Modeling and Calibration of the Galvanometric Laser Scanning Three- Dimensional Measurement System

Transcription

1 Nanomanuacturing and Metrology (8) :8 9 (45689().,-volV)(45689().,-volV) ORIGINAL ARTICLES Modeling and Calibration o the Galvanometric Laser Scanning Three- Dimensional Measurement System Shuming Yang Linlin Yang Guoeng Zhang Tong Wang Xiaokai Yang Received: March 8 / Revised: 8 June 8 / Accepted: June 8 / Published online: 9 June 8 Ó International Society or Nanomanuacturing and Tianjin University and Springer Nature 8 Abstract The traditional three-dimensional measurement system based on linear-structured light usually involves mechanical scanning platorms to perorm the linear scanning, which make the structure o the system huge and complicated. The emergence o the galvanometric laser scanner solves the problem by using a galvanometer to replace the mechanical scanning platorm. The employment o the galvanometer can improve the speed o scanning and simpliy the structure o the system. However, there are ew approaches available to calibrate this kind o system. In this paper, a high-precision calibration method is proposed to calibrate the galvanometric laser scanning three-dimensional measurement system. A precision motorized linear stage and a planar target are applied in this method. The planar target is used or the camera calibration based on Zhang s method and the precision motorized linear stage or the laser plane calibration. The validity and accuracy o this method are evaluated by scanning the standard component. The experiments conducted suggest that the proposed method is valid and accurate or the calibration o the galvanometric laser scanning system. Keywords Galvanometer Linear-structured light System calibration Laser scanner Introduction In recent years, the laser scanning systems have been widely applied in many industrial applications, such as medical imaging [], quality assurance [], machine vision [, 4] and reverse engineering [5, 6]. The ast acquisition speed, the high accuracy, the good stability and the low cost make it one o the most promising techniques to obtain the geometric surace inormation. Usually, a laser scanning system is composed o a line laser, a CCD camera and & Shuming Yang shuming.yang@mail.xjtu.edu.cn Linlin Yang llyang@stu.xjtu.edu.cn Guoeng Zhang zg@stu.xjtu.edu.cn Tong Wang wt@stu.xjtu.edu.cn Xiaokai Yang yangxiaokai@stu.xjtu.edu.cn State Key Laboratory or Manuacturing System Engineering, Xi an Jiaotong University, Xi an 49, China a mechanical scanning platorm. The line laser projects a laser stripe onto the object surace, and then the mechanical scanning platorm drives the laser stripe or the object to perorm D scanning, and at the same time, the CCD camera captures the images with the modulated laser stripe. Ater calibration, the D geometry inormation can be obtained by the rule o triangulation. The traditional laser scanners usually use a mechanical scanning platorm [, 8] to perorm D scanning. However, the employment o the mechanical scanning platorm makes the structure o the scanning system huge and complicated, which leads to the limitation in industrial application. Li [9] designed a portable laser -D scanner by mounting the laser stripe sensor to the robot end-eector. The accuracy o the -D scanner is limited by accuracy o the robot end-eector, and the scanning speed is quite slow. In recent years, the galvanometric laser scanning system has been proposed and it provides a good solution to those problems by using galvanometers to replace the mechanical device. A galvanometer includes a galvanometer-based scanning motor with an optical mirror and a detector which can send eedback signal to the system. The galvanometer has a high accuracy and

2 Nanomanuacturing and Metrology (8) :8 9 8 repeatability so it can provide a guarantee or the measurement accuracy. But there are ew approaches available to calibrate this kind o system. Manakov [] introduced a method to calibrate the two-mirror galvanometric laser scanner but it s hard to deal with the optimization o the system s mathematical model. Wissel [] proposed a datadriven learning calibration method which needs to collect a large number o data. Wagner [] used statistical learning methods such as artiicial neural networks (ANNS) and linear regression to calibrate the system, but this approach is easy to cause over-itting problem and oten has a high computational cost. Yu [] designed a novel one-mirror galvanometric laser scanner. However, the calibration procedure is quite complex and the objective unction has independent unknown parameters to be optimized. Similarly, Chi [4] designed a laser line auto-scanning system using a galvanometer and proposed the calibration method, but the calibration accuracy is not good enough. One-mirror galvanometric laser scanner is more eicient than the two-mirror galvanometric laser scanner and will not cause image