Analysis of an Extended Kalman Filter Based Orbit Determination System
Quang Lam, Daniel Junker, David J. Anhalt, David Vallado · 2010
This paper revisits the underlying mathematics behind Differential Correction, also called Orbit Determination (OD) and the derivation of the Extended Kalman Filter (EKF). Specifically, we explore the formulation in the context of satellite OD using either ground based or space based sensor measurements to update the satellite state vector. We use a slightly different approach for the partial derivative measurement matrix H. It is observed that for a six state, non-maneuvering satellite, the state transition matrix F of the EKF can be implemented as a truly linear model (derived from pure kinematics) or a linearized model whose elements are the partial derivatives of the nonlinear equation of motion with respect to the EKF state. However, for the measurement matrix H, the numerical accuracy of its elements (as partial derivatives of measurements with respect to the EKF state vector) plays a critical role in the overall accuracy. Two primary factors affecting the “quality” of the H matrix condition are the information dimension (i.e., 2 or 3 rows depending on the observations) and relative dynamics (i.e., from the sensor platform to the satellite platform) observability information captured by the sensor. The theoretical observability matrix required for the ODS filter to converge is evaluated and illustrated from simulated measurements. A combination of Matlab based simulation for the EKF implementation and AGI’s Orbit Determination Tool Kit (ODTK) are used to investigate the observability issue and evaluate the EKF based OD performance. For the ODTK environment, actual ground based measurements data were employed to reconstruct the orbit of a commercial GEO satellite. The ODS solution accuracies are observed to be acceptable under these selected testing conditions/scenarios.