Abstract

Insufficient vision information, such as occlusion, low resolvability, and a small field of view, represent important issues in microassembly and micromanipulation. We propose an active optical system to solve problems related to insufficient vision information through the integration of robotics and optics technologies. The proposed system integrates a double-prism system and a scanning mirror system to supply a compact flexible view. The kinematics of the imaging system is analyzed based on a simplified model initially to investigate the workspace and identify the kinematic performance. A more rigorous analysis of kinematics of the system is then made based on the ray tracing method. The simulation results based on the preliminary design are provided for investigating the workspace and demonstrating the capability of the system in imaging with variable views.

© 2008 Optical Society of America

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References

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  1. B. Nelson and P. K. Khosla, “Integrating sensor placement and visual tracking strategies,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 1994), pp. 1351-1356.
  2. M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.
  3. X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
    [CrossRef]
  4. B. Potsaid, Y. Bellouard, and J. Wen, “Adaptive scanning optical microscope (ASOM): a multidisciplinary optical microscope design for large field of view and high resolution imaging,” Opt. Express 13, 6504-6518 (2005).
    [CrossRef] [PubMed]
  5. X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).
  6. X. Tao and H. S. Cho, “Design of variable view imaging system for active observation of micro object,” in Proceedings of International Conference on Control, Automation, and Systems (ICROS, 2007), pp. 1785-1789.
  7. X. Tao and H. S. Cho, “Variable view imaging system and its application in vision-based microassembly,” Proc. SPIE 6719, 67190L-1-67190L-12 (2007).
  8. S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
    [CrossRef]
  9. M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).
  10. H. S. Cho, Optomechatronic: Fusion of Optical and Mechatronic Engineering (CRC, 2005).
    [CrossRef]
  11. J. E. Greivenkamp, SPIE Field Guides (SPIE, 2004).
  12. E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.
  13. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall, 1995).
  14. R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
    [CrossRef]
  15. L. Zhu, P. C. Sun, D. U. Bartsch, W. R. Freeman, and Y. Fainman, “Wave-front generation of Zernike polynomial modes with a micromachined membrane deformable mirror,” Appl. Opt. 38, 6019-6026 (1999).
    [CrossRef]
  16. C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
    [CrossRef]

2007

X. Tao and H. S. Cho, “Design of variable view imaging system for active observation of micro object,” in Proceedings of International Conference on Control, Automation, and Systems (ICROS, 2007), pp. 1785-1789.

X. Tao and H. S. Cho, “Variable view imaging system and its application in vision-based microassembly,” Proc. SPIE 6719, 67190L-1-67190L-12 (2007).

X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
[CrossRef]

2006

X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).

2005

M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).

H. S. Cho, Optomechatronic: Fusion of Optical and Mechatronic Engineering (CRC, 2005).
[CrossRef]

B. Potsaid, Y. Bellouard, and J. Wen, “Adaptive scanning optical microscope (ASOM): a multidisciplinary optical microscope design for large field of view and high resolution imaging,” Opt. Express 13, 6504-6518 (2005).
[CrossRef] [PubMed]

2004

J. E. Greivenkamp, SPIE Field Guides (SPIE, 2004).

2000

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

1999

1996

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
[CrossRef]

S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
[CrossRef]

1995

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall, 1995).

1994

B. Nelson and P. K. Khosla, “Integrating sensor placement and visual tracking strategies,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 1994), pp. 1351-1356.

1992

R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
[CrossRef]

Barber, C. B.

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
[CrossRef]

Bartsch, D. U.

Bellouard, Y.

Cho, H. S.

X. Tao and H. S. Cho, “Design of variable view imaging system for active observation of micro object,” in Proceedings of International Conference on Control, Automation, and Systems (ICROS, 2007), pp. 1785-1789.

X. Tao and H. S. Cho, “Variable view imaging system and its application in vision-based microassembly,” Proc. SPIE 6719, 67190L-1-67190L-12 (2007).

X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).

H. S. Cho, Optomechatronic: Fusion of Optical and Mechatronic Engineering (CRC, 2005).
[CrossRef]

Corke, P. I.

S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
[CrossRef]

Dobkin, D. P.

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
[CrossRef]

Dohi, T.

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

Fainman, Y.

Freeman, W. R.

Greivenkamp, J. E.

J. E. Greivenkamp, SPIE Field Guides (SPIE, 2004).

Hager, G. D.

S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
[CrossRef]

Hong, D. H.

X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).

Huhdanpaa, H. T.

