Abstract

We have developed an alternative approach to optical design which operates in the analytical domain so that an optical designer works directly with rays as analytical functions of system parameters rather than as discretely sampled polylines. This is made possible by a generalization of the proximate ray tracing technique which obtains the analytical dependence of the rays at the image surface (and ray path lengths at the exit pupil) on each system parameter. The resulting method provides an alternative direction from which to approach system optimization and supplies information which is not typically available to the system designer. In addition, we have further expanded the procedure to allow asymmetric systems and arbitrary order of approximation, and have illustrated the performance of the method through three lens design examples.

© 2010 Optical Society of America

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References

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  1. D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
  2. A. E. W. Jones and G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–37 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
    [CrossRef]
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    [CrossRef]
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2009 (1)

D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17, 10,659–10,673 (2009).
[CrossRef]

2005 (1)

J. B. Almeida, “Programming matrix optics into Mathematica,” Optik (Stuttgart) 116, 270–276 (2005).
[CrossRef]

1999 (1)

1997 (3)

V. Y. Pan, “Solving a polynomial equation: some history and recent progress,” SIAM Rev. 39, 187–220 (1997).
[CrossRef]

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

V. Lakshminarayanan and S. Varadharajan, “Expressions for aberrations coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

1996 (3)

1995 (1)

A. E. W. Jones and G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–37 (1995).
[CrossRef]

1994 (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

1988 (1)

1986 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1976 (2)

Almeida, J. B.

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bortz, J. C.

N. Shatz and J. C. Bortz, “Global optimization of high-performance concentrators,” in Nonimaging Optics (Elsevier, 2005), Chap. 11, pp. 265–304.

Brady, D. J.

D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17, 10,659–10,673 (2009).
[CrossRef]

Chen, X.

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

Forbes, G. W.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gerhard, J.

J. von zur Gathen and J. Gerhard, Modern Computer Algebra (Cambridge U. Press, 1999).

Hagen, N.

D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17, 10,659–10,673 (2009).
[CrossRef]

Hopkins, G. W.

Horst, R.

R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, 2nd ed. (Springer, 2000).

Jones, A. E. W.

A. E. W. Jones and G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–37 (1995).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kondo, M.

Kryszczynski, T.

T. Kryszczyński, “First steps towards an algebraic method of the optical design in the range of all aberration orders,” in 11th Slovak-Czech-Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics (1999), pp. 336–342.

Lakshminarayanan, V.

V. Lakshminarayanan and S. Varadharajan, “Expressions for aberrations coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Pan, V. Y.

V. Y. Pan, “Solving a polynomial equation: some history and recent progress,” SIAM Rev. 39, 187–220 (1997).
[CrossRef]

Pardalos, P. M.

R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, 2nd ed. (Springer, 2000).

Shafer, D.

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

Shatz, N.

N. Shatz and J. C. Bortz, “Global optimization of high-performance concentrators,” in Nonimaging Optics (Elsevier, 2005), Chap. 11, pp. 265–304.

Takeuchi, Y.

Thoai, N. V.

R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, 2nd ed. (Springer, 2000).

Varadharajan, S.

V. Lakshminarayanan and S. Varadharajan, “Expressions for aberrations coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

von zur Gathen, J.

J. von zur Gathen and J. Gerhard, Modern Computer Algebra (Cambridge U. Press, 1999).

Walther, A.

Winston, R.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yamamoto, K.

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

Comput. Phys. (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

J. Global Optim. (1)

A. E. W. Jones and G. W. Forbes, “An adaptive simulated annealing algorithm for global optimization over continuous variables,” J. Global Optim. 6, 1–37 (1995).
[CrossRef]

J. Mod. Opt. (1)

X. Chen and K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Express (1)

D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17, 10,659–10,673 (2009).
[CrossRef]

Optik (Stuttgart) (1)

J. B. Almeida, “Programming matrix optics into Mathematica,” Optik (Stuttgart) 116, 270–276 (2005).
[CrossRef]

Optom. Vision Sci. (1)

V. Lakshminarayanan and S. Varadharajan, “Expressions for aberrations coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

SIAM Rev. (1)

V. Y. Pan, “Solving a polynomial equation: some history and recent progress,” SIAM Rev. 39, 187–220 (1997).
[CrossRef]

Other (9)

Wolfram Research, Inc., www.wolfram.com.

