Abstract

An exact analytical approach is proposed for measuring the six-degree-of-freedom (6-DOF) motion of an object using the image-orientation-change (IOC) method. The proposed measurement system comprises two reflector systems, where each system consists of two reflectors and one position sensing detector (PSD). The IOCs of the object in the two reflector systems are described using merit functions determined from the respective PSD readings before and after motion occurs, respectively. The three rotation variables are then determined analytically from the eigenvectors of the corresponding merit functions. After determining the three rotation variables, the order of the translation equations is downgraded to a linear form. Consequently, the solution for the three translation variables can also be analytically determined. As a result, the motion transformation matrix describing the 6-DOF motion of the object is fully determined. The validity of the proposed approach is demonstrated by means of an illustrative example.

© 2012 Optical Society of America

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2010

2009

2007

2006

2005

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).

1997

1996

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst. Meas. Control 118, 425–433 (1996).
[CrossRef]

1994

E. H. Bokelberg, H. J. Sommer, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

1993

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, pp. 85–92 (1993).

1988

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

1987

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Robot. Autom. RA-3, 323–344 (1987).

Bokelberg, E. H.

E. H. Bokelberg, H. J. Sommer, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

Chen, C. J.

Ehmann, K. F.

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst. Meas. Control 118, 425–433 (1996).
[CrossRef]

Eman, K. F.

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

Holmes, R. B.

Jywe1, W. Y.

Kim, K. H.

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics, 9th ed.(Wiley, 2006), pp. 350–351.

Lee, S. W.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).

Lin, P. D.

Liu, C. S.

Lu, C. H.

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).

Mayor, R.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).

Nagle, M. G.

Ni, J.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, pp. 85–92 (1993).

Park, C. W.

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

Patrick, D.

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Robot. Autom. RA-3, 323–344 (1987).

Paul, R. P.

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

Sommer, H. J.

E. H. Bokelberg, H. J. Sommer, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

Srinivasan, M. V.

Tretheway, M. W.

E. H. Bokelberg, H. J. Sommer, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

Trethewey, M. W.

E. H. Bokelberg, H. J. Sommer, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

Tsai, C. Y.

Wilson, D. L.

Wu, S. M.

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, pp. 85–92 (1993).

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

Appl. Opt.

IEEE J. Robot. Autom.

D. Patrick, “Sandia lab’s laser tracking system: a single station solution for position and attitude data,” IEEE J. Robot. Autom. RA-3, 323–344 (1987).

J. Dyn. Syst. Meas. Control

P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” J. Dyn. Syst. Meas. Control 118, 425–433 (1996).
[CrossRef]

J. Eng. Ind.

K. H. Kim, K. F. Eman, and S. M. Wu, “Analysis alignment errors in a laser-based in-process cylindricity measurement system,” J. Eng. Ind. 109, 321–329 (1987).
[CrossRef]

C. W. Park, K. F. Eman, and S. M. Wu, “An in-process flatness error measurement and compensatory control system,” J. Eng. Ind. 110, 263–270 (1988).
[CrossRef]

J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” J. Eng. Ind. 115, pp. 85–92 (1993).

J. Manuf. Sci. Eng.

S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool,” J. Manuf. Sci. Eng. 127, 857–865 (2005).

P. D. Lin and C. H. Lu, “Modeling and sensitivity analysis of laser tracking systems by skew-ray tracing method,” J. Manuf. Sci. Eng. 127, 654–662 (2005).

J. Opt. Soc. Am. A

J. Sound Vib.

E. H. Bokelberg, H. J. Sommer, and M. W. Tretheway, “A six-degree-of-freedom laser vibrometer, part I: theoretical development,” J. Sound Vib. 178, 643–654 (1994).
[CrossRef]

E. H. Bokelberg, H. J. Sommer, and M. W. Trethewey, “A six-degree-of-freedom laser vibrometer, part II: experimental validation,” J. Sound Vib. 178, 655–667 (1994).
[CrossRef]

Opt. Express

Other

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

E. Kreyszig, Advanced Engineering Mathematics, 9th ed.(Wiley, 2006), pp. 350–351.

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of 6-DOF measurement system with reflector systems.

