# Opportunistic In-Flight INS Alignment Using LEO Satellites and a Rotatory IMU Platform
## 2 Positioning Architecture
### Slave EKF-Based Doppler Positioning Model
#### 2.2.2 LEO Downlink Measurement Model  
Signals of opportunity transmitted from different ground and satellite networks were deeply investigated. LEO-SOPs attracted the interest of different fields in engineering. Among them, navigation systems in GNSS-denied environments are in the main interest of the community, which was considered in this study. In the following section, we present a description of SOPs propagated from LEO satellites by considering the pseudorange and the pseudorange rate as measurements extracted from their downlink signals. By using software-defined radios (SDR), and by developing the acquisition and tracking algorithmsfor burst-based and continuous downlinks, the phase and Doppler frequency of different downlink channels could be tracked. Doppler frequency measurements \(f_{D}\) could then be derived and estimated using least-squares algorithms applied on the phase parameter estimation of LEO signals. In this work, the Orbcomm and the Iridium-Next downlinks were considered as a LEO-SOP source. Thus, pseudorange rate measurement was possible to obtain following Equation (13). The pseudorange rate measurement for the \(n\)th satellite of the \(m\)th constellation at the \(k\)th time-step was modeled, according to [33]. From the extracted Doppler shifts, the pseudorange rate measurements could be obtained for each observed satellite by a simple formula of \(\rho_{m_{n}}^{\cdot}=-c\frac{f_{D_{m_{n}}}}{f_{m_{n}}}\), while \(f_{D_{m_{n}}}\) and \(f_{m_{n},n}\) were the Doppler shift and the carrier frequency of the \(n\)th satellite of the \(m\)th constellation [33], respectively. \[\dot{\rho}_{m_{n}}=[v_{m_{n}}(k)-v_{r}(k)]^{T}\frac{[p_{r}(k)-p_{m_{n}}(k)]}{ \left\|p_{r}(k)-p_{m_{n}}(k)\right\|_{2}}+c.\Big{[}\dot{\delta}t_{r}(k)-\dot{ \delta}t_{m,n}(k)\Big{]}+c.\Big{[}\dot{\delta}t_{tono}(k)+\dot{\delta}t_{tropo} (k)\Big{]}+n_{m_{n}}(k) \tag{13}\]  
where \(v_{m_{n}}\) and \(p_{m_{n}}\) are the velocity and position of the \(n\)th satellite of the \(m\)th LEO constellation, respectively. Additionally, \(v_{r}\) and \(p_{r}\) were defined as the velocity and the position of the receiver, respectively, and \(n_{m_{n}}\) is the measurement noise modeled as a white Gaussian random sequence with variance \(\sigma_{rate}\). The variation in the ionospheric and tropospheric delays, \(\delta t_{iano}\) and \(\delta t_{trope}\), during LEO satellite visibility was negligible compared to the errors in the satellite's estimated velocities [34]; so, these effects were not mentioned in this article. As depicted in Figure 2, one can observe a different downlink burst from iridium-Next LEO satellites during experimental measures. This experience could also be repeated for Orbcomm, Globalstar, and other VHF/UHF and L-Band LEO satellites. The goal was to observe Doppler shifts and to measure accurately the Doppler frequencies before calculating related pseudorange rates equivalent to the respective measures. It is important to pay attention to the clock bias and the clock drift on the SDR side because they have a direct impact on the estimation accuracy as well as the alignment quality. Although the kinematic and clock states of each observed satellite could be included in the EKF system, our concentration in this study was on deriving the accurate estimation of the receiver's states. The measurement model for the slave EKF system was defined as Equation (14), where \(v(k)\) is the measurement noise with the zero-mean white Gaussian model. The modeled measurement noise had a variance of \(\sigma_{m,n}^{2}\), \(m=1,\ldots,M\), and \(n=1,\ldots,N\) where \(M\) is the total number of constellations and \(N\) is number of the satellites in each one. Additionally, \(y(k)\) is the estimated measurement vector, and z is the pseudorange rate measurement vector obtained from the MC-SDR, given by Equation (15).