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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.
Tedrake et al. (2015) studied this question.