We propose a linear MPC scheme for online computation of reactive walking motions, necessary for fast interactions such as physical collaboration with humans or collision avoidance in crowds. Unlike other existing schemes, it provides fully adaptable height, adaptable step placement and complete kinematic and dynamic feasibility guarantees, making it possible to walk perfectly safely on a piecewise horizontal ground such as stairs. A linear formulation is proposed, based on efficiently bounding the nonlinear term introduced by vertical motion, considering two linear constraints instead of one nonlinear constraint. Balance and Passive Safety guarantees are secured by enforcing a 3D capturability constraint. Based on a comparison between CoM and CoP trajectories involving exponentials instead of polynomials, this capturability constraint involves a CoM motion stopping along a segment of line, always maintaining complete kinematic and dynamic feasibility.

We study the effect of soft (i.e. compliant) soles in humanoid walking. Adding compliance at the foot sole is important for shock absorption during touchdown and to better cast the ground roughness. But, subsequent deformation needs to be considered in the attitude stabilization. We propose a sole deformation estimator used in the controller design, to ensure that the desired ZMP for stability requirement is fulfilled. We performed two comparison experiments on the HRP-4 humanoid robot with/without the estimator. Experiments showed how the estimator enhances the ZMP stabilization during walking and prevents the robot from falling down.