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
[CrossRef]

Hui, X.

X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
[CrossRef]

Hutchinson, S.

M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).

S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
[CrossRef]

Khosla, P. K.

B. Nelson and P. K. Khosla, “Integrating sensor placement and visual tracking strategies,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 1994), pp. 1351-1356.

Kobayashi, E.

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

Kratochvil, B. E.

M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.

Lining, S.

X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
[CrossRef]

Masamune, K.

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

Mayorga, R. V.

R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
[CrossRef]

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall, 1995).

Milano, N.

R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
[CrossRef]

Nelson, B.

B. Nelson and P. K. Khosla, “Integrating sensor placement and visual tracking strategies,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 1994), pp. 1351-1356.

Nelson, B. J.

M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.

Potsaid, B.

Probst, M.

M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.

Sakuma, I.

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

Spong, M. W.

M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).

Sun, P. C.

Tao, X.

X. Tao and H. S. Cho, “Design of variable view imaging system for active observation of micro object,” in Proceedings of International Conference on Control, Automation, and Systems (ICROS, 2007), pp. 1785-1789.

X. Tao and H. S. Cho, “Variable view imaging system and its application in vision-based microassembly,” Proc. SPIE 6719, 67190L-1-67190L-12 (2007).

X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).

Vidyasagar, M.

M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).

Vollmers, K.

M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.

Weibin, R.

X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
[CrossRef]

Wen, J.

Wong, A. K. C.

R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
[CrossRef]

Zhu, L.

ACM Trans. Math. Softw.

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469-483 (1996).
[CrossRef]

Appl. Opt.

IEEE Trans. Robot. Autom.

S. Hutchinson, G. D. Hager, and P. I. Corke, “A tutorial on visual servo control,” IEEE Trans. Robot. Autom. 12, 651-670 (1996).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

R. V. Mayorga, A. K. C. Wong, and N. Milano, “A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness,” IEEE Trans. Syst. Man Cybern. 22, 790-798 (1992).
[CrossRef]

Int. J. Optomechatronics

X. Hui, R. Weibin, and S. Lining, “A flexible experimental system for complex microassembly under microscale force and vision-based control,” Int. J. Optomechatronics 1, 81-102 (2007).
[CrossRef]

Opt. Express

Proc. SPIE

X. Tao and H. S. Cho, “Variable view imaging system and its application in vision-based microassembly,” Proc. SPIE 6719, 67190L-1-67190L-12 (2007).

X. Tao, D. H. Hong, and H. S. Cho, “The design of active vision system for variable view imaging of micro objects,” Proc. SPIE 6376, 637608-1-637608-12 (2006).

Other

X. Tao and H. S. Cho, “Design of variable view imaging system for active observation of micro object,” in Proceedings of International Conference on Control, Automation, and Systems (ICROS, 2007), pp. 1785-1789.

B. Nelson and P. K. Khosla, “Integrating sensor placement and visual tracking strategies,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 1994), pp. 1351-1356.

M. Probst, K. Vollmers, B. E. Kratochvil, and B. J. Nelson, “Design of an advanced microassembly system for the automated assembly of bio-microrobots,” presented at 5th International Workshop on Microfactories, Besancon, France, 25-27 October 2006.

M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control (Wiley, 2005).

H. S. Cho, Optomechatronic: Fusion of Optical and Mechatronic Engineering (CRC, 2005).
[CrossRef]

J. E. Greivenkamp, SPIE Field Guides (SPIE, 2004).

E. Kobayashi, K. Masamune, I. Sakuma, and T. Dohi, “A wide-angle view endoscope system using wedge prisms,” in Proceedings of Medical Image Computing and Computer-Aided Intervention (MICCAI, 2000), pp. 661-668.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice Hall, 1995).

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Figures (16)

Fig. 1
Fig. 1

Examples of insufficient vision information: (a) occlusion, (b) low resolvability, and (c) a small FOV in the microassembly.

Fig. 2
Fig. 2

Parameter definition: (a) view state, (b) joint space for scanning mirror, and (c) wedge prism.

Fig. 3
Fig. 3

Double-wedge prisms configuration: (a) system diagram and (b) FOV.

Fig. 4
Fig. 4

Operation of the proposed system by changing (a) the view orientation and (b) the view position.

Fig. 5
Fig. 5

Proposed system layout.

Fig. 6
Fig. 6

Workspace of the simplified system in the ( x w , y w , γ ) space and the ( x w , y w ) plane: (a), (b)  φ = 5 ° and (c), (d)  φ = 20 ° .

