## 2 Bearing-Only Localization in 3D Space  
As shown in Figure 1, the location of target \(\mathbf{T}=\begin{bmatrix}x_{t}&y_{t}&z_{t}\end{bmatrix}^{\prime}\) is unknown in 3D space. \(\mathbf{T}\) needs to be estimated based on the azimuth \(\hat{\phi}_{i}\) and elevation angle \(\hat{\theta}_{i}\) measurements collected from a mobile sensor \(\mathbf{S}_{i}=\begin{bmatrix}x_{s_{i}}&y_{s_{i}}&z_{s_{i}}\end{bmatrix}^{ \prime}\) (\(i=1,\ldots,n\)) at \(n\) locations. \(\mathbf{t}\) and \(\mathbf{s_{i}}\) are the coordinates of the target and the sensor on \(xy\) plane, respectively. The true bearing information \(\phi_{i}\) and \(\theta_{i}\) received by the sensor from the target are:  
\[\phi_{i}=\tan^{-1}\left(\frac{y_{t}-y_{s_{i}}}{x_{t}-x_{s_{i}}}\right), \tag{1}\]  
\[\theta_{i}=\tan^{-1}\left(\frac{z_{t}-z_{s_{i}}}{\sqrt{(x_{t}-x_{s_{i}})^{2}+(y _{t}-y_{s_{i}})^{2}}}\right). \tag{2}\]  
The bearing information \(\hat{\phi}_{i}\) and \(\hat{\theta}_{i}\) are the sensor measurements corrupted by the noise \(e_{i}\) and \(\delta_{i}\) at n locations, i.e.,  
\[\hat{\phi}_{i}=\phi_{i}+e_{i}, \tag{3}\]\[\hat{\theta}_{i}=\theta_{i}+\delta_{i}, \tag{4}\]  
where \(e_{i}\) and \(\delta_{i}\) are independent identically distributed white Gaussian noise with zero mean and unknown variance of \(\sigma^{2}\) and \(\xi^{2}\), respectively.  
The problem of target localization in 3D space is solved in two steps. Firstly, the bearing line is projected onto the \(xy\) plane. The corresponding 2D estimator is used to solve the resulting 2D positioning problem. Then the estimated position of \(\mathbf{T}\) along the \(z\) axis is obtained from the estimated \(x\) and \(y\) coordinates. The 2D pseudolinear estimator (PLE) algorithm is used to estimate \(\mathbf{t}\) by using \(n\) azimuth measurements \(\hat{\phi}_{i}\). To turn a nonlinear problem into a linear one, small noise assumption also appears in the development and analysis of the Stansfield algorithm [26] and the total-least-squares solution [27] for localization. The pseudolinear equation from (1) and (3) between \(\mathbf{t}\) and the given information is described as:  
\[\sin\hat{\phi}_{i}x_{t}-\cos\hat{\phi}_{i}y_{t}=\sin\hat{\phi}_{i}x_{s_{i}}- \cos\hat{\phi}_{i}y_{s_{i}}+||\mathbf{t}-\mathbf{s}_{i}||\sin e_{i}, \tag{5}\]  
which can be expressed in matrix form as:  
\[\mathbf{A}_{n}\mathbf{t}=\mathbf{B}_{n}+\mathbf{n}_{n} \tag{6}\]  
where  
\[\mathbf{A}_{n} =\begin{bmatrix}\sin\hat{\phi}_{1}&-\cos\hat{\phi}_{1}\\ \vdots&\vdots\\ \sin\hat{\phi}_{n}&-\cos\hat{\phi}_{n}\end{bmatrix},\mathbf{B}_{n}=\begin{bmatrix} \sin\hat{\phi}_{1}x_{s_{1}}-\cos\hat{\phi}_{1}y_{s_{1}}\\ \vdots\\ \sin\hat{\phi}_{n}x_{s_{n}}-\cos\hat{\phi}_{n}y_{s_{n}}\end{bmatrix},\] \[\mathbf{n}_{n} =\begin{bmatrix}||\mathbf{t}-\mathbf{s}_{1}||&0&0\\ 0&\ddots&0\\ 0&0&||\mathbf{t}-\mathbf{s}_{n}||\end{bmatrix}\begin{bmatrix}\sin e_{1}\\ \vdots\\ \sin e_{n}\end{bmatrix}. \tag{7}\]  
The 2D PLE estimator of (6) is:  
\[\mathbf{\hat{t}_{ple}}=(\mathbf{A}_{n}^{{}^{\prime}}\mathbf{A}_{n})^{-1} \mathbf{A}_{n}^{{}^{\prime}}\mathbf{B}_{n}, \tag{8}\]  
where \(\mathbf{A}_{n}^{{}^{\prime}}\) represents the transpose of \(\mathbf{A}_{n}\).  
Figure 1: Target and sensor in \(\mathbb{R}^{3}\).  
The target position of the projection in 2D space has been estimated. According to the geometric relationship, the target position in the \(z\) axis direction can be estimated according to the elevation angle measurements and other information as:  
\[z_{0}=\frac{1}{n}\sum_{i=1}^{n}\left(z_{i}+\sqrt{\left(x_{0}-x_{s_{i}}\right)^{2 }+\left(y_{0}-y_{s_{i}}\right)^{2}}\tan\hat{\theta}_{i}\right), \tag{9}\]  
where \(x_{0}\) and \(y_{0}\) are the target position estimated by different methods in 2D space.