Fig. 2.
Fig. 2.

Ray tracing at flat boundary surface.

Fig. 3.
Fig. 3.

IOC produced by the j th reflector system containing two reflectors.

Fig. 4.
Fig. 4.

Illustration of ray paths in j th reflector system in motion measurement system.

Fig. 5.
Fig. 5.

Illustration of motion measurement system comprising two reflector systems.

Equations (53)

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A b b = Trans ( t b ) Rot ( u b , θ ) = [ a x b x c x t x a y b y c y t y a z b z c z t z 0 0 0 1 ] ,
Trans ( t b ) = [ 1 0 0 t x 0 1 0 t y 0 0 1 t z 0 0 0 1 ] ,
Rot ( u b , θ ) = [ a x b x c x 0 a y b y c y 0 a z b z c z 0 0 0 0 1 ] = [ u x 2 ( 1 C θ ) + C θ u x u y ( 1 C θ ) u z S θ u x u z ( 1 C θ ) + u y S θ 0 u x u y ( 1 C θ ) + u z S θ u y 2 ( 1 C θ ) + C θ u y u z ( 1 C θ ) u x S θ 0 u x u z ( 1 C θ ) u y S θ u y u z ( 1 C θ ) + u x S θ u z 2 ( 1 C θ ) + C θ 0 0 0 0 1 ] ,
A 0 i , j = [ I i x , j J i x , j K i x , j t i x , j I i y , j J i y , j K i y , j t i y , j I i z , j J i z , j K i z , j t i z , j 0 0 0 1 ] .
n i , j = [ n i x , j n i y , j n i z , j 0 ] T = A i , j 0 n i , j i , j = ( A 0 i , j ) 1 n i , j i , j = s i , j [ I i x , j J i y , j K i z , j 0 ] T .
P i , j , k = [ P i x , j , k P i y , j , k P i z , j , k 1 ] T = P i 1 , j , k + i 1 , j , k λ i , j , k = [ P i 1 x , j , k + i 1 x , j , k λ i , j , k P i 1 y , j , k + i 1 y , j , k λ i , j , k P i 1 z , j , k + i 1 z , j , k λ i , j , k 1 ] T ,
λ i , j , k = ( I i y , j P i 1 x , j , k + J i y , j P i 1 y , j , k + K i y , j P i 1 z , j , k + t i y , j ) I i y , j i 1 x , j , k + J i y , j i 1 y , j , k + K i y , j i 1 z , j , k .
C θ i , j , k = i 1 , j , k T n i , j = s i , j , k ( I i y , j i 1 x , j , k + J i y , j i 1 y , j , k + K i y , j i 1 z , j , k ) .
i , j , k = [ i x , j , k i y , j , k i z , j , k 0 ] T = i 1 , j , k 2 n i , j ( i 1 , j , k T n i , j ) = [ i 1 x , j , k + 2 n i x , j C θ i , j , k i 1 y , j , k + 2 n i y , j C θ i , j , k i 1 z , j , k + 2 n i z , j C θ i , j , k 0 ] ,
i , j , k ( n i , j ) i 1 , j , k = I 2 n i , j n i , j T ,
Γ j = n j , j , k 0 , j , k = n j , j , k ( n n j , j ) n j 1 , j , k n j 1 , j , k ( n n j 1 , j ) n j 2 , j , k i , j , k ( n i , j ) i 1 , j , k 2 , j , k ( n 2 , j ) 1 , j , k 1 , j , k ( n 1 , j ) 0 , j , k .
