Dynamic Formulations and Computational Algorithms

Abstract

Humanoids are complex dynamic systems operating in partially uncertain environments. Reliable control of humanoid robots is only possible with model-based control schemes. Also the generation of walking patterns that ensures the overall stability and respects technological limitations of the mechanical setup requires a nonlinear mechanical model, i.e., the equations of motions (EoM). This chapter addresses the derivation of the EoM in a way that allows for a systematic generation of the EoM and a computationally efficient evaluation at the same time. Starting with the description of the kinematic topology of a multibody system (MBS), a recursive formulation of the MBS kinematics is introduced. A synthetic approach to the EoM generation is presented. The latter departs from the Newton-Euler equations of the individual bodies of the MBS. The formulation is extended to account for subsystems representing the modules of the humanoid. The EoM are complemented by the contact forces due to the contact of feet and ground. In order to facilitate the actual evaluation of the EoM on the controller hardware of the humanoid robot, a recursive $O\left ( n\right )$ inverse dynamics formulation is presented. Also presented is a recursive $O\left ( n\right )$ forward dynamics algorithm making use of the submodule modeling. The latter is necessary for the off-line dynamics simulation with the humanoid design study.