# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 2 The Navigation Filter
### Dynamic Model  
The filter dynamics can be written as [18]:  
\[\begin{array}{c}\delta\dot{\mathbf{r}}\ =\delta\mathbf{v}\\ \delta\dot{\mathbf{e}}=\mathbf{R}_{b}^{\dagger}\delta\mathbf{b}_{g}+\mathbf{R}_{b}^{l}\mathbf{\eta}_ {g}\\ \delta\dot{\mathbf{v}}=-\left(\mathbf{R}_{b}^{\dagger}\mathbf{\hat{\mathbf{r}}}\times\right) \delta\mathbf{e}+\mathbf{R}_{b}^{l}\delta\mathbf{b}_{\mathbf{a}}+\mathbf{R}_{b}^{l}\mathbf{\eta}_{\mathbf{a} }\\ \delta\mathbf{b}_{\mathbf{a}}=-\tau_{a}^{-1}\delta\mathbf{b}_{\mathbf{a}}+\mathbf{w}_{\mathbf{a}}\\ \delta\mathbf{b}_{\mathbf{g}}=-\tau_{g}^{-1}\delta\mathbf{b}_{\mathbf{g}}+\mathbf{w}_{\mathbf{g}}\\ \delta(c\delta t)=\delta\left(c\dot{\delta t}\right)+w_{b}\\ \delta\left(c\ddot{\delta t}\right)=w_{d}\end{array} \tag{2}\]  
where \(\mathbf{R}_{b}^{l}\) is the matrix that allows the transformation between the body frame and the local frame.  
\[\mathbf{R}_{b}^{l}=\left(\begin{array}{ccc}\mathbf{c}\mathbf{\phi}\ \mathbf{c}\mathbf{\psi}+\mathbf{s}\mathbf{ \phi}\ \mathbf{s}\ \mathbf{\phi}\ \mathbf{\psi}&\mathbf{c}\mathbf{\theta}\ \mathbf{\psi}&\mathbf{s}\mathbf{\phi}\ \mathbf{c}\mathbf{\psi}-\mathbf{c}\mathbf{\phi}\ \mathbf{s}\ \mathbf{\psi}\\ -\mathbf{c}\mathbf{\phi}\ \mathbf{s}\mathbf{\psi}+\mathbf{s}\mathbf{\phi}\ \mathbf{s}\ \mathbf{\phi}\ \mathbf{\psi}&\mathbf{c}\mathbf{\theta}\ \mathbf{\psi}&-\mathbf{s}\mathbf{\phi}\ \mathbf{s}\mathbf{\psi}-\mathbf{c}\mathbf{\phi}\ \mathbf{s}\ \mathbf{\phi}\ \mathbf{\psi}\\ -\mathbf{s}\mathbf{\phi}\ \mathbf{c}\mathbf{\theta}&\mathbf{s}\mathbf{\theta}&\mathbf{c}\mathbf{\phi}\ \mathbf{c}\mathbf{\theta}\end{array}\right) \tag{3}\]  
where \(\mathbf{c}\) and \(\mathbf{s}\) are the cosine and sine functions, respectively.  
\(\mathbf{\hat{\mathbf{r}}}\) is the specific force measured by the accelerometer and corrected for the accelerometer biases \(\mathbf{\hat{b}}_{\mathbf{a}}\). (\(\times\)) is the skew-symmetric matrix operator. \(\mathbf{\eta}_{\mathbf{a}}\) and \(\mathbf{\eta}_{\mathbf{g}}\) are, respectively, the accelerometer and gyroscope noise processes. Sensor biases are modelled by a first order Gauss-Markov process with correlation times \(\tau_{a}\) and \(\tau_{g}\) and noises \(\mathbf{w}_{\mathbf{a}}\) and \(\mathbf{w}_{\mathbf{g}}\), respectively, for the accelerometer and the gyroscope. The time derivative of the receiver clock bias error equals the clock drift error plus a Gaussian noise \(w_{b}\). The receiver clock drift error is a constant random process, and its time derivative is a Gaussian noise \(w_{d}\).  
The system defined in (2) can be written as \(\dot{\mathbf{\alpha}}=\mathbf{F}\delta\mathbf{x}+\mathbf{W}\), with \(\mathbf{F}\) being the dynamic matrix and \(\mathbf{W}\) being the system noise matrix with covariance \(\mathbf{Q}=\mathbf{cov}(\mathbf{W})\). The attitude is predicted by the integration of the differential equation \(\dot{\mathbf{R}}_{b}^{l}\cong\mathbf{R}_{b}^{l}(\mathbf{\omega}^{\mathbf{b}}\times)\), where \(\mathbf{\omega}^{\mathbf{b}}=\left(\ddot{\mathbf{\omega}}^{\mathbf{b}}-\dot{\mathbf{b}}_{\mathbf{g}}\right)\) is the angular velocity vector in the body frame provided by the gyroscope and corrected for the gyroscope biases. The user acceleration in the local frame is obtained by \(\mathbf{a}=\mathbf{R}_{b}^{l}\left(\dot{\mathbf{\hat{r}}}-\dot{\mathbf{b}}_{\mathbf{a}}\right)+ \mathbf{g}\), with \(\mathbf{g}\) being the local gravity vector. The receiver clock bias \(\delta t\) is predicted by the integration of the receiver clock drift \(\dot{\delta t}\). The velocity in the local frame \(\mathbf{v}\) is obtained by the integration of the acceleration \(\mathbf{a}\). Then, the position \(\mathbf{r}\) in the local frame is obtained by the integration of the velocity \(\mathbf{v}\). Finally, the state covariance matrix \(\mathbf{P}\) is computed according to \(\mathbf{P}=\mathbf{\Phi}\mathbf{P}\mathbf{\Phi}^{T}+\Delta t\mathbf{Q}\), where \(\mathbf{\Phi}=\mathbf{I}+\Delta t\mathbf{F}\) is the state transition matrix and \(\Delta t\) is the time step of the prediction.