A MPC Walking Framework With External Contact Forces

In this work, we present an extension to a linear Model Predictive Control (MPC) scheme that plans external contact forces for the robot when given multiple contact locations and their corresponding friction cone. To accomplish this we set up a two step optimization problem. In the first optimization, we compute the Center of Mass (CoM) trajectory, foot step locations, and introduce slack variables to account for violating the imposed constraints on the Center of Pressure (CoP). We then use the slack variables to trigger the second optimization, in which we calculate the optimal external force that compensates for the CoP tracking error. This optimization considers multiple contacts with the environment by formulating the problem as a Mixed Integer Quadratic Program (MIQP) that can be solved at the order of 100 Hz. Once contact is created, the MIQP collapses to a single Quadratic Program (QP) that can be solved in real time < 1kHz. Simulations show that the presented control scheme can withstand disturbances 2-5x larger with the additional force provide by a hand contact when considering delays and 3-6x larger when contact is already made.