distortion, so it has broader prospect in industrial application. The biggest dierence between the galvanometric laser scanning system and the traditional line-structured light scanning system is that the relative position between the laser plane and the camera in galvanometric laser scanning system is changing all the time. In the galvanometric laser scanning system, neither the measured objects nor the measurement device is moved and the only moving part is the laser plane. The change o the scanning mode leads to the change o the system mathematical model, the same to the calibration method. But the existing calibration methods are either only useul or the traditional linear scanning or too complex. A method to calibrate the one-mirror galvanometric laser scanning system is proposed in this paper. A planar chessboard target and a precision motorized linear stage are used. The chessboard target is used or the CCD camera calibration, and the precision motorized linear stage or the light plane calibration. The usage o the precision motorized linear stage ensures that the calibration result has a high accuracy. O W Y W CCD Camera Z W X W Line Laser The Measured Objects Galvo-System galvanometer (GVS/M) is single-axis scanning galvosystem. The employment o the galvanometer makes the whole scanning system small and easily to be integrated. The galvanometer can be driven to scan ± range by the control voltage ranging rom - to? V. The maximum scan requency is khz with the angular resolution o.8, so the system promises to have a high accuracy and resolution. When the system starts to work, the line laser projects a laser stripe onto the mirror o the galvanometer, and then the laser stripe is relected onto the object surace immediately. The shat o the galvanometer rotates as the control voltage changes, carrying the laser stripe to scan the object. At the same time, the CCD camera captures the modulated laser stripe images. Ater perorming the system calibration, the D geometry inormation o the object can be derived rom the captured images. The scanning speed o the galvanometric scanning system is ast so it has great potential or industrial applications, especially or online detection and real-time measurement. The accuracy o the system is mainly depending on the calibration method, so a convenient and accurate calibration method is too signiicant or the system. X G ZG Y G O G Fig. The diagram o the galvanometric scanning system System Construction and Working Principle As shown in Fig., the galvanometric laser scanner mainly consists o three parts: a line laser, a CCD camera and a galvanometer. They are mounted on a base structure to maintain their relative position stationary. The CCD camera (DMK 4AUC) is a monochrome industrial camera with a mm lens (Computer 4-MP). The line laser is a normal red line laser with a wavelength o 65 nm. The Mathematical Model and Calibration Method The system calibration method we proposed is separated into three parts: the irst part is to calibrate the CCD camera; the second part is to calibrate the laser plane; and the third part is to correct the galvanometric laser scanning system. Ater the camera calibration, the intrinsic parameters o the camera and the mapping relationship between the image coordinate rame and the world coordinate rame

3 8 Nanomanuacturing and Metrology (8) :8 9 are identiied. And then the mapping relationship between the world coordinate rame and the galvanometer coordinate rame is determined by the laser plane calibration. These two parts are combined together to inish the system calibration. Besides, the calibration results are used to correct the galvanometric laser scanning system. Ater all the work is done, the system can be used or three-dimensional measurement.. System Mathematical Model The system mathematical model is set up as shown in Fig.. In this model, ive dierent coordinate rames are created: the image pixel coordinate rame O uv, the image physical coordinate rame O XY, the camera coordinate rame O C X C Y C Z C, the world coordinate rame O W X W- Y W Z W and the galvanometer coordinate rame O G X G Y G Z G. The world coordinate rame is set on the surace o the planar chessboard target, in which O W is the upper-let corner o the planar target, X W and Y W are consistent with the direction o the chessboard array, and Z W is perpendicular to the O W X W Y W plane according to the right-hand coordinate system. The galvanometer coordinate rame is set on the galvanometer, in which Y G axis is upward along the rotating shat o the galvanometer, O G is the intersection point o the Y G axis and the O W X W Z W plane, X G is accordant with the direction o the normal vector o the laser plane when the control voltage is U, and Z G is set according to the right-hand coordinate rame. According to the pinhole model o the camera [5], the mapping relationship between the image pixel coordinate (u, v) and the world coordinate (X W,Y W,Z W ) is, u u dx X W q4 v 5 ¼ R T 6 4 v 6 Y W 5 dy T 4 Z W 5 Þ where (u,v ) is the principal point o the image, dx and dy are the pixel size o the camera which can be derived rom the speciication given by the CCD manuacturer, q is a scale actor, is the ocal length o the camera, the R (a 9 matrix) and T (a 9 vector) represent the mapping relationship between the camera coordinate rame and the world coordinate rame, and all those parameters can be Fig. The setup o system mathematical model

