Analytical Differentiation of Manipulator Jacobian

In order to increase the control performance, many control algorithms utilize the acceleration information as a reference signal. An end-effector's velocity is mapped from joint velocity through multiplication with a Jacobian matrix. Therefore, in order to derive the joint acceleration corresponding to the desired trajectory of an end-effector, Jacobian differentiation should be calculated. Numerical methods for Jacobian differentiation are easy to implement and provide sufficiently accurate approximations. However, they incur high computational costs due to the iterative calculation of element-wise derivations. Also, numerical error is inevitable and it can no longer be ignored in near the singular point. Therefore, it is rather hard to compute control algorithm accurately in real-time. To resolve this problem, an analytical method for the differentiation is introduced in this paper. Since it does not need any approximation, it gives accurate result. In order to verify the effectiveness, the method is compared to the numerical derivation through computer simulations.