Abstract

This paper presents an optoelectronic measurement system for measuring 6 degree-of-freedom (DOF) motion error of rotary parts. It comprises a pyramid-polygon-mirror, three laser diodes and three 2-axis position sensing detectors (PSD). The laser/PSD pairs are arranged evenly around the pyramid-polygon-mirror, which is mounted rigidly on and aligned axially with the rotary part to be measured. Laser rays from the laser diodes are reflected off the respective mirrors to the respective PSDs. The incidence point of the laser ray on the PSD’s surface varies with the pose of the pyramid-polygon-mirror, allowing the PSD to register variation in the mirror and, thereby, the rotary part. With appropriate orientation of the lasers and PSDs, this system can measure variation (error) during rotation of a rotary part. By use of skew-ray tracing and first order Taylor series expansion, the system achieves measurement of translational and rotational motion errors for each Cartesian axis. To validate the proposed methodology, a laboratory prototype system is built. System verification and stability tests are conducted to evaluate its performance. Stability test results show that measurement errors and maximum crosstalk are within ±1 μm in translation and ±1.5 arc sec in rotation.

©2007 Optical Society of America

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References

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  1. J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” ASME J. Eng. Indus. 115, 85–92 (1993).
  2. P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” ASME J. Dyn. Syst. 118, 425–433 (1996).
    [Crossref]
  3. 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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
    [Crossref]
  4. E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
    [Crossref]
  5. J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
    [Crossref]
  6. C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
    [Crossref]
  7. W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
    [Crossref]
  8. E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
    [Crossref]
  9. P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003).
    [Crossref]

2007 (1)

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

2005 (2)

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
[Crossref]

2003 (1)

P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003).
[Crossref]

2002 (1)

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

2001 (1)

E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
[Crossref]

1996 (1)

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

1994 (1)

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
[Crossref]

1993 (1)

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

Bae, E. W.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
[Crossref]

Bokelberg, E. H.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
[Crossref]

Chen, C. J.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

Ehmann, K. F.

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

Hsieh, W. H.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

Jwo, H. H.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

Jywe, W. Y.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

Kim, J. A.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
[Crossref]

Kim, K. C.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

Kim, S. H.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
[Crossref]

Kwak, Y. K.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
[Crossref]

Liao, T. T.

P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003).
[Crossref]

Lin, P. D.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003).
[Crossref]

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

Liu, C. H.

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
[Crossref]

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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
[Crossref]

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

Shyu, L. H.

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

Sommer III, H. J.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
[Crossref]

Trethewey, M. W.

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
[Crossref]

Wu, S. M.

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

Yang, T. Y.

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

ASME J. Dyn. Syst. (1)

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

ASME J. Eng. Indus. (1)

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

ASME J. Manu. Sci. Eng. (1)

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,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005).
[Crossref]

Int. J. Mach. Tool Manu. (1)

W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007).
[Crossref]

J. Appl. Opt. (1)

P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003).
[Crossref]

J. Sound Vib. (1)

E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994).
[Crossref]

Mea. Sci. Technol. (1)

E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001).
[Crossref]

Precis. Eng. (2)

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002).
[Crossref]

C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005).
[Crossref]

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Figures (14)

Fig. 1.
Fig. 1. Light ray at a flat reflective boundary surface.

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Fig. 2.
Fig. 2. Schematic diagram of a 6-DOF motion measurement system with a 6-sided pyramid-polygon-mirror.

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Fig. 3.
Fig. 3. A laser/PSD sub-system of the Fig. 2 measurement system.

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Fig. 4.
Fig. 4. Photograph of lab-built 6-DOF motion error measurement system using the mirrored exterior of the commercial corner cube as a 3-sided pyramid-polygon-mirror: (a) set up for verification of angular motion; (b) set up for verification of linear motion.

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Fig. 5.(a).
Fig. 5.(a). Verification results of δx .

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Fig. 5.(b).
Fig. 5.(b). Verification results of δy .

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Fig. 5.(c).
Fig. 5.(c). Verification results of δz .

