Abstract

A methodology based on the worst-case approach is proposed for solving the tolerance analysis and tolerance allocation problems for optical systems. Compared with existing methods, the proposed tolerance allocation method has two principal advantages, namely, (1) it is based on an optical geometry gradient matrix and therefore provides means of obtaining the allowable tolerance limits, and (2) it yields the allowable tolerance limits of all the independent variables in the optical system simultaneously. The validity of the proposed methodology is demonstrated using a Dove prism for illustration purposes.

© 2010 Optical Society of America

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    [CrossRef]
  10. A. Saltelli, K. Chan, and E. M. Scott, Sensitivity Analysis(Wiley, 2000).
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    [CrossRef]
  12. P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628(2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  14. C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
    [CrossRef]
  15. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).
  16. E. T. Fortini, Dimensioning for Interchangeable Manufacturing (Industrial Press, 1967), p. 48.
  17. M. F. Spotts, “Dimensioning stacked assemblies,” Machine Design , 50, 60–63 (1978).
  18. P. D. Lin and K. F. Ehmann, “Inverse error analysis for multi-axis machines,” ASME J. Eng. Ind. Trans. ASME 118, 88–94 (1996).
    [CrossRef]
  19. I. Moreno, G. Paez, and M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4520 (2003).
    [CrossRef] [PubMed]
  20. B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
    [CrossRef]
  21. M. F. Spotts, “Allocation of tolerances to minimize cost of assembly,” J. Eng. Ind. Trans. ASME 95, 762–764 (1973).
    [CrossRef]
  22. F. H. Speckhart, “Calculation of tolerance based on a minimum cost approach,” J. Eng. Ind. Trans. ASME 94, 447–453 (1972).
    [CrossRef]
  23. R. N. Youngworth and B. D. Stone, “Cost-based tolerancing of optical systems,” Appl. Opt. 39, 4501–4512 (2000).
    [CrossRef]

2010 (1)

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

2009 (1)

2008 (2)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628(2008).
[CrossRef]

E. G. Herrera and M. Strojnik, “Interferometric tolerance determination for a Dove prism using exact ray trace,” Opt. Commun. 281, 897–905 (2008).
[CrossRef]

2007 (1)

2006 (1)

E. Gutierrez, M. Strojnik, and G. Paez, “Tolerance determination for a Dove prism using exact ray trace,” Proc. SPIE 6307, 63070K (2006).
[CrossRef]

2004 (1)

A. M. Ghodgaonkar and R. D. Tewari, “Measurement of apex angle of the prism using total internal reflection,” Opt. Laser Technol. 36, 617–624 (2004).
[CrossRef]

2003 (1)

2002 (1)

2000 (2)

1999 (1)

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

1997 (1)

1996 (1)

P. D. Lin and K. F. Ehmann, “Inverse error analysis for multi-axis machines,” ASME J. Eng. Ind. Trans. ASME 118, 88–94 (1996).
[CrossRef]

1989 (1)

1987 (1)

B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
[CrossRef]

1983 (1)

1982 (1)

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

1978 (1)

M. F. Spotts, “Dimensioning stacked assemblies,” Machine Design , 50, 60–63 (1978).

1973 (1)

M. F. Spotts, “Allocation of tolerances to minimize cost of assembly,” J. Eng. Ind. Trans. ASME 95, 762–764 (1973).
[CrossRef]

1972 (1)

F. H. Speckhart, “Calculation of tolerance based on a minimum cost approach,” J. Eng. Ind. Trans. ASME 94, 447–453 (1972).
[CrossRef]

1970 (1)

1967 (1)

E. T. Fortini, Dimensioning for Interchangeable Manufacturing (Industrial Press, 1967), p. 48.

Benhabib, B.

B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
[CrossRef]

Chan, K.

A. Saltelli, K. Chan, and E. M. Scott, Sensitivity Analysis(Wiley, 2000).

Ehmann, K. F.

P. D. Lin and K. F. Ehmann, “Inverse error analysis for multi-axis machines,” ASME J. Eng. Ind. Trans. ASME 118, 88–94 (1996).
[CrossRef]

Esener, S. C.

