Lie group variational integrators for the full body problem in orbital mechanics
Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.
KeywordsSymplectic integrator Variational integrator Lie group method Full rigid body problem
Unable to display preview. Download preview PDF.
- 3.Cendra, H., Marsden, J.E., Ratiu, T.S.: Lagrangian reduction by stages. Mem. Am. Math. Soc. 152 (2001)Google Scholar
- 4.Fahnestock, E., Lee, T., Leok, M., McClamroch, N.H., Scheeres, D.J.: Polyhedral potential and variational integrator computation of the full two body problem. In: AIAA/AAS Astrodynamics Specialist Meeting. AIAA-2006-6289 (2006)Google Scholar
- 7.Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer (2006)Google Scholar
- 12.Lee, T., Leok, M., McClamroch, N.H.: A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications, pp. 962–967 (2005)Google Scholar
- 13.Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods App. Mech. Eng. in press (2007)Google Scholar
- 14.Leok, M.: Foundations of Computational Geometric Mechanics. Ph.D. thesis, California Instittute of Technology (2004)Google Scholar