Abstract

A computational scheme based on differential geometry is proposed for determining the first- and second-order derivative matrices of a skew ray as it is reflected/refracted at a flat boundary surface. In the proposed approach, the position and orientation of the boundary surface in 3D space are described using just four variables. As a result, the proposed method is more computationally efficient than existing schemes based on all six variables. The derivative matrices enable the cross-coupling effects of the system variables on the exit ray to be fully understood. Furthermore, the proposed method provides a convenient means of determining the search direction used by existing gradient-based schemes to minimize the merit function during the optimization stage of the optical system design process. The validity of the proposed approach as an analysis and design tool is demonstrated using a corner-cube mirror and laser tracking system for illustration purposes.

© 2013 Optical Society of America

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2013 (1)

2011 (2)

2008 (1)

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

2004 (1)

1999 (1)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

1997 (2)

1988 (2)

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

1985 (1)

1982 (1)

1978 (1)

1976 (1)

1968 (1)

1963 (1)

1957 (1)

D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
[CrossRef]

Andersen, T. B.

Arora, J. S.

J. S. Arora, Introduction to Optimum Design, 3rd ed. (Elsevier, 2012), p. 482.

Dilworth, D. C.

Feder, D. P.

Forbes, G. W.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

Lin, P. D.

Lu, C. H.

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

Olson, C.

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Stavroudis, O.

Stone, B. D.

Wu, W.

Youngworth, R. N.

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Proc. SPIE (4)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

Other (2)

J. S. Arora, Introduction to Optimum Design, 3rd ed. (Elsevier, 2012), p. 482.

Leica Inc., Geodesy and Industrial Systems Center, Norcross, GA.

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Figures (8)

Fig. 1.
Fig. 1.

Definition of flat boundary surface in 3D space by means of just four variables.

Fig. 2.
Fig. 2.

Skew-ray tracing at flat boundary surface.

Fig. 3.
Fig. 3.

Interpretation of pose matrix A ¯ i 0 describing pose of coordinate frame ( x y z ) i with respect to coordinate frame ( x y z ) o .

Fig. 4.
Fig. 4.

Schematic illustration of perfect corner-cube mirror in which reflected beam is parallel to incident beam irrespective of beam/corner-cube alignment.

Fig. 5.
Fig. 5.

Determination of Δ R ¯ n for an optical system.

Fig. 6.
Fig. 6.

Use of Hessian matrix 2 R ¯ 3 / X ¯ sys 2 in determining cross-coupling effects of system variable vector on 3 z of the exit ray of corner-cube mirror.

Fig. 7.
Fig. 7.

General laser surface tracking system.

Fig. 8.
Fig. 8.

Leica LTD 500 laser tracking system.

Tables (3)

Tables Icon

Table 1. Jacobian Matrix of Exit Ray R ¯ 3 with Respect to System Variable Vector X ¯ sys of Corner-Cube Mirror

Tables Icon

Table 2. Merit Function and System Variable Values as Function of Number of Iterations for Design No. 1 Laser Tracking System Given Use of Steepest-descent Optimization Method

Tables Icon

Table 3. Merit Function and System Variable Values as Function of Number of Iterations for Design No. 2 Laser Tracking System Given Use of Classical Newton Optimization Method

Equations (82)

