Abstract

An extrinsic parameters calibration method of multi-cameras with non-overlapping fields of view (FOV) using laser scanning is presented. Firstly, two lasers are mounted on a multi-degree-of-freedom manipulator and can scan objects freely by the projected line-structured light. Then, controlling the movement of the manipulator, the line-structured light is projected into the field of view of one of the multi-cameras, and the light plane equation in the camera coordinate frame is calibrated by the target. The manipulator is moved several times in small amplitude to change the position of structured light in the field of vision of the camera and to continue to calibrate the light plane. The light plane equation of line-structured light in the manipulator coordinate frame are solved by the hand-eye calibration method. Secondly, with the help of the light planes, projected into the field of vision of other cameras to be calibrated, the light plane equation in the camera coordinate frame is calibrated, and the external parameters between the camera coordinate frame and the manipulator coordinate frame are calculated, so that the calibration of the external parameters of multiple cameras can be realized. The proposed method connects the non-overlapping multi-cameras by the laser scanning. It can effectively solve the problem of multi-camera extrinsic parameter calibration under the conditions of long working distance and complex environment light.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (3)

2017 (2)

2016 (3)

2015 (2)

Z. Y. Xu, Y. Wang, and C. Yang, “Multi-camera global calibration for large-scale measurement based on plane mirror,” Optik 126(23), 4149–4154 (2015).
[Crossref]

Z. Liu, X. J. Li, and Y. Yin, “On-site calibration of line-structured light vision sensor in complex light environments,” Opt. Express 23(23), 129896–129911 (2015).
[Crossref]

2014 (3)

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Z. Liu, F. J. Li, and G. J. Zhang, “An external parameter calibration method for multiple cameras based on laser rangefinder,” Measurement 47, 954–962 (2014).
[Crossref]

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

2013 (1)

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

2012 (1)

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

2011 (2)

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

2008 (1)

Z. Liu, G. Zhang, and Z. Z. Wei, “Global calibration of multi-vision sensor based on one dimensional target,” Opt. Precis. Eng. 16(1), 2274–2280 (2008).

2004 (1)

R. S. Lu and Y. F. Li, “A global calibration method for large-scale multisensor visual measurement systems,” Sens. Actuators A Phys. 116(3), 384–393 (2004).
[Crossref]

2000 (1)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1994 (1)

F. C. Park and B. J. Martin, “Robot Sensor Calibration: Solving AX=XB on the Euclidean Group,” IEEE Trans. Rob. Autom. 10(5), 717–722 (1994).
[Crossref]

1989 (1)

R. Y. Tsai and R. K. Lenz, “A new technique for fully autonomous and efficient 3D robotics hand/eye calibration,” IEEE Trans. Rob. Autom. 5(3), 345–358 (1989).
[Crossref]

Ait-Aider, O.

P. Lébraly, C. Deymier, and O. Ait-Aider, “Flexible extrinsic calibration of non-overlapping cameras using a planar mirror: application to visionbased robotics,” in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2010), pp. 5640–5647.

Akimichi, S.

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Brasselet, E.

Cai, Z. W.

Chen, K.

Deetjen, M.

Deymier, C.

P. Lébraly, C. Deymier, and O. Ait-Aider, “Flexible extrinsic calibration of non-overlapping cameras using a planar mirror: application to visionbased robotics,” in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2010), pp. 5640–5647.

Dong, S.

Frahm, J. M.

R. K. Kumar, A. llie, J. M. Frahm, and M. Pollefeys, “Simple calibration of non-overlapping cameras with a mirror,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008).

Gao, B. Z.

Gu, Y. G.

Guan, Y.

He, X. Y.

Huang, B.

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

Jiang, C. F.

Jin, Y.

Kanade, Takeo

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Kang, X.

Kitahara, I.

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Kumar, R. K.

R. K. Kumar, A. llie, J. M. Frahm, and M. Pollefeys, “Simple calibration of non-overlapping cameras with a mirror,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008).

