Title: Nora Ayanian
1Controller Synthesis in Complex Environments
- Nora Ayanian
- March 20, 2006
2Introduction
- 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
3Continuous 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.
4Continuous 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
5Navigation 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
6Combined 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.
7Combined 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.
8Combined 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
9Applying the Combined Method
- A 1-dimensional integrator problem
10Thank You