Fenton, R. G.

B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
[CrossRef]

Forbes, G. W.

Fortini, E. T.

E. T. Fortini, Dimensioning for Interchangeable Manufacturing (Industrial Press, 1967), p. 48.

Fortunato, G.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Ghodgaonkar, A. M.

A. M. Ghodgaonkar and R. D. Tewari, “Measurement of apex angle of the prism using total internal reflection,” Opt. Laser Technol. 36, 617–624 (2004).
[CrossRef]

Goldenberg, A. A.

B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
[CrossRef]

Gupta, S. K.

Gutierrez, E.

E. Gutierrez, M. Strojnik, and G. Paez, “Tolerance determination for a Dove prism using exact ray trace,” Proc. SPIE 6307, 63070K (2006).
[CrossRef]

Hendrick, W. L.

Herrera, E. G.

E. G. Herrera and M. Strojnik, “Interferometric tolerance determination for a Dove prism using exact ray trace,” Opt. Commun. 281, 897–905 (2008).
[CrossRef]

Hradaynath, R.

Hsueh, C. C.

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Computationally-efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009).
[CrossRef] [PubMed]

Journet, B.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Lee, J. F.

Leung, C. Y.

Lin, P. D.

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Computationally-efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009).
[CrossRef] [PubMed]

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628(2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

P. D. Lin and K. F. Ehmann, “Inverse error analysis for multi-axis machines,” ASME J. Eng. Ind. Trans. ASME 118, 88–94 (1996).
[CrossRef]

Marchand, P. J.

Moreno, I.

Ozkan, N. S. F.

Paez, G.

E. Gutierrez, M. Strojnik, and G. Paez, “Tolerance determination for a Dove prism using exact ray trace,” Proc. SPIE 6307, 63070K (2006).
[CrossRef]

I. Moreno, G. Paez, and M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4520 (2003).
[CrossRef] [PubMed]

Paul, R. P.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

Purnet, S.

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Rimmer, M.

Saltelli, A.

A. Saltelli, K. Chan, and E. M. Scott, Sensitivity Analysis(Wiley, 2000).

Scott, E. M.

A. Saltelli, K. Chan, and E. M. Scott, Sensitivity Analysis(Wiley, 2000).

Speckhart, F. H.

F. H. Speckhart, “Calculation of tolerance based on a minimum cost approach,” J. Eng. Ind. Trans. ASME 94, 447–453 (1972).
[CrossRef]

Spotts, M. F.

M. F. Spotts, “Dimensioning stacked assemblies,” Machine Design , 50, 60–63 (1978).

M. F. Spotts, “Allocation of tolerances to minimize cost of assembly,” J. Eng. Ind. Trans. ASME 95, 762–764 (1973).
[CrossRef]

Stone, B. D.

Strojnik, M.

E. G. Herrera and M. Strojnik, “Interferometric tolerance determination for a Dove prism using exact ray trace,” Opt. Commun. 281, 897–905 (2008).
[CrossRef]

E. Gutierrez, M. Strojnik, and G. Paez, “Tolerance determination for a Dove prism using exact ray trace,” Proc. SPIE 6307, 63070K (2006).
[CrossRef]

I. Moreno, G. Paez, and M. Strojnik, “Dove prism with increased throughput for implementation in a rotational-shearing interferometer,” Appl. Opt. 42, 4514–4520 (2003).
[CrossRef] [PubMed]

Tewari, R. D.

A. M. Ghodgaonkar and R. D. Tewari, “Measurement of apex angle of the prism using total internal reflection,” Opt. Laser Technol. 36, 617–624 (2004).
[CrossRef]

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628(2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Youngworth, R. N.

