# A Comprehensive Review of GNSS/INS Integration Techniques for Land and Air Vehicle Applications
## 3 GNSS Solution. Strapdown INS Approach. Strapdown INS Mechanization. Inertial Error Propagation
In the most basic case, the measurement error vector \(\delta\mathbf{x}\) usually includes the error in determining the position of the object \(\delta\mathbf{p}=[\delta n,\delta e,\delta d]\) in the local coordinate system (NED/ENU), the error in determining the velocity of the object \(\delta\mathbf{v}=[\delta v_{n},\delta v_{e},\delta v_{d}]\) in the local coordinate system (NED/ENU) and the error in determining the attitude of the object \(\delta\mathbf{e}=[\delta\mathbf{\phi},\delta\theta,\delta\psi]\). Each SINS developer can formulate distinct model for SINS error. The attitude error model can be obtained based on the fact that there is a relationship between the attitude of the object calculated on the basis of measurements in the INS coordinate system \(\widetilde{\mathbf{C}}_{b}^{n}\) and the true attitude of the object \(\mathbf{C}_{b}^{n}\):  
\[\widetilde{\mathbf{C}}_{b}^{n}=\mathbf{B}\mathbf{C}_{b}^{n} \tag{12}\]where \(\mathbf{B}=(\mathbf{I}-\delta\mathbf{C})\) is the error matrix for determining the attitude, \(\mathbf{I}\) is the identity matrix, the matrix \(\delta\mathbf{C}\) is determined as follows:  
\[\delta\mathbf{C}= \begin{pmatrix}0&-\delta\psi&\delta\theta\\ \delta\psi&0&-\delta\phi\\ -\delta\theta&\delta\phi&0\end{pmatrix}  \tag{13}\]  
Differentiating (12) with respect to time and expressing \(\frac{d\delta\mathbf{C}}{dt}\) from there, we obtain an INS attitude error model. The velocity error model can be obtained on the basis of the following statements. Let the model for predicting the true velocity of an object be determined by one of the Equations (9)-(11) depending on the coordinate system of INS. The model for predicting the velocity of the object using INS is also based on one of the Equations (9-11), where all components are estimates of the object motion parameters and the matrix \(\widetilde{\mathbf{C}}_{b}^{n}\) is determined based on Expression (12). The difference between the obtained velocity prediction equations can be used as the basis for determining the velocity error model. A more detailed derivation of the equations for the INS measurement error models can be found in [60]. As a result of all operations, we obtain the equation of the INS measurement error model in the state space:  
\[\delta\dot{\mathbf{x}}=\mathbf{F}\,\delta\mathbf{x}+\mathbf{G}\mathbf{u} \tag{14}\]  
where \(\delta\mathbf{x}=[\delta\phi,\delta\theta,\delta\psi,\delta v_{n},\delta v_{e}, \delta v_{d},\delta n,\delta e,\delta d]\), \(\mathbf{u}=\big{[}\delta\omega_{x},\delta\omega_{y},\delta v_{z},\delta f_{x}, \delta f_{y},\delta f_{z}\big{]}\), \(\delta\omega_{x},\delta\omega_{y}\), \(\delta\omega_{z}\) are the errors of determining the angular velocity by the gyro sensor, \(\delta f_{x},\delta f_{y},\delta f_{z}\) are the errors of determining the specific force by the accelerometer. The inclusion of INS errors in the state vector of the Kalman filter leads to the need of development a model for predicting the errors of the gyro sensor and the accelerometer. All types of accelerometers and gyro sensors are subject to bias, crosslinks errors and random noise [70]. Bias is a constant error exhibited by all accelerometers and gyro sensors. It does not depend on the specific force and angular velocity. Since this is a constant value, its prediction model for gyro sensors and accelerometers has the form:  
\[\dot{\mathbf{b}}_{a}=0,\;\dot{\mathbf{b}}_{g}=0. \tag{15}\]  
where \(\mathbf{b}_{a}\) is the constant bias of the accelerometer, \(\mathbf{b}_{g}\) is the constant bias of the gyro sensor. Cross-coupling errors occur due to misalignment of the sensitive axes of the sensors and the axes of the BCS due to manufacturing defects. However, these errors are minor. Random noise or dynamic errors can come from various sources.