Orbit Determination with Improved Covariance Fidelity, Including Sensor Measurement Biases

O RBIT determination algorithms based on the minimum variance (or Kalman) filter provide unbiased position and velocity estimates when sensor measurements contain only random errors. TheKalman filter covariancematrix is usually consistent with the bias-free estimation error statistics, indicating good covariance fidelity, when random error statistics are accurately characterized. However, sensor measurements often contain unmodeled static and dynamic measurement biases, or low-frequency systematic errors, that cause biases in the position and velocity estimates. In this situation, the Kalman filter covariance usually underestimates the error statistics, indicating poor covariance fidelity, because the measurement bias statistics are not characterized during the filtering process. Covariancefidelity is important for correlation and fusion of tracks provided by multiple sensors [1,2]. Optimistic covariances often cause failures in track-to-track association because covariance uncertainty regions do not overlap. Covariance fidelity may be improved by direct estimation (and removal) of bias parameters in the system dynamics, in the measurements, or in both [3–8]. Depending on the number of bias parameters, algorithms that jointly estimate bias parameters with position and velocity states are computationally intensive, and the system covariance matrix can become numerically ill conditioned, especially when the bias states are poorly observable. Covariance inflation is a simpler alternative to joint bias estimation. As its description suggests, the uncertainty region is increased (inflated) using mathematical models that approximate the measurement bias effect on position and velocity estimates, thereby improving chances for successful track-to-track association. However, a posteriori covariance inflation does not characterize state-bias correlations that occur during the filtering process. A bias characterization filter, which is an application of the Schmidt–Kalman filter [9–11], improves covariance fidelity in the presence of sensor measurement biases. As it explicitly models biases in the estimation process, its covariance matrix is more consistent with the statistics of the biased errors in the estimates comparedwith an extendedKalman filter covariance. It is shown that the bias characterization filter provides the most consistent filter covariance, or best covariance fidelity, compared with several excursions in the Kalman filter, including elevated measurement noise, velocity process noise, and a posteriori covariance inflation. Although similar to considering filtering techniques [12,13], bias characterization is optimal, whereas the former is not. This Note is organized as follows. A nonlinear bias characterization filter with measurement biases is formulated (Sec. II). Radar measurement and bias models are provided for orbit determination (Sec. III). Nonlinear prediction models for the state, bias, and covariances are formulated from the radar models (Sec. IV). The recursive filter is initialized with an iterated batch filter version of the same algorithm (Sec. V). Satellite orbit determination accuracy and covariance fidelity are demonstrated using Monte Carlo simulations (Sec.VI). Conclusions regarding estimation accuracy and covariance fidelity are provided (Sec. VII).