Relative Semimajor Axis Uncertainty in High Earth Orbits

Abstract

S EVERAL proposed missions have identified satellite formation flying in high Earth orbit (HEO) as an enabling technology for increasing the science return while decreasing total mission risk [1,2]. Precision formationflying inHEO requires accurate estimation of the semimajor axis difference between the vehicles for state prediction, formation control, and formation maneuver planning [3– 5]. This Note derives the complete linearized mapping relating relative position and velocity error to semimajor axis uncertainty for HEO formations. The majority of relative navigation filters estimate the state of the secondary vehicle with respect to the primary in Cartesian position and velocity coordinates [6–13]. The uncertainty in the semimajor axis is found by rotating the estimated position and velocity covariance into orbital element space. Carpenter and Schiesser [3] presented the complete linearized relationship between absolute position and velocity uncertainty and semimajor axis error; however, the form of their result is not particularly intuitive. Carpenter and Alfriend [4] used a similar method to develop a simplified expression for the semimajor axis uncertainty that assumes the radial position and velocity and intrack velocity are the primary contributions. However, this assumption is not justified for eccentric orbits. The generalized relationship between relative position and velocity uncertainty and semimajor axis uncertainty is derived here explicitly. The resulting equation shows that the semimajor axis error is dependent on all of the coplanar position and velocity uncertainties in highly eccentric orbits. The coordinate frames that form the foundation of this research are described in Sec. II. The generalized relationship between position and velocity error and semimajor axis uncertainty is derived in Sec. III, and the conclusions are summarized in Sec. IV.