## 2 One-Way TOA Measurement Model  
The WSN consists of \(M\) ANs and one mobile UN in a \(N\)-dimensional space, where \(N=2\) or \(3\), as depicted in Figure 1. The true position of AN \(\#i\) is \(\mathbf{s}_{i}^{o}\), \(i=1,2,\cdots,M\), which is static. ANs are synchronized so the clock offset \(v_{i}\), \(i=1,2,\cdots,M\), among the ANs that are known. As discussed in the Introduction, the system considered in this paper employs a TD broadcasting at ANs that transmit packets in a synchronized and \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Paper** & **Measurements** & **Methods** & **Complexity** \\ \hline
[7] & One-way TOA & TSWLS & Low \\
[8,27] & One-way TOA & GN & Medium \\
[11] & One-way TOA & WLS & Low \\
[19] & One-way TOA & Distributed state estimation & High \\
[20] & Two-way TOA & WLS & Low \\
[23] & Two-way TOA & Exhaustive search and a multi-grid search & High \\
[25] & Two-way TOA & WLS & Low \\
[26] & Two-way TOA & SDP & Medium \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Category of the related works. pre-arranged sequential order. The UN passively receives these packets to obtain sequential TOA measurements, enabling JLAS. Since the transmission is one-way from the ANs to the UN, only one-way TOA measurements are available to the UN. The position and velocity of the mobile UN are unknown, denoted by \(\mathbf{p}^{o}\) and \(\mathbf{v}^{o}\). Assume that the velocity is constant within a TD broadcasting cycle. Since the clock of the UN is not synchronized to that of the ANs, the clock offset and clock skew are unknown. The one-way TOA from AN #_i_ to the UN is  
\[\tau_{i}^{o}=\frac{\|\mathbf{p}^{o}+\mathbf{v}^{o}t_{i}-\mathbf{s}_{i}^{o}\|}{ c}+\tau_{off}^{o}+\tau_{dft}^{o}t_{i}-v_{i}, \tag{1}\]  
where \(t_{i}\) represents the known time interval between the start time of the TD broadcasting cycle and AN #_i_'s launch time, \(c\) is the light speed, \(\tau_{off}^{o}\) is the clock offset, \(\tau_{dft}^{o}\) is the clock skew, and \(v_{i}\) is the clock offset of AN #_i_. In assuming that \(c\) is a known constant for the WSN, the TOA is interchangeable with the range-of-arrival:  
\[r_{i}^{o}=\|\mathbf{p}^{o}+\mathbf{v}^{o}t_{i}-\mathbf{s}_{i}^{o}\|+\gamma^{o} +t^{o}t_{i}-\gamma_{i}, \tag{2}\]  
where \(r_{i}^{o}=c\tau_{i}^{o}\), \(\gamma^{o}=c\tau_{off}^{o}\), \(\ell^{o}=c\tau_{dft}^{o}\), \(\gamma_{i}=cv_{i}\), and \(\gamma^{o}\) and \(t^{o}\) are the distance form of the clock offset and clock skew. So, the measured TOA is  
\[r_{i}=r_{i}^{o}+\varepsilon_{i},\quad i=1,\ldots,M, \tag{3}\]  
where \(\varepsilon_{i}\) represents the measurement noise, following an independent zero-mean Gaussian distribution with a variance of \(\sigma_{i}^{2}\), i.e., \(\varepsilon_{i}\sim\mathcal{N}(0,\sigma_{i}^{2})\). The measurements from all ANs are  
\[\mathbf{r}=\mathbf{r}^{o}+\varepsilon, \tag{4}\]  
where \(\mathbf{r}=[r_{1},r_{2},\cdots,r_{M}]^{T},\mathbf{r}^{o}=[r_{1}^{o},r_{2}^{o},\cdots,r_{M}^{o}]^{T}\), and \(\varepsilon=[\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{M}]^{T}\). The covariance matrix of \(\varepsilon\) is \(\mathbf{Q}_{\varepsilon}=E[\varepsilon\varepsilon^{T}]\). Typically, the true position of AN #_i_ is not known, which is usually obtained through measurement or estimation; hence, it may contain an error represented by \(\Delta\mathbf{s}_{i}\). The erroneous position of AN #_i_ is  
\[\mathbf{s}_{i}=\mathbf{s}_{i}^{o}-\Delta\mathbf{s}_{i},\quad i=1,\ldots,M. \tag{5}\]  
where the error \(\Delta\mathbf{s}_{i}\) follows a zero-mean Gaussian distribution, \(\Delta\mathbf{s}_{i}\sim\mathcal{N}(0,\sigma_{\mathbf{s}_{i}}^{2})\). In collecting all position errors in vector form, \(\Delta\mathbf{s}=[\Delta\mathbf{s}_{1}^{T},\Delta\mathbf{s}_{2}^{T},\cdots, \Delta\mathbf{s}_{M}^{T}]^{T}\), the covariance matrix of position errors is \(\mathbf{Q}_{s}=E[\Delta\mathbf{s}\Delta\mathbf{s}^{T}]\).