  Reliable quasi-static object manuipulation and robotic locomotion require
verification of the stability of equilibria under rigid contacts and friction.
In a recent paper, M. Posa, M. Tobenkin, and R. Tedrake demonstrated that
sums-of-squares (SOS) programming can be used to verify Lyapunov stability via
Lyapunov's direct method. This test was successfully applied to several simple
problems with a single point contact. At the same time it has been found that
this method is too conservative for several multi-contact systems. In this
paper, an extension of Lyapunov's direct method is proposed, which makes use of
several Lyapunov functions, and which allows \emph{temporary} increase of those
Lyapunov function along a motion trajectory. The proposed method remains
compatible with SOS programming techniques. The improved stability test is
successfully applied to a rigid body with 2 point contacts, for which the exact
conditions of Lyapunov stability are unknown.