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r ¯ i e j = [ J i x e j J i y e j J i z e j t i e j ] T .
r ¯ i = ( ( A ¯ e j 0 ) 1 ) T r ¯ i e j = A ¯ e j 0 r ¯ i e j ,
A ¯ e j 0 = [ I e j x J e j x K e j x t e j x I e j y J e j y K e j y t e j y I e j z J e j z K e j z t e j z 0 0 0 1 ] .
A ¯ e j 0 = [ I e j x J e j x K e j x 0 I e j y J e j y K e j y 0 I e j z J e j z K e j z 0 d e j x d e j y d e j z 1 ] ,
2 R ¯ i X ¯ sys 2 = [ R ¯ i 1 X ¯ sys X ¯ i X ¯ sys ] T [ 2 R ¯ i R ¯ i 1 2 2 R ¯ i R ¯ i 1 X ¯ i 2 R ¯ i X ¯ i R ¯ i 1 2 R ¯ i X ¯ i 2 ] [ R ¯ i 1 X ¯ sys X ¯ i X ¯ sys ] + [ R ¯ i R ¯ i 1 R ¯ i X ¯ i ] [ 2 R ¯ i 1 X ¯ sys 2 2 X ¯ i X ¯ sys 2 ] .
P ¯ 0 = [ P 0 x P 0 y P 0 z 1 ] T
¯ 0 = [ 0 x 0 y 0 z 0 ] T = [ C β 0 C ( 90 ° + α 0 ) C β 0 S ( 90 ° + α 0 ) S β 0 0 ] T ,
X ¯ 0 = [ P 0 x P 0 y P 0 z α 0 β 0 ] T .
R ¯ 0 X ¯ 0 = S ¯ 0 = [ P 0 x / X ¯ 0 P 0 y / X ¯ 0 P 0 z / X ¯ 0 0 x / X ¯ 0 0 y / X ¯ 0 0 z / X ¯ 0 ] = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 C β 0 S ( 90 ° + α 0 ) S β 0 C ( 90 ° + α 0 ) 0 0 0 C β 0 C ( 90 ° + α 0 ) S β 0 S ( 90 ° + α 0 ) 0 0 0 0 C β 0 ] .
A ¯ i 0 = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] .
P ¯ i = [ P i x P i y P i z 1 ] T = [ P i 1 x + i 1 x λ i P i 1 y + i 1 y λ i P i 1 z + i 1 z λ i 1 ] T ,
λ i = [ J i x P i 1 x + J i y P i 1 y + J i z P i 1 z ( J i x t i x + J i y t i y + J i z t i z ) ] J i x i 1 x + J i y i 1 y + J i z i 1 z = ( J i x P i 1 x + J i y P i 1 y + J i z P i 1 z + t i ) J i x i 1 x + J i y i 1 y + J i z i 1 z = D i E i .
¯ i = [ i x i y i z 0 ] = [ i 1 x 2 J i x E i i 1 y 2 J i y E i i 1 z 2 J i z E i 0 ] ,
¯ i = [ i x i y i z 0 ] = ( s i 1 N i 2 + ( N i E i ) 2 N i E i ) [ J i x J i y J i z 0 ] + N i [ i 1 x i 1 y i 1 z 0 ] ,
X ¯ i = [ J i x J i y J i z t i ξ i 1 ξ i ] T ,
S ¯ i = R ¯ i X ¯ i = [ P ¯ i / X ¯ i ¯ i / X ¯ i ] 6 × 6 .
P ¯ i X ¯ i = [ P i x / x v P i y / x v P i z / x v ] 3 × 6 = [ i 1 x i 1 y i 1 z ] [ λ i x v ] ,
λ i X ¯ i = [ λ i x v ] 1 × 6 = 1 E i [ D i x v ] + D i E i 2 [ E i x v ] ,
D i X ¯ i = [ D i x v ] 1 × 6 = [ P i 1 x P i 1 y P i 1 z 1 0 0 ] ,
E i X ¯ i = [ E i x v ] 1 × 6 = [ i 1 x i 1 y i 1 z 0 0 0 ] .
¯ i X ¯ i = [ i x / x v i y / x v i z / x v ] 3 × 6 = 2 E i [ J i x / x v J i y / x v J i z / x v ] 2 [ J i x J i y J i z ] [ E i x v ] ,
J i x X ¯ i = [ J i x x v ] 1 × 6 = [ 1 0 0 0 0 0 ] ,
J i y X ¯ i = [ J i y x v ] 1 × 6 = [ 0 1 0 0 0 0 ] ,
J i z X ¯ i = [ J i z x v ] 1 × 6 = [ 0 0 1 0 0 0 ] .
