Abstract

We present an enhanced multiposition method (EMM) to suppress the north finding error caused by bias drift with fiber optic gyroscopes (FOGs). The new proposal is a static method to find the true north, and it employs a differential strategy and a rotation scheme of increasing step angle. Using the noise model of Allan variance, we analyze the north finding errors caused by angle random walk (ARW), rate random walk (RRW), and rate ramp (RR) theoretically, where RRW and RR are two main noise sources of bias drift, and ARW is the rate white noise. Theoretical analysis indicates that, in the traditional multiposition method (TMM), as the position number increases, the error caused by ARW decreases while that by bias drift increases. Therefore, the suppressions of ARW and bias drift are conflicted with each other. The north finding accuracy is limited by bias drift. In contrast, in EMM, both errors caused by ARW and bias drift will decrease as the position number increases. Experimental results with two specific FOGs verify our theoretical analysis. In our experiments, we study the effect of position number and step angle on the north finding accuracy. Utilizing the proposed EMM, for FOG-1, the north finding error has been reduced by 76.60%, and for FOG-2, a 36.33% reduction has been achieved. Our proposal provides a more effective and stable way to find true north, and it can also be applied to other rate gyroscopes.

© 2013 Optical Society of America

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References

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  1. O. Celikel, “Application of the vector modulation method to the north finder capability gyroscope as a directional sensor,” Meas. Sci. Technol. 22, 035203 (2011).
    [CrossRef]
  2. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
    [CrossRef]
  3. G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009).
    [CrossRef]
  4. G. Eduardo Sandoval-Romero and V. Argueta-Díaz, “A simple theoretical comparison between two basic schemes in function of the Earth’s north pole detection: the static method,” J. Sens. 2010, 253642 (2010).
    [CrossRef]
  5. T. Tanaka, Y. Igarashi, M. Nara, and T. Yoshino, “Automatic north sensor using a fiber-optic gyroscope,” Appl. Opt. 33, 120–123 (1994).
    [CrossRef]
  6. R. B. Dyott, “Method for finding true north using a fibre-optic gyroscope,” Electron. Lett. 30, 1087–1088 (1994).
    [CrossRef]
  7. Z. Zhang, J. Sun, and K. Wu, “Study on technology of orientation and north-finding based on fiber optic gyroscope,” in Proceedings of International Conference on Mechatronics and Automation (IEEE, 2007), pp. 2252–2257.
  8. C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.
  9. I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
    [CrossRef]
  10. R. J. Vaccaro and A. S. Zaki, “Statistical modeling of rate Gyros,” IEEE Trans. Instrum. Meas. 61, 673–684 (2012).
    [CrossRef]
  11. N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
    [CrossRef]
  12. L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
    [CrossRef]
  13. D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
    [CrossRef]
  14. F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986).
    [CrossRef]
  15. “IEEE standard specification format guide and test procedure for single-axis interferometric fiber optic gyros,” , Reaff (2008).
  16. A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
    [CrossRef]
  17. C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
    [CrossRef]
  18. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011).
    [CrossRef]
  19. C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
    [CrossRef]
  20. X. Chen and C. Shen, “Study on error calibration of fiber optic gyroscope under intense ambient temperature variation,” Appl. Opt. 51, 3755–3762 (2012).
    [CrossRef]
  21. X. Chen and C. Shen, “Study on temperature error processing technique for fiber optic gyroscope,” Optik 124, 784–792 (2013).
    [CrossRef]

2013

X. Chen and C. Shen, “Study on temperature error processing technique for fiber optic gyroscope,” Optik 124, 784–792 (2013).
[CrossRef]

2012

X. Chen and C. Shen, “Study on error calibration of fiber optic gyroscope under intense ambient temperature variation,” Appl. Opt. 51, 3755–3762 (2012).
[CrossRef]

C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
[CrossRef]

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

R. J. Vaccaro and A. S. Zaki, “Statistical modeling of rate Gyros,” IEEE Trans. Instrum. Meas. 61, 673–684 (2012).
[CrossRef]