Appl. Opt. (7)

Appl. Phys. B (2)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628(2008).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

ASME J. Eng. Ind. Trans. ASME (1)

P. D. Lin and K. F. Ehmann, “Inverse error analysis for multi-axis machines,” ASME J. Eng. Ind. Trans. ASME 118, 88–94 (1996).
[CrossRef]

IEEE J. Robot. Automat. (1)

B. Benhabib, R. G. Fenton, and A. A. Goldenberg, “Computer-aided joint error analysis of robots,” IEEE J. Robot. Automat. 3, 317–322 (1987).
[CrossRef]

J. Eng. Ind. Trans. ASME (2)

M. F. Spotts, “Allocation of tolerances to minimize cost of assembly,” J. Eng. Ind. Trans. ASME 95, 762–764 (1973).
[CrossRef]

F. H. Speckhart, “Calculation of tolerance based on a minimum cost approach,” J. Eng. Ind. Trans. ASME 94, 447–453 (1972).
[CrossRef]

J. Opt. Soc. Am. A (2)

Machine Design (1)

M. F. Spotts, “Dimensioning stacked assemblies,” Machine Design , 50, 60–63 (1978).

Opt. Commun. (1)

E. G. Herrera and M. Strojnik, “Interferometric tolerance determination for a Dove prism using exact ray trace,” Opt. Commun. 281, 897–905 (2008).
[CrossRef]

Opt. Eng. (1)

S. Purnet, B. Journet, and G. Fortunato, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38, 983–990 (1999).
[CrossRef]

Opt. Laser Technol. (1)

A. M. Ghodgaonkar and R. D. Tewari, “Measurement of apex angle of the prism using total internal reflection,” Opt. Laser Technol. 36, 617–624 (2004).
[CrossRef]

Proc. SPIE (1)

E. Gutierrez, M. Strojnik, and G. Paez, “Tolerance determination for a Dove prism using exact ray trace,” Proc. SPIE 6307, 63070K (2006).
[CrossRef]

Other (3)

A. Saltelli, K. Chan, and E. M. Scott, Sensitivity Analysis(Wiley, 2000).

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

E. T. Fortini, Dimensioning for Interchangeable Manufacturing (Industrial Press, 1967), p. 48.

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Figures (3)

Fig. 1
Fig. 1

Definition of ray in terms of its position P i g and unit directional vector i g .

Fig. 2
Fig. 2

jth element with L j boundary surfaces and exit ray [ P m j 0 m j 0 ] T .

Fig. 3
Fig. 3

Optical system containing Dove prism and source ray.

Tables (2)

Tables Icon

Table 1 Ideal Values of the System Variables for the Dove Prism Shown in Fig. 3

Tables Icon

Table 2 Tolerances for the Dove Prism

Equations (43)

