Approximate Solutions for Tethered Satellite Motion

Abstract

The formulation of approximate solutions to equations that embody the dominant characteristics of the orbital motion of a two-satellite tethered system are studied. The orbital motion of the system is viewed as perturbed two-body motion, and a restricted tether problem is obtained by neglecting librational motion. An exact analytical solution to this restricted problem in terms of elliptic functions is presented. An approximate solution to the restricted tether problem obtained by applying the method of averaging is also provided. An approximation for small-amplitude librational motion is formulated, whose solution is based on methods for solving equations with variable coef cients. The analytical solutions are good approximations to the orbital motion of the tetherperturbed satellite and the librational motion of the system when the libration is small. The restricted tether motion approximation is then utilized to solve the identi cation problem of a tethered satellite system. Nomenclature a ¤ = apparent semimajor axis e ¤ = apparent eccentricity F = incomplete elliptic integral of the rst kind h = nondimensionalangular momentum K = complete elliptic integral of the rst kind K 0 = associate complete elliptic integral of the rst kind k = modulus of Jacobian elliptic functions and integral k 0 = complementary modulus of Jacobian elliptic functions and integral m; m p = masses of satellites q = Jacobi's nome r = nondimensionalradial distance, radial distance/r E r E = radius of the Earth, 6,378,000 m sn = Jacobian elliptic function t = time t ¤ = nondimensionaltime, t p .¹=r 3 E / D t £ 0:0012394 u = 1=r X = state vector ®;¯;°; » = variable coef cients "= tether parameter, m p ½=.m C m p /r E µ = orbital angle (true anomaly) µ 2 = out-of-plane libration angle µ 3 = in-plane libration angle ¹ = gravitational constant of the Earth ½ = tether length '; ' ¤ = amplitudes ! = orbital frequency

A symplectic Runge-Kutta method for the analysis of the tethered satellite system

“…Kim and Hall (2004) found that by controlling the length of the tether, the required control efforts for a TSS decrease greatly. Cho et al (2001) studied the dynamic behavior of a two-satellite tethered system and obtained the approximate analytical solutions for the planar motion equations of the TSS. Gao et al (2012) investigated the dynamic response of a TSS flying around collinear vibration points, and a two-step numerical method was used to generate the periodical orbits.…”

Coupled Librational and Orbital Motions of a Large-Scale Spacecraft

“…In such situations, the effects of coupling between librational and orbital motions are not negligible. In previous studies including coupling, orbital motions are mainly focused, [1][2][3][4] and the orbit controls through the attitude motions of spacecraft are also investigated. [5][6][7][8] In these studies, however, the stability of librational motion is not considered.…”