Constructing Kinematic Confidence Regions With Double Quaternions

A spatial displacement as an element of $\mathrm{SE}\left(3\right)$ can be approximated by a 4D rotation, which is an element of $\mathrm{SO}\left(4\right)$ . In this way, the problem of constructing confidence regions of uncertain spatial displacements may be studied as that of constructing confidence ellipsoids in $\mathrm{SO}\left(4\right)$ . In this light, a double-quaternion formulation of kinematic confidence regions is presented that approximately preserve the geometry of $\mathrm{SE}\left(3\right)$ . Examples are provided to demonstrate the efficacy of this approach in comparison with the dual-quaternion formulation.