The quaternion-based spatial-coordinate and orientation-frame alignment problems

Hanson, A.J.

doi:10.1107/S2053273320002648

Figure 6
3D components of the rotation-average transformation of the quaternion orientation data set, with each point denoting the displacement between each pair of frames as a single quaternion, corresponding to the rotation taking the test frame to the reference frame. (a) The cluster of points $[t_{{k}} = r_{{k}}\star\bar{p}_{{k}}\to{\widetilde{t}}_{{k}}]$ derived from the frame-matching problem using just the curved arcs in Fig. 5

(b). If there were no alignment errors introduced in the simulation, these would all be a single point. The yellow arrow is the quaternion solution to the chord-distance centroid of this cluster and is identical to the optimal quaternion rotation transforming the test data to have the minimal chord measure relative to the reference data. (b) Choosing a less-cluttered subset of the data in (a), we display the geodesic paths from the initial quaternion displacements $[{\widetilde{t}}_{{k}}]$ to the origin-centered set with minimal chord-measure distance relative to the origin. This is the result of applying the inverse of the quaternion q_opt to each $[{\widetilde{t}}_{{k}}]$ . Note that the paths are curved geodesics lying properly within the quaternion sphere. (c,d) Rotating the cluster using a slerp between the quaternion barycenter of the initial misaligned data and the optimally aligned position, which is centered at the origin.

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ISSN: 2053-2733

Volume 76| Part 4| July 2020| Pages 432-457

https://doi.org/10.1107/S2053273320002648

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