State transition matrix of relative motion for the perturbed noncircular reference orbit
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A precise analytic solution that includes the effects of the reference orbit eccentricity and differential perturbations is needed for the relative motion of formation-flying satellites. As a result of the spherical Earth and circular reference orbit assumptions, Hill's equations, which have often been used for describing relative motion, are insufficient for the long-term prediction of the relative motion. A new approach, called the geometric method, is developed to obtain the state transition matrix for the relative motion that includes the effects caused by the reference orbit eccentricity and the differential gravitational perturbations. The geometric method uses the relationship between the relative states and differential orbital elements to obtain the state transition matrix instead of directly solving the complex relative motion differential equations. The state transition matrices are derived for both mean and osculating elements with the primary gravitational perturbation that results from the equatorial bulge term J2. Although the results are based on the J2 effects, the approach can be extended easily to include other perturbing forces.
JOURNAL OF GUIDANCE CONTROL AND DYNAMICS
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Gim, D. W., & Alfriend, K. T.
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