Fig. 7
Fig. 7

Configurations of the system in various view states at (a)  P 1 , (b)  P 2 , (c)  P 3 , and (d)  P 4 .

Fig. 8
Fig. 8

Ray tracing of an actual system.

Fig. 9
Fig. 9

Definition of (a)  φ max and (b)  φ 2 π , max for an actual system.

Fig. 10
Fig. 10

Variation of (a)  r 2 π , min and (b)  φ 2 π , max with respect to θ v and r p .

Fig. 11
Fig. 11

System layout of the preliminary design result.

Fig. 12
Fig. 12

Workspace of the preliminary design in the ( x w , y w , γ ) space and the ( x w , y w ) plane: (a), (b)  φ = 2 ° and (c), (d)  φ = 13.8 ° .

Fig. 13
Fig. 13

Variation of A 2 π and A full : (a)  r 2 π versus φ and (b)  r full versus φ.

Fig. 14
Fig. 14

Simulation result for different azimuth angles: (a) planned view state, (b) solution of joint variables, (c) view state in the ( x w , y w , z w ) space, and (d) view state in the ( x w , y w , γ ) space.

Fig. 15
Fig. 15

Simulation result for different zenith angles: (a) desired view state, (b) solution of joint variables, (c) view state in the ( x w , y w , z w ) space, and (d) view state in the ( x w , y w , φ ) space.

Fig. 16
Fig. 16

Simulation results for a variable view pose: (a) desired view state, (b) solution of joint variables, (c) view states in the ( x w , y w , z w ) space, (d) the ( x w , y w , φ ) space, and (e) the ( x w , y w , γ ) space.

Tables (2)

Tables Icon

Table 1 Configurations of Active Optical Components

Tables Icon

Table 2 Specifications of the Preliminary Design

Equations (36)

