Distributed Reactive Collision Avoidance - University of Washington

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6 1.2 Literature Review Much work has been done on the collision avoidance problem to date, and a wide variety of solutions have been proposed. While each of these answers solves a piece of the puzzle, they also have limitations that keep them from solving the wide range of applications that the work in this dissertation seeks to address. However, they also form the foundation of tools and ideas on which this work has been built. An early theoretical work on collision avoidance [6] showed a method of designing avoidance controllers for a general class of vehicle models and constraints. The mathematical foundation is solid in that it presents a proof to avoid not just another vehicle, but an adversary. The method is limited because it requires finding an appropriate Lyapunov function, which is often a nontrivial task, and it only works for two-vehicle systems. The collision cone concept, introduced in [7] and subsequently used in the deconfliction literature [8, 9, 10, 11, 12], is a first order look ahead for detecting conflicts. The collision cone is a set of velocities for one vehicle that will cause it to collide with another, assuming each of their velocities are constant. While most algorithms using the collision cone define collision by the distance between two points (as though each vehicle is a disk or sphere), [7] shows how the method also works for irregularly shaped objects. An extension of the collision cone concept to accelerating vehicles was shown in [9], but the resulting conflict region is nonlinear and difficult to describe analytically, which limits its use for collision avoidance. A three-dimensional version of the original collision cone was used in [10] to avoid conflicts pairwise. Separate solutions were found using speed changes, heading changes, and flight path angle changes, but not in combination. An extension of that work to handle simultaneous speed, heading, and flight path angle changes was shown in [11]. Extensive real-world simulations were shown for two aircraft, but the algorithm is not designed for larger numbers of interacting vehicles. An application of the collision cone to three-dimensional air traffic control is presented
7 in [12], where the rates of change of the collision cones are used to prioritize the evasion of potentially adversarial vehicles. However, the protocols for handling multiple vehicles are not completely described, and no guarantees are given for safety, even in the two vehicle case. A combination of repulsive and vortex functions are used as force fields in [13], causing vehicles to form roundabout avoidance maneuvers, as tend to crop up in many of the collision avoidance methods. The algorithm provides no guarantees regarding safe separation, especially in the presence of control authority limits. Often parameters of the functions can be adjusted to give adequate results for a given situation, but it is unclear how to choose these parameters in general. A function involving more look-ahead is presented in [14], which shows good qualitative performance, but again lacks guarantees. Their simulations consistently show a significant percentage of the vehicles violating their separation bounds. A somewhat similar approach was taken by this author in [15], which avoided three dimensional conflicts by maneuvering out of the plane of the conflict. Again, simulations showed promise, but no guarantees could be found for safety. Gyroscopic and braking forces are used for collision avoidance in [16], resulting in a decentralized control law. However, safety can only be guaranteed so long as each vehicle only sees one other vehicle at a time within the detection range. Therefore no guarantees exist for systems with more than two vehicles, though simulations are shown for manyvehicle cases. A simple potential function method is employed for collision avoidance in [17], while simultaneously attaining a desired formation of the agents involved. The guarantee of collision avoidance given proves that the vehicles never attain the exact same position at the same time by using unbounded control authority. Problematically, no minimum separation distance or any concept of vehicle size is used in this method. Another force field approach that guarantees collision avoidance is presented in [18]. Unfortunately, the guarantee is based on a potential function that goes to infinity at the mini-
Page 1 and 2: Distributed Reactive Collision Avoi
Page 3 and 4: In presenting this dissertation in
Page 5 and 6: TABLE OF CONTENTS Page List of Figu
Page 7 and 8: LIST OF FIGURES Figure Number Page
Page 9 and 10: ACKNOWLEDGMENTS The author wishes t
Page 11 and 12: 1 Chapter 1 INTRODUCTION TO COLLISI
Page 13 and 14: 3 system state as time goes to infi
Page 15: 5 imum speed. The constant-speed un
Page 19 and 20: 9 hybrid system that describes each
Page 21 and 22: 11 at first to be the holy grail of
Page 23 and 24: 13 Chapter 2 PROBLEM STATEMENT The
Page 25 and 26: 15 ⎡ ⎤ ⎡ ⎤ r sˆt d ⎢ s
Page 27 and 28: 17 Chapter 3 CONFLICTS AND COLLISIO
Page 29 and 30: 19 where t is the time to closest a
Page 31 and 32: 21 Enter Desired Control Deconflict
Page 33 and 34: 23 must know a bound on the maximum
Page 35 and 36: 25 Figure 4.2: Example optimization
Page 37 and 38: 27 cone is enlarged from (3.9) to (
Page 39 and 40: 29 Theorem 2. For vehicle i of an n
Page 41 and 42: 31 Figure 4.4: Example optimization
Page 43 and 44: 33 Theorem 3. For vehicle i of an n
Page 45 and 46: 35 fashion. The goal then is to hav
Page 47 and 48: 37 v ĩ v e p t,ijˆt i v j p n,ij
Page 49 and 50: 39 Figure 4.8: Example of the contr
Page 51 and 52: 41 Because of the symmetry of the p
Page 53 and 54: 43 also ensures that there is a nei
Page 55 and 56: 45 to adjust their trajectories to
Page 57 and 58: 47 controllers. The goal of these c
Page 59 and 60: 49 of a complex multiplicative unce
Page 61 and 62: 51 Proof. A multiplicative uncertai
Page 63 and 64: 53 u 1 u 2 θ γ v 2,max v 1,max Fi
Page 65 and 66: 55 speed (m/s) (a) 0.5 0.4 0.3 0.2
Page 67 and 68:
57 un (m/s 2 ) 0.2 0 (a) -0.2 0 5 1
Page 69 and 70:
59 10 8 ‖˜r‖ − dsep (m) 6 4
Page 71 and 72:
61 ‖˜r‖ − dsep (m) 4 2 0 (a)
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63 Table 6.1: Comparison of computa
Page 75 and 76:
65 be stabilized with any of a vari
Page 77 and 78:
67 the heuristic performance will b
Page 79 and 80:
69 Chapter 7 CONCLUSION The primary
Page 81 and 82:
71 system of projecting everything
Page 83 and 84:
73 [11] G. Fasano, D. Accardo, and
Page 85 and 86:
75 [34] R. Skjetne, S. Moi, and T.
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