¯ i X ¯ i = [ i x / x v i y / x v i z / x v ] 3 × 6 = ( s i 1 N i 2 + ( N i E i ) 2 N i E i ) [ J i x / x v J i y / x v J i z / x v ] + [ i 1 x i 1 y i 1 z ] [ N i x v ] + ( s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 E i ) [ J i x J i y J i z ] [ N i x v ] + ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ J i x J i y J i z ] [ E i x v ] ,
N i X ¯ i = [ N i x v ] 1 × 6 = [ 0 0 0 0 1 ξ i N i ξ i ] ,
Δ R ¯ n = M ¯ n M ¯ n 1 M ¯ 1 S ¯ 0 Δ X ¯ 0 + M ¯ n M ¯ n 1 M ¯ 2 S ¯ 1 Δ X ¯ 1 + + M ¯ n S ¯ n 1 Δ X ¯ n 1 + S ¯ n Δ X ¯ n ,
Δ R ¯ n = R ¯ n X ¯ sys Δ X ¯ sys = M ¯ n M ¯ n 1 M ¯ 1 S ¯ 0 X ¯ 0 X ¯ sys Δ X ¯ sys + M ¯ n M ¯ n 1 M ¯ 2 S ¯ 1 X ¯ 1 X ¯ sys Δ X ¯ sys + + M ¯ n S ¯ n 1 X ¯ n 1 X ¯ sys Δ X ¯ sys + S ¯ n X ¯ n X ¯ sys Δ X ¯ sys ,
R ¯ n X ¯ sys = M ¯ n M ¯ n 1 M ¯ 1 S ¯ 0 X ¯ 0 X ¯ sys + M ¯ n M ¯ n 1 M ¯ 2 S ¯ 1 X ¯ 1 X ¯ sys + + M ¯ n S ¯ n 1 X ¯ n 1 X ¯ sys + S ¯ n X ¯ n X ¯ sys .
2 R ¯ i X ¯ i 2 = [ 2 P i x / X ¯ i 2 2 P i y / X ¯ i 2 2 P i z / X ¯ i 2 2 i x / X ¯ i 2 2 i y / X ¯ i 2 2 i z / X ¯ i 2 ] = [ 2 P i x / x w x v 2 P i y / x w x v 2 P i z / x w x v 2 i x / x w x v 2 i y / x w x v 2 i z / x w x v ] 6 × 6 × 6 .
2 P ¯ i X ¯ i 2 = [ 2 P ¯ i x w x v ] = [ 2 P i x / X ¯ i 2 2 P i y / X ¯ i 2 2 P i z / X ¯ i 2 ] = [ i 1 x i 1 y i 1 z ] 2 λ i X ¯ i 2 .
2 λ i X ¯ i 2 = [ 2 λ i x w x v ] 6 × 6 = 1 E i 2 [ E i x w ] T [ D i x v ] + 1 E i 2 [ D i x w ] T [ E i x v ] 2 D i E i 3 [ E i x w ] T [ E i x v ] .
2 i x X ¯ i 2 = [ 2 i x x w x v ] 5 × 5 = 2 [ J i x x w ] T [ E i x v ] 2 [ E i x w ] T [ J i x x v ] ,
2 i y X ¯ i 2 = [ 2 i y x w x v ] 5 × 5 = 2 [ J i y x w ] T [ E i x v ] 2 [ E i x w ] T [ J i y x v ] ,
2 i z X ¯ i 2 = [ 2 i z x w x v ] 5 × 5 = 2 [ J i z x w ] T [ E i x v ] 2 [ E i x w ] T [ J i z x v ] ,
2 R ¯ i R ¯ i 1 X ¯ i = [ 2 P i x / R ¯ i 1 X ¯ i 2 P i y / R ¯ i 1 X ¯ i 2 P i z / R ¯ i 1 X ¯ i 2 i x / R ¯ i 1 X ¯ i 2 i y / R ¯ i 1 X ¯ i 2 i z / R ¯ i 1 X ¯ i ] = [ 2 P i x / x w x v 2 P i y / x w x v 2 P i z / x w x v 2 i x / x w x v 2 i y / x w x v 2 i z / x w x v ] 6 × 6 × 6 .
2 P ¯ i R ¯ i 1 X ¯ i = [ 2 P i x / x w x v 2 P i y / x w x v 2 P i z / x w x v ] 3 × 6 × 6 ,
2 P i x R ¯ i 1 X ¯ i = [ 2 P i x x w x v ] 6 × 6 = [ 0 0 0 1 0 0 ] T [ λ i x v ] + i 1 x [ 2 λ i x w x v ] ,
2 P i y R ¯ i 1 X ¯ i = [ 2 P i x x w x v ] 6 × 6 = [ 0 0 0 0 1 0 ] T [ λ i x v ] + i 1 y [ 2 λ i x w x v ] ,
2 P i z R ¯ i 1 X ¯ i = [ 2 P i x x w x v ] 6 × 6 = [ 0 0 0 0 0 1 ] T [ λ i x v ] + i 1 y [ 2 λ i x w x v ] ,
