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

Hanson, A.J.

doi:10.1107/S2053273320002648

Figure 9
The behavior of the basins of attraction for the $[t_{{k}}\to{\widetilde{t}}_{{k}}]$ map is shown here, starting in (a) with the (q, −q) pairs for three movable clusters of quaternion-frame data, each having a well defined local quaternion average $[q_{\rm opt} = V(\,{\widetilde{t}}\,)/\| V\|]$ shown as the yellow arrows with their q → −q equivalents. Next we merge all three samples into one data set that can be smoothly interpolated between the data being outside the α = π/4 safe zone to all being together within that geometric boundary in quaternion space. Part (b) shows the results of taking 500 uniform samples of q and computing the set $[\{\,{\widetilde{t}}_{{k}}\,\}]$ for each sample q, placed at the magenta dots, and then computing the resulting q_opt; the black arrows follow the line from the sample point to the resultant q_opt. Clearly in (b), where the clusters are in their initial widely dispersed configuration, the black arrows (the `votes' for the best q_opt) collect in several different basins of attraction, signifying the absence of a global solution. We then interpolate all the clusters close to each other, and show the new results of the voting in (c). Now almost all of the samplings of the full quaternion space converge to point their arrows densely to the two opposite values of q_opt, and there is just one effective basin of attraction.

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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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