First-order stability cells of active multi-rigid-body systems

A stability cell is a subset of the configuration space (C-space) of a set of actively controlled rigid bodies (e.g., a manipulator) in contact with a passive body in which the contact state is guaranteed to be stable under Coulomb friction and external forces. A first-order stability cell is a subset of a stability cell with the following two properties: the state of contact uniquely determines the rate of change of the object's configuration given the rate of change of the manipulator's configuration; and the contact state cannot be altered by any infinitesimal variation in the generalized applied force. First-order stability cells can be used in planning whole-arm manipulation tasks in a manner analogous to the use of free-space cells in planning collision-free paths: a connectivity graph is constructed and searched for a path connecting the initial and goal configurations. A path through a free-space connectivity graph represents a motion plan that can be executed without fear of collisions, while a path through a stability-cell connectivity graph represents a whole-arm manipulation plan that can be executed without fear of "dropping" the object. The paper gives a conceptual and analytical development of first-order stability cells of 3D rigid-body systems as conjunctions of equations and inequalities in the C-space variables. Additionally, our derivation leads to a new quasi-static jamming condition that takes into account the planned motion and kinematic structure of the active bodies.>