Structure Preserving Optimal Control of a Three-Dimensional Upright Gait

The optimal control of human locomotion requires simulation techniques, which handle the contact's establishing and releasing between foot and ground. In this work, our aim is to optimally control the human upright gait using a structure preserving variational integrator, whereby different physiologically motivated cost functions are chosen and the obtained results are analysed with regard to the gait of humans. Thereby, the implemented three-dimensional rigid multibody system enables us to model forefoot as well as heel contact. The contacts between feet and ground are modelled as perfectly plastic impact and the orientation of the contact forces prevent penetration of the ground. To guarantee the structure preservation and the geometrical correctness, the non-smooth problem is solved including the contact configuration, time and force, in contrast to relying on a smooth approximation of the contact problem via a penalty potential. The applied mechanical integrator is based on a discrete constrained version of Lagrange-d'Alembert principle, which yields a symplectic momentum preserving method (see Leyendecker et al., Optim Control Appl Methods 31: 505-528, 2009, [31] for details).