# A Novel Compliant 2-DOF Ejector Pin Mechanism for the Mass Transfer of Robotic Mini-LED Chips
## 4 Controller Design  
In this section, the dynamics characteristics of the compliant 2-DOF ejector pin mechanism are first analyzed, then an iterative learning control (ILC) is employed to improve the tracking performances. Furthermore, the approach to regulating the parameters of the ILC in the frequency domain is demonstrated. In order to reduce the initial state error of the ILC, the input is combined with the reference as compensation. As the following analysis is conducted in the frequency domain, a transfer function to describe the relationship between the input and output of the plant is required. The output force of the piezoelectric actuator is linearly related to the input voltage in the linear area. The displacement of piezoelectric actuators is linearly related to the force when working in the micro range. Thus, it can be obtained that:  
\[\mathbf{F}=[F_{x},F_{y}]^{T}=T_{e}\cdot[u_{x},u_{y}]^{T} \tag{17}\]  
where \(T_{e}\) is the DC gain from the input to the output of the piezoelectric actuator, \(u_{x}\) and \(u_{y}\) are the input displacements of the \(x\)-axes and \(y\)-axes, respectively. Based on the electromechanical model (14) and Equation (17), the transfer function of the plant can be obtained via the Laplace transform:  
\[\left\{ \begin{array}{l}\mathrm{G_{x}}(s)=\frac{X(s)}{U_{x}(s)}=\frac{K_{e}} {s^{2}+\frac{C_{x}}{m}+s+\frac{U_{x}}{m}}\\ \mathrm{G_{y}}(s)=\frac{Y(s)}{U_{y}(s)}=\frac{K_{e}}{s^{2}+\frac{C_{y}}{m}+s+ \frac{U_{y}}{m}}\end{array} \right. \tag{18}\]  
where \(K_{e}=T_{e}/m\) is the gain. Since \(x\)-axes and \(y\)-axes are decoupled in the process of structural design and incey share the same dynamic characteristics with each other, the one-dimensional motion of the compliant 2-DOF ejector pin mechanism is studied in the following analysis. Due to the presentation of the hysteresis behavior of the piezoelectric actuator, the tracking accuracy of the device is greatly restricted [29]. Even worse, the sensor noise increases the measurement error, which is used to correct the input. On the other hand, the repetitive trajectories with the same initial state are used in the device during the process of chip transfer, which makes the hysteresis dynamics present as periodic. Therefore, ILC can be adopted to cancel the hysteresis, where the ILC corrects the input by iteratively learning and requires the lowest prior information. The controller is required to be robust to the measurement error, owing to the unknown dynamics of the piezoelectric actuator and the noise of sensors. Thus, the classical P-type ILC is adopted [30], where P is the abbreviation for proportional. Its control law can be written as:  
\[u_{j+1}(k)=Q\cdot[u_{j}(k)+\gamma\cdot e_{j}(k+1)] \tag{19}\]  
where \(Q\) is the low-pass filter, \(\gamma\) is the learning gain, and \(u_{j}(k)\) and \(e_{j}(k)\) are the input and error at the \(k\)-th time step in the \(j\)-th repetition, respectively. As the following analysis is conducted in the discrete time, the output can be obtained via the \(z\) transform:  
\[Y_{j}(z)=G(z)\cdot U_{j}(z)+W(z) \tag{20}\]  
where the \(G(z)\) is the transfer function of the plant, \(W(z)\) signifies the disturbances, \(Y_{j}(z)\) and \(U_{j}(z)\) are the output and input, respectively. According to [31], the tracking error can be denoted as: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Compliance**} & \multicolumn{2}{c}{**Frequency**} \\ \hline  & \(C_{\delta_{x},F_{z}}^{out}(\mathrm{\mu m/N})\) & \(C_{\delta_{x},F_{z}}^{out}(\mathrm{\mu m/N})\) & \(f_{x}(\mathrm{Hz})\) & \(f_{y}(\mathrm{Hz})\) \\ Theoretical & 14.51 & 14.51 & 267.16 & 267.16 \\ FEA & 14.74 & 15.22 & 250.85 & 256.76 \\ Error & 1.59\% & 4.89\% & 6.10\% & 3.89\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons between FEA and the theoretical model.