Motion Addition and Motion Optimization

We introduce rotation addition and motion addition. In this way, motions replace unit dual quaternions to represent rigid body movements in the 3D space. The infinitesimal unit is no longer needed. By means of motion addition, we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as motion optimization problems, which are actually real unconstrained optimization problems. In particular, it avoids to go through the unit dual quaternion operations.