## 4 TTL Model  
This section provides a model for TTL coupling due to the CM-VP offset, which is similar to the work in [14]. For the sake of clarity, Equation (16) can be rewritten as  
\[\rho_{\text{LRI}}=\rho_{\text{inst}}+\delta\rho_{\text{TTL}}+\delta\rho_{ \text{others}}, \tag{19}\]  
where \(\delta\rho_{\text{others}}\) denotes the contributions of all other error sources. As shown in Figure 3, we use the subscript \(i\) (\(i=1,2\)) to indicate the SAT1 and SAT2. The measured range can be regarded as the distance of \(\mathbf{VP}_{1}\) and \(\mathbf{VP}_{2}\) with additional bias and noise, and the instantaneous range is the distance between \(\mathbf{CM}_{1}\) and \(\mathbf{CM}_{2}\) of the two satellites. Therefore, we have  
\[ \begin{split}\rho_{\text{LRI}}&=|\mathbf{VP}_{2}-\mathbf{VP} _{1}|+\delta\rho_{\text{others}},\\ \rho_{\text{inst}}&=|\mathbf{CM}_{2}-\mathbf{CM}_{1}|.\end{split}  \tag{20}\]The TTL error term is then expressed as  
\[\delta\rho_{\text{TTL}} =\rho_{\text{LRI}}-\rho_{\text{inst}}-\delta\rho_{\text{others}} \tag{21}\] \[=|\mathbf{VP}_{2}-\mathbf{VP}_{1}|-|\mathbf{CM}_{2}-\mathbf{CM}_{1}|\] \[\approx|\mathbf{P}_{2}-\mathbf{P}_{1}|-|\mathbf{CM}_{2}-\mathbf{CM}_{1}|\] \[=-\epsilon_{\text{LOS},1}\cdot\mathbf{V}_{1}-\mathbf{e}_{\text{LOS},2} \cdot\mathbf{V}_{2},\]  
where the approximation \(|\mathbf{VP}_{2}-\mathbf{VP}_{1}|\approx|\mathbf{P}_{2}-\mathbf{P}_{1}|\) is applied since the distance between \(\mathbf{VP}_{i}\) and \(\mathbf{VP}_{i}\) (\(i=1,2\)) is much smaller than the distance between \(\mathbf{VP}_{1}\) and \(\mathbf{VP}_{2}\). The absolute value of the TTL error of each satellite is approximately the projection of the vector \(\mathbf{V}_{i}\) on the line of sight (LOS). One can compute the TTL error in the line-of-sight frame (LOSF) shown in Figure 3 for each satellite:  
\[\delta\rho_{\text{TTL},i}\approx-\mathbf{e}_{\text{LOS},i}^{\text{LOSF}_{i}}\cdot \mathbf{V}_{i}^{\text{LOSF}_{i}}=- \begin{bmatrix}1&0&0\end{bmatrix} \cdot\mathbf{V}_{i}^ {\text{LOSF}_{i}}, \tag{22}\]  
where the superscripts denote the frames of the vectors. Since the \(\mathbf{VP}_{i}\) co-rotates with the satellite around the \(\mathbf{CM}_{i}\), the components of the vector \(\mathbf{V}_{i}^{\text{SF}_{i}}=(\Delta x_{i},\Delta y_{i},\Delta z_{i})^{\text{T}}\) expressed in the satellite frame (SF) for each satellite remain unchanged, unless a disturbance to the center of mass (e.g., a mass trim maneuver aimed at moving the center of mass back to the ACC reference point [14]) happens. With the rotation matrix \(\mathbf{R}_{\text{SF}}^{\text{LOSF}}\), which denotes the rotation transforming a vector from SF to LOSF, we have  
\[\delta\rho_{\text{TTL},i}\approx- \begin{bmatrix}1&0&0\end{bmatrix} \cdot\mathbf{R }_{\text{SF}_{i}}^{\text{LOSF}_{i}}\mathbf{V}_{i}^{\text{SF}_{i}}. \tag{23}\]  
The roll angle \(\theta_{x}\), pitch angle \(\theta_{z}\), and yaw angle \(\theta_{z}\) used in this paper are defined by  
\[\mathbf{R}_{\text{SF}}^{\text{LOSF}} =\mathbf{R}_{x}(\theta_{x})\mathbf{R}_{y}(\theta_{y})\mathbf{R}_{z}(\theta_{ z}) \tag{24}\] \[= \begin{bmatrix}c_{y}c_{z}&c_{y}s_{z}&-s_{y}\\ s_{x}s_{y}c_{z}-c_{x}s_{z}&s_{x}s_{y}s_{z}-c_{x}c_{z}&s_{x}c_{y}\\ c_{x}s_{y}c_{z}-s_{x}s_{z}&c_{x}s_{y}s_{z}-s_{x}c_{z}&c_{x}c_{y}\end{bmatrix} ,\]  
Figure 3: Geometric relationship of reference points and reference frames. The dashed line between \(\mathbf{CM}_{1}\) and \(\mathbf{CM}_{2}\) is called the line of sight (LOS) and \(\epsilon_{\text{LOS},i}\) is the unit vector pointing from one \(\mathbf{CM}_{i}\) (\(i=1,2\)) to the other one. \(\mathbf{V}_{i}\) is the vector pointing from \(\mathbf{CM}_{i}\) to \(\mathbf{VP}_{i}\). The point \(\mathbf{P}_{i}\) is on the LOS and the line segment between \(\mathbf{CM}_{i}\) and \(\mathbf{P}_{i}\) is the projection of \(\mathbf{V}_{i}\). The origin of the satellite frame (SF) and the line-of-sight frame (LOSF) is the \(\mathbf{CM}\) of the corresponding satellite.