## 4 Experimental Investigation  
In this paper, various experiments of a single rod pneumatic cylinder system are considered. As shown in Figure 3, the middle branch of 3-RPS platform was considered as the object. The single pneumatic system was with time-varying inertia as it tracks the designed trajectory. A PXI-8106 RT controller with a data acquisition card (DAQ) was set up as the control system. A potentiometer and pressure sensors were used to measure the displacement of piston and pressure inside the cylinder. Velocity and acceleration of piston were estimated by a nonlinear tracking differentiator [35]. Experimental setup specifications are listed in Table 1. The sampling frequency of the controller was set to 1 kHz. For better testifying the proposed method, we compared the following three control algorithms: deterministic robust control (DRC), ARC, ARNNC controllers. For simplicity, define \(k_{1}=\lambda_{1}\), \(k_{2}=\lambda_{2}+\lambda_{2s},k_{3}=\lambda_{3}+\lambda_{3s}\)[36]. All the parameters of the controllers and system are shown in Table 2. The ARNNC controller had 48 neurons for each neuron network. The center \(c_{j}\) vector was fixed and distributed evenly on the reference trajectory and \(d\) was set to 0.015 for both networks. Several commonly used error indexes were used to qualify the tracking performance:  
\(s_{1 rms}=\left(\frac{1}{12}\int_{T_{f-12}}^{T_{f}}|s_{1}|_{2}dt\right)^{\frac {1}{2}}(\text{mm})\), the root mean square error during the last 12 s. \(T_{f}\) is the total experimental running time. \(s_{1m}=\max[|s_{1}|]_{T_{f}-12<t<T_{f}}(\text{mm})\), the maximal absolute value of tracking errors during the last 12 s. \(s_{1r}=s_{1 rms}/A_{f}\) is the relative error where \(A_{f}\) is the amplitude of sinusoidal trajectory. \begin{table}
\begin{tabular}{c c} \hline
**Device** & **Description** \\ \hline \multirow{4}{*}{Cylinder} & FESTO DNC-63-200-P \\  & Stroke: 200 mm \\  & Insider Diameter: 63 mm \\ Proportional directional valve & FESTO MPYE-5-1/4010B \\ Controller & NI PXI-8106 \\ DAQ & NI PXI-6251 \\ Potentiometer & KTC-200 \\  & Resolution: 0.075 mm \\ Pressure sensor & MPX-5700-AP \\ Tank & \(320\times 1200\) \\  & Volume: 100 L \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental Specifications. \begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **ARNNC** & **ARC** & **DRC** \\ \hline Initial Value of \(\left[\hat{0}_{1},\hat{0}_{2},\hat{0}_{3},\hat{0}_{4}\right]\) & [0.66, \(-9.8,0,0\)] & [0.66, \(-9.8,0,0\)] & [0.66, \(-9.8,0,0\)] \\ \(\left[\delta_{1},\delta_{2},\delta_{3},\delta_{4}\right]\) & [0.91, 10.5, 7.5, 7.5] & [0.91,10.5, 7.5, 7.5] & [0.91, 10.5, 7.5, 7.5] \\ \(\left[\delta_{1 mm}/\delta_{1 mm}\right]\) & [0.41, 0.91] & [0.41, 0.91] & [0.41, 0.91] \\ \(\hat{W}_{0}(0)\) & 0 for all neurons & & \\ \(\hat{W}_{4}(0)\) & 0 for all neurons & & \\ \(\hat{\delta}_{p}\) & 0.35 & & \\ \(\delta_{q}\) & 30 & & \\ \(\left[k_{1},k_{2},k_{3}\right]\) & [25, 1000, 100] & [25, 1000, 100] & [25, 1000, 100] \\ \(\text{\emph{u}}_{2},\text{\emph{u}}_{3}\) & 11, 1 & [1, 1] & \\ _Diagonal of \(\sum\)_ & [0.015, 5, 150, 150] & [0.015, 5, 150, 150] & \\ \(\Lambda_{p}\) & 0.5148 & & \\ \(\Lambda_{q}\) & 50148 & & \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter setting of controllers. Figure 3: Experimental apparatus of the pneumatic system. Case study I--Smoothed step: Experiments were carried out, for a single cylinder of smoothed step trajectory from \(-90\) mm to \(90\) mm back and forth, and the results are described in Figure 4. As seen in the figure, all three controllers worked well for tracking the smoothed step trajectory. The steady-state error of the DRC controller was about \(1\) mm from \(-90\) mm to \(90\) mm and \(0.5\) mm for the return stroke. Due to the inability to adapt to the parameters, both the transient response and steady-state error of the DRC controller were the largest of the three controllers.