# Single-Baseline RTK Positioning Using Dual-Frequency GNSS Receivers Inside Smartphones
## 3 RTK Positioning Using Smartphones  
Common trends and research in the domain of centimetric RTK positioning have been already been presented in the introduction and outline the possibility of using carrier-phase measurements to perform RTK smartphone positioning and navigation with the low-grade smartphone antenna (the main error source) and power saving option of the Android OS. [PERSON] and [PERSON] [12] have demonstrated the possibility to obtain decimetre-level accuracy in smartphone NRTK positioning by processing undifferenced and uncombined observations from a single frequency GNSS chipset. The requirement in this research was to apply differential corrections provided by a GNSS CORS network for a smartphone GNSS receiver in real-time. On the other hand, the objective of the present work is to address the problem of RTK positioning using smartphones in situations where the differential corrections are not available for many reasons (e.g. ; because there is no internet connection or there are no CORS networks). For geodetic surveying, this problem can be addressed using the traditional single-base master-rover configuration of two GNSS receivers: as described in references [13; 19], generally one master station (indicated with subscript A in the Equations (1) and (2)) is settled on a known point that can broadcast pseudorange (PRC, Equation (1)) or carrier-phase corrections (CPC, Equation (2)) to a rover receiver through internet connection or radio modem, if pseudorange and carrier-phase measurements are considered, respectively. \[PRC^{p}(t)=\rho_{A}^{p}(t)-R_{A}^{p}(t)-cdt^{p}(t)-cdT_{A}(t)=\alpha I_{A}^{p}( t)-T_{A}^{p}(t)-E_{A}^{p}(t) \tag{1}\]  
\[CPC(t)=\rho_{A}^{p}(t)-\phi_{A}^{p}(t)-\lambda N_{A}^{p}-cdt^{p}(t)-cdT_{A}(t) =-\alpha I_{A}^{p}(t)-T_{A}^{p}(t)-E_{A}^{p}(t) \tag{2}\]  
In equations (1) and (2) \(\rho_{A}^{p}\) represents the geometric range, \(R_{A}^{p}(t)\) and \(\phi_{A}^{p}(t)\) the pseudorange, and carrier-phase measurements, respectively, \(cdT_{k}\) and \(cdt^{p}\) are the biases related to receiver and satellite clocks multiplied by the speed of light, \(\alpha i_{k}^{p}\) is the ionospheric propagation delay with a known coefficient \(\alpha_{i}=f_{1}^{2}/f_{i}^{2}\) that depends on the i-th frequency, \(T_{k}^{p}\) represents the tropospheric propagation delay, \(M_{k}^{p}\) and \(E_{k}^{p}\) the multipath and the ephemeris error, and finally \(\lambda_{i}N_{k}^{p}\) the carrier-phase ambiguity multiplied by the wavelength, under the assumption that the random errors \(\varepsilon_{k}^{p}\) are not shown in these equations. After these estimations, the master station can broadcast the PRC and CPC values to the rover receiver, (defined as B) which in turn can exploit them through Equations (3) and (4). \[R_{B}^{p}(t)_{correct}=R_{B}^{p}(t)+PRC(t)=\rho_{B}^{p}(t)-cdT_{AB}(t)+\Delta E _{AB}^{p}(t)-\Delta I_{AB}^{p}(t)+\Delta T_{AB}^{p}(t) \tag{3}\]  
\[\varphi_{B}^{p}(t)_{correct}=\rho_{B}^{p}(t)+CPC(t)=\rho_{B}^{p}(t)-cdT_{AB}(t) -\lambda N_{AB}^{p}+\Delta I_{AB}^{p}(t)+\Delta T_{AB}^{p}(t)+\Delta E_{AB}^{ p}(t) \tag{4}\]  
The subscript AB means that the bias is referred to the master-rover combination. Considering both geodetic and mass-market GNSS receivers [20], if the distance between master and rover receivers is lower than 10 km, it is possible to affirm that the propagation of atmospheric delays and the ephemeris errors is almost the same in both places [21]. Thus, it is possible to eliminate them by differencing measurements of the two receivers: in the case of using carrier-phase measurements, it is possible to reach a centimetric level of accuracy if the phase ambiguity \(N_{AB}^{p}\) is estimated (or fixed) as an integer number [11; 22; 23; 24; 25]. Starting from these considerations, the present work investigates two different situations of real-time positioning: firstly, it is interesting to verify the accuracy and precision of a smartphone device used as rover if a CORS is used as a master.