeasurementmodeldenitionsfortheSLAMproblemareprobabilisticallydenedasthefollowingdistributions 65 PAGE 66 53 .Additionally,itisassumedthatvehiclelinearvelocity[u;v;w]andangularvelocity[p;q;r]informationisprovidedviaanIMUorvisionbasedstateestimationalgorithm(Sections 2.3.2 and 2.4 ).Therefore,theinputtothevehiclemodelisselectedtobeu=[u;v;w;p;q;r]T Fixedwingaircraftmodeldenitions.TheaircraftlinearandangularratesaredenedwithrespecttoabodyxedbasisB.TheinertialpositionandorientationofthevehicleismeasuredwithrespecttoaNorthEastDown(NED)inertialframeE. Thevehiclemodelf()acceptsnoisyestimatesofvehiclevelocitiesasinputs.Positionandorientationstateinformationisgeneratedaccordingtotheaircraftkinematicequations _x=ucoscos+v(cossin+sinsincos)+w(sinsin+cossincos)_y=ucossin+v(coscos+sinsinsin)+w(sincos+cossinsin)_z=usinvsincoswcoscos_=p+tan(qsin+rcos)_=qcosrsin_=(qsin+rcos)=cos(5{3) 66 PAGE 67 67 PAGE 68 19 ].TheairborneSLAMproblemhasbeeninvestigatedwiththeExtendedKalmanFitler(EKF)[ 18 ]andtheUnscentedKalmanFilter(UKF)[ 20 ].TheEKFapproachhasbeenshowntobesomewhatsuccessfulinimplementationforaverylimitednumberoflandmarks.Additionally,UKFapproachhasproventobesuperiortotheEKFintermsofestimationaccuracy,butsuersfromincreasedcomputationalcomplexity.ThisdissertationinvestigatesthreeadditionalSLAMltersbasedontheRaoBlackwellizedParticleFilter(RBPF).ThegoalistondasolutiontoairborneSLAMthatismorecomputationallyecientthanboththeEKFandUKFapproachesandhandlesthenonlinearitiesaswellastheUKF.Thecompleteformulationoftheselters,includingtheequationsnecessaryforimplementation,isdescribedinthissection.TheSLAMestimationtaskisframedasaprobabilisticprocesswiththegoalofcalculatingthejointposteriordistributionofthevehicleandlandmarkstates 23 87 ].TheRBPFisasamplingbasedapproachwhichfactorsthejointSLAMposteriordistributionaccordingtotheobservationthatlandmarkpositionscanbeestimatedindependentlyifthevehiclepathisknown.TheresultingRaoBlackwellizedformcontainsasampledtermrepresentingtheprobabilityofvehicle 68 PAGE 69 1. GenerateNvehicleposesamples^xikandassociatedobservations^zik;j: 2. Calculateimportanceweightswikandresampleaccordingtoweightstopromoteparticlediversity: 69 PAGE 70 2(zk^zik)T(Piwk)1(zk^zik)(5{13)where 3. CalculatetheproposaldistributionanddrawNsamplesxkdeningtheupdatedvehiclestatewherethedistributionisGaussianwithparameters: 4. UpdatedenitionsofobservedlandmarksaccordingtotheEKFmeasurementupdate: 70 PAGE 71 4.2.2 )totheRBPFSLAMformulation[ 88 ].TheURBPFisanimprovementovertheRBPFintermsofestimationperformancesimilartothecomparisonbetweentheEKFandUKFdiscussedinSection 4.2.2 .Additionally,theURBPFiseasiertoimplementbecauseitdoesnotrequirelinearizationofthestatetransitionandmeasurementmodels. 1. GenerateNvehicleposesamples^xikandassociatedobservations^zik;jaccordingtotheUKFpredictionrule: 2. Calculateimportanceweightswikandresampleaccordingtoweightstopromoteparticlediversity: 2(zk^zik)T(Piwk)1(zk^zik)(5{27) 71 PAGE 72 3. CalculatetheproposaldistributionanddrawNsamplestodenethenewvehiclestatexkwherethedistributionisGaussianwithparameters: 4. Calculatesigmapointsfortheobservedlandmarksandtransformaccordingtotheobservationmodel: 5. UpdatedenitionsofobservedlandmarksaccordingtotheUKFupdaterule: 72 PAGE 73 4.2.4 ).TheSRUKFisanecientvariantoftheUKFinwhichthematrixsquarerootrequiredtogeneratethesigmapointsiseliminatedbypropagatingthesquarerootofthecovariancematrixinlieuofthefullcovariance.Similarly,theSquareRootUnscentedRaoBlackwellizedParticleFilter(SRURBPF)isformulatedbyreplacingtheUKFstylecalculationsintheURBPFwiththeSRUKFapproach. 1. GenerateNvehicleposesamples^xikandassociatedobservations^zik;jaccordingtoSRUKFpredictionrule: 2. Calculateimportanceweightswikandresampletopromoteparticlediversity: 2(zk^zik)T(Piwk)1(zk^zik)(5{48) 73 PAGE 74 3. CalculatetheproposaldistributionanddrawNsamplestodenethenewvehiclestatexkwherethedistributionisGaussianwithparameters: 4. Calculatesigmapointsfortheobservedlandmarksandtransformaccordingtotheobservationmodel: 5. UpdatedenitionsofobservedlandmarksaccordingtotheSRUKFmeasurementupdaterule: 74 PAGE 75 7 78 ].Theobservedlandmarkinformationprovidedbythedimensionalityreductionstep,however,issparseanddistinguishable.Furthermore,texturevariationsonbuildingfacadescanleadtomultipleobservationsofthesamelandmarkduringasinglesensingoperation.Therefore,anearestneighborapproachisusedtoallowformultipleassociationstothesametarget.Itshouldbenotedthatindividualgatingtechniquessuchasthenearestneighbormethodfailinenvironmentsthatarenotsparselypopulatedorstructured.Thisfailureisaresultoftheindividualgatingmethod'sinvestigationofasinglelandmarkandsensormeasurementpairatatime. 