## 5 Experimental Results
### Optimization and Performance Indices  
MATLAB and Simulink are used to complete the simulations. Simulations are conducted based on the identified dynamic model (8)-(12), which represents the gantry platform. All four optimization methods are tested: (1) PSO, (2) IWO, (3) GWO, and (4) BBO. These optimization algorithms are employed to tune the parameters in the FOFPID controller based on the ITAE criterion. To evaluate the ITAE for the four contour types, the average is calculated by  
\[\mathrm{ITAE}=\frac{\mathrm{ITAE}_{\mathrm{C}}+\mathrm{ITAE}_{\mathrm{B}}+ \mathrm{ITAE}_{\mathrm{S}}+\mathrm{ITAE}_{\mathrm{H}}}{\mathbf{4}} \tag{29}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Trajectory Type** & **NURBS Parameters** \\ \hline Circle & \(d=2\), \(P=\) [(0,0),(0, 25,000), (\(-\)50,000, 25,000),(\(-\)50,000, 0), (\(-\)50,000, \(-\)25,000), (0, \(-\)25,000), (0, 0)], \(k=\) [0, 0, 0, 0, 0.25, 0.5, 0.5, 0.75, 1, 1, 1], \(w\) [1, 0.5, 0.5, 1, 0.5, 0.5, 1] \\  & \(d=2\), \(P=\) [(0,0),(-30,000, 20,000), (\(-\)20,000, 50,000), (0, 36,000), (20,000, 50,000), (30,000, 10,000), (0,0)], \(k=\) [0, 0, 0, 0, 0.25, 0.5, 0.5, 0.75, 1, 1, 1], \(w\) [1, 1, 1, 1, 1, 1, 1] \\ Bow & \(d=2\), \(P=\) [(0,0),(-15,000, -15,000), (\(-\)15,000, 15), (0, 0), (15,000, \(-\)15,000, 15,000), (15,000, 150,000), (0, 0)], \(k=\) [0, 0, 0, 0, 0.25, 0.5, 0.5, 0.75, 1, 1, 1], \(w\) [1, 2, 5.5, 2, 5.5, 1] \\  & \(d=2\), \(P=\) [(0, 30,000), (\(-\)2500, -30,000), (\(-\)7500, 20,000), (\(-\)20,000, 20,000), (\(-\)1000, 10,000), (\(-\)12,500, 0), (0, 0)], \(k=\) [0, 0, 0, 0.1, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.8, 1, 1, 1], \(w\) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] \\ \hline \hline \end{tabular}
\end{table}
Table 2: NURBS parameters for the four contours.  
where the subscripts of \(\rm ITAE_{X}\) denote the contour types circle, bow, star, and heart, respectively. The number of iterations for each scheme is limited to 200. Table 3 shows the parameters used in the optimization process.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Optimization Algorithm** & **Parameters** \\ \hline PSO & \begin{tabular}{l} population size: \(N=30\) \\ leaning parameters: \(c_{1}=1.2\), \(c_{2}=1.2\) \\ upper/lower bounds of the random velocity weight: \(v_{max}^{j}=0.72\), \(v_{min}^{j}=-0.72\) \\ upper/lower bound of the position: \(x_{max}^{j}=1.9\), \(x_{max}^{j}=0.1\) \\ population size: \(N=30\) \\ maximum group size \(iter_{max}=30\) \\ \end{tabular} \\ IWO & \begin{tabular}{l} nonlinear harmonic parameter \(n=2\) \\ initial standard deviation \(\sigma_{l}=0.5\) \\ final standard deviation \(\sigma_{l}=0.001\) \\ upper/lower bound of weed seeds: \(1.9/0.1\) \\ \end{tabular} \\ GWO & \
\begin{tabular}{l} population size: \(N=30\) \\ upper/lower bound of wolf group(alpha): \(1.9/0.1\) \\ number of habitats: \(N=30\) \\ \end{tabular} \\ BBO &
\begin{tabular}{l} max/min migration rates: \(E=I=1\), \(m_{max}=0.1\) \\ upper/lower bound of habitat species: \(1.9/0.1\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the optimization algorithms.  
Figure 8: Reference contours: (**a**) star, (**b**) bow, (**c**) circle, and (**d**) heart (\(\rm\upmu\)m).  
Two indexes, the tracking error standard deviation (TESD) and average tracking error (ATE) are employed to evaluate the performances of both the simulations and experiments. These indexes are defined by  
1. ATE: \[m=\sum_{k=1}^{n}\frac{e(k)}{n}\] (30) with \(e(k)=\sqrt{e_{x}^{2}(k)+e_{y}^{2}(k)}\) where \(e_{x}(k)\) and \(e_{y}(k)\) are the tracking errors in the X- and Y-axis. As there are two slides on the Y-axis, the tracking error becomes \(e_{y}(k)=\frac{e_{m}(k)+e_{y}(k)}{2}\), and \(n\) denotes the number of all contour points.
2. TESD: \[T_{s}=\sqrt{\frac{\sum_{k=1}^{n}\left(T(k)-m\right)^{2}}{n}}\] (31)  
The ATE and TESD indices are often used to measure contour tracking performance. The former is used to measure the capability of trajectory tracking, while the latter can evaluate the oscillation.