Distributed Reactive Collision Avoidance - University of Washington

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2 Figure 1.1: A snapshot of the US airspace showing the magnitude of the air traffic control task (photo courtesy of NASA). automated algorithms to decrease the workload of human operators. Collision avoidance is a difficult and interesting problem in the study of control theory because the goal is fundamentally different from that of a classical control system. The vast majority of classical control problems (and their current nonlinear counterparts) demand the system attain a particular point in the state space (be it stationary or moving). Most of the methods for this type of task use asymptotic analysis, whereby the system can be said to approach the goal regardless of the starting position or the route taken. Collision avoidance is completely different in that there is no inherent end goal. The goal instead is to not enter particular regions of the state space at any time. Collision avoidance therefore has the effect of making the allowable set in the state space nonconvex, and in the case of avoiding moving vehicles, the set is also time varying. Adding these restrictions to the state space makes otherwise simple problems extremely difficult. One of the primary reasons is that transient response is difficult to completely characterize using classical control, yet it is exactly these transients that cause most collisions. Now, instead of only considering the
3 system state as time goes to infinity, one has to consider the system state at all times. When one factors in the highly nonlinear nature of the nonholonomic dynamics that govern most real vehicles, the problem truly balloons in complexity. Control authority limitations become far more important than in most classical control problems because control saturation primarily affects transient behavior. When two vehicles are on a collision course, it is exactly that transient behavior that determines their ability to avoid each other. If saturation is not accounted for explicitly, then collision avoidance becomes a trivial problem where a vehicle simply applies infinite control in the direction away from a collision. Of course this concept is unrealistic in practice, but nonetheless, many papers have been written on collision avoidance which rely on exactly this principle for their proofs. The typical goal for designing a nonlinear control algorithm is to prove global asymptotic stability. Unfortunately, guaranteed global collision avoidance does not exist because all real vehicles have inertia and control authority limits, which place fundamental limits on their safe set of initial conditions. Two vehicles heading towards each other can always be placed close enough that nothing can be done to avoid a collision. Therefore no approach can possibly guarantee collision avoidance for arbitrary initial conditions. Determining the exact set of initial conditions for which avoidance is impossible becomes extremely complex as the number of vehicles increases, so generally the best that can be done is to define a bound or test, which if passed, yields a conservative guarantee. 1.1 Definitions of Terms A wide range of methods have attempted to solve the collision avoidance problem, and the authors of a survey paper [5] split those methods into three categories: prescribed, optimized, and force field. In prescribed maneuver approaches, all vehicles follow a set protocol, not unlike the rules of the road. This approach tends to yield a discrete event controller, which when combined with the vehicle’s continuous dynamics, forms a hybrid system. Optimization methods attempt to find the best route for all the vehicles to take to avoid each other while minimizing a cost function. Generally, these methods use a look-
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: 1 Chapter 1 INTRODUCTION TO COLLISI
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 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
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53 u 1 u 2 θ γ v 2,max v 1,max Fi
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55 speed (m/s) (a) 0.5 0.4 0.3 0.2
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57 un (m/s 2 ) 0.2 0 (a) -0.2 0 5 1
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59 10 8 ‖˜r‖ − dsep (m) 6 4
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61 ‖˜r‖ − dsep (m) 4 2 0 (a)
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63 Table 6.1: Comparison of computa
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65 be stabilized with any of a vari
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67 the heuristic performance will b
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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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