<%BANNER%>

Artificial Potential-Function Guidance for Autonomous In-Space Operations

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ARTIFICIAL POTENTIAL FUNCTI ON GUIDANCE FOR AUTONOMOUS INSPACE OPERATIONS By ANDREW R TATSCH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Andrew R Tatsch

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Dedicated to My parents who brought me into this world and My friends, family, and colleague s who opened my eyes to it.

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iv ACKNOWLEDGMENTS To my advisor, Dr. Norman Fitz-Coy, I would like to convey my gratitude. Few graduate students have the opportunity to work with, not for, an advisor that encourages freedom and creativity in the search for a pragmatic solution and uniquely challenges them beyond what they think is possible. He also taught me many lessons beyond the curriculum and for those I am grateful. W ithout his guidance this dissertation would never have come to fruition. Many thanks also go out to my supervisory committee, Drs. Rick Lind, Andrew Kurdila, Gloria Wiens, BJ Fregly, and Stanle y Dermott, for their invaluable assistance and patience. Lastly, I would like to acknowledge my br other J Bradley Tatsch for everything that he has done to make this dream a re ality. Without his assi stance my undergraduate and graduate experiences would have never go tten off the ground. I am forever grateful for all that he has done for me.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... ..x CHAPTER 1 INTRODUCTION........................................................................................................1 State of the Art for In-Space Op erations (ISO) Functionalities...................................4 Research Scope...........................................................................................................11 Dissertation Outline....................................................................................................12 2 TRAJECTORY DETERMINATION: ORBIT MECHANICS APPROACH...........14 Midcourse Trajectory Determination.........................................................................14 Lamberts Problem..............................................................................................16 Maneuver Command Determination...................................................................19 Numerical Exampl e: Midcourse..........................................................................21 Endgame Trajectory Determination...........................................................................22 CW Maneuver Command Determination............................................................24 Numerical Example: Endgame............................................................................25 Alternative Orbit Mechanics Based Methods.............................................................26 Conclusions.................................................................................................................28 3 ARTIFICIAL POTENTIAL FU NCTION POSITION GUIDANCE.........................29 Robot Motion Planning...............................................................................................29 Artificial Potential Func tion Position Guidance.........................................................32 Example: Spacecraft Unconstrained Rendezvous.....................................................35 Obstructions................................................................................................................37 Numerical Example: Pa th-Constrained Rendezvous.................................................42 Conclusions.................................................................................................................44 4 ARTIFICIAL POTENTIAL FUNCTI ON GUIDANCE: A NEW APPROACH.......45

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viDynamic Artificial Potential Func tion Obstacle Avoidance Algorithm.....................45 Attractive Potential..............................................................................................46 Repulsive Potential..............................................................................................47 Total Force Interactions.......................................................................................49 Simulation Results...............................................................................................50 Constant velocity scenario...........................................................................50 On-orbit rendezvous scenario.......................................................................52 Optimized Harmonic Attractive Potential..................................................................54 Conclusions.................................................................................................................57 5 ARTIFICIAL POTENTIAL FUNC TION ATTITUDE GUIDANCE.......................60 Attitude Equations of Motion.....................................................................................63 Continuous APF Attitude Guidance...........................................................................64 Stability Analysis.................................................................................................67 Stability of Dynamical System............................................................................72 Optimization........................................................................................................74 Numerical Example.............................................................................................75 Impulsive APF Attitude Guidance..............................................................................79 Numerical Example: Impulse-b ased APF Attitude Guidance...................................82 Conclusions.................................................................................................................85 6 CONCLUSION AND FUTURE WORK...................................................................87 Conclusion..................................................................................................................87 Future Work................................................................................................................88 LIST OF REFERENCES...................................................................................................90 BIOGRAPHICAL SKETCH.............................................................................................96

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vii LIST OF TABLES Table page 1 Orbit parameters for rendezvous simulation............................................................52 2 Genetic algorithm statistics......................................................................................57 3 Mean and standard deviations for the two parameters in the GA optimization.......57 4 Genetic algorithm statistics......................................................................................74 5 Mean and standard deviation for the six parameters in the GA optimization..........75

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viii LIST OF FIGURES Figure page 1 Model of the ROTEX manipulator.............................................................................5 2 Artist's rendition of ETS-VII components.................................................................5 3 CMU's Skyworker prototype......................................................................................6 4 NASA's Robonaut......................................................................................................6 5 University of Maryland's Ranger...............................................................................6 6 AERCam Sprint..........................................................................................................9 7 Mini AERCam............................................................................................................9 8 Artist's depiction of the Orbiter Boom Sensor System..............................................9 9 The SCAMP in the neutral buoyancy tank.................................................................9 10 Schematic of Lamberts Problem.............................................................................15 11 Transfer trajectory from midcourse trajectory determination aglorithm..................23 12 Depiction of orbital rendezvous...............................................................................23 13 Endgame maneuver trajectory in th e xy-plane (R-bar V-bar plane)........................26 14 Glideslope solution for 5 impulse rendezvous.........................................................27 15 Planar rendezvous in the CW frame under APF guidance.......................................36 16 Control efforts in x and y directions for the rendezvous..........................................36 17 Difference between IC potential a nd potential on surface of obstructions..............40 18 Contour plot of th e potential function......................................................................40 19 Zoomed view of contour plot...................................................................................41 20 Stem plot of obstruction surface potential error.......................................................41

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ix21 Trajectory traversed by robot for th e path-constrained rendezvous example..........42 22 Control action in the x-direction for the constrained rendezvous............................43 23 Control action in the y-direction for the constrained rendezvous............................43 24 2D Constant velocity simulation results...................................................................51 25 Planar on-orbit rendezvous results...........................................................................53 26 Norm of applied chaser accelerations......................................................................54 27 Optimized sink-vortex pair vs CW based vector field............................................58 28 Inequality of Eq. (64)...............................................................................................72 29 Behavior of q4...........................................................................................................73 30 Time response of vector part of quaternion.............................................................76 31 Time response of scalar part of quaternion..............................................................77 32 Time response of angular velocity...........................................................................78 33 Time response of control effort................................................................................79 34 Components of the vector part of the quaternion vs. time.......................................82 35 Components of the angular velocity vs. time...........................................................83 36 Scalar part of the quaternion vs. time.......................................................................83 37 Angular velocity impulses vs. time for the numerical example...............................84 38 Scalar part of the quaternion vs time for minus one convergence case..................85

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x Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ARTIFICIAL POTENTIAL FUNCTI ON GUIDANCE FOR AUTONOMOUS INSPACE OPERATIONS By Andrew R Tatsch May 2006 Chair: Norman Fitz-Coy Major Department: Mechanic al and Aerospace Engineering For most in-space operations (ISO), e.g., a ssembly, repair, refueling, the target location and its orbit parameters are known a pr iori. For instance in the case of an assembly mission, the general lo cation of the construction si te and the parts depot are known parameters in mission planning. Utilizing this known information, we propose a novel approach to collision-free operations fo r a heterogeneous team of spacecraft robots performing the ISO mission based on artifici al potential function (APF) guidance. In order to utilize APF gui dance two prominent deficien cies of APF methodologies need to be circumvented: premature termina tion due to local minima in the operational space and suboptimal performance. It is wi dely known that utilization of harmonic potential functions eliminates the introduction of local minima in the operational space; thus we employ harmonic functions to conquer the first of the aforementioned deficiencies. In order to surmount the subop timal performance, we propose the use of a sum of harmonic function primitives (simple func tions such as sinks and vortices in the

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xi hydrodynamic analog) to shape the velocity fi eld of the operational space to harness the underlying dynamics of the system (i.e., or bit mechanics). Furthermore, given the location and orbit parameters of the target site, a random heuristic search algorithm can be employed to optimize the velocity field shaping for maximizing the performance gains achieved from harnessing these underlying dynamics.

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1 CHAPTER 1 INTRODUCTION I believe that this nation should commit its elf to achieving the goal, before this decade is out, of landing a man on the Moon and returning him safely to the Earth.1 President John F. Kennedy, May 25, 1961 Just as the speech given forty-three y ears ago on a Thursday in late May by President John F. Kennedy presented the Na tional Aeronautics and Space Administration (NASA) a new directive, on January 14, 2004, President George W. Bush delivered a speech titled Vision for Space Exploration,2 whose intent was to again unify NASA around a set of common goals. These goals are centered around sending man and/or robotic missions to the Moon, Mars, and deeper into our solar syst em. Implicit in the robotic missions proposed by this new space ex ploration mandate are the utilization and development of autonomous space robotic sy stems that will serve as trailblazers, advance guard to the unknown.2 Based on these new directives, autonomous space robotic systems will be imperative for the success of NASAs new vision. Space robotics, in the context of this wor k, are robotic systems whose applications can be roughly dichotomized into two broa d classes of missions: in-space operations (ISO), and planetary surface exploration (PSE ). Both of these mission profiles share common functional requirements and possess functional requirements unique to each mission. The functionalities typically associat ed with in-space operations are assembly (including maintenance of pre-ex isting structures) and inspecti on. It is important to note that although treated separa tely, the assembly and insp ection functionalities share

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2 common subtasks, such as navigation and mane uvering. Planetary surface exploration (PSE) functionalities include surface mobility (which include s navigation and collision avoidance), instrument deployment and sa mple manipulation, science planning and execution, and human exploration assistance. Even though space robotics applications are categorized into these tw o mission classes, generalizati on of the functi onalities listed above allows for other mission profiles such as terrestrial assembly or in-space science. The research presented in the subsequent chap ters is primarily applicable for in-space operations; therefore those will be the primary focus here. The area of in-space robotics operation is far less mature than robotic planetary surface exploration systems, both in terms of real world implementations and theoretical research. Real world implementations of planetary surface exploration have been significantly more numerous, and more successful, than in-space robotics implementations with examples of teleopera ted surface exploration robots dating back to the 1970s with the Soviet Lunakhod Rover.3 The sophistication of more recent PSE system successes is evidenced by the Mars Exploration Rovers4 developed at NASAs Jet Propulsion Laboratory (JPL). In 1997 the Sojo urner, although not designed for advanced capabilities, was the first Mars rover to succes sfully demonstrate the ability to teleoperate a vehicle on Mars. Most recently the two mo re sophisticated Mars rovers, Spirit and Opportunity, have demonstrated effective te leoperational capabilities in surface mobility and terrestrial science. The raison d'tre that led to this disparity between PSE and ISO implementations is twofold. First, the theory and experience of te rrestrial robotics are di rectly applicable to PSE scenarios. Volumes of research si nce the 1950s have been devoted to the

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3 kinematics and control of terre strial robot manipulators with achievements in autonomous navigation, perception, and mobility. Therefor e, implementation of PSE robots springboarded off the terrestrial robotics knowle dge base and needed only to address the technical issues associated with high degr ees of mobility, science planning, perception, and exploration capabilities, the harshness of space and space transport, and the high latency communication links implicit in space teleoperation. Conversely, with the exception of autonomous rendezvous and docking (ARD) investigations5 that date back to October 1967 in Russia6 and preliminary investigation of in-space proximity operations,7,8 it has only been in the last decade that autonomous, free-flying robotics applications have appeared in the literature.9,10 Therefore, both the underlying theory and its implementation need much further investig ation in order to achieve the maturity of PSE robotics systems to date. Second, most terrestrial and PSE agents ar e developed without c onsideration of the system dynamics: they are typically kinematica lly driven, and the cases that do consider some dynamics do so only in the sense of constraints on the motion, e.g., non-holonomic constraints. For ISO applications, considera tion of the system dynamics is essential. This fundamental difference is illustrated by the local operator in each case. In terrestrial or PSE applications, the local operator ma y be zero. This means that under zero command, the robot remains stationary until it is commanded to move. The local operator in ISO systems, however is in general not zero. Inst ead, the local operator is the system dynamics, meaning under zero comma nd, an ISO system will not remain stationary but move according to the topology shaped by the system dynamics. Therefore, consideration of the dynami cs of ISO systems is imperative.

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4 The sections that follow outline the state of the art for each of the functionalities of ISO systems for both current implementati ons and experimental test beds. The expectations for the sophisticat ion of these functionalities over the next decade are also discussed. State of the Art for In-Space Op erations (ISO) Functionalities In this section both current state of the art and experime ntal test bed sophistication are assessed for assembly and inspection functio nalities of ISO systems. The relatively few implementations of ISO systems to da te are emphasized while noting the progress possible from the experimental investigations. The current state of assembly functi onality, with respect to real world implementations, is restricted to the Shuttle Remote Manipulator System (SRMS) and the Space Station Remote Manipulator System (SSRMS) aboard the International Space Station (ISS). Neither of the current system s possesses any autonomy as they are solely teleoperated on-board or from the ground. Ho wever, to date there have been on-orbit demonstrations of assembly subsystems such as the Robot Technology Experiment11 (ROTEX), a German experiment flown by NASA, and Engineering Test Satellite9 (ETSVII), flown by the Japanese Aerospace Expl oration Agency (JAXA) (formerly National Space Development Agency of Japan (NASDA)). The ROTEX was a robotic arm (see Figure 1) that flew in 1993 on Columbia as part of STS-55, and successfully completed multiple tasks that include replacement of a simulated Or bital Replacement Unit (ORU) and capture of a free-flying object vi a on-board and ground teleoperation and autonomous scripts. By accomplishing tasks from autonomous scripts during the experiment, ROTEX became the first autonomous space robotic system. The ETS-VII mission was composed of a passive target sa tellite, Orihime, and a chaser satellite,

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5 Hikoboshi, with a robotic manipulator arm (see Figure 2). ETS-VII successfully demonstrated cooperative contro l of a robotic arm and satellite attitude, and simple examples of visual inspection, equipment excha nge, refueling, and handling of a satellite. Figure 1. Model of the ROTEX manipulator Figure 2. Artist's rendition of ETS-VII components The state of the art in assembly robotics, implemented via experimental test beds only, is the Skyworker12 robot at Carnegie Mellon University (see Figure 3), NASAs Robonaut13 (see Figure 4), and Ranger14 at the University of Maryland (see Figure 5). Skyworker is an 11 DOF robot that walks ac ross the structure it is assembling to mate new components to the existing structure. The current prototype allows for high-level command inputs that are then parsed and implemented on-board as motion control commands. Robonaut is a collaborative e ffort between DARPA and NASA aimed at developing a humanoid robot capable of meeting the increasing requirements for extravehicular activity (EVA). Robonaut, comp osed of two dexterous arms and two fivefingered hands with teleoperational a nd autonomous capabilities, has already Orihime Hikoboshi

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6 demonstrated assembly of complicated EVA electrical connectors and delicate capabilities such as soldering. Ranger is a tele operated robot at the University of Maryland that completes assembly, maintena nce, and human EVA assistance tasks in a neutral buoyancy tank. Ranger has demonstr ated robotic replacement of an Orbital Replacement Unit (ORU), complete end-to-end electrical connector mate/demate, and two-arm coordinated control. Figure 3. CMU's Skyworker prototype Figure 4. NASA's Robonaut Figure 5. University of Maryland's Ranger

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7 According to Space Robotics Technology Assessment Report15 published by NASA in December of 2002, the expectation for in-space assembly under nominal research efforts is: Robots that can autonomously mate components and do fine assembly, including making connections under careful human supervision.16 It is perceived, given the current state of the ar t of robots like Robonaut, that robots will possess the mechanical capabilities equivale nt to a space-suited human, but barring a breakthrough in communications architectures more aggressive e xpectation of space robotic assembly cannot be met for teleope ration, due to current low bandwidth/high latency communication. On the other hand, automating Robonaut and other highly dexterous robots is possible, but to make highly dexter ous robots effective under autonomous operation, a system level design ne eds to be considered that designs the small components specifically for assembly by autonomous robotic systems. This requires significant redesign in current infrastructure, which was perceived to be financially impractical at the time of the Space Robotics Technology Assessment Report. However, the new NASA mandate ou tlines strategic allocation of funding to achieve its aggressive goal set, and as a result it now may be possible to a lter the current infrastructure with the intent to simplify automated robotic assembly. In-space inspection consists of examini ng space structures for anomaly detection. The robots performing the inspection can be eith er free-flyers or ma nipulators capable of performing the subtasks of moving, such as navigation and obstacle avoidance, to examine the entire exterior of the stru cture and anomaly detection via sensor interpretation. Currently, there are no ope rational space inspection systems. However, there have been on-orbit i nvestigations of subsystems, such as AERCam Sprint10 (see

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8 Fig. 6), which flew aboard Columbia (STS87) in 1997. AERCam is a teleoperated EVA camera whose purpose was to merely investigat e the feasibility of an autonomous, freeflying EVA camera system. The state of the art in the area of inspection is the mini AERCam17 (see Fig. 7), Orbiter Boom Sensor System, 18 OBSS (see Fig. 8), and the Supplemental Camera and Maneuvering Platform, 19 SCAMP (see Fig. 9), at the University of Maryland. The mini AERCam project is a second generation versi on of the Sprint aimed at adding more complex capabilities while reducing the overa ll size of the prototype. A nanosatelliteclass spherical free-flyer, the mini AERCam is only 7.5 inches in diameter and weighs a mere 10 pounds. Even though mini AERCam cutting edge hardware is not flight certified, making it far from implementable, the results obtained from orbital simulations and 5 DOF experimental test bed validations are invaluable to advancing the field of autonomous inspection agents. The Orbiter Boom Sensor System (OBSS) is a manipulator based concept for inspection of the thermal protection surface (T PS) of the Shuttle. Consisting of a camera and a laser range sensor mounted on a 50 boom, the OBSS attaches directly to the SRMS, providing teleoperational inspection of nearly 75% of the Shuttles TPS. In the Space Systems Laboratory at the University of Maryland, the Supplemental Camera and Maneuvering Platform (SCAMP) allows for investigations in free-flying camera applications. Operated in a neutral buoya ncy tank, SCAMP provide s near micro-gravity conditions for research into free-flying applic ations. SCAMP has demonstrated effective stereo video data interface and 3D navi gation in the neutral buoyancy test bed.

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9 Figure 6. AERCam Sprint Figure 7. Mini AERCam Figure 8. Artist's depiction of the Orbiter Boom Sensor System Figure 9. The SCAMP in the neutral buoyancy tank For in-space inspection, NASA perceives the expectation for the next decade to be summarized as autonomous robotic inspection of some of the exte rior surfaces with

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10 sensory data filtered for potential anomaly before being stored or sent.20 With more aggressive research effort, autonomous inspec tion and anomaly detection of most exterior surfaces of a target are realizable. The addition of autonomous capabilities to ISO systems is a timely issue. The new NASA mandate2 sets aggressive goals for space robotics missions, and autonomous capabilities for robotic systems are vital to th e achievement of these goals. Furthermore, the international community, initiated by Deutsches Zentrum fur Luftund Raumfahrt (DLR), also known as the German Aerospace Center, has been proactive in forming a community whose sole focus is on-orbit servic ing (OOS). In their own words, the spirit of the OOS initiative aims at developing a sound understanding of OOS and its issues, and at creating a dedicated community on global level21 with the overa ll objective being to develop a structured longterm approach of how to d eal with OOS by providing highquality information and latest insights to this new, complex and innovative field of space activities, and to initiate new collaborations.21 OOS has organized two workshops, in 2002 and 2004, dedicated solely to all aspects of on-orbi t servicing, and numerous sessions for on-orbit servicing in othe r conferences worldwide, e.g., the 8th International Symposium on Artificial Intelligence, Robotic s and Automation in Space (I-sairas 2005), providing further evidence to th e point that ISO systems development is a timely, global issue. With the state of the art in ISO systems, as outlined above, reflecting few autonomous capabilities, and the absence of a mature knowledge base in free-flying robotics theory, adding autonomy to ISO system s is a monumental task as evidenced by the requests for information22,23 (RFI) issued from NASA in the months directly

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11 following the issuing of the new mandate, the recent formation of the Institute for Space Robotics, and the OOS initiative. To this e nd, this dissertation expl ores strategies for onboard guidance and develops a guidance algorithm for proximity operations for autonomous ISO robotics systems in an attempt to improve the current state of the art of autonomous, in-space robotic systems. Research Scope For most ISO missions (assembly, repair, refu eling) the target location and its orbit parameters are known a priori, with the exception being maneuvering non-cooperative targets which we will not consider in the present work. For instance, in the case of an assembly mission, the location of the constr uction site and the parts depot are known parameters from mission planning. Utilizi ng this known information, we propose a novel approach to collision free operations of the spacecraft ba sed on artificial potential function (APF) guidance. In order to utilize APF based guidance schemes, two prominent deficiencies inherent to this method need conquering: prem ature termination due to local minima in the operational space, and suboptimal performance. It is widely known that utilization of harmonic potential functions eliminates th e introduction of local minima in the operational space; thus we employ harmonic fu nctions to eliminate the introduction of local minima. In order to overcome the suboptimal performance, a sum of harmonic function primitives (simple functions such as sinks and vortices in the hydrodynamic analog) is employed to shape the velocity fi eld of the operational space to harness the underlying dynamics of the system (i.e., or bit mechanics). Furthermore, given the location and orbit parameters of the target site, a random heuristic search algorithm can be employed to optimize the velocity field shaping for maximizing the performance gains

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12 achieved from harnessing these underlying dynam ics. This dissertation presents this novel approach for proximity ope rations of multiple space asse ts operating in relatively unstructured environments. Dissertation Outline This dissertation is organized as follows. Chapter 1 provides an introduction to the current trends in the space i ndustry, details the st ate of the art for ISO systems in both real world implementations a nd test bed environments, and concludes with the research scope that precisely defines the problem and solution presented in this work. Chapter 2 explores the orbit mechanics based approaches to trajectory generati on for spacecraft, and illustrates the deficiencies that eliminate th ese algorithms for the specified application considered in this dissertation. In Chapter 3 the motion planning approaches from the robotics field are surveyed as alternatives to the cla ssical orbit mechanics based a pproaches. From the robotics literature, the artificial potential function (APF) guidance approach is singled out as a promising solution approach for the problem considered here, and examples of APF guidance algorithms applied to the classi cal orbital rendezvous with and without obstacles are presented. Chapter 4 develops the novel alterations that overcome the two predominant deficiencies of APF algorith ms that would prohibit their application for ISO missions. The first part of this chapter develops an original, truly general dynamic obstacle avoidance algorithm that is a critical subsystem for the trajectory generator for ISO missions. The second part of Chapter 4 deal s with the mitigation of local minima and suboptimal performance by utilization of harm onic function primitives. The chapter is

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13 concluded with the application of the A PF guidance scheme to a representative ISO mission scenario. Eventually the spacecraft will not be cons idered to be a point mass and attitude considerations must be included. In Chapter 5 the classical APF framework is utilized to develop an attitude guidance al gorithm that can be shown to be a subclass of controllers that can be derived using a Lyapunov analysis Chapter 6 will present conclusions and future work.

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14 CHAPTER 2 TRAJECTORY DETERMINATION: ORBIT MECHANICS APPROACH The general mission scenario for all autonom ous in-space operations (ISO) systems requires translating from an initial position to a goal position where the system fulfills its designed functional purpose, e.g., assembling a stru cture. It is assumed that the initial position of the system is either a parking or bit provided by the launc h bus or an orbiting space platform (e.g., ISS), both of which imply the need for midcourse and endgame trajectory design. Midcourse trajectory determin ation refers to the transfer from initial location to close proximity, and endgame traject ory determination refers to the proximity maneuvering. In the sections that follow, classical orbit mechanics theory for the midcourse and endgame trajectory determina tion phases will be presented along with more recently developed methods based on orbit mechanics. Numerical examples illustrating the key aspects are included, and the limitations on each for autonomous ISO applications are illustrated. Midcourse Trajectory Determination The most general framework for midcourse trajectory determination is the n-body problem of astrodynamics (see Refs. 24-27 or any text on Astrodynamics or Celestial Mechanics). The solution of which requires 6n integrals, of which there are ten known,24-27 and a general closed form solution is not obtai nable. However, the artificial satellites are well within the sphere of influence (Ref. 24, Chapter 7) of the primary central body, and the mass of the satellites is negligible with respect to the ma ss of the central body. Therefore, the trajectory of any robot in th is context can be approximated effectively by

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15 the solution of a two-body problem, assuming the position of the goal is associated with an orbiting target. Figure 10. Schematic of Lamberts Problem Restricting consideration to two-body problems, the dete rmination of the orbital trajectory given two points in space and a time of flight is known as the two-body orbital two-point boundary value problem24 (see Figure 10). The first stri des in the solution of this boundary value problem came from Johann Heinrich Lambert24,25 (1728-1779). Lamberts Theorem states that the time 21tt required to traverse an elliptical arc between specified endpoints, P1 and P2, depends only on the semi-major axis aand two geometric properties of the space triangle12FPP the chord cand the sum of the radii 12rr Since Lambert arrived at this conclu sion from strictly geometric means, the mathematical statement of Lamberts Theore m, Lamberts Equation, actually comes from Joseph-Louis Lagrange24 (1736-1813) and is given by Eq. (1) in terms of Lagranges variables and In order to show that Eq. (1) indeed proves Lamberts Theorem, and can be expressed as Eq. (2) and (3), where 121 2 srrc is the semi-

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16 perimeter of the space triangle. Hence, the tr ansfer time is a function of the semi-major axis, the radii sum 12rr and the chord c. Obtaining solutions of Eq. (1) became known as Lamberts problem. Historically its soluti on was essential in determining the orbits of celestial bodies from observation, and today it is applicable for artificial satellite intercept and rendezvous problems. 3 2 21sinsin tta (1) 1 2sin 22 s a (2) 1 2sin 22 sc a (3) It is important to note that the soluti on of Lamberts Problem only defines the geometry of the intercep t problem between points P1 and P2. To effectively execute the trajectory specified by the solution geometry, th e terminal velocity vectors need to be calculated in order to obtain the necessary guidance command to initiate the transfer. Therefore, determination of the maneuver command is the essential quantity. Lamberts Problem The determination of an orbit from a specified transfer time 21tt connecting two position vectors, r 1 and r 2, is called Lamberts Problem24-27 (see Figure 10). A variety of techniques for the solution of Lamberts Problem have been developed over the years24,28,29 with each being classified by the partic ular form of the transfer-time equation and the independent variable chosen in an iterative solution technique. Classically two transfer-time equations24 were used, one from Lagranges equations used in his proof of Lamberts Theorem and the other derived from Gauss work. Each of these methods has

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17 their advantages and disadvantages, and a thir d transfer-time equation that utilizes these advantages while mitigating the disadvantages can be developed as in Battin.24 Battins combined equations technique is utilized fo r the present work. For a complete history and development of Lamberts Problem and the two-body orbital boundary value problem see Battin.24 Before attempting to solve Lamberts Problem it is instructive to consider inherent properties of Eq. (1) that will influence the techniques employed for its solution. Lamberts Equation is: 1. Transcendental equation 2. Double valued function of a(allows for conjugate orbits with same semimajor axis) 3. Derivative with respect toais infinite for the minimum energy orbit 2 s a (the minimum energy orbit corresponds to the smallest possible value of semimajor axis a ) 4. Four possible solution for each case due to quadrant ambiguities in and Property 1 requires the use of an iterative so lution approach to find the semi-major axisa. From properties 2 and 3 it is obvious that the semi-major axis is not amenable to iterative solution techniques. Furthermore, Property 4 mandates the use of a method for selecting the unique solution from the multiple solution possibilities. These difficulties can be mitigated by utilizing a transformation of Eq. (1) by Battin24 that replaces the semi-major axis with an independent vari able that produces m onotonically decreasing and single valued solution curves, which is more appropriate for an iterative solution approach. Furthermore, utilization of Prussings geometric interpretation25 provides the framework necessary to de termine a unique solution. Battins combined equations technique24 will now be presented. This procedure first transforms the independent variable in Lagranges Equation to one with properties

