# AOA-Based Three-Dimensional Multi-Target Localization in Industrial WSNs for LOS Conditions
## 3 Network Model  
In the ATML scheme, we employ two ANs to determine the 3D relative positions of a number of MSNs in a SIWSN. Let \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) denote the two ANs. Suppose that there are \(K\) MSNs in the network, denoted by \(\mathcal{M}_{k}(k=1,2,\cdots,K)\). The positions of the ANs are fixed and pre-determined. The network model is illustrated in Figure 1.  
The architectures and signal flow of the MT-SIMO scheme on the MSNs and ANs are shown in Figure 2. Each MSN uses an omnidirectional antenna to broadcast its signals, while an AN receives the signals from all the MSNs with an antenna array. Suppose that each AN array comprises \(M\) antenna elements (for easy presentation, we assume that the antenna arrays on both ANs have the same structure, but this is unnecessary and different 2D or crown arrays may be employed). In order to perform 3D localization, the AN antennas should be sector/directional antennas and placed in 2D structures (e.g., rectangular or \"L\"-shaped planar arrays) or 3D structures (e.g., cylindrical or crown arrays). The planar arrays can cover a semisphere because the main lobes of the antennas are all pointing to one side of the array. The cylindrical or crown arrays can cover a whole sphere because the main lobes point to all the directions. The structure of the antenna arrays can be selected according to the deployment of the network. If all the MSNs are located on one side of an AN, we can adopt the planar arrays. However, if the MSNs are located in all directions to an AN, the cylindrical or crown arrays should be adopted. Thus, all the MSNs are visible to the AN antennas and in main lobes. Figure 1: The network model of the ATML scheme for node localization in a SIWSN. There is a common orthogonal coordinate system for the network, which is denoted by \(\mathbb{C}_{N}\). The position of \(\mathcal{M}_{k}\) in \(\mathbb{C}_{N}\) is \((x_{M,k},y_{M,k},z_{M,k})_{C_{N}}\). The positions of the two ANs are \((x_{A,i},y_{A,i},z_{A,i})_{C_{N}}\) for \(\mathcal{A}_{i}\) (\(i=1,2\)). Furthermore, we adopt the spherical coordinate systems for the MSN positions with respect to \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\), which are denoted by \(\mathbb{C}_{1}\) and \(\mathbb{C}_{2}\), respectively. The coordinate systems are defined based on the body frames of the two ANs. The origins are located at the centers of the antenna arrays of the ANs. The \(x\)-\(y\) and \(x\)-\(z\) planes correspond to the azimuth and elevation dimensions, respectively, as shown in Figure 1.  
In complicated scattering industrial environments, signals generally propagate from an MSN to an AN through multiple paths, resulting in the _multipath components_ (MPCs). Suppose that there are \(L_{i,k}\) propagation paths from \(\mathcal{M}_{k}\) to \(\mathcal{A}_{i}\). The parameter set of the \(l\)-th (\(l=1,2,\cdots,L_{i,k}\)) component received on the antennas on \(\mathcal{A}_{i}\) is [26]  
\[\omega_{i,k,l}=\left\{\alpha_{i,k,l},\tau_{i,k,l},\phi_{i,k,l},\theta_{i,k,l}, \
u_{i,k,l}\right\}, \tag{1}\]  
where \(\alpha_{i,k,l}\), \(\tau_{i,k,l}\), \(\phi_{i,k,l}\), \(\theta_{i,k,l}\), and \(\
u_{i,k,l}\) are the complex amplitude (including the path loss and phase shift), TOA (i.e., propagation delay), azimuth of arrival, elevation of arrival, and Doppler frequency, respectively. It is important to note that the ATML scheme requires propagation along the connection directions from each MSN to both ANs. This condition can be satisfied in real industrial environments if there are clear LOS paths or LOS paths blocked by non-metallic materials such as concrete walls and plastic or wood cargoes. The radio signals can penetrate the obstacles and propagate along the LOS directions. For example, the radio propagation penetrating a building is illustrated in [26]. If the LOS paths are blocked by metal materials, the radio signals cannot penetrate the objects. Then, we can deploy more ANs in the environment such that all the MSNs have LOS propagation to two ANs.