# The First Result of Relative Positioning and Velocity Estimation Based on CAPS
## 2 CAPS Structure and Mathematical Model of Observations
### Raw Observation Model  
Based on the CAPS navigation signal from each individual CAPS satellite tracked by the receiver, the user can obtain the propagation time from the master control station to the user through each satellite. The satellite orbit, virtual clock model, and atmospheric model can be decoded from the received ephemeris message. The total propagation time of the navigation signal from the master control station to the satellite and then to the user can be divided into five parts shown by Figure 3 and Equation (1) with a time unit. \[t_{\Sigma_{i}}=\tau^{T}+t^{Uli}+\tau^{i}+t^{Di}+t_{R}+\varepsilon_{i} \tag{1}\]  
where \(t_{\Sigma_{i}}\) is the total propagation time of the CAPS navigation signal from the ground control station to the satellite and then to the user, and can be transformed to a code measurement by multiplying it by the speed of light; \(i\) is the index of the satellite tracked by the user receiver; \(\tau^{T}\) is the hardware delay of the transmit device at the ground control station; \(t^{Uli}\) is the propagation time from the transmit antenna at the master control station to the satellite antenna, including the ionospheric and tropospheric delays;  
Figure 3: Total propagation time of the CAPS navigation signal from the ground control station to the user. Figure 2: Ground track of CAPS satellites in orbit. \(\tau^{ii}\) is the time delay of the satellite transponder, which differs from satellite to satellite; \(t^{Di}\) is the propagation time from the satellite transponder antenna to the user antenna, including the ionospheric and tropospheric delays; \(t_{k}\) is the hardware delay and the clock offset of the user receiver; and \(\varepsilon_{i}\) is the sum of the measurement noise and other unmodeled errors. \(\tau^{T}\) and \(\tau^{si}\) can be generally calibrated in the laboratory and \(t^{Uli}\) can be calculated once the coordinates of ground the control station and the satellite are precisely known and the atmospheric delay in the uplink path could be estimated precisely. Equation (1) can therefore be rewritten as:  
\[\left\{ \begin{array}{c}t^{Di}=t_{\Sigma_{i}}-T-t_{k}-\varepsilon_{i}\\ T=\tau^{T}+t^{Uli}+\tau^{si}\end{array} \right. \tag{2}\]  
In CAPS, the virtual clock approach proposed by [PERSON] et al. (2011, 2012) is used and a polynomial function of 4 th order is introduced to characterize the value and its variation of \(T\) in the time domain. The coefficients of this polynomial function with a 30-s update interval are broadcast to users by the ephemeris message. The CAPS receiver measures the \(t_{\Sigma_{i}}\) with a tracked navigation signal, calculates the value of \(T\) with the model coefficients and removes it from \(t_{\Sigma_{i}}\). In this way, the receiver can measure the propagation time from the satellite transponder antenna to the receiver antenna and translate it into code pseudo-range. Therefore, the navigation signal can be considered as being sent directly from the satellite, there is a virtual satellite clock on the satellite, and \(T\) can be considered as the offset of that virtual satellite clock. To remove the effect of ionospheric and tropospheric delays in the uploading path, \(T\) was, in practice, calculated as a whole. The calculation of \(T\) is shown in Figure 4. The antenna with a large aperture uploading the navigation signal at the master control station can simultaneously receive the downlink signal of the C-band frequency from the satellite transponder. The master control station measures the total propagation time \(t^{\prime}_{\Sigma_{i}}\). Obviously, \(t^{\prime}_{\Sigma_{i}}\) contains the ionospheric delay in the uplink and downlink paths, the tropospheric delay in the uplink and downlink paths, the time delay of the satellite transponder, the hardware delay of the transmit and receive devices at the master control station.