## 2 MDPO Positioning Method
From Equations (3)-(6), it could be written as:  
\[\Delta\
abla\rho(t_{i})=\left(\rho_{1}^{1}(t_{i})-\rho_{2}^{1}(t_{i})\right)- \left(\rho_{1}^{2}(t_{i})-\rho_{2}^{2}(t_{i})\right)=\Delta\
abla r(t_{i})- \Delta\
abla\omega(t_{i}), \tag{7}\]  
where \(\Delta\
abla\) denotes the double-difference operation. The resulting double-differenced pseudorange is composed of only the double-differenced true range and pseudorange noise. Although the noise is increased by the double-difference operation, both the receiver clock and the satellite-related bias errors are efficiently eliminated. In order to eliminate the satellite and the receiver clock bias errors from the double-differenced measurements, time synchronization between the two receivers is essential. In the common GPS positioning, it can be achieved in the position calculation process by estimating the receiver clock bias at the same time. However, the receiver clock bias is removed in the double-differenced observation and cannot be estimated. Since the maximum range rate of the pseudorange observation obtained from a low lunar orbiting satellite is about 1.3 km/s, the resulting range error is no larger than 1.3 m if the time synchronization error is maintained under 1 ms. This could be achieved by the frame synchronization of the navigation signals transmitted from the lunar orbiting satellites or the communication link signal between the rover and the lander. Since the baseline is much shorter than the distance to a satellite, the following relationship is obtained:  
\[\Delta\
abla\tau(t_{i})= \left(r_{1}^{1}(t_{i})-r_{2}^{1}(t_{i})\right)-\left(r_{1}^{2}(t_ {i})-r_{2}^{2}(t_{i})\right) \tag{8}\] \[= -\left(I_{1}^{1}(t_{i})-I_{1}^{2}(t_{i})\right)\cdot\mathbf{b}(t_{i})\]  
where \(\mathbf{b}(t_{i})\) is the relative position vector between the two receivers, defined as the following:  
\[\mathbf{b}(t_{i})=\mathbf{x}_{2}(t_{i})-\mathbf{x}_{1}(t_{i}), \tag{9}\]  
and \(I_{1}^{S}(t_{i})\) is the unit direction vector from the base station receiver, which is the lander in this case, to the satellite. It is called the line-of-sight (LOS) vector and could be written as:  
\[r_{R}^{S}(t_{i})=\big{\|}\mathbf{x}^{S}(t_{i}^{S})-\mathbf{x}_{R}(t_{i})\big{\|}, \tag{2}\]0  
Since the lander location \(\mathbf{x}_{1}(t_{i})=\mathbf{x}_{1}\) is stationary and is assumed to be well known, the rover position \(\mathbf{x}_{2}(t_{i})\) in Equation (9) can be written as the following:  
\[r_{R}^{S}(t_{i})=\big{\|}\mathbf{x}^{S}(t_{i}^{S})-\mathbf{x}_{R}(t_{i})\big{\|}, \tag{2}\]1  
where the relative position \(\mathbf{b}(t_{i})\) is the only unknown vector composed of three unknown parameters. Let us now consider the relative position estimation based on Equation (8). In order to estimate \(\mathbf{b}(t_{i})\), we need at least three linearly independent observations. With the two satellites in view, however, only one double-differenced pseudorange measurement is available at each epoch. To overcome this issue, we assume the rover position \(\mathbf{b}(t_{i})=\mathbf{b}\) is also stationary during multiple observation periods \(t_{i}\)\((i=1,\ 2,\dots,\ N)\), where \(N\) must be greater than or equal to the number of unknown parameters in \(\mathbf{b}\). Then the double -differenced pseudorange measurements from multiple observation periods can be written in vector-matrix notation as below:  
\[ \begin{bmatrix}\Delta\
abla\rho(t_{1})\\ \Delta\
abla\rho(t_{2})\\ \vdots\\ \Delta\
abla\rho(t_{N})\end{bmatrix} = \begin{bmatrix}-\left(\dot{\mathbf{f}}_{1} ^{2}(t_{1})-\dot{\mathbf{f}}_{1}^{2}(t_{1})\right)^{T}\\ -\left(\dot{\mathbf{f}}_{1}^{2}(t_{2})-\dot{\mathbf{f}}_{1}^{2}(t_{2})\right)^ {T}\\ \vdots\\ -\left(\dot{\mathbf{f}}_{1}^{2}(t_{N})-\dot{\mathbf{f}}_{1}^{2}(t_{N})\right) ^{T}\end{bmatrix} \mathbf{b}- \begin{bmatrix}\Delta\
abla\omega(t_{1})\\ \Delta\
abla\omega(t_{2})\\ \vdots\\ \Delta\
abla\omega(t_{N})\end{bmatrix} .