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Nora Ayanian

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Combined continuous and discrete: Decomposition of state space ... Objective: steer the state of an affine system to a specific facet. Focus is on simplices ... – PowerPoint PPT presentation

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Title: Nora Ayanian


1
Controller Synthesis in Complex Environments
  • Nora Ayanian
  • March 20, 2006

2
Introduction
  • Many different approaches to robot motion
    planning and control
  • Continuous Navigation function
  • Configuration space must be a generalized sphere
    world
  • Any vehicle dynamics
  • Combined continuous and discrete Decomposition
    of state space
  • Can handle more complex configuration space
  • Difficulty with complex dynamics

3
Continuous Method
  • Rimon and Koditschek 1 present a method to
    guide a bounded torque robot to a goal
    configuration from almost any initial
    configuration in an environment that is
  • Completely known
  • Static
  • Deformable to a sphere world
  • Admits a navigation function
  • Create an artificial potential field that solves
    the three separate steps of robot navigation
  • Path planning
  • Trajectory planning
  • Control

1 E. Rimon and D.E. Koditschek, Exact Robot
Navigation Using Artificial Potential Functions,
IEEE Transactions on Robotics and Automation,
vol. 8, no. 5, pp. 501-518, 1992.
4
Continuous Method
  • Let V be a map
  • With a unique minimum at the goal configuration,
    qd
  • That is uniformly maximal over the boundary of
    the free space, F
  • V determines a feedback control law of the form
  • The robot copies the qualitative behavior of Vs
    gradient 2

5
Navigation Function Method
  • Star shaped sets
  • Star shaped sets contain a distinguished center
    point from which all rays cross the boundary of
    the set only once.
  • Map the star onto a disk diffeomorphically
    translated scaling map
  • Scales each ray starting at qi by ni, then
    translates along pi

6
Combined Continuous and Discrete Method
  • Habets van Schuppen 7 decompose the state
    space into polytopes
  • Each polytope is a different discrete mode of the
    system
  • Objective steer the state of an affine system to
    a specific facet
  • Focus is on simplices
  • Points contained in a simplex are described by a
    unique linear combination of the vertices

7 L.C.G.J.M. Habets and J.H. van Schuppen, A
Control Problem for Affine Dynamical Systems on a
Full-Dimensional Polytope, Automatica, no. 40,
pp. 2135, 2004.
7
Combined Method Problem Definition
  • Consider the affine system
    on PN
  • For any initial state x0 ? PN, find a time
    instant T0 0 and an input function u 0,T0 ?
    U, such that
  • ?t ? 0,T0 x(t) ? PN ,
  • x(T0) ? Fj, and T0 is the smallest time-instant
    in the interval 0,8) for which the state
    reaches the exit facet Fj
  • , i.e. the velocity vector at
    the point x(T0) ? Fj has a positive component in
    the direction of nj. This implies that in the
    point x(T0), the velocity vector points out of
    the polytope PN.

8
Combined Method Necessary Conditions
  • If the control problem is solvable by a
    continuous state feedback f, then there exist
    inputs u1,,uM ? U such that
  • ? j ? V1
  • n1T(Avj Buj a) gt 0,
  • ? i ? Wj \ 1 niT(Avj Buj a) 0.
  • ? j ? 1,,M \ V1
  • ? i ? Wj n1T(Avj Buj a) 0,

Illustration of Polyhedral Cones
Habets van Schuppen,2004
9
Applying the Combined Method
  • A 1-dimensional integrator problem

10
Thank You
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