A New Approach for Computing the Probability of Correct Fix and Integrity Risk for JPALS

Sasha Draganov, Donna Smith, Greg Johnson, and Mark Cooper

Abstract:	This paper suggests a new method for computing the Probability of Correct Fix (PCF) and the associated integrity risk for carrier-phase aircraft navigation applications. Two algorithms are presented. The first algorithm is used to compute PCF; the second one provides an example for estimating the VPL integrity risk for the null hypothesis. (Computations for other components of the integrity risk can be done similarly, but are not explicitly presented in the paper.) The PCF computation should be preceded by a standard processing of measurements and by estimation of integer ambiguities. A set of available measurements may include double-differenced code, carrier, smoothed code, etc.; a set of unknowns is assumed to include the baseline vector and integer ambiguities. (Generalization to include other parameters in the vector of unknowns is straightforward.) Measurements are processed by conventional methods, such as a Kalman filter and the LAMBDA method to estimate integer ambiguities. This procedure yields two or more candidate integer ambiguity sets. PCF is conditioned on the available measurement data and computation assumes Gaussian measurement errors. This approach yields a closed-form, exact solution for PCF and a step-by-step algorithm for estimating it. For each candidate set, the PCF algorithm computes the probability that this set is the true one, using measurements as conditions. To decouple the PCF computation from the probability density of having a particular baseline vector, we integrate over all possible baselines. The resulting equations show some superficial similarity to those for the ratio test, but do provide a quantitative estimate for a posteriori PCF in a closed form. The PCF algorithm was tested using the JPALS Performance Model developed by ARINC Engineering Services, LLC. GPS measurements were generated by using the ARINC simulation capability. The simulation assumed static antenna positions with nominal GPS error models. Results show that the PCF algorithm performed as expected as was able to perform accurately according to the error state of the measurements utilized. Finally, the paper presents a new algorithm for computing the integrity risk, which is associated with measurement errors. This algorithm is also based on a posteriori probability definition, and requires integration of a probability density function over the area, where the position error exceeds the alert limit. This paper presents an example of such a computation for VPL integrity risk, assuming the null hypothesis. In this case, the integration must be performed over two half-spaces, where VAL is exceeded. The paper presents a (somewhat algebraically cumbersome) closed-form solution. This algorithm may serve as a part of overall estimation of system integrity/availability.
Published in:	Proceedings of the 2007 National Technical Meeting of The Institute of Navigation January 22 - 24, 2007 The Catamaran Resort Hotel San Diego, CA
Pages:	1255 - 1263
Cite this article:	Draganov, Sasha, Smith, Donna, Johnson, Greg, Cooper, Mark, "A New Approach for Computing the Probability of Correct Fix and Integrity Risk for JPALS," Proceedings of the 2007 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2007, pp. 1255-1263.
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