Divergence in the Kalman filter.

Under certain conditions, the orbit estimated by a Kalman filter has errors that are much greater than predicted by theory. This phenomenon is called divergence, and renders the operation of the Kalman filter unsatisfactory. This paper investigates the control of divergence in a Kalman filter used for autonomous navigation in a low earth orbit. The system studied utilizes stellar-referenced angle sightings to a sequence of known terrestrial landmarks. A Kalman filter is used to compute differential corrections to spacecraft position, spacecraft velocity, and landmark location. A variety of filter modifications for the control of divergence was investigated. These included the Schmidt-Pines analytical modification and an empirical modification based upon Pines' machine noise treatment. Several simplified approximations to the theoretically optimum analytical modifications were also investigated. The principal numerical results are presented in graphs of the magnitude of the error in estimated position and velocity vs time for sixteen orbits. These graphs compare actual position and velocity errors with the theoretical estimates furnished by the trace of the position and velocity covariance matrices. Numerical results indicate that a properly modified filter achieves a steady-state operating level.