[ 2 λ i x w x v ] 6 × 6 = 2 λ i R ¯ i 1 X ¯ i = 1 E i 2 [ E i x w ] T [ D i x v ] 1 E i [ 2 D i x w x v ] + 1 E i 2 [ D i x w ] T [ E i x v ] + D i E i 2 [ 2 E i x w x v ] 2 D i E i 3 [ E i x w ] T [ E i x v ] ,
D i / R ¯ i 1 = [ D i / x w ] 1 × 6 = [ J i x J i y J i z 0 0 0 ] ,
E i / R ¯ i 1 = [ E i / x w ] 1 × 6 = [ 0 0 0 J i x J i y J i z ] ,
2 D i R ¯ i 1 X ¯ i = [ 2 D i x w x v ] 6 × 6 = [ 1 0 0 0 ¯ 1 × 3 0 1 0 0 ¯ 1 × 3 0 0 1 0 ¯ 1 × 3 0 ¯ 3 × 1 0 ¯ 3 × 1 0 ¯ 3 × 1 0 ¯ 3 × 3 ] ,
2 E i R ¯ i 1 X ¯ i = [ 2 E i x w x v ] 6 × 6 = [ 0 ¯ 3 × 1 0 ¯ 3 × 1 0 ¯ 3 × 1 0 ¯ 3 × 3 1 0 0 0 ¯ 1 × 3 0 1 0 0 ¯ 1 × 3 0 0 1 0 ¯ 1 × 3 ] .
2 ¯ i R ¯ i 1 X ¯ i = [ 2 i x / x w x v 2 i y / x w x v 2 i z / x w x v ] 3 × 6 × 6 ,
2 i x R ¯ i 1 X ¯ i = [ 2 i x x w x v ] 6 × 6 = 2 [ E i x w ] T [ J i x x v ] 2 J i x [ 2 E i x w x v ] ,
2 i y R ¯ i 1 X ¯ i = [ 2 i y x w x v ] 6 × 6 = 2 [ E i x w ] T [ J i y x v ] 2 J i y [ 2 E i x w x v ] ,
2 i z R ¯ i 1 X ¯ i = [ 2 i z x w x v ] 6 × 6 = 2 [ E i x w ] T [ J i z x v ] 2 J i z [ 2 E i x w x v ] ,
A ¯ e 1 0 = tran ( t e 1 x , 0 , 0 ) tran ( 0 , t e 1 y , 0 ) tran ( 0 , 0 , t e 1 z ) rot ( z , ω e 1 z ) rot ( y , ω e 1 y ) rot ( x , ω e 1 x ) = [ C ω e 1 z C ω e 1 y C ω e 1 z S ω e 1 y S ω e 1 x S ω e 1 z C ω e 1 x C ω e 1 z S ω e 1 y C ω e 1 x + S ω e 1 z S ω e 1 x t e 1 x S ω e 1 z C ω e 1 y S ω e 1 z S ω e 1 y S ω e 1 x + C ω e 1 z C ω e 1 x S ω e 1 z S ω e 1 y C ω e 1 x C ω e 1 z S ω e 1 x t e 1 y S ω e 1 y C ω e 1 y S ω e 1 x C ω e 1 y C ω e 1 x t e 1 z 0 0 0 1 ] ,
X ¯ e 1 = [ ξ air t e 1 x t e 1 y t e 1 z ω e 1 x ω e 1 y ω e 1 z ] T .
r ¯ 1 e 1 = [ J 1 x e 1 J 1 y e 1 J 1 z e 1 t 1 e 1 ] T = [ 0 0 1 0 ] T ,
r ¯ 2 e 1 = [ J 2 x e 1 J 2 y e 1 J 2 z e 1 t 2 e 1 ] T = [ 0 1 0 0 ] T ,
r ¯ 3 e 1 = [ J 3 x e 1 J 3 y e 1 J 3 z e 1 t 3 e 1 ] T = [ 1 0 0 0 ] T .
X ¯ 1 = [ J 1 x J 1 y J 1 z t 1 ξ 0 ξ 1 ] T ,
X ¯ 2 = [ J 2 x J 2 y J 2 z t 2 ξ 1 ξ 2 ] T ,
X ¯ 3 = [ J 3 x J 3 y J 3 z t 3 ξ 2 ξ 3 ] T .
X ¯ sys = [ P 0 x P 0 y P 0 z α 0 β 0 ξ air t e 1 x t e 1 y t e 1 z ω e 1 x ω e 1 y ω e 1 z J 1 x J 1 y J 1 z t 1 J 2 x J 2 y J 2 z t 2 J 3 x J 3 y J 3 z t 3 ] T .
X ¯ sys = [ P 0 x P 0 y P 0 z α 0 β 0 ξ air t e 1 x t e 1 y t e 1 z ω e 1 x ω e 1 y ω e 1 z J 1 x e 1 J 1 y e 1 J 1 z e 1 t 1 e 1 J 2 x e 1 J 2 y e 1 J 2 z e 1 t 2 e 1 J 3 x e 1 J 3 y e 1 J 3 z e 1 t 3 e 1 ] T .
