# The Energy Consumption and Robust Case Torque Control of a Rehabilitation Hip Exoskeleton
## 4 Results and Discussion  
For the robustness analyses of the system, we checked the gain (\(GM\)) and phase margins (\(PM\)), as the margin is one way to specify how much uncertainty the controller design can handle before the system becomes unstable [59]. Therefore, we checked the pure gain margin/classical gain margin (\(CGM\)) and pure phase margin/classical phase margin (\(CPM\)) for all controller cases that examined the controller's sensitivity to different uncertainties. Table 4 shows, for instance, that the controller Case 1 was still stable if the gain variations in a real system were up to 33.2 dB higher than those in our designed model. The same was valid for stability if the phase variations in the real system were up to 178.9 deg higher than those in the designed model. However, this variation was valid only if there was a \(CGM\) or \(CPM\) only, which is uncommon in real scenarios; thus, we analysed the controller cases with a combination of both \(GM\) and \(PM\) called \(DM\) (see Table 4). The \(GM\) & \(PM-C\) showed that the border of \(DM\) in Case 1 was within (0.2-4.5) around the nominal point 1 of the Nyquist plane, while the tolerance for phase variation was within \(\pm\)64.9 deg. Another way to analyse the robustness of a system to uncertainty and variation was studying the Nyquist plot (see Figure 4) for each controller case. As shown in Figure 4 and Table 4, the most robust controller was Case 4 with (\(DM=1.4\)), which tolerated more uncertainties based not only on the Nyquist plot but also on the combined gain and phase disk margin (\(GM:[0.1\;\;6.1]\Rightarrow DGM=16\) dB) and disk phase margin (\(PM:[-71.6\;\;71.6]\Rightarrow DPM=72\) deg) with respect to the other three cases as also shown in Figure 5.  
We then examined how each controller case tracked the hip torque reference signal for a healthy person in STS and walking activities at five torque levels, which represented various angular velocities for the joint's movements. The energy consumption per step activity with the four controller cases and five torque levels (\(TL1\) to \(TL5\)) have also been studied. Lastly, the number of times for the rehabilitation hip exoskeleton in STS was compared with the numbers of times for different controller cases and torques. The distances reached with a fully charged battery for each hip joint in the walking activity were also calculated. The results for STS and walking activity--based on a simulation--were as follows. \begin{table}
\begin{tabular}{c c c c} \hline
**Controller** & **CGM [dB]** & **CPM [deg]** & **GM \& PM-C** & **DM** \\ \hline Case 1 & 33.2 & 178.9 & GM \(:[\;\;0.2\;\;4.5\;]\) & 1.3 \\  & & PM \(:[\;\;-64.9\;\;64.9\;]\) & \\ Case 2 & 12 & 179.3 & GM \(:[\;\;0.3\;\;2.6\;]\) & 0.9 \\  & & PM \(:[\;\;-47.8\;\;47.8\;]\) & 0.9 \\ Case 3 & 12 & 21.4 & GM \(:[\;\;0.8\;\;1.3\;]\) & 0.2 \\  & & PM \(:[\;\;-13.1\;\;13.1\;]\) & \\ Case 4 & 27 & 179.3 & GM \(:[\;\;0.1\;\;6.1\;]\) & 1.4 \\  & & PM \(:[\;\;-71.6\;\;71.6\;]\) & \\ \hline \end{tabular}
\begin{tabular}{c} \hline CGM = Classical Gain Margin, CPM = Classical Phase Margin, GM \& PM-C = Gain and Phase Margin Combined, DM = Disk Margin. \\ \end{tabular}
\end{table} 
Table 4: Gain and phase disk margins for four different controller cases. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Case 1** & \multicolumn{3}{c}{**Case 2**} & **Case 3** & **Case 4** \\ \hline Q = \(\left[\begin{array}{ccc}0.025&100&100\\ 0&0.001&0\\ 0&0&0.001\\ \end{array}\right]\) & Q = \(\left[\begin{array}{ccc}0.025&100&100\\ 0&0.001&0\\ 0&0&0.001\\ \end{array}\right]\) & Q = \(\left[\begin{array}{ccc}0.025&100&100\\ 0&0.001&0\\ 0&0&0.001\\ \end{array}\right]\) & Q = \(\left[\begin{array}{ccc}0.025&100&100\\ 0&0.001&0\\ 0&0&100\\ \end{array}\right]\) & Q = \(\left[\begin{array}{ccc}100&0&0\\ 0&100&0\\ 0&0&100\\ \end{array}\right]\) \\ \hline \end{tabular}
\begin{tabular}{c} \(R=1\times 10^{-6}\) \\ \end{tabular} & \multicolumn{3}{c}{\(R=1\times 10^{-5}\)} & \multicolumn{3}{c}{\(R=1\times 10^{-3}\)} \\ \hline \end{tabular}
\end{table}