ZEMAX Development Corp., www.zemax.com.

N. Shatz and J. C. Bortz, “Global optimization of high-performance concentrators,” in Nonimaging Optics (Elsevier, 2005), Chap. 11, pp. 265–304.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

T. Kryszczyński, “First steps towards an algebraic method of the optical design in the range of all aberration orders,” in 11th Slovak-Czech-Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics (1999), pp. 336–342.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, 2nd ed. (Springer, 2000).

G. W. Hopkins, “Aberrational analysis of optical systems: a proximate ray trace approach,” Ph.D. thesis (University of Arizona, 1976).

J. von zur Gathen and J. Gerhard, Modern Computer Algebra (Cambridge U. Press, 1999).

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Figures (8)

Fig. 1
Fig. 1

The transfer operation involves taking low-order polynomial approximations of surfaces and modifying the surface intersection coordinates to increasing accuracy as higher orders are traced. Shown here are only the first- and third-order approximations of a spherical surface. The ray path length w from the left to right surfaces depends on the order of approximation, as indicated in Eqs. (5).

Fig. 2
Fig. 2

2D lens model used for the example. Once the analytical model is completed, we substitute the following parameter values to give the lens shown: t 1 = 5   mm , t 2 = 76.6667   mm , and the refractive index is n = 1.5 . The first lens surface is convex, and the second surface is planar. The resulting f / 2.2 design has an entrance pupil diameter of 32 mm and a focal length of 80 mm.

Fig. 3
Fig. 3

The lens used in the first design example: a freeform x - y polynomial singlet lens with an off-axis field. The system parameters are similar to those of Fig. 2: t 1 = 5   mm , t 2 = 76.6667   mm , and the refractive index is n = 1.5 , while the surface parameters are the 14 polynomial coefficients (seven for each surface) out to fourth order. The field of view for this design is 5 ° θ y 7 ° and 1 ° θ x + 1 ° .

Fig. 4
Fig. 4

The multiscale lens design example layout and prescription. The objective is fixed, while we attempt to design the lenslet to perform aberration correction on the nominal image (shown by the curved surface between the lenses) and re-image onto a tilted detector array (shown at the far right). The square pupil is 8   mm × 8   mm in size, and the objective lens focal length is 64.45 mm. The lenslet re-images a 2.5 ° × 2.5 ° square field of view.

Fig. 5
Fig. 5

Phase space of the incident rays.

Fig. 6
Fig. 6

The approximate ray trace used by the edge ray principle for designing a concentrator lens. The labeled rays a, a , and b are the phase-space points given in Fig. 5. Ray a maps to the edge of the concentration region, at a distance y m from the optical axis, while ray b is bent downward so that it reaches the image plane at y < y m .

Fig. 7
Fig. 7

(a) Layout of the concentrator lens obtained via the edge ray principle design method; (b) the profile of ray position y at the image plane as a function of entrance pupil position y ep for θ = 20 ° .

Fig. 8
Fig. 8

Concentrator example’s performance is illustrated by showing the portion of transmitted rays reaching the concentration region as a function of incidence angle. The example shown here (solid line) compares well with that shown in Fig. 8.9 of [24] (dashed line). A vertical line at 20° illustrates the maximum angle of incidence used in the design.