Γ j = [ a 0 , j b 0 , j c 0 , j 0 ] = Rot ( m 0 , j , Φ 0 , j ) [ 1 0 0 0 0 δ j 0 0 0 0 1 0 0 0 0 1 ] ,
Φ 0 , j = tan 1 { [ ( δ b 0 z , j c 0 y , j ) 2 + ( c 0 x , j a 0 z , j ) 2 + ( a 0 y , j δ b 0 x , j ) 2 ] 1 / 2 a 0 x , j + δ j b 0 y , j + c 0 z , j 1 } ,
{ m 0 x , j = ( δ j b 0 z , j c 0 y , j ) / ( 2 S Φ 0 , j ) m 0 y , j = ( c 0 x , j a 0 z , j ) / ( 2 S Φ 0 , j ) m 0 z , j = ( a 0 y , j δ j b 0 x , j ) / ( 2 S Φ 0 , j ) ,
Γ j = Γ j Rot ( ( 0 , 1 , 0 ) , ϕ j ) Rot ( ( 1 , 0 , 0 ) , ψ 1 , j ) Rot ( ( 0 , S ψ 2 , j , C ψ 2 , j ) , γ j ) ,
ϕ j = tan 1 ( w 2 x , j w 2 z , j ) ,
ψ 1 , j = cos 1 ( 1 | w 1 , j n j , j | ) ,
ψ 2 , j = cos 1 ( 1 | w 2 , j n j , j | ) ,
w 1 , j n j , j = [ w 1 x , j w 1 y , j w 1 z , j 1 ] T = P n j , j , 1 P n j , j , 2 ,
w 2 , j n j , j = [ w 2 x , j w 2 y , j w 2 z , j 1 ] T = P n j , j , 1 P n j , j , 3 ,
γ j = sin 1 ( [ ( 0 , S ψ 2 , j , C ψ 2 , j ) × ( C ψ 1 , j , S ψ 1 , j , 0 ) ] × [ ( 0 , S ψ 2 , j , C ψ 2 , j ) × ( 1 , 0 , 0 ) ] | [ ( 0 , S ψ 2 , j , C ψ 2 , j ) × ( C ψ 1 , j , S ψ 1 , j , 0 ) ] × [ ( 0 , S ψ 2 , j , C ψ 2 , j ) × ( 1 , 0 , 0 ) ] | ) .
Γ j = Rot ( u 0 , j , θ ) Γ j Rot ( u 0 , j , θ ) 1 ,
X j 0 , j = Rot ( u 0 , j , θ ) X j 0 , j .
u b = ( X 2 b X 2 b ) × ( X 1 b X 1 b ) | ( X 2 b X 2 b ) × ( X 1 b X 1 b ) | .
θ = tan 1 { [ X 1 x X 1 y + X 1 y 2 u x u y ( X 1 y u x + ( X 1 x X 1 x ) u y ) ( X 1 x u x + X 1 z u z ) X 1 y ( X 1 x ( 1 u y 2 ) X 1 x ( u x 2 u y 2 ) + u x ( X 1 y u y X 1 z u z ) ) ] / [ X 1 z 2 u z ( 1 u z 2 ) + ( X 1 x 2 u x 2 X 1 x ( X 1 x 2 X 1 y u x u y ) + X 1 y ( X 1 y u y 2 X 1 y ) ) u z + X 1 z ( X 1 x u x + X 1 y u y ( X 1 x u x + X 1 y u y ) ( 1 2 u z 2 ) ) ] } .
[ G 1 , 1 , x n G 1 , 1 , y n G 1 , 1 , z n G 2 , 1 , x n G 2 , 1 , y n G 2 , 1 , z n G 1 , 2 , x n G 1 , 2 , y n G 1 , 2 , z n G 2 , 2 , x n G 2 , 2 , y n G 2 , 2 , z n ] [ t x t y t z ] = [ Ω 1 , 1 Ω 2 , 1 Ω 1 , 2 Ω 2 , 2 ] ,
A b 1 , 1 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
A b 2 , 1 = [ 0.0158 0.9808 0.1945 17.5026 0.4537 0.1663 0.8755 14.2675 0.8910 0.1021 0.4424 2.0246 0 0 0 1 ] ,
A b 3 , 1 = [ 0.1348 0.4850 0.8641 11.5815 0.1301 0.8731 0.4698 27.1322 0.9823 0.0491 , 0.1807 1.2588 0 0 0 1 ] ,
A b 1 , 2 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
