58 D.-M. Chen and W.D. Zhu known to transform measured velocities to the SCS as one out-of-plane motion and two in-plane motions. The orientation of a SLDV is often expressed using a rotation matrix. Zeng et al. [8] presented an algorithm to calculate the rotation matrix of a SLDV. This algorithm has an iterative process and uncertainty of the rotation matrix needs to be identified in the calculation. Hence, it may not be convenient to implement it. Later, Xu and Miles [9] summarized an identification technique to obtain the rotation matrix by calculating the inverse of a coordinate matrix. However, the rotation matrix is not guaranteed to be an orthogonal matrix with this technique and the inverse of the coordinate matrix does not exist when the structure is two-dimensional (2D) because two rows of the matrix are much close to linear dependent. In this work, a mathematical model based on the scan mirrors configuration is presented and coordinates of a scan point in a scan mirrors coordinate system (SMCS) are derived from it. The relation of coordinates of the same scan point in the SMCS and SCS is obtained using the rigid transformation theory. A rotation matrix and a translation vector from the SMCS to SCS are obtained using the least squares method and singular value decomposition (SVD). Experiments to scan a 3D structure and a 2D clamped plate with three SLDVs placed at three different locations were performed to validate the mathematical model. The experimental results show that the proposed methodology is reliable in calculating the rotation matrix. Rotation angles of two scan mirrors calculated using the rotation matrix and translation vector are in excellent agreement with corresponding measured angles. 6.2 Mathematical Model of a SLDV A galvanometer scanner has two orthogonal scan mirrors called X mirror and Y mirror, as shown in Fig. 6.1. In a SLDV, X mirror and Y mirror control the horizontal movement and vertical movement of a laser beam, respectively. Hence, the laser beam can be directed to any visible position on a structure surface by rotating the mirrors; the rotations of the mirrors are driven by input voltages. The rotation angle range of either mirror in Fig. 6.1 is ˙10ı, which corresponds to an input voltage of ˙10 V; the sensitivity of the scanner is 1ı/V. Initial positions of the two mirrors are 45ı with respect to the horizontal plane when the input voltages are zero. A mathematical model of the SLDV that describes the geometric relation for a scan point between its coordinates and rotation angles of two mirrors is shown in Fig. 6.2. There are two coordinate systems that need to be defined. The SCSo-xyz is usually fixed on the structure to be scanned so that coordinates of all scan points on it can be easily determined. The SMCS o0-x0y0z0 is specified as follows: the x00 axis andy0 axis are along the rotation axes of X mirror and Y mirror, respectively. The positive directions of the x00 axis and y0 axis are directed to motors of X mirror and Y mirror, respectively (Fig. 6.1). The x0 axis is parallel to the x00 axis and has the intersection point o0 with the y0 axis. The z0 axis is perpendicular to the plane formed by the x0 axis and y0 axis; it goes through o0 and has the intersection point o00 with the x00 axis. Both the SCS and SMCS are right-handed coordinate systems. Fig. 6.1 A galvanometer; the motor of X mirror is on the opposite of X mirror, which cannot be seen in the picture
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