Nonlinear Fixed-Point Smoothers and their Application for GPS Orbit Determination

Chun Yang, Thao Nguyen, Mikel Miller

Abstract:	For well-behaved yet highly nonlinear dynamic systems, their state estimation typically resorts to fixed-point smoothing (followed by prediction) vs. filtering. One example is GPS orbit determination wherein ranging errors to satellites over an observation interval are used to estimate the initial orbit errors (at a reference epoch). Either a batch or recursive least square (LS) method is used to obtain the solution, which requires linearization of the nonlinear state and measurement equations along the nominal trajectory. In this paper, we formulate satellite orbit determination as a fixed-point smoothing problem and investigate various nonlinear fixed-point smoothers. This includes classical batch LS method, the recursive LS method (i.e., the extended Kalman filter) and emerging unscented Kalman filter (UKF) and particle filter (PF) and their variants. The LS methods make minimal assumptions about the state and measurement error distributions. The EKF propagates and updates the distributions in terms of their mean vectors and covariance matrices under linear (or linearized) Gaussian assumptions. In contrast, the fixedpoint smoothers based on the UKF and PF propagate and update deterministic and random samples of the distributions using the exact nonlinearities, respectively. To facilitate GPS orbit determination and other nonlinear estimation applications, a nonlinear estimation software toolkit is developed in C# with Microsoft Visual Studio 2005. It contains a graphic user interface (GUI) allowing a user to specify simulation scenarios, select force models, design estimators, and visualize the results for comparative analysis of performance and computation loading. With this tool, the various nonlinear fixed-point smoothers presented in this paper are illustrated via simulation examples for their operation and performance in GPS orbit determination
Published in:	Proceedings of the 21st International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2008) September 16 - 19, 2008 Savannah International Convention Center Savannah, GA
Pages:	775 - 785
Cite this article:	Yang, Chun, Nguyen, Thao, Miller, Mikel, "Nonlinear Fixed-Point Smoothers and their Application for GPS Orbit Determination," Proceedings of the 21st International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2008), Savannah, GA, September 2008, pp. 775-785.
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