The force distribution problem (FDP) in robotics requires the determination of multiple contact forces to match a desired net contact wrench. For the double support case encountered in humanoids, this problem is underspecified, and provides the opportunity to optimize desired foot centers of pressure (CoPs) and forces. In different contexts, we may seek CoPs and contact forces that optimize actuator effort or decrease the tendency for foot roll. In this work, we present two formulations of the FDP for humanoids in double support, and propose objective functions within a general framework to address the variety of competing requirements for the realization of balance. As a key feature, the framework is capable to optimize contact forces for motions on uneven terrain. Solutions for the formulations developed are obtained with a commercial nonlinear optimization package and through analytical approaches on a simplified problem. Results are shown for a highly dynamic whole-body humanoid reaching motion performed on even terrain and on a ramp. A convex formulation of the FDP provides real-time solutions with computation times of a few milliseconds. While the convex formulation does not include CoPs explicitly as optimization variables, a novel objective function is developed which penalizes foot CoP solutions that approach the foot boundaries.
Dynamic Walking on Compliant and Uneven Terrain using DCM and Passivity-based Whole-body Control