4 Nanomanuacturing and Metrology (8) :8 9 8 derived rom the camera calibration. Considering the radial distortion o the lens, the distorted images can be ixed by, ( u ¼ u d þ k q Þ v ¼ v d þ k q Þ Þ q ¼ Xd þ Y d ¼ ½ u d u ÞdxŠ þv ½ d v ÞdyŠ Plane i Z G Plane Y G where (u, v) is the ideal image pixel coordinate, (u d,v d )is the actual image pixel coordinate, and k is the lens distortion coeicient. The world coordinate rame and the galvanometer coordinate rame are both three-dimensional coordinate system. According to the rigid transormation relationship between the Euclidean coordinate system, the transormation between the world coordinate (X W,Y W,Z W ) and the galvanometer coordinate (X G,Y G,Z G ) is, X W Y W 6 4 Z W X G Y G 6 4 Z G 5 Þ 5 ¼ R T T where R is a 9 rotation matrix, and T is a 9 translation vector. R and T show the mapping relationship between the galvanometer coordinate rame and the world coordinate rame which can be obtained rom the laser plane calibration. Combining Eqs. () and (), the mapping relationship between the image pixel coordinate (u, v) and the galvanometer coordinate (X G,Y G,Z G ) can be obtained, u u 6 dx X G RT þ T 6 q4 v 5 ¼ 6 v 4 5 dy RR T 6 4 Y G Z G 5 4Þ Let M =[RR RT? T], so M is a 9 4 matrix, that is, m m m m 4 M ¼ 4 m 5 m 6 m m 8 5 5Þ m 9 m m m Substitute Eq. (5) into Eq. (4), the ollowing equation can be obtained, X G θ i O G Fig. The schematic diagram o determining the laser plane equation Equation (6) establishes the mapping relationship between the image coordinate rame and the galvanometric coordinate rame. However, we cannot get the three-dimensional point (X G,Y G,Z G ) rom the two-dimensional point (u, v) by two equations, so we need ind more constraints. According to the deinition o the galvanometer coordinate rame, the Y G axis is the upward direction o the galvanometer shat, and the X G axis is the positive direction o the initial laser plane normal vector when the control voltage is U, so the initial equation o the laser plane is, X G ¼ Þ The laser plane changes along with the rotation o the mirror just as shown in Fig.. I the rotational angle o the laser plane is h, the laser plane unction can be determined, X G coshþ Z G sinhþ ¼ 8Þ According to the galvanometer manual, there is a linear relationship between the rotational angle o the laser plane and the control voltage, assuming the scale actor is a, so the relationship between the rotational angle o the laser plane and the change o the control voltage can be expressed as: 8 u u Þm 9 dx m X G þ u u Þm dx m Y G þ u u Þm dx m Z G ¼ m 4 u u Þm dx >< v v Þm 9 dy m 5 X G þ v v Þm dy m 6 Y G þ v v Þm dy m Z G ¼ m 8 v v Þm dy >: 6Þ

5 84 Nanomanuacturing and Metrology (8) :8 9 Dh ¼ a DU 9Þ Substitute Eq. (9) into Eq. (8), we can get the ollowing equation, X G cosau U ÞÞ Z G sinau U ÞÞ ¼ Þ where a is the scale actor between the control voltage and the mirror rational angle, U is the initial control voltage, and U is the inal control voltage. This equation shows how the laser plane changes along with the change o the galvanometer control voltage. Combining Eq. () with Eq. (6), the mapping relationship between the image pixel coordinate rame and the galvanometer coordinate rame along with the control voltage can be obtained, target, and images o the chessboard target were captured rom dierent orientations. The last image was captured when the target was placed on the precision motorized linear stage, where the moving direction o the linear stage is perpendicular to the target plane. The captured images are used or the camera calibration. Ater camera calibration, the intrinsic parameters and the extrinsic parameters are gained, and the pose o each calibration target with respect to the camera coordinate system is determined. The world coordinate rame is set on the last pose o the calibration target where the target is placed on the linear stage. According to the deinition o the world coordinate rame, the extrinsic parameters (R and T) corresponding to the last image are regarded as the mapping 8 u u Þm 9 dx m X G þ u u Þm dx m Y G þ u u Þm dx m Z G ¼ m 4 u u Þm dx >< v v Þm 9 dy m 5 X G þ v v Þm dy m 6 Y G þ v v Þm dy m Z G ¼ m 8 v v Þm dy >: X G cosau U ÞÞ Z G sinau U ÞÞ ¼ Þ All the unknown parameters in Eq. () can be determined by the system calibration. We separate system calibration into three parts. Actually, the camera calibration is to determine