75 PAGE 76 5{64 forallpossiblepairsoftrueandpredictedmeasurements.AllassociatedpairsareusedtoupdatetheSLAMlter. 76 PAGE 77 7 77 PAGE 78 26 ].Thetrajectoryforsystemcanbederivedtoavoidobstacles,regulatethesystemtoadesiredstate,optimizeanobjectivefunction,oracombinationofthesetasks.Trajectoryplanningisdistinctfrompathplanningbecauseitaccountsforsystemdynamicsbyplanningmotionswithphysicallyrealizablevelocitiesandaccelerations.Thisdistinctionmeansthatthecurveq(t)istwicedierentiablewithrespecttotime.Pathplanning,however,islimitedtoconstructingakinematicallyfeasiblecurve,q(s),inthesystem'scongurationspace.Thegoaloftrajectoryplanningaddressedinthisresearchistogeneratecollisionfreetrajectoriesforanaircraftnavigatinginanunknownenvironment.Thetrajectoriesaredesignedtobuildacompleteenvironmentmapinminimaltime.TheemployedmappingprocedurewhichincrementallyconstructsamapofplanarfeatureswasdiscussedinChapter 5 .Thismapisusedasfeedbackforobstacleavoidanceinthetrajectoryplanningprocess.Optimaltrajectoryplanningforaircraftinthepresenceofobstaclesisadicultproblembecausetheoptimizationisnonconvex[ 89 ].Thisnonconvexityleadstomultiplelocallyoptimalsolutionswhichmakesmoststandardoptimizationroutinescomputationallyintractable.Severalapproachesfordirectlysolvingthetrajectoryoptimizationproblemforaircrafthavebeenproposed.TwoprominentmethodsbasedonMixedIntegerLinearProgramming(MILP)[ 90 91 ]andtheManeuverAutomaton[ 35 { 37 ]aredescribedinSection 6.2 .Thesemethodsdonot,ingeneral,guaranteeoptimality,aresensitivetoinitialconditions,andcanbecomecomputationallyintractableinrealisticsituations. 78 PAGE 79 61 .First,acoarsepathisconstructedtoaselectedgoallocation.Theportionoftheglobalpaththatoccurswithinthecurrentsensoreldofviewisrenedtoprovideadetailedlocalpath.Asanalstep,thedetailedlocalpathisconvertedtoatrajectorywithoptimalcontrol. Figure61. Anoverviewofthetrajectoryplanningprocess.Acoarseglobalpathisplannedtoaprescribedgoallocationdenedtoexploretheenvironment.Adetailedlocalpathisplannedtoprovidelocalsensorcoverage.Lastly,thelocalpathisconvertedtoatrajectoryusingoptimalcontroltheory. Theemployedpathplanningmethodologyisbasedonwellestablishedsamplingbasedstrategies[ 27 32 ]describedinSection 6.4 .Section 6.5 describesacurvatureconstrainedkinematicmodelusedtogeneratekinematicallyfeasiblepaths.TheenvironmentcoveragecriteriawhichselectsthebestpathforenvironmentcoverageareintroduceinSection 6.6 .TheoptimalcontrolformulationemployedforconvertingthekinematicallyfeasiblepathtoadynamicallyfeasibletrajectoryisdescribedinSection 6.7 .Thenalsection,Section 6.8 ,describesthefullalgorithmfortrajectoryplanningforoptimalmapbuilding. 79 PAGE 80 92 ],optimalcontrol[ 89 ],andsamplingbasedmethods[ 93 ].Thelattertwomethodshavebeendirectlyappliedtotheaircraftmotionplanningproblemandaredescribedbelow. 91 ],butcanbeextendedtothreedimensions[ 90 ].Thegoalofthetrajectoryplanningproblemistocomputeacontrolhistoryfuk+i2R2:i=0;1;:::;L1gthatdenesatrajectoryfxk+i2R2:i=1;:::;Lg.Theoptimizationseekstondaminimumtimetrajectorythatterminatesataspeciedgoallocationxg.Thisminimumtime,terminallyconstrainedproblemcanbeformulated 80 PAGE 81 _x_xmaxukumax(6{4)Lastly,collisionavoidanceisincludedintheoptimizationasaconstraintstatingthattrajectorypointsmustnotresidewithinanobstacle.Thefollowingformoftheconstraintiswrittenforrectangularobstacles 81 PAGE 82 62 .TheMAisadirectedgraph,asseeninFigure 62 ,consistingofasetofnodesandconnectingarcs.Thenodesrepresentthetrimtrajectoriesthatdeneequilibriumstatesforthevehicle.Forexample,thethreetrimtrajectorymodelshowncouldrepresentanaircraftoperatingatconstantaltitudewithonetrimtrajectoryrepresentingsteadylevelightandthetwoothertrimtrajectoriesrepresentingleftandrightturns.Themaneuvers 82 PAGE 83 AnexampleManeuverAutomaton.AManeuverAutomatondenedbyamaneuverlibraryofthreetrimtrajectoriesandninemaneuvers.M0denestheinitialstatetransitionbywhichtheaircraftentersthedirectedgraph. (w;)=w1(1)Mw1w2(2)Mw2:::MwNwN+1(wN+1);i0;8i2f1;N+1g(6{6)wherewisavectordeterminingthemaneuversequenceandisavectorofcoastingtimesthatdenehowlongthetrimprimitivesareexecuted.Thetrimprimitivesareuniquelydenedbythemaneuversequenceandtheconstraintonensuresthatallcoastingtimesarepositive.Thetrimtrajectoriesandmaneuverscanbemultipliedinthisfashionbecausetheyrepresenttransformationsofthevehiclestatewritteninahomogeneousrepresentation.Therefore,generalrigidbodymotiondenotedbyarotationRandtranslationT,suchasamaneuver,iswrittenintheform 83 PAGE 84 (w;)=argmin(w;)J(w;)=PNi=1Mwi+wiwi+wN+1wN+1s.t.:gwQNw+1i=1exp(ii)=g10gfw2L(MA)i0;8i2f1;N+1g(6{9)wherethecostfunctionJisdenedbyasummationofallmaneuvercostsMwiandprimitivecostsTwiwhicharescaledlinearlybytheirdurationofexecutionwi.TherstconstraintensuresthesystemisdriventoagoalstateandsecondconstraintassuresthatthesolutionmaneuverexistsinthesetofalladmissiblemaneuversequencesL(MA)denedbytheMA.Theposedoptimizationbecomescomputationallyintractableasthelengthofthemaneuversequenceandthenumberofavailablemotionprimitivesincreases.Therefore,theproblemisdecomposedintoasearchfortheoptimalmaneuversequencewandanoptimizationtodeterminetheoptimalcoastingtimes.Thisformulation,however,doesnotincludeconstraintsforobstacleavoidance.Forobstacleavoidance,theMAmotionplanningstrategyiscoupledwithsamplingbasedmethods[ 94 ](Section 6.4 ). 84 PAGE 85 40 41 ].Thisdissertationemploysadecoupledtrajectoryplanningstrategyinordertoincorporateamorecomplicatedcostfunctionintotheplanningprocessthancanbeemployedecientlyintoadirectmethod.Additionally,thefullvehicledynamicsareused.Themaindrawbackofthismethodisthatthecoveragecostisnotincludedintheoptimization.However,sincetheenvironmentisunknowntheplanningstrategymustbeimplementedinaRHfashionsooptimalitycanneverbeguaranteed. 27 ].ThePRMalgorithmconstructsaroadmapintheworkspaceconsistingofnodesandconnectingarcsasshowninFigure 63 .Thenodesarevehiclecongurationssampledfromthevehicle'sfreecongurationspaceQfree.Theconnectingarcsarecollisionfreepathsgeneratedbyalocalplannerthatconnectstwosampledcongurationsandsatisesthekinematicconstraintsofthevehicle.Therefore,containsacollisiondetectionstepalertingtheplannerifacollisionfreepathcannotbeconstructed.Theroadmapisintendedtocapturetheconnectivityofthevehicle'scongurationspaceinordertoecientlyplanfeasiblepathswithoutanexhaustivesearch.ThePRMprocessinvolvesalearningphaseinwhichtheroadmapisconstructedandaqueryphaseinwhichmultiplepathplanningproblemsaresolvedwiththeconstructedroadmap.Therefore,theroadmaponlyneedstobeconstructedonce. 85 PAGE 86 63A .Oncetheroadmaphasbeenconstructedpathplanningqueries,intheformofaninitialcongurationqinitandgoalcongurationqgoal,areanswered.AsshowninFigure 63B ,qinitandqgoalareconnectedtothetotheirnearestneighborintheroadmapq0andq00,respectively.Lastly,agraphsearchisperformedtondtheshortestpathconnectingtheinitialandgoallocations. B ProbabilisticRoadmapplanninginatwodimensionalenvironment.A)Aconstructedroadmapwherethecirclesarenodesdesignatingvehiclecongurationsandthestraightlinesarearcsconnectingthenodeswithkinematicallyfeasiblepaths.B)Thequeryphasewheretheshortestpathisfoundthroughtheroadmapgivenaninitialandgoalconguration. ThePRMwasdesignedasacomputationallyecientstrategyforcapturingtheconnectivityofQfreeinordertoanswermultiplepathplanningqueriesinregardstothesameenvironment.Manyplanningstrategies,however,requiretheanswertoasinglequeryanddonotrequireexplorationoftheentireQfree.TwoexamplesofsamplingbasedsinglequeryplannersaretheExpansiveSpacesTree(EST)plannerandRapidlyExploringRandomTree(RRT)planner.Thedierencebetweenthetwoapproachesisthemannerinwhichtheroadmap,ortree,isconstructed. 86 PAGE 87 4 64 continuesuntilaterminationcriteriasuchasthemaximumnumberofnodesinthetreeornodalproximitytothegoalissatised.ThemostimportantaspectoftheESTistheabilitytodirectthetreegrowthaccordingtoaprescribedmetric.GuidedsamplingofQfreeisachievedbyassigningaprobabilitydensityfunctionTtothenodesofthetree.Theprobabilitydensityfunctionbiasestheselectionoftreenodesfromwhichtoextendthetree.Thisbiasenablestreegrowthtowardspeciedregionssuchasareasthataresparselypopulatedbynodesorthegoallocation. 87 PAGE 88 ExpansiveSpacesTreeplanning.TheESTinvolvesconstructingatreeTbyextendingthetreefromrandomlysampledcongurationinthetreeq.Arandomsampleqrandlocaltoqisconnectedtothetreebyalocalplanner.Bothasuccessfulq0randandunsuccessfulconnectionq00randareshown. ThelaststepoftheESTplanningprocessistoconnectthetreetothegoallocationpgoal.Thisconnectingisaccomplishedbycheckingforconnectionswiththelocalplanner.Ifnoconnectionismade,furtherexpansionofthetreeisrequired.AnothervariationinvolvessimultaneouslygrowingatreeTgoalrootedatpgoalandproceedingwithaconnectionprocessbetweenq2Tinitandq2Tgoaltomergethetrees. 