# Background and Recent Advances in the Locata Terrestrial Positioning and Timing Technology
## 3 Locata Development History and Mathematical Models
### Locata Mathematical Model  
#### 3.2.1 Locata Measurements  
Similar to GNSS, the range measurements of Locata are of two types: code pseudorange and carrier phase. The carrier phase measurements are more precise than the code pseudorange measurements. The basic Locata observation equation between receiver and LocataLite channel \(i\) can be written as:  
\[\left\{ \begin{array}{c}P_{j}^{i}=\rho^{i}+\tau_{\textit{trop}}^{i}+c\delta t +\varepsilon_{p,j}^{i}\\ \Phi_{j}^{i}=\rho^{i}+\tau_{\textit{trop}}^{i}+c\delta t+\lambda_{j}N_{j}^{i}+ \varepsilon_{\varphi,j}^{i}\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **First Generation System** & **Second Generation System** \\  & & **(Prototype Since 2002)** & **(Commercial Deployment** \\ \hline \multirow{4}{*}{Signal structure} & Frequencies & Single-frequency at GPS L1 & Dual-frequency 2.4 GHz (80 MHz bandwidth) \\ \cline{2-4}  & PRN code & C/A (1.023 MHz chipping rate) & Proprietary (10 MHz chipping rate) \\ \cline{2-4}  & Licence requirements & Licensing issues \& problem for wide area deployment & None required, FCC compliant \\ \hline \multirow{4}{*}{Locata(transceiver)} & Hardware & FPGA \& DDS technology & FPGA \& DDS technology with a modular design \\ \cline{2-4}  & Output power & Several microwatts & Maximum of 1 watt \\ \cline{2-4}  & Range & \(\sim\)600 metres & \(\sim\)10 km line-of-sight \\ \cline{2-4}  & Antenna & RHCP patch \& \(\%\) wave & Antenna design dependent on application \\ \cline{2-4}  & Size & \(260\times 200\times 45\) mm & \(240\times 135\times 30\) mm \\ \cline{2-4}  & Weight & 2.1 kg & 1 kg \\ \hline \multirow{4}{*}{Locata(receiver)} & Hardware & \begin{tabular}{c} Zarlink/Mitel based GPS \\ receiver chipset \\ \end{tabular} & \begin{tabular}{c} FPGA technology, \\ modular design \\ \end{tabular} \\ \cline{2-4}  & Measurement rate & \(1\) Hz & 25 Hz \\ \cline{2-4}  & RT positioning & \(1\) Hz on-board & \begin{tabular}{c} 25 Hz through LINE \\ software, 10 Hz onboard \\ \end{tabular} \\ \cline{2-4}  & AR & \
\begin{tabular}{c} Known point initialisation \\ (KPI) \\ \end{tabular} & On-the-fly (OTF) \\ \cline{2-4}  & Antenna &
\begin{tabular}{c} Various types tested including \\ RHCP patch and \(\%\) wave \\ \end{tabular} & Antenna design will depend on application \\ \cline{2-4}  & Size & \(200\times 100\times 40\) mm & \(130\times 135\times 30\) mm \\ \cline{2-4}  & Weight & 300 g & 500 g \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification summary of Locata’s systems [16]. where \(P_{j}^{i}\) and \(\Phi_{j}^{i}\) are the code pseudorange and carrier phase measurements, respectively (in units of metres), \(\rho^{i}\) is the geometric distance between the Locata Rover and the transmitting antenna \(i\), \(\tau_{\textit{trap}}^{i}\) is the tropospheric delay, \(c\) is the speed of microwave signals in a vacuum, \(\delta t\) is the receiver clock bias, \(\lambda_{j}\) is the wavelength of the \(j^{\text{th}}\) frequency, \(N_{j}^{i}\) is the carrier phase ambiguity term, and \(\varepsilon_{p,j}^{i}\) and \(\varepsilon_{q,j}^{i}\) are the lumped sum of unmodelled residual errors (including noise) on code pseudorange and carrier phase measurements, respectively. Note that there is no transmitter clock bias in the observation equation because of the tight time synchronisation of the LocataLites. Nor is there an ionospheric delay term. Unlike GNSS, Locata's ambiguities are typically estimated as the floating-point values [33]. The receiver clock bias can be estimated or eliminated using Single-Differenced (SD) measurements between the two LocataLite channels. Similar to GNSS SD between satellites, the SD measurements for the Locata system are:  
\[\left\{ \begin{array}{c}\Delta P_{j}^{ik}=P_{j}^{i}-P_{j}^{k}=\Delta\rho^{ik} +\Delta\tau_{\textit{trap}}^{ik}+\Delta\varepsilon_{p,j}^{ik}\\ \Delta\Phi_{j}^{ik}=\Phi_{j}^{i}-\Phi_{j}^{k}=\Delta\rho^{ik}+\tau_{\textit{ trap}}^{ik}+\lambda_{j}\cdot\Delta N_{j}^{ik}+\Delta\varepsilon_{\varphi,j}^{ik} \end{array} \right.