Dual-number transformation and its applications to robotics

In robotics, to deal with coordinate transformation in three-dimensional (3D) Cartesian space, the homogeneous transformation is usually applied. It is defined in the four-dimensional space, and its matrix multiplication performs the simultaneous rotation and translation. The homogeneous transformation, however, is a point transformation. In contrast, a line transformation can also naturally be defined in 3D Cartesian space, in which the transformed element is a line in 3D space instead of a point. In robotic kinematics and dynamics, the velocity and acceleration vectors are often the direct targets of analysis. The line transformation will have advantages over the ordinary point transformation, since the combination of the linear and angular quantities can be represented by lines in 3D space. Since a line in 3D space is determined by four independent parameters, finding an appropriate type of "number representation" which combines two real variables is the first key prerequisite. The dual number is chosen for the line representation, and lemmas and theorems indicating relavent properties of the dual number, dual vector, and dual matrix are proposed. This is followed by the transformation and manipulation for the robotic applications. The presented procedure offers an algorithm which deals with the symbolic analysis for both rotation and translation. In particular, it can effectively be used for direct determination of Jacobian matrices and their derivatives. It is shown that the proposed procedure contributes a simplified approach to the formulation of the robotic kinematics, dynamics, and control system modeling.