5 .InsteadofextendingthetreefromrandomlyselectedpointsinthetreeT,asintheESTapproach,theRRTextendsthetreetowardsrandomlyselectedpointsqrandinthefreecongurationspaceQfree.TreeconstructionbeginsbyselectingqrandandndingthenearestcongurationqnearinTtothesample.Thetreeisextendfromqnearadistancetowardqrand.Theextensionisperformedbyalocaloperatorterminatingatqnew.Iftheconstructedpathiscollisionfree,itisaddedtoT.Treeconstruction,depictedinFigure 88 PAGE 89 ,continuesuntilaterminationcriteriasuchasthemaximumnumberofnodesinthetreeornodalproximitytothegoalissatised. Figure65. RapidlyExploringTreeplanning.TheRRTinvolvesconstructingatreeTbyextendingthetreetowardarandomlysampledcongurationqrandinQfree.Thetreeisextendedadistancetowardqrandterminatingatqnew. GuidedsamplingcanbeincorporatedintotheRRTinasimilarfashiontotheEST.Toguidesampling,qrandissampledaccordingtoaprobabilitydistributionQ.Therefore,themaindierencebetweentheESTandRRTplannersisthemechanismforguidingthesampling.TheESTguidesthesamplingbyselectingattractivenodesofthetreetoextendwhiletheRRTexpandsbyselectingregionsintheenvironmenttomovetowards.Though 89 PAGE 90 6.4 relyonalocalplannertoconnectvehiclecongurations.Thelocalplannersaredenedbyvehiclekinematicconstraintssothattheconstructedpathiskinematicallyfeasible.Fornonholonomicvehiclemotionplanning,apopularlocalplanningstrategyistheDubinsoptimalpath[ 34 95 96 ].TheDubinsoptimalpathisashortestpathsolutionforconnectingtwovehiclecongurationssubjecttoaminimumcurvatureconstraint.Thesolutionpertainstoacarlikevehicleoperatinginathreedimensionalcongurationspacedeninglocationandorientationqi=[xi;yi;i]T2R3.Thecurvatureconstraintrelatestotheminimumturningradiusofthevehiclemin.TheseminalworkbyDubins[ 33 ]provedthattheoptimalcurvatureconstrainedpathconsistsofthreepathsegmentsofsequenceCCCorCSCwhereCisacirculararcofradiusminandSisastraightlinesegment.Therefore,byenumeratingallpossibilitiestheDubinssetisdenedbysixpathsD=fLSL;RSR;RSL;LSR;RLR;LRLgwhereLdenotesaleftturnandRdenotesarightturn.TheoptimalDubinspathq(s)istheshortestpathq(s)2D.Foramobilerobot,theoptimalDubinspathisthetimeoptimalpathfornonacceleratingmotionwhereminisuniquetothegivenconstantvelocity. 90 PAGE 91 66 .Thetransformedcongurationshavetheformqinit=[0;0;i]Tandqgoal=[d;0;g]Twhereqinitislocatedatoriginandqgoalislocatedontheyaxis. Figure66. ThecoordinatesystemforcalculatingthepathsintheDubinssetD.Theinitialcongurationisqinitistranslatedtotheoriginandrotatedsothatqgoalislocatedonyaxis. TheDubinspathsaretheconcatenationofthreecanonicaltransformationsscaledsothatmin=1andthevehiclemoveswithunitspeed.Thisformulationallowsthegeneralsolutiontobescaledbythevehicle'sminbecausethemaximumheadingrateisequaltoV min.Thetransformationsarewrittenintermsthedistanceofmotionvalongthepathsegmentsforanarbitrarycongurationq=[x;y;]Tasfollows 91 PAGE 92 1. DubinspathLSL v1=i+tan1cosgcosi 2. DubinspathRSR v1=itan1cosicosg 3. DubinspathLSR v1=i+tan1cosicosg 4. DubinspathRSL v1=itan1cosi+cosg 5. DubinspathRLR v1=itan1cosicosg 8(6d2+2cos(ig)+2d(sinising))v3=igv1+v2(6{16) 6. DubinspathLRL v1=i+tan1cosgcosi 8(6d2+2cos(ig)+2d(singsini))v3=i+gv1+v2(6{17) 92 PAGE 93 67 Figure67. TheDubinssetofpathsfortwocongurations.NoticethattheLRLpathdoesnotgiveafeasiblesolutionforthisset.Thetimeoptimalisdenotedasq(s). 97 ].AmethodsimilartotheMA(Section 6.2.2 )plansaninitialpathwhichplacesthevehicleatapositionandorientationthatformsaplanewiththenalpositionandorientation[ 98 ].A2DDubinspathisthenplannedbetweenthetwoinplanecongurations.Becausethedynamicsofthevehiclewillbeincorporatedafterthepathplanningtask,the3DDubinssolutionemployedinthisdissertationassumesthatchangesinight 93 PAGE 94 68 B Examplesof3DDubinspaths.Thepathsoriginatefromforthesamestartcongurationandterminateatrandomlygeneratedgoalcongurations.A)Perspectiveview.B)Topdownview. 6.5.2.1GlobalpathcollisiondetectionThe3DDubinspathdenesthelocalplannertobeusedforconnectingaircraftcongurations.EachpathmustbecheckedforcollisionsbeforeitcanbeaddedtothetreeTaccordingtothesamplingbasedplanningstrategies(Section 6.4 ).Theresultofisapathcomposedofpiecewiselinearsegments.Thesensedenvironmentisrepresentedbyasetofboundedplanes.Therefore,thecollisiondetectionalgorithmmustsearchforcollisionsalongtheentirepathbydetectingintersectionsbetweenlinearpathsegmentsandplanesasdepictedinFigure 69 94 PAGE 95 Collisiondetectionfortheglobalpath.Thecollisiondetectionalgorithmsearchesforanyintersectionsbetweenthepiecewiselinearpathsofatreeplannerandtheplanarenvironmentobstacles. Therststepofthecollisiondetectionalgorithmistosearchforintersectionsbetweeneachpiecewiselinearsegmentofthepathandallinniteplanesinthecurrentmap.Theintersectionpointpintiscalculatedby 95 PAGE 96 Interiortoboundarytest.ApointthatintersectswiththeinniteplaneiswithintheboundarydenedbytheconvexhullHifthesumoftheinterioranglesisequalto2. planeboundary.Chapter 3 denedeachplanarboundarybyaconvexhullH=fh1;h1;:::;hng2R3nofnboundarypoints.Therefore,thetestfordeterminingifacollisionhasoccurredbecomesdeterminingiftheintersectionpointpintisinteriortothethreedimensionalboundaryH.ThisdeterminationisaccomplishedbyinspectingthetwodimensionalprojectionsontotheplanarlandmarkofpintandHdenotedbypint2R2andH2R2n.AsshowninFigure 