On the geometric analysis of optimum trajectories for cooperating robots using dual quaternion coordinates

A geometric method for the dynamic analysis of trajectories for cooperating robot systems is presented. The technique uses the algebraic manifolds that arise from the image space of spatial displacements to do trajectory analysis. The geometric structure of these manifolds offers a convenient framework to study robot motion. In this technique, the dynamics of the robots and the workpiece are formulated in the operational space of the robot. The position of the robot is written in the image space of dual quaternion coordinates. Thus, the analysis is performed in the operational image space that provides an algebraically defined geometric structure upon which to examine the trajectories that the robot system follows. This analysis offers a tool for determining optimal trajectories for the robot system. The technique is demonstrated through the analysis of two planar three-revolute-joint (3R) robots manipulating a common object. A planar example was chosen because the results can be demonstrated graphically. The algebraic properties of the constraint manifolds can be used to extend the work to more general, multidimensional spatial robot problems