# Target Localization via Integrated and Segregated Ranging Based on RSS and TOA Measurements
## 2 Problem Formulation
### R-GTRS Algorithm  
Unlike [21], where the authors first treated all links as LOS to later apply an alternating optimization approach and improve the location estimate and the mean NLOS bias estimate in an iterative fashion, the authors in [14] took a different (single-step) approach. By treating all links as NLOS and NLOS biases as nuisance parameters whose upper bound on the magnitude is assumed (imperfectly) known, they mitigated their negative influence by resorting to a robust, worst-case scenario, approach. First, \(\frac{b_{\max}}{2}\) was added to both sides of (1a) and \(\beta_{\max}/2\) was subtracted from both sides of (1b). \[\tilde{P}_{i}=P_{0}-\tilde{b}_{i}-10\gamma\log_{10}\frac{\|\mathbf{x}-\mathbf{a}_{i} \|}{d_{0}}+n_{i}, \tag{17a}\] \[\tilde{d}_{i}=\|\mathbf{x}-\mathbf{a}_{i}\|+\tilde{\beta}_{i}+m_{i}, \tag{17b}\]  
with \(\tilde{P}_{i}=P_{i}+b_{\max}/2\), \(\tilde{b}_{i}=b_{i}-b_{\max}/2\), \(\tilde{d}_{i}=d_{i}-\frac{\beta_{\max}}{2}\) and \(\tilde{\beta}_{i}=\beta_{i}-\frac{\beta_{\max}}{2}\). After doing some simple manipulations with (17a), the first order Taylor series approximation for small noise was applied to this equation. The so derived equation was then squared, together with (17b). After disregarding the second-order noise terms and applying a WLS criterion, the following min-max problem was obtained:  
\[\underset{\mathbf{x}}{\text{minimize}}\ \underset{\rho_{i}}{ \text{maximize}}\ \sum_{i=1}^{N}\left(\frac{v_{i}^{2}\|\mathbf{x}-\mathbf{a}_{i}\|^{2}-\rho_{i}^{2}}{2 v_{i}\|\mathbf{x}-\mathbf{a}_{i}\|}\right)^{2}, \tag{18a}\] \[\underset{\mathbf{x}}{\text{minimize}}\ \underset{\tilde{\beta}_{i}}{ \text{maximize}}\ \sum_{i=1}^{N}\left(\frac{(\tilde{d}_{i}-\tilde{\beta}_{i})^{2}-\|\mathbf{x}-\mathbf{ a}_{i}\|^{2}}{2\|\mathbf{x}-\mathbf{a}_{i}\|}\right)^{2}, \tag{18b}\]  
which can be written as  
\[\underset{\mathbf{x}}{\text{minimize}}\ \underset{\rho_{i}}{ \text{maximize}}\sum_{i=1}^{N}f^{2}(\rho_{i}),\ \ \text{where}\ \ f(\rho_{i})=\frac{\left|v_{i}^{2}\|\mathbf{x}-\mathbf{a}_{i}\|^{2}-\rho_{i}^{2} \right|}{2v_{i}\|\mathbf{x}-\mathbf{a}_{i}\|}, \tag{19a}\] \[\underset{\mathbf{x}}{\text{minimize}}\ \underset{\tilde{\beta}_{i}}{ \text{maximize}}\sum_{i=1}^{N}f^{2}(\tilde{\beta}_{i}),\ \ \text{where}\ \ f(\tilde{\beta}_{i})=\frac{\left|(\tilde{d}_{i}-\tilde{\beta}_{i})^{2}-\| \mathbf{x}-\mathbf{a}_{i}\|^{2}\right|}{2\|\mathbf{x}-\mathbf{a}_{i}\|}. \tag{19b}\]  
By noticing that  
\[\left|\tilde{b}_{i}\right|=\left|b_{i}-\frac{b_{\max}}{2}\right|\leq\frac{b_{ \max}}{2},\ \ \left|\tilde{\beta}_{i}\right|=\left|\beta_{i}-\frac{\beta_{\max}}{2}\right|\leq \frac{\beta_{\max}}{2}, \tag{20}\]  
and that  
\[\underset{\rho_{i}}{\text{maximize}}\sum_{i=1}^{N}f^{2}(\rho_{i}) =\sum_{i=1}^{N}\left[\underset{\rho_{i}}{\text{maximize}}f(\rho_{i })\right]^{2}, \tag{21a}\] \[\underset{\tilde{\beta}_{i}}{\text{maximize}}\sum_{i=1}^{N}f^{2}( \tilde{\beta}_{i}) =\sum_{i=1}^{N}\left[\underset{\tilde{\beta}_{i}}{\text{maximize}}f( \tilde{\beta}_{i})\right]^{2}, \tag{21b}\]the authors in [14] first solve the maximization problems under certain conditions, where they got two possible solutions. By applying these two solutions, and joining the RSS and TOA branches together into a single estimator, the following robust estimation problem was derived:  
\[ \begin{split}&\underset{x}{\text{minimize}}\sum_{i=1}^{N}\left( \frac{\upsilon_{i}^{2}\|\mathbf{x}-\mathbf{a}_{i}\|^{2}-\mu}{2\upsilon_{i}\|\mathbf{x}- \mathbf{a}_{i}\|}\right)^{2}+\sum_{i=1}^{N}\left(\frac{\upsilon_{i}^{2}\|\mathbf{x}- \mathbf{a}_{i}\|^{2}-\eta}{2\upsilon_{i}\|\mathbf{x}-\mathbf{a}_{i}\|}\right)^{2}\\ &+\sum_{i=1}^{N}\left(\frac{d_{i}^{2}-\|\mathbf{x}-\mathbf{a}_{i}\|^{2}}{2 \|\mathbf{x}-\mathbf{a}_{i}\|}\right)^{2}+\sum_{i=1}^{N}\left(\frac{(\tilde{d}_{i}- \frac{\beta_{\max}}{2})^{2}-\|\mathbf{x}-\mathbf{a}_{i}\|^{2}}{2\|\mathbf{x}-\mathbf{a}_{i}\|} \right)^{2}.\end{split}