X ¯ s y s = [ P 0 x P 0 y P 0 z α 0 β 0 ξ air t e 1 x t e 1 y t e 1 z ω e 1 x ω e 1 y ω e 1 z J 1 x e 1 J 1 y e 1 t 1 e 1 J 2 x e 1 J 2 z e 1 t 2 e 1 J 3 y e 1 J 3 z e 1 t 3 e 1 ] T .
r ¯ 1 = [ J 1 x J 1 y J 1 z t 1 ] T = [ 0.17299 0.01513 0.98481 0 ] T ,
r ¯ 2 = [ J 2 x J 2 y J 2 z t 2 ] T = [ 0.20679 0.96297 0.17299 0 ] T .
Φ = 1 2 [ ( P 3 x P 3 x / target ) 2 + ( P 3 y P 3 y / target ) 2 ] .
X ¯ sys / next = X ¯ sys / current + Δ X ¯ sys ,
Δ X ¯ sys = Φ X ¯ sys .
Δ X ¯ sys = ( 2 Φ X ¯ sys 2 ) 1 Φ X ¯ sys .
Φ X ¯ sys = [ Φ x v ] = ( P 3 x P 3 x / target ) [ P 3 x x v ] + ( P 3 y P 3 y / target ) [ P 3 y x v ] ,
2 Φ X ¯ sys 2 = [ 2 Φ x w x v ] = ( P 3 x P 3 x / target ) [ 2 P 3 x x w x v ] + ( P 3 y P 3 y / target ) [ 2 P 3 y x w x v ] + [ P 3 x x w ] T [ P 3 x x v ] + [ P 3 y x w ] T [ P 3 y x v ] .
R ¯ 0 = [ x 1 x 2 500 C ( 45 ° ) C ( 210 ° ) C ( 45 ° ) S ( 210 ° ) S ( 45 ° ) ] T .
¯ = 0 [ C x 2 C ( 90 ° + x 1 ) C x 2 S ( 90 ° + x 1 ) S x 2 0 ] T ,
R ¯ 0 = [ 0 0 0 0 1 0 ] T ,
r ¯ 1 = [ J 1 x J 1 y J 1 z t 1 ] T = [ 1 / 2 1 / 2 0 35.35534 ] T ,
r ¯ 2 = [ J 2 x J 2 y J 2 z t 2 ] T = [ 1 / 2 1 / 2 0 37.35534 ] T ,
r ¯ 3 = [ J 3 x J 3 y J 3 z t 3 ] T = [ C ( x 2 ) C ( x 1 ) C ( x 2 ) S ( x 1 ) S ( x 2 ) 100 C ( x 2 ) S ( x 1 ) ] T .
Φ = 1 2 [ ( P 4 x 20 ) 2 + ( P 4 y 130 ) 2 ] .
2 i x X ¯ i 2 = [ 2 i x x w x v ] = i 1 x [ 2 N i x w x v ] + J i x ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) [ 2 N i x w x v ] + ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) ( [ N i x w ] T [ J i x x v ] + [ J i x x w ] T [ N i x v ] ) + ( s i E i N i 2 1 N i 2 + ( N i E i ) 2 N i ) ( [ E i x w ] T [ J i x x v ] + [ J i x x w ] T [ E i x v ] ) + s i N i 2 ( 1 N i 2 ) J i x ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + s i ( E i 2 1 ) J i x ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ N i x w ] T [ N i x v ] + J i x ( s i N i E i ( 2 N i 2 + ( N i E i ) 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) ( [ E i x w ] T [ N i x v ] + [ N i x w ] T [ E i x v ] ) ,
2 i y X ¯ i 2 = [ 2 i y x w x v ] = i 1 y [ 2 N i x w x v ] + J i y ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) [ 2 N i x w x v ] + ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) ( [ N i x w ] T [ J i y x v ] + [ J i y x w ] T [ N i x v ] ) + ( s i E i N i 2 1 N i 2 + ( N i E i ) 2 N i ) ( [ E i x w ] T [ J i y x v ] + [ J i y x w ] T [ E i x v ] ) + s i N i 2 ( 1 N i 2 ) J i y ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + s i ( E i 2 1 ) J i y ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ N i x w ] T [ N i x v ] + J i y ( s i N i E i ( 2 N i 2 + ( N i E i ) 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) ( [ E i x w ] T [ N i x v ] + [ N i x w ] T [ E i x v ] ) ,
