# Design of an Active and Passive Control System of Hand Exoskeleton for Rehabilitation
## 5 Experiments
### Experimental Results  
* Passive rehabilitation mode:  
From Equation (1) the trajectories of \(\mathbf{\theta}_{d}\) and \(\mathbf{\theta}\) in Cartesian space were calculated by forward kinematics, as shown in Figure 9. The trajectories in \(x_{t}\) direction and \(y_{t}\) direction are shown as the curves in Figure 9a,b respectively, and the mean \(\pm\) standard deviation of errors are \(0.17\pm 0.62\) mm and \(-0.28\pm 0.85\) mm, respectively. Therefore, the results show that the motion of the exoskeleton is stable,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link \(i\)** & \(\mathbf{m_{i}}\) **(kg)** & \(\mathbf{I_{xx}}\) **(kg:mm2)** & \(\mathbf{I_{yy}}\) **(kg:mm2)** & \(\mathbf{I_{zz}}\) **(kg:mm2)** \\ \hline
1 & 0.048 & 10.286 & 31.093 & 31.720 \\
2 & 0.029 & 12.325 & 37.057 & 43.224 \\
3 & 0.037 & 14.489 & 32.140 & 36.589 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dynamic parameters of the single finger of the exoskeleton.  
and the actual trajectory generally accords with the desired trajectory, tracking the planned trajectory well to do the rehabilitation motion, which can meet the need of passive rehabilitation.  
Active rehabilitation mode:  
The real-time simulation results are shown as the desired trajectory (dash lines) in Figure 10a,b. The actual trajectories in \(x_{t}\) direction and \(y_{t}\) direction are shown as the solid line in Figure 10a,b, and the mean \(\pm\) standard deviation of errors are \(-0.33\pm 2.04\) mm and \(0.16\pm 1.92\) mm, respectively. The curve of force on the tip collected in the experiment is shown in Figure 10c.  
From the figures, when the finger is relaxed, the output of the force sensor on the fingertip of the exoskeleton is close to 0 and the exoskeleton keeps its attitude. When the subject exerts force on the finger, the force sensor can perceive the contact force changes immediately. The exoskeleton can follow the finger motion well to fulfill the control strategy of active rehabilitation mode with resistance at around 3.8 N. The experiment results show that the designed controller has the ability in position tracking and it can automatically track and adapt to the motion status of the patient during active rehabilitation training.  
Figure 9: Passive rehabilitation experiment: (**a**) \(x_{t}\) direction displacement; (**b**) \(y_{t}\) direction displacement.  
Figure 10: Active rehabilitation experiment: (**a**) \(x_{t}\) direction displacement; (**b**) \(y_{t}\) direction displacement; (**c**) contact force.  
Active plus passive rehabilitation mode:  
The dash lines in Figure 11b,d show the desired trajectories of the exoskeleton in \(x_{t}\) and \(y_{t}\) directions, and the mean \(\pm\) standard deviation of errors are 0.97 \(\pm\) 1.64 mm and \(-\)0.12 \(\pm\) 1.04 mm, respectively. Figure 11a shows the force on the fingertip collected during the rehabilitation process, and Figure 11b,d shows the actual trajectories of the exoskeleton in \(x_{t}\) and \(y_{t}\) directions.  
The hand rehabilitation exoskeleton can judge the rehabilitation intention of the patient by detecting the force on the tip. At about \(t\) = 2.4 s, when the subject cannot exert enough force, the force on the tip is smaller than the threshold value. Hence the exoskeleton switches to the passive mode, and the angular velocities are calculated by the passive control, as shown in Figure 11c. At about \(t\) = 2.9 s and \(t\) = 4 s, when the subject can exert force, the force on the tip of the exoskeleton finger is larger than the threshold value. Hence the exoskeleton is in active rehabilitation mode and follows the subject's motion. The experiment results verify the feasibility of the control strategy and the stability of the system.