# The First Result of Relative Positioning and Velocity Estimation Based on CAPS
## 2 CAPS Structure and Mathematical Model of Observations
### Relative Positioning Model  
The accuracy of the code and carrier-phase measurements are significantly affected by the errors from the orbit, virtual clock, ionospheric delay and tropospheric delay. Those errors can generally be eliminated or mitigated as much as possible by the differential observation between two neighboring receivers. Therefore, the differential observation model of the code and carrier-phase measurements between the user and base receivers can be expressed as:  
\[\left\{ \begin{array}{c}\Delta\rho^{i}=\Delta r^{i}+c(t_{u}-t_{b})+\Delta\tau _{mul,\rho_{i}}+\varepsilon_{\Delta\rho_{i}}\\ \Delta\phi^{i}=\Delta r+c(t_{u}-t_{b})-\lambda\cdot\Delta N_{i}+\Delta\tau_{ mul,\phi_{i}}+\varepsilon_{\Delta\phi_{i}}\end{array} \right. \tag{7}\]  
where \(\Delta\) denotes the differential operation; and \(t_{b}\) is the sum of the hardware delay and the clock error at the base receiver. \(\Delta\tau_{mul,\rho_{i}}\) and \(\Delta\tau_{mul,\phi_{i}}\) are the differential multipath errors for the code and carrier-phase measurements, respectively. The errors from the orbit and virtual clock can be completely eliminated via the differential operation, while the ionospheric and tropospheric errors can only be mitigated as much as possible. The remaining ionospheric and tropospheric delay generally increase when the distance between two receivers increases. As discussed in Section 2.2, a smaller differential ionospheric delay error is very advantageous for extending the length of the baseline between the rover and base station. Moreover, the carrier-phase observation can be considered as the high precise range measurement when the integer ambiguity \(\Delta N^{i}\) can be reliably resolved. The number of CAPS satellites was no more than 5 when we carried out our experiment, and the geometric distribution was not very good due to the 4 GEO satellites being located in east-west directions. Additionally, the receiver can only obtain the observation of a single frequency. These make it quite difficult to fix the ambiguities reliably and quickly based on the geometry-based algorithm in current GNSSs like LAMBDA [14] since this relies on the geometrical change caused by the satellites' movement [15; 16], while the GEO satellites are almost stationary relative to users in CAPS. In this study, the ambiguities were directly estimated using the geometry-free method via Equations (8) and (9). \[\Delta\hat{N}^{i}=mean\big{(}\frac{\Delta\rho^{i}-\Delta\phi^{i}}{\lambda} \big{)}_{L} \tag{8}\]  
\[\Delta N^{i}=\big{[}\Delta\hat{N}^{i}\big{]} \tag{9}\]  
where, \(\Delta N^{i}\) and \(\Delta\hat{N}^{i}\) are the fixed and float ambiguity, respectively; \(\Delta\rho^{i}\) and \(\Delta\phi^{i}\) means the series of differential code and carrier-phase observations during a continuous period; _mean_ represents the average operation over a period of \(L\) epochs; and the square brackets [ ] represent rounding to the nearest integer. For Equations (8) and (9), the premise for obtaining the correct ambiguity is that the series \(\Delta\rho^{i}-\Delta\phi^{i}\) is zero-mean. For the differential observations \(\Delta\rho^{i}\) and \(\Delta\phi^{i}\), the ionospheric delay, tropospheric delay, the model error of the virtual clock and the ephemeris error were almost completely eliminated under a zero-baseline or very short baseline. In this case, the main error sources which effect the fix of ambiguities are the multipath error and the observation noise of the code and carrier phase measurements.