610 ,pintcanbeconnectedwitheverypointinH.Ifthesumoftheanglesbetweentheconnectingvectorsisequalto2thenthepointisinteriortotheboundary.Thisconstraintcanbewrittenfornboundarypointsas 96 PAGE 97 6.7 .Therefore,thelocalcollisioncheckingalgorithmsearchesforintersectionsbetweenthepiecewiselinearpathsegmentsrepresentedbynitelengthcylindersandobstaclesrepresentedbyboundedplanes(seeFigure 6.5.2.2 ).ThissearchisaccomplishedbyrstndingtheintersectionpointpintbetweenthecentralaxisofthecylinderandtheobstacleusingEquation 6{18 .Theintersectionbetweenthecylinderandplaneisanellipsewithcenterpint.AsshowninFigure 6.5.2.2 ,themajoraxisoftheellipsewillbeR 6{19 6.4 and 6.5.2 describedamethodforconstructingcollisionfreepathsinaclutteredenvironment.Thissectiondenesamethodforselectingpathswiththegoalofconstructingacompleteenvironmentmapinminimumtime.Forthissituation,optimality 97 PAGE 98 B Collisiondetectionforthelocalpath.Thelocalcollisiondetectionalgorithmincorporatesasafetyregionwhichencompassesthepathsegmentsradially.A)ThepathsegmentrepresentedbyacylinderwithradiusRintersectsanobstacledescribedasaplanewithnormalvectorn.B)Atwodimensionalviewrotatedorthogonaltotheplanecontainingthemajoraxisoftheintersectingellipse. cannotbeprovenbecauseinitiallytheenvironmentisentirelyunknown.Therefore,anantagonisticexamplecanbeconstructedforanyplannertoprovideanonoptimalresult.Theroboticcoverageproblemreferstooptimalcoverageofanenvironmentwithknownobstacles[ 99 ].Solutionstotheroboticcoverageproblemgenerallyinvolvedecomposingthefreespaceoftheenvironmentintoconnectedregions.Eachregionoftheenvironmentissensedwithabackandforthlawnmower,orboustrophedon,patternasshowninFigure 6.6 .Theboustrophedonpathisoptimalforatwodimensionalholonomicvehicleundertheassumptionthatpassingthroughasmallregionoftheenvironmentisequivalenttosensingtheregion.Thespacedecompositionprocessreducestheplanningtasktondingtheoptimalrouteforwhichtoconnecttheindividualpaths.Whentheenvironmentisunknownandtheplanningtaskmustbedirectedbygatheredsensorinformation,theproblemiscalledsensorbasedcoverageorsensorbasedexploration.Sensorbasedcoveragetaskstypicallyoperateunderseveralstrictassumptionsincludingatwodimensionalenvironment,anomnidirectionalrangesensor,anda 98 PAGE 99 B Motionplanningforcoveragestrategies.Traditionalmotionplanningstrategiesforenvironmentalcoveragearenotwellsuitedforvehicleswithdynamicconstraintsanddirectionalsensors.A)Roboticcoveragetasksassumetheenvironmentisknownanddecomposethefreespaceintomanageablesearchspaces.B)Sensorbasedcoveragestrategiesoftenassumeomnidirectionalsensors. robotcapableofnonholonomicmotion[ 100 101 ](seeFigure 6.6 ).Therefore,currentsensorbasedcoveragealgorithmsdoaddresstheproblemsinherentinmappingathreedimensionalenvironmentwithadynamicallyconstrainedvehicleequippedwithadirectionalsenor.Thecoveragemethodologyusedinthisdissertationaectsboththeglobalandlocalplanningstrategies.Thestrategyisbasedontrackingalllocationstowhichthevehiclehastraveledandtheattitudesfromwhichindividuallocationshavebeensensed.Thegoaloftheplannerismaximizethevarianceoftheseindividualquantities.Therefore,thediscretevehiclelocationsandthevectorsdeninglocationandorientationofsensedregionsaretrackedovertimeasshowninFigure 613 .Thevectorsdeningsensedregionareconstructedfromtheattitudeofthevehicleduringthesensingoperationandthecenteroftheeldofviewofthesensor.Thetotalvarianceofthevehiclepositioniscalculatedasthetraceofthecovarianceofthevehiclepositionsqposasgivenby 99 PAGE 100 Environmentalcoveragemetric.Environmentcoverageisquantiedbythevarianceofthepositionofthevehicleandsensinglocationsovertheentirepath.Sensinglocationsaredenedbythevectororiginatingfromtheendofthesensoreldofvieworientedbacktowardsthesensor. whereposisthesamplecovarianceoftheallvehiclepositionvectors.Thesameequationcancomputethevarianceofthethesensorposevectors2sengiventhecovarianceofthesensorposevectorssen.Thevarianceofthevehicleposition2posisusedtodeterminethegloballocationbyselectingthepointintheenvironmentthat,whenaddedtothecurrentvehiclelocationhistory,maximizes2pos.Thiscalculationiscomputedonacoarsegridofthevehicleenvironmentforthepurposeofcomputationaleciency.Theglobalpathplannerconstructsapathtothislocationwithagreedysearchandacoarsediscretizationofthepath.Thelocalpathisdeterminedbymaximizingthevarianceofthesensorposevectors2senoverallcandidatepaths.Asopposedtothequick,greedysearchusedforglobalpathplanning,thelocalsamplingbasedstrategygeneratesatreethatmorecompletelyexplorestheenvironment.Whenaspeciednumberofcandidatepathsarefoundthepaththatmaximizes2senisselected. 