# High-Order Analytical Solution of Relative Motion Equation for Satellite Formation Flying in Elliptical Orbit

### Ting Wang, Hanlun Lei, Bo Xu, Jun Guo

#### Abstract

The paper studied relative motion equation for satellite formation flying with large separations. The configuration is traditionally designed by the periodic solutions of the C-W equation in circle reference orbit or Lawden equation in elliptic reference orbit. Hence, the linear solutions are more suitable for the configuration with small scale formation than large scale formation. However, in some specific situations, it is necessary to use satellites with large separations. Then the paper studied relative motion based on the nonlinear equations in an elliptic reference orbit. The solution is expanded as series form with respect to the eccentricity of the reference orbit, in-plane amplitude and out-of-plane amplitude. Taking the Lawden periodic solution as starting point, the high-order analytical solution is constructed by Lindstedt-Poincare method. Particularly, as the eccentricity is zero, the analytical solution degenerated to express the relative motion in circle reference orbit. Finally, the practical convergence of the analytical solution is discussed in order to examine its validity and applicability.

#### References

- Richardson D L. Analytic construction of periodic orbits about the collinear points. Celest Mech, 22: 241C253, 1980.
- Jorba A' , Masdemont J. Dynamics in the center manifold of the collinear points of the restricted three body problem. Physica D, 132:189C213, 1999.
- Masdemont J J. High-order expansions of invariant manifolds of libration point orbits with application to mission design. Dyn Syst, 20:59C113, 2005.
- Lei H L, Xu B. High-order analytical solutions around triangular libration points in circular restricted three-body problem. Mon Not R Astron Soc,434: 1376C1386, 2013.
- Richardson D L, Mitchell J W. A third-Order Analytical Solution for Relative Motion with a Circular Reference Orbit. J Astron Sci, 51:1C12, 2003.
- Gomez G, Marcote M. High-order analytical solutions of Hills equations. Celest Mech Dyn Astron, 94: 197C211, 2006.
- Ren Y, Masdemont J J, MarcoteMet al. Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit. Acta Astronaut, 81: 186C199, 2012.
- Szebehely V. Theory of orbits. New York: Academic Press, 1967.

#### Paper Citation

#### in Harvard Style

Wang T., Lei H., Xu B. and Guo J. (2014). **High-Order Analytical Solution of Relative Motion Equation for Satellite Formation Flying in Elliptical Orbit** . In *Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,* ISBN 978-989-758-040-6, pages 256-263. DOI: 10.5220/0004996902560263

#### in Bibtex Style

@conference{icinco14,

author={Ting Wang and Hanlun Lei and Bo Xu and Jun Guo},

title={High-Order Analytical Solution of Relative Motion Equation for Satellite Formation Flying in Elliptical Orbit},

booktitle={Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,},

year={2014},

pages={256-263},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004996902560263},

isbn={978-989-758-040-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,

TI - High-Order Analytical Solution of Relative Motion Equation for Satellite Formation Flying in Elliptical Orbit

SN - 978-989-758-040-6

AU - Wang T.

AU - Lei H.

AU - Xu B.

AU - Guo J.

PY - 2014

SP - 256

EP - 263

DO - 10.5220/0004996902560263