# Asymmetric Constrained Control of a Cervical Orthotic Device Based on Barrier Sliding Modes
## 6 Numerical Simulations  
### Orthosis Specifications  
Figure 1 describes the robotic orthosis that is made up of three principal sections: (1) the head support that holds the head of the patient preventing falling and (2) the collar structure which supports the whole mechanism. This structure should be positioned on the shoulders of the patient to prevent any cervical damage. Finally, (3) the mechanical actuators that forces the movement of the orthosis.  
The movement of actuators allows to control the range, and the trajectory of the neck during rehabilitation therapies. The corresponding motion trajectories should be suggested by professional physicians. Figure 2 shows the class of actuators selected for the orthosis. The orthosis employs linear actuators that generates a movement in the x-y direction.  
Figure 1: Orthosis design in a CAD software. (**a**) Isometric view; (**b**) frontal view; and (**c**) lateral view.  
Figure 2: The mechanism of movement generated by a set of linear actuators. The closer view describes the position and orientation of the actuators.  
Once the proposal design was approved, the CAD design is exported to Simulink in Matlab to prove the orthosis in a virtual environment. The SimMechanics environment allows to prove virtually the functions of the orthosis. With the measurements of position and velocity obtained from SimMechanics, it is possible to evaluate the control performance for a future implementation. The control must satisfy the neck motion for the rehabilitation according to the trajectories suggested by physicians.  
A comparison is done employing a classical PID controller. The control gains employed in this controller are \(k_{P}=10\), \(k_{D}=2.5\) and \(k_{I}=0.5\). For the FOSM controller the gains are \(\gamma=0.002\), \(\mu_{i}=0.1\). The positive restriction in position is \(5.795\times 10^{-4}\) m, while the negative restrictions is \(-5.725\times 10^{-4}\) m, the restrictions in velocity were \(1\) cm/s and \(-1.3\) cm/s. Figure 3 shows the results obtained with the proposed asymmetric FOSM controller. Notice that the position of the orthosis follows the desired trajectories without violating the state restrictions. A comparison against a classical proportional-integral-derivative (PID) controller demonstrates the advantages using the analysis based on BLF. Even though, the results obtained with the PID strategy do not violates the asymmetric state restrictions imposed on the orthosis, the trajectories do not reach the desired ones. Figure 4 describes the tracking trajectory task for the velocity. Similar to previous results the trajectories remain inside the constraints. The initial overshoot presented by the PID controller does not fulfill the state restrictions, while the overshoot provided by the asymmetric FOSM that is smaller than the PID remains inside the restrictions. Notice that the convergence is improved applying the controller proposed in this manuscript. Figure 5 describes the control signal obtained with each approach. Finally, Figure 6 shows the comparison of the tracking error obtained with the PID and the AFOSM. The tracking error resulted from applying the PID is bigger than the one obtained with the AFOSM. These numerical simulations corroborate the theoretical results showed in Theorem 1.  
Figure 3: Tracking trajectory of position.