On the Controllability of the Pendubot *

We consider the question of whether the pendubot system is controllable about the kinematic singularity equilibrium point, where linear controllability is lost. Although accessibility is easily confirmed using Lie brackets, conclusions about the controllability of the pendubot system appear to be quite challenging for the non-specialist–-the system is not driftless and its homogeneity properties are unclear. In this paper, we tackle the question head on and prove controllability of a second order approximation to the system by constructing a family of linearly controllable trajectory loops based at the equilibrium. The structure of the approximation and the selected control input allow us to explicitly integrate the approximate system equations of motion. The trajectory loop construction exploits a particular time reversal symmetry shared by the pendubot and its second order approximation. The trajectory loop construction also works for the (full) pendubot since the corresponding pendubot and second order pendubot trajectories agree to seventh order in the horizon length. Example transition trajectories are shown.