100 PAGE 101 6.5.2.2 )ensuresthattheplannedpathstraversetheenvironmentinsideofasafecorridor.Theoptimalpathtrackingstrategyseekstondatimeoptimaltrajectorythatfollowstheplannedpathwithinadistancer PAGE 102 =!2n;(u)2!n;__=u (6{23)where!n;isthethenaturalfrequencyfortheresponse,isthedampingratiocorrespondingto,isthetimeconstantfortheresponse,andthenewcontrolinputstothesystemareuandu.Theoptimizationproblemisformulatedasthefollowingnonlinearprogram(NLP) (u)=argminuJ(u)=t2f+PNk=1(xk;q(s))s.t.:xk+1=f(xk;uk)dtxN=qfk PAGE 103 5 5 6{24 103 PAGE 104 6{24 .TheexibilityoftheplanningframeworkallowsfortheoptimalcontrolformulationtobesubstitutedwithaMAimplementationasdescribedin 6.2.2 .Bothmethodologieswouldincorporatedynamicsintothenalsolution.ThenalplannedtrajectoryisimplementedandtheplanningprocessisrepeatedwhenthevehiclearrivesatqL.Thischapterhaspresentedaframeworkthatplansaircrafttrajectoriesforcompletesensorcoverageofanunknown,uncertainenvironment.ResultsoftheplanningstrategyarepresentedinChapter 7 104 PAGE 105 7.2 givesresultsoftheplanettingalgorithmintroducedinChapter 3 .TheplanettingalgorithmisemployedintheSLAMalgorithm(Chapter 5 )toproducetheresultsinSection 7.3 .Finally,resultsofthetrajectoryplanningalgorithm(Chapter 6 )arepresentedinSection 7.4 3 generatesalowdimensionalenvironmentrepresentationofthepointwisesensordataprovidedbySFM(Chapter 2 ).ThequalityofthetisinvestigatedwithrespecttofeaturepointnoiseinSection 7.2.1 andtheinclusionofnonplanarobstaclesinSection 7.2.2 .ThesimulationenvironmentemployedforanalyzingtheplanettingalgorithmisshowninFigure 71 Figure71. Resultoftheplanettingalgorithmwithoutimagenoise. 105 PAGE 106 71 consistsofpolyhedralobstaclesviewedbyacameraatagivenvantagepointdenotedbyavisionfrustum.TheobstaclesrepresentanurbanenvironmentandthefrustumindicatestheeldofviewforacameraattachedtoaUAVnavigatingtheenvironment.Featurepointsareaddedtotheobstaclefacadesandcheckedforvisibilityaccordingtoapinholecameramodel.Thevisibilityconstraintsaccountforenvironmentalocclusion,sensoreldofview,andangleofincidence.TheangleofincidenceconstraintforafeaturepointisdepictedinFigure 72 wherefeaturepointswithanangleofincidence,#aredeemedvisibleif15#75.Lastly,allvisiblefeaturepointsareinputintotheplanettingalgorithmtogeneratetheplanartofthepointwisedata. Figure72. Angleofincidenceconstraint.Theangleofincidence,#,forfeaturepointfistheanglebetweenthefeaturepoint'sposition,~f,measuredinthecameracoordinateframeandthenormaldirection,n,oftheplaneonwhichthefeaturepointissituated. 106 PAGE 107 2{14 ).ResultsoftheplanettingalgorithminthepresenceofnoiseareshowninFigure 73 B C D Resultsoftheplanettingalgorithminthepresenceofimagenoise.A)Planettingresultwith=0:5pixelimagenoiseandacameradisplacementof25ft.B)Planettingresultwith=1:0pixelimagenoiseandacameradisplacementof25ft.C)Planettingresultwith=0:5pixelimagenoiseandacameradisplacementof15ft.D)Planettingresultwith=1:0pixelimagenoiseandacameradisplacementof15ft. Theplanettingresults,showninFigure 73 ,weregeneratedwithGaussiannoiseofstandarddeviation=0:5and=1:0pixelsappliedtotheimages.Additionally,the 107 PAGE 108 73B and 73D .Thesetworesultsshowthatincreasingthebaselinebetweencameravantagepointsproducesmoreaccuratetriangulationresultsleadingtoamorecompleteplanarrepresentation. 74 presentsplanettingresultswhennonstructuredobstaclesareincludedintheformofseveraltreelikeobjects.Theadditionalfeaturepointsgeneratespuriousplanarfeatures.SuchfeaturesareinaccuraterepresentationsofthephysicalworldandprovidepoorlandmarksfortheSLAMalgorithm.Specically,thespuriousplanesarenotnecessarilyobservablefrommultiplevantagepoints.Inordertoreducethenumberofspuriousplanarobstaclesinthemap,theSLAMalgorithmonlyincorporatesobstaclesthathavebeenreobservedaminimumnumberoftimes.Aminimumvalueof3reobservationswasfoundtoprovidesuccessfulmapresultsinthepresenceofspuriousplanarfeatures. 5 areperformedforthethreelterspresentedinthisdissertation:theRBPF,URBPF,andSRURBPF. 108 PAGE 109 Resultoftheplanettingalgorithmwhennonplanarobstaclesareincluded. ThelterswerecomparedbyperformingMonteCarlo(MC)simulationsinwhichthevehiclestatemeasurementnoiseandsensornoisewererandomlygeneratedwithintheconnesofthecovariancematricesqandr,respectively.Thisprocessteststheaverageperformanceoftheltersovermanyestimationprocesses. 