Method to determine uncertain configurations of 6R manipulators

In this paper, a method is proposed to analytically determine the "uncertain" configurations of a general 6R manipulator. A configuration is called uncertain if in that configuration the manipulator’s joint freedom is greater than its end-effector freedom. In uncertain configurations, the manipulator acts like a redundant system where the endeffector remains in the same pose even if one (or more) of the joint angles is varied. In uncertain configurations the 6R manipulator acts as an mR (m<6) chain that is either an open-loop degenerate manipulator or a closed-loop overconstrained manipulator. The method to analytically determine uncertain configurations is based on a novel formulation for the loop-closure equation of a general 6R manipulator. In this formulation a 6R manipulator is converted to an mR manipulator, and new structural parameters are defined for this virtual mR chain. The conditions on the structural parameters of the mR chain, so that it forms a degenerate open-loop chain or an overconstrained mechanism, yield a set of equations that lead to the manipulator’s uncertain configurations.