A b 2 , 2 = [ 0.3019 0.9490 0.0939 32.9426 0.8917 0.3159 0.3243 26.8538 0.3373 0.0141 0.9413 3.8107 0 0 0 1 ] ,
A b 3 , 2 = [ 0.3179 0.1978 0.9273 30.7836 0.0015 0.9781 0.2081 47.5129 0.9481 0.0648 0.3112 4.7455 0 0 0 1 ] .
Γ 1 = [ 0.8590 0.0857 0.5048 0 0.0910 0.9447 0.3151 0 0.5039 0.3166 0.8036 0 0 0 0 1 ] ,
Γ 2 = [ 0.7652 0.5691 0.3009 0 0.3895 0.7814 0.4875 0 0.5126 0.2558 0.8196 0 0 0 0 1 ] .
{ P 3 , 1 , 1 3 , 1 = [ 5.9843 0 3.1082 1 ] T P 3 , 1 , 2 3 , 1 = [ 4.9686 0 2.9985 1 ] T P 3 , 1 , 3 3 , 1 = [ 5.8858 0 4.1044 1 ] T
{ P 3 , 2 , 1 3 , 2 = [ 10.7985 0 5.5122 1 ] T P 3 , 2 , 2 3 , 2 = [ 9.7778 0 5.4319 1 ] T P 3 , 2 , 3 3 , 2 = [ 10.7342 0 6.5124 1 ] T .
Γ 1 = [ 0.9741 0.1988 0.1073 0 0.2044 0.9779 0.0437 0 0.0963 0.0646 0.9933 0 0 0 0 1 ]
Γ 2 = [ 0.9951 0.0522 0.0840 0 0.0487 0.9979 0.0430 0 0.0860 0.0387 0.9955 0 0 0 0 1 ] .
{ X 1 b = [ 0.1036 0 0.9946 0 ] T X 2 b = [ 0.0149 0 0.9999 0 ] T X 1 b = [ 0.0768 0.1156 0.9903 0 ] T X 2 b = [ 0.0118 0.1215 0.9925 0 ] T .
Rot ( u b , θ ) = [ 0.9977 0.0626 0.0267 0 0.0589 0.9907 0.1224 0 0.0341 0.1206 0.9921 0 0 0 0 1 ] .
Trans ( t b ) = [ 1 0 0 0.1019 0 1 0 0.2200 0 0 1 0.2231 0 0 0 1 ] .
A b b = [ 0.9977 0.0626 0.0267 0.1019 0.0589 0.9907 0.1224 0.2200 0.0341 0.1206 0.9921 0.2231 0 0 0 1 ] .
Ω 1 , j = ( G x 1 , j H x 1 , 1 + G y 1 , j H y 1 , j + G z 1 , j H z 1 , j ) + { J 3 y , j [ G x 2 , j ( 2 R 1 , j ( G z 1 , j P 0 y , j , 1 G y 1 , j ( P 0 z , j , 1 P 3 z , j , 1 ) ) 0 z , j , 1 ( P 0 y , j , 1 2 G y 1 , j R P 1 , j ) + 0 y , j , 1 ( ( P 0 z , j , 1 P 3 z , j , 1 ) 2 G z 1 , j R P 1 , j ) ) G y 2 , j ( 2 R 1 , j ( G z 1 , j ( P 0 x , j , 1 P 3 x , j , 1 ) G x 1 , j ( P 0 z , j , 1 P 3 z , j , 1 ) ) 0 z , j , 1 ( ( P 0 x , j , 1 P 3 x , j , 1 ) 2 G x 1 , j R P 1 , j ) + 0 x , j , 1 ( ( P 0 z , j , 1 P 3 z , j , 1 ) 2 G z 1 , j R P 1 , j ) ) + G z 2 , j ( 2 R 1 , j ( G y 1 , j ( P 0 x , j , 1 P 3 x , j , 1 ) G x 1 , j P 0 y , j , 1 ) 0 y , j , 1 ( ( P 0 x , j , 1 P 3 x , j , 1 ) 2 G x 1 , j R P 1 , j ) + 0 x , j , 1 ( P 0 y , j , 1 2 G y 1 , j R P 1 , j ) ) ] + [ G x 2 , j ( 0 z , j , 1 2 G z 1 , j R 1 , j ) G z 2 , j ( 0 x , j , 1 2 G x 1 , j R 1 , j ) ] ( T t 3 , j I 3 y , j P 3 x , j , 1 K 3 y . j P 3 z , j , 1 ) } / { 2 J 3 y , j [ ( G y 2 , j G z 1 , j G y 1 , j G z 2 , j ) 0 x , j , 1 + ( G x 1 , j G z 2 , j G x 1 , j G z 1 , j ) 0 y , j , 1 + ( G x 2 , j G y 1 , j G x 1 , j G y 2 , j ) 0 z , j , 1 ] } ,
