Structure preserving simulation of compass gait and monopedal jumping

This work discusses two different structure preserving mechanical integrators in the framework of optimal control simulations. The first one is a variational integrator and is based on a constrained version of the Lagrange-d’Alembert Principle. The resulting integrator preserves the symplecticity and momentum maps of the simulated multibody dynamics. The second one is an energy momentum scheme, which is based on a discrete derivative. Both integrators are used to simulate the optimal control a of bipedal walker, which is actuated in the hip. The optimal control problem is solved by a direct transcription method, which transforms it into a constrained optimisation problem. The second example studies the forward dynamics with contact of a monopedal jumper and the corresponding algebraic equations are solved by the aforementioned symplectic momentum method. The derivation via a discrete variational principle includes the variation of the contact time and configuration.