70 D.-M. Chen and W.D. Zhu Table 6.7 Euclidean-norm differences (unit: mm) between exact and estimated coordinates for the 3D structure Exact coordinate (mm) Location 1 Location 2 Location 3 x y z Set 1 Set 2 Set 1 Set 2 Set 1 Set 2 150 150 0 0.391 0.553 0.347 0.245 0.378 0.626 150 150 0 0.351 0.535 0.296 0.740 0.799 0.552 150 150 0 1.022 1.371 0.472 0.432 0.816 0.530 150 150 0 0.787 0.955 0.307 0.229 0.443 0.343 5 35 80 0.865 1.231 0.035 0.310 0.277 0.285 5 35 80 0.670 1.020 0.362 0.346 0.497 0.190 5 25 80 0.484 0.807 0.424 0.493 0.332 0.393 5 25 80 0.906 1.269 0.437 0.346 0.483 0.401 5 25 150 1.847 2.603 0.134 0.715 1.240 0.798 5 25 150 1.880 2.642 0.328 0.909 1.113 0.665 5 35 150 1.755 2.514 0.416 0.865 1.323 0.897 5 35 150 1.782 2.542 0.271 0.794 0.959 0.497 120 120 0 0.368 0.501 0.272 0.175 0.673 0.873 120 120 0 0.388 0.540 0.289 0.605 0.723 0.739 120 120 0 0.610 0.867 0.214 0.128 0.568 0.467 120 120 0 0.395 0.548 0.089 0.187 0.434 0.337 80 80 0 0.205 0.243 0.108 0.105 0.239 0.342 80 80 0 0.209 0.324 0.100 0.269 0.596 0.641 80 80 0 0.584 0.792 0.144 0.329 0.372 0.392 80 80 0 0.114 0.244 0.466 0.484 0.397 0.443 Table 6.8 Coordinates and measured rotation angles of nine scan points on the clamped plate Coordinates (mm) Location 1 Location 2 Location3 Index x y z ˛ (ı) ˇ(ı) r 0 (mm) ˛ ( ı) ˇ(ı) r 0 (mm) ˛ ( ı) ˇ(ı) r 0 (mm) 1 78.5 62 0 2.778 2.159 1400 0.864 3.907 1320 4.707 5.808 1340 2 65.5 57.5 0 5.635 2.099 1410 1.600 3.698 1400 7.487 5.701 1260 3 67.8 55.4 0 5.685 0.095 1410 1.654 1.486 1400 7.408 3.269 1260 4 82.8 63 0 2.673 0.251 1410 0.943 1.349 1320 4.514 3.288 1340 5 44.5 31.2 0 3.444 1.568 1400 0.257 3.253 1320 5.301 5.174 1340 6 23.5 39.2 0 4.813 1.748 1410 0.921 3.384 1400 6.637 5.339 1260 7 40.9 22.8 0 5.154 0.529 1410 1.206 2.127 1400 6.907 3.987 1260 8 40.8 36.5 0 3.546 0.246 1410 0.159 1.863 1320 5.336 3.764 1340 9 0 0 0 4.338 0.961 1410 0.530 2.593 1320 6.129 4.501 1300 6.5.2 2D Clamped Plate Scanning Figure 6.10a shows the second experiment where a clamped plate was scanned by three SLDVs. The locations and orientations of the three SLDVs are the same as those in the first experiment. Since there would be no marked reference points in a practical structure as in Fig. 6.8b, nine scan points are selected here and a SCS is built to determine coordinates of these points, as shown in Fig. 6.10b. The coordinates of the nine scan points measured by a ruler and their corresponding rotation angles of two mirrors are listed in Table 6.8. Based on the rectangular shape of the plate, four scan points with indices 2, 4, 5 and 7 are selected as the reference points to calculate rotation matrices and translation vectors. The rotation matrices with respect to three different locations are listed in Table 6.9. Since the orientations of the three SLDVs are the same as those in the first experiment, there are little differences between the rotation matrices in Tables 6.2 and 6.9. Figure 6.11 shows comparison between exact and estimated coordinates that are calculated using the same method as described in Sect. 6.5.1. Table 6.10 lists the corresponding Euclidean-norm differences between the exact and estimated coordinates. All the Euclidean-norm differences are less than 1.7 mm and are acceptable for an engineering application. This experiment indicates that the methodology can also be used to obtain the rotation matrix of a SLDV with high accuracy when a structure with less precise coordinates of reference points in the SCS is scanned.
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