ION GNSS 2010
Session A2: Marine Navigation

Title: Unscented Kalman Filter Robustness Assessment for Orbit Determination Using GPS Signals
Author(s): P.C.P.M. Pardal, H.K. Kuga, INPE, Brazil; R. Vilhena de Morales, Unifesp, Brazil

The purpose of this work is to evaluate the nonlinear unscented Kalman filter (UKF) for the satellite orbit determination problem, using GPS measurements. The assessment is based on the robustness of the filter. The main subjects for the evaluation are convergence speed and computational implementation complexity, which is based on comparing the UKF results against the extended Kalman filter (EKF) results for the solution of the same problem. Based on the analysis of such criteria, the advantages and drawbacks of the implementations are presented. In this work, the orbit of an artificial satellite is determined using real data from the Global Positioning System (GPS) receivers. This is a nonlinear problem, with respect to the dynamics and measurements equations, in which the disturbing forces are not easily modeled. The problem of orbit determination consists essentially of estimating values that completely specify the body trajectory in the space, processing a set of measurements related to this body. Such observations can be collected through a tracking network grounded on Earth or through sensors, like space GPS receivers onboard the satellite. The GPS is a wide spread system that allows computation of orbits for artificial Earth satellites by providing many redundant measurements. Throughout an onboard GPS receiver it is possible to obtain nonlinear measurements (pseudoranges) that can be processed to estimate the orbital state. As it is known, the EKF is probably the most widely used real time estimation algorithm for nonlinear systems. Nevertheless, many difficulties arise due to the linearizations needed by the EKF method. Specifically for the orbit estimation problem, under inaccurate initial conditions and scattered measurements, the EKF implementation can lead to unstable or diverging solutions. To overcome this limitation, the unscented transformation was developed as a technique to more accurately propagate mean and covariance information through nonlinear transformations. The UKF is an estimator of the nonlinear sigma point Kalman filter family that claims to yield equivalent or better performance than the EKF and elegantly extends to nonlinear systems without the linearization steps. This algorithm is an approach to generalize the Kalman filter for nonlinear process and observation models. In this orbit determination problem the focus is to analyze UKF convergence behavior in two situations. The first one deals with using different sampling rates for the GPS signals, where scattering of measurements will be taken into account. The second one uses inaccurate initial conditions, introducing since small up to larger errors in the initial position and velocity conditions. Another aim is to know how adversely such situations affect the performance of the estimators. Therefore, a performance comparison between EKF (the wide spread used estimation algorithm) and UKF (supposedly more adequate algorithm for non-linear systems) is justified. In this work, the standard differential equations describing the orbital motion and the GPS measurements equations used in the EKF algorithm need to be placed in a suitable form. They are adapted for the unscented transformation application, so that the UKF algorithm is also used for estimating the orbital state. After solving the real time satellite orbit determination problem using actual GPS measurements, through both the EKF and the UKF algorithms, the results obtained are compared in computational terms such as burden (complexity), convergence, accuracy, and relative CPU time.

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