Practical guide to solve the minimum-effort problem with geometric algorithms and B-Splines

This paper focuses on important implementation issues of numerical optimal control that are often overlooked. In particular, transcription methods should be carefully implemented for obtaining a discrete representation of the problem. For this purpose, we explain the algorithms to solve the minimum-effort problem by applying a direct collocation method based on B-Splines. In addition, we describe how to compute the gradient of the objective function as well as the Jacobian of the constraints without the use of finite differences and automatic differentiation. Geometric algorithms based on Lie groups and Lie algebra are examined to efficiently compute the analytical derivatives of the equations of motion of articulated robots. These ingredients allow the fast computation of dynamically feasible robot motions. We provide numerical comparisons with a biped robot to validate our recipe against classical direct collocation methods.