An efficient quantitative test for form closure valid for any number of contact points is formulated as a linear program, the optimal objective value of which provides a measure of how far a grasp is from losing form closure. When the grasp does not have form closure, manipulation planning requires a means for predicting the object's stability and instantaneous velocity, given the joint velocities of the hand. The classical approach to computing these quantities is to solve the systems of kinematic inequalities corresponding to all possible combinations of separating or sliding at the contacts. All combinations resulting in the interpenetration of bodies or the infeasibility of the equilibrium equations are rejected. The remaining combination is consistent with all the constraints and is used to compute the velocity of the manipulated object and the contact forces, which indicate whether or not the object is stable. A linear program whose solution yields the same information as the classical approach, usually without explicit testing of all possible combinations of contact interactions, is formulated.
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