Generalized Kalman filtering applied to real-time high-precision GNSS

Real-time estimation of user positioning parameters forms the backbone of high-precision GNSS navigation. The recursive Kalman filter is the most widely adopted estimation method for this task, providing optimal estimates in the minimum variance sense under the assumption that the underlying models are correctly specified. In many practical GNSS applications, however, this assumption may be violated, as measurement models commonly involve state-vector elements that are not naturally linked in time. Imposing incorrect dynamic modeling on such parameters may lead to sub-optimal solutions.

In this contribution, we examine the mechanics of the generalized Kalman filter (Teunissen et al., 2021), which offers a statistically rigorous alternative to the standard Kalman-filter practice of inflating the process-noise variances or assigning arbitrary initial values to states of newly-tracked satellites. Rather than enforcing all state-vector elements to vary in time, the generalized formulation permits only some functions of the state-vector to be linked in time, while others remain unlinked in time. This relaxed dynamic model offers a flexible framework for recursive parameter estimation when limited or insufficient knowledge is available on the temporal behaviour of the involved parameters. Typical applications include purely kinematic precise positioning, network-based satellite clock estimation, kinematic precise orbit determination in low Earth orbit, and GNSS precise positioning during periods with enhanced ionospheric activity.

As any real-time estimation process inevitably requires validation of the underlying models, recursive quality control of the measurement model needs to be executed in parallel with the filter. A direct consequence of the generalized filter is that the classical predicted residuals used in quality control procedures are no longer applicable. It is shown here how these residuals are generalized to predictable functions of the measurements, while practical methods are demonstrated for constructing them in real time for different choices of unlinked-in-time states.

Supported by real-world multi-GNSS simulated kinematic and vehicle-borne datasets, the performance of the generalized Kalman filter using the carrier-phase ambiguity resolution-enabled precise point positioning (PPP-RTK) concept is presented. Next to the positioning performances, the required adaptations to the recursive data quality control procedure, involving both the detection and the identification of mismodeling biases, are illustrated.

 

Teunissen, P.J.G., Khodabandeh, A. & Psychas, D. A generalized Kalman filter with its precision in recursive form when the stochastic model is misspecified. J Geod 95, 108 (2021). https://doi.org/10.1007/s00190-021-01562-0