Direct Calculation of the Jacobian Matrix: Leveraging Dual Numbers in Robot Forward Kinematics

In this paper, an innovative method to compute Jacobian matrices in robot kinematics using dual numbers is introduced. By representing each joint with a dual number—comprising a real scalar and a dual vector component, thus forming a dual expression for the entire set of articulations comprised of a vector of joint values as the real part and a matrix of base vectors as the dual part. The validation of this approach is performed on a 6-degree-of-freedom KUKA KR 500 robot, comparing it to traditional geometric methods via Python simulations for 1000 randomized joint configurations. The proposed method demonstrates an average computational speed improvement of 12%, underscoring its efficiency and efficacy in Jacobian matrix computation for robot kinematics.