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

2. 2 GRASP 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. 3 GRASP 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. 4 GRASP Continuous Method Let V be a map  With a unique minimum at the goal configuration, q d  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 V ’s gradient [2]

5. 5 GRASP 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 q i by i, then translates along p i *[1] Rimon & Koditschek pipi qiqi D S

6. 6 GRASP 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. 21–35, 2004.

7. 7 GRASP Combined Method: Problem Definition Consider the affine system on P N For any initial state x 0  P N, find a time instant T 0 ≥ 0 and an input function u: [0,T 0 ]  U, such that   t  [0, T 0 ]: x(t)  P N,  x(T 0 )  F j, and T 0 is the smallest time-instant in the interval [0,∞) for which the state reaches the exit facet F j , i.e. the velocity vector at the point x(T 0 )  F j has a positive component in the direction of n j. This implies that in the point x(T 0 ), the velocity vector points out of the polytope P N.

8. 8 GRASP Combined Method: Necessary Conditions If the control problem is solvable by a continuous state feedback f, then there exist inputs u 1, …,u M  U such that   j  V 1 : n 1 T (Av j + Bu j + a) > 0,  i  W j \ {1}: n i T (Av j + Bu j + a) ≤ 0.   j  {1, …,M} \ V 1 :  i  W j : n 1 T (Av j + Bu j + a) ≤ 0, Illustration of Polyhedral Cones Habets & van Schuppen,2004

9. 9 GRASP Applying the Combined Method A 1-dimensional integrator problem x x'

10. 10 GRASP Thank You