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18 amenable to iterative solution techniques. Th en, utilizing another fo rm of the transfertime equation attributed to Carl Frie drich Gauss (1777-1855), Battin improves the computational efficiency by reducing the num ber of hypergeometric function evaluations from two to one, however the independent vari able in this transfer-time equation lacks the attractive properties obtai ned previously. By developi ng a relationship between the two independent variables, Battin combines the two transfer-time equations in order to obtain the combined equation form utilized in the current work. Battins combined equations transfer-time equation is given by Eqs. (4a) through (4f), where ,;;Fx is the hypergeometric function. This form of the transfer-time equation is devoid of the difficulties encountered from using the semi-major axis as the independent variable, which renders it readily adaptable to an iterative solution method, such as Newtons Method. The parameter is a function of the known geometry and y is an intermediate variable expressible in terms of x as Eq. (4c). Furthermore, is essentially a function of x only since y can be given by Eq. (4c), and 1Sis a function of x only, which implies that QQx only. Therefore Eq. (4a) is a function of the independent variable x only. It is also important to point out that the solution obtained for x is unique, a property not apparent by insp ection. The reason for the uniqueness can be attributed to the use of variable x and y which can be given by alternate expressions 1 cos 2x and 1 cos 2y Therefore, ambiguities arising from the trigonometric functions involving and are avoided. 3 21 34mttQ a (4a)

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19 12cos 2 rr s (4b) 2211 yx (4c) y x (4d) 11 1 2Sx (4e) 145 3,1;; 32 QFS (4f) Maneuver Command Determination In order to calculate the maneuver command the solution x from the Battin algorithm must be converted to geometric properties a, and The equations utilized for this purpose are given by Eqs. (2), (3), and (5), where 2ms a is the value of the semimajor axis for the minimum energy transfer between P1 and P2. From the inverse trigonometric operations the conversion always yields the principal values0 and0 characterized by 000 However, four solutions for and are possible arising from quadrant ambiguities from inve rse trigonometric operations. In order to obtain the correct solution from the four possi bilities, Prussing geometric interpretation25 is employed. 21ma x a (5) Prussing geometric interpretation invol ves transforming the geometry of Figure 10 to a rectilinear ellipse, thereby introducing geometric significance to the angles and The results of this procedure provides a fram ework for the logical determination of the

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20 appropriate values of and from the principle values0 and0 The assignment procedure is given below by Eqs. (6) and (7), where 21tt is the transfer time, mt is the transfer time on the minimum energy transfer, and is the transfer angle. 021 021, 2 ,m mttt ttt (6) 0 0,0 ,2 (7) Once and have been obtained from the above procedure the terminal velocity vector 1v at 1r(and 2v at 2r for a rendezvous) is calcula ted. To do so, introduce a skewed set of unit vectors, defined in Eqs. (8a) through (8c), where iu i= 1, 2, are unit vectors in the direction of ir, and cu is a unit vector parallel to the chord. It can be shown25 that in this coordinate bases 1v has the elegantly simple expression given by Eq. (9), with A and B given by Eqs. (10) and (11). Therefore, the final objective of obtaining the maneuver command necessary to initiate the trajectory between the start and goal configuration can be calculated by taking the vector differenc e of the satellites current velocity with 1v. 1 1 1 r u r (8a) 2 2 2 r u r (8b) 21crr u c (8c) 11cvBAuBAu (9)

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21 1 2cot 42A a (10) 1 2cot 42B a (11) Numerical Example: Midcourse To illustrate the midcourse trajectory determination and maneuver command algorithms, a numerical example is presented. The problem is as follows: given position vectors 1956700Tr and 237394107830Tr (units are kilometers) coordinatized in the Perifocal (PQW) frame (x-axis pointed in the direction of the eccentricity vector) and a time of flight equal to four hours, determine the solution trajectory and the maneuver command. The init ial position was chosen to be the periapse which allows for simple analytical verificati on of the terminal velo city vector produced by the midcourse maneuver command algorithm. Using algorithms based on the theory of th e previous sections and developed by the author the solution is obtained. The iterative solution for x in Lamberts Problem using Newtons method required 19 iterations to c onverge with a relative error less than 10-8. The terminal velocity at 1P was found as 10 18.73108.2280Tv kilometers per second. The CPU runtime on a dual 2.4 GHz Xeon processor workstation for the midcourse maneuver command determination algorithm was approximately 0.15 0.05 seconds. The terminal velocity 1v produces by the algorithm can be verified by checking against the analytical expression for the velocity at periapse passage 1 1pe v ae Performing this calculation produces the an alytical terminal velocity vector as

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22 08.2280T pv km/s, which shows good agreement with the results from the algorithm. Figure 11 depicts the tran sfer trajectory, obtained from a Keplerian motion simulator using the initial conditions given by the position and terminal velocity vector solutions. It is important to note a caveat for th e numerical example above. The position vectors were given in the Perifocal (PQW) fr ame, which is an orbit based coordinate frame. However, it is unlikely that the position information will be known in the PQW frame, instead position information is more likely to be obtained in the Earth-Center Inertial (ECI) frame or the Local-Vertical Local-Horizontal (LVLH) frame. The implication of this is, unlike the example above the initial value of the true anomaly is unknown and cannot be determined by insp ection from the position vectors. For instance, if the position vectors in the exam ple above were given in the ECI frame (with = 10 i = 28.5 = 45 ) the position vector would be 15629.47694.7792.9Tr and 114080.135350.38166.2Tr km, which does not exhibit any evidence that the true anomaly is zero at 1P Therefore, in most cases only the difference in true anomaly is known from geometry and the elegan t calculations used to verify the terminal velocity vector here are not applicable. Endgame Trajectory Determination Endgame conditions for this work will be de fined such that the dynamics of relative motion between two bodies in orbit can be accurately approximated by linear orbit theory. The limit on the applic ability of linear orbit theory is dependent on both the desired level of fidelity and the class of or bit operation, e.g., Low-Earth orbit (LEO) or Geostationary orbit (GEO). For an in-depth analysis on the applicability of linear orbit

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23 theory see Refs. 30 and 31. For the analysis that follows it is assumed that endgame conditions are met. Figure 11. Transfer trajectory from midc ourse trajectory determination algorithm Figure 12. Depiction of orbital rendezvous The basic scenario for the development of linear orbit theory is depicted in Figure 12, where both the Target and Chaser are assumed to be in two-body motion about the

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24 central body, denoted by M. The most familiar equations from linear orbit theory and the ones employed in the present work are the Clohessy-Wiltshire (CW) equations,32 originally developed for the first orbital re ndezvous flight demonstration programs. The CW equations are given in state transition matrix form by Eqs. (12) (14), where x y ,z (zout of the page) are the coordinates of the Chaser with respect to the Target in the local frame depicted in Figure 12 and nis the orbital mean motion of the Target. It is vitally important to understand the implic it assumption of the CW equations that ultimately limits their applicability and fidelity The reference (Target) orbit is assumed to be circular. Therefore, for any scenario wh ere the reference orbit is not circular errors in addition to the ones from linearization are introduced, thus, further degrading the fidelity of the model. 0rttr (12) Trtxtytztxtytzt (13) 2 43cos001cos0 24sin3 6sin101cos0 00cos00 3sin00cos2sin0 61cos002sin4cos30 00sin00cos s ntnt nn ntnt ntntnt nn s t nt n nntntnt nntntnt nntnt (14) CW Maneuver Command Determination The primary advantage of the CW equations is the computational simplicity of determining the transfer velocity necessary for performing a maneuv er. The objective of the maneuver is to bring the Chaser with initial conditions 0 rand 0 rto the origin

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25 of the CW frame (depicted in Figure 12) given a time of flight *t. Partitioning the state transition matrix into four 3 x 3 matrices as given by Eq. (15), the top partition leads to the equation for the relative position 00 rttrtr MN. Substitute *tt and *0rt giving the transfer velocity 0r the expression shown by Eq. (16). tt t tt MN ST (15) 1 ** 00rttr NM (16) Numerical Example: Endgame Continuing the maneuver begun in the numer ical example of the midcourse section, the problem scenario is as follows. The robo t completes the midcourse trajectory to find the goal in a circular orbit with radius 38916.7Gr km. The discrepancy in the positions of the goal and robot at the end of the mi dcourse maneuver can be viewed as errors incurred from unmodeled perturbations such as drag, oblateness, or solar radiation pressure effects. The objective of this ex ample is to use the endgame algorithm outlined in the previous section to generate the mane uver command that will bring the Chaser to the Targets position with a time of flight arbitrarily specified as one-quarter of period of the Target orbit. All vector quantities are coordinatized in an R-bar V-bar frame attached to the Target, with x in the direction of R-bar and y in the direction of V-bar. The endgame maneuver command generated by the authors algorithm is given by 0.00011.40280Tv km/s with a CPU time less than a few milliseconds on the workstation used previously. The mane uver was simulated, as depicted by Figure 13, and the absolute value of the errors in position at the end of the mane uver were obtained as

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26 991.059101.206100T which illustrates the convergence of the robot to the goal position in the specified time. Figure 13. Endgame maneuver trajector y in the xy-plane (R-bar V-bar plane) Alternative Orbit Mechanics Based Methods Having illustrated the classical method s of orbit mechanics for trajectory determination, we turn to a more recent algorithm33 based on an adapta tion of glideslope guidance that has been utilized for rendezv ous and proximity operations for the space shuttle and other vehicles with astronauts in the loop. To date this is the most widely used approach from a real worl d implementation perspective. The glideslope approach utilizes the linearized equations of motion Eqs. (12) (14) but completes the maneuver with N thruster firings. The algorithm divides the range-togo into N equally spaced segments along the ch ord connecting the initial and final Goal Initial Endgame Position Endgame trajectory

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27 relative positions. At equally spaced time inte rvals an impulse is calculated using the CW equations that drive the current pos ition to, the next intermediate position rm. An example of the algorithm is shown in Figure 14. Figure 14. Glideslope solution for 5 impulse rendezvous. The computational simplicity of the glideslope algorithm lends itself readily to application to ISO mission. However, this computational simplicity is sacrificed for situations where obstacle avoida nce is necessary. In order to use a glideslope approach with obstacle avoidance, a chord must be drawn between the centers of mass of all the spacecraft in the operating environment. Therefore, (n-1)! chords, each with N impulse to be calculated are necessary. For inst ant for a five spacecraft mission, requires 24N

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28 online impulse calculations! For this reas on alone the glideslope approach is an unacceptable alternative for the ISO mission s considered in this dissertation. Conclusions The existence of two trajectory determin ation algorithms, one for midcourse and one for end-game, implies that a multi-tier architecture is necessary. Therefore, not only are two algorithms necessary but the adde d complexity of switching algorithms to determine when the linear control is valid is introduced. To mitigate a muti-tier control structure, the midcourse maneuver commands could be utilized for both midcourse and endgame scenarios. However, the midcour se maneuver command determination is a computationally intensive algorithm that requires an iterative solution, which does not lend itself to on-orbit implementation due to the limited computational power that exists for current space systems. Anot her alternative is to modify the linear orbit theory in order to expand its range of applicability to midcourse trajectory determination as well. This subject has received intense research,34-37 including investigations by the author, with minimal achievements in expand ing the range of applicability. An additional disadvantage of classical orbit mechanics theory was obvious to Stern and Fowler30 twenty years ago; classical orbit mechanics theory does not admit the possibility of the incorporation of physical trajectory constraints, introduced by either goal structure or obstacles in the path. Si nce the ability to dynamically incorporate obstacles and trajectory constrai nts is vital for all autonomous ISO systems, and given the additional disadvantages mentioned above, othe r sources of trajectory planning theory for autonomous ISO systems must be explored. Chapter 3 will survey the robotics literature for motion planning algorithms and will motivate and develop artificial potential function guidance algorithms for autonomous ISO applications.

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29 CHAPTER 3 ARTIFICIAL POTENTIAL FU NCTION POSITION GUIDANCE In the preceding chapter it was demonstrated that trajectory determination for autonomous in-space operation (ISO) systems cannot be easily handled by classical orbit mechanics theory. A logical avenue to explor e for trajectory determination is the robotics literature on robot motion planning (MP). A brief outline of the robot MP problem is discussed in the first section of this ch apter with an emphasis on motion planning approaches. Given that space-born robotics applications require motion planning subject to properties distinct from classical robot MP problems, for instance the difference in local operator mentioned in Chapter 1, only some of the common motion planning approaches are feasible for in-space applica tions. Based on the unique needs of in-space applications, a potential field approach to MP is perceived to be most appropriate for autonomous ISO systems, and this chapter c oncludes with a presentation of classical artificial potential field (APF) guidance algorithm for ISO applications. Robot Motion Planning In order to precisely define the robot MP problem, it is first necessary to define two terms, configuration and configuration space (Cspace). A configuration of a robot is defined as the set of independent parameters degrees of freedom (DOF), necessary to specify the location of every point of that ro bot. Cspace, a concept introduced in an influential paper by Lozano-Perez and Welsey38 in 1979, is defined as the set of all possible configurations of a robot, i.e., the Cspace represents all possible motions of a robot. Essentially all MP problems are equi valent when cast in the Cspace, and can be

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30 stated as follows. Find a connected sequen ce of point in the Csp ace between an initial configuration and goal configuration. Confi gurations that result in collision with the environment, known as configuration obstacles,39 must be computed, and the connected sequence of points must not co ntain any points associated wi th configuration obstacles. Computation of the configuration obstacles is difficult39,40 and the dimension of the Cspace is equivalent to the DOF of the system, both of which makes motion planning challenging for robots with multiple degrees of freedom. Classically the MP problem has been broken down into the steps below. Concentration on the approaches in the litera ture for Steps 3 and 4 are the focus here for determining an applicable approach for ISO systems. For a complete investigation on methodologies for the steps in the MP pr oblem see the excellent surveys in Refs. 39-43. 1. Parameterize the configur ation of the robot(s) 2. Choose a representation scheme for robot(s) and the environment 3. Select a motion planning approach 4. Select a search method to find f easible path through environment 5. Optimize the solution path based on va rious constraints (e.g., smoothness) Citing the landmark survey by Hwang and Ahuja,39 the most common motion planning approaches found in the literat ure are; cell decomposition, mathematical programming, skeletons, and potential fi eld methods. Cell decomposition techniques involve computation of the entire Cspace and all configuration obstac les in addition to requiring large memory allocations. Therefor e, given that on-orbit computational power is limited, this approach is not feasible for ISO systems. Mathematical programming methods represent the requirements for obstacl e avoidance as a set of inequalities on the configuration parameters. The motion planni ng problem is then formulated as an optimization problem that finds a solution pa th between start and goal configurations

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31 based on minimization of a scalar constr aint. This motion planning approach is essentially a nonlinear optimization problem su bject to multiple inequality constraints, which generally requires a numerical method for its solution. Again this process is computationally intensive and therefore does not lend itself to on-orbit implementation. Skeletons, otherwise known as roadmap or highway approaches, contract the environment into a network of 1-D lines and the motion planning problem becomes a graph search problem restricted to the netw ork. Common examples of the skeleton method are Voronoi diagrams and subgoal networks. The skeleton methods have low memory requirements, and if an accurate repr esentation of the environment and obstacles is known a priori, this method is feasible for ISO applications. In fact a Voronoi diagram approach is currently being inve stigated for the AERCam project.44 Potential field methods for motion planni ng are the simplest, computationally speaking, of all the motion planning approach es presented here. This method creates a scalar function called the potential that has the following properties. Potential is at a minimum when the robot is at the goal configuration Potential is high on the surface of obstacles Everywhere in the environment away fr om the obstacles, the potential slopes toward the goal configuration Typically the potential field method is combin ed with a specified local operator, such as go-straight and the path taken to the goal is recorded. Potential field methods also allow for simple consideration of non-point robots by evaluating the potential at representative points on the robot. Ad ditionally, no explicit Cspace computation is necessary. It is the opinion of the author that the utilization of an artificial potential field

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32 motion planning approach with the local ope rator being the dynami cs of motion is the most effective motion planning approach for ISO applications. Artificial Potential Function Position Guidance Artificial potential function (APF) guidance for satellite systems45-49 has appeared in the literature recently. However, the appl ication of real-time proximity operations of an autonomous ISO system has yet to be inves tigated. In this sect ion the robot will be considered to be a point-mass moving in th ree dimensions. This restriction will be removed for attitude considerations in Chapter 5. The objective of the APF position guidance algorithm is to drive the robot to the origin of the state sp ace through a series of impulsive maneuvers. It is instructive at this point to state th e equation of motion of the dynamical system considered in this section. Since this section is focused on the relative motion between a robot and target in a two-body orbit, the motion of each is governed by Eq. (17), where gr is a vector function describi ng the gravitational field and D ais the disturbing accelerations due to perturbations such as atmo spheric drag or oblateness. It will be assumed that the perturbations are zero and the both satellites are in an inverse gravitational field given by 3grr r for the present analysis. Therefore, the relative motion between the robot and target RTrrr is given by the vector difference of the governing equation of the robot an d the target, as given by Eq. (18). However, these are the equations of motion in a Newtonian reference frame. It is preferable to transform these equations into a local vert ical local horizontal (LVLH) fr ame attached to the target. This is accomplished by the cla ssical transformation given in Eq. (19) where primes

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33 denote differentiation with respect to the local coordinate frame and and are the angular velocity and accelerati on of the local frame, respectively. Thus, the equations of motion for the relative motion of the robot and target are given by Eq. (20). Furthermore, it is obvious that the origin of the LVLH frame 0r is indeed an equilibrium point of the dynamical system. () D rgra (17) RTRTrrrgrgr (18) 2 rrrrr (19) 332RT RTrrrrrr rr (20) The general theory for APF guidance is based on the 2nd method of Lyapunov50 which can be described as follows : Consider the dynamical system x fxt with nx An equilibrium point of the system e x is such that ,0efxt For simplicity it will be assumed that through a change of coor dinates any equilibrium point can be shifted to the origin, and therefore, Lyapunovs theo rem will be presented for equilibrium states at the origin only. Essentially, Lyapunovs 2nd Method is a means of asse ssing the stab ility of the equilibrium points of the dynamical system.50 The theorem states that if there is a scalar function ,Vxtsuch that Eqs. (21) (23) are satisfied, where x fxtV is the derivative evaluated along the solution traject ories, then the origin of the state space is a globally attractive point. ,00Vxtx (21)

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34 ,Vxtasx (22) ,00xfxtVx (23) The first two conditions are satisfied by choosing V to be an appropriate positive definite function of x (it will be assumed for now that the potential function is not an explicit function of time). Th e third condition in Lyapunovs 2nd theorem is enforced by an appropriate choi ce of control action.51 The total derivative of the potential function is given by T xVVx and the control is defined by Eq. (24) where is such that the velocity after a control impulse 0 x is directly opposite to th e local gradient of the potential 0 x x kV This proper51 control constrains the mo tion of the robot after an impulse to be tangent to the local gradient of the potential. Defini ng the control in this manner ensures the derivative of the potential after a control impulse is negative definite, as given by Eq. (25), and therefore Eq. (23) is satisfied. Thus the origin is a globally attractive point under the control action specified. 0,0 ,0V Control V (24) ()0T xxVxkVV (25) A pseudo-code for the APF guidance algorith m is given below, and an example of APF guidance for an unconstrai ned (no obstacles) rendezvous is given in the next section to demonstrate the implementation of the algorithm. Algorithm 1. Obtain the current relative position a nd velocity between robot and target 2. Check the switching condition T xVVx

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35 3. If 0V give an impulsive velocity input such that 0 x x kV 4. If 0V coast under the influence of the dynamics of motion 5. Sample position and velocities at desi red frequencies and apply control based on switching conditions 6. Repeat until terminal error bound is satisfied Example: Spacecraft Unco nstrained Rendezvous Define the potential function 1 () 2TVxxx P, where P is assumed to be a positive definite, diagonal matrix for simplicity and T x rxyzare the coordinates of the relative position in the LVLH frame. This gives xVx Pand the total derivative TVxx P For the rendezvous, the relativ e velocities of the spacecraft x must also converge to zero. The converg ence is actually guaranteed by the choice of control presented above. Consider the norm of the velocity after an impulse given by Eq. (26), thus 0xas 0xand the rendezvous is guaranteed. 1 2 x kx P (26) The numerical example considers the rendez vous between a robot in a circular orbit with radius 6705Rr km and a target satellite also in a circular orbit with radius 6700Tr The robot leads the target radius vector by a phase of 0.001 degrees. The APF guidance parameters used were 1k and PI, where Iequal to the 3 x 3 identity matrix. Figure 15 depicts the trajectory taken by the robot in the xy-plane, where x and y are the coordinates of th e LVLH frame in the orbit plane wi th x in the radial direction. The sharp corners on the trajectory mark wh ere an impulsive control was applied. Figure 16 shows the impulses in the x and y dire ctions, respectively, applied by the APF guidance algorithm as a function of time. In all a total 27.10v m/s was necessary to

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36 bring the robot to within a meter of the ta rget. The duration of the maneuver was 86.2 minutes. It is important to note that th e APF guidance algorithm ensures asymptotic convergence to the origin, thus a majority of the duration of the maneuver came from closing the final meters. Figure 15. Planar rendezvous in the CW frame under APF guidance Figure 16. Control efforts in x a nd y directions for the rendezvous

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37 Obstructions In proximity operating conditions, instances exist where certain regions of the state space need to be avoided; for example, solid obstructions in the flight path. Therefore, the potential function needs to be altered in order to ensure the robot avoids these forbidden areas. For the current work, all obstructions will be approximated by spheres (or circles in the planar case). This is a reasonable assumption for two reasons. Firstly, for now the objective of the guidance algorithm is to provide safe trajectories between a start to goal location, not to generate traj ectories for close proximity fly-bys, yet. Therefore, an approximate map of objects is acceptable. Secondly, the big picture for autonomous ISO systems is perceived to st art with a rough estimate of the working environment, with perhaps this estimated en vironment approximating objects initially as spheres and updating the map through sens or data obtained from fly-bys. The obstructions will be represented as re gions of high artificial potential in the potential field in the form of Gaussian function as in Eq. (27). The parameters k andk define the width and skewness of the Ga ussian function whose center is located at k x and the matrix M defines the shape (since all obstacles are assumed spherical MI). Representing obstacles by Gaussian f unctions ensures that controls remain bounded, no singularities are intr oduced into the potential fi eld, and there are no local minima. It is instructive to note here the units of the width and skewness are [L2] which will be important due to the f act that all simulations are run in nondimensional units for completeness. 1(,)expT kkk kkk x xxxxxM (27)

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38 Once the APF for the obstacles has been se lected, the parameters of the obstacle APF must be determined such that the robot cannot cross into the forbidden regions. At first glance it would appear that constraini ng the potential everywhere on the surface of the obstruction to be equal to the potential at the initial conditions, stated mathematically in Eq. (28), would be sufficient. However, th is requires the width and skewness to be function of x which would degrade the most attr active property of the APF approach, simplicity. An alternative constraint such that the 0()()surfVxVx still guarantees the robot cannot cross the obstruction surface, and constant values of k andk can be found to satisfy this constraint. 0()()()surf surfaceVxVxVx (28) Utilizing the assumption of spherical obstruc tions only, a simple analytical method for the determination of the obstruction APF parameters k andk can be obtained. The potential on the surface of the obstruction is given by Eq. (29). For spherical obstruction 2T surfksurfk k x xxxR where k R is the radius of the spherical obstruction. Effectively there are two parameters to be c hosen based on one constr aint equation, thus only one independent parameter. The prescription for obtaining k is to choose a value for k and vary s urf x over all the points on the surface of the obstruction calculating a k at each value via Eq. (30). Then looking at the result ing potential fields for the various k select the ones that do not violate 0()()surfVxVx. A simulation was implemented to investigate which values of s urf x produced k such that the constraint holds, the results of which are illustrated by Figure 17 through Figure 20.

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39 1()expT T surfsurfsurfsurfksurfk kkVxxxxxxx P (29) 2 0exp()T k s urfsurf k kR Vxxx P (30) Figure 17 shows percent error between the potential at th e initial conditions and the potential on the surface of the obstacle. Th e black line represents the percent error incurred when using the value of k calculated using s urf x for which s urfx is a maximum over all points on the surface, and the red line corresponds to percent error using the value of k computed from s urf x such that s urfx is a minimum over all points on the surface. It is clear from th is diagram that the latter value of s urf x must be used to compute k in order for 0()()surfVxVx to be satisfied. For spherical obstructions finding s urf x whose norm is a minimum over all points on the surface s urf x is a simple exercise in geometry, and can be computed by Eq. (31) were x is the vector connecting the center of the obstruction to the origin. *1obs surfR x x x (31) Figure 18 and Figure 19 depict contour plots of the potential in the xy-plane using k calculated by Eq. (31). Figure 18 shows the level of cont our at the initial conditions as labeled. Figure 19 is a zoomed view of the pr evious figure in the region of the obstacle (for this case the obstacle center was given by 200T kx ). The blue doted line corresponds to the positions on the surface of the obstacle, and it is evident that the potential is greater than the potential at the initial conditi ons at all the points on the

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40 boundary. Figure 20 shows the percentage error between the potential at the initial conditions and potential on the surface of the obstruction fork as calculated by Eq. (31). Figure 17. Difference between IC potentia l and potential on surface of obstructions Figure 18. Contour plot of the potential function

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41 Figure 19. Zoomed view of contour plot Figure 20. Stem plot of obs truction surface potential error

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42 Figure 21. Trajectory trav ersed by robot for the path-c onstrained rendezvous example Numerical Example: Path -Constrained Rendezvous The initial conditions for the previous exampl e will be used again in this example. The APF guidance algorithm parameters used were 0.25k and 3PI, the different value of kwas selected to enhance the illustration of the behavior of the robot around the obstacle. The obstacle is a circle in the plane with center 200Tx and radius 0.5oR km. The parameter 0.01k which leads to a51.085510k both given in nondimensional units (the nondimensionalization em ployed the radius of the target as the length scale factor and the m ean motion of the target as the time scale factor). Figure 21 shows the trajectory traverse d by the robot during the constr ained rendezvous maneuver. The duration of the maneuver was a pproximately 245 minutes and a total 61.52vm/s to bring the robot to within a meter of the target. Figure 22 (a) and (b) and Figure 23 (a)

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43 and (b) depict the impulses in the x and y directions, respective ly, applied by the APF guidance algorithm as a function of time. Th e second plot in both figures is zoomed around the times the robot en counters the obstacle. Figure 22. Control action in the x-directi on for the constrained rendezvous (a) entire simulation (b) control actions in vicinity of obstacle Figure 23. Control action in the y-directi on for the constrained rendezvous (a) entire simulation (b) control actions in vicinity of obstacle (a)

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44 Conclusions In this chapter the motivation for employing artificial potential function (APF) guidance for autonomous ISO applications was presented. Furthermore, position guidance based on APF was developed fo r unconstrained rendezvous and pathconstrained rendezvous, with a numerical example presented for each. Through the numerical examples it was illustrated that A PF position guidance can be an effective tool for autonomous proximity operations, a vital element for all ISO systems. Even though the classical APF examples fr om above were shown to be effective, they suffer from two predominant drawback s of APF methods: local minima causing premature termination of algorithm, and sub-optimal performance. Furthermore, the absence of a truly general APF-based obs tacle avoidance algorithm needs to be addressed. In the next chapter, the A PF guidance framework will be augmented with novel alterations that both conquer these dr awbacks and extrapolate the obstacle avoidance algorithm to the most general dynamic case.

