Our overall research interest is in synthesizing human like reaching and grasping using anthropomorphic robot hand-arm systems, as well as understanding the principles underlying human control of these actions. When one needs to define the control and task requirements in the Cartesian space, the problem of inverse kinematics needs to be solved. For non-redundant manipulators, a desired end-effector position and orientation can be achieved by a finite number of solutions. For redundant manipulators however, there are in general infinitely many solutions where the cardinality of the solution set must be made finite by imposing certain constraints. In this paper, we consider the Mitsubishi PA10 manipulator which is similar to the human arm, in the sense that both wrist and shoulder joints can be considered to emulate a 3DOF ball joint. We explicitly derive the analytic solution for the inverse kinematics using quaternions. Then, we derive a parameterization in terms of a pure quaternion called the swivel quaternion. The swivel quaternion is similar to the elbow swivel angle used in most approaches, but avoid the computation of inverse trigonometric functions. This parameterization of the self-motion manifold is continuous with any end-effector motion. Given the pose of the end-effector and the swivel quaternion (or swivel angle), the algorithm derives all solution of the inverse kinematics (finite number). We then show how the parameterization of the elbow self-motion can be used for the real-time control of the PA10 manipulator in the presence of obstacles.

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