  We introduce motions as real six-dimensional vectors. A motion means a
rotation and a translation. We define a motion operator which maps unit dual
quaternions to motions, and a UDQ operator which maps motions to unit dual
quaternions. By these operators, we present the formulation of motion
optimization, which is actually a real unconstrained optimization formulation.
Then 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. This opens a new way to solve these
problems via real unconstrained optimization.
