Rigid body trajectories in different 6D spaces. (English) Zbl 1360.70004
Summary: The objective of this paper is to show that the group \(\mathrm{SE}(3)\) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.
MSC:
| 70B10 | Kinematics of a rigid body |
References:
| [1] | D. D. Holm, Geometric Mechanics. Part II: Rotating, Translating and Rolling, Imperial College Press, London, UK, 2008. · Zbl 1149.70001 |
| [2] | J. E. Marsden and T. S. Ratiu, Mechanics and Symmetry: Reduction Theory, 1998. |
| [3] | K. R. Etzel and J. M. McCarthy, “A metric for spatial displacements using biquaternions on SO(4),” in Proceedings of the IEEE International Conference on Robotics and Automation, pp. 3185-3190, 1996. |
| [4] | P. M. Larochelle, A. P. Murray, and J. Angeles, “SVD and PD based projection metrics on SE(n),” in On Advances in Robot Kinematics, pp. 13-22, Kluwer Academic Publsihers, 2004. |
| [5] | M. Craioveanu, C. Pop, A. Aron, and C. Petri\csor, “An optimal control problem on the special Euclidean group SE (3,R),” in Proceedings of the International Conference of Differential Geometry and Dynamical Systems (DGDS ’09), vol. 17, pp. 68-78, 2010. · Zbl 1227.53088 |
| [6] | B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, vol. 222 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2003. · Zbl 1026.22001 |
| [7] | V. Jurdjevic, Geometric Control Theory, vol. 52 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1997. · Zbl 0940.93005 |
| [8] | D. D. Holm, Geometric Mechanics. Part I: Dynamics and Symmetry, Imperial College Press, London, 2nd edition, 2011. · Zbl 1227.70001 |
| [9] | A. M. Bloch, Nonholonomic Mechanics and Control: With the Collaboration of J. Baillieul, P. Crouch, and J. Marsden, vol. 24 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 2003. · Zbl 1182.68329 · doi:10.1007/b94107 |
| [10] | J. M. Selig, Geometric Fundamentals of Robotics, Monographs in Computer Science, Springer, New York, NY, USA, 2nd edition, 2005. · Zbl 1062.93002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