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45 CHAPTER 4 ARTIFICIAL POTENTIAL FUNCTION GUIDANCE: A NEW APPROACH The earliest implementation of APF algorithms is attributed to Khatib and Mampey52 in 1978, who used the approach for force control applications. Since then, APF methods have been applied to a wide variety of research areas, most predominantly robot motion planning5355 and computer graphics.56 Recently, APF guidance has been utilized in on-orbit applicatio ns such as autonomous rendezvous,57 position,58 and attitude59,60 guidance for inspection, maintenance, and assembly operations; and formation control. Despite the proliferation of APF methods in the robotic literatur e and their diverse utilization in robotic s research, the two primary drawb acks of these methods, spurious local minima and suboptimal performance, have received sparse atten tion. Furthermore, in the vast body of literature on APF methods a void exists for algorithms capable of handling a general dynamic environment. The first section of this chapter will address the vacancy in the literature pertaining to APF based obstacle avoidance algorithms for truly general dynamic environments. Section two of this chapter will be charged with overcoming the two predominant shortcomings by utilizing a sum of harmonic function primitives to compose the attractive potential function. Dynamic Artificial Potential Functi on Obstacle Avoidance Algorithm Despite the numerous applications of A PF methods, most of these instances deal with static environments (potentials). However, a few examples of published works5963 dealing with an APF approach in dynamic environments exist with the most prominent

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46 being the work of Ge and Cui.63 Nevertheless, these works introduce prohibitive restriction as to the nature of the motions of the target and obst acles (e.g., constant velocities for both). Furthermore, they only consider the situati on were the robot is required to take obstacle avoida nce measures for one obstacle at a time. To the authors knowledge, a truly general APF algorithm fo r dynamic environments without these restrictions has not been addressed. The dynamic artificial potential function gui dance scheme employed in this section consists of two components: an attractive and a repulsive potential. The attractive potential is a familiar form seen in previous studies on artificial potential field applications. However, the repulsive potential has been augmented with a priority weighting scheme that implements simulta neous obstacle avoidance for all obstacles requiring such. This weighting scheme enab les APF guidance to be extended to dynamic environments, with the caveats that the robot possesses sufficient actuation to complete the maneuver, and the refresh rate for updati ng environment data is sufficiently fast. Attractive Potential The attractive potential, Uatt, given by Eq. (1) includes both the position, tr, and velocity, tv of the chaser and position and velocity of the target, ,tartarttrv respectively. The constants m n > 1 and ,pv are design parameters. ()()mn attptarvtarUttttrrvv (1) Similarly to the classical APF approach, we define the virtual attractive force, Fatt, as the sum of the gradients of the attractiv e potential with respect to the position and velocity of the chaser, as given in Eq. (2). Evaluation of two gradient operations yields the expression for the virtual attractive force shown by Eq. (3), where nRT is a unit vector

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47 pointing from chaser (robot) to the target (goal) and v RTn is a unit vector in the direction of relative velocity between chaser and target. It is clear to see that the virtual attractive force goes to zero as the chaser simultaneousl y obtains the same position and velocity as the target. attpattvattUU F (2) 11()()()()mn v attptarRTvtarRTmttnttFrrnvvn (3) Repulsive Potential In order to define the repulsive potential, parameters need to be introduced to quantify the necessity for obstacle avoidance as it pertains to each obstacle in the environment. Define the parameters given by Eqs. (4) (8) for each obstacle i = 1n. With these parameters in mind, we define the repulsive potential as given by Eq. (9). Qualitatively, the potential is zero when the chaser is moving away from the obstacle, as indicated by 0i ROvt, while the potential is nonzero fo r situations when the chase is approaching an obstacle and is within that obs tacles range of influence. The repulsive potential is not defined when minimum distance between chaser and the obstacle is less than m meaning a collision cannot be avoided. The total repulsive pot ential is obtained by summing each repulsive potential over all the obstacles, i reprep iUU maxmaximum acceleration of chaser a (4) min distance between robot and obstaclei ii s obsttr rr (5) 2 max maxdistance under 0 2i ii RO mROv av a (6) 00influence range of obstacle,iii ROvtT (7) 1...T iii ROobsROvtttin vvn (8)

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48 0 0 00, if or 0 11 if 0 and 0 not defined, if 0 and iii sRO iiiii repsmRO iii sm iii ROsmvt Uvt vt (9) Just as in the attractive force case, the vi rtual repulsive force is defined as the sum of the gradients of the repulsi ve potential with respect to the position and velocity of the chaser. After performing the operations and simplifying, the virtual repulsive force, which invokes the necessary obstacl e avoidance, is given by Eqs. (10) (12), where i ROv is the magnitude of the component of the re lative velocity of chas er with respect to the ith obstacle in the direction perpendicular to th e line connecting the chaser and that obstacle. 0 120 0 if or 0 if 0and 0 undefined ifand 0 iii sRO iiiiiii repreprepsmRO iii smROvt vt vt FFF (10) 2 1 max1i RO iii reprepORvt U a Fn (11) 2 2 max iii repRORO ii repOR i sUvtv a Fn (12) Given the repulsive force definition above, the priority weighted scheme is elucidated. Introduce the priority index for the ith obstacle (iPI), calculated as shown by Eq. (13). Again, qualitatively the priority index is a measure of how close are the minimum distance under action of maximum acceleration and the shortest distance between the robot and ob stacle. As any one iPI approaches some predefined factor of safety, FOS, (one would prefer not to tempt fate by waiting until the PI 1 and the robot

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49 was forced to do a critical avoidance maneuver) the obstacle avoidance algorithm switches from the simultaneous mode to a single obstacle avoidance mode for the obstacle with critical valued PI. Note that we did not consider the case of simultaneous obstacles with critical valued PIs. This logic guarantees a collision free trajectory through the operational space. It is important to note at this juncture that the priority index implicitly depends on relative velocity along the chord connecting the two centers of mass of the robot and obstacle. This im plicit dependence is cruc ial to the situation with non-constant velocity targets and obstacles. 00 if or 0 otherwiseiiii smRO i i m i sv PI (13) Under normal operating conditions, i.e., no iPI FOS, the total repulsive force felt by the robot is calculated as the PI -weighted sum of the indivi dual repulsive forces for each obstacle in the environment, as shown in Eq. (14). ii reprep iPI FF (14) Total Force Interactions The total virtual force acting on the chaser is the sum of the attractive and repulsive forces,totalattrepFFF. Through Newtons 2nd Law we relate this force to the applied acceleration of the chaser, given by Eq. (15). Notice that if the applied acceleration is greater than the maximum acceleration of the chaser, the maximum acceleration is applied in the direction of th e total virtual force. With the applied acceleration, the position and velocity can be obtained via inte gration of the equation of motion for the chaser.

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50 max max, if otherwisetotaltotal total totala mm t a FF a F F (15) Simulation Results To illustrate the efficacy of the algorithm we present two simple 2D simulation scenarios. As the first example, we demons trate the differences generated by the priority weighted avoidance of the new algo rithm with that of Ge and Cui,63 whose work is the baseline for the present analysis, which constr ains the target and obstacles to constant velocities. In a second simulation we present a scenario more in line with the intended application, that being the Heterogene ous Expert Robots for On-orbit Servicing (HEROS) architecture,64 demonstrating a planar, on-or bit rendezvous with obstacle avoidance. This simulation showcases a situ ation where target a nd obstacles have nonconstant velocities. Constant velocity scenario The 2D constant velocity simulation presented below is a comparison between the algorithm present in Ge and Cui63 and the algorithm derived in the present work. In the former work the repulsive potential is rest ricted to single obstacle interactions. The parameters 0(,,,,,)pvmn for both algorithms are equal in order to isolate the effect of the PI -weighting scheme. The pertinent parameters of the simulation are as follows. The target has initial position [10 10]T and constant velocity [0.1 -0.05]T. Two obstacles are in the operational space with initial positions [5 0]T and [21 11]T and constant velocities [0 0.1]T and [-0.05 -0.065]T, respectively. The chase is of unit mass with the maximum acceleration, Amax, also equal to unity. The chase is assigned initial position [1

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51 1]T and velocity [0.1 0]T. A FOS of 0.9 was used for the PI -weighting algorithm. The results of the simulation are shown in Figure 24. 0 5 10 15 20 25 30 0 5 10 15 20 XY Target Obst1 Obst2 Tatsch Ge&Cui Figure 24. 2D Constant velocity simulation results. Figure 24 shows the locus of poi nts for all entities in the operational space for the 200 seconds. Most important to note is the mo re direct route traversed by the chase using the PI -weighting scheme when in the influence of both obstacles. As chase approaches the first obstacle in both situati ons, the differences in the algorithms is illustrated. The chase under Ges and Cuis algorithm makes a more prominent maneuver with respect to the PI -weighting algorithm because the former doe s not consider the influence of the second obstacle at all. As the chase en counters the second obstacle, the algorithms converge to the same trajectory; which is to be expected consideri ng that in the single obstacle case, the algorithms only differ by the fact or of safety. Finally we see that in the

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52 absence of obstacles the algorithms are equi valent given that they employ identical attractive potential/force interactions. On-orbit rendezvous scenario The on-orbit rendezvous simulation consis ts of four spacecraft in Keplerian motion: target, chase, and two obstacles. The initial conditions for all the vehicles are summarized in Table 1 The objective of this scenario is to drive the chase spacecraft using the APF guidance presented above to the target spacecraft while avoiding the obstacles. Table 1. Orbit parameters for rendezvous simulation. Vehicle a [km] e i f Target 8000 0.1 0 0 0 Chase 9005 0.1 0 0 0 Obstacle 1 10288 0.3 0 -0.0718 0 Obstacle 2 12001 0.4 0 -0.7171 0 Figure 25 illustrates the result of this simu lation again in the form of the locus of points for all entities in the operational space. The chase spacecraft st arts radially aligned with the target at a 3.6 kilometers higher altitude. Initially the chase proceeds by thrusting in the radial direction to lower its radial velocity (a nd altitude), thereby increasing its orbital speed. However, it enc ounters the first obstacle which requires it to thrust in the negative radial direction (raisi ng the chases altitude) to avoid a collision. Post-avoidance, the chase ag ain thrusts as before. The second obstacle encounter requires a less drastic trajectory change. Once past all obstacles in the environment, the chase finally achieves an orbit altitude lower than that of the target and thus achieves a

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53 positive relative velocity (meani ng it is catching the target without thrusting). However, the dependence on the relative position results in small accelerations still being applied to the chase until the rendezvous is achieved. 7150 7160 7170 7180 7190 7200 0 100 200 300 400 500 600 700 800 900 1000 IJ Target Obst 1 Obst 2 Chase Figure 25. Planar on-orbit rendezvous results. Figure 26 is a graph of the norm of the appl ied acceleration of the chase spacecraft. This simulation also took into account the maximum allowable acceleration that could be achieved by the vehicle. We see that initiall y the chase acceleration is saturated until we encounter the first obstacle (indicated by the di p in the graph). After passing the obstacle we again see a saturation of the acceleration due to increasing lag between the chase and target spacecraft. The second dip in the gr aph indicates the encount er with the second

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54 obstacle. Beyond the second obstacle encounter the chase acceleration does not saturate again but rather smoothly decays to zero as it approaches the target spacecraft. 0 20 40 60 80 100 120 0 0.01 0.02 0.03 0.04 0.05 0.06 Time [s]Acceleration [kN] Figure 26.. Norm of applie d chaser accelerations. Optimized Harmonic Attractive Potential Recall the two prominent flaws mentione d in the introduction: spurious local minima and suboptimal performance. The first of these is easily mitigated by utilization of harmonic potential functions, i.e., functions that satisfy La places equation. By using harmonic functions for constructi on of the artificial potential field, the resulting potential field is guaranteed to be free of local minima. The first u tilization of ha rmonic potential field for motion control is ge nerally attributed to Satoh66 in the mid-1980s. However the work was published in Japanese and did not receive exposure until it was reprinted in

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55 English in 1993. Masoud67 has explored both scalar and vector potential field methods using harmonic functions. For an extensive survey on these methods we refer you to the introduction of their work. More recently Waydo68 has utilized harmonic potential field methods, in the vein of potential flow an alogs in hydrodynamics, for motion planning on the Cornell RoboFlag test bed. Conquering the suboptimal penalties inhere nt to APF methodologies without sacrificing its most attractive characteristic, that being computational simplicity, is a daunting task. However we proposed a solu tion using a library of harmonic function primitives (i.e., simple harmonic functions such as source, sinks, doublets in the hydrodynamic analog) to shape the force/accele ration/velocity fiel d around the spacecraft such that it takes into acc ount the system dynamics. A high degree of similarity can be achi eved by using just two primitives, a sinkvortex pair from the hydrodynamic analog, that have been optimized for sink strength and vortex circulation to mimic the true velocity field of a dynamic system using a genetic algorithm (GA). The reference velocity field is that for single impulse orbital intercept maneuvers (utilizing the solution of the Clohessy-Wiltshire equations). The parameters and were optimized over a 50 x 50 grid of points in terms of directional accuracy in the velocity fields. A program that generates the velocity vectors for CW transfers (with time of flight, TOF = /4 in this case) at all 25 00 points in the grid and the velocity field for the sink-vortex pair was generated by specifying a value for and subsequently quantifying the error between th e two resulting vector fields. The error between the two velocity fields is qu antified by the scalar error measure, given by Eq. (16), where i varies over all points in the grid, and v and v are the velocities for the

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56 CW transfer and the sink-vortex field, respectiv ely. In the ideal case (velocity fields are perfectly aligned) the angles i are all zero, and cosi equal one. Therefore, in the case of perfectly aligned vector fields, will equal the number of points in the grid, n in this case 2500. Using this value of as the reference value, REF we define the cost function, F employed in the GA as a simple difference REF i.e., the difference between REF and the actual value of the summation of the cosines of the angle between the velocity vectors at each point in the grid, as given by Eq. (17). The GA then minimized the value of the cost functional F for all values of and in their specified ranges. The genetic algorithm parameters used for the optimization are summarized inTable 2. cos 'i ii ivv vv (16) 2500cosi iF (17) The two parameters of and were restricted to range between 0.0001 and 5.0. The output of the GA for el even runs is shown in Table 3. As is illustrated by the small standard deviations in Table 3(highlighted in blue), both parame ters converged to a unique value in their respective ranges. The be st fitness score achieved was 43.1029 with corresponding values of 0.1992 and 0.2941 (highlighted in red in Table 3). Figure 27 plots a comparison of the velocity fields for both the optimized sink-vortex pair (blue) a nd the CW field (red). Figure 27 illustrates the fidelity in the velocity field that can be achieved with just two harmonic function primitives.

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57 Table 2. Genetic algorithm statistics Parameter Name PARAMETER VALUE Population size 25 Numberof Generations 25 Probability of Mutation 0.05 Probability of Crossover 0.9 Crossover type Uniform Fitness Function Rosenbrock Number of Bits 20 Table 3. Mean and standard deviations for the two parameters in the GA optimization Run Cost Function Score 1 0.397 0.5309 55.2659 2 0.397 0.5309 55.2659 3 0.397 0.5309 55.2659 4 0.397 0.5309 55.2659 5 0.298 0.4451 48.8113 6 0.9852 1.3229 98.0802 7 0.1878 0.2784 43.2809 8 0.3011 0.4162 47.5251 9 0.2099 0.3171 43.5354 10 0.2838 0.3799 47.0728 11 0.1992 0.2941 43.1029 MEAN 0.3685 0.507 53.86110909 STD 0.2206 0.2881 15.52175451 Conclusions To summarize, with the results of the previous section, the guidance algorithm presented in this work is the combina tion of a fully dynamic obstacle avoidance algorithm coupled with an attractive potential based on harmonic functions. This form of the attractive potential is guaranteed to be local minima free. Furthermore, by using a sum of the harmonic function primitives to shape the velocity/force/acceleration field, with the intent to harness the underlying dynami cs of the system, the suboptimal penalties of artificial potential fiel d methods are mitigated. The new artificial potential functio n-based guidance algorithm for dynamic environments introduced a prio rity-based queuing approach to obstacle avoidance called

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58 the PI -weighting scheme. The PI -weighting scheme gives the ability to do multiple obstacle avoidance simultaneously, and give n its implicit dependence on the relative velocity between the chase (robot) and the obstacles, extends the APF guidance algorithm to general dynamic environments (non-cons tant velocity targ ets and obstacles). -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 XY Figure 27. Optimized sink-vortex pa ir vs. CW based vector field Through a simple 2D simulation we isolated the effects of the PI -weighting scheme and demonstrated the effectiveness of the si multaneous avoidance in situations involving multiple obstacles. Furthermore, a second simulation was presented to demonstrate the efficacy of the overall APF guidance algorithm in terms of the ultimate application, the HEROS architecture. This simulation demonstrated the effectiveness of the algorithm to

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59 handle an on-orbit rendezvous with multiple obstacles, all of which were experiencing Keplerian motion.

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60 CHAPTER 5 ARTIFICIAL POTENTIAL FUNCTION ATTITUDE GUIDANCE Thus far the spacecraft have been modeled as point masses, however at some stage we must admit the vehicles as rigid bodies ( due to directionality c onstraints, e.g., docking axes aligment) and account fo r both attitude and position dyna mics of the true system. Even though the attitude and position dynamics are highly coupled in the most general case, there are situations where decoupled dyna mics are acceptable, and we will deal with uncoupled cases in this work, which allows independent development of an attitude guidance design. In this vein, Chapter 5 develops a novel attitude guidance algorithm utilizing the APF. The motivation for employing artificial po tential functions (APF) for guidance algorithms came from the robotics literature. The literature contains volumes of position (translational) guidance algorithms, and implem entations of them, but very few examples of algorithms for rigid body (6 DOF, position and attitude) guidance. Primarily the scarcity of literature on the rigid body motion planning can be attributed to two factors: the configuration space (Cspace) has dimens ion equal to the DOF of the robot, and configuration obstacles, configur ations that result in collis ions with the environment, do not have compact representation for high di mensional Cspace. Thus for all but the simplest cases, motion pla nning is challenging. Classically, in order to circumvent th e complexities of dealing with a high dimensional Cspace, for robots with multiple DOF the attitude, or rotational, DOF have been approximated or constrained. The simplest solution for mitigating high

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61 dimensionality of the Cspace is to ignore the at titude states all together, and merely grow the robot to a sphere with radius equal to its maximum dimension. Although this does reduce the dimensionality of the Cspace, it also reduces the capabilities of the robot; for example, for in-space operations (ISO) systems, the ability to point at objects is crucial, and this approximation eliminates this possi bility. An alternative employed in the literature is to hold all but one of the attitude states fixed. Ag ain, this alleviates the issues with dimensionality, but still si gnificantly limits the capabilities of the robot. Therefore, a new approach that allows for incorporation of all attitude states is necessary for ISO systems, preferably one that avoids calcul ating the Cspace and configuration obstacles. APF guidance for attitude is ut ilized to accomplish this task. At this point one might interject that the two deficiencies of A PF methods that were discussed in the introduction to Chapter 4 would prohibit the use of the classical approach. However, in the case of attitude guidance, as opposed to position guidance, these deficiencies do not arise. The introduc tion of local minima which causes premature termination of the algorithm is not an issue fo r the attitude case b ecause the equations of motion, as will be shown later, possess only two extrema in the state space (which are identical points in the physic al space) at the equilibrium poi nts. Suboptimal performance concerns can be dismissed by showing that th e guidance law obtained from this approach belongs to a subclass of cont rollers obtained from a more general Lyapunov analysis. Most obviously an obstacle avoidance al gorithm, dynamic or static, is no longer necessary in the attitude guidance case. In this chapter the common attitude re presentations and their properties are presented in order to justify the selection of the quaternions of ro tation as the attitude

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62 representation scheme. Then, paralleling th e procedure for deve loping the APF position guidance algorithm in Chapter 3, APF guidance for attitude is developed. Attitude Representation Before proceeding with the developmen t of the APF guidance algorithms for attitude, an attitude representa tion scheme must be selected. For attitude representations, it is commonly known that there are no globa lly nonsingular 3-parameter representations of the rotation group.69,70 Therefore, we look to the 4-pa rameter sets in order to obtain global nonsingularity, and in the present work the quaternions are chosen as the attitude representation. More precise ly, we are using the term qua ternion as a quaternion of rotation, i.e., the norm is constrained to unity, which is precisely the column vector of the Euler-Rodrigues symmetric parameters.69 The primary advantage of the quaternions is that successive rotations result in successive multiplication of 4 x 4 matrices which are commutative.69,70 As a result of this computational simplicity, the quaternion attitude representation has become commo nplace in computer animation71 and spacecraft contro.l72,73 Furthermore, the kinematic relationshi p between the attitude states and the angular velocity can be writt en as a linear relationship,69 as given by Eq. (33). For a comprehensive survey on attit ude representations see Shuster.69 The development of the attitude APF gui dance is identical to the position APF guidance algorithm. However, hidden in th e details is a critical difference in state variables between the two cases. For the posi tion case, the generalized coordinates were x and the generalized velocities were x representing the relati ve position and velocities in the LVLH frame, respectively. This yiel ds the relationship between the derivative of the generalized coordinates and the gene ralized velocities as unity, shown by Eq. (32).

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63 For the attitude case the generalized coordi nates are the elements of the quaternion q and the generalized velocities are the com ponents of the angular velocity of the robot The kinematic equation relating the coordinates to the velocities is no longer unity, as illustrated by Eqs. (33) and (34). Although not immediately obvious at this point what complication to the development of the guidance algorithm for attitude this presents, it will be made apparent in the subsequent development. dx x dt (32) ~1 2 0Tqq (33) 32 ~ 31 210 0 0 (34) Attitude Equations of Motion The rotational dynamics of a rigid body are governed by the Euler Equations, Eq. (35) where is the external torque and u is the vector of control torques. For the present work it will be assumed that th e external torque is zero. u JJ (35) The quaternion q was selected to represent the att itude, which is given in terms of the unit vector in the direction of the eigen-axis of rotation, a, and angle of rotation, in Eqs. (36) (38) The kinematic equations re lating the angular velocity and the attitude are given by Eqs. (33) and (34) above.

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64 4T Tqqq (36) sin* 2 qa (37) 4cos 2 q (38) Again since APF guidance is based on enfo rcing stability about the equilibrium of the dynamical system, it is instructive to find the equilibrium solutions for the uncontrolled ( 0u) attitude equations of motion. The equilibrium states can be found by solving the following system of equations, derived from Eqs. (33) and (35): 0uJ (39) 40 qq (40) 0Tq (41) From Eqs. (39) (41) we observe that the uncontro lled equilibrium state is defined as 0e regardless of the st eady state values of 4, eq and eq. Continuous APF Attitude Guidance The development of continuous attitude gui dance algorithms app lies to situations where the actuation on board is from contro l moment gyros (CMG) clusters, reaction wheels, or similar continuous input devices. The control is derived utili zing the general procedure fo r APF methods discussed in Chapter 3. This general procedure will be repeated here for completeness. Choose a scalar function ()Vx of the states x such that V is positive definite for all e x x where e x is the equilibrium state. Furt hermore, we constrain the control u such that its action

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65 is always in the direction dire ctly opposite the gradient of ()Vx. Employing this control ensures the total derivative of V is negative definite for all e x x thereby we are guaranteed by Lyapunovs 2nd theorem that e x x as t Following the above procedure, consider the positive definite scalar potential function 1 (,) 2TTVqqq PQ, where q is the vector component of the quaternion and P and Q are positive definite and diagona l (for simplicity) matrices. The utilization of the vector component of the quaternion only is justified by the existence of the unity norm constraint, by wh ich the scalar part is computable from knowledge of the vector component. Implem enting a continuous control by constraining the attitude rates, under th e action of the control ( 0q ) to be proportional and directly opposite to the resultant of the gradient of V with respect to q and this can be thought of as a pseudo-steepes t decent approach, given by 0qqkVV (42) where qVqP and V Q. In order to obtai n the control input q necessary to achieve this condition, bring the attitu de rates before control application, 0q, to the right hand side of Eq. (42) resulting in Eq. (43). Since the attitude rates q are non-causal, the control must be implemented via the angular velocity. Using the kinematic equation to relate 0q and 0 and assuming the control input to the attitude rates is equivalent to the control input in the angular velocity states, i.e., q I yields the control u in Eq. (71), where vGis given by Eq. (45).

