In analyzing robotic manipulator kinematics, a Jacobian matrix is derived from a set of transition matrices. The Jacobian matrix relates the velocities at joints to those at the end-effector. For the systems in which the axes of joint rotations intersect and are orthogonal to each other, there are efficient traditional methods, such as the Denavit-Hartenberg parameter technique. However, in biomechanical systems, such as the human body, the axes of joint rotations often do not intersect and are not orthogonal to each other. For such systems, the traditional methods lead to large and complex symbolic Jacobian matrices that are difficult to simplify, and thus the computational time becomes prohibitive.

The authors of this paper present a new approach to the development of the Jacobian matrices, based on the fact that the total transformation is a product of a sequence of transition matrices, each of which is regarded as a function of the joint angle. Leibniz’s law is then applied to the product to derive the Jacobian matrix. The major advantage of this approach is that the Jacobian matrix can be derived one column at a time, rather than one entry at a time. Consequently, the Jacobian matrix can be efficiently computed. Their small and simple example shows that their method produces the same symbolic Jacobian matrix as the one produced by the traditional Denavit-Hartenberg parameter technique. Their large and complex example verifies that the manipulator Jacobian matrix computed by their method is numerically accurate, while still simple and efficient to compute.

Their method is easy to implement and very useful for the analysis of complex spatial manipulators.