TY - CHAP
T1 - Analytical Expressions of Serial Manipulator Jacobians and their High-Order Derivatives based on Lie Theory∗
AU - Fu, Zhongtao
AU - Spyrakos-Papastavridis, Emmanouil
AU - Lin, Yen Hua
AU - Dai, Jian S.
PY - 2020/5
Y1 - 2020/5
N2 - 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 would 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.
AB - 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 would 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.
UR - http://www.scopus.com/inward/record.url?scp=85092746163&partnerID=8YFLogxK
U2 - 10.1109/ICRA40945.2020.9197131
DO - 10.1109/ICRA40945.2020.9197131
M3 - Conference paper
AN - SCOPUS:85092746163
T3 - Proceedings - IEEE International Conference on Robotics and Automation
SP - 7095
EP - 7100
BT - 2020 IEEE International Conference on Robotics and Automation, ICRA 2020
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 2020 IEEE International Conference on Robotics and Automation, ICRA 2020
Y2 - 31 May 2020 through 31 August 2020
ER -