Reciprocity-based resolution of velocity degeneracies (singularities) for redundant manipulators

Abstract

A methodology for determining the velocity-degenerate (singular) configurations of joint-redundant manipulators is presented. The developed methodology uses the properties of reciprocal screws to determine all degenerate configurations resulting in a single degree-of-freedom (DOF) motion loss. Six joint screws ($ sub 1 ,$ sub 2 ,…,$ sub 6 ) are chosen to form one six-joint sub-group Jacobian, J sub . Setting the determinant of J sub to zero ( | J sub |=0 ) allows all conditions (say n in total) that cause the six-joint sub-group to become velocity degenerate to be identified. Reciprocal screws, W recip 1 , W recip 2 ,…, W recip n , can be found for each of the n velocity-degeneracy conditions. The reciprocal screw, W recip i , is reciprocal to the six joints that comprise J sub when the ith J sub degeneracy condition is true, but will not necessarily be reciprocal to the redundant joints $ r 1 ,$ r 2 ,…,$ r k−6 (where k is the total number of joints and k>6). Taking reciprocal products of W recip i and each redundant joint $ r 1 ,$ r 2 ,…,$ r k−6 and setting the results to zero yields all additional conditions necessary for W recip i to be simultaneously reciprocal to all of the redundant joints. This procedure is repeated for all of the reciprocal screws. The method identifies all unique sets of conditions that result in the redundant-manipulator joints $ sub 1 ,$ sub 2 ,…,$ sub 6 ,$ r 1 ,$ r 2 ,…,$ r k−6 becoming degenerate. A by-product of the methodology is that reciprocal screws for all degenerate configurations are derived. Examples are presented to demonstrate the effectiveness of the methodology at determining the velocity-degenerate configurations for joint-redundant manipulators.