Semimajor Axis Estimation Strategies

This paper extends previous analysis on the impact of sensing noise for the navigation and control aspects of formation flying spacecraft. We analyze the use of Carrier-phase Differential GPS (CDGPS) in relative navigation filters, with a particular focus on the filter correlation coefficient. This work was motivated by previous publications which suggested that a "good" navigation filter would have a strong correlation (i.e., coefficient near -1) to reduce the semimajor axis (SMA) error, and therefore, the overall fuel use. However, practical experience with CDGPS-based filters has shown this strong correlation seldom occurs (typical correlations approx. -0.1), even when the estimation accuracies are very good. We derive an analytic estimate of the filter correlation coefficient and demonstrate that, for the process and sensor noises levels expected with CDGPS, the expected value will be very low. It is also demonstrated that this correlation can be improved by increasing the time step of the discrete Kalman filter, but since the balance condition is not satisfied, the SMA error also increases. These observations are verified with several linear simulations. The combination of these simulations and analysis provide new insights on the crucial role of the process noise in determining the semimajor axis knowledge.