the ocal length, principal point (u,v ), rotation matrix R and translation vector T; the laser plane calibration is to determine the rotation matrix R and translation vector T ; and the galvanometric laser scanning system correction is to get the actual value o a instead o using the galvanometer standard scale actor. Ater the system calibration, we can get the galvanometric coordinate (X G,Y G,Z G ) by the corresponding image coordinate (u, v, h).. Camera Calibration The cameral calibration is to determine the intrinsic parameters and the extrinsic parameters o the camera, which include the ocal length, the principle point (u,v ), the rotation matrix R and the translation vector T in Eq. (). The camera calibration theory [6 ] has developed rapidly, and lots o calibration method are proposed in recent years. Zhang [8] provides a lexible camera calibration algorithm by observing a planar target rom multiple points o view, which is easible with quite high accuracy. So in this paper, we used this method to calibrate the camera. A 9 8 chessboard was used as the planar relationship between the camera coordinate rame and the world coordinate rame. The camera calibration results will also serve or the laser plane calibration.. Laser Plane Calibration As mentioned beore, the laser plane calibration is to determine the mapping relationship between the world coordinate rame and the galvanometer coordinate rame, that is, to get the R and T in Eq. (). The world coordinate rame and the galvanometer coordinate rame are both three-dimensional coordinate rame in Euclidean space. I the galvanometer coordinate origin O G and the unit direction vectors o X G axis, Y G axis and Z G axis in the world coordinate rame are known, then the R and T can be obtained. A precision motorized linear stage is used to calibrate the laser plane. By changing the voltage o galvanometer and moving the precision motorized linear stage in the direction which is perpendicular to the plane o the planar target, a large number o laser stripe images o dierent laser planes can be captured. The mapping relationship between the world coordinate rame and the galvanometer coordinate rame can be determined by dealing with those laser stripes, as shown in Fig. 4. In act, the world coordinate rame is established on the chessboard target when the linear stage is at position Z,so

6 Nanomanuacturing and Metrology (8) : Fig. 4 The schematic diagram o the laser plane calibration procedure all the laser stripes on the same target plane share the same value o Z W, that is, Z W =. Similarly, when the linear stage is in position Z, Z,, Z n, all the laser stripes have the same value o Z W at each position, that is, Z W- = Z - Z, Z W = Z - Z,, Z Wn = Z n - Z. The image pixel coordinates (u, v) o all the laser stripes can be extracted rom the laser stripe images by the grayscale barycenter method. Combining with the value o Z W, the world coordinates (X W,Y W,Z W ) o the laser stripes can be obtained by, u u dx X W q4 v 5 ¼ R T 6 4 v 6 Y W 5 T 4 Z dy W 5 Þ All the parameters in the above ormula can be gained by the camera calibration. The laser stripes o the same control voltage belong to the same laser plane, so these laser stripes can be used to it the laser plane equation by least-square method. The equation o the laser plane can be written as, Ax þ By þ Cz þ D ¼ Þ The unit normal vector o a certain laser plane can be gained rom the above equation, and we can make it nj; k; lþ. For all the control voltage U, U, U,, U m (m C 4), we can igure out the unit normal vectors n j ; k ; l Þ, n j ; k ; l Þ, n j ; k ; l Þ,, n m j m ; k m ; l m Þ (m C 4). Ideally, all the laser planes intersect into one line, which is the shat o the galvanometer. From the deinition o the galvanometer coordinate rame, Y G axis is upward along the shat o the galvanometer, so Y G axis is perpendicular to all the unit normal vectors. Assuming the unit direction vector the Y G axis in the world coordinate rame is n y a y ; b y ; c y Þ, or each laser plane, n y n i ¼ i ¼ ; ; ;...; mþ 4Þ In reality, due to the impact o the environment and system assembly deviation, the dot product o n y and n i is not equal zero, and the error can be deined as, error ¼ n y n i 5Þ ¼ n y n i For all laser planes, the objective unction can be deined as,! F ¼ min Xm errorþ i¼! ¼ min Xm a y j i þ b y k i þ c y l i Þ m4þ 6Þ i¼ The vector n y a y ; b y ; c y can be obtained by minimizing the objective unction. According to the deinition o the galvanometer coordinate rame, the unit direction vector o X G n x a x ; b x ; c x Þ in the world coordinate rame is in accordance with n j ; k ; l Þ, namely n x = n. So the unit direction vector o Z G n z a z ; b z ; c z Þ can be gained by n x n y. The Y G axis equation in the world coordinate rame is, x x ¼ y y ¼ z z Þ a y b y c y where P(x,y,z ) is a point on the Y G axis, gained by itting all laser plane by the least-square method.