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Fig. 5.(d).
Fig. 5.(d). Verification results of ηx

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Fig. 5. (e).
Fig. 5. (e). Verification results of ηy

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Fig. 5. (f).
Fig. 5. (f). Verification results of ηz .

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Fig. 6.
Fig. 6. Results of system stability test (sample rate 1000 Hz): (a) translational parameters δx , δy and δz ; (b) rotational parameters ηx , ηy and ηz .

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Fig. 7.
Fig. 7. Photograph of lab-built 6-DOF rotary table measurement system.

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Fig. 8.
Fig. 8. The measured results of the 6-DOF motion error of a rotary table from the lab-built system (square, circle and triangle each equal one rotation of the rotary table).

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Fig. 9.
Fig. 9. Autocollimator results (with 24-sided mirror) for rotary table measurement (flattened curve is artifact caused by curve exceeding measurement range).

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Tables (2)

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Table 1: The differences of δ̱ and η̱ calculated from Eqs. (23) and (21) (units: deg. or mm)

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Table 2: Component details

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Equations (56)

Equations on this page are rendered with MathJax. Learn more.

i r i = Rot ( y i , α i ) [ β i 0 0 1 ] T = [ β i Cα i 0 − β i Sα i 1 ] T
i n i = s i ( ∂ ( i r i ) ∂ α i × ∂ ( i r i ) ∂ β i ) ( ∂ ( i r i ) ∂ α i × ∂ ( i r i ) ∂ β i ) = s i [ 0 0 − 1 0 ] T
i A 0 = [ I ix J ix K ix t ix I iy J iy K iy t iy I iz J iz K iz t iz 0 0 0 1 ] .
n i = [ n ix n iy n iz 0 ] T = 0 A i i n i = s i [ − I iy − J iy − K iy 0 ] T .
P i = [ P ix P iy P iz 1 ] T = [ P i − 1 x + ℓ i − 1 x λ i P i − 1 y + ℓ i − 1 y λ i P i − 1 z + ℓ i − 1 z λ i 1 ] T
λ i = − ( I iy P i − 1 x + J iy P i − 1 y + K iy P i − 1 z + t iy ) I iy ℓ i − 1 x + J iy ℓ i − 1 y + K iy ℓ i − 1 z = − G i B i .
