Optimizing Motion Sequences with Projective Dual Quaternions

This paper builds upon a previous study of rotation sequences with four factors, in which the additional parameter is used for optimization, extending the result to generic rigid motions in three-dimensional Euclidean space. To do that in practice, one uses dual projective quaternions (dual Rodrigues’ vectors) describing screw motions and applies the well-known principle of transference in a rather straightforward manner. There are, however, some technicalities worth discussing, like the famous gimbal lock problem emerging in Euler-type decompositions. Also, there is ambiguity in the cost function which in this case includes both spherical and Euclidean distance—for pure rotations and translations, respectively. Since the relative cost of each counterpart depends on engineering details, we consider them separately. Explicit closed-form solutions are derived, based only on geometry, but numerical examples are also provided for illustration.