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

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

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

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

Cataldo 5 Collision Avoidance Problem (Continuous Time)

Cataldo 6 Collision Avoidance Problem (Discrete Time)

Cataldo 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

Reachability-Based Avoidance (Mitchell, Tomlin) compact

Cataldo 9 Hamilton Jacobi Equation (Mitchell, Tomlin)

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

Cataldo 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.

Cataldo 12 Analytic Computation: Soft Walls Example

Cataldo 13 Change of Variables

Cataldo 14 Lyapunov Function

Cataldo 15 A Sufficient Condition (Leitmann)

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

One Possible Extension

Cataldo 18 One Possible Extension

Cataldo 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?

Cataldo 20 Bisimilarity and Collision Avoidance unsafe state disable this transition 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?

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

Cataldo 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

Cataldo 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

Cataldo 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?

Cataldo 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

Cataldo 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

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

Cataldo 28 Additional Slides

Cataldo 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

Cataldo 30 Potential Functions (Rimon & Koditschek)

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

Cataldo 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

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

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

Viscosity Solutions (Crandall, Evans, Lions)

Cataldo 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

Cataldo 37 Applying Optimal Control: Soft Walls Example safeunsafe

Cataldo 38 Lyapunov-Like Condition (Leitmann) Given a C 1 Lyapunov function V:  S , A is avoidable under control law k if Note that this can be generalized when V is piecewise C 1

Cataldo 39 Lyapunov-Like Condition (Leitmann) Let {Y i } be a countable partition of  S, and let {W i } be a collection of open supersets of {Y i }, that is, W i  Y i

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

Cataldo 41 Transition System

Cataldo 42 Bisimulation

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

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

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