On the Jacobian matrix of 3−RPS and triangular 6−UPS linkages

The Jacobian matrix of a linkage relates actuated velocities to its end-effector velocity. Deriving this matrix for serial mechanisms is straightforward, but usually more challenging for parallel linkages. The velocity equation of these linkages typically comprises two distinct Jacobian matrices. An additional step required to establish the velocity of the end-effector of the linkage as a function of velocities of its actuators consists in inverting the first matrix and multiplying the result with the second, yielding a unique matrix which is here referred to as the Jacobian matrix. The previous inversion is almost universally conducted numerically except for the simplest of linkages. Here, we demonstrate that, even for non-trivial linkages such as the $3-RPS$ and triangular $6-UPS$ this inversion can also be done analytically. Similar to serial linkages, the Jacobian matrix is presented as a series of twists, each reflecting the impact of an actuator on the end-effector’s motion. Their geometric interpretation is provided and studied. Finally, how the method applies to other topologies of mechanisms is also discussed.