Modeling and Calibration of a Structured-light Optical - Michigan ...

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Fig. 1 Integration of a laser stripe sensor to a CMM The multiple-axis integrated system allows the sensor to be rotated to any angle for optimal view of the part surface to be measured. The same sensor can also be attached to the endeffector of a robot. The robot itself cannot be used for scanning due to its poor positioning accuracy. However, when combined with a precision X-Y motion stage, the robot integrated system has its advantages: the robot arm can easily position the sensor to reach and measure large parts from many positions and orientations. Since the laser scanner is only a two dimensional sensor, a third dimension needs to be integrated to the system to realize 3-D measurements. When scanning a part, the CMM shaft moves the sensor along a certain axis, e.g., the X axis, in small increments at a fixed speed. The sensor provides Ys and Zs data while the optical (or magnetic) scale of the CMM X axis provides the Xs information (Fig. 2). If the part instead of the sensor moves, the scanning mechanism is the same. If the laser plane, on which the sensor coordinate frame Ys- Zs lies, is always perpendicular to the surface to be measured (i.e., the laser plane is placed perpendicular to the CMM Xaxis), the coordinate system thus formed is Cartesian. However, this severely limits the application of the system, practically precluding the measurement of complex parts requiring multiple-axis scanner motions and multiple orientations of the laser sensor to achieve an optimal projecting angle. During scanning, the laser scanning coordinate system usually is non-orthogonal, i.e., skewed. If data obtained from this mechanical driving direction(Xs) Structured laser line lighting / Sensing optics >• Xs: added axis Fig. 2 2D sensor used for 3D measurement 596 / Vol. 118, NOVEMBER 1996 imaginary Cartesian sensor Zs(Zs') coordinate system {S'} world coordinate system {W} Xw YsCYs 1 ) real skewed sensor coordinate system (S} Fig. 3 Characterization of the laser stripe scanning system skewed coordinate system are to be used for dimensional measurement, they have to be transformed to a Cartesian frame. Otherwise, the dimensions on the part will be distorted, e.g., after non-Cartesian scanning, a sphere in real space will look like an ellipsoid and a cube will look like an inclined parallelepiped. 3 System Modeling 3.1 Skewed Frame Representation. Three frames, the world coordinate frame {W}, an imaginary sensor coordinate system {S'} and the sensor's body frame {S}, as shown in Figs. 3 and 4, are used in the following discussions. Both {W} and {S'} are Cartesian frames, but {S} frame is not. {S} and {S'} frames share the same origin. Let b aTbe the 4 X 4 matrix representation of coordinate transformation from {a} to {b}, a, b e {W,S',S). Then T is a homogeneous transform of which various equivalent representations exist (e.g., Craig, 1992; Wang, 1992). This paper adopts the matrix representation using the roll-pitch-yaw angles (a, /3, y): "T = cacfi sac (3 -sp 0 cas0sy — sacy sasfisy + cacy cfisy 0 caspcy + sasy sas/3cy — easy c0cy 0 : where sin a sa, cos a •• = COL and (qx,qy,qz, 1 ) vector Neither fT nor form. Their representations are explored below T is a translation W T constitutes a homogeneous trans- Ys(Ys A x^ Fig. 4 Spatial relationship between the real skewed sensor coordinate system {S} and the imaginary Cartesian coordinate system {S } (1) (2) Transactions of the ASME Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
In Figs. 3 and 4, the orientation of the real sensor Xs axis in the imaginary Cartesian sensor coordinate system {S'} can be defined by two angles, 9 and ip. 9 is defined as the angle between axes Xs and Xs'. ip is defined as the angle between Xprj (the orthogonal projection of Xs axis onto the Ys-ZslYs'-Zs' plane) and Ys. Point P is used to find the transformation from {S} to {S'}. P has two sets of coordinates, {x, y, z) in {S} and (x', y', z') m{S'}. Since the laser plane [Ys - Zs] moves along the Xs axis during scanning, the (x, y, z) coordinates of P in sensor frame are not all perpendicular projections onto the three axes of {S}. P' is the intersection of line PP' and the Ys-Zs plane, and PP' the second or the third column of the corresponding transformation matrices of Eq. (4) and Eq. (6). From Eq. (6), when 9 = 0°, fTbecomes an identical