## 3 Distributed Positioning  
### The Position Estimation  
The numerical estimation of the mobile tag position is based on signal magnitude level measurements, and is generally pursued by minimizing an objective function. Typical objective functions are expressed by sums of squared residuals, likelihood functions, posterior density functions, risk functions, as well as robust loss functions, with the two most common being the Least Squares (LS) and the Maximum Likelihood (ML) functions [1].  
In this paper, we focus on an algorithm based on a derivation of the ML method, whose estimation only requires the assumption of a first order knowledge of the range measurement probability distribution function. It is based on the minimization of the calculated variance for both measured and expected data from an unknown position, [16]. Its operating principle assumes the acquisition of a  
Figure 5: Experimental target node (**a**) and its radiation pattern (**b**).  
coherent set of signal levels during network communication between the mobile tag and a generic number, N, of available anchors. The collected data consist of a steering vector, containing all the RSSI values obtained from each antenna element of each anchor composing the constellation; the steering vector is defined as:  
\[\hat{S}=\left(\begin{array}{c}\hat{S_{1}}\\ \hat{S_{2}}\\ \vdots\\ \hat{S_{N}}\end{array}\right) \tag{1}\]  
where \(\hat{S_{i}}\) denotes the steering vector of the \(i\)-th anchor, with:  
\[\hat{S_{i}}=\left(\begin{array}{c}s_{1i}\\ s_{2i}\\ \vdots\\ s_{Mi}\end{array}\right) \tag{2}\]  
and where \(s_{ki}\) is the RSSI collected by the \(k\)-th antenna of the \(i\)-th anchor, with \(M\) the number of SBA antenna elements. The positioning technique adopted in this work is hereinafter descripted, considering the \(i\)-th steering vector composed by the RSSI data obtained from the \(M\) antenna elements composing the single anchor node. The position estimation relies on the assumption of a known mobile tag height \(z_{TAG}\) (\(z\)-axis), used to define the steering vector reference map, \(M\)(\(x\),\(y\)). The latter is obtained as the projection of the SBA patterns throughout the (\(x\),\(y\)) localization domain calculated by a first-order propagation model, [16], between a generic tag position (\(x\),\(y\)) and the anchor known position. This is characterized by the path-loss model 20 \(\log\left(\frac{\mathrm{D_{i}}}{\lambda_{0}}\right)\), with \(\mathrm{D_{i}}\) the distance between the mobile tag at (\(x\),\(y\)) and the \(i\)-th anchor node, and \(\lambda_{0}\) the free-space wave length at the operative frequency. Furthermore, it is assumed that between the anchors and the mobile tag persists a line-of sight. Under these realistic hypothesis, the tag position estimation \((\hat{x},\hat{y})\) is obtained by minimizing the variance of the difference between the steering vector and the expected signal map vector across the entire analysis domain:  
\[(\hat{x},\hat{y})=\mathrm{argmin}_{(x,y)}\mathrm{C}(x,y), \tag{3}\]  
with  
\[\mathrm{C}(x,y)=\mathrm{var}_{i}\Big{\{}\hat{S}-\mathrm{M}(x,y)\Big{\}}. \tag{4}\]  
In Equation (4), the variance of the difference between the steering vector and the expected signal map for the actual position is given by:  
\[\mathrm{C}(x,y)=\frac{1}{MN}\sum_{i=1}^{MN}\!\!\left[(s_{i}-m_{i})-\frac{1}{ MN}\sum_{j=1}^{MN}\!\!\left(s_{j}-m_{j}\right)\right]^{2} \tag{5}\]  
As demonstrated in [16], the robustness of the estimator in Equation (5) is able to counterbalance the approximation introduced by the map model and the large variability of possible user node heights.