Collision Avoidance Systems: Computing Controllers which Prevent Collisions

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Collision Avoidance SystemsComputing
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
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Talk Outline

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

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MotivationSoft 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
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Collision Avoidance Problem(Continuous Time)
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Collision Avoidance Problem(Discrete Time)
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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
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Reachability-Based Avoidance(Mitchell, Tomlin)
compact
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Hamilton Jacobi Equation(Mitchell, Tomlin)
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Computing Safe Control laws(Mitchell, Tomlin)
offline
online
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Applied to Soft Walls(Masters 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.

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Analytic ComputationSoft Walls Example
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Change of Variables
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Lyapunov Function
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A Sufficient Condition(Leitmann)
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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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One Possible Extension
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One Possible Extension
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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
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Example Controllable Linear Systems (Tabuada,
Pappas)
semilinear sets on W
LTL formula
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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 wont 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

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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

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Acknowledgements

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

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Additional Slides
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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

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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 n3k-m
  parameters
• Obtain dynamics through Lagrange's equation

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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 ControlSoft Walls Example
safe
unsafe
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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

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Lyapunov-Like Condition(Leitmann)

• Given a continuous Lyapunov function VS??, A is
  avoidable under control k if

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Transition System
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Bisimulation
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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
Hybrid, Discrete-Time State-Feedback Control Law
Finite-State Supervisor
Finite Transition System
Discrete-Time System