When two spacecraft have a close formation flying or reach the terminal phase of rendezvous maneuvers, a reference frame centered on the primary body of the system (Sun or Earth) is typically inappropriate to investigate their relative motions. The primary reason for that is the differential acceleration on each spacecraft is small and fine relative motion may be masked by the orbital motion. In this case, it is standard to describe the relative motion of one spacecraft with respect to a non - inertial reference frame centered on the other spacecraft. See the diagram below.

\fOnce you fix the primary or target spacecraft [the selection of primary spacecraft is only a matter of context}, the relative motion of the secondary spacecraft is governed by the following initial value problem: 9:" 2103f 31022: = f3; 33(0):.110, 29(0) = 9:3 9\" + 2wm' = fyi 9(0) = yo, M0] = 91] 2'\" +10% = fz;z(0) = zmz'UJ) = 23 Where It: stands for the mean angular velocity of the primary spacecraft within its orbit plane. Is Also. fy represents the differential acceleration on the secondary spacecraft. f2 Assuming there is no external force acting on the secondary spacecraft. use Laplace Transform to determine its in\" (t) position y (t) in the space. Your answer will be in terms of w and the set of intial conditions. 2 (t) Hint: The .3 part of the given IVP is a second order IVP by itself. Therefore it can be solved for 2: using Laplace Transform. For each of the first two differential equations (from the top). begin by taking Laplace Transform on both sides. This would eventually result in an algebraic system of equations in X (s) and Y{s)