# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 2 The Navigation Filter
### Observation Model
#### 2.2.1 Doppler Shift Model
 \tag{7}\]  
By inserting (7) into (6), we obtain  
\[\hat{\rho}^{k}=\frac{\left(\mathbf{r}+\delta\mathbf{r}-\mathbf{r}^{k}\left(t+\delta t- \tau^{k}\right)\right)^{T}}{\|\mathbf{r}+\delta\mathbf{r}-\mathbf{r}^{k}\left(t+\delta t- \tau^{k}\right)\|}\left(\mathbf{v}+\delta\mathbf{v}-\mathbf{v}^{k}\left(t+\delta t-\tau^{k }\right)\right)+c\dot{\delta}t+\delta\left(c\dot{\delta}t\right) \tag{8}\]  
The satellite position and velocity at the biased emission time can be written as:  
\[\mathbf{r}^{k}\left(t+\delta t-\tau^{k}\right)=\mathbf{r}^{k}\left(t-\tau^{k}\right)+ \delta t\hat{\mathbf{r}}^{k}\left(t-\tau^{k}\right) \tag{9}\]  
\[\mathbf{v}^{k}\left(t+\delta t-\tau^{k}\right)=\mathbf{v}^{k}\left(t-\tau^{k}\right)+ \delta t\hat{\mathbf{v}}^{k}\left(t-\tau^{k}\right) \tag{10}\]Inserting (9) and (10) into (8) yields  
\[\hat{\hat{\rho}}^{k}=\frac{\Big{(}\mathbf{r}-\mathbf{r}^{k}+\delta\mathbf{r}-\delta t\hat{ \mathbf{r}}^{k}\Big{)}^{T}}{\|\mathbf{r}-\mathbf{r}^{k}+\delta\mathbf{r}-\delta t\hat{\mathbf{r}}^{k }\|}\Big{(}\mathbf{v}-\mathbf{v}^{k}+\delta\mathbf{v}-\delta t\hat{\mathbf{v}}^{k}\Big{)}+c\dot{ \delta t}+\delta\Big{(}c\dot{\delta t}\Big{)} \tag{11}\]  
In (11), the reference to time \(t-\tau^{k}\) in \(\hat{\mathbf{r}}^{k}\), \(\hat{\mathbf{r}}^{k}\), \(\mathbf{v}^{k}\), and \(\hat{\mathbf{v}}^{k}\) is omitted for clarity. To develop the predicted range rate (11), we can write  
\[\frac{1}{\|\mathbf{r}-\mathbf{r}^{k}+\delta\mathbf{r}-\delta t\hat{\mathbf{r}}^{k}\|}\cong \frac{1}{\|\mathbf{r}-\mathbf{r}^{k}\|}\left(1-\frac{(\mathbf{r}-\mathbf{r}_{k})^{T}\delta\bm {r}}{\|\mathbf{r}-\mathbf{r}^{k}\|^{2}}+\frac{(\mathbf{r}-\mathbf{r}_{k})^{T}\hat{\mathbf{r}}^{k} }{\|\mathbf{r}-\mathbf{r}^{k}\|^{2}}\delta t\right) \tag{12}\]  
In (12), we neglected the quadratic terms (i.e., \(\delta t^{2}\|\hat{\mathbf{r}}^{k}\|^{2}\approx 0\), \(\|\delta\mathbf{r}\|^{2}\approx 0\), and \(\delta t\delta\mathbf{r}^{T}\hat{\mathbf{r}}^{k}\approx 0\)) and we limited the development to the first order for small values of \(\delta\mathbf{r}\) and \(\delta t\). Substituting \(\frac{\mathbf{r}-\mathbf{r}^{k}}{\|\mathbf{r}-\mathbf{r}^{k}\|}\) by \(\mathbf{u}^{k}\) in (12), and then inserting (12) into (11), we obtain the following approximation of the range rate  
\[ \begin{array}{rl}\hat{\dot{\rho}}^{k}\cong&\frac{\Big{(}\mathbf{r}-\mathbf{r}^{k}+ \delta\mathbf{r}-\delta t\hat{\mathbf{r}}^{k}\Big{)}^{T}}{\|\mathbf{r}-\mathbf{r}^{k}\|}\bigg{(} 1-\frac{\mathbf{u}^{k}\hat{\sigma}\mathbf{r}}{\|\mathbf{r}-\mathbf{r}^{k}\|}+\frac{\mathbf{u}^{k} \hat{\mathbf{r}}^{k}\hat{\mathbf{r}}^{k}}{\|\mathbf{r}-\mathbf{r}^{k}\|}\delta t\Big{)}\Big{(} \mathbf{v}-\mathbf{v}^{k}+\delta\mathbf{v}-\delta t\hat{\mathbf{v}}^{k}\Big{)}\\ &+c\dot{\delta t}+\delta\Big{(}c\dot{\delta t}\Big{)}\end{array}  \tag{13}\]  
The development of (13) by neglecting the second order terms yields  
\[ \begin{array}{rl}\hat{\dot{\rho}}^{k}\cong\dot{\rho}^{k}+\bigg{(}\frac{(\mathbf{ v}-\mathbf{v}^{k})^{T}}{\|\mathbf{r}-\mathbf{r}^{k}\|}-\mathbf{u}^{k}\mathbf{T}\big{(}\mathbf{v}-\mathbf{v}^{k} \big{)}\frac{\mathbf{u}^{k}\mathbf{T}}{\|\mathbf{r}-\mathbf{r}^{k}\|}\bigg{)}\delta\mathbf{r}+\mathbf{ u}^{k}\mathbf{T}\delta\mathbf{v}+\\ \bigg{(}\frac{\Big{(}\mathbf{u}^{k}\mathbf{T}^{k}\Big{)}\mathbf{u}^{k}-\hat{\mathbf{r}}^{k} \Big{)}^{T}}{\|\mathbf{r}-\mathbf{r}^{k}\|}\Big{(}\mathbf{v}-\mathbf{v}^{k}\Big{)}-\mathbf{u}^{k} \mathbf{T}\hat{\mathbf{v}}^{k}\bigg{)}\delta t+\delta\Big{(}c\dot{\delta t}\Big{)}+ \eta_{\dot{\rho}}\end{array}  \tag{14}\]  
Therefore, the range rate error defined by \(\Delta\hat{\rho}^{k}=\hat{\rho}^{k}-\dot{\rho}^{k}\) can be written as:  
\[