7 86 Nanomanuacturing and Metrology (8) :8 9 The point O G is the intersection point o the Y G axis and the O W X W Z W plane, so the coordinate o O G in the world coordinate rame can be igured out as O G (x w,,z w ), 8 x w ¼ x a y y >< b y y w ¼ z w ¼ z c y >: y b y 8Þ So the transormation between the world coordinate rame and the galvanometer coordinate rame can be obtained by solve the ollowing equations, 8 ½x w ; ; z w Š T ¼ T >< ½a x ; b x ; c x Š T ¼ R ½; ; Š T T¼ a y ; b y ; c y R ½; ; Š T 9Þ >: ½a z ; b z ; c z Š T ¼ R ½; ; Š T Finally, the parameters R and T in Eq. () are obtained and the laser plane calibration is completed..4 Galvanometric Laser Scanning System Correction In the previous section, we have mentioned the galvanometer scale actor a which represent the linear relationship between the rotational angle o the laser plane and the increase o the control voltage. This scale actor can be obtained by the galvanometer manual. In act, the galvanometer is driven by a PCI data acquisition card (the PCI D/A unction). We set an output digital voltage signal to PCI data acquisition card and then the PCI data acquisition card converts it to an analog voltage signal to drive the galvanometer. Due to the limitation o the conversion accuracy o the PCI data acquisition card, the rotation angle o the laser plane does not meet the instructions o the galvanometer, which means there is an error between the ideal value and the true value o the scale actor. In this part, our aim is to get the true value o the scale actor a to correct the galvanometric laser scanning system. To correcting the galvanometric laser scanning system, we would use the results o the laser plane calibration. In the laser plane calibration part, dierent laser plane equations have been obtained, and the corresponding control voltages o the galvanometer are also known. The actual rotation angle o the laser plane can be calculated by the ollowing equation, n n i h ¼ acos Þ jn jjn i j where n is the unit normal vector o the irst laser plane, n i is the unit normal vector o the ith laser plane. Ater obtaining all the dierent rotation angle o the laser plane and the corresponding control voltage o the galvanometer, we can it the relationship between the rotation angle and the control voltage, that is, to correct the scale actor a. Ater the camera calibration, the laser plane calibration and the galvanometric laser scanning system correction, all the parameters in Eq. () are obtained and the system calibration is inally completed. The system can be used or D geometry measurement. 4 Experiments The proposed calibration method is used to calibrate the galvanometric laser scanning system to evaluate the easibility and accuracy. Ater the calibration, standard workpieces (a standard ceramic ball and two gauge blocks) were measured or several times. By comparing the measure value with the true value o the workpieces, the eiciency and accuracy o the proposed calibration method are veriied. 4. System Calibration The structure o the system is displayed in Fig., and it is composed o a CCD camera, a line laser and the galvanometer system. First o all, the camera was calibrated. The pixel size o the CCD is.5 lm 9.5 lm, and the resolution o the image is 4(H) 9 (V). Twenty images o the chessboard target were captured rom dierent positions, and the last image was captured when the planar target was placed on the precision motorized linear stage. The precision motorized linear stage (PSA--x) is rom Zolix. It has a high resolution up to.5 lm. The images were used to calibrate the camera by MATLAB camera calibrator. Ater the calibration, the intrinsic parameters were =.4 mm, u = 6.598, v = 45.56, k = -.. The extrinsic parameters o the last image were the translation between the camera coordinate rame and the world coordinate rame, that is, :9898 : :4 6 R ¼ 4 : :998 :666 5 :4 :656 :98 T ¼½ 9:998; 4:969; 55:948Š The chessboard target was mounted on the precision motorized linear stage. The precision motorized linear stage drive the target to move six times and move 5 mm each time. At each position, six laser stripes were captured.

