# A Robust Controller for Upper Limb Rehabilitation Exoskeleton
## 4 Simulation Results
### Results with the Robust GPI Controller  
The response of the exoskeleton with the robust GPI controller (6) with trajectory tracking to achieve a 90\({}^{\circ}\) bending movement in 5 s is shown in Figure 9. The Bezier polynomial was used to achieve a smooth movement, as shown for the angle \(\theta_{2}\). In this simulation, the weight derived from the limb of the patient to be rehabilitated was considered. Compared with the response using the PID type controller, it is observed that this response follows the desired trajectory without increasing the control effort. The advantage of this controller is that it only requires knowledge of the output (angular position), reducing cost and data processing.  
Figure 8: Response for a flexion movement with the PID controller considering the weight of the patient’s arm.  
Figure 9: Response for a flexion movement with the GPI controller, given by Equation (6), considering the weight of the patient’s arm.  
The response of the exoskeleton with the robust GPI controller (13) with trajectory tracking to achieve a 90\({}^{\circ}\) bending movement in 5 s is shown in Figure 10. In this simulation, the weight derived from the limb of the patient to be rehabilitated was also considered. As shown in the results, it achieved a better performance, minimizing the error and still maintaining a control effort similar to the PID controller (1) and the robust controller GPI (6) for ramp-type disturbances.  
Next, the exoskeleton response is shown for abduction movement (45\({}^{\circ}\)) and flexion (45\({}^{\circ}\)) of the arm, and for flexion (90\({}^{\circ}\)) of the forearm (Figure 11), considering the masses of the arm (Table 2) as forces acting on the exoskeleton (Figure 12). The GPI controller (13) was used with the trajectory tracking defined by the Bezier polynomial (17). The movements were performed at the same time, for 5 s, to reach the desired value. It can be seen that the controller achieved a good performance in tracking the trajectory, and the control effort of all links was kept below 15 Nm when performing the combined movements. Figure 12 shows the exoskeleton in the desired final position considering the combined movements.  
Figure 11: Response for a combined movement with the GPI controller, given by Equation (13), considering the weight of the patient’s arm.  
Figure 10: Response for a flexion movement with the GPI controller, given by Equation (13), considering the weight of the patient’s arm.  
Muscle spasticity was quantified by muscle displacements and compliance of myotonometer measurements and resistive torques from repeated passive stretching at speeds of \(5\ \frac{\circ}{\mathrm{s}}\) and \(10\ \frac{\circ}{\mathrm{s}}\), where stiffness (\(0.17\) Nm/\({}^{\circ}\)) total muscle was estimated, and muscle spasticity was analyzed in chronic stroke survivors using myotonometry and conventional passive stretching techniques [27]. The elbow flexor torque \(\tau_{\mathrm{s}}\) is a function of forearm flexion \(\theta_{3}\).  
The control of the combined motion of the exoskeleton was performed using a GPI control (13), with trajectory tracking defined by the Bezier polynomial (17), for the abduction (45\({}^{\circ}\)) and flexion (45\({}^{\circ}\)) motions of the arm and the flexion (90\({}^{\circ}\)) of the forearm (Figure 13), also considering the arm masses held in Table 2 and the forces acting on the exoskeleton and the elbow flexor torque \(\tau_{\mathrm{s}}\). The movements were performed at the same time, for 5 s, to reach the desired value. It is observed that the controller achieved a good performance in trajectory tracking, and only the control effort in the second link (\(\tau_{2}\)) increased to 26 Nm (Figure 13) when performing the combined movements to compensate for the elbow flexor torque (Figure 14).  
Figure 12: CAD view showing the upper limb rehabilitation exoskeleton.  
Figure 13: Response for a combined movement with the GPI controller, given by Equation (13), considering the weight of the patient’s arm and elbow flexor torque (spasticity).