Tables (4)

Tables Icon

Table 1 Prescription for the Lens Shown in Fig. 2

Tables Icon

Table 2 Surface Parameters Obtained for the First Design Example (an Off-Axis Singlet) a

Tables Icon

Table 3 Multiscale Lens Design Example Layout and Prescription

Tables Icon

Table 4 Surface Parameters Obtained for the Second Design Example (Multiscale Lenslet) a

Equations (119)

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r s + 1 = r s + w s c s ,
n ( c × N ) = n ( c × N ) ,
N = f ( r ) ,
x = 0 + x ( 1 ) + x ( 2 ) + x ( 3 ) + x ( 4 ) + ,
y = 0 + y ( 1 ) + y ( 2 ) + y ( 3 ) + y ( 4 ) + ,
z = z ( 0 ) + z ( 1 ) + z ( 2 ) + z ( 3 ) + z ( 4 ) + ,
c x = 0 + c x ( 1 ) + c x ( 2 ) + c x ( 3 ) + c x ( 4 ) + ,
c y = 0 + c y ( 1 ) + c y ( 2 ) + c y ( 3 ) + c y ( 4 ) + ,
c z = 1 + c z ( 1 ) + c z ( 2 ) + c z ( 3 ) + c z ( 4 ) + ,
w = w ( 0 ) + w ( 1 ) + w ( 2 ) + w ( 3 ) + w ( 4 ) + .
( c x ( 1 ) + c x ( 3 ) + ) = θ x 1 3 ! θ x 3 + ,
( c y ( 1 ) + c y ( 3 ) + ) = θ y 1 3 ! θ y 3 + ,
( c z ( 0 ) + c z ( 2 ) + c z ( 4 ) + ) = 1 1 2 ( θ x 2 + θ y 2 ) + [ 1 16 ( θ x 2 + θ y 2 ) 2 + 1 6 ( θ x 4 + θ y 4 ) ] + .
0 = z 0 ( 0 ) + w ( 0 ) ,
r s ( 1 ) = r 0 ( 1 ) + c ( 1 ) w ( 0 ) + c ( 0 ) w ( 1 ) ,
r s ( 2 ) = r 0 ( 2 ) + c ( 2 ) w ( 0 ) + c ( 1 ) w ( 1 ) + c ( 0 ) w ( 2 ) ,
r s ( 3 ) = r 0 ( 3 ) + c ( 3 ) w ( 0 ) + c ( 2 ) w ( 1 ) + c ( 1 ) w ( 2 ) + c ( 0 ) w ( 3 ) ,
w ( 0 ) = z 0 ( 0 ) ,
w ( 1 ) = z s ( 1 ) z 0 ( 1 ) c z ( 1 ) w ( 0 ) ,
w ( 2 ) = z s ( 2 ) z 0 ( 2 ) c z ( 2 ) w ( 0 ) c z ( 1 ) w ( 1 ) ,
x s ( 1 ) = x 0 ( 1 ) + c x ( 1 ) w ( 0 ) ,
x s ( 2 ) = x 0 ( 2 ) + c x ( 2 ) w ( 0 ) + c x ( 1 ) w ( 1 ) ,
y s ( 1 ) = y 0 ( 1 ) + c y ( 1 ) w ( 0 ) ,
y s ( 2 ) = y 0 ( 2 ) + c y ( 2 ) w ( 0 ) + c y ( 1 ) w ( 1 ) ,