Ω 2 , j = ( G x 2 , j H x 2 , j + G y 2 , j H y 2 , j + G z 2 , j H z 2 , j ) + { J 3 y , j [ + G x 1 , j ( 0 z , j , 1 ( P 0 y , j , 1 + 2 G y 2 , j ( G x 2 , j P 3 x , j , 1 + G z 3 , j P 3 z , j , 1 ) ) + 0 y , j , 1 ( ( P 0 z , j , 1 P 3 z , j , 1 ) + 2 G z 2 , j ( G x 2 , j P 3 x , j , 1 + G z 2 , j P 3 z , j , 1 ) ) ) + G y 1 , j ( 0 z , j , 1 ( ( P 0 x , j , 1 P 3 x , j , 1 ) + 2 G x 2 , j ( G x 2 , j P 3 x , j , 1 + G z 2 , j P 3 z , j , 1 ) ) 0 x , j , 1 ( ( P 0 z , j , 1 P 3 z , j , 1 ) + 2 G z 2 , j ( G x 2 , j P 3 x , j , 1 + G z 2 , j P 3 z , j , 1 ) ) ) + G z 1 , j ( 0 y , j , 1 ( ( P 0 x , j , 1 P 3 x , j , 1 ) + 2 G x 2 , j 2 P 3 x , j , 1 + 2 G x 2 , j G z 3 , j P 3 z , j , 1 ) + 0 x , j , 1 ( P 0 y , j , 1 + 2 G y 2 , j ( G x 2 , j P 3 x , j , 1 + G z 2 , j P 3 z , j , 1 ) ) ) ] + [ G z 1 , j ( ( 1 2 G y 2 , j 2 ) 0 x , j , 1 + 2 G x 2 , j G y 2 , j 0 y , j , 1 ) + 2 G y 1 , j G y 2 , j ( G z 2 , j 0 x , j , 1 G x 2 , j 0 z , j , 1 ) + G x 1 , j ( 2 G y 2 , j G z 2 , j 0 y , j , 1 + ( 1 + 2 G y 2 , j 2 ) 0 z , j , 1 ) ] ( T t 3 , j + I 3 y , j P 3 x , j , 1 + K 3 y , j P 3 z , j , 1 ) } / { 2 J 3 y , j [ ( G y 2 , j G z 1 , j G y 1 , j G z 2 , j ) 0 x , j , 1 + ( G x 1 , j G z 2 , j G x 1 , j G z 1 , j ) 0 y , j , 1 + ( G x 2 , j G y 1 , j G x 1 , j G y 2 , j ) 0 z , j , 1 ] } ,
G x i , j = a 0 x , j I i y , j + b 0 x , j J i y , j + c 0 x , j K i y , j ,
G y i , j = a 0 y , j I i y , j + b 0 y , j J i y , j + c 0 y , j K i y , j ,
G z i , j = a 0 z , j I i y , j + b 0 z , j J i y , j + c 0 z , j K i y , j ,
H x i , j = a 0 x , j t i x , j + b 0 x , j t i y , j + c 0 x , j t i z , j ,
H y i , j = a 0 y , j t i x , j + b 0 y , j t i y , j + c 0 y , j t i z , j ,
H z i , j = a 0 z , j t i x , j + b 0 z , j t i y , j + c 0 z , j t i z , j ,
T t i , j = I i y , j t i x , j + J i y , j t i y , j + K i y , j t i z , j ,
R i , j = 0 x , j , 1 G i , j , x n + 0 y , j , 1 G i , j , y n + 0 z , j , 1 G i , j , z n ,
R P i , j = P 0 x , j , 1 G i , j , x n + P 0 y , j , 1 G i , j , y n + P 0 z , j , 1 G i , j , z n .

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