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66 0qkqkq PQ (43) ukq vPG (44) 41 2 kqq vGQ1 (45) It is instructive at this point to addres s the issue of suboptimality for APF methods. The control defined in Eq. (45) is a special case of a more general quaternion feedback regulator, shown in Eq. (46), developed in a seminal paper by Bong Wie,75 et al. This larger class of quaternion feedback regulators was shown, with proper gain selection, to provide near-eigenaxis (near-optimal) rotatio ns with guaranteed global stability. For a precise treatment of the genera l problem we refer you to Ref. 75. We will prove the stability of the special case in the next section. uq JDK (46) Before proceeding with the stability analys is, it is instructive at this point to evaluate the closed-loop equili brium states now that we ha ve defined a control law. Returning to the equilibrium equations in Eqs.(39) (41), we find that for 0 u to be satisfied at the closed-loop equi librium state, both the angular velocity and the vector part of the quaternion must be zero (i.e., 0eeq ). Therefore, the closed-loop system has two equilibrium states as shown below; howev er, they represent the same state in the physical space. 0e (47) 0eq (48) 41eq (49)

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67 Stability Analysis The following theorems, along with conditions set forth in the subsequent lemmas, establish the global asymptotic stability of the equilibrium state, as defined previously. THEOREM 1 : The closed-loop system defined by Eqs.(35) and (44) is globally asymptotically stable if M is positive definite, where M is defined by Eq. (50). 22 2 k k k vGQ1P M Q1PP (50) PROOF : Consider the candidate Lyapunov function 1 (,) 2TTVqqq PJ, where J is the inertia dyadic of the rigid body. It is important to note that the inertia matrix is used in the place of Q due to an elegant simplification that results during the stability proof. The expressi on for the total derivative of V is shown in Eq. (51). Substituting for J from Eq. (35), with the external torque 0 and for q from the control constraint given by Eq. (42) we obtain Eq. (52). Substituting the control given by Eq. (44) and simplifying yields Eq. (53) (NOTE: the simplification resulting from the use of J as the derivative matrix gain is the elimination of the cross product term in Eq. (52). TTVqq PJ (51) TTVqkqku PPQJ (52) 2TTTTVkqqkqkq vPPQPG (53) Since V is a scalar function, we may trans pose any term in the above expression while maintaining its value. Therefore, transposing the second term in Eq. (53) and

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68 combining with the third term leads to the expression in Eq. (54). Rearranging yields a compact expression for the derivative in a quadratic form given by Eq. (55). 2TTTVkqqkq vPQPPG (54) 22 2Tk V qq k k vGQ1P Q1PP (55) Now, observe that the matrix in Eq. (55) is precisely -M. Therefore, if M is shown to be positive definite, then V is negative definite and by Lyapunovs 2nd theorem global asymptotic stability is guaranteed. Note, vG being positive definite is a necessary condition for the positive definiteness of M. LEMMA 1 : The derivative gain matrix, vG, is positive definite if and only if 1 2ikQ PROOF : The derivative gain matrix is precisely given by Eq. (56) below. Positive definiteness requires that the principle minors and the determinant are positive. The ith principle minor of vG is shown in Eq. (57) with 1,2,3 i and ijk The principle minors are positive if Eq. (58) holds. Given that 411 q Eq. (58) implies that the necessary condition for the positive definiteness of vG is 1 2ikQ 3 42 1 3 41 2 214 3222 222 222 q qq k q qq k qqq k vQ GQ Q (56)

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69 2 44222i ijkq qq PMkQkQ (57) 40 2iq kQ (58) The determinant of vG, given by Eq. (59), is always positive for positive diagonal elements of vG. Therefore, 1 2ikQ is a necessary and sufficient condition for the positive definiteness of vG. 2 3 3 ,, 1 1det 2j vviivjj j iq GG G (59) It is instructive to point out that, in general, that al l diagonal elements 42iq kQ could be negative and still satisfy the positivity of the principal minors as required by Eq. (57). Since Q was assume diagonal and positive definite, this implies that 0 k It will be shown that 0 k is required for M to be positive definite, a nd therefore, the condition 1 2ikQ will be used. LEMMA 2 : A block matrix S, given by Eq. (60) is positive definite if and only if the matrices A and1 RDBAB are positive definite. DB S BA (60) PROOF : Start by executing block elementary column operation on S in order to make the resulting matrix, T shown in Eq. (61), block upper triangular. From Eq. (61) we see that the upper left block entry is exactly R. Also, from its upper triangular form, T is positive definite if and only if A and 1 RDBAB are positive definite. Since

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70 elementary operations, similarity transformatio ns in general, do not alter the eigenvalues, it is implied that if T is positive definite only if S is positive definite and vice versa. -1 DBABB T 0A (61) THEOREM 2 : For 1 2ikQ M is positive definite. PROOF : In Lemma 2 let S=M (Note: M has the necessary structure) which gives the matrices A, B, and D from Eq. (60) as 22vk k AP BQ1P DG (62) Using the result of Lemma 2 if A and 1 RDBAB are positive definite then M is positive definite. In this case k2A=P which needs to be positive definite, which implies that 0 k Furthermore, since P is a diagonal matrix so is A, which implies that the inverse of A is just the reciprocal of the diagonal elements 1 123111 ,, diag AAA A. This gives R the form shown in Eq. (63). 2 1 11213 1 2 2 12223 2 2 3 13233 3B DDD A B DDD A B DDD A R (63) R has the same structure as vG, hence it follows from Lemma 1 that R will be positive definite if the condition 20i i iB D A is satisfied. Assuming that 1iA (i.e.,

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71 21/iPk ) and substituting for A, B, and D the condition for positive definiteness of R becomes the expression in Eq. (64). 2 22 4(1)0 24iiiq k kQQP (64) Let the left hand side of the inequality in Eq. (64) be represented by 2()iiWPcaP as shown in Figure 1, where 422ickQq and 224(1)iakQ Therefore, R is positive definite for all values of iP such that W is positive. W is the equation of a parabola, with c characterizing the W-intercept and a governing the width. Examining the range of the two constants a and c, we find that 0 a always, which implies W is always opening downward, and 0 c from 1/2ikQ and 411q This implies that for all choices of 1/2ikQ a portion of W lies above the iP axis as can be seen in Figure 1 below. Therefore, for 1/2ikQ we can find a range of iP such that R is positive definite and therefore M is positive definite. Furthermore, the range on iP is given by //icaPca however P was assumed positive definite previously, meaning iP must be selected from the positive side of this range. It is important to note that k is a free parameter in the sense that it is not explicitly specified in the selection of the control gains. Consequently, the gains may be chosen within their range of stability specified by lemmas and theorems and a k found after the fact to guarantee the stability. The restriction that 0/iPca along with the other constraints imposed on the controller parameters from Theorems 1 and 2 and Lemmas 1 and 2 make selection of the control gains pG and DG cumbersome. To circumvent th is difficulty we restrict our

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72 control gains to a region of interest and employ an optimi zation algorithm, specifically a genetic algorithm discussed in the next section, to determine the control gains pG and DG. Figure 28. Inequality of Eq. (64) Stability of Dynamical System The convergence of the vector component of the quaterni on and the angular velocity to zero for the controller derived in the present work is guaranteed from the analysis of the previous section. One woul d ask at this point as to the behavior of 4q in the dynamical system. For a general perspectiv e on the nature of the equilibrium states given by Eqs. (39) (41), look at the linearization of the closed loop dynamical system,

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73 given by Eqs. (35) and (44), around both equilibrium stat es. The behavior of the dynamical system in the neighborhood of these st ates is elucidated by the eigenvalues of these linearizations. It can be shown that for 41q the linearization is stable, implying that the spectrum of the eigenvalues, 0 For 41q the dynamical system is unstable, meaning there exist 0 Therefore, regardless of the initial value of 4q, it will converge to 41q which depending on the application may be viewed as a blemish on the efficiency of the controller A plot illustrating the behavior of 4q when the initial states are ve ry close to desired (equilibrium) st ate in the physical space, but the unstable equilibrium in the state space is shown below. Figure 29. Behavior of q4

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74 Optimization To aid in the selection of the control gains pG and DG a genetic algorithm was used to optimize the gains with respect to a cost function, F, given by Eq. (17) that is a function of the error in the states T T TXq and the control effort u. The matrices x H and D H are weight functions for the state erro r and control states respectively, and for the optimization performed here both we re assigned the appr opriated dimensioned identity matrix, meaning equal weighting was used for the control and state errors. Since the objective of the control is to bring the st ate vector to the equi librium state defined by Eq. (75) (76), the error states are equivalent to states themselves TT TT TT err errqq 11 22TT errerr x uFXXuu HH (65) The genetic algorithm parameters used fo r the optimization are summarized in Table 4. Table 4. Genetic algorithm statistics Parameter Name PARAMETER VALUE Population size 25 Numberof Generations 25 Probability of Mutation 0.05 Probability of Crossover 0.9 Crossover type Uniform Fitness Function Rosenbrock Number of Bits 20

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75 The fitness function evalua ted the contribution of the error in the states, q, 4q and and the control effort over the first 50 seconds of the simulation given by the numerical example in the next section. The control matrices pG and DG are 3x3 diagonal matrices, and therefore six parameters were optimized. The mean and standard deviation for these six parameters for ten runs of the genetic algorithm are summarized in Table 5. Table 5. Mean and standard deviation fo r the six parameters in the GA optimization Parameter Mean Std. Deviation Gp1 1.1083 0.1083 Gp2 1.0111 0.0101 Gp3 1.1703 0.1293 GD1 0.5253 0.0218 GD2 0.5527 0.0433 GD3 0.8038 0.1094 Cost Function 49.5967 0.4197 The three parameters of PG were restricted to range be tween 1.0 and 2.0, and those of DG were restricted to 0.5 to 1.0. As is illu strated by the small standard deviations in Table 5, in general all parameters converged to a unique value in their respective ranges. Numerical Example In this section, we use the numerical example given in Wen and Kreutz-Delgado70 to compare our controller to their classical PD controller. The objective of the numerical example is to maneuver a spacecraft, represented by its inertia matrix, J, from an initial orientation and angular velocity to the equilibrium state given by Eqs. (75) (76),, with 41q The controller in Ref. 70 is defined as shown in Eq. (66), where 4pk and 8vk

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76 Figure 30. Time response of vector part of quaternion From Ref. 70, the spacecrafts inertia matrix is given by by Eq. (77) and the initial orientation in axis-angle representation is 0.4896 0.2052 0.8480Ta and 2.4648 radians. The spacecrafts in itial angular velocity is 0.10.30.5T. The gains for the controller developed in the present work are listed in Eqs. (68) (69). pvukqk (66) 1.000 00.630 000.85 J (67)

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77 1.108300 01.01110 001.1703p G (68) 0.525300 00.55270 000.8038D G (69) Figure 30 illustrates the response of the thre e components of the vector part of the quaternion, q, and Figure 31 shows the response of the scalar part of the quaternion, q4. Figure 32 and Figure 33 show the response of the three components of the angular velocity and the control input, respectively As demonstrated by these plots, the controller developed in this pa per is capable of performing th e stated maneuver. In fact, the maneuver was accomplished at a significan t reduction in control effort. It is important to note however, that the gains were arbitrarily selected in Ref. 70 and no optimization was investigated. Figure 31. Time response of scalar part of quaternion

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78 Figure 32. Time response of angular velocity The effective performance of the contro ller was demonstrated through comparison with a standard PD controller by a numerical ex ample. It is importa nt to note that even though the gain matrices components were in ge neral smaller that the gains used in the PD controller simulation, the performance achieved was superior, as is illustrated by Figure 30 through Figure 32. Furthermore, the contro l effort to achieve this improved performance was considerably less, as can be seen in Figure 33. It should be stated here that the convergence of q4 to +1 introduces a possible ineffi ciency for attitude maneuvers. If initially q4 is closer to -1 than +1, then it would be more efficient to drive the attitude toward the equilibrium state at q4 = -1. However, our controller drives the attitude to the equilibrium state with q4 = +1, regardless of the initial value of the scalar part of the

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79 quaternion. One possibility to mitigate this inefficiency is to make the gain matrix PG in Eq. (71) proportional to the 4()signumq, as done in Wie, et. al.73 Figure 33. Time response of control effort Impulsive APF Attitude Guidance Impulsive attitude guidance is applicable in those situations where the actuation is provided by thrusters as opposed to the device s used in the continuous case, which with the growing sophistication a nd maturity of micro-Newton capable micro-thrusters is becoming a popular option for attitude cont rol systems (ACS) on board micro and nano class satellites.

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80 Paralleling the development in previous section, let the artificial potential function for this case be 1 2TVqqq Q, where qis the vector component of the quaterion q. Note, it is acceptable to use the vector portion of qonly due to the constraint on quaternions of rotation, i.e., 1 q The total derivative of the potential function is given by T qVVq and the control is defined by Eq. (71) where is such that the rate of the vector part of the quaternion after a control impulse 0q is directly opposite to the local gradient of the potential 0qqkV Defining the control in this manner ensures the derivative of the pot ential after a control impulse is negative definite, as given by Eq. (70), and therefore, by Lyapunovs 2nd method qis guaranteed to converge to its equilibrium value. ()0T qqVxkVV (70) 0,0 ,0V Control V (71) 0 00q Tqq qkV q (72) At this point, the difference in the relationship between the generalized coordinates and velocities become apparent. Since qare not causal states, the control must be applied to the angular velocity states. Utilizing the kinematic relationship given by Eq. (33) the control constraint can be transformed to a form in terms of the angular velocity given by Eq. (72), while maintaining the constraint on 0q Substituting the gradient

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81 qVqQand premultiplying both sides of Eq. (72) by the inverse of the coefficient matrix of 0 yields Eq. (73). Thus the control is applie d to the angular velocity states such that an impulsive a ngular velocity correction 00 that constrains the rate of the vector component of the quaternion to be tangent to the lo cal gradient of the potential. 1 00Tqq kq q Q (73) Another difference for APF guidance de velopment between the position and attitude cases is the proof of the stability of the equilibrium states. The equilibrium was the origin in the position case, with the equilibrium derived from the orbital relative motion equations of motion. For the attitude we must return to Eqs. (39) (41) now that the control has been defined in order to obtain the controlled ( 0u ) equilibria. Just as before, the equilibrium equations show that for 0u to be satisfied at the equilibrium solutions both the angular velocity and the vector part of the qua ternion must be zero (i.e., 0e eq ). Again there are two equilibriu m states (both the same point in the physical space) for the controll ed system, restated in Eqs. (74) (76). Since no differentiation between the equilibrium solutions is necessary in the convergence analysis above, the states will converge to the equilibrium solution to whic h it is initially closest. 0e (74) 0eq (75) 4,1eq (76)

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82 Numerical Example: Impulse-b ased APF Attitude Guidance To illustrate the efficacy of the impulse based APF guidance algorithm for attitude, a numerical example is considered. A satellite in orbit has an initial orientation given in axis/angle representation as 0.48960.20320.8480Ta and 2.4648 radians, and its initial angular velocity is given by 0.50.80.9Tradians per second. The central principle inertia matrix repr esenting the spacecraft is given by Eq. (77). The APF parameters used were 0.1k and QI, where Iis the 3 x 3 identity matrix. The objective of the APF guidance al gorithm is to bring the sp acecraft with its initial parameters to the equilibrium state 4010T T T TTTqq 1.000 00.630 000.85 J (77) Figure 34. Components of the vector part of the quaternion vs. time

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83 Figure 35. Components of the angular velocity vs. time Figure 36. Scalar part of the quaternion vs. time

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84 Figure 34 and Figure 35 show q and versus time, respectively. Figure 36 illustrates the trajectory followed by 4qas a function of time. The maneuver duration was 25 seconds and a total impulsive angular velocity change 1.4034total radian per second was necessary to bring the 3110q The impulsive angular velocity inputs are illustrated by Figure 37, and it can be observed th at only three impulses totaling 1.4034 radians per second were necessary for the maneuver. Solely for the purpose of illustrating the convergence to the equilibrium point initially closer, the numerical example was repeated with identical pa rameters, except the initial value of 44.4648q radians. Figure 38 depicts the trajectory fo llowed by the scalar part of the quaternion in this case, and it is obse rved that indeed it converges to -1. Figure 37. Angular velocity impulse s vs. time for the numerical example

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85 An important caveat to the results presente d here the relatively fast convergence of the attitude and angular velocities are dir ect consequences of a small inertia matrix in Eq. (77) and small initial angular velocities. Ho wever, the results here are for illustration of the efficacy of the algorithms to bring the states to their equilibrium values. A more precise analysis of the performance of thes e algorithms is necessary utilizing more practical inertias and initial parameters. Figure 38. Scalar part of the quatern ion vs. time for minus one convergence case Conclusions In this chapter two original attitude gui dance algorithms were derived from an artificial potential function approach. In the continuous guidance case, the algorithm was shown to belong to a more general class of quaternion feedback regulator to eliminate concerns about suboptimal performance inhe rent in APF methods, and the algorithm was

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86 proven to be globally asympto tically stable. The controlle r parameters were obtained using a genetic algorithm in order to mitigate th e difficulty in gain selection and to obtain a degree of optimization with respect to stat e error and control effort. The GA obtained convergence for all six parameters in their respective ranges, as was illustrated by the small standard deviation of each parameter. Furthermore, an impulsive guidance approach was developed and shown to be e ffective for spacecraft with thruster based ACS. With these algorithms in place, a more complete development of APF guidance that synthesized both position and attitude APF algor ithms for rigid bodies is pertinent. These are the topics under current inves tigation and will be the direc tions pursued in the future. In the next chapter, we conclude this disse rtation with some recommendations for future work and our concluding remarks.

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87 CHAPTER 6 CONCLUSION AND FUTURE WORK Conclusion The current direction of th e space industry is toward utilization of autonomous space robotic systems for in-space operations (ISO) and planetary surface exploration (PSE). The state of the art outlined in Ch apter 1 illustrates few achievements in real world implementations and test bed environm ents for ISO space robotics. As a result research in autonomous space robotic systems is both timely and necessitated. With this as the stimulus we explore autonomous guida nce techniques for a dvancing the state of the art ISO space robotic systems. As a first attempt the classi cal orbit mechanics methods were explored in Chapter 2. These methods were shown to be insuffi cient, primarily due to the inability to incorporate obstacle avoidance, for a large clas s of ISO mission. In Chapter 3 we turned to the robotics literature for alternative a pproaches and discovered artificial potential function (APF) guidance to be a feasible solu tion, provided the defici encies inherent in these methods are overcome; the lack of dynamic obstacle avoidance, spurious local minima, and suboptimal performance. Chapter 4 developed the key elements that allow APF guidance to be an effective solution for ISO missions. A novel dynamic obs tacle avoidance algorithm is generated that is capable of handling general dynamic environments. Furthermore, through the use of a sum of harmonic function primitives we can guarantee no introduction of local minima and mitigate the suboptimal penalties of APF methods.

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88 At some point the point mass assumption for the spacecraft becomes unacceptable and instead must be admitted as rigid bodies. As a result, consideration of the attitude in introduced. In Chapter 5 the classical APF framework is utilized to develop two novel attitude guidance algorithms, one for continuous and one for impulsive type actuation. From characteristics of the attitude c ontrol problem and proper potential function selection eliminates the inherent deficiencies of an APF approach. Both controller were shown to perform effectively for their appropriate application. Future Work The solution method presented in this di ssertation is an eff ective solution for autonomous guidance of in-space operations mission, such as assembly, inspection, and servicing. However, further development of th is approach is possible in several areas, as outlined below. Harmonic Functions o Utilization of more complex primitives o Use additional primitives in sum o Eliminate the use of offline optimi zation so algorithm could be used for real time guidance Reference Velocity Field o Generate full nonlinear velocity field (Lambert solution field) o Include perturbing effects in reference field Drag J2 Solar radiation

PAGE 100

89 System Dynamics o Fuse attitude and position guidance algorithms for couple attitude/position dynamics Research in these areas is the focus of ongoing post graduate study.

PAGE 101

90 LIST OF REFERENCES 1. Kennedy, J.F., Urgent National Needs, Sp eech to US Congress, Washington, DC, May 25, 1961. 2. Bush, G.W., Vision for US Space Expl oration, Speech to US Congress, Washington, DC, January 14, 2004. 3. Stathopoulos, V., Lunakhod Spacecraft, http://www.aerospaceguide.net /spacecraft/lunakhod.html December 5, 2005 (last accessed). 4. Jet Propulsion Laboratory, Mars Exploration Rover Missions, http://marsrovers.jpl.nasa.gov/home/ December 5, 2005 (last accessed). 5. Polites, M, Technology of Automated Rend ezvous and Capture in Space, Journal of Spacecraft and Rockets, Vol. 36, No. 2, March-April 1999, pp. 280-291. 6. Legostayev,V.P. and Raushenbakh, B.V ., Automatic Rendezvous in Space, Foreign Technology Division, FT D-HT-23-21346-68, December 1968. 7. Stern, S.A., and Fowler, W.T., Path-C onstrained Maneuvering Near Large Space Structures, Journal of Spacecraft and Ro ckets, Vol. 22, Sept.-Oct. 1985, pp. 548553. 8. Soileau, K.M., and Stern, S.A., Path-Constrained Rendezvous: Necessary and Sufficient Conditions, Journal of Spacecraf t and Rockets, Vol. 23, No. 5, pp. 492498. 9. Mitsushige, O., ETS-VII: Achievements, Troubles and Future, Proceedings of the 6th International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-SAI RAS), 2001, pp. 914-919. 10. AERCam Sprint, http://vesuvius.jsc.nasa.gov/er_er/html/sprint/index.htm December 5, 2005 (last accessed). 11. ROTEX, http://www.cosimir.com/VR/English/rotex.htm January 11, 2005 (last accessed). 12. Skyworker, http://www.frc.ri.cmu.edu/projects/skyworker/ December 5, 2005 (last accessed).

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91 13. Necessary, R., Curator, Robonaut, http://vesuvius.jsc.nasa.gov/ er_er/html/robonaut/robonaut.html December 5, 2005 (last accessed). 14. Space Systems Laboratory, University of Maryland, Dexterous Robotics at the Space Systems Laboratory, http://robotics.ssl.umd.edu/ranger/index.shtml December 5, 2005 (last accessed). 15. Pedersen, L., Kortenkamp, D., Wettergreen D., and Nourbakhsh, I., editors, NASA Exploration Team (NEXT), "Space R obotics Technology Assessment Report," Moffet Field, CA, December 2002. 16. Pedersen, L., Kortenkamp, D., Wettergreen D., and Nourbakhsh, I., editors, NASA Exploration Team (NEXT), "Space R obotics Technology Assessment Report," Moffet Field, CA, December 2002, pp. 17. 17. Mini AERCam, http://aercam.nasa.gov/ December 5, 2005 (last accessed). 18. NASAs Implementation Plan for Space S huttle Return to Flight and Beyond, http://www.nasa.gov/pdf/58541main_RTF_rev2.pdf Vol. 1, Revision 2, April 26, 2004, pp. 1-21 1-31. 19. Space Systems Laboratory, University of Maryland, The Supplemental Camera and Maneuvering Platform (SCAMP), http://www.ssl.umd.edu/projects/SCAMP/Project_overview.html December 5, 2005 (last accessed). 20. Pedersen, L., Kortenkamp, D., Wettergreen D., and Nourbakhsh, I., editors, NASA Exploration Team (NEXT), "Space R obotics Technology Assessment Report," Moffet Field, CA, December 2002, pp. 31. 21. On-Orbit Servicing (OOS), http://www.on-orbit-servicing.com/index.html December 5, 2005 (last accessed). 22. RFI 2004-500, Intramural Call for Human and Robotic (H&RT) Proposals, http://www.moontoday.net/news/viewsr.html?pid=12890 December 5, 2005 (last accessed). 23. RFI04212004, Intramural Call for Huma n and Robotic (H&RT) Proposals, http://www.moontoday.net/news/viewsr.html?pid=12890 December 5, 2005 (last accessed). 24. Battin, R.H., An Introduction to the Mathematic s and Methods of Astrodynamics, Revised Edition, AIAA, Reston, VA, 1999. 25. Prussian, J.E. and Conway, Bruce A., Orbit Mechanics, Oxford University Press, New York, 1993, pp.139-152.

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92 26. Bate, R., Mueller, D., and White, J., Fundamentals of Astrodynamics, Dover Publications Inc., New York, 1971 27. Vallado, D., Fundamentals of Astrodynamics and Applications, The McGraw-Hill Companies, Inc., New York, 1997. 28. Battin, R.H., and Vaughan, R.M., An Elegant Lambert Algorithm, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 6, Nov-Dec 1984, pp. 662-670. 29. Gooding, R.H., A Procedure for the Solution of Lamberts Orbital Value Problem, Celestial Mechanics, Vol. 48, 1990, pp. 145-165. 30. Stern, S.A., and Fowler, W.T., Path-C onstrained Maneuvering Near Large Space Structures, Journal of Spacecraft and Ro ckets, Vol. 22, Sept.-Oct. 1985, pp. 548553. 31. Soileau, K.M., and Stern, S.A., Path-Constrained Rendezvous: Necessary and Sufficient Conditions, Journal of Spacecraf t and Rockets, Vol. 23, No. 5, pp. 492498. 32. Clohessy, W. H., and Wiltshire, R. S., Terminal Guidance Systems for Satellite Rendezvous, Journal of the Aerospace Sciences, Vol. 27, No. 9, 1960, pp. 653658. 33. Hablani, H., Tapper, M., and Dana-B ashian, D., Guidance and Relative Navigation for Autonomous Rendezvous in a Circluar Orbit, Journal of Guidance, Control, and Dynamics, Vol. 25, No. 3, May-June 2002, pp. 553 562. 34. Carter, T.E., State Transition Matrices for Terminal Rendezvous Studies: Brief Survey and New Example, Journal of Guid ance, Control, and Dynamics, Vol. 21, No.1, 1998, pp. 148-155. 35. Carter, T.E., New Form for the Optima l Rendezvous Equations Near a Keplerian Orbit, Journal of Guidance, Vol. 13, No. 1, 1990, pp. 183-186. 36. Tschauner, J., "Elliptic Orbit Rendezvous," AIAA Journal, Vol. 5, No. 6, 1967, pp. 1110-1113. 37. Humi, M., and Carter, T., Rendezvous E quations in a Central-Force Field with Linear Drag, Journal of Guidance, Cont rol, and Dynamics, Vol. 25, No. 1, 2002, pp. 74-79. 38. Lozano-Perez, T., and Wesley, M.A., A n algorithm for planning collision-free paths among polyhedral obstacl es, Communications of th e ACM, October 10, pp. 560-570.

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93 39. Hwang, Y.K. and Ahuja, N., Gross Motion Planning A Survey, ACM Computing Surveys, Vol. 24, No. 3; ABI/INFORM Global, Sep. 1992, pp.219 280. 40. Rao,N., Hareti, S., Shi, W., and Iye ngar, S., Robot Navigation in Unknown Terrains: Introductory Survey of Nonheuristic Algorithms, Technical Report ORNL/TM-12410, Oak Ridge National Laboratory, 1993. 41. Wise, K.D., and Bowyer, A., A Survey of Global Configuration-Space Mapping Techniques for a Single Robot in a Static Environment, Intern ational Journal of Robotics Research, Vol. 19, No. 8, August 2000, pp. 762-779. 42. Wise, K.D., and Bowyer, A., A Surv ey of Configuration-Space Mapping Techniques and Applications, Part II: Fr om Dynamic Obstacles to Mechanisms, Technical Report 09/98, Department of M echanical Engineering, University of Bath, February, 1997. 43. Johnson, C.G., Robot Motion Planning Survey, Annual Technical Report, Department of Mathematics, Napier University, Edinburg, May 3,1995. 44. Choset, H., and Kortenkamp, D., Path Planning and Control for Free-Flying Inspection Robot in Space, Journal of Aerospace Engineering, Vol. 12, No. 2, April 1999, pp. 74-81. 45. Lopez, I., and McInnes, C., Autonomous Rendezvous Using Artificial Potential Function Guidance, Journal of Guidance, C ontrol, and Dynamics, Vol. 18, No. 2, Mar.-Apr. 1995, pp. 237-241. 46. McQuade, F., and McInnes, C.R., Aut onomous control for on-orbit assembly using potential function met hods, The Aeronautical Jour nal, June/July 1997, pp. 255-262. 47. Roger, A.B., and McInnes, C.R., Safety Constrained Free-Flyer Path Planning at the International Space Station, Journal of Guidance Control and Dynamics, Vol. 23, No. 6, November-December 2000, pp. 971-979. 48. McQuade, F., Ward, R., and McInnes, C. R.: The Autonomous Configuration of Satellite Formations using a Generic Potential Function, 5th International Symposium on Spaceflight Dynamics, Biarr itz, France, June 26-30, pp. 255 262. 49. Nagarajan, N, Tan, S.H., Lee, B.T., and Ta n, B.L., Suitable strategies for In-Plane Orbit Acquisition Using Mi crothrusters, Proc-T he 15th Annual AIAA/USU Conference on Small Satellites, USA, August 2001, SSC01-X1-4. 50. Murray, R. and Li, Z., A Mathematical Introducti on to Robotic Manipulators, CRC Press, Boca Raton, FL, March 22, 1994.

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94 51. Grantham, W., and Chingcuanco, A., Lyapunov Steepest Descent Control of Constrained Linear Systems, Transactions on Automatic Control, Vol. AC-29, No. 8, 1984, pp. 740-743. 52. Khatib, O and Mampey, L. M., Fonction decision-commande dun robot manipulateur, Report 2/7. DERA /CERT, Toulouse, France, 1978. 53. Khatib, O., Real-time obstacle avoidance for robot manipulator and mobile robots, The International J ournal of Robotics Research, Vol. 5, No. 1, 1986, pp. 90 98. 54. Rimon, E., and Koditschek, D., Exact R obot Navigation using Artificial Potential Functions, IEEE Transactions on Robotics and Automation, Vol. 8, No. 5, 1992. 55. Planas, R.M., Fuertes, J.M., and Martinez, A.B., Qualitative Approach for Mobile Robot Path Planning based on Potential Field Methods, 16 th International Workshop on Qualitative Reasoning, 2002, CD-ROM. 56. Kiupers, J.B., Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press, Princeton, New Jersey, 1999. 57. Lopez, I., and McInnes, C., Autonomous Rendezvous Using Artificial Potential Function Guidance, Journal of Guidance, C ontrol, and Dynamics, Vol. 18, No. 2, Mar.-Apr. 1995. 58. McQuade, F., and McInnes, C., Autonom ous Control of on-Orbit assembly using Potential Function Methods, The Aeronautic al Journal of the Royal Aeronautical Society, June-July 1997. 59. Tatsch, A., Xu, Y., and Fitz-Coy, N., A Nonlinear Controller via Artificial Potential Function for Spacecraft Attitude Maneuvers, 28th AAS Guidance and Control Conference, Brekcenridge, CO, Feb. 2005, AAS 05-008. 60. Tatsch, A., Xu, Y., and Fitz-Coy, N., A Nonlinear Controller via Artificial Potential Functions for Impulsive Spacecr aft Maneuvers, 2004 Core Technologies for Space Systems Conference, Colorado Springs, CO, CD-ROM. 61. Ko., N.Y., and Lee, B.H., Avoidability Measure in Moving Obstacle Avoidance Problem and Its Use for Robot Moti on Planning, Proceedings of the 1996 IEEE/RSJ International Conf erence on Intelligent Robots and Systems, Vol. 3, 1996, pp. 1296-1303. 62. Hussein, B., Robot Path Planning and Obstacle Avoidance by Means of Potential Function Method, Ph.D Dissertation, Universi ty of Missouri-Columbia, 1989.

