An Analytical Solution for Relative Motion of Satellites

An analytical solution to the relative motion of satellites in eccentric orbits, under the effects of the nonlinearity of the gravitational field and the 2 J -perturbation, is presented. The most notable feature of the solution is the use of the unit sphere for the description of the motion. Solutions to the relative motion coordinates and velocities are obtained by using a geometric approach, without making use of any small-angle approximations. The results obtained from a nonlinear simulation model and the analytical solutions are compared and shown to agree remarkably well with each other. Examples involving eccentric reference orbits and large relative distances are presented. Introduction Relative motion between satellites is a subject of interest to rendezvous and formation flying. There exists a rich body of literature on relative motion between two satellites [1]. The solutions available can be classified along the following lines: 1) The independent variable used: time, true anomaly, or eccentric anomaly. 2) Circular or eccentric orbit of the reference satellite. 3) Linearization or higher order expansion of the gravitational acceleration. 4) 2 J -perturbation accounted for or not. The starting point for most of the results using linearized gravitational acceleration models are the Clohessy-Wiltshire equations for circular reference orbits [2] and the Tschauner-Hempel equations for elliptic reference orbits [3]. Melton [4] presents a time-explicit solution for relative motion for elliptic orbits using a perturbation method. Vaddi et al. [5] also present a time-explicit perturbation solution to the relative motion in elliptic orbits by retaining quadratic nonlinearities of the gravitational field. Alternatively, Gim and Alfriend [6] and Garrison et al. [7] consider a geometric method for deriving the state transition matrix, utilizing small differences in orbital elements between two satellites. The results presented in [6] account for the 2 J -perturbation. Time-explicit solutions have the advantage that Kepler’s equation need not be solved. However, geometric methods utilizing orbital elements are more accurate and a series solution for Kepler’s equation can be utilized for many practical applications. This paper also uses the geometric method to obtain relative motion solutions for eccentric orbits under the influence of the 2 J -perturbation. The solution, presented in terms of differences in the orbital elements, is exact and valid for large relative distances. However, eccentricity expansions and mean orbital elements are resorted to, in order to express the solution in a time-explicit manner. The enabling feature of the elegant solution is the use of the unit sphere to study the relative motion. The actual relative motion is obtained by a scaling transformation. Several examples are presented for eccentric orbits with and without the 2 J -perturbation. Relative Motion on a Unit Sphere A satellite’ s motion can be projected onto a unit sphere by normalizing its Cartesian coordinates by its distance from the Earth. The point obtained thus is called the subsatellite point. The relative motion between two subsatellite points shown in Fig. 1, is the focus of this paper. Figure 1 Projections of Two Satellites on the Unit Sphere Let 0 C and 1 C , indicate the direction cosine matrices of the Local-Vertical-LocalHorizontal (LVLH) frames of the two satellites with respect to the inertial frame. Satellite # 0 will be designated as the reference satellite or Chief and satellite # 1 will be the Deputy. The relative motion can be expressed in the LVLH frame of the Chief, as follows: 0 1 1 [ ] 0 0 T x y C C I z D Î Þ Î Þ Ï ß Ï ß D = Ï ß Ï ß Ï ß Ï ß D Ð à Ð à (1) where , , and x y z D D D , are respectively, the radial, along-track, and cross-track relative positions on the unit sphere and I stands for a 3X3 identity matrix. The direction cosine matrices can be parameterized by a set of orbital elements. The above equation can then be used to solve for the relative displacement variables explicitly as given below: Satellite # 0 Satellite # 1