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Model the Relative Motion between Satellites in Orbit

The Clohessy–Wiltshire equations model the relative motion between two satellites orbiting a central body. This is a linear model for the Cartesian position of one satellite relative to another. Two simulations are done here with different initial conditions for each simulation.

    
Wolfram Language code: sat = StateSpaceModel[{(| | | | | | | | ----- | ---- | - | --- | ---- | - | | 0 | 1 | 0 | 0 | 0 | 0 | | 3 n^2 | 0 | 0 | 2 n | 0 | 0 | | 0 | 0 | 0 | 1 | 0 | 0 | | 0 | -2 n | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 1 | | 0 | 0 | 0 | 0 | -n^2 | 0 |), (| | | | | - | - | - | | 0 | 0 | 0 | | 1 | 0 | 0 | | 0 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 0 | | 0 | 0 | 1 |), (| | | | | | | | - | - | - | - | - | - | | 1 | 0 | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 0 | 0 |)}];
Use StateResponse to obtain the closed relative orbits from a particular set of launch conditions:
Wolfram Language code: n = 0.001;Subscript[x, 0] = 1;Subscript[xd, 0] = 0.0005; x = StateResponse[{sat, {Subscript[x, 0], Subscript[xd, 0], (2Subscript[xd, 0]/n), -2n Subscript[x, 0], Sequence@@##}}, {0, 0, 0}, {t, 0, 5 (2Pi/n)}]& /@ {{-2 Subscript[x, 0], -2 Subscript[xd, 0]}, {2 Subscript[x, 0], 2 Subscript[xd, 0]}};
Wolfram Language code: ParametricPlot3D[{x[[1, 1 ;; 5 ;; 2]], x[[2, 1 ;; 5 ;; 2]]}, {t, 0, 5 (2Pi/n)}, BoxRatios -> {1, 1, 1}, AxesLabel -> {"X", "Y", "Z"}]