## 2 Localization Scenario  
Consider a scenario as shown in Figure 1 where \(M\) moving sensors are distributed in an \(N\)-dimensional space (\(N=3\) or \(N=2\)). The known initial positions and velocities of these sensors are denoted as \(\mathbf{s}_{i,1}\in\mathbb{R}^{N}\) and \(\mathbf{\dot{s}}_{i}\in\mathbb{R}^{N}\), respectively. These sensors are deployed to determine the position and velocity of a moving source. Assuming synchronization in time and frequency among the sensors, they can produce TDOA and FDOA measurements between receiving sensors and a reference sensor. Without loss of generality, the reference sensor is chosen as \(\mathbf{s}_{1}\). The unknown initial position and velocity of the source are denoted as \(\mathbf{u}^{o}\in\mathbb{R}^{N}\) and \(\mathbf{\dot{u}}^{o}\in\mathbb{R}^{N}\), respectively, which will be estimated using TDOA and FDOA measurements. The velocities of the source and sensors are constant. During the motion of the sensors, the source is observed a total of \(K\) times, with a time interval of \(\tau\) between the adjacent observation frame. Assuming the observation time is short, the sensors and source are considered as quasi-static during one observation frame. At the \(k\)-th observation frame, the positions of the sensors are denoted as  
\[\mathbf{s}_{i,k}=\mathbf{s}_{i,1}+\mathbf{\dot{s}}_{i}(k-1)\tau,\quad k=1,2, \cdots,K, \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Acronym** & **Description** \\ \hline MLE & Maximum Likelihood Estimation \\ CFS & Closed-form solution \\
2 SWLS & Two-Step Weighted Least Squares \\ WLS & Weighted Least Squares \\ CWLS & Constrained Weighted Least Squares \\ CRLB & Cramer–Rao Lower Bound \\ MSE & Mean-Square Error \\ MDS & Multidimensional Scaling \\ SDP & Semidefinite programming \\ SDR & Semidefinite Relaxation \\ MIMO & Multiple-Input Multiple-Output \\ TMA & Target Motion Analysis \\ BO & Bearing-only \\ GN & Gauss–Newton \\ QN & Quasi-Newton \\ LM & Levenberg–Marquardt \\ LMI & Linear Matrix Inequality \\ SOC & Second-order cone \\ FIM & Fisher Information Matrix \\ RMSE & Root mean-squared error \\ UAV & Unmanned aerial vehicle \\ GDOP & Geometric Dilution of Precision \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Method** & **Paper** \\ \hline TDOA and FDOA & CFS & [18; 19; 20; 21; 22; 23; 24; 25; 26] \\ \hline TDOA and FDOA & Iterative solution & [27] \\ \hline TDOA and FDOA & SDP solution & [28; 29; 30] \\ \hline BO-TMA & Particle filter & [40] \\ \hline BO-TMA & CFS & [41; 42; 43; 44] \\ \hline TMA for TDOA only & Iterative solution & [45] \\ \hline TMA for FDOA only & CFS and SDP & [46] \\ \hline TMA for TDOA and FDOA & SDP and GN & Proposed \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Categories of the related works. and the position of the source is  
\[\mathbf{u}_{k}^{o}=\mathbf{u}^{o}+\dot{\mathbf{u}}^{o}(k-1)\tau. \tag{2}\]  
Collecting the initial position and velocity of the source together, the unknown vector is \(\mathbf{\beta}^{o}=[\mathbf{u}^{oT},\dot{\mathbf{u}}^{oT}]^{T}\). The TDOAs between the \(i\)-th and \(1\)st sensors are interchangeable with range differences by multiplying with the signal propagation speed,  
\[r_{i,k}^{o}=d_{i,k}^{o}-d_{1,k}^{o},\quad i=2,\cdots,M. \tag{3}\]  
where  
\[d_{i,k}^{o}=\|\mathbf{u}_{k}^{o}-\mathbf{s}_{i,k}\|,\quad i=1,\cdots,M, \tag{4}\]  
is the true distance from the source to the \(i\)-th sensor during the \(k\)-th observation frame.