Abstract
The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abad A., Arribas M., Elipe A. (1989) On the attitude of a spacecraft near a Lagrangian point. Bull. Astron. Inst. Czechosl. 40, 302–307
Burdet C.A. (1968) Regularization of the two-body problem. Z. Angew. Math. Phys. 18, 434–438
Cid R., Sansaturio M.E. (1988) Motion of rigid bodies in a set of redundant variables. Celest. Mech. Dyn. Astron. 42, 263–274
Danby J.M.A. (1992) Fundamentals of Celestial Mechanics. Willmann-Bell Inc., Richmond
Deprit A., Elipe A. (1993) Complete reduction of the Euler-Poinsot problem. J. Astron. Sci. 41, 603–628
Deprit A., Elipe A., Ferrer S. (1994) Linearization: Laplace versus Stiefel. Celest. Mech. Dyn. Astron. 58, 151–201
Ferrándiz J.M. (1988) A general canonical transformation increasing the number of variables with application to the two-body problem. Celest. Mech. 41, 343–357
Hamilton W.R. (1844) On Quaternions; or a new system of imaginaries in algebra. Philos. Mag. 25, 489–495
Henrard J. (2005) The rotation of Europa. Celest. Mech. Dyn. Astron. 91, 131–149
Junkins J.L., Singla P. (2004) How nonlinear is it? A tutorial on nonlinearity of orbit and attitude dynamics. J. Astron. Sci. 52, 7–60
Maciejewski A. (1985) Hamiltonian formalism for Euler parameters. Celest. Mech. Dyn. Astron. 37, 47–57
Morton H.S. (1993) Hamiltonian and Lagrangian formulation of rigid-body rotational motion based on Euler parameters. J. Astron. Sci. 41, 439–517
Scheeres D.J. (2004) Bounds of rotation periods of disrupted binaries in the full 2-body problem. Celest. Mech. Dyn. Astron. 89, 127–140
Shuster M.D. (1993) A survey on attitude representation. J. Astron. Sci. 41, 439–517
Shuster M.D., Natanson G.A. (1993) Quaternion computation from a geometric point of view. J. Astron. Sci. 41, 545–556
Stiefel E., Scheifele G. (1971) Linear and Regular Celestial Mechanics. Springer-Verlag, New York
Waldvogel J. (2006) Quaternions and the perturbed Kepler problem. Celest. Mech. Dyn. Astron. 95, 201–212
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arribas, M., Elipe, A. & Palacios, M. Quaternions and the rotation of a rigid body. Celestial Mech Dyn Astr 96, 239–251 (2006). https://doi.org/10.1007/s10569-006-9037-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-006-9037-6