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Attitude and Position Tracking

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Guide to Geometric Algebra in Practice

Abstract

Several applications require the tracking of attitude and position of a body based on velocity data. It is tempting to use direction cosine matrices (DCM), for example, to track attitude based on angular velocity data, and to integrate the linear velocity data separately in a suitable frame. In this chapter we make the case for using bivectors as the attitude tracking method of choice since several features make their performance and flexibility superior to that of DCMs, Euler angles or even rotors. We also discuss potential advantages in using CGA to combine the integration of angular and linear velocities in one step, as the features that make bivectors attractive for tracking rotations extend to bivectors that represent general displacements.

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Notes

  1. 1.

    Miller’s algorithm is used for computing the update rotation Φ directly from gyroscope data. This algorithm is of order 5 and only has to be evaluated on every third sample.

  2. 2.

    Modern gyroscope output bandwidths are typically 100–400 Hz.

  3. 3.

    Which is effectively Chasles’ theorem; see also Chap. 5 in this volume.

References

  1. Bar-Itzhack, I.Y.: Navigation computation in terrestrial strapdown inertial navigation systems. IEEE Trans. Aerosp. Electron. Syst. 13, 679–689 (1977)

    Article  Google Scholar 

  2. Bar-Itzhack, I.Y.: Extension of Eulers theorem to n-dimensional spaces. IEEE Trans. Aerosp. Electron. Syst. 25, 903–909 (1989)

    Article  MathSciNet  Google Scholar 

  3. Bortz, J.E.: A new mathematical formulation for strapdown inertial navigation. IEEE Trans. Aerosp. Electron. Syst. 7, 61–66 (1971)

    Article  Google Scholar 

  4. Goodman, L.E., Robinson, A.R.: Effect of finite rotations on gyroscopic sensing devices. J. Appl. Mech. 210–213 (June 1958)

    Google Scholar 

  5. Gusinsky, V.Z., Lesyuchevsky, V.M., Litmanovich, Yu.A.: New procedure for deriving optimized strapdown attitude algorithms. AIAA J. Guid. Control Dyn. 20, 673–680 (1997)

    Article  MATH  Google Scholar 

  6. Hestenes, D.: New Foundations for Classical Mechanics. Fundamental Theories of Physics. Springer, Berlin (1999)

    MATH  Google Scholar 

  7. Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Reidel, Dordrecht (1984)

    MATH  Google Scholar 

  8. Ignani, M.B.: On the orientation vector differential equation in strapdown inertial navigation systems. IEEE Trans. Aerosp. Electron. Syst. 30, 1076–1081 (1994)

    Article  Google Scholar 

  9. Ignani, M.B.: New procedure for deriving optimized strapdown attitude algorithms. AIAA J. Guid. Control Dyn. 20, 673–680 (1997)

    Article  Google Scholar 

  10. Jiang, Y.F., Lin, Y.P.: On the rotation vector differential equation. IEEE Trans. Aerosp. Electron. Syst. 27, 181–183 (1991)

    Article  Google Scholar 

  11. Jiang, Y.F., Lin, Y.P.: Improved strapdown coning algorithms. IEEE Trans. Aerosp. Electron. Syst. 28, 448–490 (1992)

    Article  Google Scholar 

  12. Miller, R.: A new strapdown attitude algorithm. AIAA J. Guid. Control Dyn. 6, 287–291 (1983)

    Article  Google Scholar 

  13. Nazaroff, G.: The orientation vector differential equation. AIAA J. Guid. Control Dyn. 2, 351–352 (1979)

    Article  Google Scholar 

  14. Onunka, C., Bright, G.: A study on Direction Cosine Matrix (DCM) for autonomous navigation. In: CAD/CAM Robotics and Factories of the Future (2010)

    Google Scholar 

  15. Phillips, W.F., Haily, C.E.: Review of attitude representations used for aircraft kinematics. J. Aircr. 38, 718–737 (2001)

    Article  Google Scholar 

  16. Savage, P.: Strapdown inertial navigation integration algorithm design, part 1: Attitude algorithms. AIAA J. Guid. Control Dyn. 21, 19–28 (1998)

    Article  MATH  Google Scholar 

  17. Titterton, D., Weston, J.: Strapdown Inertial Navigation Technology. 2nd revised edn. IEEE Press, New York (2005)

    Google Scholar 

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Acknowledgements

We would like to thank the Council for Scientific and Industrial Research in South Africa for sponsoring this research.

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Authors and Affiliations

  1. The Council for Scientific and Industrial Research (CSIR), Meiring Naude Rd, Pretoria, South Africa

    Liam Candy

  2. Engineering Department, Cambridge University, Trumpington Street, Cambridge, UK

    Joan Lasenby

Authors
  1. Liam Candy
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  2. Joan Lasenby
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Correspondence to Liam Candy .

Editor information

Editors and Affiliations

  1. Informatics Institute, University of Amsterdam, Science Park 107, Amsterdam, 1098 XG, Netherlands

    Leo Dorst

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom

    Joan Lasenby

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© 2011 Springer-Verlag London Limited

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Candy, L., Lasenby, J. (2011). Attitude and Position Tracking. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_6

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  • DOI: https://doi.org/10.1007/978-0-85729-811-9_6

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-810-2

  • Online ISBN: 978-0-85729-811-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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