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

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

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

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

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

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

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

Collocation Method t u t0tf t x t0tf

Collocation Method t u t0tf t x t0tf subject to

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

Simple Example

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

max_isometric_force = 10excitation, activation ~= 1

max_isometric_force = 100excitation, activation ~= 0.5

max_isometric_force = 1000excitation, activation ~= 0.05

max_isometric_force = 10000excitation, activation ~= 0.01

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

mesh refinement iteration = 2,9 secs

mesh refinement iteration = 3,2.5 mins

mesh refinement iteration = 4,3 mins

mesh refinement iteration = 10,15 mins

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

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

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

Simulation of one muscle

Next Combine with optimization solver Parallel processing

Thank you