Dynamics and balance control of the reaction mass pendulum: A three-dimensional multibody pendulum with variable body inertia

Pendulum models have been studied as benchmark problems for development of nonlinear control schemes, as well as reduced-order models for the dynamics analysis of gait, balance and fall for humanoid robots. We have earlier introduced the reaction mass pendulum (RMP), an extension of the traditional inverted pendulum models, which explicitly captures the variable rotational inertia and angular momentum of a human or humanoid. The RMP consists of an extensible "leg", and a "body" with moving proof masses that gives rise to the variable rotational inertia. In this paper, we present a thorough analysis of the RMP, which is treated as a three-dimensional (3D) multibody system in its own right. We derive the complete kinematics and dynamics equations of the RMP system and obtain its equilibrium conditions. We show that the equilibria of this system consist of an unstable equilibrium manifold and a stable equilibrium manifold. Next, we present a nonlinear control scheme for the RMP, which is an underactuated system with three unactuated degrees of freedom (DOFs). This scheme asymptotically stabilizes this underactuated system at its unstable equilibrium manifold, with a vertically upright configuration for the "leg" of the RMP. The domain of convergence of this stabilization scheme is shown to be almost global in the state space of the RMP. Numerical simulation results verify this stability property of the control scheme and demonstrate its effectiveness in stabilizing the unstable equilibrium manifold.