z s ( x s , y s ) = z s ( 0 , 0 ) + x s [ z s x s ] ( 0 , 0 ) + y s [ z s y s ] ( 0 , 0 ) + 1 2 x s 2 [ 2 z s x s 2 ] ( 0 , 0 ) + x s y s [ 2 z s x s y s ] ( 0 , 0 ) + 1 2 y s 2 [ 2 z s y s 2 ] ( 0 , 0 ) + ,
z s ( 0 ) = z s ( 0 , 0 ) ,
z s ( 1 ) = x s ( 1 ) [ z s x s ] ( 0 , 0 ) + y s ( 1 ) [ z s y s ] ( 0 , 0 ) ,
z s ( 2 ) = x s ( 2 ) [ z s x s ] ( 0 , 0 ) + y s ( 2 ) [ z s y s ] ( 0 , 0 ) + 1 2 [ x s ( 1 ) ] 2 [ 2 z s x s 2 ] ( 0 , 0 ) + 1 2 [ y s ( 1 ) ] 2 [ 2 z s y s 2 ] ( 0 , 0 ) + x s ( 1 ) y s ( 1 ) [ 2 z s x s y s ] ( 0 , 0 ) ,
z s ( 0 ) = 0 ,
z s ( 1 ) = 0 ,
z s ( 2 ) = [ r s ( 1 ) ] 2 2 R ,
z s ( 3 ) = 0 ,
z s ( 4 ) = r s ( 1 ) r s ( 3 ) R + [ r s ( 1 ) ] 4 8 R 3 ,
z s = α 1 x s + α 2 y s + α 3 x s 2 + α 4 x s y s + α 5 y s 2 + α 6 x s 3 + α 7 x s 2 y s + α 8 x s y s 2 + α 9 y s 3 + .
z s ( 0 ) = 0 ,
z s ( 1 ) = α 1 x s ( 1 ) + α 2 y s ( 1 ) ,
z s ( 2 ) = α 1 x s ( 2 ) + α 2 y s ( 2 ) + 2 α 3 [ x s ( 1 ) ] 2 + α 4 x s ( 1 ) y s ( 1 ) + 2 α 5 [ y s ( 1 ) ] 2 ,
c x = B x ± N x D A x y ,
c y = B y ± N y D A x y ,
c z = B z ± D A z ,
A x y = n 2 2 N z ( N x 2 + N y 2 + N z 2 ) ,
A z = n 2 2 ( N x 2 + N y 2 + N z 2 ) ,
B x = n 1 n 2 N z [ N y 2 c x N x N y c y + N z ( N z c x N x c z ) ] ,
B y = n 1 n 2 N z [ N x 2 c y N x N y c x + N z ( N z c y N y c z ) ] ,
B z = n 1 n 2 [ N z ( N x c x + N y c y ) + ( N x 2 + N y 2 ) c z ] ,
D = n 2 2 N z 2 [ n 2 2 ( N x 2 + N y 2 + N z 2 ) n 1 2 ( N z 2 ( c x 2 + c y 2 ) 2 N x N z c x c z 2 N y c y ( N x c x + N z c z ) + N y 2 ( c x 2 + c z 2 ) + N x 2 ( c y 2 + c z 2 ) ) ] ,
( α ( 0 ) + α ( 1 ) + ) ( α ( 0 ) + α ( 1 ) + ) = ( D ( 0 ) + D ( 1 ) + ) .
α ( 0 ) α ( 0 ) = D ( 0 ) ,
α ( 0 ) α ( 1 ) + α ( 1 ) α ( 0 ) = D ( 1 ) ,
α ( 0 ) α ( 2 ) + α ( 1 ) α ( 1 ) + α ( 2 ) α ( 0 ) = D ( 2 ) ,
( c ( 0 ) + c ( 1 ) + ) ( a ( 0 ) + a ( 1 ) + ) = ( b ( 0 ) + b ( 1 ) + ) ,
c ( 0 ) a ( 0 ) = b ( 0 ) ,
c ( 0 ) a ( 1 ) + c ( 1 ) a ( 0 ) = b ( 1 ) ,
