# 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 particular, if the navigation system is implemented relative to the inertial coordinate system (ECI) associated with the Earth, then the mechanization equations for determining the velocity of an object relative to the Earth have the form [60]:  
\[\frac{dv_{e}}{dt}=\mathbf{C}_{b}^{i}\mathbf{f}^{b}-\mathbf{\omega}_{ie}\times\mathbf{v}_{e}-\mathbf{ \omega}_{ie}\times[\mathbf{\omega}_{ie}\times\mathbf{r}]+\mathbf{g} \tag{9}\]  
where \(\mathbf{v}_{e}\) is the velocity of the object in the inertial coordinate system, \(\mathbf{C}_{b}^{i}\) is the matrix of the affine transformation between the ECI and the body coordinate system (BCS), \(\mathbf{f}^{b}\) is the specific force measured in the BCS, \(\mathbf{\omega}_{ie}\) is the rotation velocity of the Earth in the ECI, \(\mathbf{\omega}_{ie}\times\mathbf{v}_{e}\) is the Coriolis acceleration due to the Earth's rotation and \(\mathbf{\omega}_{ie}\times[\mathbf{\omega}_{ie}\times\mathbf{r}]\) is the centripetal acceleration. If the navigation system is implemented relative to the fixed coordinate system associated with the Earth (ECEF), then the velocity is determined from the following mechanization equations [60]:  
\[\frac{dv_{e}}{dt}=\mathbf{C}_{b}^{i}\mathbf{f}^{b}-2\mathbf{\omega}_{ie}\times\mathbf{v}_{e}- \mathbf{\omega}_{ie}\times[\mathbf{\omega}_{ie}\times\mathbf{r}]+\mathbf{g} \tag{10}\]  
where \(\mathbf{v}_{e}\) is the velocity of the object in ECEF, \(\mathbf{C}_{b}^{i}\) is the affine transformation matrix between BCS and ECEF, \(\mathbf{\omega}_{ie}\) is the velocity of the Earth's rotation in ECEF. In the case of using the local geographic coordinate system NED, the velocity is determined from the following equations [8,60]:  
\[\frac{dv_{e}}{dt}=\mathbf{C}_{b}^{n}\mathbf{f}^{b}-[2\mathbf{\omega}_{ie}+\mathbf{\omega}_{en} ]\times\mathbf{v}_{e}-\mathbf{\omega}_{ie}\times[\mathbf{\omega}_{ie}\times\mathbf{r}]+\mathbf{g} \tag{11}\]  
where \(\mathbf{v}_{e}\) is the velocity of the object in ECI, \(\mathbf{\omega}_{en}\) is the angular velocity of rotation of local NED coordinate system relative ECI, \(\mathbf{\omega}_{ie}\) is the rotational velocity of the Earth in NED and \(\mathbf{C}_{b}^{n}\) is the matrix of the affine transformation between NED and BCS. The SINS computing device operates in the following sequence: at the first stage, the attitude of the object is determined based on the measurements of the angular velocity of the object with gyro sensors and the angular velocity of the INS coordinate system by applying the appropriate mechanization equations. Based on the obtained angular position, a transformation matrix is formed for the specific force measured by the accelerometers and used in the SINS equations provided above: (9), (10) or (11). At the same time it is necessary to consider the choice of the coordinate system for the INS and the parameters for attitude representation: Euler angles, direction cosine matrix or quaternion. The local geographic coordinate system and quaternions are used to determine the attitude of an object in most of the works in this review. The main algorithms of INS operation for determining the position and velocity of an object are detailed in [60]. As noted above, in order to form the Kalman filter algorithm that evaluates the INS error in the integrated INS/GNSS systems, an INS error prediction model considering all possible sources of errors or only a part of them depending on the complexity of the algorithm is obviously needed. Alignment errors, biases and computation errors are usually the sources of INS error [60].