Load Balancing and Closed Chain Multiple Arm Control

We give the general dynamical equations for several rigid link manipulators rigidly grasping a commonly held rigid object. It is shown that the number of lost arm configuration degrees of freedom due to imposing the closed loop kinematic constraints is the same as the number of degrees of freedom gained for controlling the internal forces of the closed chain system. This number is equal to the dimension of the kernel of the Jacobian operator which transforms contact forces to the net forces acting on the held object, and it is shown that this kernel can be identified with the subspace of controllable internal forces of the closed chain system. Control of these forces allows one to regulate the grasping forces imparted to the held object or to control the load taken by each arm. It is shown that the internal forces can be influenced without affecting the control of the configuration degrees of freedom, and that this fact is independent of control law choice and involves only kinematical information about each arm. Control laws of feedback linearization type are shown to be useful for controlling the location and attitude of a frame fixed with respect to the held object, while simultaneously controlling the internal forces of the closed chain system. Since the kernel describing the internal forces subspace depends only on the relative location of arm tip contact points, force feedback can be used to feedback linearize and control the system even when the held object has unknown mass properties.