Manipulator Inverse Kinematics for Untimed End-Effector Trajectories With Ordinary Singularities

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Abstract
This article addresses the following inverse kinematics problem: given an untimed spatial end-effector trajectory, determine joint trajectories that are consistent with its execution. An algorithm for the continuous iterative solution of this problem for six-degree-of-freedom manipulators of arbitrary structure is presented. The main idea of this algorithm is that it converges with equal ease at both regular configurations and certain ( "or dinary ") singularities. Ordinary singularities (herein defined) depend on the given end-effector trajectory, as well as the ma nipulator configuration, and represent "dead points" where the end effector (but not the manipulator) must pause during trajectory execution. The algorithm is based on the predictor-corrector method of path following using (1) a newly developed second-order predictor, (2) a first-order Newton method corrector, and (3) the idea of including the end effector's position (along its trajec tory) as a dependent (rather than independent) variable in the formulation. Both the predictor and the corrector are derived from Taylor series expansion of the 4 x 4 matrix equation of closure and require only the solution of linear systems of equations. Examples are given that demonstrate the algorithm's ability to pass through ordinary singularities to determine alternate joint trajectories for the same end-effector trajectory. These examples also show how ordinary singularities can be included in manipulator motions that are similar to a boxer's jab or a runner's kick.