Advertisement

Rotations, Transformations, Left Quaternions, Right Quaternions?

  • Renato ZanettiEmail author
Article
  • 43 Downloads

Abstract

This paper surveys the two fundamental possible choices in representing the attitude of an aerospace vehicle: active and passive rotations. The consequences of the choice between the two are detailed for the two most common attitude parameterizations, a three-by-three orthogonal matrix and the quaternion. Successive rotations are also reviewed in this context as well as the attitude kinematic equations.

Keywords

Rotations Spacecraft attitude Quaternion 

Notes

References

  1. 1.
    Shuster, M.D.: A survey of attitude representations. J. Astronaut. Sci. 41(4), 439–517 (1993)MathSciNetGoogle Scholar
  2. 2.
    Markley, F.L., Crassidis, J.L.: Fundamentals of Spacecraft Attitude Determination and Control. Space Technology Library. Springer (2014)Google Scholar
  3. 3.
    Yazell, D.J.: Origins of the unusual space shuttle quaternion definition. In: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, No. AIAA 2009–43. Orlando, FL. AIAA, pp. 5–8 (2009)Google Scholar
  4. 4.
    Euler, L.: Formulae generales pro translatione quacunque corporum rigidorum. Novi Commentarii academiae scientiarum Petropolitanae 20, 189–207 (1776). E478Google Scholar
  5. 5.
    Euler, L.: Nova methodus motum corporum rigidorum degerminandi. Novi Commentarii academiae scientiarum Petropolitanae 20, 208–238 (1776). E479Google Scholar
  6. 6.
    Grubin, C.: Derivation of the quaternion scheme via euler axis and angle. J. Spacecrafts Rockets 7, 1261–1263 (1970)CrossRefGoogle Scholar
  7. 7.
    Hurtado, J.E.: Cayley family of attitude coordinates. J. Guid. Control Dyn. 33, 246–249 (2009)CrossRefGoogle Scholar
  8. 8.
    Paul, B.: On the composition of finite rotations. Am. Math. Mon. 70, 859–862 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rodrigues, O.: Des lois géométriques que régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire. J. de Mathématiques Pures et Appliquées 5, 380–440 (1840)Google Scholar
  10. 10.
    Euler, L.: Problema algebraicum ob affectiones prorsus singulares memorabile. Novi Commentarii academiae scientiarum Petropolitanae 15, 75–106 (1771). E407Google Scholar
  11. 11.
    Hamilton, W.R.: On a new species of imaginary quantities connected with a theory of quaternions. Proc. R. Ir. Acad. 2, 424–434 (1844)Google Scholar
  12. 12.
    Altmann, S.: Hamilton, Rodrigues, and the quaternion scandal. Math. Mag. 62, 291–308 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cayley, A.: On certain results relating to quaternions. Philos. Mag. 26, 141–145 (1845)Google Scholar
  14. 14.
    Pittelkau, M.E.: Kalman filtering for spacecraft system alignment calibration. J. Guid. Control Dyn. 24(6), 1187–1195 (2001)CrossRefGoogle Scholar
  15. 15.
    Sommer, H., Gilitschenski, I., Bloesch, M., Weiss, S., Siegwart, R., Nieto, J.: Why and how to avoid the flipped quaternion multiplication. Aerospace 5(3), 72 (2018).  https://doi.org/10.3390/aerospace5030072 CrossRefGoogle Scholar
  16. 16.
    Caparrini, S.: The discovery of the vector representation of moments and angular velocity. Arch. History Exact Sci. 56, 151–181 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn. Dover, New York (1944)zbMATHGoogle Scholar

Copyright information

© American Astronautical Society 2019

Authors and Affiliations

  1. 1.Department of Aerospace Engineering and Engineering MechanicsThe University of Texas at AustinAustinUSA

Personalised recommendations