# A Novel Compliant 2-DOF Ejector Pin Mechanism for the Mass Transfer of Robotic Mini-LED Chips
## 2 Mechanical Design and Modeling
### Compliance Modeling  
As shown in Figure 3a, the displacement guidance of the designed compliant mechanism mainly depends on the deformation of four L-shaped flexures (as shown in Figure 3b). An L-shaped flexure has a highly compact structure and multiple degrees of freedom, and is used for mechanism guidance. In principle, the decoupling output of the compliant mechanism can be realized by using the parallelogram structure of the flexible hinge and the orthogonal placement of two piezoelectric actuators. As shown in Figure 3c, there are six types of forces on the free end of a flexure hinge, which are \(F_{x}\), \(F_{y}\), \(F_{z}\), \(M_{x}\), \(M_{y}\) and \(M_{z}\). Similarly, there are six forms of displacement of the free end of the flexure: linear displacements \(\delta_{x}\), \(\delta_{y}\), \(\delta_{z}\) and angular displacements \(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\). The free end translates \(\delta_{x}\) along the \(x\)-axis under the action of force \(F_{x}\). Both \(F_{y}\) and \(M_{z}\) result in a linear displacement \(\delta_{y}\) along the \(y\)-axis and an angular displacement \(\theta_{z}\) on the \(z\)-axis. Moreover, \(F_{z}\) and \(M_{y}\) both result in \(\delta_{z}\) and \(\theta_{y}\). When the flexure works in the elastic stage, the relationship between the displacement and force is linearized:  
\[\mathbf{\Delta}=\mathbf{C}\mathbf{F} \tag{1}\]  
where \(\mathbf{\Delta}=\left[ \begin{array}{cccccc}\delta x&\delta y&\delta z&\theta_{x }&\theta_{y}&\theta_{z}\end{array} \right]^{T}\) and \(\mathbf{F}=\left[ \begin{array}{cccccc}F_{x}&F_{y}&F_{z}&M_{x}&M_{y}&M_{z} \end{array} \right]^{T}\) are the force and displacement at the free end, respectively. \(\mathbf{C}\) denotes the compliance matrix. Based on the above analysis, \(\mathbf{C}\) can be written in the following form [28]:  
\[\mathbf{C}=\left( \begin{array}{cccccc}C_{\delta_{x},F_{x}}&0&0&0&0&0\\ 0&C_{\delta_{y},F_{y}}&0&0&0&C_{\delta_{y},M_{z}}\\ 0&0&C_{\delta_{z},F_{z}}&0&-C_{\delta_{z},M_{y}}&0\\ 0&0&0&C_{\theta_{x},M_{x}}&0&0\\ 0&0&-C_{\delta_{y},F_{z}}&0&C_{\delta_{y},M_{y}}&0\\ 0&C_{\theta_{z},F_{y}}&0&0&0&C_{\theta_{z},M_{z}}\end{array} \right) \tag{2}\]  
Figure 2: Compliant 2-DOF ejector pin mechanism for the mass transfer of robotic mini-LED chips. (**a**) Schematic diagram of the designed mini-LED transfer device. (**b**) Three-dimensional diagram of the designed device. (**c**) Schematic diagram of continuous chip ejection. Moreover, the compliance matrix of a flexure can be transformed from its local coordinate \(o_{i}-xyz\) to another coordinate \(o_{j}-xyz\) by  
\[\mathbf{C}_{j}=\mathbf{R}_{i}\mathbf{P}(\mathbf{r}_{i})\mathbf{C}_{i}\mathbf{P} (\mathbf{r}_{i})^{T}\mathbf{R}_{i}^{T} \tag{3}\]  
where \(\mathbf{P}(\mathbf{r}_{i})\) is defined as the orientation matrix of \(o_{i}-xyz\) with respect to \(o_{j}-xyz\); \(\mathbf{R}_{i}\) is also known as the direction cosine matrix; and \(\mathbf{P}(\mathbf{r}_{i})\) denotes the translation matrix and can be described as follows:  
\[\mathbf{P}(\mathbf{r}_{i})=\left[ \begin{array}{cccc}&0&-z_{i}&y_{i}\\ \mathbf{I}_{3\times 3}&z_{i}&0&-x_{i}\\ &-y_{i}&x_{i}&0\\ \mathbf{O}_{3\times 3}&&\mathbf{I}_{3\times 3}&\end{array} \right] \tag{4}\]  
where \(\mathbf{I}_{3\times 3}\) and \(\mathbf{O}_{3\times 3}\) are the identity matrix and zero matrix, respectively, and \(\mathbf{r}_{i}=[x_{i},y_{i},z_{i}]^{T}\) is the position vector of the point \(O_{j}\) that is expressed in the reference frame of \(o_{i}-xyz\). \(\mathbf{R}(\alpha,\beta,\gamma)=\mathbf{R}_{z}(\gamma)\mathbf{R}_{y}(\beta) \mathbf{R}_{x}(\alpha)\) is the rotation transformation matrix. \(\alpha\), \(\beta\), and \(\gamma\) are the rotation angles of the coordinate system \(o_{j}-xyz\) around the \(x\)-axis, \(y\)-axis and \(z\)-axis of the coordinate system \(o_{i}-xyz\), respectively. This can be expressed as:  
\[\mathbf{R}_{x}(\alpha)=\left[ \begin{array}{cc}\mathbf{r}_{x}&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{r}_{x}\end{array} \right],\mathbf{R}_{y}(\beta)= \left[