A Sufficient Condition for Manipulation of Envelope Family

This paper discusses a sufficient condition for manipulation of Envelope Family, where multiple contacts are allowed between object and chains (or environment). All chains are assigned by either position controlled chain (P-chain) or torque controlled chain (T-chain). While the object motion under multiple contacts can not be uniquely specified by T-chains only, we show a sufficient condition ensuring that a given set of torque commands for T-chains always move the object toward the goal along the surface of P-chain (or a fixed environment). Experiments as well as simulations are also shown to verify the basic idea. K e y w o r d s : Enveloping Grasp, Envelope Family, Manipulation of Object, Envelope Walk. 1 I n t r o d u c t i o n Fig.1 shows a couple of examples of Envelope Family where Fig.l(a), (b), and (c) are envelope grasp by a robot hand, manipulation of a huge and heavy object by a humanoid robot, and envelope walk by a legged robot, respectively. Each robot in Envelope Family has a common working style where multiple contacts are allowed between chains and object (or environment). For a manipulation of object or body, so far, many researchers have focused on the link tip level manipulation or locomotion, since a dexterous motion can be anticipated by utilizing many degrees of freedom existing in the system. Such manipulation or locomotion, however, may easily fail in grasping an object under an external disturbance or in standing over the ground with a steap slope. On the other hand, due to multiple contacts, manipulation or locomotion in enveloping style can be expected even more robustness than that in tip level one. Also, this type of manipulation or locomotion greatly contributes to suppressing the peak torque by distributing the gravitational force to each contact point. In a humanoid robot as shown in Fig.l(b), the manipulation in enveloping style may provide the only feasible solution under joint torque limitation. While we can find such advantages for the manipulation 0 7 8 0 3 5 8 8 6 4 / 0 0 / $ 1 0 . 0 0 © 2000 IEEE 10 ems Engineering niversity 739-8527 JAPAN (a) Robot hand ~,..-~' 1~2:J P-chail (b) Humanoid robot (c) Legged robot Figure 1: Examples of Envelope Family in enveloping style, there are a couple of barriers to break through for achieving it. The biggest one is that the total force produced by all contact forces is not uniquely determined, due to the one-to-multiple mapping from the joint torque to the contact force. As a result, the moving direction of object is not uniquely specified either. To make us release from the mapping problem, we give up manipulating an enveloped object by pure torque control. Instead, we assign all chains (fingers or legs) into position controlled chain (P-chain) and torque controlled chain (Tchain), respectively, where P-chain might be replaced by an environment. Now, let us consider a sphere enveloped by a robot hand as shown in Fig.l(a), where four fingers are categorized into T-chains and P-chains. T-chains and P-chains can be regarded as pushers and supports, respectively. The object will move along the surface of P-chains, if we impart an appropriate set of torque commands for the T-chains. Under such multiple contacts, a rolling motion may occur at a particular contact point, while sliding motions happen at most contact points. We do not care what kinds of motion actually happen during the manipulation process. The goal of this paper is to obtain a sufficient condition ensuring that a given set of torque for T-chains always move the object toward the designated direction along the surface of P-chains (or environment). Assuming that each contact force set is bounded and