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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).
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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 u J = xx t t0 tf
Numerical Methods for Optimal control • Indirect method • Direct method : convert to NLP • Shooting • Collocation u J = xx t t0 tf
Shooting Method u x ordinary differential eq. integration t t t0 t0 tf tf
Shooting Method u x ordinary differential eq. integration t t t0 t0 tf tf s.t.
Collocation Method u x t t t0 t0 tf tf
Collocation Method u x subject to t t t0 t0 tf tf
Solver • GPOPS (General PseudospectralOPtimal Control Software) • Colloation (Gauss pseudospectral method)
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 = 10 excitation, activation ~= 1
max_isometric_force = 100 excitation, activation ~= 0.5
max_isometric_force = 1000 excitation, activation ~= 0.05
max_isometric_force = 10000 excitation, activation ~= 0.01
Rotation Example • Minimize (torque – Mf) • Change # of collocation points, optimality tolerance
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 4thorder integrator • Evaluation of cost function means simulation of muscle dynamics during one gait cycle
Next • Combine with optimization solver • Parallel processing