Grounding the three-dimensional divergent component of motion: Geometric analysis of contact and dynamic stability and its application to humanoid push recovery

The three-dimensional divergent component of motion (3D-DCM) framework has been successfully utilized to generate center of mass (CoM) trajectories for various locomotion types. While the 3D-DCM encodes the CoM dynamics, it relies on the contact forces between the robot’s end effectors and the environment to realize the intended motion. In the original formulation of the 3D-DCM, the feasibility of contact forces concerning contact constraints is assumed, but a comprehensive analysis of this assumption is lacking. In this work, we address this gap by extending the 3D-DCM framework to incorporate contact constraints and dynamic stability of the system. This is achieved by encoding feasible CoM forces as geometric sets. We derive an analytical relationship that characterizes how these sets can be modulated by humanoid push recovery strategies. Building on these insights, we propose a push recovery algorithm that integrates ankle, hip, height-variation, and stepping strategies. The proposed method is evaluated through extensive experiments with the humanoid robot TORO, including scenarios of force-disturbed balancing, walking, and multi-contact configurations.