# Artificial Neural Networks for Navigation Systems: A Review of Recent Research
## 2 The Use of Neural Networks in the Global Positioning System
### Processing for Navigation States for GPS Receivers  
Blind signal separation, picture registration, and blind deconvolution are just a few examples of situations for which neural networks have been proposed as nonlinear filters. The receiver's position and clock bias solutions are obtained from four or more GPS pseudorange measurements. One of the widely used methods aims at linearizing the equations and solving them with the least squares (LS) method based on an iteration technique. However, a digital computer frequently needs to adhere to the necessary computation time for real-time applications, where the solution can be acquired within a hundred nanoseconds. The continuous inversion of the matrix, which is typically necessary for finding LS solutions and GDOP computations in classic GPS receivers, is mostly carried out by the circuits of basic analog processors resembling neurons. Finally, the suggested system's features and performance will be evaluated and contrasted with those offered by the traditional approach, which involves a digital computer in the matrix inversion process. Nonlinear quadratic equations are used to solve for the location and bias of the receiver's clock using multiple GPS pseudorange readings. Three-layer neural networks were used by [PERSON] [23] to offer a novel method for resolving the GPS pseudorange equations. In contrast to the linear least squares or EKF techniques used in conventional GPS receivers, the three-layer radial basis function (RBF) neural network in Figure 6 was created to solve the nonlinear GPS pseudorange equations directly. [PERSON]'s simulations exhibit consistent behavior even in poor geometry situations, in contrast to the conventional recursive least squares and EKF techniques, which are very sensitive to measurement mistakes. Input, hidden, and output layers make up the three-layer structure of an RBFNN. In addition, the neural network solution exhibits somewhat better noise performance under favorable geometry conditions than the anticipated iteration of the conventional least-squares solution. To assess the effectiveness of the trained neural network, computations with interrelated noisy models and additive white Gaussian noise have been analyzed. Figure 5: General GRNN architecture. One common method aims to linearize the variables and use a recursive gradient-based LS technique to solve those. A digital computer frequently falls short of the required computation time for real-world implications--finding the answer is required in less than 100 nanoseconds, or its utilization is quite more costly. The conceptualization of two standard differential equation approaches and related circuits of analog processors that resemble neurons was used in [31; 32] for the processing of GPS navigation. The study covers various ordinary differential equation formulation approaches and accompanying circuits of neuron-like analog processors. On the basis of the mean square error minimization criterion, a structure with a linear system of equations is solved and is frequently employed to determine positioning solutions. In GPS receivers, the circuits of basic neuron-like analog processors are primarily used. Second, computer simulation studies on single epoch and dynamic positioning were performed to confirm the effectiveness of the suggested strategy. Because it reduces the mean squared error of the observation, the LS estimation scheme may not be the ideal strategy when outliers are a concern. The L\({}_{1}\) and L\({}_{\infty}\) criteria's effectiveness in terms of outlier tolerance and GPS location are discussed in this research. For the purpose of GPS navigation processing, three ordinary differential equation derivation approaches and accompanying circuits of neuron-like analog L\({}_{1}\) (least-absolute), L\({}_{\infty}\) (minimax), and L\({}_{2}\) (least-squares) architectures will be used.