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95 63. Ge, S.S., and Cui, Y.J., Dynamic Mo tion Planning for Mobile Robots using Potential Field Method, Proceedings of the 8th IEEE Mediterranean Conference on Control and Automation, Rio, Patras, Greece, 2000, CD-ROM. 64. Tatsch, A., Fitz-Coy, N., and Edmonson, W., Artificial Potential Function Guidance for On-orbit Inspection, Space Automation and Robotics Symposium, Naval Research Laboratory, March 30 31, 2005, CD-ROM. 65. Connolly, C. and Grupen, R., The applica tion of harmonic functions to robotics, Journal of Robotic Systems, Vol. 10, No. 7, pp. 931, 1993. 66. Satoh, K., Collision avoidance in multi-dimensional space using laplace potential, in Preprint of 5th Annual Conference of the Robotics Society of Japan, 1987, pp. 155-156. 67. Masoud, A., Techniques in Potential-Based Path Planning, Ph.D. Dissertation, Electrical and Computer Engineering Depa rtment, Queen's University, Kingston, Ontario, Canada, Mar. 1995. 68. Waydo, S., Vehicle motion planning us ing stream functions, CDS Technical Report 2003-001, California Institute of Technology, 2003. Available online: http://caltechcdstr.library.caltech.edu/. 69. Shuster, M., "A Survey of Attitude Representations," Journal of the Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993, pp. 439-517. 70. Wen, J., and Kreutz-Delgado, K., The Attitude Control Problem, IEEE Transactions on Automatic Control, Vo l. 36, No. 10, October 1991, pp. 1148-1162. 71. Shoemaker, K., Animating Rotation with Quaternion Curves, SIGGRAPH 1985, San Francisco, CA, July 22-26, Vol. 19, No. 3, pp. 245-254. 72. Joshi, S.M., Kelkar, A.G., and Wen, J ., Robust Attitude Stab ilization of Spacecraft Using Nonlinear Quaternion Feedback, I EEE Transactions on Automatic Control, Vol. 40, No. 10, Oct. 1995, pp. 1800-1803. 73. Wie, B., Weiss, H., and Arapostathis, A., Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotations, Journal of Guidance, Control, and Dynamics, Vol. 12, No. 3, May-June 1989, pp. 375-380. 74. Grantham, W., and Chingcuanco, A., Lyapunov Steepest Descent Control of Constrained Linear Systems, Transactions on Automatic Control, Vol. AC-29, No. 8, 1984, pp. 740-743. 75. Wie, B., Weiss, J., and Arapostathis, A., Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotations, Journal Guidance, Control, and Dynamics, Vol. 12, No. 3, May-June 1989, pp. 375-380.

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96 BIOGRAPHICAL SKETCH Andrew R Tatsch was born in Fairmount Indiana, on November 24, 1978 but has resided in the state of Florida since 1981. In August of 1997 he was accepted to the University of Floridas College of Engi neering in the Department of Aerospace Engineering. He received his bachelors degree in aerospace engineering in May of 2001. In August of 2001 Andrew was awarded the Alumni Fellowship for pursuing a doctoral degree in aerospace engineering at the University of Florida. He completed his masters degree in 2004 and was subsequently admitted to candidacy for his PhD. His research interests include guidance, na vigation, and control of autonomous inspace operational system.


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ARTIFICIAL POTENTIAL FUNCTION GUIDANCE FOR AUTONOMOUS IN-
SPACE OPERATIONS















By

ANDREW R TATSCH


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


2006

































Copyright 2006

by

Andrew R Tatsch

































Dedicated to






My parents who brought me into this world and
My friends, family, and colleagues who opened my eyes to it.















ACKNOWLEDGMENTS

To my advisor, Dr. Norman Fitz-Coy, I would like to convey my gratitude. Few

graduate students have the opportunity to work with, not for, an advisor that encourages

freedom and creativity in the search for a pragmatic solution and uniquely challenges

them beyond what they think is possible. He also taught me many lessons beyond the

curriculum and for those I am grateful. Without his guidance this dissertation would

never have come to fruition.

Many thanks also go out to my supervisory committee, Drs. Rick Lind, Andrew

Kurdila, Gloria Wiens, BJ Fregly, and Stanley Dermott, for their invaluable assistance

and patience.

Lastly, I would like to acknowledge my brother J Bradley Tatsch for everything

that he has done to make this dream a reality. Without his assistance my undergraduate

and graduate experiences would have never gotten off the ground. I am forever grateful

for all that he has done for me.















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ............ ........... .. ......... ................. ........... vii

LIST OF FIGU RE S ............... .... .......... ............... ...... ......... .. .......... viii

A B ST R A C T ................. .......................................................................................... x

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

State of the Art for In-Space Operations (ISO) Functionalities .............. ...............4
R research Scope ...................................... ............................. ................. 11
D issertation O outline .................. ............................................. .............. .. 12

2 TRAJECTORY DETERMINATION: ORBIT MECHANICS APPROACH........... 14

M idcourse Trajectory D eterm nation .............................................. .....................14
Lam bert's Problem ............................................... ........ ............... 16
M maneuver Command Determination .... .......... ........................................ 19
N um erical Exam ple: M idcourse................................... .................................... 21
Endgam e Trajectory D eterm nation ...................................... ........... .................. 22
CW Maneuver Command Determination............ ...... ..............24
N um erical Exam ple: Endgam e.................................... ..................................... 25
Alternative Orbit Mechanics Based Methods..........................................................26
C o n clu sio n s.................................................... .................. 2 8

3 ARTIFICIAL POTENTIAL FUNCTION POSITION GUIDANCE.........................29

R ob ot M option P planning .................................................................... .....................2 9
Artificial Potential Function Position Guidance...................................................32
Example: Spacecraft Unconstrained Rendezvous ............................................... 35
O b struction s .................................................................................. ......... 37
Numerical Example: Path-Constrained Rendezvous..............................................42
C o n clu sio n s..................................................... ................ 4 4

4 ARTIFICIAL POTENTIAL FUNCTION GUIDANCE: A NEW APPROACH.......45









Dynamic Artificial Potential Function Obstacle Avoidance Algorithm...................45
A attractive Potential .................. .............................. .... .. .. ......... .... 46
R epulsive Potential .................. ............................. .. ... ... .. ........ .... 47
T otal F orce Interactions........................................................................... 49
Sim ulation R results ............................................... ........ .. ............ 50
C constant velocity scenario ........................................ ....................... 50
O n-orbit rendezvous scenario.................................... ........ ............... 52
Optimized Harmonic Attractive Potential ...................................... ............... 54
C o n c lu sio n s..................................................... ................ 5 7

5 ARTIFICIAL POTENTIAL FUNCTION ATTITUDE GUIDANCE.....................60

A attitude E quations of M otion ......................................................... .....................63
Continuous APF Attitude Guidance .................................. ............... ............... 64
Stab ility A n aly sis............ ...... ............ ............................................ .... .... ..... 6 7
Stability of Dynam ical System ................................ ......................... ........ 72
O ptim ization ...................................... ............................ ............... 74
N um erical Exam ple .............................................. .... .... .. ........ .... 75
Im pulsive A PF A attitude G uidance................................................... ......... .......... .. 79
Numerical Example: Impulse-based APF Attitude Guidance.............. ................ 82
C o n c lu sio n s.................................................... .................. 8 5

6 CONCLUSION AND FUTURE WORK ........................................ .....................87

C o n clu sio n ...................................... ................................. ................ 8 7
F u tu re W o rk .......................................................................................................... 8 8

L IST O F R E FE R E N C E S ....................................................................... ... ................... 90

BIOGRAPH ICAL SKETCH ...................................................... 96
















LIST OF TABLES

Table page

1 Orbit parameters for rendezvous simulation. ................................ .................52

2 G enetic algorithm statistics ............................................. ............................. 57

3 Mean and standard deviations for the two parameters in the GA optimization .......57

4 G genetic algorithm statistics ........................................................... .....................74

5 Mean and standard deviation for the six parameters in the GA optimization ..........75
















LIST OF FIGURES


Figure page

1 M odel of the ROTEX m anipulator................................ ........................ ......... 5

2 Artist's rendition of ETS-VII components ..... ......... ....................................... 5

3 CM U 's Skyw orker prototype........................................................... ............... 6

4 N A SA 's Robonaut ....................... ................................................. .6

5 University of Maryland's Ranger ............ ........ ...........................6

6 AERCam Sprint.................. ......................... ........... .... .............

7 M ini A ER Cam .................................................... 9

8 Artist's depiction of the Orbiter Boom Sensor System .................. ..................

9 The SCAMP in the neutral buoyancy tank..............................................9

10 Schematic of Lambert's Problem.................... ................. ... ............... 15

11 Transfer trajectory from midcourse trajectory determination aglorithm................23

12 D epiction of orbital rendezvous ........................................ .......................... 23

13 Endgame maneuver trajectory in the xy-plane (R-bar V-bar plane).....................26

14 Glideslope solution for 5 impulse rendezvous. .............................. ................27

15 Planar rendezvous in the CW frame under APF guidance............................. 36

16 Control efforts in x and y directions for the rendezvous........................................36

17 Difference between IC potential and potential on surface of obstructions ..............40

18 Contour plot of the potential function.......................................................... 40

19 Zoom ed view of contour plot......................................................... ... ............ 41

20 Stem plot of obstruction surface potential error................................................41









21 Trajectory traversed by robot for the path-constrained rendezvous example ..........42

22 Control action in the x-direction for the constrained rendezvous ..........................43

23 Control action in the y-direction for the constrained rendezvous ..........................43

24 2D Constant velocity simulation results........................................ ............... 51

25 Planar on-orbit rendezvous results. ........................................ ....... ............... 53

26 Norm of applied chaser accelerations. ........................................ ............... 54

27 Optimized sink-vortex pair vs. CW based vector field.............. .............. 58

28 Inequality of E q. (64) ...................... .................. ................... .. ...... 72

2 9 B eh av ior of q4 ................................................. ............... ....................73

30 Time response of vector part of quaternion .........................................................76

31 Tim e response of scalar part of quaternion................................... .................77

32 Tim e response of angular velocity ........................................ ....................... 78

33 Tim e response of control effort.................................................................... ..... 79

34 Components of the vector part of the quaternion vs. time .................. .............82

35 Components of the angular velocity vs. time .......................................................83

36 Scalar part of the quaternion vs. tim e.................................................................... 83

37 Angular velocity impulses vs. time for the numerical example..............................84

38 Scalar part of the quaternion vs. time for minus one convergence case ..................85















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

ARTIFICIAL POTENTIAL FUNCTION GUIDANCE FOR AUTONOMOUS IN-
SPACE OPERATIONS

By

Andrew R Tatsch

May 2006

Chair: Norman Fitz-Coy
Major Department: Mechanical and Aerospace Engineering

For most in-space operations (ISO), e.g., assembly, repair, refueling, the target

location and its orbit parameters are known a priori. For instance in the case of an

assembly mission, the general location of the construction site and the parts depot are

known parameters in mission planning. Utilizing this known information, we propose a

novel approach to collision-free operations for a heterogeneous team of spacecraft robots

performing the ISO mission based on artificial potential function (APF) guidance.

In order to utilize APF guidance two prominent deficiencies of APF methodologies

need to be circumvented: premature termination due to local minima in the operational

space and suboptimal performance. It is widely known that utilization of harmonic

potential functions eliminates the introduction of local minima in the operational space;

thus we employ harmonic functions to conquer the first of the aforementioned

deficiencies. In order to surmount the suboptimal performance, we propose the use of a

sum of harmonic function primitives (simple functions such as sinks and vortices in the









hydrodynamic analog) to shape the velocity field of the operational space to harness the

underlying dynamics of the system (i.e., orbit mechanics). Furthermore, given the

location and orbit parameters of the target site, a random heuristic search algorithm can

be employed to optimize the velocity field shaping for maximizing the performance gains

achieved from harnessing these underlying dynamics.














CHAPTER 1
INTRODUCTION

I believe that this nation should commit itself to achieving the goal, before this
decade is out, of landing a man on the Moon and returning him safely to the Earth.1

President John F. Kennedy, May 25, 1961

Just as the speech given forty-three years ago on a Thursday in late May by

President John F. Kennedy presented the National Aeronautics and Space Administration

(NASA) a new directive, on January 14, 2004, President George W. Bush delivered a

speech titled "Vision for Space Exploration,"2 whose intent was to again unify NASA

around a set of common goals. These goals are centered around sending man and/or

robotic missions to the Moon, Mars, and deeper into our solar system. Implicit in the

robotic missions proposed by this new space exploration mandate are the utilization and

development of autonomous space robotic systems that "will serve as trailblazers,

advance guard to the unknown."2 Based on these new directives, autonomous space

robotic systems will be imperative for the success of NASA's new vision.

Space robotics, in the context of this work, are robotic systems whose applications

can be roughly dichotomized into two broad classes of missions: in-space operations

(ISO), and planetary surface exploration (PSE). Both of these mission profiles share

common functional requirements and possess functional requirements unique to each

mission. The functionalities typically associated with in-space operations are assembly

(including maintenance of pre-existing structures) and inspection. It is important to note

that although treated separately, the assembly and inspection functionalities share









common subtasks, such as navigation and maneuvering. Planetary surface exploration

(PSE) functionalities include surface mobility (which includes navigation and collision

avoidance), instrument deployment and sample manipulation, science planning and

execution, and human exploration assistance. Even though space robotics applications

are categorized into these two mission classes, generalization of the functionalities listed

above allows for other mission profiles such as terrestrial assembly or in-space science.

The research presented in the subsequent chapters is primarily applicable for in-space

operations; therefore those will be the primary focus here.

The area of in-space robotics operation is far less mature than robotic planetary

surface exploration systems, both in terms of real world implementations and theoretical

research. Real world implementations of planetary surface exploration have been

significantly more numerous, and more successful, than in-space robotics

implementations with examples of teleoperated surface exploration robots dating back to

the 1970's with the Soviet Lunakhod Rover.3 The sophistication of more recent PSE

system successes is evidenced by the Mars Exploration Rovers4 developed at NASA's Jet

Propulsion Laboratory (JPL). In 1997 the Sojourner, although not designed for advanced

capabilities, was the first Mars rover to successfully demonstrate the ability to teleoperate

a vehicle on Mars. Most recently the two more sophisticated Mars rovers, Spirit and

Opportunity, have demonstrated effective teleoperational capabilities in surface mobility

and terrestrial science.

The raison d'etre that led to this disparity between PSE and ISO implementations is

twofold. First, the theory and experience of terrestrial robotics are directly applicable to

PSE scenarios. Volumes of research since the 1950's have been devoted to the









kinematics and control of terrestrial robot manipulators with achievements in autonomous

navigation, perception, and mobility. Therefore, implementation of PSE robots spring-

boarded off the terrestrial robotics knowledge base and needed only to address the

technical issues associated with high degrees of mobility, science planning, perception,

and exploration capabilities, the harshness of space and space transport, and the high

latency communication links implicit in space teleoperation. Conversely, with the

exception of autonomous rendezvous and docking (ARD) investigations5 that date back

to October 1967 in Russia6 and preliminary investigation of in-space proximity

operations,7'8 it has only been in the last decade that autonomous, free-flying robotics

applications have appeared in the literature.9'10 Therefore, both the underlying theory and

its implementation need much further investigation in order to achieve the maturity of

PSE robotics systems to date.

Second, most terrestrial and PSE agents are developed without consideration of the

system dynamics: they are typically kinematically driven, and the cases that do consider

some dynamics do so only in the sense of constraints on the motion, e.g., non-holonomic

constraints. For ISO applications, consideration of the system dynamics is essential.

This fundamental difference is illustrated by the local operator in each case. In terrestrial

or PSE applications, the local operator may be zero. This means that under zero

command, the robot remains stationary until it is commanded to move. The local

operator in ISO systems, however, is in general not zero. Instead, the local operator is the

system dynamics, meaning under zero command, an ISO system will not remain

stationary but move according to the topology shaped by the system dynamics.

Therefore, consideration of the dynamics of ISO systems is imperative.









The sections that follow outline the state of the art for each of the functionalities of

ISO systems for both current implementations and experimental test beds. The

expectations for the sophistication of these functionalities over the next decade are also

discussed.

State of the Art for In-Space Operations (ISO) Functionalities

In this section both current state of the art and experimental test bed sophistication

are assessed for assembly and inspection functionalities of ISO systems. The relatively

few implementations of ISO systems to date are emphasized while noting the progress

possible from the experimental investigations.

The current state of assembly functionality, with respect to real world

implementations, is restricted to the Shuttle Remote Manipulator System (SRMS) and the

Space Station Remote Manipulator System (SSRMS) aboard the International Space

Station (ISS). Neither of the current systems possesses any autonomy as they are solely

teleoperated on-board or from the ground. However, to date there have been on-orbit

demonstrations of assembly subsystems such as the Robot Technology Experiment1

(ROTEX), a German experiment flown by NASA, and Engineering Test Satellite9 (ETS-

VII), flown by the Japanese Aerospace Exploration Agency (JAXA) (formerly National

Space Development Agency of Japan (NASDA)). The ROTEX was a robotic arm (see

Figure 1) that flew in 1993 on Columbia as part of STS-55, and successfully completed

multiple tasks that include replacement of a simulated Orbital Replacement Unit (ORU)

and capture of a free-flying object via on-board and ground teleoperation and

autonomous scripts. By accomplishing tasks from autonomous scripts during the

experiment, ROTEX became the first autonomous space robotic system. The ETS-VII

mission was composed of a passive target satellite, Orihime, and a chaser satellite,









Hikoboshi, with a robotic manipulator arm (see Figure 2). ETS-VII successfully

demonstrated cooperative control of a robotic arm and satellite attitude, and simple

examples of visual inspection, equipment exchange, refueling, and handling of a satellite.


















Figure 1. Model of the ROTEX Figure 2. Artist's rendition of ETS-VII
manipulator components



The state of the art in assembly robotics, implemented via experimental test beds

only, is the Skyworker12 robot at Carnegie Mellon University (see Figure 3), NASA's

Robonaut13 (see Figure 4), and Ranger14 at the University of Maryland (see Figure 5).

Skyworker is an 11 DOF robot that walks across the structure it is assembling to mate

new components to the existing structure. The current prototype allows for high-level

command inputs that are then parsed and implemented on-board as motion control

commands. Robonaut is a collaborative effort between DARPA and NASA aimed at

developing a humanoid robot capable of meeting the increasing requirements for

extravehicular activity (EVA). Robonaut, composed of two dexterous arms and two five-

fingered hands with teleoperational and autonomous capabilities, has already









demonstrated assembly of complicated EVA electrical connectors and delicate

capabilities such as soldering. Ranger is a teleoperated robot at the University of

Maryland that completes assembly, maintenance, and human EVA assistance tasks in a

neutral buoyancy tank. Ranger has demonstrated robotic replacement of an Orbital

Replacement Unit (ORU), complete end-to-end electrical connector mate/demate, and

two-arm coordinated control.


Figure 3. CMU's Skyworker prototype Figure 4. NASA's Robonaut


Figure 5. University of Maryland's Ranger









According to "Space Robotics Technology Assessment Report""1 published by

NASA in December of 2002, the expectation for in-space assembly under nominal

research efforts is: "Robots that can autonomously mate components and do fine

assembly, including making connections under careful human supervision."16 It is

perceived, given the current state of the art of robots like Robonaut, that robots will

possess the mechanical capabilities equivalent to a space-suited human, but barring a

breakthrough in communications architectures, more aggressive expectation of space

robotic assembly cannot be met for teleoperation, due to current low bandwidth/high

latency communication. On the other hand, automating Robonaut and other highly

dexterous robots is possible, but to make highly dexterous robots effective under

autonomous operation, a system level design needs to be considered that designs the

small components specifically for assembly by autonomous robotic systems. This

requires significant redesign in current infrastructure, which was perceived to be

financially impractical at the time of the "Space Robotics Technology Assessment

Report." However, the new NASA mandate outlines strategic allocation of funding to

achieve its aggressive goal set, and as a result it now may be possible to alter the current

infrastructure with the intent to simplify automated robotic assembly.

In-space inspection consists of examining space structures for anomaly detection.

The robots performing the inspection can be either free-flyers or manipulators capable of

performing the subtasks of moving, such as navigation and obstacle avoidance, to

examine the entire exterior of the structure and anomaly detection via sensor

interpretation. Currently, there are no operational space inspection systems. However,

there have been on-orbit investigations of subsystems, such as AERCam Sprinto1 (see









Fig. 6), which flew aboard Columbia (STS-87) in 1997. AERCam is a teleoperated EVA

camera whose purpose was to merely investigate the feasibility of an autonomous, free-

flying EVA camera system.

The state of the art in the area of inspection is the mini AERCam17 (see Fig. 7),

Orbiter Boom Sensor System, 18 OBSS (see Fig. 8), and the Supplemental Camera and

Maneuvering Platform, 19 SCAMP (see Fig. 9), at the University of Maryland. The mini

AERCam project is a second generation version of the Sprint aimed at adding more

complex capabilities while reducing the overall size of the prototype. A nanosatellite-

class spherical free-flyer, the mini AERCam is only 7.5 inches in diameter and weighs a

mere 10 pounds. Even though mini AERCam cutting edge hardware is not flight

certified, making it far from implementable, the results obtained from orbital simulations

and 5 DOF experimental test bed validations are invaluable to advancing the field of

autonomous inspection agents.

The Orbiter Boom Sensor System (OBSS) is a manipulator based concept for

inspection of the thermal protection surface (TPS) of the Shuttle. Consisting of a camera

and a laser range sensor mounted on a 50' boom, the OBSS attaches directly to the

SRMS, providing teleoperational inspection of nearly 75% of the Shuttle's TPS. In the

Space Systems Laboratory at the University of Maryland, the Supplemental Camera and

Maneuvering Platform (SCAMP) allows for investigations in free-flying camera

applications. Operated in a neutral buoyancy tank, SCAMP provides near micro-gravity

conditions for research into free-flying applications. SCAMP has demonstrated effective

stereo video data interface and 3D navigation in the neutral buoyancy test bed.



























Figure 6. AERCam Sprint Figure 7. Mini AERCam


Figure 8. Artist's depiction of the Orbiter
Boom Sensor System


; "

Figure 9. The SCAMP in the neutral
buoyancy tank


For in-space inspection, NASA perceives the expectation for the next decade to be

summarized as "autonomous robotic inspection of some of the exterior surfaces with









sensory data filtered for potential anomaly before being stored or sent."20 With more

aggressive research effort, autonomous inspection and anomaly detection of most exterior

surfaces of a target are realizable.

The addition of autonomous capabilities to ISO systems is a timely issue. The new

NASA mandate2 sets aggressive goals for space robotics missions, and autonomous

capabilities for robotic systems are vital to the achievement of these goals. Furthermore,

the international community, initiated by Deutsches Zentrum fur Luft- und Raumfahrt

(DLR), also known as the German Aerospace Center, has been proactive in forming a

community whose sole focus is on-orbit servicing (OOS). In their own words, the spirit

of the OOS initiative "aims at developing a sound understanding of OOS and its issues,

and at creating a dedicated community on global level"21 with the overall objective being

"to develop a structured long-term approach of how to deal with OOS by providing high-

quality information and latest insights to this new, complex and innovative field of space

activities, and to initiate new collaborations."21 OOS has organized two workshops, in

2002 and 2004, dedicated solely to all aspects of on-orbit servicing, and numerous

sessions for on-orbit servicing in other conferences worldwide, e.g., the 8th International

Symposium on Artificial Intelligence, Robotics and Automation in Space (I-sairas 2005),

providing further evidence to the point that ISO systems development is a timely, global

issue.

With the state of the art in ISO systems, as outlined above, reflecting few

autonomous capabilities, and the absence of a mature knowledge base in free-flying

robotics theory, adding autonomy to ISO systems is a monumental task as evidenced by

the requests for information22,23 (RFI) issued from NASA in the months directly









following the issuing of the new mandate, the recent formation of the Institute for Space

Robotics, and the OOS initiative. To this end, this dissertation explores strategies for on-

board guidance and develops a guidance algorithm for proximity operations for

autonomous ISO robotics systems in an attempt to improve the current state of the art of

autonomous, in-space robotic systems.

Research Scope

For most ISO missions (assembly, repair, refueling) the target location and its orbit

parameters are known a priori, with the exception being maneuvering non-cooperative

targets which we will not consider in the present work. For instance, in the case of an

assembly mission, the location of the construction site and the parts depot are known

parameters from mission planning. Utilizing this known information, we propose a novel

approach to collision free operations of the spacecraft based on artificial potential

function (APF) guidance.

In order to utilize APF based guidance schemes, two prominent deficiencies

inherent to this method need conquering: premature termination due to local minima in

the operational space, and suboptimal performance. It is widely known that utilization of

harmonic potential functions eliminates the introduction of local minima in the

operational space; thus we employ harmonic functions to eliminate the introduction of

local minima. In order to overcome the suboptimal performance, a sum of harmonic

function primitives (simple functions such as sinks and vortices in the hydrodynamic

analog) is employed to shape the velocity field of the operational space to harness the

underlying dynamics of the system (i.e., orbit mechanics). Furthermore, given the

location and orbit parameters of the target site, a random heuristic search algorithm can

be employed to optimize the velocity field shaping for maximizing the performance gains









achieved from harnessing these underlying dynamics. This dissertation presents this

novel approach for proximity operations of multiple space assets operating in relatively

unstructured environments.

Dissertation Outline

This dissertation is organized as follows. Chapter 1 provides an introduction to the

current trends in the space industry, details the state of the art for ISO systems in both

real world implementations and test bed environments, and concludes with the research

scope that precisely defines the problem and solution presented in this work. Chapter 2

explores the orbit mechanics based approaches to trajectory generation for spacecraft, and

illustrates the deficiencies that eliminate these algorithms for the specified application

considered in this dissertation.

In Chapter 3 the motion planning approaches from the robotics field are surveyed

as alternatives to the classical orbit mechanics based approaches. From the robotics

literature, the artificial potential function (APF) guidance approach is singled out as a

promising solution approach for the problem considered here, and examples of APF

guidance algorithms applied to the classical orbital rendezvous with and without

obstacles are presented.

Chapter 4 develops the novel alterations that overcome the two predominant

deficiencies of APF algorithms that would prohibit their application for ISO missions.

The first part of this chapter develops an original, truly general dynamic obstacle

avoidance algorithm that is a critical subsystem for the trajectory generator for ISO

missions. The second part of Chapter 4 deals with the mitigation of local minima and

suboptimal performance by utilization of harmonic function primitives. The chapter is






13


concluded with the application of the APF guidance scheme to a representative ISO

mission scenario.

Eventually the spacecraft will not be considered to be a point mass and attitude

considerations must be included. In Chapter 5 the classical APF framework is utilized to

develop an attitude guidance algorithm that can be shown to be a subclass of controllers

that can be derived using a Lyapunov analysis. Chapter 6 will present conclusions and

future work.