Kuo, T.

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Lébraly, P.

P. Lébraly, C. Deymier, and O. Ait-Aider, “Flexible extrinsic calibration of non-overlapping cameras using a planar mirror: application to visionbased robotics,” in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2010), pp. 5640–5647.

Lentink, D.

Lenz, R. K.

R. Y. Tsai and R. K. Lenz, “A new technique for fully autonomous and efficient 3D robotics hand/eye calibration,” IEEE Trans. Rob. Autom. 5(3), 345–358 (1989).
[Crossref]

Li, F.

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

Li, F. J.

Z. Liu, F. J. Li, and G. J. Zhang, “An external parameter calibration method for multiple cameras based on laser rangefinder,” Measurement 47, 954–962 (2014).
[Crossref]

Li, X. J.

Z. Liu, X. J. Li, and Y. Yin, “On-site calibration of line-structured light vision sensor in complex light environments,” Opt. Express 23(23), 129896–129911 (2015).
[Crossref]

Li, Y. F.

R. S. Lu and Y. F. Li, “A global calibration method for large-scale multisensor visual measurement systems,” Sens. Actuators A Phys. 116(3), 384–393 (2004).
[Crossref]

Lim, B.

Litchinitser, N. M.

Liu, C.

Liu, Q. Z.

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

Liu, X. L.

Liu, Z.

Z. Liu, X. J. Li, and Y. Yin, “On-site calibration of line-structured light vision sensor in complex light environments,” Opt. Express 23(23), 129896–129911 (2015).
[Crossref]

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

Z. Liu, F. J. Li, and G. J. Zhang, “An external parameter calibration method for multiple cameras based on laser rangefinder,” Measurement 47, 954–962 (2014).
[Crossref]

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Z. Liu, G. Zhang, and Z. Z. Wei, “Global calibration of multi-vision sensor based on one dimensional target,” Opt. Precis. Eng. 16(1), 2274–2280 (2008).

llie, A.

R. K. Kumar, A. llie, J. M. Frahm, and M. Pollefeys, “Simple calibration of non-overlapping cameras with a mirror,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008).

Lu, J. Y.

Lu, R. S.

R. S. Lu and Y. F. Li, “A global calibration method for large-scale multisensor visual measurement systems,” Sens. Actuators A Phys. 116(3), 384–393 (2004).
[Crossref]

Manjunath, B. S.

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Marc, E.

Martin, B. J.

F. C. Park and B. J. Martin, “Robot Sensor Calibration: Solving AX=XB on the Euclidean Group,” IEEE Trans. Rob. Autom. 10(5), 717–722 (1994).
[Crossref]

Mokhtar, M.

Morita, R.

Ni, Z.

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Ohta, Y.

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Omatsu, T.

Onno, T.

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Park, F. C.

F. C. Park and B. J. Martin, “Robot Sensor Calibration: Solving AX=XB on the Euclidean Group,” IEEE Trans. Rob. Autom. 10(5), 717–722 (1994).
[Crossref]

Peng, X.

Pollefeys, M.

R. K. Kumar, A. llie, J. M. Frahm, and M. Pollefeys, “Simple calibration of non-overlapping cameras with a mirror,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008).

Saito, H.

I. Kitahara, H. Saito, S. Akimichi, T. Onno, Y. Ohta, and Takeo Kanade, “large-scale virtualized reality,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001).

Shao, X.

Sun, J.

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

Sun, J. H.

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

Sunderrajan, S.

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Tsai, R. Y.

R. Y. Tsai and R. K. Lenz, “A new technique for fully autonomous and efficient 3D robotics hand/eye calibration,” IEEE Trans. Rob. Autom. 5(3), 345–358 (1989).
[Crossref]

Wang, J.

Wang, J. M.

Wang, P.

Wang, Y.

Z. Y. Xu, Y. Wang, and C. Yang, “Multi-camera global calibration for large-scale measurement based on plane mirror,” Optik 126(23), 4149–4154 (2015).
[Crossref]

Wei, Z. Z.

