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Convex Properties of Center-of-Mass Trajectories for Locomotion Based on Divergent Component of Motion

This letter presents an in-depth analysis of the convex properties of center-of-mass (CoM) trajectories for legged robot locomotion based on the concept of Divergent Component of Motion (DCM). In particular, we show that the union of all possible trajectories forms a bounded convex set under appropriate boundary conditions. Additionally, we describe in detail our approach of generating closed-form CoM trajectories through piecewise interpolation over a sequence of waypoints and show how to compute the CoM trajectory efficiently through equations given in a matrix form. Applying the convex properties to our trajectory-generation approach, we present an algorithm for computing convex overapproximations of the CoM waypoints. Finally, we provide an example of usage in placing waypoints that lead to feasible CoM trajectories with respect to kinematic and dynamic constraints. The approach is validated with a multi-contact scenario in simulation with the humanoid robot TORO.