2011

O. Celikel, “Application of the vector modulation method to the north finder capability gyroscope as a directional sensor,” Meas. Sci. Technol. 22, 035203 (2011).
[CrossRef]

X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011).
[CrossRef]

2010

G. Eduardo Sandoval-Romero and V. Argueta-Díaz, “A simple theoretical comparison between two basic schemes in function of the Earth’s north pole detection: the static method,” J. Sens. 2010, 253642 (2010).
[CrossRef]

2009

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009).
[CrossRef]

2008

N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
[CrossRef]

2006

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[CrossRef]

2004

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

2001

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

1997

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

1994

T. Tanaka, Y. Igarashi, M. Nara, and T. Yoshino, “Automatic north sensor using a fiber-optic gyroscope,” Appl. Opt. 33, 120–123 (1994).
[CrossRef]

R. B. Dyott, “Method for finding true north using a fibre-optic gyroscope,” Electron. Lett. 30, 1087–1088 (1994).
[CrossRef]

1986

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986).
[CrossRef]

1966

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Allan, D. W.

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986).
[CrossRef]

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Argueta-Díaz, V.

G. Eduardo Sandoval-Romero and V. Argueta-Díaz, “A simple theoretical comparison between two basic schemes in function of the Earth’s north pole detection: the static method,” J. Sens. 2010, 253642 (2010).
[CrossRef]

Celikel, O.

O. Celikel, “Application of the vector modulation method to the north finder capability gyroscope as a directional sensor,” Meas. Sci. Technol. 22, 035203 (2011).
[CrossRef]

Chen, X.

X. Chen and C. Shen, “Study on temperature error processing technique for fiber optic gyroscope,” Optik 124, 784–792 (2013).
[CrossRef]

X. Chen and C. Shen, “Study on error calibration of fiber optic gyroscope under intense ambient temperature variation,” Appl. Opt. 51, 3755–3762 (2012).
[CrossRef]

Collin, J.

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

Culshaw, B.

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[CrossRef]

Dyott, R. B.

R. B. Dyott, “Method for finding true north using a fibre-optic gyroscope,” Electron. Lett. 30, 1087–1088 (1994).
[CrossRef]

Eduardo Sandoval-Romero, G.

G. Eduardo Sandoval-Romero and V. Argueta-Díaz, “A simple theoretical comparison between two basic schemes in function of the Earth’s north pole detection: the static method,” J. Sens. 2010, 253642 (2010).
[CrossRef]

El-Sheimy, N.

N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
[CrossRef]

Fan, C.

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

He, C.

C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
[CrossRef]

X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011).
[CrossRef]

Hou, H.

N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
[CrossRef]

Igarashi, Y.

Iozan, L. I.

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

Irvine-Halliday, D.

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

Ivanov, Yu. V.

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

Jin, Z.

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

Kirkko-Jaakkola, M.

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

Li, J.

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

Liu, C.

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

Malykin, G. B.

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009).
[CrossRef]

Matisov, I. A.

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

Mintchev, M. P.

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

Nara, M.

Nikolaev, V. A.

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

Niu, X.

N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
[CrossRef]

Noureldin, A.

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

Qian, F.

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

Rusu, C.

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

Shen, C.

X. Chen and C. Shen, “Study on temperature error processing technique for fiber optic gyroscope,” Optik 124, 784–792 (2013).
[CrossRef]

X. Chen and C. Shen, “Study on error calibration of fiber optic gyroscope under intense ambient temperature variation,” Appl. Opt. 51, 3755–3762 (2012).
[CrossRef]

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

Strigalev, V. E.

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

Sun, J.

Z. Zhang, J. Sun, and K. Wu, “Study on technology of orientation and north-finding based on fiber optic gyroscope,” in Proceedings of International Conference on Mechatronics and Automation (IEEE, 2007), pp. 2252–2257.

Tabler, H.

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

Takala, J.

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

Tanaka, T.

Tian, W.

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

Vaccaro, R. J.

R. J. Vaccaro and A. S. Zaki, “Statistical modeling of rate Gyros,” IEEE Trans. Instrum. Meas. 61, 673–684 (2012).
[CrossRef]

Walls, F. L.

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986).
[CrossRef]

Wang, X.

