## 2 System Concept and Dual One-Way Measurements
In detail, let us have two spacecraft \(i\) and \(j\) in position \(\mathbf{r}_{i}=(x_{i},y_{i},z_{i})^{T}\) and \(\mathbf{r}_{j}=(x_{j},y_{j},z_{j})^{T}\) and let us assume a common reference time \(t\). The spacecraft can have different clock biases with respect to the reference time; \(\delta t_{i}\), \(\delta t_{j}\), and the time of flight of signal can be measured by the use of the TOAs taken by both of the satellites, in the following way:  
\[(TOA_{i}-ToT_{j}) =\frac{\rho_{i,j}}{c}=\frac{\|\mathbf{r}_{i}-\mathbf{r}_{j}\|}{ c}+(\delta t_{i}-\delta t_{j})+\epsilon_{i,j} \tag{1}\] \[(TOA_{j}-ToT_{i}) =\frac{\rho_{j,i}}{c}=\frac{\|\mathbf{r}_{j}-\mathbf{r}_{i}\|}{ c}+(\delta t_{j}-\delta t_{i})+\epsilon_{j,i} \tag{2}\]  
where \(ToT_{j}\) is the Time of Transmission on satellite \(j\) (encoded in the transmitted signal), and \(\rho_{i,j}\) is called \"pseudorange\" from satellite \(i\) to satellite \(j\). Figure 1: Illustrative example of inter-satellite measurements between two spacecraft deployed in an orbit around a Lagrange point. Finally, \(\epsilon_{ij,j}\) is the measurement error. The pseudo-ranges, \(\rho_{ij,j}\) and \(\rho_{j,j}\) do not coincide with the real range, but, if both of the spacecraft produce a measurement, it is possible to compute the range making the average of the two pseudo-ranges because, by definition, \((\delta t_{i}-\delta t_{j})=-(\delta t_{j}-\delta t_{i})\):  
\[r_{ij}=\|\mathbf{r}_{i}-\mathbf{r}_{j}\|=\frac{\rho_{i,j}+\rho_{j,j}}{2}+\frac {c(\epsilon_{i}+\epsilon_{j})}{2} \tag{3}\]  
Moreover, if the difference between the pseudo ranges also computed, an estimation of the clock bias between the two spacecraft can be obtained:  
\[\delta t_{i,j}=\delta t_{i}-\delta t_{j}=\frac{\rho_{i,j}-\rho_{j,j}}{2c}+ \frac{(\epsilon_{i,j}-\epsilon_{j,i})}{2} \tag{4}\]  
As mentioned before, the time of arrival is measured in the receiver exploiting a Delay Locked Loop as in the case of GNSS [18]. Exploiting the same signals, it is also possible to measure the range rate between two spacecraft, with \(i\) and \(j\) exploring the Doppler effect. Let us recall that the Doppler shift for a radio frequency signal is given by:  
\[f_{d}=-(\delta v)/c\cdot f_{0} \tag{5}\]  
where \(\delta v\) is the radial velocity of the spacecraft \(j\) with respect to the spacecraft \(i\), \(c\) is the speed of light, and \(f_{0}\) is the carrier frequency of the data link. It follows that \(\delta v\) can be seen as the derivative of the geometrical distance of the satellites and that the Doppler shift can be expressed as follows:  
\[f_{i,j}=\frac{c}{f_{0}}\cdot\dot{\rho}_{i,j}=\frac{c}{f_{0}}\cdot\left(\frac{ [(x_{i}-x_{j})(\dot{x}_{i}-\dot{x}_{j}]+[(y_{i}-y_{j})(\dot{y}_{i}-\dot{y}_{j} )]+[(z_{i}-z_{j})(\dot{z}_{i}-\dot{z}_{j}]}{r}\right)+n_{ij} \tag{6}\]  
where \(n_{ij}\) represents the measurement error in measuring the Doppler shift of satellite \(i\) from satellite \(j\). The Doppler shift can be measured by the Phase Lock Loop implemented in the receivers [18]. Figure 2: Block diagram of the proposed main orbiter and floating anchors. Concerning the measurement errors, measuring the TOA of a given signal is a well-known problem, and various methods have been proposed in the literature to find the Cramer-Rao limit, and the most used expression is given by [19]:  
\[\sigma_{TOA}\cong\frac{1}{\beta\sqrt{2E/N_{0}}} \tag{7}\]  
where \(E\) is the signal energy, \(N_{0}\) is the noise power spectral density, and \(\beta\) is the Gabor bandwidth, which is defined as:  
\[\beta^{2}=\frac{\int_{-\infty}^{+\infty}(2\pi f)^{2}\Big{|}S(f)\Big{|}^{2}\,df }{\int_{-\infty}^{+\infty}\Big{|}S(f)\Big{|}^{2}\,df} \tag{8}\]  
where \(S(f)\) is the power spectrum of the received signal.