Screw-based Relative Jacobian for Manipulators Cooperating in a Task

A current trend in Robotics is the analysis and the design of cooperative manipulation system, i.e. systems composed of multiple manipulator units interacting one another in a coordinated way. A cooperative system is characterized by higher manipulability and load capacity with respect to single-manipulator systems. The concept of the relative Jacobian is used to generate trajectories in the joint space, for two robots in cooperation executing a specified task. This paper proposes an alternative method to derive the relative Jacobian, here designated screw-based relative Jacobian, that relates the velocities of a tool (attached to one manipulator), relative to a blank (attached to another manipulator) as a function of the manipulators joint velocities, using the screw representation of differential kinematics, and the method of successive screw displacements. It is presented a systematic method to calculate the relative Jacobian in a compact, direct and simple form. The presented method is specially suitable when the geometry of manipulators becomes more general and for systems with spatial manipulators. A Cartesian-space tool path, defined relative to the blank, is resolved in the joint space.