# The Energy Consumption and Robust Case Torque Control of a Rehabilitation Hip Exoskeleton
## 4 Results and Discussion
### Walking Activity  
The RMSE, the energy consumption for the walking activity and the distance when using each controller at different torque levels are illustrated in Tables 8-10, respectively. Despite the fact that the regulator in Case 4 was the most robust controller, the RMSEs for STS and walking activity at the various torques were still not the minimum values compared with the other cases. The RMSEs were the highest for STS and walking in Case 4 at all torque levels compared with those for other cases, while the RMSEs for Case 1 were the lowest for STS and walking movements. The reason controller Case 1 had the lowest RMSE values was the longer distance of the poles on the left half of complex planes to the origin. The faster response of the controller occurred when the poles lay far away from the origin or the controller had a large input signal [60]. For this reason, controller Case 1 provided a faster response to the dynamics of the system because its poles lay on (\(P_{1,2}=-49.4\pm 46.3i\), \(P_{3}=-47.8\)) and its \(R=1\times 10^{-6}\), compared with controller Case 4, which had its poles on (\(P_{1,2}=-50.1\pm 93.6i\), \(P_{3}=-19.4\)) with a minimum input signal due to the high penalty value of \(R=1\times 10^{-2}\). Although \(P_{1,2}\) Case 4 was farther away from the origin than Case 1, it still had its dominant pole at (\(P_{3}=-19.4\)), which was closer to the origin than the dominant pole of Case 1 at (\(P_{3}=-47.8\)). Consequently, the system slowed in Case 4, which affected the system's behaviour and speed [61]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Torque Level** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline TL 1 & 1.4 & 0.4 & 0.4 & 0.4 \\ TL 2 & 5.1 & 2.2 & 2.4 & 2.1 \\ TL 3 & 10.9 & 6.6 & 6.9 & 6.5 \\ TL 4 & 19.2 & 15.6 & 16.1 & 15.4 \\ TL 5 & 30.0 & 30.1 & 31.2 & 29.9 \\ \hline \end{tabular}
\end{table} 
Table 6: The energy consumption for sit to stand (milliwatts). \begin{table}
\begin{tabular}{c c c c c} \hline
**Torque Level** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline TL 1 & 108430 & 350970 & 342710 & 356800 \\ TL 2 & 30052 & 69720 & 67194 & 70621 \\ TL 3 & 13727 & 22575 & 21824 & 22929 \\ TL 4 & 7804 & 9630 & 9379 & 9747 \\ TL 5 & 5005 & 4981 & 4813 & 5014 \\ \hline \end{tabular}
\end{table} 
Table 7: The number of sit-to-stand movements with a fully charged battery (step). \begin{table}
\begin{tabular}{c c c c c} \hline
**Torque Level** & **Case 1 \({}^{*}\)** & **Case 2** & **Case 3** & **Case 4** \\ \hline TL 1 & 26.5 & 44.7 & 43.3 & 68.8 \\ TL 2 & 106.3 & 178.9 & 173.5 & 275.2 \\ TL 3 & 239.1 & 402.5 & 390.5 & 619.3 \\ TL 4 & 425.1 & 715.5 & 694.2 & 1100.1 \\ TL 5 & 664.1 & 1100.1 & 1100.2 & 1700.1 \\ \hline \end{tabular}
\end{table} 
Table 5: The RMSE for sit to stand (\(\times 10^{-6}\) Nm/kg). Interestingly, controller Case 4 in the walking activity did not have the lowest consumed energy at all torque levels as it showed in STS. The differences are obvious in Figure 7 and Table 9; in particular, the controller Case 1 and Case 4 had the lowest energy consumption at the speed of 0.5 m/s = \(TL1\)) and the second-lowest energy consumption at a speed of 2.6 m/s = \(TL5\) for Case 4; still, the controller Case 1 had the lowest energy consumption at speeds of 0.5 m/s = \(TL1\) and 2.6 m/s = \(TL5\), respectively. Therefore, we obtained the longest distance for the walking activity in Case 1 at speeds of 0.5 m/s = \(TL1\) and 2.6 m/s = \(TL5\), as shown in Table 10, compared to in other cases. Another notable feature of the walking activity was that the most extended distances were reached in Case 2 at speeds of 1.1-2.1 m/s = \(TL2\)-\(TL4\) compared with the other cases for these speeds. The benefit of obtaining the longest distances was caused by the minimal energy consumed in Case 2 at these speeds (see Tables 9 and 10). This finding supported the hypothesis that the highest penalty of the \(R\) matrix in the quadratic cost function led to instability in walking activity at certain speeds.