## 2 INS/GNSS Concepts
### Extended Kalman Filter (EKF)
#### 2.4.2 Indirect Mode (Error State Model)  
In the Indirect EKF (IEKF), also known as the error state model, the EKF estimates the errors in the navigation states rather than the states themselves. This approach involves maintaining a reference trajectory and estimating deviations from this trajectory. Therefore, in this work, the navigation error states are represented by the error state vector:  
\[\delta\mathbf{x}_{k}=[\delta\mathbf{p}^{T},\ \delta\mathbf{v}^{T},\ \delta \boldsymbol{e}^{T}]^{T}. \tag{17}\]  
The continuous-time system dynamic model and measurement model of the IEKF are defined as follows:  
\[\delta\dot{\mathbf{x}}(t)=\mathbf{f}(\delta\mathbf{x}(t),t)+\mathbf{G}\mathbf{ w}(t), \tag{18}\]\[\mathbf{z}(t)=\mathbf{h}(\delta\mathbf{x}(t),t)+\mathbf{v}(t), \tag{19}\]  
where \(\mathbf{f}\) and \(\mathbf{h}\) are the nonlinear functions of the error state vector, \(\mathbf{G}\) is the process noise gain matrix, and \(\mathbf{w}\) and \(\mathbf{v}\) represent process and measurement noise, respectively. The discrete state of process and measurement models of the IEKF are defined as follows:  
\[\delta\mathbf{x}_{k}=\mathbf{\Phi}_{k-1}\delta\mathbf{x}_{k-1}+\mathbf{G}_{k- 1}\mathbf{w}_{k-1}, \tag{20}\]  
\[\mathbf{z}_{k}=\mathbf{H}_{k}\delta\mathbf{x}_{k}+\mathbf{v}_{k}, \tag{21}\]  
where \(\delta\mathbf{x}\) represents the error state vector as defined in Vector (17), \(\mathbf{z}\) is the measurement vector, \(\mathbf{\Phi}\) is the transition matrix, \(\mathbf{H}\) is the observation matrix, and and \(k\) denotes the discrete time step. The transition matrix \(\mathbf{\Phi}_{k-1}\) can be found based on the first-order Taylor series approximation [10; 42]:  
\[\mathbf{\Phi}_{k-1}\approx\mathbf{I}_{9}+\Delta t\left[ \begin{array}{cccccc} 0&0&\frac{\rho_{\mathrm{E}}}{R_{e}}&\frac{1}{R_{e}}&0&0&0&0&0\\ \frac{-\rho_{D}}{\cos(\phi)}&0&\frac{-\rho_{N}}{R_{e}\cos(\phi)}&0&\frac{1}{R _{e}\cos(\phi)}&0&0&0\\ 0&0&0&0&-1&0&0\\ F_{41}&0&F_{43}&k_{D}&2\omega_{D}&-\rho_{E}&0&f_{D}&-f_{E}\\ F_{51}&0&F_{53}&F_{54}&F_{55}&F_{56}&-f_{D}&0&f_{N}\\ -2v_{E}\Omega_{D}&0&F_{63}&2\rho_{E}&-2\omega_{N}&0&f_{E}&-f_{N}&0\\ -\Omega_{D}&0&\frac{\rho_{N}}{R_{e}}&0&-\frac{1}{R_{e}}&0&0&\omega_{D}&-\omega _{E}\\ 0&0&\frac{\rho_{\mathrm{E}}}{R_{e}}&\frac{1}{R_{e}}&0&0&-\omega_{D}&0&\omega_{ N}\\ F_{91}&0&\frac{\rho_{D}}{R_{e}}&0&\frac{\tan(\phi)}{R_{e}}&0&\omega_{E}&-\omega_{N}&0 \end{array} \right], \tag{22}\]  
\[\Omega_{N} =\omega_{ie}\cos(\phi) F_{41} =-2\Omega_{N}v_{e}-\frac{\rho_{N}v_{e}}{\cos^{2}(\phi)}\] \[\Omega_{D} =-\omega_{ie}\sin(\phi) F_{43} =\rho_{E}k_{D}-\rho_{N}\rho_{D}\] \[\rho_{N} =\frac{v_{e}}{R_{e}} F_{51} =2(\Omega_{N}v_{n}+\Omega_{D}v_{d})+\frac{\rho_{N}v_{n}}{\cos^{2}(\phi)}\] \[\rho_{E} =-\frac{v_{n}}{R_{e}} F_{53} =-\rho_{E}\rho_{D}-k_{D}\rho_{N}\] \[\rho_{D} =-\frac{v_{e}\tan(\phi)}{R_{e}} F_{54} =-(\omega_{D}+\Omega_{D})\] \[\omega_{N} =\Omega_{N}+\rho_{N} F_{55} =k_{D}-\rho_{E}\tan(\phi)\] \[\omega_{E} =\rho_{E} F_{56} =\omega_{N}+\Omega_{N}\] \[\omega_{D} =\Omega_{D}+\rho_{D} F_{63} =\rho_{N}^{2}+\rho_{E}^{2}-\frac{2g}{R_{e}}\] \[k_{D} =\frac{v_{d}}{R_{e}} F_{91} =\Omega_{N}+\frac{\rho_{N}}{\cos(\phi)^{2}}\]  
where  
* \(R_{e}\) is the Earth's mean radius;
* \(\omega_{ie}\) is the Earth's rotation rate;
* \(\Omega_{N}\) and \(\Omega_{D}\) are components of the angular velocity due to Earth's rotation;
* \(\rho_{N}\), \(\rho_{E}\), and \(\rho_{D}\) are components of the relative velocity with respect to the Earth's surface;
* \(v_{e}\), \(v_{n}\), and \(v_{d}\) are the eastward, northward, and downward components of the velocity, respectively;
* \(\phi\) is the latitude;
* \(\omega_{N}\), \(\omega_{E}\), and \(\omega_{D}\) are the angular velocities incorporating both the Earth's rotation and the relative velocities;
* \(f_{N}\), \(f_{E}\), and \(f_{D}\) are components of specific force;* \(k_{D}\) is a component of the velocity. The matrix \(\mathbf{G}\) projects the process noise \(\mathbf{w}\) into the state space, accounting for uncertainty in the state estimation [10], thus: \[\mathbf{G}=