Wang, Z.

C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
[CrossRef]

X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011).
[CrossRef]

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

Wu, K.

Z. Zhang, J. Sun, and K. Wu, “Study on technology of orientation and north-finding based on fiber optic gyroscope,” in Proceedings of International Conference on Mechatronics and Automation (IEEE, 2007), pp. 2252–2257.

Yang, C.

C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
[CrossRef]

Yoshino, T.

Yu, S.

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

Zaki, A. S.

R. J. Vaccaro and A. S. Zaki, “Statistical modeling of rate Gyros,” IEEE Trans. Instrum. Meas. 61, 673–684 (2012).
[CrossRef]

Zhang, Z.

Z. Zhang, J. Sun, and K. Wu, “Study on technology of orientation and north-finding based on fiber optic gyroscope,” in Proceedings of International Conference on Mechatronics and Automation (IEEE, 2007), pp. 2252–2257.

Appl. Opt.

Electron. Lett.

R. B. Dyott, “Method for finding true north using a fibre-optic gyroscope,” Electron. Lett. 30, 1087–1088 (1994).
[CrossRef]

IEEE Trans. Instrum. Meas.

R. J. Vaccaro and A. S. Zaki, “Statistical modeling of rate Gyros,” IEEE Trans. Instrum. Meas. 61, 673–684 (2012).
[CrossRef]

N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertial sensors using Allan variance,” IEEE Trans. Instrum. Meas. 57, 140–149 (2008).
[CrossRef]

J. Sens.

G. Eduardo Sandoval-Romero and V. Argueta-Díaz, “A simple theoretical comparison between two basic schemes in function of the Earth’s north pole detection: the static method,” J. Sens. 2010, 253642 (2010).
[CrossRef]

Meas. Sci. Technol.

O. Celikel, “Application of the vector modulation method to the north finder capability gyroscope as a directional sensor,” Meas. Sci. Technol. 22, 035203 (2011).
[CrossRef]

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[CrossRef]

L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications,” Meas. Sci. Technol. 23, 025005 (2012).
[CrossRef]

C. Fan, Z. Jin, W. Tian, and F. Qian, “Temperature drift modeling of fibre optic gyroscopes based on a grey radial basis function neural network,” Meas. Sci. Technol. 15, 119–126 (2004).
[CrossRef]

Opt. Eng.

A. Noureldin, D. Irvine-Halliday, H. Tabler, and M. P. Mintchev, “New technique for reducing the angle random walk at the output of fiber optic gyroscopes during alignment processes of inertial navigation systems,” Opt. Eng. 40, 2097–2106 (2001).
[CrossRef]

C. He, C. Yang, and Z. Wang, “Fusion of finite impulse response filter and adaptive Kalman filter to suppress angle random walk of fiber optic gyroscopes,” Opt. Eng. 51, 124401 (2012).
[CrossRef]

Opt. Lett.

Optik

X. Chen and C. Shen, “Study on temperature error processing technique for fiber optic gyroscope,” Optik 124, 784–792 (2013).
[CrossRef]

Proc. IEEE

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986).
[CrossRef]

Tech. Phys.

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009).
[CrossRef]

I. A. Matisov, V. E. Strigalev, V. A. Nikolaev, and Yu. V. Ivanov, “A method for determining errors of a static fiber-optic gyrocompass,” Tech. Phys. 42, 1081–1084 (1997).
[CrossRef]

Other

Z. Zhang, J. Sun, and K. Wu, “Study on technology of orientation and north-finding based on fiber optic gyroscope,” in Proceedings of International Conference on Mechatronics and Automation (IEEE, 2007), pp. 2252–2257.

C. Shen, Z. Wang, C. Liu, J. Li, and S. Yu, “Data processing method of multi-position strap-down north seeking system based on SVD,” in International Conference on Mechatronic Science, Electric Engineering and Computer (IEEE, 2011), pp. 994–997.

“IEEE standard specification format guide and test procedure for single-axis interferometric fiber optic gyros,” , Reaff (2008).

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Figures (8)

Fig. 1.
Fig. 1.

