The Momentum Equilibrium Principle: Foot Contact Stabilization with Relative Angular Momentum/Velocity

Abstract

The spatial momentum relation of an underac-tuated articulated multibody system on a floating base is a dynamic equilibrium relation between its coupling and relative momenta. The relative momentum is the difference between the system momentum and the momentum of the composite-rigid-body (CRB) that is obtained when the joints are locked. This relation is referred to as the momentum equilibrium principle. The focus in this work is on the angular momentum component of the momentum equilibrium principle. It is clarified that the relative angular momentum component can be represented in terms of the so-called relative angular velocity that is used as a control input in a balance controller. The balance controller proposed here is a whole-body controller that has independent inputs for center of mass (CoM) velocity and base-link angular velocity control. In addition, the relative angular velocity control input endows the controller with the unique property of generating an appropriate upper-limb motion that can stabilize the system momentum. More specifically, it is shown that when the relative angular velocity is derived from the reaction null-space (RNS) of the system, it becomes possible to stabilize the unstable states with a rolling foot/feet. The formulation is simple and yet quite efficient — there is no need to modify the contact model to account for the transitions between the stable and unstable contact states. There is also no need to command the upper-limb motion directly. A few simulation examples are presented to demonstrate and discuss the properties of the controller.