c ( 0 ) a ( 2 ) + c ( 1 ) a ( 1 ) + c ( 2 ) a ( 0 ) = b ( 2 ) ,
N x = x s z s ( x s , y s ) ,
N y = y s z s ( x s , y s ) ,
N z = 1.
N x = N x ( 0 ) + N x ( 1 ) + N x ( 2 ) + N x ( 3 ) + N x ( 4 ) + ,
N y = N y ( 0 ) + N y ( 1 ) + N y ( 2 ) + N y ( 3 ) + N y ( 4 ) + ,
N z = N z ( 0 ) + N z ( 1 ) + N z ( 2 ) + N z ( 3 ) + N z ( 4 ) + ,
x z ( x , y ) = x R 2 x 2 y 2 ,
y z ( x , y ) = y R 2 x 2 y 2 ,
N x ( 0 ) = α 1 ,
N y ( 0 ) = α 2 ,
N x ( 1 ) = 2 α 3 x s ( 1 ) α 4 y s ( 1 ) ,
N y ( 1 ) = α 4 x s ( 1 ) 2 α 5 y s ( 1 ) ,
N x ( 2 ) = 2 α 3 x s ( 2 ) α 4 y s ( 2 ) 3 α 6 [ x s ( 1 ) ] 2 2 α 7 x s ( 1 ) y s ( 1 ) α 8 [ y s ( 1 ) ] 2 ,
N y ( 2 ) = α 4 x s ( 2 ) 2 α 5 y s ( 2 ) α 7 [ x s ( 1 ) ] 2 2 α 8 x s ( 1 ) y s ( 1 ) 3 α 9 [ y s ( 1 ) ] 2 ,
M = [ x ( ) x G ( ) ] 2 + [ y ( ) y G ( ) ] 2 d x ep d y ep d θ x d θ y ,
M = [ x ( ) x G ( ) ] 2 + [ y ( ) y G ( ) ] 2 d ϕ ρ d ρ d θ ,
x ¯ ( ) = x ( ) d x ep d y ep d θ x d θ y ,
y ¯ ( ) = y ( ) d x ep d y ep d θ x d θ y .
z ( y ) = y 2 2 R + y 4 8 R 3 + ,
z = y 2 2 R + y 4 8 R 3 .
w 0 ( 0 ) = 0 ,     w 0 ( 2 ) = y ep 2 2 R ,
y 1 ( 1 ) = y ep ,     y 1 ( 3 ) = y ep 2 θ y 2 R ,
z 1 ( 0 ) = 0 ,     z 1 ( 2 ) = y ep 2 2 R ,
[ c y ( 1 ) ] 1 = [ 1 R ( n 1 ) y ep + θ y n ] ,
[ c y ( 3 ) ] 1 = 1 6 R 3 [ θ y 3 R 3 n 3 θ y 2 y ep R 2 n ( n 1 ) 3 y ep 3 n ( n 1 ) 3 θ y y ep 2 ( n 2 n 1 ) ] ,
[ c z ( 0 ) ] 1 = 1 ,
[ c z ( 2 ) ] 1 = 1 2 R 2 [ y ep 2 ( 1 2 n + n 2 ) + y ep 2 n ( θ y R n θ y R ) + θ y 2 R 2 n 2 ] .
w 1 ( 0 ) = t 1 ,
w 1 ( 2 ) = 1 2 R 2 [ t 1 y ep 2 ( n 1 ) 2 + θ y 2 R 2 t 1 n 2 y ep 2 R + 2 θ y y ep R t 1 n ( n 1 ) ] ,
y 2 ( 1 ) = t 1 R y ep ( n 1 ) + y ep + θ y t 1 n ,
y 2 ( 3 ) = 1 6 R 3 ( 3 θ y 2 y ep R 2 t 1 n ( n 1 ) ( 1 + 3 n ) + θ y 3 R 3 t 1 n ( 3 n 2 1 ) + 3 y ep 3 ( n 1 ) [ R + t 1 + t 1 n ( n 1 ) ] 3 θ y y ep 2 R ( n 1 ) [ R + t 1 ( 3 n 2 + n 1 ) ] ) ,
