Open-loop stability -: A new paradigm for periodic optimal control and analysis of walking mechanisms

The present paper deals with the numerical solution of periodic optimal design and control problems, such that the resulting optimal trajectories are open-loop stable. This means that under small perturbations the real process will asymptotically converge back to the periodic orbit without any feedback corrections. Open-loop stability is de ned in terms of the spectral radius of the monodromy matrix which is a dif cult non-standard optimization criterion. In particular we consider applications in one- and two-legged walking robots with point feet, which can not stand still stably, but will be shown to be able to walk or run in an open-loop stable way given the right con guration and control. This class of problems is characterized by discontinuous dynamics with impact, multiple motion phases and changing degrees of freedom. The numerical results which are obtained by a two-level optimization approach not only suggest that walking robots can be built without sophisticated feedback devices, but may also have an impact on understanding biomechanical motions and their stability.