6 A Mathematical Model for Determining the Pose of a SLDV 71 Fig. 6.10 (a) A 2D clamped plate scanned by three SLDVs placed at three different locations, and (b) the SCS of the clamped plate with nine scan points on it Table 6.9 Rotation matrices with respect to three different locations for the clamped plate Unit: ı x y z Location1 x0 90.436 171.829 81.840 y0 85.120 81.907 9.467 z0 175.101 88.878 85.231 Location2 x0 91.374 177.873 88.376 y0 123.447 87.887 33.531 z0 146.518 89.750 123.481 Location3 x0 83.791 171.370 84.030 y0 50.520 81.419 40.774 z0 139.839 90.904 49.853 6.6 Conclusion A mathematical model based on the scan mirrors configuration is developed in this work to determine the orientation and position of a SLDV. Coordinates of a scan point in the SMCS is derived from the mathematical model and the relation of coordinates of the same scan point in the SMCS and SCS is derived using the rigid transformation theory. The least squares method and SVD are used to calculate the rotation matrix and translation vector from the SMCS to SCS. A procedure to determine the orientation and position of a SLDV with respect to the SCS is presented. Experiments to scan a 3D structure and a 2D clamped plate are performed to validate the methodology. The 3D structure is scanned by three SLDVs placed at three different locations and two different sets of reference points are selected to obtain the rotation matrices and translation vectors. Results indicate that the proposed methodology is reliable and has high accuracy in estimating the orientation and position of a SLDV. The 2D clamped plate is a special case of a 3D structure and the rotation matrix and translation vector can also be obtained with high accuracy; this shows universal applicability of the proposed methodology and extends its application scope from 3D structures to 2D structures, which is a major improvement over previous techniques. Based on the methodology presented in this work, measured velocities by a SLDV can be transformed to the SCS in which one can better describe the ODS of a structure and analyze its dynamic response.
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