75A ,consistsofgeometricallyregularobstaclesindicativeofanurbanenvironment.Theaircraftnavigatesalongaprecomputedtrajectorygeneratedbyahighdelitynonlinearaircraftmodel.SFM,asdiscussedinChapter 2 ,issimulatedbydistributingthreedimensionalfeaturepointsonthebuildingfacadesandground.Thefeaturepointsprovidepointwisestructureinformationtotheplanettingalgorithm.Theplanettingalgorithm,introducedinChapter 3 ,generatesobservationsasthevehiclenavigatesasshowninFigure 75B .Thevehiclestatepredictionoccursat30Hz.ThoughanIMUcouldprovideafasterstateestimationrate,itisassumedthattheIMUisfusedwithimageprocessing 109 PAGE 110 B ThevisionbasedightsimulationenvironmentfortestingtheSLAMalgorithm.A)Theurbansimulationenvironmentiscomposedofgeometricallyregularobstaclesrepresentingbuildings.Aprecomputedtrajectorygeneratedfromanonlinearaircraftmodelisshowncirclingtheenvironment.B)Astheaircraftnavigatesalongtheprecomputedtrajectorysensorinformationisgathered.Thegatheredsensorinformationisrestrictedtoasenorconedenedbyamaximumdepthofsensingandhorizontalandverticaleldsofview.Threedimensionalfeaturepointsaredistributedonthebuildingfacesandplanesarettothepointwisedata. 110 PAGE 111 76 .Theresult,obtainedwiththeURBPFapproach,simulatesaknownenvironmentbyinitializingthelandmarkstotheiractuallocations.ThisisequivalenttousingtheSLAMframeworkstrictlyforlocalizingthevehicleinthepresenceofnoisystatemeasurementsandnoisysensormeasurements.Thenoisevalues,denedinEquations 5{3 and 5{4 ,weregiventhefollowingstandarddeviations 76 )showsthelteriscapableofproducingverygoodlocalizationresultsinthepresenceofconsiderablenoise.TheestimatedtrajectorygivenbytheSLAMalgorithmisdenedbytheparticlewiththelargestweightandthe 111 PAGE 112 B C D ExamplelocalizationresultfortheairborneSLAMsimulation.A)Thecompletelocalizationresultwithconstructedmapandestimatedtrajectory.B)Theconstructedmapafterasingleloopthroughtheenvironment.C)Thelocalizationresultwithouttheconstructedmap.D)Atopdownviewoftheactualvehicletrajectory,trajectorypredictedfrommeasuredvehiclestatesonly,andthetrajectoryestimatedbytheSLAMalgorithm. measuredtrajectoryiscalculatedfromthenoisystatemeasurementswithoutupdatesfromthevisionsensorinformation.Theestimatedtrajectoryquicklyconvergestothegroundtruthtrajectoryasisconsiderablymoreaccuratethanthemodelbasedpredictionovertheentiretrajectory.ThisshowstheversatilityoftheSLAMalgorithminthatitcanbeemployedifthestructureoftheenvironmentisknowntoprovideaccurateestimatesofvehiclelocationdespiteverynoisysensormeasurements. 112 PAGE 113 77 showstheaveragepositionandorientationerrorsforthethreeSLAMltersalongwiththeaverageerrorofthemeasuredtrajectory.Themeasuredtrajectoryisthetrajectorycalculatedpurelyfromtheaircraftkinematicmodelandvelocitymeasurementswithoutupdatesfromtheenvironmentalsensing.Thesegraphsindicatetheaccuracyofthepositionandattitudeestimation.TheresultsshowthattheSLAMalgorithmsprovideasignicantdecreaseinestimationerrorsforaircraftposition,rollangle,andpitchangle.Theheadingangleestimates,however,begintodivergehalfwaythroughthesimulation.Theapparentsharpdecreasesinerrorforthemeasuredtrajectorycanbeattributedtheerraticbehaviorofthemeasuredtrajectory.Thepresentederrorplotsshowtheabsolutevalueoftheerrors,therefore,whentheerrorchangesfrompositivetonegativeitcrossesthezeroerroraxisandappearsasadecreaseinerror.Thissharpdecreaseshouldnotbemistakenforacceptableestimationperformance. 113 PAGE 114 B C D E F AverageestimationerrorforthethreeSLAMltersandthenominaltrajectoryfor50MonteCarloruns.A)EstimationerrorintheNorthdirection.B)EstimationerrorintheEastdirection.C)Estimationerrorinaltitude.D)Estimationerrorinrollangle.E)Estimationerrorinpitchangle.F)Estimationerrorinheadingangle. 114 PAGE 115 79 ]calculatedaccordingto 103 104 ].Theconsistencyofthelters,however,isanacceptablemetricforassessingtherobustnessofthelterbecausethelengthoftimethatalterremainsconsistentrelatestohowwellthelterperformsoverlongtimeperiods.TheresultspresentedinFigure 78 showthattheURBPFprovidesthebestresultsintermsoflterconsistency.ThisislikelyduetothefactthattheURBPFdoesnotrequirelinearizationasdoestheRBPF.Also,itisbelieved 115 PAGE 116 B Filterconsistencyresults.A)AverageNEESforallthreeSLAMltersover50MonteCarloruns.B)AverageNEESfortheURBPFlterfor50MonteCarlorunswith100and1000particles. 