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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

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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)


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)



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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

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A Sufficient Condition(Leitmann)

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

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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?

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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

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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

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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?

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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

  • 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

  • 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)

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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

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Relation to Isaacs Equation

  • Isaacs Equation:

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

    (terminal value only)

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Relation to Isaacs Equation

  • Isaacs Equation:

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

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Viscosity Solutions(Crandall, Evans, Lions)

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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

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Applying Optimal Control:Soft Walls Example



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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

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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:

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Semilinear Sets

  • The complement, finite intersection, finite union, or of semilinear sets is a semilinear set

  • The following are semilinear sets

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Computing Safe Control Laws(Tabuada, Pappas)

LTL Formula

Buchi Automaton




Control Law





Finite Transition