ℓ i = [ ℓ ix ℓ iy ℓ iz 0 ] T = [ ℓ i − 1 x − 2 I iy B i ℓ i − 1 y − 2 J iy B i ℓ i − 1 z − 2 K iy B i 0 ] T .
0 ' A 1 a = 0 A 1 a = Trans ( t ax , t ay , t az ) Rot z 90° Rot x − ϕ Rot y ω ay
0 ' A 1 b = 0 A 1 b = Trans ( t bx , t by , t bz ) Rot ( z , 210° ) Rot x − ϕ Rot y ω by
0 ' A 1 c = 0 A 1 c = Trans ( t cx , t cy , t cz ) Rot ( z , 330 ° ) Rot x − ϕ Rot y ω cy
P 0 a = 0 A 1 a 1 P 0 = [ f ( Sϕω ay + Cϕ ) 2 + t ax fCω ay 2 + t ay f ( − CϕSω ay + Sϕ ) 2 + t az 1 ] T
P 0 b = 0 A 1 b 1 P 0
= [ f − ( − 3 Cω by − SϕSω by − Cϕ ) 2 2 + t bx f ( − Cω by + 3 SϕSω by + 3 Cϕ ) 2 2 + t by f ( − CϕSω by + Sϕ ) 2 + t bz 1 ] T
P 0 c = 0 A 1 c 1 P 0
= [ f − ( − 3 Cω cy − SϕSω cy − Cϕ ) 2 2 + t cz f ( − Cω cy + 3 SϕSω cy + 3 Cϕ ) 2 2 + t cy f ( − CϕSω cy + Sϕ ) 2 + t cz 1 ] T
P 1 a = 0 A 1 a 1 P 1 = [ t ax t ay t az 1 ] T
P 1 b = 0 A 1 b 1 P 1 = [ t bx t by t bz 1 ] T
P 1 c = 0 A 1 c 1 P 1 = [ t cx t cy t cz 1 ] T .
ℓ 0 a = 0 A 1 a 1 ℓ 0 = [ − Sϕ Sω ay − Cϕ 2 − Cω ay 2 CϕSω ay − Sϕ 2 0 ] T
ℓ 0 b = 0 A 1 b 1 ℓ 0 = [ SϕSω by + Cϕ + 3 Cω by 2 2 Cω by − 3 SϕSω by − 3 Cϕ 2 2 CϕSω by − Sϕ 2 0 ] T
ℓ 0 c = 0 A 1 c 1 ℓ 0 = [ SϕSω cy + Cϕ − 3 Cω cy 2 2 Cω cy + 3 SϕSω cy + 3 Cϕ 2 2 CϕSω cy − Sϕ 2 0 ] T .
ℓ 0 b = Rot z 120° ℓ 0 a
ℓ 0 c = Rot z 120° ℓ 0 a .
P 1 b = Rot z 120° P 1 a
P 1 c = Rot z 120° P 1 a
0 A 0 ' = Trans ( δ x , δ y , δ z ) Rot ( z , η z ) Rot ( y , η y ) Rot ( x , η x )
1 a A 0 ' = Rot ( y , − ω ay ) Rot x ϕ Rot z − 90° Trans − t ax − t ay − t az
= Rot ( y , − ω ay ) Rot x ϕ Rot z − 90° Trans − t ax 0 − t az
1 b A 0 ' = Rot ( y , − ω by ) Rot x ϕ Rot z − 210 ° Trans − t bx − t by − t bz
= Rot ( y , − ω ay ) Rot x ϕ Rot z − 210 ° Trans ( t ax 2 , − 3 t ax 2 , − t az )
1 c A 0 ' = Rot ( y , − ω cy ) Rot x ϕ Rot z − 330 ° Trans − t cx − t cy − t cz
= Rot ( y , − ω ay ) Rot x ϕ Rot z − 330 ° Trans ( t ax 2 , − 3 t ax 2 , − t az ) .
P 1 a = [ P 1 ax P 1 ay P 1 az 1 ] T = P 0 a + ℓ 0 a λ 1 a
P 1 b = [ P 1 bx P 1 by P 1 bz 1 ] T = P 0 b + ℓ 0 b λ 1 b
P 1 c = [ P 1 cx P 1 cy P 1 cz 1 ] T = P 0 c + ℓ 0 c λ 1 c
ℓ 1 a = [ ℓ 1 ax ℓ 1 ay ℓ 1 az 0 ] T = [ ℓ 0 ax − 2 I 1 ay B 1 a ℓ 0 ay − 2 I 1 ay B 1 a ℓ 0 az − 2 I 1 ay B 1 a 0 ]
ℓ 1 b = [ ℓ 1 bx ℓ 1 by ℓ 1 bz 0 ] T = [ ℓ 0 bx − 2 I 1 by B 1 b ℓ 0 by − 2 I 1 by B 1 b ℓ 0 bz − 2 I 1 by B 1 b 0 ]
