Optimal control problems have become popular in recent years in biomechanical movement predictions mainly due to an increase of computational capacities and development of new optimization software . The convergence of optimal control problems can be influenced by the type of coordinates used to describe the model, as well as by the dynamic formulation used to introduce the equations of motion. In this study we present a comparison of optimal control problems with different complexity, solved using different types of coordinates and dynamic formulations.
Four planar skeleton models were used to test dynamic formulations: an inverted double pendulum (model A), an inverted triple pendulum (model B), an inverted quadruple pendulum (model C) and a humanoid with five degrees of freedom (two shanks, two thighs and a torso) with a fixed foot to the ground (model D). Each model was defined using absolute (Ab), relative (Re) and natural coordinates (NC). Note that the latter leads to a constant mass matrix .
For each model and type of coordinate definition an optimal control problem was solved twice: using implicit dynamics formulation and using explicit dynamics formulation, which led to a total of 24 optimal control problems. Each problem consisted in predicting the movement from an initial to a final state minimizing the integral of squared joint torque values. For models A to C, the movement consisted in standing up from a sitting position and for model D, the movement represented a single-support stance phase of gait. These optimizations were solved formulating the problems using direct collocation schemes in CasADi .
The optimal solution obtained for each model was similar among formulations. Mean and maximum differences of optimal cost function values respect to Ab-explicit formulation were 7.7% and 24.6%. In all cases (except for the quadruple pendulum with Ab and NC coordinates), the optimal cost function value was lower using implicit formulations. Optimal control problems using relative coordinates tended to provide optimal solutions with lower cost function values (Fig. 1), whereas using absolute coordinates, optimal solutions with a lower number of iterations were obtained.
Fig. 1. Number of iterations with respect to logarithm of optimal cost function values for implicit formulations, using absolute (Ab), relative (Re) and natural (NC) coordinates. Rectangles group solutions of the same skeletal model.
Comparing convergence among optimal control problem solutions, we observed that overall implicit dynamic formulations and absolute coordinates provide solutions earlier, while relative coordinates provide a solution with a lower cost function. Further analysis will be carried out to investigate the benefits of having a constant mass matrix in more complex problems.