Noise characteristics of two specific FOGs: (a) raw noise, (b) corresponding Allan variance curve.

Fig. 2.
Fig. 2.

Schematic diagram of the north finding system using FOGs.

Fig. 3.
Fig. 3.

Examples of the TMM and EMM to find the true north (the numbers in the figure indicate the measurement sequence): (a) TMM—position number n=9 and step angle Δθ=40°; (b) EMM—n+1=10, step size m=1, and step angle Δθ=40°; (c) EMM—n+1=10, step size m=2, and step angle Δθ=80°; (d) EMM—n+1=16, step size m=7, and step angle Δθ=168°.

Fig. 4.
Fig. 4.

Numerical analysis for comparison of the TMM and EMM with two FOGs: (a) FOG-1—S=1.29×104deg/s1/2, K=3.21×105deg/s3/2, and R=9.16×107deg/s2; (b) FOG-2—S=1.23×104deg/s1/2, K=3.27×106deg/s3/2, and R=0deg/s2.

Fig. 5.
Fig. 5.

Numerical analysis of the effect of the step angle on the error caused by ARW and RRW in the scheme of the EMM: (a) FOG-1 with n=9 and 15, (b) FOG-2 with n=9 and 15.

Fig. 6.
Fig. 6.

Experimental setup to find the true north.

Fig. 7.
Fig. 7.

Experimental and theoretical results of the north finding error versus position number in the TMM and EMM with two FOGs; the curves present the theoretical results, and the discrete markers present the experimental results.

Fig. 8.
Fig. 8.

Experimental and theoretical results of the north finding error versus step angle in the EMM with two FOGs; the curves present the theoretical results, and the discrete markers present the experimental results.

Equations (28)

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

σΩ2(aτ0)=R2a2τ022+K2aτ03+B2[2π]ln(2)+S2aτ0+3Q2a2τ02,
{yk=Ωk+bk+vkbk=bk1+wk1+rk1,
{E{vk}=0E{wk}=0rk1=Rτ0,E{[vkwk][vjwj]}=[S2/τ000K2τ0]δkj,
y¯i=ωecosαcos(ϕθi)+b¯i+v¯i,
y¯i=acosθi+csinθi+b¯i+v¯i,
ϕ^=arctan(c^/a^),
Δθ=2π/n,
a^=(2/n)i=1i=ny¯icosθi,c^=(2/n)i=1i=ny¯isinθi.
Δθ=2mπ/n,
y¯iy¯i1=a(cosθicosθi1)+c(sinθisinθi1)+(b¯ib¯i1)+(v¯iv¯i1).
a^=(1/(nncosΔθ))i=2i=n+1(y¯iy¯i1)(cosθicosθi1),
c^=(1/(nncosΔθ))i=2i=n+1(y¯iy¯i1)(sinθisinθi1).
ϕ^=ϕ+Δϕ,
a^=a+Δa,
c^=c+Δc,
tanΔϕ=cosϕΔcsinϕΔaωecosα+cosϕΔa+sinϕΔc.
tanΔϕcosϕΔcsinϕΔaωecosα=Δv+Δw+Δr,
σΔv2=2S2(ωecosα)2nts,
σΔw2K2(tr+ts)[2+cos(2ϕ+Δθ)]+K2ts(cosΔθ1)/3(ωecosα)2(1cosΔθ)n,
σΔr2=R2(ts+tr)2(ωecosα)2(cosϕsinΔθcosΔθ1+sinϕ)2,
σΔw2K2(ts+tr)n2π2(ωecosα)2(32sin2ϕ),
σΔr2R2(ts+tr)2(ωecosα)2(ncosϕπ+sinϕ)2.
σΔv2=2S2[11/n+2cos2ϕ/(1cosΔθ)/n](ωecosα)2nts,
σΔw2K2ts[2+cosΔθ(11/n)(cos2ϕ)/n]+3K2tr3(ωecosα)2(1cosΔθ)n,
σΔr2=0,
σΔv22S2(ωecosα)2nts,
σΔw2K2[ts(2+cosΔθ)+3tr]3(ωecosα)2(1cosΔθ)n.
tr=Δθ/ωs+tc,

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