CHAPTER 2
TRAJECTORY DETERMINATION: ORBIT MECHANICS APPROACH

The general mission scenario for all autonomous in-space operations (ISO) systems

requires translating from an initial position to a goal position where the system fulfills its

designed functional purpose, e.g., assembling a structure. It is assumed that the initial

position of the system is either a parking orbit provided by the launch bus or an orbiting

space platform (e.g., ISS), both of which imply the need for midcourse and endgame

trajectory design. Midcourse trajectory determination refers to the transfer from initial

location to close proximity, and endgame trajectory determination refers to the proximity

maneuvering. In the sections that follow, classical orbit mechanics theory for the

midcourse and endgame trajectory determination phases will be presented along with

more recently developed methods based on orbit mechanics. Numerical examples

illustrating the key aspects are included, and the limitations on each for autonomous ISO

applications are illustrated.

Midcourse Trajectory Determination

The most general framework for midcourse trajectory determination is the n-body

problem of astrodynamics (see Refs. 24-27 or any text on Astrodynamics or Celestial

Mechanics). The solution of which requires 6n integrals, of which there are ten known,24-

27 and a general closed form solution is not obtainable. However, the artificial satellites

are well within the sphere of influence (Ref 24, Chapter 7) of the primary central body,

and the mass of the satellites is negligible with respect to the mass of the central body.

Therefore, the trajectory of any robot in this context can be approximated effectively by









the solution of a two-body problem, assuming the position of the goal is associated with

an orbiting target.



P:










Figure 10. Schematic of Lambert's Problem



Restricting consideration to two-body problems, the determination of the orbital

trajectory given two points in space and a time of flight is known as the two-body orbital

two-point boundary value problem24 (see Figure 10). The first strides in the solution of

this boundary value problem came from Johann Heinrich Lambert24'25 (1728-1779).

Lambert's Theorem states that the time (t, t) required to traverse an elliptical arc

between specified endpoints, P1 and P2, depends only on the semi-major axis a and two

geometric properties of the space triangle FPP2, the chord c and the sum of the radii

r, + r Since Lambert arrived at this conclusion from strictly geometric means, the

mathematical statement of Lambert's Theorem, Lambert's Equation, actually comes from

Joseph-Louis Lagrange24 (1736-1813) and is given by Eq. (1) in terms of Lagrange's

variables a and/f. In order to show that Eq. (1) indeed proves Lambert's Theorem,


a and / can be expressed as Eq. (2) and (3), where s = -(r, + + c) is the semi-
2









perimeter of the space triangle. Hence, the transfer time is a function of the semi-major

axis, the radii sum r, + r, and the chord c. Obtaining solutions of Eq. (1) became known

as Lambert's problem. Historically its solution was essential in determining the orbits of

celestial bodies from observation, and today it is applicable for artificial satellite intercept

and rendezvous problems.

3
-(t 2 ,-t) =a2[a,8- (sin ca- sin)] (1)


sin (2)

2 2a (3)

sin K8 CJ (3)


It is important to note that the solution of Lambert's Problem only defines the

geometry of the intercept problem between points P1 and P2. To effectively execute the

trajectory specified by the solution geometry, the terminal velocity vectors need to be

calculated in order to obtain the necessary guidance command to initiate the transfer.

Therefore, determination of the maneuver command is the essential quantity.

Lambert's Problem

The determination of an orbit from a specified transfer time (t2 t) connecting two

position vectors, ri and r2, is called Lambert's Problem2427 (see Figure 10). A variety of

techniques for the solution of Lambert's Problem have been developed over the

years24,28,29 with each being classified by the particular form of the transfer-time equation

and the independent variable chosen in an iterative solution technique. Classically two

transfer-time equations24 were used, one from Lagrange's equations used in his proof of

Lambert's Theorem and the other derived from Gauss' work. Each of these methods has









their advantages and disadvantages, and a third transfer-time equation that utilizes these

advantages while mitigating the disadvantages can be developed as in Battin.24 Battin's

combined equations technique is utilized for the present work. For a complete history

and development of Lambert's Problem and the two-body orbital boundary value

problem see Battin.24

Before attempting to solve Lambert's Problem it is instructive to consider inherent

properties of Eq. (1) that will influence the techniques employed for its solution.

Lambert's Equation is:

1. Transcendental equation
2. Double valued function of a (allows for conjugate orbits with same semi-
major axis)
s
3. Derivative with respect to a is infinite for the minimum energy orbit a = -
2
(the minimum energy orbit corresponds to the smallest possible value of
semimajor axis a)
4. Four possible solution for each case due to quadrant ambiguities in a and/7

Property 1 requires the use of an iterative solution approach to find the semi-major

axisa. From properties 2 and 3 it is obvious that the semi-major axis is not amenable to

iterative solution techniques. Furthermore, Property 4 mandates the use of a method for

selecting the unique solution from the multiple solution possibilities. These difficulties

can be mitigated by utilizing a transformation of Eq. (1) by Battin24 that replaces the

semi-major axis with an independent variable that produces monotonically decreasing

and single valued solution curves, which is more appropriate for an iterative solution

approach. Furthermore, utilization of Prussing's geometric interpretation25 provides the

framework necessary to determine a unique solution.

Battin's combined equations technique24 will now be presented. This procedure

first transforms the independent variable in Lagrange's Equation to one with properties









amenable to iterative solution techniques. Then, utilizing another form of the transfer-

time equation attributed to Carl Friedrich Gauss (1777-1855), Battin improves the

computational efficiency by reducing the number of hypergeometric function evaluations

from two to one, however the independent variable in this transfer-time equation lacks

the attractive properties obtained previously. By developing a relationship between the

two independent variables, Battin combines the two transfer-time equations in order to

obtain the combined equation form utilized in the current work.

Battin's combined equations transfer-time equation is given by Eqs. (4a) through

(4f), where F [a, f; y; x] is the hypergeometric function. This form of the transfer-time

equation is devoid of the difficulties encountered from using the semi-major axis as the

independent variable, which renders it readily adaptable to an iterative solution method,

such as Newton's Method. The parameter A is a function of the known geometry and y

is an intermediate variable expressible in terms ofx as Eq. (4c). Furthermore, ris

essentially a function of x only since y can be given by Eq. (4c), and S, is a function ofx

only, which implies that Q = Q(x) only. Therefore Eq. (4a) is a function of the

independent variable x only. It is also important to point out that the solution obtained

for x is unique, a property not apparent by inspection. The reason for the uniqueness can

be attributed to the use of variable x andy, which can be given by alternate expressions

1 1
x = cos-a and y = cos-/ Therefore, ambiguities arising from the trigonometric
2 2

functions involving a and / are avoided.


I(t -tl)= 3Q +4A)7 (4a)
V am










c= -cos (4b)
s 2


y 1= -2 -x2) (4c)

S= y Ax (4d)


S = (1- A- 77x) (4e)
2


Q =F 3,1; 5-; S (4f)
3 2(

Maneuver Command Determination

In order to calculate the maneuver command the solution x from the Battin

algorithm must be converted to geometric properties a, a, and f. The equations utilized

S
for this purpose are given by Eqs. (2), (3), and (5), where a --is the value of the semi-
2

major axis for the minimum energy transfer between Pi and P2. From the inverse

trigonometric operations the conversion always yields the principal values ac and /0,

characterized by 0 < /0 < o a <
arising from quadrant ambiguities from inverse trigonometric operations. In order to

obtain the correct solution from the four possibilities, Prussing geometric interpretation25

is employed.

x2 am (5)
a

Pressing geometric interpretation involves transforming the geometry of Figure 10

to a rectilinear ellipse, thereby introducing geometric significance to the angles a and .

The results of this procedure provides a framework for the logical determination of the









appropriate values of a and / from the principle values oa and /0. The assignment

procedure is given below by Eqs. (6) and (7), where (t2 t1) is the transfer time, tm is the

transfer time on the minimum energy transfer, and 0 is the transfer angle.


a {2 a ,( S aor,(t2 t )>tm

= a 6, o 2 ) 8= 0 < 2 (7)
\- 0,O
Once a and / have been obtained from the above procedure the terminal velocity

vector v, at r1 (andv2 at r2 for a rendezvous) is calculated. To do so, introduce a

skewed set of unit vectors, defined in Eqs. (8a) through (8c), where i~, i = 1, 2, are unit

vectors in the direction of r,, and ui is a unit vector parallel to the chord. It can be

shown25 that in this coordinate bases v, has the elegantly simple expression given by Eq.

(9), with A and B given by Eqs. (10) and (11). Therefore, the final objective of obtaining

the maneuver command necessary to initiate the trajectory between the start and goal

configuration can be calculated by taking the vector difference of the satellite's current

velocity with v,.



r
1 (sa)
'1


u2 = 2 (8b)
r2


S= 2((8c)
c


EI = (B+ A) fc ( -A)fi










A = cot- (10)
(4a) 2


B = C cot (11)
4a 2

Numerical Example: Midcourse

To illustrate the midcourse trajectory determination and maneuver command

algorithms, a numerical example is presented. The problem is as follows: given position

vectors r = [9567 0 0] and r2 =[-37394 10783 0] (units are kilometers)

coordinatized in the Perifocal (PQW) frame (x-axis pointed in the direction of the

eccentricity vector) and a time of flight equal to four hours, determine the solution

trajectory and the maneuver command. The initial position was chosen to be the periapse

which allows for simple analytical verification of the terminal velocity vector produced

by the midcourse maneuver command algorithm.

Using algorithms based on the theory of the previous sections and developed by the

author the solution is obtained. The iterative solution for x in Lambert's Problem using

Newton's method required 19 iterations to converge with a relative error less than 108.

The terminal velocity at P] was found as v, = -8.73 xlO10 8.228 01T kilometers per

second. The CPU runtime on a dual 2.4 GHz Xeon processor workstation for the

midcourse maneuver command determination algorithm was approximately 0.15 0.05

seconds. The terminal velocity v, produces by the algorithm can be verified by checking


i (I+ e)
against the analytical expression for the velocity at periapse passage v = (1e
Performing this calculation produces the analytical terminal velocity vector as
Performing this calculation produces the analytical terminal velocity vector as









p = [0 8.228 0] km/s, which shows good agreement with the results from the

algorithm. Figure 11 depicts the transfer trajectory, obtained from a Keplerian motion

simulator using the initial conditions given by the position and terminal velocity vector

solutions.

It is important to note a caveat for the numerical example above. The position

vectors were given in the Perifocal (PQW) frame, which is an orbit based coordinate

frame. However, it is unlikely that the position information will be known in the PQW

frame, instead position information is more likely to be obtained in the Earth-Center

Inertial (ECI) frame or the Local-Vertical Local-Horizontal (LVLH) frame. The

implication of this is, unlike the example above, the initial value of the true anomaly is

unknown and cannot be determined by inspection from the position vectors. For

instance, if the position vectors in the example above were given in the ECI frame (with

Q = 10,i = 28.5, co = 45) the position vector would be r1 = [5629.4 -7694.7 792.9]'


and ri = [-14080.1 35350.3 -8166.2] km, which does not exhibit any evidence that

the true anomaly is zero at P]. Therefore, in most cases only the difference in true

anomaly is known from geometry and the elegant calculations used to verify the terminal

velocity vector here are not applicable.

Endgame Trajectory Determination

Endgame conditions for this work will be defined such that the dynamics of relative

motion between two bodies in orbit can be accurately approximated by linear orbit

theory. The limit on the applicability of linear orbit theory is dependent on both the

desired level of fidelity and the class of orbit operation, e.g., Low-Earth orbit (LEO) or

Geostationary orbit (GEO). For an in-depth analysis on the applicability of linear orbit










theory see Refs. 30 and 31. For the analysis that follows it is assumed that endgame

conditions are met.


x 104

2 Solution
STr ajectory









: p r,= =9567 Anm



-1:

1.5
-0. 1 ........ .
: =392 8160
o


x [km]


Figure 11. Transfer trajectory from midcourse trajectory determination algorithm


Chaser


Figure 12. Depiction of orbital rendezvous




The basic scenario for the development of linear orbit theory is depicted in Figure

12, where both the Target and Chaser are assumed to be in two-body motion about the


x 10









central body, denoted by M. The most familiar equations from linear orbit theory and the

ones employed in the present work are the Clohessy-Wiltshire (CW) equations,32

originally developed for the first orbital rendezvous flight demonstration programs. The

CW equations are given in state transition matrix form by Eqs. (12) (14), where x, y, z

(z out of the page) are the coordinates of the Chaser with respect to the Target in the

local frame depicted in Figure 12 and n is the orbital mean motion of the Target. It is

vitally important to understand the implicit assumption of the CW equations that

ultimately limits their applicability and fidelity. The reference (Target) orbit is assumed

to be circular. Therefore, for any scenario where the reference orbit is not circular errors

in addition to the ones from linearization are introduced, thus, further degrading the

fidelity of the model.

r() = ()r(0) (12)


r (t) =x(t) y(t) z(t) x(t) y(t) z (t) (13)

s 2
4-3cosnt 0 0 -(-cosnt) 0
n n
2 4 sin nt 3nt
6(sinnt -nt) 1 0 -(1-cosnt) 0
n n
S(t) 0 0 cost 0 0 (14)
n
3n sinnt 0 0 cost 2 sinnt 0
-6n(1-cosnt) 0 0 -2 sinnt 4cosnt-3 0
0 0 -n sinnt 0 0 cost

CW Maneuver Command Determination

The primary advantage of the CW equations is the computational simplicity of

determining the transfer velocity necessary for performing a maneuver. The objective of

the maneuver is to bring the Chaser with initial conditions 8r(0) and 8r(0) to the origin








of the CW frame (depicted in Figure 12) given a time of flight t*. Partitioning the state

transition matrix D into four 3 x 3 matrices as given by Eq. (15), the top partition leads to

the equation for the relative position r(t) = M(t)dr(0)+ N(t)d(0) Substitute

t =t*and dr(t*)= 0, giving the transfer velocity r,0 the expression shown by Eq. (16).

M(t) (t) N(t) (15)
(t ) =( (15)
( S(t) T(t)J

+ =-N(t*) M(t*)dr(O) (16)

Numerical Example: Endgame
Continuing the maneuver begun in the numerical example of the midcourse section,

the problem scenario is as follows. The robot completes the midcourse trajectory to find

the goal in a circular orbit with radius rG = 38916.7 km. The discrepancy in the positions

of the goal and robot at the end of the midcourse maneuver can be viewed as errors

incurred from unmodeled perturbations such as drag, oblateness, or solar radiation

pressure effects. The objective of this example is to use the endgame algorithm outlined

in the previous section to generate the maneuver command that will bring the Chaser to

the Target's position with a time of flight arbitrarily specified as one-quarter of period of

the Target orbit. All vector quantities are coordinatized in an R-bar V-bar frame attached

to the Target, with x in the direction of R-bar and y in the direction of V-bar.

The endgame maneuver command generated by the author's algorithm is given by

Av= [0.0001 1.4028 0] km/s with a CPU time less than a few milliseconds on the

workstation used previously. The maneuver was simulated, as depicted by Figure 13, and

the absolute value of the errors in position at the end of the maneuver were obtained as










I = 1.059x 10 9 1.206 x 109 0 which illustrates the convergence of the robot to

the goal position in the specified time.


0.05

Ini ial Endgame
Goal Position


-0.05

SEndgame
0.1 trajectory






-0,2
0 0. 02 0.3 0.4 05 0.6 0.7 0.8 0.9 1
x [kml


Figure 13. Endgame maneuver trajectory in the xy-plane (R-bar V-bar plane)



Alternative Orbit Mechanics Based Methods

Having illustrated the classical methods of orbit mechanics for trajectory

determination, we turn to a more recent algorithm33 based on an adaptation of glideslope

guidance that has been utilized for rendezvous and proximity operations for the space

shuttle and other vehicles with astronauts in the loop. To date this is the most widely

used approach from a real world implementation perspective.

The glideslope approach utilizes the linearized equations of motion Eqs. (12) (14)

but completes the maneuver with Nthruster firings. The algorithm divides the range-to-

go into N equally spaced segments along the chord connecting the initial and final









relative positions. At equally spaced time intervals an impulse is calculated using the

CW equations that drive the current position to, the next intermediate position rm. An

example of the algorithm is shown in Figure 14.


x [km]

Figure 14. Glideslope solution for 5 impulse rendezvous.



The computational simplicity of the glideslope algorithm lends itself readily to

application to ISO mission. However, this computational simplicity is sacrificed for

situations where obstacle avoidance is necessary. In order to use a glideslope approach

with obstacle avoidance, a chord must be drawn between the centers of mass of all the

spacecraft in the operating environment. Therefore, (n-1)! chords, each with Nimpulse

to be calculated are necessary. For instant for a five spacecraft mission, requires 24N









online impulse calculations! For this reason alone the glideslope approach is an

unacceptable alternative for the ISO missions considered in this dissertation.

Conclusions

The existence of two trajectory determination algorithms, one for midcourse and

one for end-game, implies that a multi-tier architecture is necessary. Therefore, not only

are two algorithms necessary but the added complexity of switching algorithms to

determine when the linear control is valid is introduced. To mitigate a muti-tier control

structure, the midcourse maneuver commands could be utilized for both midcourse and

endgame scenarios. However, the midcourse maneuver command determination is a

computationally intensive algorithm that requires an iterative solution, which does not

lend itself to on-orbit implementation due to the limited computational power that exists

for current space systems. Another alternative is to modify the linear orbit theory in

order to expand its range of applicability to midcourse trajectory determination as well.

This subject has received intense research,34-37 including investigations by the author,

with minimal achievements in expanding the range of applicability.

An additional disadvantage of classical orbit mechanics theory was obvious to

Stern and Fowler30 twenty years ago; classical orbit mechanics theory does not admit the

possibility of the incorporation of physical trajectory constraints, introduced by either

goal structure or obstacles in the path. Since the ability to dynamically incorporate

obstacles and trajectory constraints is vital for all autonomous ISO systems, and given the

additional disadvantages mentioned above, other sources of trajectory planning theory for

autonomous ISO systems must be explored. Chapter 3 will survey the robotics literature

for motion planning algorithms and will motivate and develop artificial potential function

guidance algorithms for autonomous ISO applications.














CHAPTER 3
ARTIFICIAL POTENTIAL FUNCTION POSITION GUIDANCE

In the preceding chapter it was demonstrated that trajectory determination for

autonomous in-space operation (ISO) systems cannot be easily handled by classical orbit

mechanics theory. A logical avenue to explore for trajectory determination is the robotics

literature on robot motion planning (MP). A brief outline of the robot MP problem is

discussed in the first section of this chapter with an emphasis on motion planning

approaches. Given that space-born robotics applications require motion planning subject

to properties distinct from classical robot MP problems, for instance the difference in

local operator mentioned in Chapter 1, only some of the common motion planning

approaches are feasible for in-space applications. Based on the unique needs of in-space

applications, a potential field approach to MP is perceived to be most appropriate for

autonomous ISO systems, and this chapter concludes with a presentation of classical

artificial potential field (APF) guidance algorithm for ISO applications.

Robot Motion Planning

In order to precisely define the robot MP problem, it is first necessary to define two

terms, configuration and configuration space (Cspace). A configuration of a robot is

defined as the set of independent parameters, degrees of freedom (DOF), necessary to

specify the location of every point of that robot. Cspace, a concept introduced in an

influential paper by Lozano-Perez and Welsey38 in 1979, is defined as the set of all

possible configurations of a robot, i.e., the Cspace represents all possible motions of a

robot. Essentially all MP problems are equivalent when cast in the Cspace, and can be









stated as follows. Find a connected sequence of point in the Cspace between an initial

configuration and goal configuration. Configurations that result in collision with the

environment, known as configuration obstacles,39 must be computed, and the connected

sequence of points must not contain any points associated with configuration obstacles.

Computation of the configuration obstacles is difficult39'40 and the dimension of the

Cspace is equivalent to the DOF of the system, both of which makes motion planning

challenging for robots with multiple degrees of freedom.

Classically the MP problem has been broken down into the steps below.

Concentration on the approaches in the literature for Steps 3 and 4 are the focus here for

determining an applicable approach for ISO systems. For a complete investigation on

methodologies for the steps in the MP problem see the excellent surveys in Refs. 39-43.

1. Parameterize the configuration of the robot(s)
2. Choose a representation scheme for robot(s) and the environment
3. Select a motion planning approach
4. Select a search method to find feasible path through environment
5. Optimize the solution path based on various constraints (e.g., smoothness)


Citing the landmark survey by Hwang and Ahuja,39 the most common motion

planning approaches found in the literature are; cell decomposition, mathematical

programming, skeletons, and potential field methods. Cell decomposition techniques

involve computation of the entire Cspace and all configuration obstacles in addition to

requiring large memory allocations. Therefore, given that on-orbit computational power

is limited, this approach is not feasible for ISO systems. Mathematical programming

methods represent the requirements for obstacle avoidance as a set of inequalities on the

configuration parameters. The motion planning problem is then formulated as an

optimization problem that finds a solution path between start and goal configurations









based on minimization of a scalar constraint. This motion planning approach is

essentially a nonlinear optimization problem subject to multiple inequality constraints,

which generally requires a numerical method for its solution. Again this process is

computationally intensive and therefore does not lend itself to on-orbit implementation.

Skeletons, otherwise known as roadmap or highway approaches, contract the

environment into a network of 1-D lines and the motion planning problem becomes a

graph search problem restricted to the network. Common examples of the skeleton

method are Voronoi diagrams and subgoal networks. The skeleton methods have low

memory requirements, and if an accurate representation of the environment and obstacles

is known a priori, this method is feasible for ISO applications. In fact a Voronoi diagram

approach is currently being investigated for the AERCam project.44

Potential field methods for motion planning are the simplest, computationally

speaking, of all the motion planning approaches presented here. This method creates a

scalar function called the potential that has the following properties.

* Potential is at a minimum when the robot is at the goal configuration
* Potential is high on the surface of obstacles
* Everywhere in the environment away from the obstacles, the potential slopes
toward the goal configuration


Typically the potential field method is combined with a specified local operator, such as

"go-straight" and the path taken to the goal is recorded. Potential field methods also

allow for simple consideration of non-point robots by evaluating the potential at

representative points on the robot. Additionally, no explicit Cspace computation is

necessary. It is the opinion of the author that the utilization of an artificial potential field









motion planning approach with the local operator being the dynamics of motion is the

most effective motion planning approach for ISO applications.

Artificial Potential Function Position Guidance

Artificial potential function (APF) guidance for satellite systems45-49 has appeared

in the literature recently. However, the application of real-time proximity operations of

an autonomous ISO system has yet to be investigated. In this section the robot will be

considered to be a point-mass moving in three dimensions. This restriction will be

removed for attitude considerations in Chapter 5. The objective of the APF position

guidance algorithm is to drive the robot to the origin of the state space through a series of

impulsive maneuvers.

It is instructive at this point to state the equation of motion of the dynamical system

considered in this section. Since this section is focused on the relative motion between a

robot and target in a two-body orbit, the motion of each is governed by Eq. (17), where

g (r) is a vector function describing the gravitational field and aD is the disturbing

accelerations due to perturbations such as atmospheric drag or oblateness. It will be

assumed that the perturbations are zero and the both satellites are in an inverse


gravitational field given by g(r) = r3 for the present analysis. Therefore, the relative


motion between the robot and target &r = rR r is given by the vector difference of the

governing equation of the robot and the target, as given by Eq. (18). However, these are

the equations of motion in a Newtonian reference frame. It is preferable to transform

these equations into a local vertical local horizontal (LVLH) frame attached to the target.

This is accomplished by the classical transformation given in Eq. (19) where primes









denote differentiation with respect to the local coordinate frame and o) and c are the

angular velocity and acceleration of the local frame, respectively. Thus, the equations of

motion for the relative motion of the robot and target are given by Eq. (20). Furthermore,

it is obvious that the origin of the LVLH frame 0r = 0 is indeed an equilibrium point of

the dynamical system.

r = g(r) + aD (17)

iL = R i = g(R)-g(rT) (18)


r = r" + 2o x r'+ o' x r +o) x o) x r (19)

-r" = -Pr +- rT- (2oxr'+co' xr+cox coxr) (20)
rR rT

The general theory for APF guidance is based on the 2nd method of Lyapunov5s

which can be described as follows: Consider the dynamical system x = f (x, t) with

x e 9 An equilibrium point of the system x, is such that f(xe,t) = 0. For simplicity it

will be assumed that through a change of coordinates any equilibrium point can be shifted

to the origin, and therefore, Lyapunov's theorem will be presented for equilibrium states

at the origin only.

Essentially, Lyapunov's 2nd Method is a means of assessing the stability of the

equilibrium points of the dynamical system.50 The theorem states that if there is a scalar

function V (x,t) such that Eqs. (21) (23) are satisfied, whereV V (,t is the derivative

evaluated along the solution trajectories, then the origin of the state space is a globally

attractive point.

V(x,t) > Vx#0 (21)









V(x_,t) -> as IIX 4- (22)

r t <0 VxO 0(23)

The first two conditions are satisfied by choosing V to be an appropriate positive

definite function of x (it will be assumed for now that the potential function is not an

explicit function of time). The third condition in Lyapunov's 2nd theorem is enforced by

an appropriate choice of control action.51 The total derivative of the potential function is

given by V = (VV)T x, and the control is defined by Eq. (24) where 8 is such that the

velocity after a control impulse x+ is directly opposite to the local gradient of the

potential _x, = -kVxV. This proper51 control constrains the motion of the robot after an

impulse to be tangent to the local gradient of the potential. Defining the control in this

manner ensures the derivative of the potential after a control impulse is negative definite,

as given by Eq. (25), and therefore Eq. (23) is satisfied. Thus the origin is a globally

attractive point under the control action specified.


Control V< (24)
\8, V>0

V'(x)= -k(VV) (VV) <0 (25)

A pseudo-code for the APF guidance algorithm is given below, and an example of

APF guidance for an unconstrained (no obstacles) rendezvous is given in the next section

to demonstrate the implementation of the algorithm.

Algorithm

1. Obtain the current relative position and velocity between robot and target
2. Check the switching condition V = (Vv)' x









3. If V > 0, give an impulsive velocity input such that x,+ = -kVxV
4. If V < 0, coast under the influence of the dynamics of motion
5. Sample position and velocities at desired frequencies and apply control based
on switching conditions
6. Repeat until terminal error bound is satisfied

Example: Spacecraft Unconstrained Rendezvous

Define the potential function V(x) = -x Px, where P is assumed to be a positive
2- -

definite, diagonal matrix for simplicity and x = &r = [x y z]T are the coordinates of the

relative position in the LVLH frame. This gives VxV = Px and the total derivative

S= (px) x For the rendezvous, the relative velocities of the spacecraft x must also

converge to zero. The convergence is actually guaranteed by the choice of control

presented above. Consider the norm of the velocity after an impulse given by Eq. (26),

thus _d -> 0 as Ix| -> 0 and the rendezvous is guaranteed.


_|l = k[Px]2 (26)

The numerical example considers the rendezvous between a robot in a circular orbit

with radius rR = 6705 km and a target satellite also in a circular orbit with radius

r, = 6700. The robot leads the target radius vector by a phase of 0.001 degrees. The

APF guidance parameters used were k = 1 and P = I, where I equal to the 3 x 3 identity

matrix. Figure 15 depicts the trajectory taken by the robot in the xy-plane, where x and y

are the coordinates of the LVLH frame in the orbit plane with x in the radial direction.

The sharp corners on the trajectory mark where an impulsive control was applied. Figure

16 shows the impulses in the x and y directions, respectively, applied by the APF

guidance algorithm as a function of time. In all a total Av = 27.10 m/s was necessary to











bring the robot to within a meter of the target. The duration of the maneuver was 86.2

minutes. It is important to note that the APF guidance algorithm ensures asymptotic

convergence to the origin, thus a majority of the duration of the maneuver came from

closing the final meters.









