A more compact expression of relative Jacobian based on individual manipulator Jacobians

This work presents a re-derivation of relative Jacobian matrix for parallel manipulators, expressed in terms of the individual manipulator Jacobians and multiplied by their corresponding transformation matrices. This is particularly useful when the individual manipulator Jacobians are given, such that one would not need to derive an entirely new expression of a relative Jacobian but will only use the existing manipulator Jacobians and perform the necessary transformations. In this work, the final result reveals a wrench transformation matrix which was not present in previous derivations, or was not explicitly expressed. The proposed Jacobian expression results in a simplified, more compact and intuitive form. It will be shown that the wrench transformation matrix is present in stationary as well as mobile combined manipulators. Simulation results show that at high angular end-effector velocities, the contribution of the wrench transformation matrix cannot be ignored. Highlights: B A relative Jacobian is derived and expressed in terms of the individual manipulator Jacobians. B Previous forms of a relative Jacobian were unsimplified, and were not expressed in terms of the wrench transformation matrix. B The simplified relative Jacobian presented in the work is more compact, intuitive, and is easier to implement when the manipulator Jacobians are given.