Distributed Reactive Collision Avoidance - University of Washington

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8 mum vehicle separation. Effectively, for the guarantee to apply, the vehicle must be capable of infinite control authority, which is an unreasonable assumption for any real vehicle. A realistic situation in which force-field collision avoidance is necessary is given in [19]. The idea is to use electrostatic charges on spacecraft to create enough repulsion force to avoid collisions in high orbit and deep space. This situation results in a different kind of underactuated system, but one for which collision avoidance is guaranteed, even in the case of control saturation, given restrictions on the initial conditions. These results use a Lyapunov function and only apply to the two-vehicle case. The work in [20] uses a navigation function to combine the tasks of collision avoidance and waypoint navigation into a single gradient-following routine. The navigation function is useful in its ability to guarantee liveness in a static environment. Unfortunately, adequate separation is not proven because the dynamics of the vehicle make it unable to follow the gradient in all cases. A vortex is added to the function to better the heuristic performance, but its effect on safety is unclear. A collision avoidance approach for a three-dimensional unicycle model that uses dipole navigation functions to avoid static obstacles while maneuvering to a goal is presented in [21]. While the dipole navigation functions provide smooth controls for the vehicle, still no way is given to choose the parameters such that particular rate limits are observed. These control limits can therefore cause collisions in certain situations. A high-performance force-field method is presented in [22], which uses a concept similar to an abbreviated collision cone to take into account how much space an avoidance maneuver will need based on the relative velocity between the vehicles. Simulations show promising results for large numbers of vehicles, and the computation time is kept low by the decentralized nature of the algorithm. While this algorithm does guarantee safety for a homogeneous system of two or three vehicles with acceleration constraints, its performance in larger systems is heuristic, and it does not allow unicycle-type dynamics in its vehicle model. The optimization scheme presented in [23] is performed over the reachable set of a
9 hybrid system that describes each aircraft involved. Safety is guaranteed, but only in the two-vehicle case. An n-vehicle optimization approach is presented in [8] that uses a relaxation approach to turn the collision avoidance problem into a semidefinite program, solved with convex optimization. This achievement is impressive given the highly nonconvex nature of collision avoidance. Unfortunately, the solution must be centralized, and the algorithm is randomized, which means that to get an optimal solution, the optimization must be run several times. While convex optimization is computationally efficient compared to general nonlinear optimization, it is still less scalable than prescribed maneuver or force field approaches. A method showing better computation time is presented in [24], which uses mixed integer programming. This method allows for either speed changes or heading changes, but not both concurrently. It is still a centralized solution, and like many of the other optimization approaches, no mention is made of restrictions on the initial conditions to ensure that a feasible solution exists. Another convex relaxation approach to optimizing collision avoidance is shown in [25] for an n-spacecraft system using 3D double integrator dynamics. The algorithm works by computing an optimal set of new velocities each time any vehicles get too close together. These new velocities cause the vehicles to bounce away from each other, and in this way collision avoidance is guaranteed for the system as is liveness of the solution. However, in addition to the optimization being centralized, the control is designed to attain the new velocity in a single time step, meaning the magnitude of the control force is unbounded as the time step goes to zero. Alternatively, one could choose the time step to enforce a control limit, but given the discrete time nature of the constraints, motion during the time step would become significant and could cause collisions. An approach to distributing an optimization scheme across the vehicles involved is presented in [26]. This approach is much broader than collision avoidance, as the optimization occurs across a whole host of constraints, from visiting targets to avoiding danger zones. The authors use market-based task trading and evolutionary optimization to find
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 and 16: 5 imum speed. The constant-speed un
Page 17: 7 in [12], where the rates of chang
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)
Page 73 and 74:
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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Distributed Reactive Collision Avoidance - University of Washington

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