41





4 -





1 0 1 2 3 4



Figure 15. Planar rendezvous in the CW frame under APF guidance










A010
-It

5011 IWoC Iwlo] OOfWj (1 Sa.a a w5 0 w0
I






Figu 15 P t in the frm un A gu


-0 500 O 1500 M2O 25M 300 3 4000 45 SOw
.M f i S;


Figure 16. Control efforts in x and y directions for the rendezvous









Obstructions

In proximity operating conditions, instances exist where certain regions of the state

space need to be avoided; for example, solid obstructions in the flight path. Therefore,

the potential function needs to be altered in order to ensure the robot avoids these

forbidden areas. For the current work, all obstructions will be approximated by spheres

(or circles in the planar case). This is a reasonable assumption for two reasons. Firstly,

for now the objective of the guidance algorithm is to provide safe trajectories between a

start to goal location, not to generate trajectories for close proximity fly-bys, yet.

Therefore, an approximate map of objects is acceptable. Secondly, the big picture for

autonomous ISO systems is perceived to start with a rough estimate of the working

environment, with perhaps this estimated environment approximating objects initially as

spheres and updating the map through sensor data obtained from fly-bys.

The obstructions will be represented as regions of high artificial potential in the

potential field in the form of Gaussian function as in Eq. (27). The parameters

Vrk and ko define the width and skewness of the Gaussian function whose center is located

atxk, and the matrix M defines the shape (since all obstacles are assumed spherical

M = I). Representing obstacles by Gaussian functions ensures that controls remain

bounded, no singularities are introduced into the potential field, and there are no local

minima. It is instructive to note here the units of the width and skewness are [L2] which

will be important due to the fact that all simulations are run in nondimensional units for

completeness.

'k (xk)= vk exp {-r (x-k )M(x- k)} (27)









Once the APF for the obstacles has been selected, the parameters of the obstacle

APF must be determined such that the robot cannot cross into the forbidden regions. At

first glance it would appear that constraining the potential everywhere on the surface of

the obstruction to be equal to the potential at the initial conditions, stated mathematically

in Eq. (28), would be sufficient. However, this requires the width and skewness to be

function ofx, which would degrade the most attractive property of the APF approach,

simplicity. An alternative constraint such that the V(x,,,) 2 V(_x) still guarantees the

robot cannot cross the obstruction surface, and constant values of y'k and o- can be found

to satisfy this constraint.

V (x) V(X ,) V(xo) (28)

Utilizing the assumption of spherical obstructions only, a simple analytical method

for the determination of the obstruction APF parameters rk and ok can be obtained. The

potential on the surface of the obstruction is given by Eq. (29). For spherical obstruction

(s_,ur -)xk surf k) = R where Rk is the radius of the spherical obstruction.

Effectively there are two parameters to be chosen based on one constraint equation, thus

only one independent parameter. The prescription for obtaining yk is to choose a value

for o- and vary x,, over all the points on the surface of the obstruction calculating a Vrk

at each value via Eq. (30). Then looking at the resulting potential fields for the various

ryk, select the ones that do not violate V(x_ ) > V(x0o). A simulation was implemented

to investigate which values of x,,s produced Vyk such that the constraint holds, the results


of which are illustrated by Figure 17 through Figure 20.









V(x,,,) = x Px,,,f +k exp {-crk1 ( x ) (x x) (29)


Vk = exp R V(x) -x Px ,,) (30)


Figure 17 shows percent error between the potential at the initial conditions and the

potential on the surface of the obstacle. The black line represents the percent error

incurred when using the value of Vfk calculated using xs,, for which ixs is a

maximum over all points on the surface, and the red line corresponds to percent error

using the value of lVk computed from x,, such that _xJ is a minimum over all points

on the surface. It is clear from this diagram that the latter value of xsuf must be used to

compute Vrk in order for V(x,, ) > V(x,) to be satisfied. For spherical obstructions

finding xs, whose norm is a minimum over all points on the surface x* is a simple

exercise in geometry, and can be computed by Eq. (31) were x is the vector connecting

the center of the obstruction to the origin.


xsurf = 1I k (31)


Figure 18 and Figure 19 depict contour plots of the potential in the xy-plane using

Vyk calculated by Eq. (31). Figure 18 shows the level of contour at the initial conditions

as labeled. Figure 19 is a zoomed view of the previous figure in the region of the

obstacle (for this case the obstacle center was given by k= [2 0 0] ). The blue doted

line corresponds to the positions on the surface of the obstacle, and it is evident that the

potential is greater than the potential at the initial conditions at all the points on the











boundary. Figure 20 shows the percentage error between the potential at the initial


conditions and potential on the surface of the obstruction foryfk as calculated by Eq. (31).


Figure 17. Difference between IC potential and potential on surface of obstructions


catws Mop of Pdn"W Fhotwf Obapmbmn W [MO.M


X lnri


Figure 18. Contour plot of the potential function


Eirar Bthu V, Wd Rad PM Obtinrtn Pote





























X khl


Figure 19. Zoomed view of contour plot



Obstrueltan &uioe PoUtit Earr vs. Suface Poaion

12,



I 2:
I T


Y Inrl 1 1 2


Figure 20. Stem plot of obstruction surface potential error


23 24




























0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x [knl


Figure 21. Trajectory traversed by robot for the path-constrained rendezvous example



Numerical Example: Path-Constrained Rendezvous

The initial conditions for the previous example will be used again in this example.

The APF guidance algorithm parameters used were k = 0.25 and P = I3, the different

value of k was selected to enhance the illustration of the behavior of the robot around the

obstacle. The obstacle is a circle in the plane with center x = [2 0 0] and radius

R = 0.5 km. The parameter rk = 0.01 which leads to ak k = 1.0855 x 105, both given in

nondimensional units (the nondimensionalization employed the radius of the target as the

length scale factor and the mean motion of the target as the time scale factor). Figure 21

shows the trajectory traversed by the robot during the constrained rendezvous maneuver.

The duration of the maneuver was approximately 245 minutes and a total Av = 61.52 m/s

to bring the robot to within a meter of the target. Figure 22 (a) and (b) and Figure 23 (a)







43


and (b) depict the impulses in the x and y directions, respectively, applied by the APF

guidance algorithm as a function of time. The second plot in both figures is zoomed

around the times the robot encounters the obstacle.


x Io(a)



"q3


Til[7


1400 1600 1800 2000 2200 2400 2600
Tiw I[s]


Figure 22. Control action in the x-direction for the constrained rendezvous (a) entire
simulation (b) control actions in vicinity of obstacle

(0) (a)


l1.0-1 -......-------...--

0 50 100oo00
I'd '.1. 515


15oo 2000


15000


Time ls]


Figure 23. Control action in the y-direction for the constrained rendezvous (a) entire
simulation (b) control actions in vicinity of obstacle


"` ~--------~-----~--









Conclusions

In this chapter the motivation for employing artificial potential function (APF)

guidance for autonomous ISO applications was presented. Furthermore, position

guidance based on APF was developed for unconstrained rendezvous and path-

constrained rendezvous, with a numerical example presented for each. Through the

numerical examples it was illustrated that APF position guidance can be an effective tool

for autonomous proximity operations, a vital element for all ISO systems.

Even though the classical APF examples from above were shown to be effective,

they suffer from two predominant drawbacks of APF methods: local minima causing

premature termination of algorithm, and sub-optimal performance. Furthermore, the

absence of a truly general APF-based obstacle avoidance algorithm needs to be

addressed. In the next chapter, the APF guidance framework will be augmented with

novel alterations that both conquer these drawbacks and extrapolate the obstacle

avoidance algorithm to the most general dynamic case.














CHAPTER 4
ARTIFICIAL POTENTIAL FUNCTION GUIDANCE: A NEW APPROACH

The earliest implementation of APF algorithms is attributed to Khatib and

Mampey52 in 1978, who used the approach for force control applications. Since then,

APF methods have been applied to a wide variety of research areas, most predominantly

robot motion planning5355 and computer graphics.56 Recently, APF guidance has been

utilized in on-orbit applications such as autonomous rendezvous,7 position,5 and

attitude59'60 guidance for inspection, maintenance, and assembly operations; and

formation control.

Despite the proliferation of APF methods in the robotic literature and their diverse

utilization in robotics research, the two primary drawbacks of these methods, spurious

local minima and suboptimal performance, have received sparse attention. Furthermore,

in the vast body of literature on APF methods a void exists for algorithms capable of

handling a general dynamic environment. The first section of this chapter will address

the vacancy in the literature pertaining to APF based obstacle avoidance algorithms for

truly general dynamic environments. Section two of this chapter will be charged with

overcoming the two predominant shortcomings by utilizing a sum of harmonic function

primitives to compose the attractive potential function.

Dynamic Artificial Potential Function Obstacle Avoidance Algorithm

Despite the numerous applications of APF methods, most of these instances deal

with static environments (potentials). However, a few examples of published works5943

dealing with an APF approach in dynamic environments exist with the most prominent









being the work of Ge and Cui.63 Nevertheless, these works introduce prohibitive

restriction as to the nature of the motions of the target and obstacles (e.g., constant

velocities for both). Furthermore, they only consider the situation were the robot is

required to take obstacle avoidance measures for one obstacle at a time. To the author's

knowledge, a truly general APF algorithm for dynamic environments without these

restrictions has not been addressed.

The dynamic artificial potential function guidance scheme employed in this section

consists of two components: an attractive and a repulsive potential. The attractive

potential is a familiar form seen in previous studies on artificial potential field

applications. However, the repulsive potential has been augmented with a priority

weighting scheme that implements simultaneous obstacle avoidance for all obstacles

requiring such. This weighting scheme enables APF guidance to be extended to dynamic

environments, with the caveats that the robot possesses sufficient actuation to complete

the maneuver, and the refresh rate for updating environment data is sufficiently fast.

Attractive Potential

The attractive potential, Uatt, given by Eq. (1) includes both the position, r (t), and

velocity, v(t) of the chaser and position and velocity of the target, rtar (t), vt (t)

respectively. The constants m, n > 1 and a,, a are design parameters.


U,, =a lr,, (t)-r(t) m + a, IIv, (t)-v(t) (1)

Similarly to the classical APF approach, we define the virtual attractive force, Fatt,

as the sum of the gradients of the attractive potential with respect to the position and

velocity of the chaser, as given in Eq. (2). Evaluation of two gradient operations yields

the expression for the virtual attractive force shown by Eq. (3), where nRT is a unit vector









pointing from chaser (robot) to the target (goal) and n', is a unit vector in the direction

of relative velocity between chaser and target. It is clear to see that the virtual attractive

force goes to zero as the chaser simultaneously obtains the same position and velocity as

the target.

F tt = -VpUa VU,, (2)
Fa, = maCp rtr(t) -r(t) lm l1 n + na-, lcVt (t) v(t)l n, (3)


Repulsive Potential

In order to define the repulsive potential, parameters need to be introduced to

quantify the necessity for obstacle avoidance as it pertains to each obstacle in the

environment. Define the parameters given by Eqs. (4) (8) for each obstacle i = 1...n.

With these parameters in mind, we define the repulsive potential as given by Eq. (9).

Qualitatively, the potential is zero when the chaser is moving away from the obstacle, as

indicated by 'vRo (t) < 0, while the potential is nonzero for situations when the chase is

approaching an obstacle and is within that obstacles range of influence. The repulsive

potential is not defined when minimum distance between chaser and the obstacle is less

than pm, meaning a collision cannot be avoided. The total repulsive potential is obtained

by summing each repulsive potential over all the obstacles, Urp = 'Ue .


amax maximum acceleration of chaser (4)
p, = min distance between robot and obstacle = r (t -' ro (bt) -'r (5)

'Pm distance under amax 'VRo -0 -RO (6)
2amax
'PA influence range of obstacle= 'pO ('vR (t), T) (7)

VRO(t)= v(t) -'vob (tT n RO, = ...n (8)









0, if 'p, > 'p0 or 'vo (t)< 0

'UI = { r 'Pm if 0 < 'p, -'pm < and 'vR (t) > 0 (9)
SIP P. -P, Po.'0)
not defined, if 'vRO (t) > 0 and 'p, < 'Pm


Just as in the attractive force case, the virtual repulsive force is defined as the sum

of the gradients of the repulsive potential with respect to the position and velocity of the

chaser. After performing the operations and simplifying, the virtual repulsive force,

which invokes the necessary obstacle avoidance, is given by Eqs. (10) (12), where

'VRO is the magnitude of the component of the relative velocity of chaser with respect to

the ith obstacle in the direction perpendicular to the line connecting the chaser and that

obstacle.

0 ,if 'p, > 'po or 'vR (t) < 0
F rep = Frep, +'Frep2 if 0 <'p 'p < oand 'vRO (t) >0 (10)
undefined if'p, <'pmand 'vRO (t) > 0

rep r, ep VRO(t) O (11)
Samax

'Fep2 q p vRO (t) vRO "O (12)
amax Ps


Given the repulsive force definition above, the priority weighted scheme is

elucidated. Introduce the priority index for the ith obstacle ('PI), calculated as shown by

Eq. (13). Again, qualitatively the priority index is a measure of how close are the

minimum distance under action of maximum acceleration and the shortest distance

between the robot and obstacle. As any one 'PI approaches some predefined factor of

safety, FOS, (one would prefer not to tempt fate by waiting until the PI 1 and the robot









was forced to do a critical avoidance maneuver) the obstacle avoidance algorithm

switches from the simultaneous mode to a single obstacle avoidance mode for the

obstacle with critical valued PI. Note that we did not consider the case of simultaneous

obstacles with critical valued PIs. This logic guarantees a collision free trajectory

through the operational space. It is important to note at this juncture that the priority

index implicitly depends on relative velocity along the chord connecting the two centers

of mass of the robot and obstacle. This implicit dependence is crucial to the situation

with non-constant velocity targets and obstacles.

S if 'ps- p > po or 'vR < 0
'PI = p (13)
Pm' otherwise (13)



Under normal operating conditions, i.e., no 'PI> FOS, the total repulsive force felt

by the robot is calculated as the PI-weighted sum of the individual repulsive forces for

each obstacle in the environment, as shown in Eq. (14).

Frep = PI* 'Frep (14)


Total Force Interactions

The total virtual force acting on the chaser is the sum of the attractive and repulsive

forces, Fo,,, = Fa + Fep Through Newton's 2nd Law we relate this force to the applied

acceleration of the chaser, given by Eq. (15). Notice that if the applied acceleration is

greater than the maximum acceleration of the chaser, the maximum acceleration is

applied in the direction of the total virtual force. With the applied acceleration, the

position and velocity can be obtained via integration of the equation of motion for the

chaser.









Total total ma

a(t)= I (15)
Ftotal
a ota' otherwise
|| total ||


Simulation Results

To illustrate the efficacy of the algorithm, we present two simple 2D simulation

scenarios. As the first example, we demonstrate the differences generated by the priority

weighted avoidance of the new algorithm with that of Ge and Cui,63 whose work is the

baseline for the present analysis, which constrains the target and obstacles to constant

velocities. In a second simulation we present a scenario more in line with the intended

application, that being the Heterogeneous Expert Robots for On-orbit Servicing

(HEROS) architecture,64 demonstrating a planar, on-orbit rendezvous with obstacle

avoidance. This simulation showcases a situation where target and obstacles have non-

constant velocities.

Constant velocity scenario

The 2D constant velocity simulation presented below is a comparison between the

algorithm present in Ge and Cui63 and the algorithm derived in the present work. In the

former work the repulsive potential is restricted to single obstacle interactions. The

parameters (a,, a,, m, n, 7, p,) for both algorithms are equal in order to isolate the effect

of the PI-weighting scheme. The pertinent parameters of the simulation are as follows.

The target has initial position [10 10]T and constant velocity [0.1 -0.05]T. Two obstacles

are in the operational space with initial positions [5 0] and [21 11]T and constant

velocities [0 0.1]' and [-0.05 -0.065]T, respectively. The chase is of unit mass with the

maximum acceleration, Amax, also equal to unity. The chase is assigned initial position [1










1]' and velocity [0.1 0]T. A FOS of 0.9 was used for the PI-weighting algorithm. The

results of the simulation are shown in Figure 24.




20 Target
Obstl
Obst2
Tatsch
15 Ge&Cui




> 10


5 -------- -- ------------


aI

0 -------- ----------j ----------------- ----------


0 5 10 15 20 25 30
X

Figure 24. 2D Constant velocity simulation results.


Figure 24 shows the locus of points for all entities in the operational space for the

200 seconds. Most important to note is the more direct route traversed by the chase using

the P1-weighting scheme when in the influence of both obstacles. As chase approaches

the first obstacle in both situations, the differences in the algorithms is illustrated. The

chase under Ge's and Cui's algorithm makes a more prominent maneuver with respect to

the PI-weighting algorithm because the former does not consider the influence of the

second obstacle at all. As the chase encounters the second obstacle, the algorithms

converge to the same trajectory; which is to be expected considering that in the single

obstacle case, the algorithms only differ by the factor of safety. Finally we see that in the









absence of obstacles the algorithms are equivalent given that they employ identical

attractive potential/force interactions.

On-orbit rendezvous scenario

The on-orbit rendezvous simulation consists of four spacecraft in Keplerian

motion: target, chase, and two obstacles. The initial conditions for all the vehicles are

summarized in Table 1. The objective of this scenario is to drive the chase spacecraft

using the APF guidance presented above to the target spacecraft while avoiding the

obstacles.


Table 1. Orbit parameters for rendezvous simulation.
Vehicle a [km] e i f (o

Target 8000 0.1 00 00 00
Chase 9005 0.1 00 00 00
Obstacle 1 10288 0.3 00 -0.07180 00

Obstacle 2 12001 0.4 00 -0.71710 00



Figure 25 illustrates the result of this simulation again in the form of the locus of

points for all entities in the operational space. The chase spacecraft starts radially aligned

with the target at a 3.6 kilometers higher altitude. Initially the chase proceeds by

thrusting in the radial direction to lower its radial velocity (and altitude), thereby

increasing its orbital speed. However, it encounters the first obstacle which requires it to

thrust in the negative radial direction (raising the chase's altitude) to avoid a collision.

Post-avoidance, the chase again thrusts as before. The second obstacle encounter

requires a less drastic trajectory change. Once past all obstacles in the environment, the

chase finally achieves an orbit altitude lower than that of the target and thus achieves a











positive relative velocity (meaning it is catching the target without thrusting). However,

the dependence on the relative position results in small accelerations still being applied to

the chase until the rendezvous is achieved.






1000 -
S "Target
900
Obst 1
CObst 2
800
Chase
700


600


400

300-

200

100

0 -
7160 7160 7170 7180 7190 7200
I

Figure 25. Planar on-orbit rendezvous results.



Figure 26 is a graph of the norm of the applied acceleration of the chase spacecraft.

This simulation also took into account the maximum allowable acceleration that could be

achieved by the vehicle. We see that initially the chase acceleration is saturated until we

encounter the first obstacle (indicated by the dip in the graph). After passing the obstacle

we again see a saturation of the acceleration due to increasing lag between the chase and

target spacecraft. The second dip in the graph indicates the encounter with the second










obstacle. Beyond the second obstacle encounter the chase acceleration does not saturate

again but rather smoothly decays to zero as it approaches the target spacecraft.


0.


0.
Z


0.01


0 20 40 60
Time [s]

Figure 26.. Norm of applied chaser accelerations.


80 100


Optimized Harmonic Attractive Potential

Recall the two prominent flaws mentioned in the introduction: spurious local

minima and suboptimal performance. The first of these is easily mitigated by utilization

of harmonic potential functions, i.e., functions that satisfy Laplace's equation. By using

harmonic functions for construction of the artificial potential field, the resulting potential

field is guaranteed to be free of local minima. The first utilization of harmonic potential

field for motion control is generally attributed to Satoh66 in the mid-1980's. However the

work was published in Japanese and did not receive exposure until it was reprinted in









English in 1993. Masoud67 has explored both scalar and vector potential field methods

using harmonic functions. For an extensive survey on these methods we refer you to the

introduction of their work. More recently Waydo68 has utilized harmonic potential field

methods, in the vein of potential flow analogs in hydrodynamics, for motion planning on

the Cornell RoboFlag test bed.

Conquering the suboptimal penalties inherent to APF methodologies without

sacrificing its most attractive characteristic, that being computational simplicity, is a

daunting task. However we proposed a solution using a library of harmonic function

primitives (i.e., simple harmonic functions such as source, sinks, doublets in the

hydrodynamic analog) to shape the force/acceleration/velocity field around the spacecraft

such that it takes into account the system dynamics.

A high degree of similarity can be achieved by using just two primitives, a sink-

vortex pair from the hydrodynamic analog, that have been optimized for sink strength a

and vortex circulation F to mimic the true velocity field of a dynamic system using a

genetic algorithm (GA). The reference velocity field is that for single impulse orbital

intercept maneuvers (utilizing the solution of the Clohessy-Wiltshire equations). The

parameters a and F were optimized over a 50 x 50 grid of points in terms of directional

accuracy in the velocity fields. A program that generates the velocity vectors for CW

transfers (with time of flight, TOF = 7/4 in this case) at all 2500 points in the grid and the

velocity field for the sink-vortex pair was generated by specifying a value for a, F and

subsequently quantifying the error between the two resulting vector fields. The error

between the two velocity fields is quantified by the scalar error measure, E, given by Eq.

(16), where i varies over all points in the grid, and v and v' are the velocities for the









CW transfer and the sink-vortex field, respectively. In the ideal case (velocity fields are

perfectly aligned) the angles O, are all zero, and cos(, ) equal one. Therefore, in the

case of perfectly aligned vector fields, e will equal the number of points in the grid, n, in

this case 2500. Using this value of E as the reference value, REF, we define the cost

function, F, employed in the GA as a simple difference REF e i.e., the difference

between REF and the actual value of the summation of the cosines of the angle between

the velocity vectors at each point in the grid, as given by Eq. (17). The GA then

minimized the value of the cost functional F for all values of a and F in their specified

ranges. The genetic algorithm parameters used for the optimization are summarized

inTable 2.


= Zcos( ')= I (16)

F = 2500 cos(O0) (17)


The two parameters of a and F were restricted to range between 0.0001 and 5.0.

The output of the GA for eleven runs is shown in

Table 3. As is illustrated by the small standard deviations in

Table 3(highlighted in blue), both parameters converged to a unique value in their

respective ranges. The best fitness score achieved was 43.1029 with corresponding

values of a = 0.1992 and F = 0.2941 (highlighted in red in

Table 3). Figure 27 plots a comparison of the velocity fields for both the optimized

sink-vortex pair (blue) and the CW field (red). Figure 27 illustrates the fidelity in the

velocity field that can be achieved with just two harmonic function primitives.









Table 2. Genetic algorithm statistics
Parameter Name PARAMETER VALUE
Population size 25
Numberof Generations 25
Probability of Mutation 0.05
Probability of Crossover 0.9
Crossover type Uniform
Fitness Function Rosenbrock
Number of Bits 20

Table 3. Mean and standard deviations for the two parameters in the GA optimization
Run a F Cost Function Score
1 0.397 0.5309 55.2659
2 0.397 0.5309 55.2659
3 0.397 0.5309 55.2659
4 0.397 0.5309 55.2659
5 0.298 0.4451 48.8113
6 0.9852 1.3229 98.0802
7 0.1878 0.2784 43.2809
8 0.3011 0.4162 47.5251
9 0.2099 0.3171 43.5354
10 0.2838 0.3799 47.0728
11 0.1992 0.2941 43.1029
MEAN 0.3685 0.507 53.86110909
STD 0.2206 0.2881 15.52175451


Conclusions

To summarize, with the results of the previous section, the guidance algorithm

presented in this work is the combination of a fully dynamic obstacle avoidance

algorithm coupled with an attractive potential based on harmonic functions. This form of

the attractive potential is guaranteed to be local minima free. Furthermore, by using a

sum of the harmonic function primitives to shape the velocity/force/acceleration field,

with the intent to harness the underlying dynamics of the system, the suboptimal penalties

of artificial potential field methods are mitigated.

The new artificial potential function-based guidance algorithm for dynamic

environments introduced a priority-based queuing approach to obstacle avoidance called







58


the PI-weighting scheme. The PI-weighting scheme gives the ability to do multiple

obstacle avoidance simultaneously, and given its implicit dependence on the relative

velocity between the chase (robot) and the obstacles, extends the APF guidance algorithm

to general dynamic environments (non-constant velocity targets and obstacles).






*8
0.8 .






0.2
0 .. ... .. .. i ...









-0.4

-0.6
-08 : \\\ \















-0.8


-1 -0.5 0 0.5 1
X
Figure 27. Optimized sink-vortex pair vs. CW based vector field



Through a simple 2D simulation we isolated the effects of the PI-weighting scheme

and demonstrated the effectiveness of the simultaneous avoidance in situations involving

multiple obstacles. Furthermore, a second simulation was presented to demonstrate the

efficacy of the overall APF guidance algorithm in terms of the ultimate application, the

HEROS architecture. This simulation demonstrated the effectiveness of the algorithm to
0.2 ,, *- t'1.4ddVGg;^





'' ',." .1 X.--. .i Y


Figur 27. Optimized sink-vort* ex pi vs. Cs v fr" JKSield






59


handle an on-orbit rendezvous with multiple obstacles, all of which were experiencing

Keplerian motion.














CHAPTER 5
ARTIFICIAL POTENTIAL FUNCTION ATTITUDE GUIDANCE

Thus far the spacecraft have been modeled as point masses, however at some stage

we must admit the vehicles as rigid bodies (due to directionality constraints, e.g., docking

axes aligment) and account for both attitude and position dynamics of the true system.

Even though the attitude and position dynamics are highly coupled in the most general

case, there are situations where decoupled dynamics are acceptable, and we will deal with

uncoupled cases in this work, which allows independent development of an attitude

guidance design. In this vein, Chapter 5 develops a novel attitude guidance algorithm

utilizing the APF.

The motivation for employing artificial potential functions (APF) for guidance

algorithms came from the robotics literature. The literature contains volumes of position

translationall) guidance algorithms, and implementations of them, but very few examples

of algorithms for rigid body (6 DOF, position and attitude) guidance. Primarily the

scarcity of literature on the rigid body motion planning can be attributed to two factors:

the configuration space (Cspace) has dimension equal to the DOF of the robot, and

configuration obstacles, configurations that result in collisions with the environment, do

not have compact representation for high dimensional Cspace. Thus for all but the

simplest cases, motion planning is challenging.

Classically, in order to circumvent the complexities of dealing with a high

dimensional Cspace, for robots with multiple DOF the attitude, or rotational, DOF have

been approximated or constrained. The simplest solution for mitigating high









dimensionality of the Cspace is to ignore the attitude states all together, and merely grow

the robot to a sphere with radius equal to its maximum dimension. Although this does

reduce the dimensionality of the Cspace, it also reduces the capabilities of the robot; for

example, for in-space operations (ISO) systems, the ability to point at objects is crucial,

and this approximation eliminates this possibility. An alternative employed in the

literature is to hold all but one of the attitude states fixed. Again, this alleviates the issues

with dimensionality, but still significantly limits the capabilities of the robot. Therefore,

a new approach that allows for incorporation of all attitude states is necessary for ISO

systems, preferably one that avoids calculating the Cspace and configuration obstacles.

APF guidance for attitude is utilized to accomplish this task.

