Controllability and Stabilizability of Planar Multibody Systems with Angular Momentum Preserving Control Torques

Control and stabilization properties for planar multibody systems consisting of a tree interconnection of rigid bodies are studied. A reduced system model is obtained by considering rotational motion only. It is assumed that there are no exogeneous effects and angular momentum preserving torques generated by joint motors are used as means of control. The angular momentum constant is assumed to be zero so that any set of absolute angles together with zero angular velocities can be studied as an equilibrium. The final form of the reduced system equations is obtained by a further reduction using the expression for the angular momentum constant which is regarded as a nonholonomic constraint. The reduced equations completely characterize the dynamics of the multibody system; these equations necessarily include drift dynamics. We first show that for three or more interconnected bodies, the dynamics are strongly accessible ; for two interconnected bodies, the dynamics are not accessible. We next show that for three or more interconnected bodies, (1) the dynamics are small-time locally controllable from any equilibrium solution and (2) any initial condition can be transferred to any equilibrium solution in an arbitrarily small time period. The latter result is proved by construction of the required control using the holonomy property. We then prove the nonexistence of smooth feedback stabilization to a single equilibrium solution. Comments are made about construction of a stabilizing nonsmooth feedback control.