New Closed-Form, Fast GPS Position Estimation

Jonathan Kipling Knight

Abstract: Presented herein is a new closed form general solution to the classic global navigation satellite system (GNSS) position fix. Previously, this has been considered a non-linear problem requiring expensive iterations, large pseudo-inverted matrices or Kalman filtering. The new formula builds on the ideas in the Generalized Delonge-Kása Estimator for the center of a sphere (Knight & Semwal, 2010) and the Bancroft method (Bancroft, 1985) for over-determined GNSS positions. The end result is a formula for the position that requires no iterations nor guesses and uses all available data. This formula is known to be statistically biased in noisy data but can be compensated by an estimate of the noise. There is a speed increase using this formula over previous solutions since it involves the inverse of a 3x3 Cholesky matrix for no matter how many space vehicles. This paper will derive the new equation and explain the complexity of the operations (27N+56) as well as memory requirements. A statistical analysis of the formula will produce the magnitude of the bias. Real world pseudo-ranges will then be tested for formula accuracy. The formula will also be compared to the more widely used solutions including the Maximum Likelihood Estimator (MLE). This formula provides direct estimation of position by means of the inverse of a 3x3 Cholesky matrix for any number of space vehicles. The Cholesky Inverse is ten times faster than the singular-value decomposition used in the Moore-Penrose pseudo-inverse. Neither guesses nor iterations are needed. Furthermore, the formula has constant memory requirements and linear time requirements on the number of space vehicles in view. Testing will show that the bias produces answers that are not significantly far from the MLE on average. The formula can be implemented easily in hardware.
Published in: Proceedings of the ION 2015 Pacific PNT Meeting
April 20 - 23, 2015
Marriott Waikiki Beach Resort & Spa
Honolulu, Hawaii
Pages: 901 - 904
Cite this article: Knight, Jonathan Kipling, "New Closed-Form, Fast GPS Position Estimation," Proceedings of the ION 2015 Pacific PNT Meeting, Honolulu, Hawaii, April 2015, pp. 901-904.
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