6 A Mathematical Model for Determining the Pose of a SLDV 63 max 3X iD1 iMii Dmax . 1M11 C 2M22 C 3M33/ (6.20) The objective function in Eq. (6.20) is a linear function of variables M11, M22 and M33. Since the domain of each variable is [ 1, 1], the objective function attains its extremum on boundaries of the domains. If Ris a reflection matrix when Mii D1, the next maximization solution is M11 D1, M22 D1, and M33 D 1 since 1 > 2 > 3 0. Ris given by RDV2 4 1 0 0 0 1 0 0 0 1 3 5UT (6.21) A general formula is derived to calculate the rotation matrix Rthat includes both cases in which the determinant of VUT is 1 or 1: RDV2 4 1 0 0 0 1 0 0 0 det VUT 3 5UT (6.22) After Ris calculated, t is obtained by Eq. (6.10). Coordinates of an arbitrary scan point Qin the SMCS are obtained by x0Q; y0Q; z0Q T DRT x Q; yQ; zQ T t (6.23) where coordinates of point Qin the SCS are known. Further, coordinates of point Qin the SMCS can be transformed to corresponding rotation angles of two mirrors and the distance rQ: ˇQ D 1 2 tan 1 z0Q x0Q! ; ˛Q D 1 2 tan 1 0 B@ y0Q x0 Q cos.2ˇQ/ d 1 CA; rQ D x0Q cos 2˛Q cos 2ˇQ (6.24) A laser beam can be automatically directed to point Qwith corresponding input voltages to the two mirrors. 6.4 Procedure to Determine the Orientation and Position of a SLDV From Sect. 6.3, Randt can be obtained with known coordinates of at least three scan points in both the SCS and SMCS. These scan points are defined as reference points, whose coordinates in the SCS can be easily obtained and corresponding rotation angles of two mirrors can be accurately measured from experiment. The parameter r in Eq. (6.5) needs to be calculated by a geometric method shown in Fig. 6.6. The distance between reference point Pi and reference point Pj remains the same when their coordinates are expressed in the SCS and SMCS: q xi xj 2 C yi yj 2 C z i zj 2 Dr x0 i x0j 2 C y0i y0j 2 C z0i z0j 2 (6.25) where i is from1 toN 1and j is fromi C1toN. It is seen from Eq. (6.6) that coordinates of scan points in the SMCS can be transformed to those in the SCS using the rotation matrix and translation vector. For a 3D structure, the reference points should be selected to cover all the different heights of the structure in order to map the whole structure from the SMCS to SCS. The frame made up by the reference points should roughly outline the geometry of the structure. The N reference points have N unknown distances r (Fig. 6.6) and can formN .N 1/ equations like Eq. (6.25). This is an over-determined nonlinear problem if N is larger than 3, which can be solved by nonlinear least squares method. The initial value of the distance ri can be estimated using a ruler to measure the distance from Y mirror to reference point Pi. This is a good approach to obtain good estimates of the initial values to solve the equations.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==