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p = [ x w y w γ φ ] T .
θ d ( n 1 ) θ v .
x w = V 1 cos θ p 1 + V 2 cos θ p 2 , y w = V 1 sin θ p 1 + V 2 sin θ p 2 , γ = arctan 2 ( y w , x w ) , φ = tan 1 ( x w 2 + y w 2 D 1 ) ,
V 1 = V 2 = D 1 tan θ d .
θ p 1 = γ + π cos 1 ( tan ϕ 2 tan θ d ) , θ p 2 = tan 1 tan φ sin γ + tan θ d sin θ p 1 tan φ cos γ + tan θ d cos θ p 1 .
x w = θ s x D 2 + V 1 cos θ p 1 + V 2 cos θ p 2 , y w = θ s y D 2 + V 1 sin θ p 1 + V 2 sin θ p 2 , γ = arctan 2 ( ( V 1 sin θ p 1 V 2 sin θ p 2 ) , ( V 1 cos θ p 1 V 2 cos θ p 2 ) ) , φ = tan 1 ( ( V 1 cos θ p 1 + V 2 cos θ p 2 ) 2 + ( V 1 sin θ p 1 + V 2 sin θ p 2 ) 2 D 1 ) .
θ s x = ( x w V 1 cos θ p 1 V 2 cos θ p 2 ) / D 2 ,
θ s y = ( y w V 1 sin θ p 1 V 2 sin θ p 2 ) / D 2 ,
θ p 1 = γ + π cos 1 ( tan φ 2 tan θ d ) ,
θ p 2 = tan 1 tan φ sin γ + tan θ d sin θ p 1 tan φ cos γ + tan θ d cos θ p 1 .
( θ s x D 2 ) 2 + ( θ s y D 2 ) 2 < ( r p η ) .
r 2 π = r p D 1 tan φ ,
r full = r p + D 1 tan φ ,
[ x w y w γ φ ] = [ f 1 ( θ p 1 , θ p 2 , θ s x , θ s y ) f 2 ( θ p 1 , θ p 2 , θ s x , θ s y ) f 3 ( θ p 1 , θ p 2 , θ s x , θ s y ) f 4 ( θ p 1 , θ p 2 , θ s x , θ s y ) ] ,
X ^ i = X i / X i for     i = 1 , , 5 .
U ^ 1 = ( sin θ v sin θ p 1 sin θ v cos θ p 1 cos θ v ) T ,
U ^ 3 = ( sin θ v sin θ p 2 sin θ v cos θ p 2 cos θ v ) T ,
U ^ 2 = U ^ 4 = ( 0 0 1 ) T ,
Φ 1 = ( U ^ 1 ( 1 ) U ^ 1 ( 2 ) U ^ 1 ( 3 ) U ^ 1 c 1 ) T = ( sin θ v sin θ p 1 sin θ v cos θ p 1 cos θ v cos θ v c 1 ( 3 ) ) T , Φ 2 = ( U ^ 2 ( 1 ) U ^ 2 ( 2 ) U ^ 2 ( 3 ) U ^ 2 c 2 ) T = ( 0 0 1 c 2 ( 3 ) ) T , Φ 3 = ( U ^ 3 ( 1 ) U ^ 3 ( 2 ) U ^ 3 ( 3 ) U ^ 3 c 3 ) T = ( sin θ v sin θ p 2 sin θ v cos θ p 2 cos θ v cos θ v c 3 ( 3 ) ) T , Φ 4 = ( U ^ 4 ( 1 ) U ^ 4 ( 2 ) U ^ 4 ( 3 ) U ^ 4 c 4 ) T = ( 0 0 1 0 ) T .
Φ 5 = ( 0 0 1 c 5 ( 3 ) ) T .
U ^ i × X ^ i = n ( U ^ i × X ^ i + 1 ) , i = 1 , 3 , n ( U ^ i × X ^ i ) = U ^ i × X ^ i + 1 , i = 2 , 4.
X ^ i + 1 = 1 n X ^ i + 1 n ( n 1 1 n 2 + 1 n 2 ( X ^ i T U ^ i ) 2 X ^ i T U ^ i ) U ^ i , i = 1 , 3 , X ^ i + 1 = n X ^ i + ( 1 n 2 + n 2 ( X ^ i T U ^ i ) 2 n X ^ i T U ^ i ) U ^ i , i = 2 , 4.
φ = cos 1 ( X ^ 5 ( 3 ) ) , γ = π arctan 2 ( X ^ 5 ( 1 ) , X ^ 5 ( 2 ) ) .
e 1 = c 0 + X ^ 1 t 1 , e 2 = e 1 + X ^ 2 t 2 , e 3 = e 2 + X ^ 3 t 3 , e 4 = e 3 + X ^ 4 t 4 , e 5 = e 4 + X ^ 5 t 5 .
t 1 = c 0 X ^ 1 + Φ 1 ( 4 ) U ^ 1 X ^ 1 , t 2 = e 1 X ^ 2 + Φ 2 ( 4 ) U ^ 2 X ^ 2 , t 3 = e 2 X ^ 3 + Φ 3 ( 4 ) U ^ 3 X ^ 3 , t 4 = e 3 X ^ 4 + Φ 4 ( 4 ) U ^ 4 X ^ 4 , t 5 = e 4 X ^ 5 + Φ 5 ( 4 ) U ^ 5 X ^ 5 .
x w = e 5 ( 1 ) , y w = e 5 ( 2 ) .
J = [ f 1 θ w 1 f 1 θ w 2 f 1 θ s x f 1 θ s y f 2 θ w 1 f 2 θ w 2 f 2 θ s x f 2 θ s y f 3 θ w 1 f 3 θ w 2 f 3 θ s x f 3 θ s y f 4 θ w 1 f 4 θ w 2 f 4 θ s x f 4 θ s y ] .
Q i + 1 = Q i + ζ J + ( P d F ( Q i ) ) .
δ = δ 0 ( 1 ω / ω 0 ) 2 if     ω < ω 0 , δ = 0 otherwise ,
θ p 1 = θ p 2 = 90 , θ s x = 0 , θ s y < 0 , e 4 ( 1 ) 2 + e 4 ( 2 ) 2 = ( r p η ) .
θ p 1 = θ p 2 = 90 ° , θ s x = 0 , θ s y > 0 , e 1 ( 1 ) 2 + e 1 ( 2 ) 2 = ( r p η ) .
NA = 0.61 λ r ,
e i ( 1 ) 2 + e i ( 2 ) 2 < ( r p η ) , i = 1 , 2 , 3 , 4.
ω 0 = 0.0045 , ζ = 0.004 , δ 0 = 0.51 , ε = ( 1 × 10 6 , 1 × 10 6 , 1 × 10 6 , 1 × 10 6 ) .
For     y 0 : arctan 2 ( y , x ) = { ψ · sgn ( y ) x > 0 π 2 · sgn ( y ) x = 0 ( π ψ ) · sgn ( y ) x < 0 ,
For   y = 0 : arctan 2 ( 0 , x ) = { 0 x > 0 undefined x = 0 π x < 0 .

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