A closed-form solution for real-time ZMP gait generation and feedback stabilization

Here we present a closed-form solution to the continuous time-varying linear-quadratic regulator problem for zero-moment point (ZMP) tracking. This generalizes previous analytical solutions for gait generation by allowing "soft" tracking (with a quadratic cost) of the desired ZMP, and by providing the feedback gains for the resulting time-varying optimal controller. This enables very fast O(n) computation, with n the number of piecewise polynomial segments in the desired ZMP trajectory. Results are presented using the Atlas humanoid robot where dynamic walking is achieved by recomputing the optimal controller online.