## 4 Performance Evaluation
### Performance Parameters  
The metrics used to evaluate the algorithms' performance using MATLAB simulation and hardware prototype using Node MCU are average position error, standard deviation of localization error, localization coverage ratio, path length, and power consumption. 1. Average localization error. Localization error is an important parameter for evaluating any proposed path model. Two metrics are applied to find the localization error of the proposed path models, average localization error, and the standard deviation of the localization error. The average localization error [30] is calculated using (9):  
\[\text{localization error}_{\text{j}}=\sqrt{\left(\text{x}_{\text{j}}-\text{ est}_{\text{j}}\right)^{2}+\left(\text{y}_{\text{j}}-\text{est}_{\text{j}} \right)^{2}} \tag{9}\]  
where \((\text{x}_{\text{j}},\text{y}_{\text{j}})\) is the real location of node j, and \(\left(\text{est}_{\text{j}},\text{est}_{\text{j}}\right)\) is the estimated location of node j using the localization models. Hence, the average localization error is:  
\[\text{localization error}_{\text{av}}{=}\sum_{\text{j}=1}^{\text{n}}(\text{ localization error}_{\text{j}})/\text{n} \tag{10}\]  
where n is the total number of unknown nodes. 2. Standard deviation of position error. We compute the standard deviation of position error to estimate how it differs from the average position error. If the value of the standard deviation of position error is low, the estimated position error values of nodes tend to vary near to the average position error of all nodes in the network. The equation of standard deviation of position error is as follows:  
\[\text{position error}_{\text{std}}{=}\sum_{\text{j}=1}^{\text{n}}\left(\text{ position error}_{\text{j}}-\text{position error}_{\text{av}}\right)^{2} \tag{11}\]  
where n is total number of unknown nodes, position error, is the position error of node j, and the position error\({}_{\text{av}}\) is the average position error. 3. Coverage ratio. Coverage ratio is the percentage of nodes localized successfully in the network. 4. Number of pausing points. The mobile node stops at each pausing point for a predefined period in each algorithm. Increasing the number of pausing points impacts on networking time and power consumption. 5. Path length. The path length is defined as the distance wrapped by the mobile anchor to follow the suggested path model in the network. The path length does not have influence on the position error. However, it impacts the power consumed by the mobile node and the overall time of the localization procedure. The unpretentious solution for decreasing the power consumption is to reduce the path length covered by the mobile anchor node. Thepath length of Arrow-Curve and R-Curve are compared to that of H-Curve trajectory [33] using the following equations:  
\[\text{Pathlength of H}-\text{Curve}=\frac{\text{A}^{2}}{\text{L}}+18\text{ L} \tag{12}\]  
where A is the network area and L is the resolution as defined in previous sections. \[\text{Path length of Arrow}-\text{Curve}=\left(2\,*\,\text{L}+2\sqrt{2}\, \text{L}\,\right)\,*\,4^{\text{1e}-1}+10\,\text{L}-10\sqrt{2}\,\text{L} \tag{13}\]  
\[\text{Path\_length of R}-\text{Curve}=\left(4\,*\,\text{L}+\sqrt{2}\,\text{L} \,\right)\,*\,4^{\text{1e}-1}+4\,*\,\text{L} \tag{14}\]  
6. Energy or power consumption. Energy or power consumption is an important metric to be taken into consideration when proposing path models. However, the path models that take energy or power consumption parameter into consideration in literature are limited [33]. Most of the energy or power consumed by the mobile anchor node is dissipated while traveling through the network.