International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395 -0056
Volume: 04 Issue: 01 | Jan -2017
p-ISSN: 2395-0072
www.irjet.net
Analytical solution of the relative orbital motion in unperturbed elliptic orbits using Laplace transformation S. M. El-Shaboury1, M. K. Ammar2, W. M. Yousef 3 1Mathematics
Dept. Faculty of Science, Ain Shams University, Cairo, Egypt Dept. Faculty of Science, Helwan University, Cairo, Egypt 3Basic Science Dept. Canadian International College for Engineering, Cairo, Egypt 2Mathematics
---------------------------------------------------------------------***--------------------------------------------------------------------present a time-explicit solution for relative motion for Abstract - This paper introduces a different approach to elliptic orbits. But, Gim and Alfriend [5] and Garrison et obtain the exact solution of the relative equations of motion of a deputy (follower) object with respect to a chief (leader) al. [6] represent a geometric method for deriving the object that both rotate about central body in elliptic orbits by state transition matrix, utilizing small differences in using Laplace transformations. We will use Kepler orbital elements between two satellites. Also Srinivas assumptions considering the unperturbed case to get our R. Vadali [7] uses the geometric method but under the equations of motion which in turn subjected to linearization influence of -perturbation. In the present work, we process. These type of equations known as Tschauner – will consider the unperturbed case, and we will use the Hempel equations or elliptic Hill – Clohessy – Wiltshire (HCW) true anomaly to be the independent variable that the equations. The solution of such equations in this work is represented in terms of the eccentricity of the chief orbit and solution will be represented, and we will apply our its true anomaly as the independent variable. After getting our solution to solve a numerical example. solution, we will apply it on numerical example to compare the results obtained by this new approach with previous results.
2. Equations of motion
Key Words: Relative motion of two satellites, Formation flying, Tschauner – Hempel equations, Elliptic Hill – Clohessy –Wiltshire equations, Laplace transformation.
Consider the chief (C) and deputy (D) space crafts that orbiting the same point mass central body. To set up the equations of motion of (D) relative to (C), we define two frames of references. The first is inertial and centred at the central body and the second is rotating chief centred (Hill’s non-inertial frame of reference) [8]. As shown in figure 1,
1. INTRODUCTION Solving and modelling the relative motion problem between satellites or space crafts is of great importance in the field of formation flying, rendezvous and disturbed satellite systems which in turn play a significant rule in space missions. Since 1960s, many researchers have contributed in this regard, but their contributions are varied from several aspects. For example, according to the independent variable, some of the researchers use the time and the others use the true or eccentric anomaly. Also according to the linearity of the obtained equations of motion, some of them make linearization and the other make higher order expansion. Also from point of view of perturbation consideration, some of them put it in their calculations and the other don’t. But most of the results depend of the same start point which is linearized gravitational acceleration represented by ClohessyWiltshire equations using circular reference orbits [1] and the Tschauner-Hempel equations using elliptic reference orbits [2]. Both Melton [3] and Vaddi et al. [4] © 2017, IRJET
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Fig. -1: Chief and deputy position vectors w.r.t the central body, and the position vector of (D) relative to (C), with the
basis of the chief centered frame
rC and rD are the position vectors of the chief and deputy with respect to the central body respectively. And the position vector of (D) relative to (C) is represented by . Also we define eˆr the unit vecior in the direction of rC , eˆh perpenduclar to the chief’s |
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