An Efficient GPS Position Determination Algorithm

Abstract

INTRODUCTIONGPS is a space-based satellite radio navigation system that provides three-dimensional (3-D) user positioning by solving a set of nonlinear trilateration equations using pseudorange measurements. The current method of solving the nonlinear equations is to linearize the pseudorange equations and calculate the user position iteratively, starting with a user-provided initial position estimate [1]. For near-earth navigation, the center of the earth is a good initial estimate, and the currently used iterative least-squares (ILS) algorithm converges to the GPS solution. An area of potential improvement that has been investigated in recent years is the use of noniterative closed-form solutions to the nonlinear pseudorange GPS equations. Closed-form solutions have been developed in [2 -9]. This paper improves on [8] and [9] by employing a more rigorous mathematical formulation. In this paper and in our previous work [8,9], an overdetermined system is treated, making use of all-in-view (n Ͼ 5) satellites as opposed to using just 4 satellites. Moreover, this work departs from a deterministic formulation of the problem [2,4,6,7] and specifically addresses the development of a reliable closed-form solution that works in the presence of measurement noise. Previous work, with the exception of [10], treated the pseudorange equations as a deterministic set of equations. In [11], the deterministic solution of [2] is adapted to account for measurement noise, and this approach is further developed in [12].In this paper, it is recognized up front that pseudorange measurements are noise corrupted. Hence, the stochastic nature of the measurements is reflected in the GPS pseudorange equations from the outset to develop a probabilistically sound GPS solution. By stochastically modeling the measurement situation at hand, solving for position becomes a stochastic estimation problem. The use of correct stochastic modeling and estimation yields a GPS solution that, in addition to the position estimate, provides an estimate of the measurement noise intensity, provided there are more than 5 satellites in view.Thus, the estimation algorithm developed here provides a data-driven position (and user clock bias) estimation error covariance prediction. This prediction introduces a new confidence factor into GPS positioning that is critical for the downstream integration of GPS and inertial navigation systems (INS) or synthetic aperture radar (SAR) sensors. Moreover, an attractive feature of our solution is its good estimation performance, achieved even under high geometric dilution of precision (GDOP) conditions and in urban environments where the number of visible satellites is reduced to 4.Moreover, a direct, or autonomous, solution that does not require an initial position estimate is attractive for space navigation and for unusual planar array configurations using pseudolites, where the iterative process is sensitive to the initial position estimate (e.g., the application discussed in [13] and [14]

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