Skip to main content
SpringerLink
Log in

Integer least squares with quadratic equality constraints and its application to GNSS attitude determination systems

  • Regular Papers
  • Control Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

In this paper we introduce the quadratically constrained integer least-squares problem and show how the LAMBDA method can be used to solve it for the purpose of GNSS attitude determination. The integer least-squares principle with quadratic equality constraints (ILSQE) is used to formulate our cost function. The solution of the ILSQE problem is derived and it is shown how the solution can be computed efficiently and rigorously with a novel LAMBDA based method. Experimental results with various single frequency GPS receivers are given to show the effectiveness of the proposed method. The method is also compared with some current methods of GNSS attitude determination. Apart from its efficiency, the proposed method is shown to dramatically improve the success rates of integer ambiguity GNSS attitude resolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ashtech, “ADU5: GNSS sensors and marine survey solutions,” http://products.thalesnavigation.com/en/products/index.asp , 2003.

  2. I. Y. Bar-Itzhack, P. Montgomery, and J. Garrick, “Algorithm for attitude determination using GPS,” Proc. of AIAA Guidance, Navigation and Control Conf., New Orleans, LA, USA Paper no. AIAA 97-3616, Aug. 1997.

  3. C. E. Cohen, Attitude Determination Using GPS, Ph.D. Thesis, Stanford University, 1992.

  4. Furuno Electric Co., “Model SC-120: Satellite compass,” http://www.furuno.co.jp/english/index.html, 2003.

  5. G. H. Golub and C. F. van Loan, Matrix Computations, 2nd edition, The Johns Hopkins University Press, Baltimore, Maryland, USA, 1998.

    Google Scholar 

  6. R. R. Hatch and H. J. Euler, “Comparison of several kinematic techniques,” Proc. of ION GPS-94, Salt Lake City, Utah, USA pp. 363–370, September 20–23, 1994.

  7. P. J. Jong de and C. C. J. M. Tiberius, The LAMBDA Method For Integer Ambiguity Estimation: Implementation Aspects, Delft Geodetic Computing Centre LGR series, no. 12, 1996.

  8. P. Joosten, The LAMBDA-Method: Matlab™ Implementation. Matlab Toolbox and Manual, Mathematical Geodesy and Positioning, TU Delft, 2001.

    Google Scholar 

  9. G. Lu, “Development of a GPS multi-antenna system for attitude determination,” UCGE Reports 20073, Dept. of Geomatics Eng., University of Calgary, 1995.

  10. C. Park, I. Kim, G. I. Jee, and J. G. Lee, “Efficient ambiguity search using constraints equation,” Proc. of IEEE Position, Location and Navigation Symposium PLANS’96, Atlanta, Georgia, USA, 1996.

  11. C. Park, I. Kim, G. I. Jee, and J. G. Lee, “Efficient technique to fix GPS carrier phase integer ambiguity on-the-fly,” IEE Proceedings, Radar, Sonar and Navigation, vol. 144, no. 3, pp. 148–155, 1997.

    Article  Google Scholar 

  12. C. Park and I. Kim, “An error analysis of 2-dimensional attitude determination using global positioning system,” IEICE Transaction on Communications, vol. E83-B, no. 6, pp. 1370–1373, 2000.

    Google Scholar 

  13. C. Park and P. J. G. Teunissen, “A baseline constrained LAMBDA method for an integer ambiguity resolution of GNSS attitude determination system,” Journal of Institute of Control, Robotics and Systems (in Korean), vol. 14, no. 6, pp. 587–594, June 2008.

    Google Scholar 

  14. P. J. G. Teunissen, Least-squares Estimation of the Integer Ambiguities, Delft Geodetic Computing Centre LGR series, no. 6, 1993.

  15. P. J. G. Teunissen, “A new method for fast carrier phase ambiguity estimation,” Proc. of IEEE Position, Location and Navigation Symposium PLANS’94, Las Vegas, NV, USA, pp. 562–573, April 11–15, 1994.

  16. P. J. G. Teunissen and A. Kleusberg editors, GPS for Geodesy, 2nd edition, Springer-Verlag, Berlin, Germany, 1998.

    Google Scholar 

  17. P. J. G. Teunissen, “Success probability of integer GPS ambiguity rounding and bootstrapping,” Journal of Geodesy, vol. 72, pp. 606–612, 1998.

    Article  MATH  Google Scholar 

  18. P. J. G. Teunissen, “GNSS ambiguity bootstrapping: theory and application,” Proc. of International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, Canada, pp. 246–254, 2001.

  19. P. J. G. Teunissen, “Theory of carrier phase ambiguity resolution,” Proc. of the 9th GNSS workshop-2002 International Symposium on GPS/GNSS, November 6–8, Wuhan, China, 2002.

  20. C. C. J. M. Tiberius, Recursive Data Processing for Kinematic GPS Surveying, Ph.D. Thesis, Mathematical Geodesy and Positioning, TU Delft, 1998.

    Google Scholar 

  21. C. H. Tu, K. Y. Wang, and W. C. Melton, “GPS compass: a novel navigation equipment,” Proc. of ION National Technical Meeting, Santa Monica, CA, USA, 1996.

  22. S. Verhagen, Visualization of GNSS-related design parameters. Manual for Matlab User Interface Visual, Mathematical Geodesy and Positioning, TU Delft, 2002.

    Google Scholar 

  23. M. Ziebart and P. Cross, “LEO GPS attitude determination algorithm for a macro-satellite using boom-arm deployed antennas,” GPS Solutions, vol. 6, pp. 242–256, 2003.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical and Computer Eng., Chungbuk National University, 410 Sungbong-ro, Heungduk-gu, Chungbuk, 361-763, Korea

    Chansik Park

  2. Delft Institute for Earth Observation and Space Systems (DEOS), Technical University of Delft, Delft, the Netherlands

    Peter J. G. Teunissen

Authors
  1. Chansik Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter J. G. Teunissen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chansik Park.

Additional information

Recommended by Editor Hyun Seok Yang. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No. R11-2008-014-02001-0). The research of the second author was done in the framework of his ARC International Linkage Professorial Fellowship, at Curtin University of Technology, Perth, Australia, with Professor Will Featherstone as his host. These supports are greatly acknowledged.

Chansik Park received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University in 1984, 1986, and 1997 respectively. He is currently with the School of Electrical and Computer Engineering, Chungbuk National University, Cheongju, Korea. His research interests include GNSS, SDR, AJ, ITS and WSN.

Peter J. G. Teunissen has 20 years of research experience in GNSS Positioning and Navigation. He is the inventor of the LAMBDA method for GNSS carrier phase ambiguity resolution. He is the Head of the Department of Earth Observation and Space Systems of the Delft University of Technology and a member of the Royal Netherlands Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Park, C., Teunissen, P.J.G. Integer least squares with quadratic equality constraints and its application to GNSS attitude determination systems. Int. J. Control Autom. Syst. 7, 566–576 (2009). https://doi.org/10.1007/s12555-009-0408-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-009-0408-0

Keywords

Search

Navigation