# Investigation of 2 DOF PID Controller for Physio-Therapeutic Application for Elbow Rehabilitation
## 2 Mechanical Design of Exoskeleton
### System Identification  
System identification is a simple and convenient method of representing a linear or nonlinear model. This approach is commonly used in control engineering to design a system where the parameters of a system are unknown and required to be controlled effectively. Both the system's inputs and outputs are considered in this procedure, and their characteristics are estimated using the variances in the inputs and outputs. In this work, a sine-by-sine method with a periodic band limit has been considered to derive the transfer function, as shown in Equation (4). In this process, time and frequency domains have been considered.  
\[I(t)=I_{0}\sin\left(\omega_{f}t\right) \tag{4}\]  
where, \(I\), \(I_{0}\) and \(\omega_{f}\) are the motor current, signal amplitude with the range of [\(-0.4\)-\(0.4\)] and frequency (that is \(4\pi\)) respectively at sampling time \(t\).  
Figure 3 depicts some amplitude-varying inputs to the system as well as the system's output. To accomplish the system responses of different sine waves with different ampli  
Figure 2: Block diagram of revolute joint of the prototype and its movement range.  
tudes, the MATLAB System Identification Toolbox was considered. In addition, several transfer functions have been computed from these responses in order to find the curve that best fits the system responses as shown in Table 1.  
Now based on the comparison, this table shows that the performance of Model 2 is satisfactory in all four different amplitudes, such as 0.1, 0.2, 0.3 and 0.4 ampere, where its best fittings are 95.47%, 95.33%, 94.56% and 93.86% respective to the amplitudes. Figure 4 demonstrates the prototype's responses and the Model 2 responses in the amplitudes for a period of time, 1 s.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Transfer Function**} & \multicolumn{5}{c}{**Best Fit (\%) in Different Amplitudes**} \\ \cline{3-6}  & & **Amplitude 0.1** & **Amplitude 0.2** & **Amplitude 0.3** & **Amplitude 0.4** \\ \hline
1 & 76.48s–140.3 & 95.91 & 94.92 & 94.05 & 93.22 \\
2 & 358.2s–97.59 & 95.47 & 95.33 & 94.56 & 93.86 \\
3 & 37  
However, it is important to highlight that the model is quite different from the motor transfer function, since it has several unknown parameters, includes a controller and is an open loop system. Whenever a controller is considered with a system, it turns into a closed loop system. Hence, the derived system is required to be designed with the proposed controller so that it can offer the same performance as the system shows. It should be noted that the prototype has its own in-built controller and, hence, this study is considering the proposed controller with motor's transfer function to achieve the prototype's performance. Figure 5 shows a block diagram that represents the actual system and derived system (using system identification toolbox), where the actual system comprises both the controller and plant in a closed loop system, whereas the derived model (using SI toolbox) includes only the plant.  
Figure 4: Prototype response and Model 2 response in amplitudes (**a**) 0.1 amp, (**b**) 0.2 amp, (**c**) 0.3 amp and (**d**) 0.4 amp.  
Figure 5: A block diagram of actual and derived (using SI toolbox) elbow revolute joint system.