Abstract-The well-known extended Kalman filter (EKF) has been widely applied to the Global Positioning System (GPS) navigation processing. The adaptive algorithm has been one of the approaches to prevent the divergence problem of the EKF when precise knowledge on the system models are not available. One of the adaptive methods is called the strong tracking Kalman filter (STKF), which is essentially a nonlinear smoother algorithm that employs suboptimal multiple fading factors, in which the softening factors are involved. Traditional approach for selecting the softening factors heavily relies on personal experience or computer simulation. In order to resolve this shortcoming, a novel scheme called the adaptive fuzzy strong tracking Kalman filter (AFSTKF) is carried out. In the AFSTKF, the fuzzy logic reasoning system based on the Takagi-Sugeno (T-S) model is incorporated into the STKF. By monitoring the degree of divergence (DOD) parameters based on the innovation information, the fuzzy logic adaptive system (FLAS) is designed for dynamically adjusting the softening factor according to the change in vehicle dynamics. GPS navigation processing using the AFSTKF will be simulated to validate the effectiveness of the proposed strategy. The performance of the proposed scheme will be assessed and compared with those of conventional EKF and STKF.Index Terms-Adaptive extended Kalman filtering, fuzzy logic adaptive system (FLAS), global positioning system (GPS), strong tracking Kalman filter (STKF).

IntroductionThe geometric dilution of precision (GDOP) is a geometrically determined factor that describes the effect of geometry on the relationship between measurement error and position error. It is used to provide an indication of the quality of the solution. Some of the GPS receivers may not be able to process all visible satellites due to limited number of channels. Consequently, it is sometimes necessary to select the satellite subset that offers the optimal or acceptable solutions. The optimal satellite subset is sometimes obtained by minimizing the GDOP factor.The most straightforward approach for obtaining GDOP is to use matrix inversion to all combinations and select the minimum one. However, the matrix inversion by computer presents a computational burden to the navigation computer. For the case of processing four satellite signals, it has been shown that GDOP is approximately inversely proportional to the volume of the tetrahedron formed by four satellites (Kihara and Okada 1984;Stein 1985). Therefore, it is optimum to select satellite such that the volume is as large as possible, which is sometimes called the maximum volume method. However, it is not universal acceptable since it does not guarantee optimum selection of satellites.The neural network (NN) approach provides a promising and very realistic computational alternative. The application of NN approach for navigation solution processing has not been widely explored yet in the GPS community. Simon and El-Sherief (1995a) initially proposed the NN approach to approximate and classify the GDOP factors for the benefit of computational efficiency, where it could be seen that a total of 160 floating Abstract In this paper, the neural network (NN)-based navigation satellite subset selection is presented. The approach is based on approximation or classification of the satellite geometry dilution of precision (GDOP) factors utilizing the NN approach. Without matrix inversion required, the NN-based approach is capable of evaluating all subsets of satellites and hence reduces the computational burden. This would enable the use of a high-integrity navigation solution without the delay required for many matrix inversions. For overcoming the problem of slow learning in the BPNN, three other NNs that feature very fast learning speed, including the optimal interpolative (OI) Net, probabilistic neural network (PNN) and general regression neural network (GRNN), are employed. The network performance and computational expense on NN-based GDOP approximation and classification are explored. All the networks are able to provide sufficiently good accuracy, given enough time (for BPNN) or enough training data (for the other three networks).

The Kalman filter (KF) is a form of optimal estimator characterized by recursive evaluation, which has been widely applied to the navigation sensor fusion. Utilizing the KF requires that all the plant dynamics and noise processes are completely known, and the noise process is zero mean white noise. If the theoretical behaviour of the filter and its actual behaviour do not agree, divergence problems tend to occur. The adaptive algorithm has been one of the approaches to prevent divergence problems in the Kalman filter when precise knowledge on the system models is not available. Two popular types of adaptive Kalman filter are the innovation-based adaptive estimation (IAE) approach and the adaptive fading Kalman filter (AFKF) approach. In this paper, an approach involving the concept of the two methods is presented. The proposed method is a synergy of the IAE and AFKF approaches. The ratio of the actual innovation covariance based on the sampled sequence to the theoretical innovation covariance will be employed for dynamically tuning two filter parameters -fading factors and measurement noise scaling factors. The method has the merits of good computational efficiency and numerical stability. The matrices in the KF loop are able to remain positive definitive. Navigation sensor fusion using the proposed scheme will be demonstrated. Performance of the proposed scheme on the loosely coupled GPS/INS navigation applications will be discussed. K E Y

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For a GPS receiver, decreasing the receiver tracking loop bandwidth reduces the probability of loss of lock if there are no vehicle dynamics. However, reduced bandwidth increases tracking errors due to dynamics. Beyond a certain limit it causes a serious degradation in the dynamic tracking performance. Therefore, there is involvement of a tradeoff between two opposing considerations: narrow tracking loop bandwidths are desired for filtering noise due to thermal effects, but wide tracking loop bandwidths are desired to permit tracking of vehicle dynamics. Optimal tracking loop bandwidths, which yield the minimum errors in a certain dynamics environment, are first investigated. The linear Kalman filter is employed as the optimal estimator. The covariance for the arbitrary gain model is solved and applied to the sensitivity analysis for investigating error growth due to incorrect noise level estimate. Theoretical results are verified by numerical simulation, and results from both approaches are in very good agreement.