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

Z. Liu, G. Zhang, and Z. Z. Wei, “Global calibration of multi-vision sensor based on one dimensional target,” Opt. Precis. Eng. 16(1), 2274–2280 (2008).

Wu, X. L.

Xu, J.

Xu, Z. Y.

Z. Y. Xu, Y. Wang, and C. Yang, “Multi-camera global calibration for large-scale measurement based on plane mirror,” Optik 126(23), 4149–4154 (2015).
[Crossref]

Yang, C.

Z. Y. Xu, Y. Wang, and C. Yang, “Multi-camera global calibration for large-scale measurement based on plane mirror,” Optik 126(23), 4149–4154 (2015).
[Crossref]

Yang, F. J.

Yin, Y.

Z. Liu, X. J. Li, and Y. Yin, “On-site calibration of line-structured light vision sensor in complex light environments,” Opt. Express 23(23), 129896–129911 (2015).
[Crossref]

Zhai, C.

Zhang, G.

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

Z. Liu, G. Zhang, and Z. Z. Wei, “Global calibration of multi-vision sensor based on one dimensional target,” Opt. Precis. Eng. 16(1), 2274–2280 (2008).

Zhang, G. J.

Z. Liu, F. J. Li, and G. J. Zhang, “An external parameter calibration method for multiple cameras based on laser rangefinder,” Measurement 47, 954–962 (2014).
[Crossref]

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

Zhang, G. L.

Zhang, S.

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Zhao, Y. T.

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

Zhu, Y.

ACM Trans. Sens. Netw. (1)

T. Kuo, Z. Ni, S. Sunderrajan, and B. S. Manjunath, “Calibrating a widearea camera network with non-overlapping views using mobile devices,” ACM Trans. Sens. Netw. 2(10), 1–24 (2014).
[Crossref]

Appl. Opt. (6)

Chin. J. Mech. Eng. (1)

Q. Z. Liu, J. H. Sun, Z. Liu, and G. J. Zhang, “Global calibration method of multi-sensor vision system using skew laser lines,” Chin. J. Mech. Eng. 2(25), 405–411 (2012).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

IEEE Trans. Rob. Autom. (2)

R. Y. Tsai and R. K. Lenz, “A new technique for fully autonomous and efficient 3D robotics hand/eye calibration,” IEEE Trans. Rob. Autom. 5(3), 345–358 (1989).
[Crossref]

F. C. Park and B. J. Martin, “Robot Sensor Calibration: Solving AX=XB on the Euclidean Group,” IEEE Trans. Rob. Autom. 10(5), 717–722 (1994).
[Crossref]

J. Opt. Soc. Am. A. (1)

Z. Liu, F. Li, B. Huang, and G. Zhang, “Real-time and accurate rail wear measurement method and experimental analysis,” J. Opt. Soc. Am. A. 31(8), 1721–1729 (2014).
[Crossref]

Meas. Sci. Technol. (1)

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “A global calibration method for multiple vision sensors based on multiple targets,” Meas. Sci. Technol. 22(12), 125102–125111 (2011).
[Crossref]

Measurement (1)

Z. Liu, F. J. Li, and G. J. Zhang, “An external parameter calibration method for multiple cameras based on laser rangefinder,” Measurement 47, 954–962 (2014).
[Crossref]

Opt. Eng. (1)

Q. Z. Liu, J. H. Sun, Y. T. Zhao, and Z. Liu, “Calibration method for geometry relationships of non-overlapping cameras using light planes,” Opt. Eng. 52(7), 074108 (2013).
[Crossref]

Opt. Express (3)

Opt. Precis. Eng. (2)

Z. Liu, G. Zhang, Z. Z. Wei, and J. Sun, “Novel calibration method for non-overlapping multiple vision sensors based on 1D target,” Opt. Precis. Eng. 49(4), 570–577 (2011).

