# The First Result of Relative Positioning and Velocity Estimation Based on CAPS
## 2 CAPS Structure and Mathematical Model of Observations
### Raw Observation Model
The distance from the satellite antenna to the antenna of the master control station can be calculated by the coordinates of the master control station which are precisely known and the coordinates of the satellite which are calculated from the ephemeris message. The distance was transformed to \(t^{\prime}_{c}\) by dividing it by the speed of light. It should be noted that the coordinates of the satellite generally contain some bias from the ephemeris estimation error. Following this, \(T\) was calculated via:  
\[T=t^{\prime}_{\Sigma_{i}}-(t^{\prime}_{c}+t_{signac}) \tag{3}\]  
where \(t_{signac}\) is the Sagnac effect in the downlink path which could be precisely calculated with the coordinates of satellite and the master control station. This means that for a receiver at the master control station, if \(T\) is removed from the measured \(t^{\prime}_{\Sigma_{i}}\), the ionospheric delay in the uplink and downlink, the tropospheric delay in uplink and downlink, the time delay of the satellite transponder, and the hardware delay of the transmit and receive devices were all removed. The remaining error contains the Sagnac and ephemeris errors. Regarding the user receiver far from the master control station, the remaining errors mainly consists of the Sagnac error, the ephemeris error, the multipath errors, and the residual errors of ionosphere and troposphere after removing \(T\) from the measured \(t_{\Sigma_{i}}\). It should be mentioned that the residual error of ionosphere and troposphere is resulted from the differences of the propagation paths of the downlink signal from the CAPS satellite respectively to the master control station and the user receiver. Generally, the residual errors of ionosphere and troposphere may become larger with the increase of the distance between the master control station and user receiver. To make it easier to understand this, we still refer to them ionospheric and tropospheric error later in this article. In this way, the code pseudo-range measurement \(\rho^{i}\) can be obtained by removing \(T\) from the measured \(t_{\Sigma_{i}}\). Taking the ephemeris error, the bias resulting from the virtual clock offset model, the multipath error, and the ionospheric and tropospheric errors into account, the code pseudo-range can be described as:  
\[\rho^{i}=r^{i}+ct_{u}-ct^{i}-\tau_{iono}-\tau_{trop}-\tau_{mul,\rho}+\varepsilon_ {i,\rho} \tag{4}\]  
where \(r^{i}\) is the geometric distance between the CAPS satellite and the user receiver, \(c\) is the speed of light; and \(t^{i}\) is the sum of the model error of the virtual clock offset and the ephemeris error. The model error of the virtual clock offset represents the estimation error of \(T\) with the polynomial model; \(t_{u}\) is the sum of the hardware delay and the clock error at the user receiver; \(\tau_{iono}\) and \(\tau_{trop}\) are the ionospheric and tropospheric errors, respectively. \(\tau_{mul,\rho}\) is the multipath error of the code measurement, and \(\varepsilon_{i,\rho}\) is the sum of the code measurement noise and other unmodeled errors. Regarding the carrier-phase measurement, which is indeed the difference between the carrier phase of the incoming signal from the CAPS satellite and the carrier signal phase replicated by the user receiver, its observation model is shown in Equation (5), where the ambiguity is introduced to the observation:  
\[\lambda\cdot\phi^{i}=r^{i}+ct_{u}-ct^{i}+\tau_{iono}-\tau_{trop}-\lambda\cdot N ^{i}-\tau_{mul,\phi}+\varepsilon_{i,\phi} \tag{5}\]  
where \(\phi^{i}\) is the carrier-phase measurement in cycles at the C1 frequency; \(\lambda\) is the wavelength of the CAPS navigation signal and \(N^{i}\) means the ambiguity of the carrier-phase; \(\tau_{mul,\phi}\) is the multipath error of the carrier-phase measurement; and \(\varepsilon_{i,\phi}\) is the sum of the carrier-phase measurement noise and other unmodeled errors.