Dynamics and Controllability of a Planar Rigid Body with aThruster

In this paper we discuss the dynamics of a planar rigid body with a thruster. We identify the position and orientation of the planar rigid body with an element of the Lie group SE(2). The invariance of the kinetic energy Lagrangian and the forcing term (thrust) under the SE(2) action of translations and rotations is used to obtain a set of Poisson-reduced equations on T SE(2)=SE(2) which is isomorphic to se(2) , the dual of the Lie algebra of SE(2). Depending on the control authority available, we distinguish two versions of the problem which we call the \jet-puck" problem and the \hovercraft" problem respectively. In the jet-puck problem the control enters linearly, and we show that, using only a scalar control we have controllability over se(2). In the hovercraft problem the controls enter nonlinearly. In the equations of motion of the above cases the drift vectorreld is periodic (and hence trivially weakly positively Poisson stable). This, along with the fact that the Lie algebra rank condition is satissed at all points on se(2) , is used to show controllability of the systems.