Distributed Reactive Collision Avoidance - University of Washington

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24 given certain bounds on the separation. The separation bounds are necessary because a wide class of initial conditions exist for which collision avoidance is impossible, usually due to a lack of sufficient control authority. While the deconfliction maintenance phase can operate with arbitrarily small control authority, the deconfliction maneuver requires control authority to get out of conflict before a collision occurs. Therefore it is important when designing a maneuver that the vehicle will have sufficient control authority to complete it. The problem with using speed changes in the maneuver is that all vehicles have a maximum speed, so their control authority to accelerate forward goes to zero as this maximum speed is approached. Therefore, turning as a means of deconfliction is applicable to a wider range of vehicles, since there is almost never a maximum heading angle. In addition, most vehicles have more acceleration available in turning than in speeding up or slowing down, since the latter requires work and former does not. Vertical acceleration can be as easy as turning for a buoyant vehicle, or require a significant change in potential energy for a vehicle such as an airplane. Therefore it may be prudent to restrict some vehicles to a 2D deconfliction maneuver in order to ensure they are capable of completing it. Two deconfliction maneuvers will be presented here, first a constant-speed maneuver that nearly any vehicle is capable of, and second a variable speed maneuver that has better performance at the cost of computational complexity. Both of these maneuvers are 2D, but deconfliction maintenance works for 3D as well. In order to use these maneuvers in a 3D application one would have to project the vehicle’s positions and velocities onto the plane, perform the maneuver, then add back in the original vertical component to the resulting velocity. The guarantees of attaining conflict-freedom will still apply (though the solutions will be more conservative) and once deconfliction maintenance takes over, the 3D nature of the system will be accounted for in a less conservative way.
25 Figure 4.2: Example optimization of a constant-speed maneuver. The black vector is v i , the red vector is v ′ i, the circle is the allowable space of solutions and the blue regions are the collision cones. 4.1.1 Constant-Speed Maneuver The optimization vehicle i performs is to find a new velocity vector, v i, ′ which is free of conflict with the other vehicles in D and minimizes ‖∆v i ‖, where ∆v i = v i ′ − v i . While this minimization is nonconvex in general (see Fig. 4.2), it can be solved relatively quickly because only a small number of possible points must be checked. The algorithm for solving this works as follows. The allowable space for v i ′ is a circle, centered at the origin, of radius ‖v i ‖. The collision cones split this space into several disjoint regions, each of which is an arc of the circle. The optimal solution to each region must lie on one of its two ends, unless there are no conflicts, in which case v i ′ = v i . The only possible end points are points where a collision cone intersects the allowable circle. Therefore the first step of the optimization is to calculate each of the intersections for each collision cone. This is done by first computing the unit-vector ĉ for each side, representing the direction of the edge of the collision cone (see
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 and 18: 7 in [12], where the rates of chang
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: 23 must know a bound on the maximum
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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