Analytical Expressions of Serial Manipulator Jacobians and Their High-Order Derivatives Based on Lie Theory

Abstract

Serial manipulator kinematics provide a mapping between joint variables in joint-space coordinates, and end-effector configurations in task-space Cartesian coordinates. Velocity mappings are represented via the manipulator Jacobian produced by direct differentiation of the forward kinematics. Acquisition of acceleration, jerk, and snap expressions, typically utilized for accurate trajectory-tracking, requires the computation of high-order Jacobian derivatives. As compared to conventional numerical/D-H approaches, this paper proposes a novel methodology to derive the Jacobians and their high-order derivatives symbolically, based on Lie theory, which requires that the derivatives are calculated with respect to each joint variable and time. Additionally, the technique described herein yields a mathematically sound solution to the high-order Jacobian derivatives, which distinguishes it from other relevant works. Performing computations with respect to the two inertial-fixed and body-fixed frames, the analytical form of the spatial and body Jacobians are derived, as well as their higher-order derivatives, without resorting to any approximations, whose expressions depend explicitly on the joint state and the choice of reference frames. The proposed method provides more tractable computation of higher-order Jacobian derivatives, while its effectiveness has been verified by conducting a comparative analysis based on experimental data extracted from a KUKA LRB iiwa7 R800 manipulator.