8 Nanomanuacturing and Metrology (8) :8 9 8 Fig. 5 The laser stripes and itted laser plane in world coordinate rame Z/mm X/mm Y/mm Table The control voltage and the rotation angle Plane no. Control voltage (V) Rotation angle ( ) Rotation Angle/ Ideal Output.5 Actual Output Control Voltage/V Fig. 6 The relation between the rotation angle and the control voltage And then the control voltage increase to get six dierent laser planes and all the laser stripes was captured. Finally, we can get 6 laser stripes and stripes belong to the same plane were itted to the plane equation. Those stripes and planes are shown in Fig. 5. All the laser stripes were used to calibrate the laser plane by the laser plane calibration method in this paper. Ater the calibration o the laser plane, the R and T were obtained, :98 : :59 6 R ¼ 4 :8 :9999 :54 :59 :4 :98 T ¼½56:5; ; 4:5Š Ater the laser plane calibration, six dierent laser planes were obtained and the unit normal vector o each laser plane was calculated. And then all the rotation angle o the laser plane can be calculated by Eq. () and the results are shown in Table. According to the speciication o the galvanometer, the ideal scale actor a is V/, which means that the laser plane ideally rotates degree while the control voltage increases V. By combining the actual rotation angle o the laser plane, we can get the relationship between the rotation angle and the control voltage, which is shown in Fig. 6. From the above igure, we can see that there is a linear relationship between the actual laser plane rotation angle and the control voltage o the galvanometer. By linear itting, we can get the actual scale actor. In the system, this 5

9 88 Nanomanuacturing and Metrology (8) :8 9 Fig. a The standard ceramic ball; b the point cloud o the ceramic ball; c the error distribution o the itting sphere and the point clouds Table The itting results o the standard sphere Ball position Fitted sphere centers (mm) Fitted diameter (mm) Error (mm) X G Y G Z G Diameter Error o Standard Sphere/mm Ball Position Fig. 8 Diameter error with the ball position actual scale actor is.958 V/, which is smaller than the ideal scale actor. So ar, all the parameters in Eq. () were ensured and the system calibration was complete. In the ollowing part, we would use the galvanometric laser scanning three-dimensional measurement system to do some experiments to test the calibration accuracy. 4. Accuracy Tests 4.. Ceramic Ball Scan Test To evaluate the easibility and accuracy, a standard ceramic ball as shown in Fig. a was measured rom dierent positions. The standard diameter o the ceramic ball is mm, and the measured value is mm. Comparing the diameter o the ceramic ball with the itting diameter o the D point clouds, the eiciency and accuracy o the proposed calibration method are veriied. The point clouds o the ceramic ball were obtained as shown in Fig. b. Ater getting the point clouds o the sphere, we put it into the Image ware sotware and use the sphere itting unction to get the diameter, and the error distribution between all the scatter points and the itted sphere is shown in Fig. c. The itted diameters were compared with the actual value o the ceramic ball, and the results are shown in Table. According to the results, the average itting diameter o the ceramic ball is.5 mm. The diameter errors are distributed rom -.99 to.69 mm and the

10 Nanomanuacturing and Metrology (8) : Fig. 9 a The measured gauge blocks; b the model o the gauge blocks 4.. Gauge Blocks Scan Test Z/mm X/mm Fig. The point clouds o the measured planes RMSE value is.9 mm, which proves that the precision o our calibration method is high enough. By analyzing the data in Table, we can see how the diameter error distributed along with the change o the standard ball position, which is shown in Fig. 8. There is a trend that the arther the ball position is, the greater the diameter error will be. Thickness Error o Gauge Block /mm Y/mm Laser Plane Fig. Thickness error with the laser plane rotation angle Apart rom the standard ceramic ball, gauge blocks were also measured by the scanning system. Two dierent gauge blocks were put together just as shown in Fig. 9a. The