1. Progress Report - Solving optimal control problem Yoonsang Lee, Movement Research Lab., Seoul National University

2. Today Several numerical approaches to solving optimal control problem Some simple & incomplete results

3. Optimization : min value = 1, at x =0 Nonlinear Programming (NLP) s.t.

4. Numerical Methods for Optimal control Indirect method Direct method : convert to NLP –Shooting –Collocation t u J = xx t0tf

5. Numerical Methods for Optimal control Indirect method Direct method : convert to NLP –Shooting –Collocation t u J = xx t0tf

6. Shooting Method t u t0tf t x t0tf ordinary differential eq. integration

7. Shooting Method t u t0tf t x t0tf ordinary differential eq. integration s.t.

8. Collocation Method t u t0tf t x t0tf

9. Collocation Method t u t0tf t x t0tf subject to

10. Solver GPOPS (General Pseudospectral OPtimal Control Software) –Colloation (Gauss pseudospectral method)

11. Simple Example

13. Static Pose Example Activation, contraction dynamics Minimize (torque – Mf) –torque : inverse dyn. solution (reference data) –M : moment arm matrix (reference data) –f : muscle force Change maximum isometric force

14. max_isometric_force = 10excitation, activation ~= 1

15. max_isometric_force = 100excitation, activation ~= 0.5

16. max_isometric_force = 1000excitation, activation ~= 0.05

17. max_isometric_force = 10000excitation, activation ~= 0.01

18. Rotation Example Minimize (torque – Mf) Change # of collocation points, optimality tolerance

19. mesh refinement iteration = 2,9 secs

20. mesh refinement iteration = 3,2.5 mins

21. mesh refinement iteration = 4,3 mins

22. mesh refinement iteration = 10,15 mins

23. mesh refinement iteration = 10, feasibility tolerance, optimality tolerance : 1e-6, 2e-6,34 hours

25. What’s wrong? Optimization solver does not guarantee find feasible solution –Equality constraints could not be satisfied –Dynamics constraint are checked only at collocation points Shooting method provides feasible solution although it accumulates error

26. Shooting Method Activation / contraction dynamics Runge-Kutta 4 th order integrator Evaluation of cost function means simulation of muscle dynamics during one gait cycle

27. Simulation of one muscle

28. Next Combine with optimization solver Parallel processing

29. Thank you