Z. Liu, G. Zhang, and Z. Z. Wei, “Global calibration of multi-vision sensor based on one dimensional target,” Opt. Precis. Eng. 16(1), 2274–2280 (2008).

Optik (1)

Z. Y. Xu, Y. Wang, and C. Yang, “Multi-camera global calibration for large-scale measurement based on plane mirror,” Optik 126(23), 4149–4154 (2015).
[Crossref]

Sens. Actuators A Phys. (1)

R. S. Lu and Y. F. Li, “A global calibration method for large-scale multisensor visual measurement systems,” Sens. Actuators A Phys. 116(3), 384–393 (2004).
[Crossref]

Other (3)

R. K. Kumar, A. llie, J. M. Frahm, and M. Pollefeys, “Simple calibration of non-overlapping cameras with a mirror,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008).

P. Lébraly, C. Deymier, and O. Ait-Aider, “Flexible extrinsic calibration of non-overlapping cameras using a planar mirror: application to visionbased robotics,” in Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2010), pp. 5640–5647.

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Figures (18)

Fig. 1
Fig. 1 The flowchart of the calibration system and the algorithm.
Fig. 2
Fig. 2 Measurement model of line-structured light vision sensor.
Fig. 3
Fig. 3 Calibration of multi-camera with non-overlapping FOV using laser scanning.
Fig. 4
Fig. 4 Calibration of the laser scanning system (that is the light plane equation in the manipulator coordinate frame.)
Fig. 5
Fig. 5 Structural drawing of the manipulator.
Fig. 6
Fig. 6 Simulation schematic diagram for calibration of the manipulator and the camera.
Fig. 7
Fig. 7 (a) The effect of rotation error of the manipulator on the extrinsic parameters rmc. (b) The effect of rotation error of the manipulator on the extrinsic parameters tmc.
Fig. 8
Fig. 8 (a) The effect of calibration error of the light plane on the extrinsic parameters rmc. (b) The effect of calibration error of the light plane on the extrinsic parameters tmc..
Fig. 9
Fig. 9 multi-camera vision measurement system. The Rotation accuracy of two-axis turntable is better than 0.01°.
Fig. 10
Fig. 10 (a) Reprojection error of the camera A. (b) Reprojection error of the camera B. (c) Reprojection error of the camera C. (d) Reprojection error of the camera D.
Fig. 11
Fig. 11 Calibration of the laser scanning system in the physical experiment.
Fig. 12
Fig. 12 Calibration images of the light plane equation.The light plane equation is calibrated by using a 7 ×7 dot array plane target with LED. The distance between dots is 10mm and the machining accuracy is 0.02mm.
Fig. 13
Fig. 13 Extrinsic parameters calibration based on hand-eye.
Fig. 14
Fig. 14 Measurement images of the standard length ruler (the length is 1021.413mm).
Fig. 15
Fig. 15 Multi-camera vision system is applied to the measurement of four-wheel alignment.
Fig. 16
Fig. 16 Calibration of the multi-camera vision measurement system based on laser scanning.
Fig. 17
Fig. 17 Plane target images mounted on wheels captured by each camera.
Fig. 18
Fig. 18 Spatial distribution of camera and target.

Tables (4)

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Table 1 The D–H parameters of the manipulator

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Table 2 Intrinsic parameters of the four cameras

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Table 3 Rotate parameters of the turntable

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Table 4 Measurement of the standard length ruler at ten different positions

Equations (28)