measured thickness o ront gauge block is 8.95 mm. Ater the surace o the two gauge blocks were scanned, the ront gauge block s thickness can be measured by calculating the distance o the two parallel planes (planes and ). Gauge blocks were also measured by the scanning system. Two dierent gauge blocks were put together just as shown in Fig. 9a. Ater the surace o the two gauge blocks was scanned, the thickness o the ront gauge block can be measured by calculating the distance o the two parallel planes (planes and ) just as shown in Fig. b. The gauge blocks were scanned at dierent positions or ten times, and the point clouds o the two planes were obtained. The point clouds belonging to plane were used to it into a plane by the least-square method. The equation o plane is the same as Eq. (). And then points belonging to plane were randomly chosen and the distance between the plane and each point was calculated by, j d i ¼ Ax i þ By i þ Cz i þ Dj piiiiiiiiiiiiiiiiiiiiiiiiiii Þ A þ B þ C where d i is the distance; (x i, y i, z i ) is the coordinate o the point; A, B, C and D are the coeicients o plane. The average distance between those points and plane was regard as the thickness o the ront gauge block. The point clouds o the measured two planes are shown in Fig., and these point clouds were used to calculate the thickness o the ront gauge block. The results are shown in Table. The measured value o the thickness is ranging rom 8.94 to 9.6 mm. The average thickness is mm and the error is only.84 mm, which means that the calibration method proposed in this paper has a quite high accuracy. We get how the thickness error distributed with the increase in the laser plane rotation angle

11 9 Nanomanuacturing and Metrology (8) :8 9 Table The measured results o the gauge blocks Position no. Average distance (mm) Deviation (mm) Maximum Minimum RMSE Fig. a The image o the plaster model; b the -D model o the plaster model in Fig., and it indicates that the greater the laser plane rotation angle is, the greater the error will be. 4.. Plaster Model Scan Test Apart rom the above two measurements, we also use the galvanometric laser scanning three-dimensional measurement system to scan a plaster model. The size o this model is about 8 mm 9 8 mm 9 5 mm, and the result is shown in Fig.. The irst image is the image o the original plaster model, and the second image is the -D model o the plaster model. The reconstructed -D model is very smooth with detail. 5 Conclusions An approach was proposed to calibrate the galvanometric laser scanning system in this paper. The mathematical model o the scanning system was established based on the principle o triangulation, and then the calibration procedure was presented. A planar chessboard target and a precision motorized linear stage were used to calibrate the scanning system. A standard ceramic ball and two gauge blocks were measured to evaluate the easibility and the accuracy o the calibration method. The experimental results showed that the proposed calibration method was easible and had a quite high accuracy. So the method presented in this paper can be used to calibrate the galvanometric laser scanning system and be used as a reerence or the calibration o similar laser scanning system. Acknowledgements The authors would like to thank the supports by National Science Fund or Excellent Young Scholars (No. 559), National Natural Science Foundation o China (No ), National Key R&D Program o China (YFB4) and Shaanxi Science and Technology Project (No. 6GY-). Reerences. Zhao Y, Xiong Y, Wang Y () Three-dimensional accuracy o acial scan or acial deormities in clinics: a new evaluation method or acial scanner accuracy. PLoS ONE ():e694

