Inverse Transformation of Elliptical Relative State Transition Matrix

Affiliation(s)

Institute of Spacecraft System Engineering, CAST, Beijing, China.

School of Astronautics, Beihang University, Beijing, China.

Institute of Spacecraft System Engineering, CAST, Beijing, China.

School of Astronautics, Beihang University, Beijing, China.

ABSTRACT

A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.

KEYWORDS

Relative Orbit Elements, Elliptical Formation Flying, Relative State Transition Matrix, Inverse Transformation, Poisson Bracket

Relative Orbit Elements, Elliptical Formation Flying, Relative State Transition Matrix, Inverse Transformation, Poisson Bracket

Cite this paper

Yin, J. , Rao, Y. and Han, C. (2014) Inverse Transformation of Elliptical Relative State Transition Matrix.*International Journal of Astronomy and Astrophysics*, **4**, 419-428. doi: 10.4236/ijaa.2014.43037.

Yin, J. , Rao, Y. and Han, C. (2014) Inverse Transformation of Elliptical Relative State Transition Matrix.

References

[1] Casotto, S. (2010) Position and Velocity Perturbations in the Orbital Frame in Terms of Classical Element Perturbations. Celestial Mechanics and Dynamical Astronomy, 55, 209-221.

http://dx.doi.org/10.1007/BF00692510

[2] Garrison, J.L., Gardner, J.L. and Axelrad, P. (1995) Relative Motion in Highly Elliptical Orbits. Paper No. AAS 95-194, Space Flight Mechanics Conference, Alburquerque.

[3] Alfriend, K.T., Schaub, H. and Gim, D.-W. (2000) Gravitational Perturbations, Nonlinearity and Circular Orbit Assumption Effects on Formation Flying Control Strategies. AAS Guidance and Control Conference, Breckenridge, CO, AAS 00-012, 139-158.

[4] Gim, D.-W. and Alfriend, K.T. (2005) Satellite Relative Motion Using Differential Equinoctial Elements. Celestial Mechanics and Dynamical Astronomy, 92, 295-336. http://dx.doi.org/10.1007/s10569-004-1799-0

[5] Sengupta, P., Vadali, S.R. and Alfriend, K.T. (2006) Second-Order State Transition for Relativemotion near Perturbed, Elliptic Orbits. Celestial Mechanics and Dynamical Astronomy, 97, 101-129.

http://dx.doi.org/10.1007/s10569-006-9054-5

[6] Fontdecaba, J. (2008) Dynamics of Formation Flying Applications to Earth and Universe Observation. Observatoire de la Cote d’Azur, Grasse.

[7] Chobotov, V. (2002) Orbital Mechanics. American Institute of Aeronautics and Astronautics, Reston.

[8] Han, C. and Yin, J. (2011) Satellite Relative Motion in Elliptical Orbit Using Relative Orbit Elements. American Astronautical Society, Paper AAS 11-497.

[9] Han, C. and Yin, J. (2012) Formation Design in Elliptical Orbit Using Relative Orbit Elements. Acta Astronautica, 77, 34-47. http://dx.doi.org/10.1016/j.actaastro.2012.02.026

[10] Yin, J. and Han, C. (2012) Perturbation Effects on Elliptical Relative Motion Based on Relative Orbit Elements. American Astronautical Society, Paper AAS 12-148.

[11] Lane, C. and Axelrad, P. (2006) Formation Design in Eccentric Orbits Using Linearized Equations of Relative Motion. Journal of Guidance, Control, and Dynamics, 29, 146-160.

http://dx.doi.org/10.2514/1.13173

[1] Casotto, S. (2010) Position and Velocity Perturbations in the Orbital Frame in Terms of Classical Element Perturbations. Celestial Mechanics and Dynamical Astronomy, 55, 209-221.

http://dx.doi.org/10.1007/BF00692510

[2] Garrison, J.L., Gardner, J.L. and Axelrad, P. (1995) Relative Motion in Highly Elliptical Orbits. Paper No. AAS 95-194, Space Flight Mechanics Conference, Alburquerque.

[3] Alfriend, K.T., Schaub, H. and Gim, D.-W. (2000) Gravitational Perturbations, Nonlinearity and Circular Orbit Assumption Effects on Formation Flying Control Strategies. AAS Guidance and Control Conference, Breckenridge, CO, AAS 00-012, 139-158.

[4] Gim, D.-W. and Alfriend, K.T. (2005) Satellite Relative Motion Using Differential Equinoctial Elements. Celestial Mechanics and Dynamical Astronomy, 92, 295-336. http://dx.doi.org/10.1007/s10569-004-1799-0

[5] Sengupta, P., Vadali, S.R. and Alfriend, K.T. (2006) Second-Order State Transition for Relativemotion near Perturbed, Elliptic Orbits. Celestial Mechanics and Dynamical Astronomy, 97, 101-129.

http://dx.doi.org/10.1007/s10569-006-9054-5

[6] Fontdecaba, J. (2008) Dynamics of Formation Flying Applications to Earth and Universe Observation. Observatoire de la Cote d’Azur, Grasse.

[7] Chobotov, V. (2002) Orbital Mechanics. American Institute of Aeronautics and Astronautics, Reston.

[8] Han, C. and Yin, J. (2011) Satellite Relative Motion in Elliptical Orbit Using Relative Orbit Elements. American Astronautical Society, Paper AAS 11-497.

[9] Han, C. and Yin, J. (2012) Formation Design in Elliptical Orbit Using Relative Orbit Elements. Acta Astronautica, 77, 34-47. http://dx.doi.org/10.1016/j.actaastro.2012.02.026

[10] Yin, J. and Han, C. (2012) Perturbation Effects on Elliptical Relative Motion Based on Relative Orbit Elements. American Astronautical Society, Paper AAS 12-148.

[11] Lane, C. and Axelrad, P. (2006) Formation Design in Eccentric Orbits Using Linearized Equations of Relative Motion. Journal of Guidance, Control, and Dynamics, 29, 146-160.

http://dx.doi.org/10.2514/1.13173