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A g h = [ I g x h J g x h K g x h t g x h I g y h J g y h K g y h t g y h I g z h J g z h K g z h t g z h 0 0 0 1 ] .
X e 0 = [ P 0 x 0 P 0 y 0 P 0 z 0 β 0 α 0 ] T .
A e j 0 = Trans ( t e j x 0 , t e j y 0 , t e j z 0 ) Rot ( z , ω e j z 0 ) Rot ( y , ω e j y 0 ) Rot ( x , ω e j x 0 ) ,
X e j = [ t e j x 0 t e j y 0 t e j z 0 ω e j x 0 ω e j y 0 ω e j z 0 ξ air ξ e j g e j γ e j ϕ e j ] T .
X system = [ P 0 x 0 P 0 y 0 P 0 z 0 β 0 α 0 t e j x 0 t e j y 0 t e j z 0 ω e j x 0 ω e j y 0 ω e j z 0 ξ air ξ e j g e j γ e j ϕ e j ... ] T = [ x 1 x 2 x 3 x m ^ ] = X system ( x r ) ,
A e 1 0 = Trans ( t e 1 x 0 , t e 1 y 0 , 0 ) ,
A 1 e 1 = Rot ( z , ( 90 γ e 1 ) ) ,
A 2 e 1 = Rot ( z , 90 ) ,
A 3 e 1 = Trans ( 0 , g e 1 , 0 ) Rot ( z , 90 ϕ e 1 ) .
X e 1 = [ t e 1 x 0 t e 1 y 0 t e 1 z 0 ω e 1 x 0 ω e 1 y 0 ω e 1 z 0 ξ air ξ e 1 g e 1 γ e 1 ϕ e 1 ] T ,
X system = X system ( x r ) = [ P 0 x 0 P 0 y 0 P 0 z 0 β 0 α 0 t e 1 x 0 t e 1 y 0 t e 1 z 0 ω e 1 x 0 ω e 1 y 0 ω e 1 z 0 ξ air ξ e 1 g e 1 γ e 1 ϕ e 1 ] T .
OPL system = i = 1 i = 3 OPL i = OPL 1 + OPL 2 + OPL 3 .
X system = [ 0 0 0 0 90 7 . 5 50 0 0 0 0 1 1.701 87 65 65 ] T .
Δ OPL system = OPL system X system Δ X system = r = 1 r = 16 OPL system x r Δ x r = 0 . 466307852 Δ P 0 x 0 Δ P 0 y 0 + 231 . 004777967 Δ α 0 + 0 . 466307852 Δ t e 1 x 0 + Δ t e 1 y 0 28 . 666244795 Δ ω e 1 z 0 + 462 . 606824063 Δ ξ air + 76 . 045812875 Δ ξ e 1 + 1 . 591533699 Δ g e 1 + 21 . 381069339 Δ γ e 1 + 14 . 519246543 Δ ϕ e 1 ,
Δ OPL system = 21 . 381069339 Δ γ e 1 + 14 . 519246543 Δ ϕ e 1 .
Δ OPL system = ( 21 . 381069339 + 14 . 519246543 ) × ( 0 . 35 / 3600 ) × ( π / 180 ) mm = 0.060917 μm .
[ Δ P 3 0 Δ 3 0 ] = [ Δ P 3 x 0 Δ P 3 y 0 Δ P 3 z 0 Δ 3 x 0 Δ 3 y 0 Δ 3 z 0 ] = [ 1 0 0 0 495.391114 0.466307 0 0 0 231.004677 0 0 1 501.352224 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 ] [ Δ P 0 x 0 Δ P 0 y 0 Δ P 0 z 0 Δ β 0 Δ α 0 ] + [ 2 0 0 0 0 128.896859 0.466307 1 0 0 0 28.659628 0 0 0 32.153539 0.006609 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 ] [ Δ t e 1 x 0 Δ t e 1 y 0 Δ t e 1 z 0 Δ ω e 1 x 0 Δ ω e 1 y 0 Δ ω e 1 z 0 ] + [ 19.535392 11.484651 0.172338 32.695324 1.572208 9.109503 5.355381 0.919637 15.246080 8.389669 0 0 0 0 0 0 0 0 0.817995 0.817995 0 0 0 0 0 0 0 0 0 0 ] [ Δ ξ air Δ ξ e 1 Δ g e 1 Δ γ e 1 Δ ϕ e 1 ] .
[ Δ P 3 x 0 Δ P 3 y 0 ] = [ 32.695324 1.572208 15.246080 8.389669 ] [ Δ γ e 1 Δ ϕ e 1 ] = [ 0.058146 μm 0.040106 μm ] .