ℓ 1 c = [ ℓ 1 cx ℓ 1 cy ℓ 1 cz 0 ] T = [ ℓ 0 cx − 2 I 1 cy B 1 c ℓ 0 cy − 2 I 1 cy B 1 c ℓ 0 cz − 2 I 1 cy B 1 c 0 ]
P 2 a = [ P 2 ax P 2 ay P 2 az 1 ] T = P 1 a + ℓ 1 a λ 2 a
P 2 b = [ P 2 bx P 2 by P 2 bz 1 ] T = P 1 b + ℓ 1 b λ 2 b
P 2 c = [ P 2 cx P 2 cy P 2 cz 1 ] T = P 1 c + ℓ 1 c λ 2 c
2 a P 2 a = [ 2 a P 2 ax ( δ ̱ , η ̱ ) 0 2 a P 2 az ( δ ̱ , η ̱ ) 1 ] T
2 b P 2 b = [ 2 b P 2 bx ( δ ̱ , η ̱ ) 0 2 b P 2 bz ( δ ̱ , η ̱ ) 1 ] T
2 c P 2 c = [ 2 c P 2 cx ( δ ̱ , η ̱ ) 0 2 c P 2 cz ( δ ̱ , η ̱ ) 1 ] T
[ 2 a P ̅ 2 ax 2 a P ̅ 2 az ] = [ 2 a P 2 ax ( 0 ̱ , 0 ̱ ) 2 a P 2 az ( 0 ̱ , 0 ̱ ) ]
[ 2 b P ̅ 2 bx 2 b P ̅ 2 bz ] = [ 2 b P 2 bx ( 0 ̱ , 0 ̱ ) 2 b P 2 bz ( 0 ̱ , 0 ̱ ) ]
[ 2 c P ̅ 2 cx 2 c P ̅ 2 cz ] = [ 2 c P 2 cx ( 0 ̱ , 0 ̱ ) 2 c P 2 cz ( 0 ̱ , 0 ̱ ) ] .
[ X a ( δ ̱ , η ̱ ) Z a ( δ ̱ , η ̱ ) ] = [ 2 a P 2 ax 2 a P 2 az ] − [ 2 a P ̅ 2 ax 2 a P ̅ 2 az ]
[ X b ( δ ̱ , η ̱ ) Z b ( δ ̱ , η ̱ ) ] = [ 2 b P 2 bx 2 b P 2 bz ] − [ 2 b P ̅ 2 bx 2 b P ̅ 2 bz ]
[ X c ( δ ̱ , η ̱ ) Z c ( δ ̱ , η ̱ ) ] = [ 2 c P 2 cx 2 c P 2 cz ] − [ 2 c P ̅ 2 cx 2 c P ̅ 2 cz ]
{ Z a = − 2 d Sω ay ( η x ) 6 − − 2 d Cω ay ( η y ) 2 + − 2 d Sω ay ( η z ) 3 Z b = d ( 3 Cω ay + Sω ay ) ( η x ) 6 + d ( Cω ay − Sω ay ) ( η y ) 2 + 2 d Sω ay ( η z ) 3 Z c = − d ( 3 Cω ay − Sω ay ) ( η x ) 6 + d ( Cω ay − Sω ay ) ( η y ) 2 + 2 d Sω ay ( η z ) 3 X a = − 2 δ x 3 − 2 δ z 3 + 2 d Sω ay ( η x ) 3 − 2 d Sω ay ( η y ) 3 − 4 d Cω ay ( η z ) 6 X b = δ x 3 − δ y − 2 δ z 3 − d ( Cω ay − 3 Sω ay ) ( η x ) 3 + d ( Cω ay + Sω ay ) ( η y ) − 4 d Cω ay ( η z ) 6 X c = δ x 3 + δ y − 2 δ z 3 − d ( Cω ay + 3 Sω ay ) ( η x ) 3 − d ( Cω ay − Sω ay ) ( η y ) − 4 Cω ay ( d η z ) 6
{ η x = ( 2 Z a − Z b − Z c ) ( 6 d ) η y = ( Z b − Z c ) ( 2 d ) η z = − ( Z a + Z b + Z c ) ( 12 d ) δ x = [ 3 ( − 2 X a + X b + X c ) + 3 6 ( Z b − Z c ) ] 6 δ y = ( X c − X b ) 2 − 2 Z a + ( Z b + Z c ) 2 δ z = − ( X a + X b + X c ) 6
Rot x θ = [ 1 0 0 0 0 Cθ − Sθ 0 0 Sθ Cθ 0 0 0 0 1 ]
Rot y θ = [ Cθ 0 Sθ 0 0 1 0 0 − Sθ 0 Cθ 0 0 0 0 1 ]
Rot z θ = [ Cθ − Sθ 0 0 Sθ Cθ 0 0 0 0 1 0 0 0 0 1 ]
Trans ( t x , t y , t z ) = [ 1 0 0 t x 0 1 0 t y 0 0 1 t z 0 0 0 1 ]

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