matrix. This implies that no skew transformations are involved, which is intuitively correct. When the Xs' axis coincides with the Xs axis, i.e., when the mechanical driving system is perpendicular to the 2D sensor plane, the coordinate system will become orthogonal. Here we limit the parameters to 0 deg s 9 < 90 deg and 0 deg s \p =s 360 deg. We have no interest in 180 deg a 6 a 90 deg because the coordinate frame will be left-handed. Combining the nonhomogeneous coordinate transformation matrix f'Tand the homogeneous coordinate transformation matrix y T, the composition gives cached + (casfisy — sacy)s9cip + (casficy + sasy)s9stp caspsy — sacy casficy + sasy qx sac0c9 + (sasfisy + cacy)s9cip + (sas/3cy — casy)s9sip sasPsy + cacy sasficy — easy qy -sficO + c{3sys9cip + cficys8sip cfisy cficy qz is parallel to axis Xs. The y and z coordinates of P, which are the lengths of OB and OA, are formed by the perpendicular projections of P' onto the Ys (Ys') and Zs (Zs') axes, respectively. Coordinate x of P is formed by a parallel projection of P onto Xs axis. The line of parallel projection PC is parallel to OP'. Thus, the x coordinate of P is formed by a parallel projection while the y and z coordinates are formed by perpendicular projections. This characteristic is very different from that of a normal Cartesian coordinate system, where all the point coordinates are formed by perpendicular projections. To derive the transformation between the two frames, the algebraic relationship between (x, y, z) and (x', y', z') needs to be established. We have, or in matrix form, s T - X = x' • sec 9 • y = y' — x' -tan 9 -cos ip . z = z' - x' -tan #-sin ip |~ sec 9 0 0 0 1 -tan 9 cos tp 1 0 0 -tan 9 sin ip 0 1 0 0 0 0 1 J Note that S S>T\s non-orthonormal. Besides, it is undefined at 9 = 90 deg which disables the data collection. From Eq. (3), the inverse transformation | Teem be obtained as or in matrix form x = x • cos 9 y' = y + x • sin 9 cos \p z' = z + x• sin 9 sin ip • cos 9 0 0 0' sin 8 cos ip 1 0 0 sin 9 sin ip 0 1 0 0 0 0 1 Note that both f. T and f' T are non-orthonormal. If the Ys or Zs axis were skewed, non-orthogonal terms would appear in 0 0 0 1 (3) (4) (5) (6) which describes the entire laser stripe scanning process. 3.2 Properties of the Composition Matrix and Identification of Extrinsic Parameters. When the angular encoder of the pan-tilt device is accurate enough, external calibration needs to be done only once when the system is initially setup. All the rest of the external re-calibrations could be avoided by knowing all the parameters of the position and orientation of the sensor. However, if the accuracy of the angular encoder is not good enough, the calibration matrix of (7) has to be identified whenever a change in sensor position or orientation occurs. In this section, the properties of the calibration matrix are explored and extrinsic parameters are derived to facilitate onetime calibration when system angular encoder is accurate. Since the three projections of the skewed sensor coordinate system are not all perpendicular projections, six parameters which are usually required of any homogeneous coordinate transformation can not fully describe the skewed frame transformation here. Two extra parameters, 9 and ip, are needed. Therefore, the number of extrinsic parameters extends to eight: five angular parameters {a, (3, y, 9, tp] and three translational parameters {qx, qy, qz]. We can also write the overall transformation as tn fl2 fl3 Pi til hi ?23 Pi hx hi ?33 P3 0 0 0 1 " R p~ 0 1 where R is a composition of rotation, shearing and scaling transformations and P = (pi, p2, PT, ) T is the translational vector. The orthonormality of a matrix governs that the norms of its three column vectors are one and that the three cross products of the vectors are zero, which gives six constraints to twelve variables. For the multiple-axis laser scanning process, due to the introduction of a skewed axis, two of the three orthogonality constraints are relaxed and it can be shown that the following four constraints exist for the overall transformation fT, Journal of Manufacturing Science and Engineering NOVEMBER 1996, Vol. 118/597 Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm (7) (8) (9)
Page 1: Chenggang Che Graduate Research Ass
Page 5 and 6: Fig. 6 Angular deformation as a fun
Page 7 and 8: Table 1 Comparative analysis of err
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