Using the Algebra of Dual Quaternions for Motion Alignment

Abstract

Whenever measurements have to be taken with respect to two different coordinate frames the problem arises how to relate these measurements to each other. When these measurements are rigid 3D-displacements we obtain descriptions of motions with respect to two different coordinate systems. These systems might for example be the motor coordinate system of a vehicle and the coordinate system of a sensor mounted on the vehicle. If we use conventional homogeneous coordinates notation we usually obtain the well known equation AX = XB where all the variables are 4x4 matrices representing rigid motions. On the other hand, we might have line measurements with respect to two coordinate frames in which case we usually have a problem of the form P = QX where P, Qare matrices containing the Pliicker coordinates of the lines and Xa matrix encoding the rigid motion which we will describe later. We will first make a short break in our motivation in order to introduce the Clifford algebra we will use. Our geometric algebra treatment is inspired by [154], [167], and [22]. Then, we will describe 3D-lines and 3D-motions in this framework and we will present two examples of the algorithmic superiority of this representation.