Series expansion-based state transition matrix for relative motion on eccentric orbits

Abstract

This paper considers the formation dynamics with the target satellite in an elliptic orbit. A new state transition matrix for the linearized equations of relative motion is presented by series expansion and some mathematical transformations. The state transition matrix is applicable to any eccentricity elliptic reference orbit. Besides, the state transition matrix is just related to the trigonometric function of the true anomaly of the target satellite, so it is easy to calculate. With the state transition matrix, the contribution of three-order nonlinearity in the differential gravitational acceleration on the relative motion is estimated by a perturbation approach. Numerical simulations are included to evaluate the proposed methods.

Keywords

Series expansion state transition matrix perturbation approach relative orbit estimation

Get full access to this article

View all access options for this article.

References

Melton

. Time-explicit representation of relative motion between elliptical orbits. J Guid Control Dyn 2000; 23(4): 604–610.

Yamanaka

Ankersen

. New transition matrix for relative motion on an arbitrary elliptical orbit. J Guid Control Dyn 2002; 25(1): 60–66.

Gim

Alfriend

. State transition matrix of relative motion for the perturbed noncircular reference orbit. J Guid Control Dyn 2003; 26(6): 956–971.

Carter

Humi

. Fuel-optimal rendezvous near a point in general keplerian orbit. J Guid Control Dyn 1987; 10(6): 567–573.

Carter

. New form for the optimal rendezvous equations near a keplerian orbit. J Guid Control Dyn 1990; 13(1): 183–186.

Inalhan

Tillerson

How

. Relative dynamics and control of spacecraft formations in eccentric orbits. J Guid Control Dyn 2002; 25(1): 48–59.

Sengupta

Sharma

Vadali

. Periodic relative motion near a keplerian elliptic orbit with nonlinear differential gravity. J Guid Control Dyn 2006; 29(5): 1110–1121.

Vaddi

Vadali

Alfriend

. Formation flying: accomodating nonlinearity and eccentricity perturbations. J Guid Control Dyn 2003; 26(2): 214–222.

Sengupta

Vadali

. Relative motion and geometry of formations in keplerian elliptic orbits. J Guid Control Dyn 2007; 30(4): 953–964.

10.

Schaub

Alfriend

. J₂-invariant relative orbits for spacecraft formations. Celest Mech Dyn Astron 2001; 79(2): 77–95.

11.

Damaren

. Almost periodic relative orbits under J₂ perturbations. Proc IMechE, Part G: J Aerosp Eng 2007; 221: 767–774.

12.

Richardson

Mitchell

. A third-order analytical solution for the relative motion with a circular reference orbit. J Astronaut Sci 2003; 51(1): 1–12.

13.

Battin

. An introduction to the mathematics and methods of astrodynamics, New York: AIAA, Inc., 1987.

14.

Miller

Michel

. Ordinary differential equations, New York: Academic Press, 1982.

15.

Apostol

. Mathematical analysis, Massachusetts: Addison-Wesley, 1974.