A sufficient condition for manipulation of Envelope Family

This paper discusses a sufficient condition for the manipulation of the Envelope Family, where multiple contacts are allowed between object and chains (or the environment). All chains are assigned to either the position controlled chain (P-chain) or the torque controlled chain (T-chain). Although the object motion under multiple contacts can not be uniquely specified only by the T-chains, we are able to provide a sufficient condition ensuring that a given set of torque commands for the T-chains always moves the object within the designated direction along the surface of a P-chain (or a fixed environment). Experiments as well as simulations are shown to verify this basic idea.