12 Nanomanuacturing and Metrology (8) : Ghahremani K, Saa M, Yeung J et al (5) Quality assurance or high-requency mechanical impact (HFMI) treatment o welds using handheld D laser scanning technology. Weld World 59():9 4. Cao JL, Luo Y, Li Z () Study on -D laser-scanning-based machine vision system or robotic construction vehicles. Adv Mater Res 59 59: Lindner L, Sergiyenko O, Rivas-López M, et al () Machine vision system errors or unmanned aerial vehicle navigation. In: IEEE, international symposium on industrial electronics, pp Javed MA, Won SHP, Khamesee MB et al () A laser scanning based reverse engineering system or D model generation. IECON - 9th Annual Conerence o the IEEE Industrial Electronics Society, Vienna, pp Sang CP, Chang M (9) Reverse engineering with a structured light system. Comput Ind Eng 5(4): 84. Yin S, Ren Y, Guo Y, Zhu J, Yang S, Ye S (4) Development and calibration o an integrated D scanning system or highaccuracy large-scale metrology. Measurement 54: Xiao J, Hu X, Lu W, Ma J, Guo X (5) A new three-dimensional laser scanner design and its perormance analysis. Optik 6: 9. Li JF, Zhu JH, Guo YK, Lin XD, Duan KL, Wang YS, Tang Q (8) Calibration o a portable laser -D scanner used by a robot and its use in measurement. Opt Eng 4:. Manakov A, Seidel HP, Ihrke I () A mathematical model and calibration procedure or galvanometric laser scanning systems. In: Vision, modeling, and visualization workshop, Berlin, 4 6 October, DBLP, pp 4. Wissel T, Wagner B, Stuber P, Schweikard A (5) Data-driven learning or calibrating galvanometric laser scanners. IEEE Sens J 5:59 5. Wagner B, Stuber P, Wissel T, Bruder R, Schweikard A, Ernst F (4) Accuracy analysis or triangulation and tracking based on time-multiplexed structured light. Med Phys 4:8. Yu C, Chen X, Xi J () Modeling and calibration o a novel one-mirror galvanometric laser scanner. Sensors ():64 4. Chi S, Xie Z, Chen W (6) A laser line auto-scanning system or underwater D reconstruction. Sensors 6(9):54 5. Xie Z, Wang X, Chi S (4) Simultaneous calibration o the intrinsic and extrinsic parameters o structured-light sensors. Opt Lasers Eng 58(4): Tsai R (98) A versatile camera calibration technique or highaccuracy D machine vision metrology using o-the-shel TV cameras and lenses. IEEE J Robot Autom (4): 44. Faugeras OD, Luong QT, Maybank SJ (99) Camera sel-calibration: theory and experiments. In: European conerence on computer vision. Springer, Berlin, pp 4 8. Zhang Z (999) Flexible camera calibration by viewing a plane rom unknown orientations. In: The proceedings o the seventh IEEE international conerence on computer vision, vol, pp Junhua Sun (9) Camera calibration based on lexible D target. Acta Opt Sin 9():4 49. Peng E, Li L () Camera calibration using one-dimensional inormation and its applications in both controlled and uncontrolled environments. Pattern Recogn 4():88 98 Shuming Yang is a proessor at School o Mechanical Engineering, Xi an Jiaotong University (XJTU), China. He achieved his BSc and MSc in mechanical engineering rom XJTU, and Ph.D. in nanotechnology and instrumentation rom University o Huddersield (UoH) o the UK. He then started to work at UoH, ater that he joined in XJTU till now. His research areas include micro-/nano-abrication and measurement, optical technology and instrumentation, precision/ultraprecision manuacturing etc. Linlin Yang is a MSc candidate at School o Mechanical Engineering, Xi an Jiaotong University, China. He achieved his BSc in mechanical design-manuacture and automation rom Sichuan University. His research interest is optical D measurement. Guoeng Zhang is a Ph.D candidate at School o Mechanical Engineering, Xi an Jiaotong University (XJTU), China. He achieved his MSc in mechanical engineering rom XJTU, and BSc in measurement and control technology and instrumentation rom Xi an Technological University. His research interests are ocused on optical intererometry and instrumentation.

13 9 Nanomanuacturing and Metrology (8) :8 9 Tong Wang is a Ph.D candidate at School o Mechanical Engineering, Xi an Jiaotong University, China. He achieved his BSc and MSc in mechanical engineering rom Zhengzhou University and Beijing University o Technology, respectively. His research interests are ocused on optical measurement and diraction optics. Xiaokai Yang is a Ph.D candidate at School o Mechanical Engineering, Xi an Jiaotong University (XJTU), China. He achieved his MSc in computer science and technology rom XJTU, and BSc in mechanical design-manuacture and automation rom Shaanxi University o Technology. His research interests are ocused on numerical simulation o electromagnetic ield and near-ield optics.