# Retractable Wireless Charging Windings for Inspection Robots
## 3 Design and Simulation of Retractable Wireless Power Transfer Coils
### Formation of Figure of Merit  
In assessing different WPT coil shapes for mobile robots, a figure of merit (FOM) is created to gauge the energy efficiency, the coil misalignments, and its feasibility for practical implementation. FOM derivation from the metric components alongside their respective weights can be presented in Equation (9). The cumulative sum of these weights equals 1,  
Figure 5: Coil group by shape. (**a**) Group 1: triangles. (**b**) Group 2: four sides. (**c**) Group 3: other polygons.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Coil Shapes** & \(\mathbf{r_{\text{coil}}}\)**, \(\mathbf{\Omega}\)** & \(\mathbf{M}\), \(\mathbf{\mu}\)**H** & \(\mathbf{k}\) & \(\mathbf{P_{\text{out}}}\), \(\mathbf{W}\) & \(\mathbf{\eta}\), \% \\ \hline \multicolumn{6}{c}{Group 1: Triangle (Number of edges, \(N=3\))} \\ \multicolumn{6}{c}{Equilateral} & 0.3510 & 0.805 & 0.0142 & 146.65 & 73.32 \\ Right-angled & 0.3703 & 0.781 & 0.0133 & 140.84 & 70.42 \\ \hline \multicolumn{6}{c}{Group 2: 4-sided (Number of edges, \(N=4\))} \\ \multicolumn{6}{c}{Square} & 0.4112 & 2.37 & 0.0318 & 181.33 & 90.67 \\ Rectangle 2:1 ratio & 0.3254 & 0.796 & 0.0145 & 150.01 & 75.01 \\ \hline \multicolumn{6}{c}{Group 3: Other polygon (Number of edges, \(N=5\), 6, 8, 16, 32)} \\ Pentagon & 0.3799 & 2.09 & 0.0299 & 181.23 & 90.61 \\ Hexagon & 0.3653 & 1.96 & 0.0290 & 180.87 & 90.44 \\ Octagon & 0.3481 & 1.85 & 0.0281 & 180.73 & 90.37 \\ Circle 16 edges & 0.3387 & 1.77 & 0.0272 & 180.33 & 90.17 \\ Circle 32 edges & 0.3363 & 1.76 & 0.0270 & 180.31 & 90.16 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulated results for geometrical coil shapes with fixed input power of 200 W, five coil turns.  
serving as a normalization technique. The pivotal factor lies within Metric A (\(M_{a}\)), wherein the WPT performance heavily relies on the fundamental parameter of the WPT followed by Metric B (\(M_{b}\)) and Metric C (\(M_{c}\)):  
\[FoM_{WPT}=\sum_{i=1}^{3}M_{i}\cdot W_{i} \tag{9}\]  
Metric A (power related): This represents the assessment on energy efficiency and output voltage:  
\[M_{a}=aW+b(P_{out})+c(\eta)+d(V_{out}) \tag{10}\]  
where \(a,b,c\), and \(d\) are weights assigned to each component. \(W\) is a measure based on the coil parameter (\(L\), \(M\), \(R\), and \(N\)). \(P_{out}\) evaluates the output power, \(\eta\) evaluates the energy efficiency, and \(V_{out}\) evaluates the output voltage relative to the target.  
Metric B (dimension related): represents the evaluation framework based on the described dimension-related aspects:  
\[M_{b}=e(NE)+f(SR)+g(CR) \tag{11}\]  
These weights assigned (\(e,f\), and \(g\)) to each component should sum up to 1. \(NE,SR\), and \(CR\) are normalized scores or factors representing the evaluation of the number of coil edges, coil shape ratio defined by dividing the area of a coil shape by its perimeter, and compression ratio of fully extended to fully retracted, respectively.  
Metric C (mass related): This captures the overall evaluation based on mass-related considerations, combining the calculated coil mass, coil weight, and a measure of coil management ease or difficulty into a single metric:  
\[M_{c}=h(CM)+k(WW)+l(CMT) \tag{12}\]  
\(h,k\) and \(l\) are weights assigned to each component. \(CM\) is the calculated mass of the coil based on the perimeter and number of turns, \(WW\) is the weight of the coil, and \(CMT\) forms are graded on a scale from 1 to 10, indicating the ease or difficulty of managing coil entanglement during extension and compression states.  
Figure 6 illustrates the relationship between the component metric concerning individual weight, highlighting the optimal choice in selecting the geometric coil shape based on the previously described linear aggregation method. The square coil shows the highest score among other coil shape variations with a score of 9.73.  
Figure 6: FOM with component metric and corresponding weights for different geometrical shapes WPT coils.