Restricted Jacobian Matrices of Redundant Manipulators in Constrained Motion Tasks

Abstract

Kinematically redundant manipulators are considered, with a prespecified end-effector task and multiple additional tasks. The number of additional tasks is assumed to be equal to or less than the degree of redundancy. This assumption justifies an equal-priority (or duality) concept for the end-effector task and additional tasks. Based on this concept, expressions for two restricted Jacobian matrices are derived: the restricted manipulator Jacobian and the restricted additional-task Jaco bian. It is shown that generalized inverses of these matrices can be utilized to derive two alternative solutions of instanta neous inverse kinematics that satisfy both tasks simultaneously. The first solution, which is based on the null space of the manipulator Jacobian, has been widely applied in various redundancy-resolution techniques; it is shown that when used in manipulator self-motion tasks, this solution yields cyclic- path motion in joint space. The second solution utilizes the null space of the additional-task Jacobian; it is dual to the first one. Hence, in the particular case of end-effector motion with additional-task maintenance, this solution yields cyclic-path motion as well. It is further shown that at algorithmic singularities, both restricted Jacobian matrices lose rank simultaneously. Algo rithmic singularities, together with the well-known manipulator singularities, and newly defined additional-task singularities are shown to play an important role in kinematic analysis. The duality concept calls for the definition of a new type of dexterity measure: the additional-task performance measure. The restricted Jacobian matrices are shown to also provide a convenient tool for deriving "restricted" dexterity measures that are directly related to the various types of singularities. Typical end-effector/additional-task combinations have been tested through computer simulation, illustrating the application of the restricted Jacobian matrices to configuration analy sis and path-planning problems of redundant manipulators in constrained-motion tasks.