6 A Mathematical Model for Determining the Pose of a SLDV 59 Fig. 6.2 Mathematical model of the SLDV, and the two coordinate systems o-xyz and o0-x0y0z0 Fig. 6.3 Direction of the laser beam when both mirrors are at their initial positions The constant distance of o0o00, denoted byd, is defined as the separation distance between two mirrors (Fig. 6.2). Variables ˛ and ˇ are rotation angles of X mirror and Y mirror measured from their initial positions, respectively. The positive directions of ˛ and ˇ are shown in Fig. 6.2. Each scan point P corresponds to a unique set of rotation angles ˛ and ˇ. Hence, a scan point Pwith coordinates in the SCS can also be expressed in the SMCS in terms of parameters ˛, ˇ, d and r, where r is the distance from point P0, which is the incident laser point of Y mirror, to point P(Fig. 6.2). Details on how to derive coordinates of the scan point Pin the SMCS in terms of parameters ˛, ˇ, d and r are described in what follows. 6.2.1 Case When Only X Mirror Rotates A laser beam directing to X mirror is perpendicular to the x00 axis, as shown in Fig. 6.3, which indicates that the laser beam comes out fromo0 in the negative direction of the x0 axis when both mirrors are at their initial positions. When ˇD0 and X mirror rotates with an angle ˛, the relation between the reflected laser beam and incident laser beam with respect to X mirror is shown in Fig. 6.4a based on the principle of light reflection, which indicates that the reflected laser beam rotates
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