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Collision Avoidance Systems: Computing Controllers which Prevent CollisionsPowerPoint Presentation

Collision Avoidance Systems: Computing Controllers which Prevent Collisions

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### 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

Talk Outline

- Motivation and Problem Statement
- Collision Avoidance Background
- Potential Field Methods
- Reachability-Based Methods

- Research Thrusts
- Continuous-Time Methods
- Discrete-Time Methods

Motivation—Soft Walls

- Enforce no-fly zones using on-board avionics
- A collision occurs if the aircraft enters a no-fly zone

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

Collision Avoidance Problem(Continuous Time)

Collision Avoidance Problem(Discrete Time)

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

Hamilton Jacobi Equation(Mitchell, Tomlin)

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.

Analytic Computation:Soft Walls Example

A Sufficient Condition(Leitmann)

A Sufficient Condition(Leitmann)

- Find a Lyapunov function over an open set encircling the collision set which ensures against collisions

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?

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

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

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

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?

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

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

Acknowledgements

- Aaron Ames
- Alex Kurzhanski
- Xiaojun Liu
- Eleftherios Matsikoudis
- Jonathan Sprinkle
- Haiyang Zheng
- Janie Zhou

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

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

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

Relation to Isaacs Equation

- Isaacs Equation:
- W(t,p) gives the optimal cost at time t
(terminal value only)

Relation to Isaacs Equation

- Isaacs Equation:
- The min with 0 term gives the minimum cost over [t,0]

Viscosity Solutions(Crandall, Evans, Lions)

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

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

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, WiYi

Lyapunov-Like Condition(Leitmann)

- Given a continuous Lyapunov function V:S, A is avoidable under control k if

Linear Temporal Logic (LTL)

- Given a set P of predicates, the following are LTL formula:

Semilinear Sets

- The complement, finite intersection, finite union, or of semilinear sets is a semilinear set
- The following are semilinear sets

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

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