Skip to main content
SpringerLink
Log in

The TRIAD algorithm as maximum likelihood estimation

  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

But what likelihood is in that? William Shakespeare (15642–1616) Measure for Measure, Act IV, scene ii

Abstract

The TRIAD algorithm is shown to be derivable as a maximum-likelihood estimator. In particular, using the QUEST measurement model, the TRIAD attitude error covariance matrix can be derived as the inverse of the Fisher information matrix. The treatment here gives a microscopic analysis of the algorithm and its connection to the QUEST algorithm. It also sheds valuable light on the origin of discrete degeneracies in deterministic attitude estimation.

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. BLACK, H. D. “A Passive System for Determining the Attitude of a Satellite,” AIAA Journal, Vol. 2, July 1964, pp. 1350–1351.

    Article  Google Scholar 

  2. SHUSTER, M. D. and OH, S. D. “Three-Axis Attitude Determination from Vector Observations,” Journal of Guidance and Control, Vol. 4, No. 1, January–February 1981, pp. 70–77.

    Article  MATH  Google Scholar 

  3. SHUSTER, M. D. “Maximum Likelihood Estimation of Spacecraft Attitude,” The Journal of the Astronautical Sciences, Vol. 37, No. 1, January–March 1989, pp. 79–88.

    Google Scholar 

  4. MARKLEY, F. L. and MORTARI M. “Quaternion Attitude Estimation Using Vector Measurements, The Journal of the Astronautical Sciences, Vol. 48, Nos. 2 and 3, April–September 2000, pp. 359–380.

    Google Scholar 

  5. MARKLEY, F. L. “Attitude Determination Using Vector Observations and the Singular Value Decomposition,” The Journal of the Astronautical Sciences, Vol. 36, No. 3, July–September 1988, pp. 245–258.

    Google Scholar 

  6. MARKLEY, F. L. “Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm,” The Journal of the Astronautical Sciences, Vol. 41, No. 2, April–June 1993, pp. 261–280.

    MathSciNet  Google Scholar 

  7. MORTARI, D. “ESOQ: A Closed-Form Solution to the Wahba Problem,” The Journal of the Astronautical Sciences, Vol. 45, No. 2, April–June 1997, pp. 195–204.

    MathSciNet  Google Scholar 

  8. MORTARI, D. “Second Estimator for the Optimal Quaternion,” Journal of Guidance, Contro and Dynamics, Vol. 23, No. 4, September–October 2000, pp. 885–888.

    Article  Google Scholar 

  9. BRUCCOLERI, C., LEE, D.-J., and Mortari, D. “Single-Point Optimal Attitude Determination Using Modified Rodrigues Parameters,” presented as paper AAS-05-459 at the AAS Astronautics Symposium, Grand Island, New York, 2005; reprinted in Advances in the Astronautical Sciences, Vol. 122, 2006, pp. 127–148.

    Google Scholar 

  10. SHUSTER, M. D. “A Survey of Attitude Representations,” The Journal of the Astronautical Sciences, Vol. 41, No. 4, October–December 1993, pp. 439–517.

    MathSciNet  Google Scholar 

  11. WAHBA, G. “Problem 65-1: A Least Squares Estimate of Spacecraft Attitude,” SIAM Review, Vol. 7, No. 3, July 1965, p. 409.

    Article  Google Scholar 

  12. SHUSTER, M. D. “SCAD 3 A Fast Algorithm for Star Camera Attitude Determination,” The Journal of the Astronautical Sciences, Vol. 52, No. 3, July–September 2004, pp. 391–404.

    MathSciNet  Google Scholar 

  13. SHUSTER, M. D. “Deterministic Three-Axis Attitude Determination,” The Journal of the Astronautical Sciences, Vol. 52, No. 3, July–September 2004, pp. 405–419.

    MathSciNet  Google Scholar 

  14. SHUSTER, M. D. “Attitude Determination in Higher Dimensions” (Engineering Note), Journal of Guidance, Control and Dynamics, Vol. 16, No. 2, March–April 1993, pp. 393–395.

    Article  MathSciNet  Google Scholar 

  15. SHUSTER, M. D. “Attitude Analysis in Flatland: The Plane Truth,” The Journal of the Astronautical Sciences, Vol. 52, Nos. 1 and 2, January–June 2004, pp. 195–209.

    MathSciNet  Google Scholar 

  16. SORENSEN, H.W. Parameter Estimation, Marcel Dekker, New York 1980.

    Google Scholar 

  17. CHENG, Y. and SHUSTER, M. D. “QUEST and the Anti-QUEST: Good and Evil Attitude Estimation,” The Journal of the Astronautical Sciences, Vol. 53, No. 3, July–September 2005, pp. 337–351.

    Google Scholar 

  18. SHUSTER, M. D. “Kalman Filtering of Spacecraft Attitude and the QUEST Model,” The Journal of the Astronautical Sciences, Vol. 38, No. 3, July–September 1990, pp. 377–393; Erratum: Vol. 51, No. 3, July–September 2003, p. 359.

    Google Scholar 

  19. SEDLAK, J. and CHU, D. “Kalman Filter Estimation of Attitude and Gyro Bias with the QUEST Observation Model,” presented as paper AAS-93-297 at the AAS/GSFC International Symposium on Space Flight Dynamics, NASA Goddard Space Flight Center, April 26–30, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Acme Spacecraft Company, 13017 Wisteria Drive, Box 328, Germantown, Maryland, 20874, USA

    Malcolm D. Shuster (Director of Research)

Authors
  1. Malcolm D. Shuster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Malcolm D. Shuster.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shuster, M.D. The TRIAD algorithm as maximum likelihood estimation. J of Astronaut Sci 54, 113–123 (2006). https://doi.org/10.1007/BF03256479

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03256479

Keywords

Search

Navigation