Δ Φ worst = r = 1 r = m ^ ( | Φ system x r | Δ x r limit ) ,
ψ 1 = ( Δ OPL worst ) 2 ( Δ OPL spec ) 2 = ( r = 1 r = m ^ | OPL system x r | Δ x r limit ) 2 ( Δ OPL spec ) 2 0 ,
ψ 2 = ( Δ rms system ) 2 ( Δ rms spec ) 2 = ( r = 1 r = m ^ | rms system x r | Δ x rlimit ) 2 ( Δ rms spec ) 2 0.
ψ 3 = ( Δ Φ worst ) 2 ( Δ Φ spec ) 2 = ( r = 1 r = m ^ | Φ system x r | Δ x r limit ) 2 ( Δ Φ spec ) 2 0.
Δ x r limit = κ r Δ x e ( r = 1 , 2 , 3 , ... , m ^ ) .
κ r = { f r | Φ system / x r | Φ system / x r 0 Φ system / x r = 0 .
Δ x r limit = { f r Δ x e | Φ system / x r | Φ system / x r 0 Φ system / x r = 0 .
Δ x e 1 Δ OPL spec / r = 1 r = m ^ f r ,
Δ x e 2 Δ rms spec / r = 1 r = m ^ f r ,
Δ x e 3 Δ Φ spec / r = 1 r = m ^ f r .
Δ x r limit = min ( f r Δ x e 1 | OPL system / x r | , f r Δ x e 2 | rms system / x r | , f r Δ x e 3 | Φ system / x r | ) .
$ = r = 1 r = m ^ c r = r = 1 r = m ^ [ u r + v r ( Δ x r limit ) 2 ] ,
ψ ^ 1 = ( Δ OPL worst ) 2 ( Δ OPL spec ) 2 + ( s ^ 1 ) 2 = 0 ,
ψ ^ 2 = ( Δ rms worst ) 2 ( Δ rms spec ) 2 + ( s ^ 2 ) 2 = 0 ,
ψ ^ 3 = ( Δ Φ worst ) 2 ( Δ Φ spec ) 2 + ( s ^ 3 ) 2 = 0 ,
L ^ ( Δ x r limit , λ ^ 1 , λ ^ 2 , λ ^ 3 , s ^ 1 , s ^ 2 , s ^ 3 ) = $ + λ ^ 1 [ ( Δ OPL worst ) 2 ( Δ OPL spec ) 2 + ( s ^ 1 ) 2 ] + λ ^ 2 [ ( Δ rms worst ) 2 ( Δ rms spec ) 2 + ( s ^ 2 ) 2 ] + λ ^ 3 [ ( Δ Φ worst ) 2 ( Δ Φ spec ) 2 + ( s ^ 3 ) 2 ] ,
L ^ Δ x r limit = $ Δ x r limit + λ ^ 1 ψ ^ 1 Δ x r limit + λ ^ 2 ψ ^ 2 Δ x r limit + λ ^ 3 ψ ^ 3 Δ x r limit = 0 , r = 1 , 2 , , m ^ ,
L ^ λ ^ 1 = ( Δ OPL worst ) 2 ( Δ OPL spec ) 2 + ( s ^ 1 ) 2 = 0 ,
L ^ λ ^ 2 = ( Δ rms worst ) 2 ( Δ rms spec ) 2 + ( s ^ 2 ) 2 = 0 ,
L ^ λ ^ 3 = ( Δ Φ worst ) 2 ( Δ Φ spec ) 2 + ( s ^ 3 ) 2 = 0 ,
L ^ s ^ 1 = 2 λ ^ 1 s ^ 1 = 0 ,
L ^ s ^ 2 = 2 λ ^ 2 s ^ 2 = 0 ,
L ^ s ^ 3 = 2 λ ^ 3 s ^ 3 = 0.
g ^ r = t r 2 - Δ x r limit = 0 ( r = 1 , 2 , 3 , ... , m ^ ) ,
c 1 = u 1 + v 1 ( Δ x 1 limit ) 2 = 1 + 1 ( Δ x 1 limit ) 2 , c 2 = u 2 + v 2 ( Δ x 2 limit ) 2 = 1 + 1 ( Δ x 2 limit ) 2 , 2 ( Δ x 1 limit ) 3 + 21 . 381069339 × λ ^ 1 ( 21 . 381069339 Δ x 1 limit + 14 . 519246543 Δ x 2 limit ) = 0 , 2 ( Δ x 2 limit ) 3 + 14 . 519246543 × λ ^ 1 ( 21 . 381069339 Δ x 1 limit + 14 . 519246543 Δ x 2 limit ) = 0 , ψ ^ 1 = ( 21 . 381069339 Δ x 1 limit + 14 . 519246543 Δ x 2 limit ) 2 ( 0.0608946 × 10 3 ) 2 + ( s ^ 1 ) 2 = 0 , 2 λ ^ 1 s ^ 1 = 0 , g ^ 1 = t 1 2 Δ x 1 limit = 0 , g ^ 2 = t 2 2 Δ x 2 limit = 0.

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