2 i z X ¯ i 2 = [ 2 i z x w x v ] = i 1 z [ 2 N i x w x v ] + J i z ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) [ 2 N i x w x v ] + ( E i + s i N i ( E i 2 1 ) 1 N i 2 + ( N i E i ) 2 ) ( [ N i x w ] T [ J i z x v ] + [ J i z x w ] T [ N i x v ] ) + ( s i E i N i 2 1 N i 2 + ( N i E i ) 2 N i ) ( [ E i x w ] T [ J i z x v ] + [ J i z x w ] T [ E i x v ] ) + s i N i 2 ( 1 N i 2 ) J i z ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + s i ( E i 2 1 ) J i z ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ N i x w ] T [ N i x v ] + J i z ( s i N i E i ( 2 N i 2 + ( N i E i ) 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) ( [ E i x w ] T [ N i x v ] + [ N i x w ] T [ E i x v ] ) ,
2 N i X ¯ i 2 = [ 2 N i x w x v ] 6 × 6 = [ 0 ¯ 4 × 4 0 ¯ 4 × 1 0 ¯ 4 × 1 0 ¯ 1 × 4 0 1 / ξ i 2 0 ¯ 1 × 4 1 / ξ i 2 2 N i / ξ i 2 ] .
2 i x R ¯ i 1 X ¯ i = [ 2 i x x w x v ] = ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ E i x w ] T [ J i x x v ] + [ 0 0 0 1 0 0 ] T [ N i x v ] + J i x ( s i N i E i ( 2 N i 2 + N i 2 E i 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) [ E i x w ] T [ N i x v ] + s i N i 2 ( 1 N i 2 ) J i x ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + J i x ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ 2 E i x w x v ] ,
2 i y R ¯ i 1 X ¯ i = [ 2 i y x w x v ] = ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ E i x w ] T [ J i y x v ] + [ 0 0 0 0 1 0 ] T [ N i x v ] + J i y ( s i N i E i ( 2 N i 2 + N i 2 E i 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) [ E i x w ] T [ N i x v ] + s i N i 2 ( 1 N i 2 ) J i y ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + J i y ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ 2 E i x w x v ] ,
2 i z R ¯ i 1 X ¯ i = [ 2 i z x w x v ] = ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ E i x w ] T [ J i z x v ] + [ 0 0 0 0 0 1 ] T [ N i x v ] + J i z ( s i N i E i ( 2 N i 2 + N i 2 E i 2 ) ( 1 N i 2 + ( N i E i ) 2 ) 3 2 1 ) [ E i x w ] T [ N i x v ] + s i N i 2 ( 1 N i 2 ) J i z ( 1 N i 2 + ( N i E i ) 2 ) 3 2 [ E i x w ] T [ E i x v ] + J i z ( s i N i 2 E i 1 N i 2 + ( N i E i ) 2 N i ) [ 2 E i x w x v ] .

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