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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

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Collision avoidance systems computing controllers which prevent collisions l.jpg

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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Collision Avoidance Problem(Continuous Time)



Potential field methods rimon koditschek khatib l.jpg
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 l.jpg
Reachability-Based Avoidance(Mitchell, Tomlin)

compact


Hamilton jacobi equation mitchell tomlin l.jpg
Hamilton Jacobi Equation(Mitchell, Tomlin)


Computing safe control laws mitchell tomlin l.jpg
Computing Safe Control laws(Mitchell, Tomlin)

offline

online


Applied to soft walls master s report l.jpg
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 l.jpg
Analytic Computation:Soft Walls Example





A sufficient condition leitmann16 l.jpg
A Sufficient Condition(Leitmann)

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




Open questions l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Acknowledgements

  • Aaron Ames

  • Alex Kurzhanski

  • Xiaojun Liu

  • Eleftherios Matsikoudis

  • Jonathan Sprinkle

  • Haiyang Zheng

  • Janie Zhou



Global existence and uniqueness sontag l.jpg
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 l.jpg
Potential Functions(Rimon & Koditschek)


Holonomic constraints murray li sastry l.jpg
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 l.jpg
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 l.jpg
Relation to Isaacs Equation

  • Isaacs Equation:

  • W(t,p) gives the optimal cost at time t

    (terminal value only)


Relation to isaacs equation34 l.jpg
Relation to Isaacs Equation

  • Isaacs Equation:

  • The min with 0 term gives the minimum cost over [t,0]


Viscosity solutions crandall evans lions l.jpg
Viscosity Solutions(Crandall, Evans, Lions)


Convergence of v l.jpg
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


Applying optimal control soft walls example l.jpg
Applying Optimal Control:Soft Walls Example

unsafe

safe


Lyapunov like condition leitmann l.jpg
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 leitmann39 l.jpg
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


Lyapunov like condition leitmann40 l.jpg
Lyapunov-Like Condition(Leitmann)

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




Linear temporal logic ltl l.jpg
Linear Temporal Logic (LTL)

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


Semilinear sets l.jpg
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 l.jpg
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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