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 u J = xx t t0 tf

5. Numerical Methods for Optimal control • Indirect method • Direct method : convert to NLP • Shooting • Collocation u J = xx t t0 tf

6. Shooting Method u x ordinary differential eq. integration t t t0 t0 tf tf

7. Shooting Method u x ordinary differential eq. integration t t t0 t0 tf tf s.t.

8. Collocation Method u x t t t0 t0 tf tf

9. Collocation Method u x subject to t t t0 t0 tf tf

10. Solver • GPOPS (General PseudospectralOPtimal Control Software) • Colloation (Gauss pseudospectral method)

11. Simple Example

12. 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 = 10 excitation, activation ~= 1

15. max_isometric_force = 100 excitation, activation ~= 0.5

16. max_isometric_force = 1000 excitation, activation ~= 0.05

17. max_isometric_force = 10000 excitation, 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 4thorder 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