# The Energy Consumption and Robust Case Torque Control of a Rehabilitation Hip Exoskeleton
## 3 Methods  
The simulation and design of the rehabilitation hip exoskeleton were modelled on the Denavit-Hartenberg (DH) parameters as shown in Figure 3 using the MATLAB and Autodesk Inventor software. The actuator selection was an SEA for the active hip joint. The hip rehabilitation consisted of a pelvis, four links for both legs and four joints that were active, except the knees, which were passive, ending up with a four-degree-of-freedom (DoF) prototype moving in the sagittal plane. The limitations and the range of motion (RoM) for each joint were based on the data in Table 1[8]. The trunk and exoskeleton masses were connected to a body-weight-supported treadmill similar to the one used with the LokomatPro exoskeleton [53].  
The control framework was based on a linear quadratic regulator of an SEA for a hip rehabilitation exoskeleton in which the torque control signals were the reference signals in the admittance control strategy. The control signals were five levels of torque (\(TL1\) to \(TL5\)) of STS and the walking activities for healthy subjects, as shown in Figures 1 and 2 and in articles [8; 41; 42].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Joint** & **DoFs-Description** & **Type** & **RoM** \\ \hline Hip & Flexion–Extension & Active & \(120^{\circ}\) to \(-122^{\circ}\) \\ Knee & Flexion–Extension & Passive & \(120^{\circ}\) to \(0^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The hip and knee joints with their range of motions (RoMs) [8].  
The kinematic and kinetic formulation, the system mathematical representation and the admittance control are thoroughly discussed in [8], where the modelling was based on the average height and weight of male subjects. The length of the thigh link was 52 and the shank link was 39 cm, as mentioned in [54]. The SEA was modelled based on [55; 56; 57; 58] and the parameters used are shown in Table 2, where \(J_{m},J_{s}\) and \(J_{l}\) are the inertia of the motor, spring and load link, respectively. \(B_{m},B_{s}\) and \(B_{l}\) are the damping coefficients for the motor, spring and load, respectively, whereas \(K_{s}\) and \(N\) are the spring stiffness and gear ratios in the SEA. Therefore, these parameters with the system links concluded with the transfer function \(G(S)\).  
\[G(s)=\frac{17.54\;s^{3}+0.02\;s^{2}+4.04\cdot 10^{8}s+6.69\cdot 10^{10}}{s^{3}+0.0 2\;s^{2}+1.11\cdot 10^{4}\;s+2.19\cdot 10^{5}}, \tag{7}\]  
The LQR control strategy was considered to operate in a linear time-invariant system for this study. The LQR was examined with four different values of the \(Q\) and \(R\) matrices (four cases), which were chosen empirically with different weights on \(Q\) and \(R\) to check the speed and robustness with different penalties on the input signal and states as shown in Table 3, which were the basis for finding the minimum quadratic cost function, \(J\), regarding (2) and, consequently, the optimal controller.  
Moreover, the result of using the regulator to follow the trajectory torque signal was investigated in the form of the RMSE. Finally, the energy consumption and distance for each step activity when using the hip exoskeleton powered by a charged battery (lithium-iron 48 V, 150 Wh, with 30 LiFePO\({}_{4}\) cells mounted in a battery tray) for each hip joint were simulated and studied.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(J_{m}\) & \(2.7\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ \(J_{s}\) & \(45\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\) \\ \(J_{l}\) & \(7\times 10^{-2}\) kg\(\cdot\)m\({}^{2}\) \\ \(B_{m}\) & \(1.5\times 10^{-3}\) Nm\(\cdot\)s/rad \\ \(B_{s}\) & \(2\times 10^{-2}\) Nm\(\cdot\)s/rad \\ \(B_{l}\) & \(5\times 10^{-3}\) Nm\(\cdot\)s/rad \\ \(K_{s}\) & \(5\times 10^{3}\) Nm\(\cdot\)rad \\ \(N\) & \(1\times 10^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The simulation parameters of the SEA [56].  
Figure 3: The DH parameters and the rehabilitation hip exoskeleton model [8].