Collision Avoidance Systems: Computing Controllers which Prevent Collisions

1. Collision Avoidance Systems:Computing Controllers which Prevent Collisions By Adam Cataldo Advisor: Edward Lee Committee: Shankar Sastry, Pravin Varaiya, Karl Hedrick PhD Qualifying Exam UC Berkeley December 6, 2004

2. Talk Outline • Motivation and Problem Statement • Collision Avoidance Background • Potential Field Methods • Reachability-Based Methods • Research Thrusts • Continuous-Time Methods • Discrete-Time Methods

3. Motivation—Soft Walls • Enforce no-fly zones using on-board avionics • A collision occurs if the aircraft enters a no-fly zone

4. The Research Question • For what systems can I compute a collision avoidance controller? • Correct by construction • Analytic Control Law, Safe Initial States System Model, Collision Set

5. Collision Avoidance Problem(Continuous Time)

6. Collision Avoidance Problem(Discrete Time)

7. Potential Field Methods(Rimon & Koditschek, Khatib) • Provide analytic solutions, derived from a virtual potential field • No disturbance is allowed • Dynamics must be holonomic Oussama Khatib: Real-time Obstacle Avoidance for Manipulators and Mobile Robots

8. Reachability-Based Avoidance(Mitchell, Tomlin) compact

9. Hamilton Jacobi Equation(Mitchell, Tomlin)

10. Computing Safe Control laws(Mitchell, Tomlin) offline online

11. Applied to Soft Walls(Master’s Report) • Works for a many systems • Storage requirements may be prohibitive • 40 Mb for the Soft Walls example • Cannot analyze qualitative system behavior under numerical control law • switching surfaces, equilibrium points, etc.

12. Analytic Computation:Soft Walls Example

13. Change of Variables

14. Lyapunov Function

15. A Sufficient Condition(Leitmann)

16. A Sufficient Condition(Leitmann) • Find a Lyapunov function over an open set encircling the collision set which ensures against collisions

17. One Possible Extension

18. One Possible Extension

19. Open Questions • When can we find our control law analytically? • When can we find the corresponding Lyapunov function analytically? • Can we build up complex models from simple ones?

20. Bisimilarity and Collision Avoidance • When is the system bisimilar to an finite-state transition system (FTS)? • If the system is bisimilar to an FTS, can I compute a control law from a controller on the FTS? unsafe state disable this transition

21. Example: Controllable Linear Systems (Tabuada, Pappas) semilinear sets on W LTL formula

22. The Result(Tabuada, Pappas) • There exists a bisimilar FTS for observations given as semilinear subsets of W • A feedback strategy k which enforces the LTL constraint exists iff a controller for the FTS which enforces the constraint exists

23. Bounded Control Inputs • If we want to extend this for disturbances, we will need to be able to bound the control inputs • Adding states won’t work; we may lose controllability

24. Research Questions • When we have bounds on the control input, when can we find a bisimilar FTS? • For systems with disturbances, when can we find a bisimilar FTS? • For nonlinear systems with disturbances, when can we find a bisimilar FTS?

25. Where is this Going? • Build a toolkit of collision avoidance methods • These methods must give correct by construct control strategies • We should be able to analyze the control strategies

26. Conclusions • I plan to develop new collision avoidance methods • Many approaches to collision avoidance have been developed, but methods which produce analytic control laws have limited scope • In the end, we would like to automate controller design for problems such as Soft Walls

27. Acknowledgements • Aaron Ames • Alex Kurzhanski • Xiaojun Liu • Eleftherios Matsikoudis • Jonathan Sprinkle • Haiyang Zheng • Janie Zhou

28. Additional Slides

29. Global Existence and Uniqueness(Sontag) • Given the initial value problem • There exists a unique global solution if • f is measurable in t for fixed x(t) • f is Lipschitz continuous in x(t) for fixed t • |f| bounded by a locally integrable function in t for fixed x

30. Potential Functions(Rimon & Koditschek)

31. Holonomic Constraints(Murray, Li, Sastry) • Given k particles, a holonomic constraint is an equation • For m constraints, dynamics depend on n=3k-m parameters • Obtain dynamics through Lagrange's equation

32. Information Patterns(Mitchell, Tomlin) • In computing the unsafe set, we assume the disturbance player knows all past and current control values (and the initial state) • The control player knows nothing (except the initial state) • This is conservative • In computing a control law, we assume the control player will at least know the current state

33. Relation to Isaacs Equation • Isaacs Equation: • W(t,p) gives the optimal cost at time t (terminal value only)

34. Relation to Isaacs Equation • Isaacs Equation: • The min with 0 term gives the minimum cost over [t,0]

35. Viscosity Solutions(Crandall, Evans, Lions)

36. Convergence of V • At each p, V can only decrease as t decreases • If g bounded below, then V converges as • It may be the case that all values are negative, that is, no safe states

37. Applying Optimal Control:Soft Walls Example unsafe safe

38. Lyapunov-Like Condition(Leitmann) • Given a C1 Lyapunov function V:S, A is avoidable under control law k if • Note that this can be generalized when V is piecewise C1

39. Lyapunov-Like Condition(Leitmann) • Let {Yi} be a countable partition of S, and let {Wi} be a collection of open supersets of {Yi}, that is, WiYi

40. Lyapunov-Like Condition(Leitmann) • Given a continuous Lyapunov function V:S, A is avoidable under control k if

41. Transition System

42. Bisimulation

43. Linear Temporal Logic (LTL) • Given a set P of predicates, the following are LTL formula:

44. Semilinear Sets • The complement, finite intersection, finite union, or of semilinear sets is a semilinear set • The following are semilinear sets

45. Computing Safe Control Laws(Tabuada, Pappas) LTL Formula Buchi Automaton Hybrid, Discrete-Time State-Feedback Control Law Finite-State Supervisor Discrete-Time System Finite Transition System