Model predictive control of biped walking with bounded uncertainties

A biped walking controller for humanoid robots has to handle together hard constraints, dynamic environments, and uncertainties. Model Predictive Control (MPC) is a suitable and widely used control method to handle the first two issues. Uncertainties on the robot imply a non-zero tracking error when trying to follow a reference motion. A standard solution for this issue is to use tighter constraints by introducing some hand tuned safety margins, for the reference motion generation to ensure that the actual robot motion will satisfy all constraints even in presence of the tracking error. In this article, we find bounds for the tracking error and we show how such safety margins can be precisely computed from the tracking error bounds. Also, a tracking control gain is proposed to reduce the restrictiveness introduced with the safety margins. MPC with these considerations ensure the correct operation of the biped robot under a given degree of uncertainties when it is implemented in open-loop. Nevertheless, the straightforward way to implement an MPC closed-loop scheme fails. We discuss the reasons for this failure and propose a robust closed-loop MPC scheme.