71 .SincetheltersareimplementedwithnonoptimizedMATLABcode,theeciencyresultshavebeenpresentedascomputationtimenormalizedtotheRBPFresultforNparticles.ThecomparisonincludesresultsforN,2N,and4Nparticlestocomparetheeectsofincreasedsamplingonltereciency.TheeciencyresultsshowthattheRBPFlterisconsiderablyfasterthantheothertwoltersandthattheltersscalelinearlywiththenumberofparticles.Theineciencyoftheunscentedltersisduetothesigmapointpropagationstepinwhich2L+1sigmapointsmustbetransformedaccordingtothesystemdynamicsandsensormodel.TheURBPF,however,isamoreconsistentltermeaningthatitismorerobustwithrespectthelengthoftheestimationtime.Therefore,fewerparticlescouldbeusedintheURBPFtoproduceresultscomparabletotheRBPFoverthesametimeperiod.Thisdecreasein 116 PAGE 117 Table71. EciencycomparisonoftheSLAMlters. Numberofparticles RBPF URBPF SRURBPF N 1.00 2.06 1.602N 2.07 3.97 2.854N 3.98 7.63 5.64 79 wheretheestimatedtrajectorygivenbytheSLAMalgorithmisdenedbytheparticlewiththelargestweightateachtimestep.TheerrorcovariancesforthissolutionaregiveninEquation 7{2 6 .ThetrajectorieswereplannedwithinthesimulationenvironmentpresentedinSection 7.3.1 .Theconstructedmapsusedintheplanningprocessareexact,representingaSLAMsolutionwithnoerror. 117 PAGE 118 B C D AnexampleresultforthecompleteairborneSLAMsolutionusingtheURBPF.A)ThecompleteSLAMresultwithconstructedmapandestimatedtrajectory.B)Theconstructedmapafterasingleloopthroughtheenvironment.C)TheSLAMresultwithouttheconstructedmap.D)Atopdownviewoftheactualvehicletrajectory,trajectorypredictedfrommeasuredvehiclestatesonly,andthetrajectoryestimatedbytheSLAMalgorithm. aregionhasbeenreachedithasbeensensed.This,however,isnotcorrectforadirectedsensorsuchasanonboardcamera.Inthiscase,directpathsthroughtheenvironmentwillnotprovidetheneededinformationtobuildacompletemap.Thelocalplanningstrategyincorporatesthesensorphysicsbytrackingvehicleposesandselectingpathswhichmaximizethevarianceinsensingvantages.Therefore,pathsareselectedaccordingtotheamountofnewinformationthatwillbeprovidedbytraversingthepath.Thelocalplanningprocessoccurswithinthevisionconedenedby 118 PAGE 119 B C D Examplepathsforenvironmentcoverage.A)Exampleresultfortheglobalplanningstrategy.B)Exampleresultforthecompleteplanningstrategy.C)Topdownviewoftheglobalplanningstrategyresult.D)Topdownviewofthecompleteplanningstrategyresult. Inordertoquantifythebenetoftheoverallplanningstrategy,Figure 710 presentstrajectoryresultsforcaseswhentheplanningprocessisrestrictedtotheglobalplannerandwhenthecompleteplanningstrategyisemployed.Therstcase,wherejustthe 119 PAGE 120 711 whichcomparesbothmethodsintermsoftotalobstaclesurfaceareaviewed.Theresultshowsthataddingthelocalplanningstrategyincreasesenvironmentcoveragedespitereducingtheoveralldistancetraversedtotherstglobalgoallocation. Figure711. Comparisonofenvironmentcoverageresults.TheamountofobstaclesurfaceareaviewedduringthetransversalofpathsgiveninFigure 710 120 PAGE 121 6 wasusedtoconvertDubinspathstoaircrafttrajectories.TheobjectivefunctionoftheNLPminimizesthedistancebetweenthetrajectoryandthegivenpaths.Therefore,theNLPdenesacontrolstrategyforoptimalpathfollowing.ResultsoftheoptimalcontrolproblemarepresentedinFigure 712 B Resultanttrajectoriesfromthepathtrackingoptimization.A)ResultsoftheoptimalcontrolstrategyusedtoconvertDubinspathstoaircrafttrajectories.B)Topdownviewoftheoptimalpathfollowingresult. TheoptimalcontrolresultsaregeneratedbysupplyingtheDubinspathastheinitialconditiontotheoptimization.TheDubinspathsspecifyinitialconditionsforthestatesV,,,x,y,h,and.Theangleofattack,,wascalculatedforeachsegmentoftheDubinspathbycalculatingthetrimangleofattackaccordingtothevehicle'sequationsofmotion(Equation 6{21 ).TheoptimalcontrolresultsshowthattheaircraftiscapableoftrackingtheDubinspathsinthelateraldirection.Thelongitudinaldirectionaltracking,however,producespoorerresults.Itwasfoundthattheoptimizationdidnotalwaysconverge,generatingpoorresultsforaggressivemaneuvers.Additionally,theoptimizationsrequiredcomputationtimeontheorderof20to60seconds.Forthesereasons,itisbelievedthatanMAstrategywouldprovideabettersolutiontoincorporatingdynamicsintotheplanningstrategy. 121 PAGE 122 122 PAGE 123 123 PAGE 124 [1] Rudakevych,P.andCiholas,M.,\PackBotEODFiringSystem,"ProceedingsoftheSPIEInternationalSocietyforOpticalEngineering,Vol.5804,2005,pp.758771. 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[104] Bailey,T.,NeitoJ.,andNebot,E.,\ConsistencyoftheEKFSLAMAlgorithm,"IEEE/RSJInternationalConferenceonIntelligentRobotsandSystems,2006. 131 PAGE 132 AdamScottWatkinswasborninOrlando,FloridaonNovember11,1980.AftergraduatingfromLakeMaryHighSchool,AdamattendedtheUniversityofFloridareceivingaBachelorofScienceinMechanicalEngineeringin2003.AdamcontinuedhiseducationattheUniversityofFloridareceivingaMasterofScienceinMechanicalEngineeringin2005.UponobtaininghisMS,AdampursuedadoctoraldegreeundertheadvisementofDr.RichardLind.Adam'sresearchinvolvesthecontrolofunmannedsystemsandheispreparedtograduatewithhisPh.D.inthesummerof2007.UpongraduationAdamwillbeginworkattheJohnsHopkinsUniversityAppliedPhysicsLabcontinuinghisresearchintheeldofunmannedsystems. 132 