# Artificial Neural Networks for Navigation Systems: A Review of Recent Research
## 3 Navigation Integration Using the State Estimators  
Navigation integration is typically carried out through the KF to estimate the system state variables and suppress the measurement noise. Different Bayesian filtering techniques, including the KF, EKF, UKF [34; 35; 36], CKF [37], and particle filter (PF) [36; 38] techniques, are typically used in the data fusion algorithms of the INS/GPS-integrated navigation system to calibrate the navigation estimation error. In linear systems with white Gaussian noise, it is generally accepted that the KF makes use of the minimum mean square error (MMSE) as the best criterion. On the other hand, if the system's dynamic and measurement models are accurately described, the KF leverages measurement updates to correct system state mistakes and restrict navigation solution errors. The noise statistics of the process provide the best estimates of a system's states. Without measurement updates, the KF's forecast diverges, and when GPS signals are not available, a combined GPS/INS navigation system's performance may suffer quickly. For nonlinear models, EKF employs an approximation technique to linearize the nonlinear system model. The state and observation equations are linearized using EKF using first-order Taylor reduction to enhance the conventional KF [38]. By constructing the linear differential equation, the nonlinear system is roughly approximated. It is straightforward to calculate the assessment in the Bayesian recursive formula since KF and EKF both turn the problem into a linear Gaussian model. The UKF is based on a method for mean and  
Figure 7: Architecture of the ARMA neural network using the BPNN. covariance propagation through a nonlinear mapping built using the unscented transform (UT) approach. The state vector is discretized as a minimum collection of sample points (sigma points), which approximates the second-order precision posterior mean and covariance of the random variable with a Gaussian distribution. To approximate the probability density distribution of a nonlinear function, UKF uses a set of predetermined samples. The EKF's linearization method can only produce first-order accuracy. When the computing complexity is the same, the UKF performs better than the EKF. The aforementioned filtering algorithms for the high-dimensional process model are susceptible to a non-positive definite covariance matrix. Non-Gaussian noises, on the other hand, are commonly present in many real-world contexts and seriously degrade their functionality. The KF, EKF, and UKF techniques have many significant flaws that prevent their application. They are unable to deal with the following two issues in particular: modeling non-Gaussian processes and observation noises and well as unidentified noise covariance matrices are the first two sources of noise. When the methods are coupled with colored noise, as is the case in several scenarios for practical implementation, the EKF cannot perform satisfactorily since the estimation outcomes are vulnerable to strong outliers and non-Gaussian noise. The CKF approximates the state mean and covariance of a nonlinear system with extra Gaussian noise using a collection of volume points based on the third-order spherical radial volume criteria. Theoretically, CKF is the approximation technique that comes closest to Bayesian filtering, and it is an effective tool for dealing with nonlinear system state estimates. For each integral point, CKF has the same weight, which is positive. Compared to UKF, CKF's numerical stability is developing more rapidly. The resilience is decreased, and even the filter fails despite being impacted by the process model's instability and the measured noise statics' variability. In recent years, numerous academics have suggested enhanced CKF algorithms to address the aforementioned issues. The PF is computationally expensive for an online implementation even if it can handle the non-Gaussian, nonlinear system effectively.