Dynamic Models of Satellite Relative Motion Around an Oblate Earth

Accurate dynamic model of relative motion is basic and critical to the study of satellite formation flying. Hence, accurate nonlinear and linear dynamic models of satellite relative motion considering J(2) perturbation are derived in this chapter. Firstly, an exact J(2) nonlinear model of satellite relative motion is developed based on the Lagrangian mechanics. Subsequently, with the aid of Gegenbauer polynomials, the nonlinear model is linearized to a complete J(2) linear relative model. Finally, by means of eliminating the second-order J(2) effect, the linear model is further approximated to a first-order J(2) linear relative model. Simulation results show that the exact J(2) nonlinear model produces exact results, and the first-order J(2) linear model also performs well under conditions that the inter-satellite distance is small and time duration is short. Many relative dynamic models have been derived in the literature under different assumptions and using different methodologies. A comparison study is necessary to select an appropriate model for a specific mission and determine what kind of perturbation should be considered for specific applications. Thus, in this chapter, a simulation method with a modeling error index is also introduced for comparing and evaluating various existing models for relative motion of satellites flying in formation. The comparison results show that, when the Earth aspherical gravity and the air drag are present, the accuracy of some models is affected adversely by eccentricity, semimajor axis, inclination, and formation size. The numerical results provide valuable information for formation design.