  Lyapunov stability of a mechanical system means that the dynamic response
stays bounded in an arbitrarily small neighborhood of a static equilibrium
configuration under small perturbations in positions and velocities. This type
of stability is highly desired in robotic applications that involve multiple
unilateral contacts. Nevertheless, Lyapunov stability analysis of such systems
is extremely difficult, because even small perturbations may result in hybrid
dynamics where the solution involves many nonsmooth transitions between
different contact states. This paper concerns with Lyapunov stability analysis
of a planar rigid body with two frictional unilateral contacts under inelastic
impacts, for a general class of equilibrium configurations under a constant
external load. The hybrid dynamics of the system under contact transitions and
impacts is formulated, and a \Poincare map at two-contact states is introduced.
Using invariance relations, this \Poincare map is reduced into two
semi-analytic scalar functions that entirely encode the dynamic behavior of
solutions under any small initial perturbation. These two functions enable
determination of Lyapunov stability or instability for almost any equilibrium
state. The results are demonstrated via simulation examples and by plotting
stability and instability regions in two-dimensional parameter spaces that
describe the contact geometry and external load.