z 2 ( 0 ) = 0 ,
z 2 ( 2 ) = 0 ,
[ c y ( 1 ) ] 2 = 1 6 n R 3 [ 6 y ep R 2 ( n 1 ) 6 θ y R 3 n ] ,
[ c y ( 3 ) ] 2 = 1 6 R 3 n [ θ y 3 R 3 n 3 θ y 2 y ep R 2 n ( n 1 ) 3 y ep 3 n ( n 1 ) + 3 θ y y ep 2 R ( 2 n 2 + n 1 ) ] ,
[ c z ( 0 ) ] 2 = 1 ,
[ c z ( 2 ) ] 2 = 1 2 R 2 n 2 ( [ y ep ( n 1 ) + θ y R n ] 2 ) .
y 3 ( 1 ) = 1 R n [ y ep ( t 2 + n R n t 1 + n t 2 + t 1 n 2 ) + θ y R ( n t 2 + n 2 t 1 ) ] ,
y 3 ( 3 ) = 1 6 R 3 n 3 { y ep 3 [ 3 t 2 ( n 4 3 n 2 + 3 n 1 ) + 3 n 3 ( n 1 ) ( R + t 1 + t 1 n ( n 1 ) ) ] + θ y y ep 2 [ 3 R t 2 n ( n 1 ) ( 2 n 2 + 4 n 3 ) 3 R n 3 ( n 1 ) ( R + t 1 ( 3 n 2 + n 1 ) ) ] + θ y 2 y ep [ 3 R 2 t 2 n 2 ( n 1 ) ( n + 3 ) + 3 R 2 n 4 t 1 ( n 1 ) ( 3 n + 1 ) ] + θ y 3 [ 2 R 3 t 2 n 3 + R 3 t 1 n 4 ( 3 n 2 1 ) ] } .
M = [ y 3 ( ) y 3 ( 1 ) ( ) ] 2 d y ep d θ y = [ y 3 ( 3 ) ( ) ] 2 d y ep d θ y .
z front = α 1 x 2 + α 2 y 2 + α 3 y 3 + α 4 x 2 y + α 5 x 4 + α 6 y 4 + α 7 x 2 y 2 ,
z back = β 1 x 2 + β 2 y 2 + β 3 y 3 + β 4 x 2 y + β 5 x 4 + β 6 y 4 + β 7 x 2 y 2 ,
z = α 1 x 2 + α 2 y 2 + α 3 y 3 + α 4 x 2 y + α 5 x 4 + α 6 y 4 + α 7 x 2 y 2 .
z front = α 1 y 2 + α 2 y 4 + α 3 y 6 ,
z rear = β 1 y 2 + β 2 y 4 + β 3 y 6 .
M = D ep / 2 D ep / 2 d y ep 20 ° 20 ° d θ y y 2 ,
M = [ ( D ep / 2 ) y a D ep / 2 [ ( y 3 y k ) 2 + ( z 3 z k ) 2 ] d y ep + D ep / 2 ( D ep / 2 ) y a ( y y m ) 2 d y ep ] θ = 20 ° ,
α 1 = 0.297 73 ,     β 1 = 0.038 88 ,
α 2 = 0.018 92 ,     β 2 = 0.023 36 ,
α 3 = 0.001 71 ,     β 3 = 0.002 53 ,
y = ( y ( 0 ) , y ( 1 ) , y ( 2 ) , y ( 3 ) , ) T ,
1 T x y T 1 ,
x y T = ( x ( 0 ) y ( 0 ) x ( 1 ) y ( 0 ) x ( 2 ) y ( 0 ) x ( 0 ) y ( 1 ) x ( 1 ) y ( 1 ) x ( 2 ) y ( 1 ) x ( 0 ) y ( 2 ) x ( 1 ) y ( 2 ) x ( 2 ) y ( 2 ) ) .
g = ( 1 , g , g 2 , g 3 , ) T .
y g = ( y ( 0 ) , g y ( 1 ) , g 2 y ( 2 ) , g 3 y ( 3 ) , ) ,

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