# A Comprehensive Review of GNSS/INS Integration Techniques for Land and Air Vehicle Applications
## 2 GNSS/INS Integration Techniques  
In most cases, INS/SNS measurements are integrated using the filtering algorithm in loosely coupled, tightly coupled and deeply coupled integration schemes. In loosely coupled and tightly coupled integration schemes, the difference between INS and SNS measurements is used to estimate the INS error, and then the INS navigation solution is corrected with resulting INS error estimate [7; 8] (Figures 1 and 2). In the case of a loosely coupled scheme, the difference between the position and velocity measurements obtained by GNSS and INS is used. In the case of a tightly coupled scheme, the difference between the pseudorange, carrier phase or Doppler shift measurements obtained by GNSS and INS is used. Figure 1: INS/GNSS loosely coupled integration scheme. In the deep integration scheme, GNSS receiver and INS are not independent devices. GNSS measurements are used to estimate INS errors and INS measurements are used to aid GNSS receiver tracking loops (Figure 3). In [19], the deep integration approach is considered that starts fusion of GPS and inertial data at the earliest processing stage possible by combining radiofrequency GNSS samples with sampled INS measurements. Obtained integrated signals are applied to estimate GNSS solution, including code phase, carrier Doppler frequency shift and carrier phase. After that, these GNSS estimates are used in Kalman filter to update INS error state. As can be noted, deep integration is performed at the level that requires the access to the GNSS receiver source code, which is usually available only for receiver manufacturers. For this reason, we have not considered this type of integration in this manuscript and have studied in detail the loosely coupled and tightly coupled integration schemes that are more available to implementation by most GNSS users. From the review, the Kalman filter is the most used integration tool. The Kalman filter is implemented according to the standard scheme. At the first stage, the estimate of the INS error vector and the covariance matrix for the INS error are predicted based on the available measurements of the INS and SNS. At the next stage, after receiving the measurements, the predicted INS measurement error vector and the covariance matrix are corrected. Since the INS and SNS operate at different frequencies, the integration in the Kalman filter is performed at the moments \(t_{k}=t_{GNSS}\) when the GNSS measurements are obtained. The Kalman filter algorithm is as follows:  
(1) Predicting the estimate of the INS measurement error vector. Since, after receiving each new SNS measurement, the INS measurements are corrected, the best current estimate of the INS error is zero \(\delta\hat{\pi}_{k/k}=0\). Then, the predicted score at \(t_{k+1}\) is:  
\[\delta\hat{\pi}_{k+1/k}=0 \tag{1}\]  
Figure 3: INS/GNSS deeply coupled integration scheme. Figure 2: INS/GNSS tightly coupled integration scheme. The predicted value of the covariance matrix at \(t_{k+1}\) is determined based on the expression:  
\[\mathbf{P}_{k+1/k}=\mathbf{\Phi}_{k}\mathbf{P}_{k/k}\mathbf{\Phi}_{k}+\mathbf{Q}_{k} \tag{2}\]  
where \(\mathbf{P}_{k/k}\) is the covariance matrix at the moment of time \(t_{k}\), \(\mathbf{\Phi}_{k}=\mathbf{I}+F\Delta t\), \(\mathbf{Q}_{k}=\mathbf{G}\mathbf{Q}\mathbf{G}^{T}\Delta t\), \(\Delta t=t_{k+1}-t_{k}\), matrices \(\mathbf{F}\) and \(\mathbf{Q}\), respectively, depend on the INS measurement error model and the type of the INS measurement error vector. (2) Correction of the predicted INS measurement error vector. To perform this step, it is necessary to update the measurements of the INS and SNS and calculate their difference \(\delta\mathbf{z}_{k+1}\).