## 3 Neural Inverse Optimal Control
### Real-Time Results  
The work [17] presents a NOIC for the Modified HD2(r) Treaded ATR Tank Robot Platform with wireless communication (Figure 3), this implementation uses an RHONN identifier as (2) to identify the unknown model of the HD2(r). The obtained model is then used as the based to synthesize the control law using the inverse optimal control approach. In [17] the values of \(\mathbf{P}_{1}\) and \(\mathbf{P}_{2}\) are defined in (21) for real-time operation. The following results show a comparison obtained with the values from [17] and \(\mathbf{P}_{1}^{*}\) and \(\mathbf{P}_{2}^{*}\) in (22) found by the GCO algorithm. In this section, there are presented two experiments named as \"test 1\" and \"test 2\", respectively.  
\[\mathbf{P}_{1}=72000\left[\begin{array}{ccc}162&1&2\\ 1&162&3\\ 2&3&162\end{array}\right]\quad\mathbf{P}_{2}=10000\left[\begin{array}{cc}20 &0\\ 0&20\end{array}\right] \tag{21}\]  
\[\mathbf{P}_{1}^{*}=1\times 10^{7}\left[\begin{array}{ccc}3.2039&0&4\\ 0&3.2039&0\\ 4&0&3.2039\end{array}\right]\quad\mathbf{P}_{2}^{*}=1\times 10^{4}\left[ \begin{array}{ccc}2.4128&0\\ 0&2.4128\end{array}\right] \tag{22}\]  
Figures 9-11 show the tracking performance and the error comparison for the position \(x\), position \(y\) and position \(\theta\) of the experimental test 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**RMSE** & \(x\) **(m)** & \(y\) **(m)** & \(\theta\) **(rad)** \\ \hline NIOC [17] & **0.0260** & 0.0362 & **0.0158** \\ Super Twisting & 0.0317 & **0.0036** & 0.0652 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Tracking error comparison of NIOC [17] and a Super Twisting controller. Bold values highlight the best result.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Control** & **RMSE \(x\) (m)** & **RMSE \(y\) (m)** & **RMSE \(\theta\) (rad)** & **Total** \\ \hline NIOC [17] & **0.0073** & 0.0043 & 0.0064 & 0.0180 \\ NIOC with GCO & **0.0073** & **0.0025** & **0.0049** & **0.0147** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Root Mean Squared Error (RMSE) in states for second simulation. Bold values highlight the best result.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control** & **RMSE \(x\) (m)** & **RMSE \(y\) (m)** & **RMSE \(\theta\) (rad)** & **Total** \\ \hline NIOC [17] & **0.0077** & 0.0055 & 0.0064 & 0.0195 \\ NIOC with GCO & 0.0079 & **0.0041** & **0.0049** & **0.0169** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Root Mean Squared Error (RMSE) in states for first simulation. Bold values highlight the best result.  
Table 5 shows the RMSE of each state variable and their total which is the evaluation of Equation (18) for the experimental test 1.  
Figure 11: Tracking of \(\theta\) (**left**) and error comparison (**right**) for the experimental test 1.  
Figure 10: Tracking of \(y\) position (**left**) and error comparison (**right**) for the experimental test 1.  
Figure 9: Tracking of \(x\) position (**left**) and error comparison (**right**) for the experimental test 1.  
Figures 12-14 show the tracking performance and the error comparison for the position \(x\), position \(y\) and position \(\theta\) of the experimental test 2.  
Table 6 shows the RMSE of each state variable and their total which is the evaluation of Equation (18) for the experimental test 2.  
Figure 14: Tracking of \(\theta\) (**left**) and error comparison (**right**) for the experimental test 2.  
Figure 12: Tracking of \(x\) position (**left**) and error comparison (**right**) for the experimental test 2.  
Figure 13: Tracking of \(y\) position (**left**) and error comparison (**right**) for the experimental test 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control** & **RMSE \(x\) (m)** & **RMSE \(y\) (m)** & **RMSE \(\theta\) (rad)** & **Total** \\ \hline NIOC [17] & 0.0238 & 0.0109 & **0.0267** & 0.0614 \\ NIOC with GCO & **0.0152** & **0.0043** & 0.0270 & **0.0465** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Root Mean Squared Error in states for Real-Time experimental test 1. Bold values highlight the best result.