# Positioning Using IRIDIUM Satellite Signals of Opportunity in Weak Signal Environment
## 4 Positioning Using Iridium Satellite SOPs in Weak Signal Environment  
The essential elements of Iridium satellite SOPs positioning are observation information acquisition, satellite orbit calculation, and positioning algorithm. The principle of Iridium SOPs positioning in weak signal environment is presented in Figure 3. The solid part denotes the hard platform and the dotted part denotes the software algorithm. A wide-band antenna with a frequency band from 380 to 20 GHz and 5 dB gain connects to a radio frequency front end (RFFE). There are typically three main modules in the software processing: Doppler-shift estimation, satellite orbit reconstruction, and positioning solution. The Doppler-shift measurements are obtained by the QSA-IDE algorithm discussed above. The two-line element set format (TLE) provided from NOARD is used as the Iridium satellite orbital ephemeris. It consists of orbital parameters and time information. We use the SGP4 prediction model to realize the purpose of satellite orbit reconstruction [16]. NOARD periodically updates the Iridium TLE sets once or twice every day. The TLE data can be received directly from the NORAD website celestrak.com. The Iridium satellite position calculated with TLE data is with an error that can be from 100 m to several kilometers. While the corresponding velocity errors are optimistic, the value can be approximately 3 m/s. The position errors and velocity errors are rather smaller in the radial direction than for cross and along direction. After obtaining the Doppler-shift measurements and Iridium orbital, the receiver positioning can be achieved by instantaneous Doppler positioning algorithm, and the position accuracy can be further improved by height aiding. The Iridium continuously transmits ring alert signal (simplex 7 channel) and primary message signal (simplex 11 channel) to the ground. The quaternary message signal (simplex 3 channel) and tertiary message signal (simplex 4 channel) are occasionally transmitted. In this paper, the simplex 7 and simplex 11 channel signals are used for receiver position, respectively. The Doppler-shift caused by the relative motion between the satellite and the receiver provides the basis for receiver positioning [17]. The instantaneous Doppler positioning uses multiple intersections of equal Doppler cones to obtain a receiver position instead of using integral Doppler-shift measurements of the hyperbolic positioning method. The Doppler-shift is expressed as [18]:  
\[\dot{\rho}_{k}=\mathbf{v}_{k}\frac{\mathbf{r}_{k}-\mathbf{r}_{u}}{\|\mathbf{r}_{k}-\mathbf{r}_{u} \|}+f_{u}+\varepsilon_{\dot{\rho}_{k}} \tag{16}\]  
where \(\dot{\rho}_{k}\) is the Doppler-shift of satellite \(k\) and its unit is m/s. \(\mathbf{r}_{u}=[r_{u_{k}}r_{u_{k}}r_{u_{k}}]\) is the receiver position vector and \(\mathbf{v}_{k}=\mathbf{v}_{st}-\mathbf{v}_{u}\) denotes the relative velocity vector, \(\mathbf{v}_{st}=[v_{si_{x}}v_{si_{y}}v_{si_{z}}]^{T}\) and \(\mathbf{v}_{u}=[v_{u_{x}}v_{u_{y}}v_{u_{z}}]^{T}\) are the satellite and receiver velocity vectors, respectively. \(f_{u}\) is the receiver clock frequency drift and \(\varepsilon_{\dot{\rho}_{k}}\) is the Doppler-shift error. The navigation equation of the instantaneous Doppler positioning for static receiver is presented here directly which is:  
\[\mathbf{d}=\left[ \begin{array}{cc}\mathbf{v}_{1}\cdot(\mathbf{r}_{1}-\mathbf{r}_{u0})\frac {(\mathbf{r}_{1}-\mathbf{r}_{u0})^{T}}{\|\mathbf{r}_{1}-\mathbf{r}_{u0}\|^{3}}-\frac{\mathbf{v}_{ 1}}{\|\mathbf{r}_{1}-\mathbf{r}_{u0}\|}E_{3\times 3}&1\\ \vdots&\vdots\\ \mathbf{v}_{\mathbf{k}}\cdot(\mathbf{r}_{k}-\mathbf{r}_{u0})\frac{(\mathbf{r}_{k}-\mathbf{r}_{u0})^{T} }{\|\mathbf{r}_{k}-\mathbf{r}_{u0}\|^{3}}-\frac{\mathbf{v}_{k}}{\|\mathbf{r}_{2}-\mathbf{r}_{u0}\| }E_{3\times 3}&1\end{array} \right]\left[ \begin{array}{c}\mathbf{\Delta r}_{\mathbf{u}_{x}} \\ \Delta r_{\mathbf{u}_{y}}\\ \Delta r_{\mathbf{u}_{z}}\\ \Delta f_{\mathbf{u}}\end{array}