1. Stabilization of Trajectories for Systems with Nonholonomic Constraints G. Walsh, D. Tilbury, S. Sastry, R. Murray, J. P. Laumond

2. Presentation Topics l Problem of stabilizing a system with nonholonomic(nonintegrable) constraints  Non-smooth feedback laws  Time-varying feedback laws Brockett’s necessary conditions for stability No nonholonomic system can be asymptotically stabilized using smooth time-invariant state feedback

3. Problem formulation Given a nonholonomic system and a feasible trajectory to follow, find a control law to stabilize the system to the trajectory

4. Proposition (Stabilizing control law)

5. Proof of the Proposition

6. Example 1 - Heisenberg Control Algebra Trajectories investigated 1) origin x 0 (t)=[0 0 0] u 0 (t)=[0 0] 2) straight line x 0 (t)=[0 t 0] u 0 (t)=[0 1] System dynamics

7. Example 1 - Heisenberg Control Algebra

8. 1) origin

9. Example 1 - Heisenberg Control Algebra 2) straight line

10. Example 1 - Heisenberg Control Algebra  Simulation result (  =1/6,  =1,  =0.5)

11. Example 2 - Hilare x3 (x1, x2) Desired trajectory: Perfect circle Nominal inputs: u 0 =[1 1]

12. Example 2 - Hilare => cannot directly compute the control law

13. Example 2 - Hilare  Numerical approach Initial values (t=0) Update laws

14. Example 2 - Hilare  Simulation result (  =0.1,  =1,  =3)

15.  Simulation result Example 2 - Hilare