Characterizing the self-motion manifolds of redundant robots of arbitrary kinematic structures

Abstract

Robot fault tolerance measures can be classified into two categories: 1) Local measures that are based on the singular value decomposition (SVD) of the robot Jacobian, and 2) Global measures that are suitable to quantify the fault tolerance more effectively in pick-and-place applications. One can use the size of the self-motion manifold of a robot as a global fault-tolerance measure. The size of the self-motion manifold at a certain end-effector location can be simply the sum of the range of the joint angles of a robot at that location. This work employs the fact that the largest self-motion manifolds occur due to merging two (or more) previously disjoint manifolds. The connection of previously disjoint manifolds occur in special configurations in the joint space called singularities. Singularities (singular configurations) occur when two or more of the robot joint axes become aligned and are linearly dependent. A significant amount of research has been performed on identifying the robot singularities but was all based on symbolically solving for when the robot Jacobian is not of full rank. In this work, an algorithm was proposed that is based on the gradient of the singular values of the robot Jacobian. This algorithm is not limited to any Degree of Freedom (DoF) nor any specific robot kinematic structure and any rank of singularity. Based on the robot singularities, one can search for the largest self-motion manifold near robot singularities. The measure of the size of the self-motion manifold was chosen to eliminate the effect of the self-motion manifold's topology and dimension. Because the SVD at singularities is indistinct, one can employ Givens rotations to define the physically meaningful singular directions, i.e., the directions where the robot is not able to move. This approach has been extensively implemented on a 4-DoF robot, different 7-DoF robot, and an 8-DoF robot. The global fault-tolerance measure might be further optimized by changing the kinematic structure of a robot. This may allow one to determine a globally fault-tolerant robot, i.e., a robot with 2π range for all of its joint angles at certain end-effector location, i.e., a location that is the most suitable for pick-and-place tasks.