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s [ u v 1 ] = K C [ I 3 × 3 , 0 3 × 1 ] [ X C Y C Z C 1 ] , K C = [ α x γ u 0 0 α y v 0 0 0 1 ] .
a X C + b Y C + c Z C + d = 0 ,
Π c i = H c T Π c i ,
Π m i = H m T Π m i ,
{ Π m i = H mc T Π c i Π m i = H mc T Π c i ,
n c i = R c n c i .
R c = [ n c 1 n c 2 n c 3 ] [ n c 1 n c 2 n c 3 ] 1 ,
R m R mc = R mc R c .
{ d m i = ( R mc T T mc ) T n c i + d c i d p i = ( R mc T T mc ) T n c i + d c i .
d m i = ( R m T T m ) T n m i + d m i = ( R m T T m ) T R mc n c i + d m i .
[ ( R mc n c i ) T ( R mc n c i ) T ] T mc + ( R m T T m ) T R mc n c i = d c i d c i ,
min f ( x ) = j , k = 1 , j k n i = 1 , 2 ( H m T j k Π c i k k Π c i H m c T Π c i j 2 ) ,
Π pr i = H mrl Π ml i ,
{ Π ml i = H mcl T Π cl i Π mr i = H mcr T Π cr i ,
{ n ml i = R mcl n cl i n mr i = R mcr n cr i ,
R lr = R mcl 1 R mcr .
{ n cl i T R mcl T T mcl = d cl i d ml i n cr i T R mcr T T mcr = d cr i d mr i .
H lr = H mcl 1 H mcr .
T lr = R mcl R mcr 1 T mcr + T mcl .
min f ( x ) j = 1 n i = 1 , 2 ( Π li j H lr Π ri j 2 ) ,
T = T 1 0 T 2 1 T 3 2 T 4 3 T 5 4 T 6 5 ,
R AC = [ 0.8200 0.0330 0.5714 0.0221 0.9994 0.0261 0.5719 0.0088 0.8202 ] , T AC = [ 448.4052 17.9286 223.5824 ] .
R BD = [ 0.8599 0.1119 0.4980 0.0926 0.9937 0.0633 0.5019 0.0083 0.8649 ] , T BD = [ 437.0094 54.919 161.6975 ] .
R mc = [ 0.9962 0.8192 0.015 0.0868 0.5714 0.0872 0.014 0.0500 0.9962 ] , T mc = [ 311.1928 10.2271 21.2129 ] .
{ 0.0868 X + 0.5714 Y 0.0087 Z 9.2211 = 0 0.1041 X + 0.9907 Y 0.0065 Z + 29.7789 = 0 .
R AB = [ 0.7710 0.0292 0.6361 0.0438 0.9991 0.0073 0.6353 0.0355 0.7716 ] , T AB = [ 1077.5112 5.5833 18.2765 ] .
R AB = [ 0.7712 0.0291 0.6351 0.0437 0.9971 0.0073 0.6349 0.0355 0.7714 ] , T AB = [ 1077.2616 5.5342 18.3168 ] .
T 1 0 = [ cos ( θ 1 ) 0 sin ( θ 1 ) 0 sin ( θ 1 ) 0 cos ( θ 1 ) 0 0 1 0 d 1 0 0 0 1 ] , T 2 1 = [ cos ( θ 2 ) sin ( θ 2 ) 0 a 2 cos ( θ 2 ) sin ( θ 2 ) cos ( θ 2 ) 0 a 2 sin ( θ 2 ) 0 0 1 0 0 0 0 1 ] , T 3 2 = [ cos ( θ 3 ) sin ( θ 3 ) 0 a 3 cos ( θ 3 ) sin ( θ 3 ) cos ( θ 3 ) 0 a 3 sin ( θ 3 ) 0 0 1 0 0 0 0 1 ] , T 4 3 = [ cos ( θ 4 ) 0 sin ( θ 4 ) 0 sin ( θ 4 ) 0 cos ( θ 4 ) 0 0 1 0 d 4 0 0 0 1 ] , T 5 4 = [ cos ( θ 5 ) 0 sin ( θ 5 ) 0 sin ( θ 5 ) 0 cos ( θ 5 ) 0 0 1 0 d 1 0 0 0 1 ] , T 6 5 = [ cos ( θ 6 ) sin ( θ 6 ) 0 0 sin ( θ 6 ) cos ( θ 6 ) 0 0 0 0 1 d 6 0 0 0 1 ] .

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