At this point one might interject that the two deficiencies of APF methods that were

discussed in the introduction to Chapter 4 would prohibit the use of the classical

approach. However, in the case of attitude guidance, as opposed to position guidance,

these deficiencies do not arise. The introduction of local minima which causes premature

termination of the algorithm is not an issue for the attitude case because the equations of

motion, as will be shown later, possess only two extrema in the state space (which are

identical points in the physical space) at the equilibrium points. Suboptimal performance

concerns can be dismissed by showing that the guidance law obtained from this approach

belongs to a subclass of controllers obtained from a more general Lyapunov analysis.

Most obviously an obstacle avoidance algorithm, dynamic or static, is no longer

necessary in the attitude guidance case.

In this chapter the common attitude representations and their properties are

presented in order to justify the selection of the quaternions of rotation as the attitude









representation scheme. Then, paralleling the procedure for developing the APF position

guidance algorithm in Chapter 3, APF guidance for attitude is developed.

Attitude Representation

Before proceeding with the development of the APF guidance algorithms for

attitude, an attitude representation scheme must be selected. For attitude representations,

it is commonly known that there are no globally nonsingular 3-parameter representations

of the rotation group.69,70 Therefore, we look to the 4-parameter sets in order to obtain

global nonsingularity, and in the present work the quaternions are chosen as the attitude

representation. More precisely, we are using the term quaternion as a quaternion of

rotation, i.e., the norm is constrained to unity, which is precisely the column vector of the

Euler-Rodrigues symmetric parameters.69 The primary advantage of the quaternions is

that successive rotations result in successive multiplication of 4 x 4 matrices which are

commutative.69,70 As a result of this computational simplicity, the quaternion attitude

representation has become commonplace in computer animation71 and spacecraft

contro.172'73 Furthermore, the kinematic relationship between the attitude states and the

angular velocity can be written as a linear relationship,69 as given by Eq. (33). For a

comprehensive survey on attitude representations see Shuster.69

The development of the attitude APF guidance is identical to the position APF

guidance algorithm. However, hidden in the details is a critical difference in state

variables between the two cases. For the position case, the generalized coordinates were

x and the generalized velocities were x, representing the relative position and velocities

in the LVLH frame, respectively. This yields the relationship between the derivative of

the generalized coordinates and the generalized velocities as unity, shown by Eq. (32).








For the attitude case the generalized coordinates are the elements of the quaternion i

and the generalized velocities are the components of the angular velocity of the robot .

The kinematic equation relating the coordinates to the velocities is no longer unity, as

illustrated by Eqs. (33) and (34). Although not immediately obvious at this point what

complication to the development of the guidance algorithm for attitude this presents, it

will be made apparent in the subsequent development.

dx
--=x (32)
dt


q=- -0 q (33)
-0_


= 0 3 0 -0 (34)
-C92 O1 0



Attitude Equations of Motion

The rotational dynamics of a rigid body are governed by the Euler Equations, Eq.

(35) where T is the external torque and u is the vector of control torques. For the present

work it will be assumed that the external torque is zero.

JO +(0 x JO = Z r+u (35)

The quaternion was selected to represent the attitude, which is given in terms of

the unit vector in the direction of the eigen-axis of rotation, a, and angle of rotation, (p,

in Eqs. (36) (38) The kinematic equations relating the angular velocity and the

attitude are given by Eqs. (33) and (34) above.









q =[q q4 ] (36)


q= sin *a (37)


q4= cos( ) (38)


Again since APF guidance is based on enforcing stability about the equilibrium of

the dynamical system, it is instructive to find the equilibrium solutions for the

uncontrolled (u = 0) attitude equations of motion. The equilibrium states can be found

by solving the following system of equations, derived from Eqs. (33) and (35):

-wx (J) + u = 0 (39)

q x + q4 =0 (40)

) q =0 (41)

From Eqs. (39) (41) we observe that the uncontrolled equilibrium state is defined

as oe = 0 regardless of the steady state values of q4e and q .

Continuous APF Attitude Guidance

The development of continuous attitude guidance algorithms applies to situations

where the actuation on board is from control moment gyros (CMG) clusters, reaction

wheels, or similar continuous input devices.

The control is derived utilizing the general procedure for APF methods discussed in

Chapter 3. This general procedure will be repeated here for completeness. Choose a

scalar function V(x) of the states x, such that V is positive definite for all x # x, where

x, is the equilibrium state. Furthermore, we constrain the control u such that its action









is always in the direction directly opposite the gradient of V(x). Employing this control

ensures the total derivative of V is negative definite for all x xe, thereby we are

guaranteed by Lyapunov's 2nd theorem that x > _x as t -> o.

Following the above procedure, consider the positive definite scalar potential

function V(q,a )= ( qTPq + Co'Q), where q is the vector component of the


quaternion and P and Q are positive definite and diagonal (for simplicity) matrices.

The utilization of the vector component of the quaternion only is justified by the

existence of the unity norm constraint, by which the scalar part is computable from

knowledge of the vector component. Implementing a continuous control by constraining

the attitude rates, under the action of the control (q ) to be proportional and directly

opposite to the resultant of the gradient of V with respect to q and c, this can be

thought of as a pseudo-steepest decent approach, given by

o+ = -k(VV +VVV) (42)

where V V = Pq and VmV = QQ. In order to obtain the control input Aq necessary to

achieve this condition, bring the attitude rates before control application, q to the right

hand side of Eq. (42) resulting in Eq. (43). Since the attitude rates q are non-causal, the

control must be implemented via the angular velocity. Using the kinematic equation to

relate 4_ and 0o and assuming the control input to the attitude rates is equivalent to the

control input in the angular velocity states, i.e., Aq4 = IAo, yields the control u in Eq.

(71), where G, is given by Eq. (45).









Aq = -kPq- kQm- q (43)

u = -kPq G, (44)


G, = kQ+(- q41 ) (45)
2-

It is instructive at this point to address the issue of suboptimality for APF methods.

The control defined in Eq. (45) is a special case of a more general quaternion feedback

regulator, shown in Eq. (46), developed in a seminal paper by Bong Wie,75 et al. This

larger class of quaternion feedback regulators was shown, with proper gain selection, to

provide near-eigenaxis (near-optimal) rotations with guaranteed global stability. For a

precise treatment of the general problem we refer you to Ref. 75. We will prove the

stability of the special case in the next section.

u= -(nJ+D)m-Kq (46)

Before proceeding with the stability analysis, it is instructive at this point to

evaluate the closed-loop equilibrium states now that we have defined a control law.

Returning to the equilibrium equations in Eqs.(39) -(41), we find that for u = 0 to be

satisfied at the closed-loop equilibrium state, both the angular velocity and the vector part

of the quaternion must be zero (i.e., e, = qe = 0). Therefore, the closed-loop system has

two equilibrium states as shown below; however, they represent the same state in the

physical space.


e = 0 (47)

q (48)

41 (49)









Stability Analysis

The following theorems, along with conditions set forth in the subsequent lemmas,

establish the global asymptotic stability of the equilibrium state, as defined previously.



THEOREM 1: The closed-loop system defined by Eqs.(35) and (44) is globally

asymptotically stable if M is positive definite, where M is defined by Eq. (50).



M = (50)
( (Q +1) P k2

PROOF: Consider the candidate Lyapunov function V(q, c) = ( Pq + or Jo),


where J is the inertia dyadic of the rigid body. It is important to note that the inertia

matrix is used in the place of Q due to an elegant simplification that results during the

stability proof. The expression for the total derivative of V is shown in Eq. (51).

Substituting for Jo from Eq. (35), with the external torque r = 0, and for q from the

control constraint given by Eq. (42) we obtain Eq. (52). Substituting the control given by

Eq. (44) and simplifying yields Eq. (53) (NOTE: the simplification resulting from the

use of J as the derivative matrix gain is the elimination of the cross product term in Eq.

(52).

V = qT pq + CJor j6(
-(51)
= q (-kPq-kQ)+T (- Jwo+u) (52)

= -kq P2q -kq PQ) OkPq -o Gv (53
(53)
Since V is a scalar function, we may transpose any term in the above expression

while maintaining its value. Therefore, transposing the second term in Eq. (53) and






68

combining with the third term leads to the expression in Eq. (54). Rearranging yields a

compact expression for the derivative in a quadratic form given by Eq. (55).

S= -kq P2q -T (kQP + P)q_ G(54)

k

q k (_Q + 1) p _- qP
10)] -kP2
k 2 (55)
Now, observe that the matrix in Eq. (55) is precisely -M. Therefore, if M is

shown to be positive definite, then V is negative definite and by Lyapunov's 2nd theorem

global asymptotic stability is guaranteed. Note, Gv being positive definite is a necessary

condition for the positive definiteness of M.


LEMMA 1: The derivative gain matrix, Gv, is positive definite if and only if
1
kQ, > -.
2

PROOF: The derivative gain matrix is precisely given by Eq. (56) below. Positive

definiteness requires that the principle minors and the determinant are positive. The ith

principle minor of Gv is shown in Eq. (57) with i = 1, 2,3 and i # j # k. The principle

minors are positive ifEq. (58) holds. Given that -1 < q4 < 1, Eq. (58) implies that the

1
necessary condition for the positive definiteness of Gv is kQ, > -.
2


kQ + 4 q3 q2
kQ _+-
2 2 2
G 1= kQ2 + 4
2 2 2
q2 q__ 2kQ3 +
2 2 2 (56)
(56)










PM, = kQ +4 kQk + q(5)+
: 2 2 2 + ) LI (57)


kQ, + q4>0
2 (58)
The determinant of G,, given by Eq. (59), is always positive for positive diagonal

1
elements of G,. Therefore, kQ, > is a necessary and sufficient condition for the
2

positive definiteness of G,.


3 3
det (G,) = Iv,, +4I Gv,, (59)
1 =1 y = 1 2

It is instructive to point out that, in general, that all diagonal elements kQ +4
2

could be negative and still satisfy the positivity of the principal minors as required by Eq.

(57). Since Q was assume diagonal and positive definite, this implies that k < 0. It will

be shown that k > 0 is required for M to be positive definite, and therefore, the condition

1
kQ, > will be used.
2

LEMMA 2: A block matrix S, given by Eq. (60) is positive definite if and only if

the matrices A and R = D BA B are positive definite.


S =[ A] (60)

PROOF: Start by executing block elementary column operation on S in order to

make the resulting matrix, T shown in Eq. (61), block upper triangular. From Eq. (61)

we see that the upper left block entry is exactly R Also, from its upper triangular form,

T is positive definite if and only if A and R = D- BA 1B are positive definite. Since









elementary operations, similarity transformations in general, do not alter the eigenvalues,

it is implied that if T is positive definite only if S is positive definite and vice versa.


T = [D-BA-'B B
T= A ] (61)
0 A


1
THEOREM2: For kQ, > -, M is positive definite.


PROOF: In Lemma 2 let S = M (Note: M has the necessary structure) which

gives the matrices A, B, and D from Eq. (60) as

A = kP2

B= (Q+1)P (62)
2
D=G,
Using the result of Lemma 2, if A and R = D BA 'B are positive definite then M

is positive definite. In this case A = kP2 which needs to be positive definite, which

implies that k > 0. Furthermore, since P is a diagonal matrix so is A, which implies

that the inverse of A is just the reciprocal of the diagonal elements

A' =diag (Y,' A 1A. Y This gives R the form shown in Eq. (63).


B2
D A D12 D13


R= -D12 D 2- D23 (63)
A2

-D3 -D23 A-
A 3
R has the same structure as GV, hence it follows from Lemma I that R will be


positive definite if the condition D I > 0 is satisfied. Assuming that A > 1 (i.e.,









p2 > /lk) and substituting for A, B, and D the condition for positive definiteness of R

becomes the expression in Eq. (64).


kQ + 4 k2( +1)22 > 0 (64)

Let the left hand side of the inequality in Eq. (64) be represented by

W(P,) = c a2 as shown in Figure 1, where 2c = 2kQ, +q4 and 4a = k2(Q, +1)2.

Therefore, R is positive definite for all values of ], such that W is positive. W is the

equation of a parabola, with c characterizing the W -intercept and a governing the

width. Examining the range of the two constants a and c, we find that a > 0 always,

which implies W is always opening downward, and c > 0 from kQ, >1/2 and

-1< q4 < 1. This implies that for all choices of kQ, > 1/2 a portion of W lies above the

]4 axis as can be seen in Figure 1 below. Therefore, for kQ > 1/2 we can find a range of

P4 such that R is positive definite and therefore M is positive definite. Furthermore, the

range on ], is given by
previously, meaning ], must be selected from the positive side of this range.

It is important to note that k is a free parameter in the sense that it is not explicitly

specified in the selection of the control gains. Consequently, the gains may be chosen

within their range of stability specified by lemmas and theorems and a k found after the

fact to guarantee the stability.

The restriction that 0 < I < c la along with the other constraints imposed on the

controller parameters from Theorems 1 and 2 and Lemmas 1 and 2 make selection of the

control gains Gp and GD cumbersome. To circumvent this difficulty we restrict our







control gains to a region of interest and employ an optimization algorithm, specifically a
genetic algorithm discussed in the next section, to determine the control gains G, and

GD.


W


-(c/a) 1 2


Figure 28. Inequality of Eq. (64)


+(c/a)'2


Pi


stable Pi


Stability of Dynamical System
The convergence of the vector component of the quaternion and the angular
velocity to zero for the controller derived in the present work is guaranteed from the
analysis of the previous section. One would ask at this point as to the behavior of q4 in

the dynamical system. For a general perspective on the nature of the equilibrium states
given by Eqs. (39) (41), look at the linearization of the closed loop dynamical system,







73


given by Eqs. (35) and (44), around both equilibrium states. The behavior of the

dynamical system in the neighborhood of these states is elucidated by the eigenvalues of

these linearizations. It can be shown that for q4 = +1 the linearization is stable, implying


that the spectrum of the eigenvalues, o-(A) < 0. For q4 = -1 the dynamical system is


unstable, meaning there exist A > 0. Therefore, regardless of the initial value of q4, it

will converge to q4 = +1, which depending on the application may be viewed as a

blemish on the efficiency of the controller. A plot illustrating the behavior of q4 when

the initial states are very close to desired (equilibrium) state in the physical space, but the

unstable equilibrium in the state space is shown below.




1

0.8 -

0.6







-0.2 -
-0.2






S20
-Time [s-
5 10 15 20 25
Time [s]


Figure 29. Behavior of q4









Optimization

To aid in the selection of the control gains G, and GD a genetic algorithm was

used to optimize the gains with respect to a cost function, F given by Eq. (17) that is a

function of the error in the states X [= ) ( ] and the control effort u. The matrices


Hx and HD are weight functions for the state error and control states respectively, and

for the optimization performed here both were assigned the appropriated dimensioned

identity matrix, meaning equal weighting was used for the control and state errors. Since

the objective of the control is to bring the state vector to the equilibrium state defined by

Eq. (75) (76), the error states are equivalent to states themselves

[-r T [-r Tr



F = XerrHX +-ul u (65)
2 -- 2- -

The genetic algorithm parameters used for the optimization are summarized in

Table 4.



Table 4. Genetic algorithm statistics
Parameter Name PARAMETER VALUE
Population size 25
Numberof Generations 25
Probability of Mutation 0.05
Probability of Crossover 0.9
Crossover type Uniform
Fitness Function Rosenbrock
Number of Bits 20









The fitness function evaluated the contribution of the error in the states, q, q4 and

ca and the control effort over the first 50 seconds of the simulation given by the

numerical example in the next section. The control matrices G, and GD are 3x3

diagonal matrices, and therefore six parameters were optimized. The mean and standard

deviation for these six parameters for ten runs of the genetic algorithm are summarized in

Table 5.



Table 5. Mean and standard deviation for the six parameters in the GA optimization
Parameter Mean Std. Deviation
Gpi 1.1083 0.1083
Gp2 1.0111 0.0101
Gp3 1.1703 0.1293
GD1 0.5253 0.0218
GD2 0.5527 0.0433
GD3 0.8038 0.1094
Cost Function 49.5967 0.4197


The three parameters of G, were restricted to range between 1.0 and 2.0, and those

of GD were restricted to 0.5 to 1.0. As is illustrated by the small standard deviations in

Table 5, in general all parameters converged to a unique value in their respective ranges.

Numerical Example

In this section, we use the numerical example given in Wen and Kreutz-Delgado70

to compare our controller to their classical PD controller. The objective of the numerical

example is to maneuver a spacecraft, represented by its inertia matrix, J, from an initial

orientation and angular velocity to the equilibrium state given by Eqs. (75) (76),, with

q4 = +1. The controller in Ref. 70 is defined as shown in Eq. (66), where kp = 4 and

k =8.
















CO)
0..4


0 5 10 15 20
Time (s)
Figure 30. Time response of vector part of quaternion



From Ref. 70, the spacecraft's inertia matrix is given by by Eq. (77) and the initial

orientation in axis-angle representation is a = [0.4896 0.2052 0.8480]T and 0 = 2.4648

radians. The spacecraft's initial angular velocity is = [-0.1 0.3 -0.5] The gains

for the controller developed in the present work are listed in Eqs. (68) (69).

u = -k q-k, (66)


1.0 0 0
J= 0 0.63 0 (67)
0 0 0.85










1.1083
GP = 0
0

0.5253
GD= 0
0


0
1.0111
0

0
0.5527
0


0
0
1.1703

0
0
0.8038


(68)


(69)


Figure 30 illustrates the response of the three components of the vector part of the

quaternion, q, and Figure 31 shows the response of the scalar part of the quaternion, q4.

Figure 32 and Figure 33 show the response of the three components of the angular

velocity and the control input, respectively. As demonstrated by these plots, the

controller developed in this paper is capable of performing the stated maneuver. In fact,

the maneuver was accomplished at a significant reduction in control effort. It is

important to note however, that the gains were arbitrarily selected in Ref. 70 and no

optimization was investigated.


5 10 15
Time (s)
Figure 31. Time response of scalar part of quaternion


20 25







78




0.6
--- w
1wen
...... W2wen
0 .4 -------------------- ---------------------- --------------------- ----.--------------2-n
w
W- 3wen
o W1


o
0 .2 --- ----------------------------------- -------------------
02
. . ..







-0.4 ------- ---
0

-0.4 ------------------------------------------
0




-0.8
0 5 10 15 20 25
Time (s)

Figure 32. Time response of angular velocity




The effective performance of the controller was demonstrated through comparison

with a standard PD controller by a numerical example. It is important to note that even

though the gain matrices components were in general smaller that the gains used in the

PD controller simulation, the performance achieved was superior, as is illustrated by

Figure 30 through Figure 32. Furthermore, the control effort to achieve this improved

performance was considerably less, as can be seen in Figure 33. It should be stated here

that the convergence of q4 to +1 introduces a possible inefficiency for attitude maneuvers.

If initially q4 is closer to -1 than +1, then it would be more efficient to drive the attitude

toward the equilibrium state at q4 = -1. However, our controller drives the attitude to the

equilibrium state with q4 = +1, regardless of the initial value of the scalar part of the









quaternion. One possibility to mitigate this inefficiency is to make the gain matrix G, in

Eq. (71) proportional to the signum(q4), as done in Wie, et. al.73


0 5 10 15
Time [sl

Figure 33. Time response of control effort


20 25


Impulsive APF Attitude Guidance

Impulsive attitude guidance is applicable in those situations where the actuation is

provided by thrusters as opposed to the devices used in the continuous case, which with

the growing sophistication and maturity of micro-Newton capable micro-thrusters is

becoming a popular option for attitude control systems (ACS) on board micro and nano

class satellites.






80

Paralleling the development in previous section, let the artificial potential function

for this case be V(q) = q Qq where q is the vector component of the quaterion q.

Note, it is acceptable to use the vector portion of i only due to the constraint on

quaternions of rotation, i.e., 11711 = 1. The total derivative of the potential function is


given by V = (VqV) q, and the control is defined by Eq. (71) where is such that the

rate of the vector part of the quaternion after a control impulse q is directly opposite to

the local gradient of the potential qo = -kVqV Defining the control in this manner

ensures the derivative of the potential after a control impulse is negative definite, as given

by Eq. (70), and therefore, by Lyapunov's 2nd method q is guaranteed to converge to its

equilibrium value.

V'(x)= -k(VV) (VV) < 0 (70)

o, r'<0
Control = (71)
\8, V> > 0


S _= 0 0 =-kVV (72)




At this point, the difference in the relationship between the generalized coordinates

and velocities become apparent. Since q are not causal states, the control must be applied

to the angular velocity states. Utilizing the kinematic relationship given by Eq. (33) the

control constraint can be transformed to a form in terms of the angular velocity 0, given

by Eq. (72), while maintaining the constraint on q Substituting the gradient









Vq = Qq and premultiplying both sides of Eq. (72) by the inverse of the coefficient

matrix of 0,, yields Eq. (73). Thus the control is applied to the angular velocity states

such that an impulsive angular velocity correction Ao = 0, g, that constrains the rate

of the vector component of the quaternion to be tangent to the local gradient of the

potential.


o0 =-k J Qq (73)
-q 0 -

Another difference for APF guidance development between the position and

attitude cases is the proof of the stability of the equilibrium states. The equilibrium was

the origin in the position case, with the equilibrium derived from the orbital relative

motion equations of motion. For the attitude we must return to Eqs. (39) (41) now that

the control has been defined in order to obtain the controlled (u_ 0) equilibria. Just as

before, the equilibrium equations show that for u = 0 to be satisfied at the equilibrium

solutions both the angular velocity and the vector part of the quaternion must be zero

(i.e., o)e = q = 0 ). Again there are two equilibrium states (both the same point in the

physical space) for the controlled system, restated in Eqs. (74) (76). Since no

differentiation between the equilibrium solutions is necessary in the convergence analysis

above, the states will converge to the equilibrium solution to which it is initially closest.

co =0 (74)

q = 0 (75)


q4. = +1 (76)










Numerical Example: Impulse-based APF Attitude Guidance

To illustrate the efficacy of the impulse based APF guidance algorithm for attitude,

a numerical example is considered. A satellite in orbit has an initial orientation given in


axis/angle representation as a = [0.4896 -0.2032 0.8480]T and 0 = 2.4648 radians, and


its initial angular velocity is given by = [0.5 -0.8 0.9]' radians per second. The


central principle inertia matrix representing the spacecraft is given by Eq. (77). The APF

parameters used were k = 0.1 and Q = I, where I is the 3 x 3 identity matrix. The

objective of the APF guidance algorithm is to bring the spacecraft with its initial


parameters to the equilibrium state [qT q4 'T T = 0 +1 01.


1.0 0 0
J= 0.63 0 (77)
0 0 0.85






0.7-
.- 0.6







0.1

S 0-

A.1 1

05 10 13 20 25
Time Is]


Figure 34. Components of the vector part of the quaternion vs. time




















.0.2-
-0. --




0.







-0.8
0 5 10 15 20 25
Time [s]


Figure 35. Components of the angular velocity vs. time








0.9


as
I 8 7






| I

R 0.5 6


0.4


0 5 10 s1 20 25
Time [s]


Figure 36. Scalar part of the quaternion vs. time







84


Figure 34 and Figure 35 show q and a versus time, respectively. Figure 36


illustrates the trajectory followed by q4 as a function of time. The maneuver duration was

25 seconds and a total impulsive angular velocity change Aca,,, = 1.4034 radian per


second was necessary to bring the <1 10 3. The impulsive angular velocity inputs


are illustrated by Figure 37, and it can be observed that only three impulses totaling

1.4034 radians per second were necessary for the maneuver. Solely for the purpose of

illustrating the convergence to the equilibrium point initially closer, the numerical

example was repeated with identical parameters, except the initial value of

q4 = 4.4648 radians. Figure 38 depicts the trajectory followed by the scalar part of the

quaternion in this case, and it is observed that indeed it converges to -1.


O 5 10 15 20 25


0 5 10 15 20 25







4-,
0- --,---T--------[---,


Time FR1

O". TI__,.

.2 II


Time [s]


Figure 37. Angular velocity impulses vs. time for the numerical example










An important caveat to the results presented here the relatively fast convergence

of the attitude and angular velocities are direct consequences of a small inertia matrix in

Eq. (77) and small initial angular velocities. However, the results here are for illustration

of the efficacy of the algorithms to bring the states to their equilibrium values. A more

precise analysis of the performance of these algorithms is necessary utilizing more

practical inertias and initial parameters.


-0.65

-0,7

-0.75

-0.8

-D.85

-0.9

-0.95


Time [s]


Figure 38. Scalar part of the quaternion vs. time for minus one convergence case




Conclusions

In this chapter two original attitude guidance algorithms were derived from an

artificial potential function approach. In the continuous guidance case, the algorithm was

shown to belong to a more general class of quaternion feedback regulator to eliminate

concerns about suboptimal performance inherent in APF methods, and the algorithm was









proven to be globally asymptotically stable. The controller parameters were obtained

using a genetic algorithm in order to mitigate the difficulty in gain selection and to obtain

a degree of optimization with respect to state error and control effort. The GA obtained

convergence for all six parameters in their respective ranges, as was illustrated by the

small standard deviation of each parameter. Furthermore, an impulsive guidance

approach was developed and shown to be effective for spacecraft with thruster based

ACS.

With these algorithms in place, a more complete development of APF guidance that

synthesized both position and attitude APF algorithms for rigid bodies is pertinent. These

are the topics under current investigation and will be the directions pursued in the future.

In the next chapter, we conclude this dissertation with some recommendations for future

work and our concluding remarks.














CHAPTER 6
CONCLUSION AND FUTURE WORK

Conclusion

The current direction of the space industry is toward utilization of autonomous

space robotic systems for in-space operations (ISO) and planetary surface exploration

(PSE). The state of the art outlined in Chapter 1 illustrates few achievements in real

world implementations and test bed environments for ISO space robotics. As a result

research in autonomous space robotic systems is both timely and necessitated. With this

as the stimulus we explore autonomous guidance techniques for advancing the state of

the art ISO space robotic systems.

As a first attempt the classical orbit mechanics methods were explored in Chapter

2. These methods were shown to be insufficient, primarily due to the inability to

incorporate obstacle avoidance, for a large class of ISO mission. In Chapter 3 we turned

to the robotics literature for alternative approaches and discovered artificial potential

function (APF) guidance to be a feasible solution, provided the deficiencies inherent in

these methods are overcome; the lack of dynamic obstacle avoidance, spurious local

minima, and suboptimal performance.

Chapter 4 developed the key elements that allow APF guidance to be an effective

solution for ISO missions. A novel dynamic obstacle avoidance algorithm is generated

that is capable of handling general dynamic environments. Furthermore, through the use

of a sum of harmonic function primitives we can guarantee no introduction of local

minima and mitigate the suboptimal penalties of APF methods.









At some point the point mass assumption for the spacecraft becomes unacceptable

and instead must be admitted as rigid bodies. As a result, consideration of the attitude in

introduced. In Chapter 5 the classical APF framework is utilized to develop two novel

attitude guidance algorithms, one for continuous and one for impulsive type actuation.

From characteristics of the attitude control problem and proper potential function

selection eliminates the inherent deficiencies of an APF approach. Both controller were

shown to perform effectively for their appropriate application.

Future Work

The solution method presented in this dissertation is an effective solution for

autonomous guidance of in-space operations mission, such as assembly, inspection, and

servicing. However, further development of this approach is possible in several areas, as

outlined below.

Harmonic Functions

o Utilization of more complex primitives

o Use additional primitives in sum

o Eliminate the use of offline optimization so algorithm could be used

for real time guidance

Reference Velocity Field

o Generate full nonlinear velocity field (Lambert solution field)

o Include perturbing effects in reference field

Drag

J2

Solar radiation






89


System Dynamics

o Fuse attitude and position guidance algorithms for couple

attitude/position dynamics

Research in these areas is the focus of ongoing post graduate study.