© 2004-2012 IEEE. This paper presents a method for efficiently testing the stability of an object under contact that accommodates empirical models of admissible forces at individual contact points. It handles a diverse range of possible geometries of the admissible force volume, including anisotropy, adhesion, and even nonconvexity. The method discretizes the contact region into patches, performs a convex decomposition of a polyhedral approximation to each admissible force volume, and then formulates the problem as a mixed integer linear program. The model can also accommodate articulated robot hands with torque limits and joint frictions. Predictions of our method are evaluated experimentally in object lifting tasks using a gripper that exploits microspines to exert strongly anisotropic forces. The method is applied to calculate gripper loading capabilities and equilibrium predictions for a quadruped climbing robot on steep and overhanging terrain.