# Variable Guide Vane Scheduling Method Based on the Kinematic Model and Dual Schedule Curves
## 3 Analytical Model of the VGV Link Mechanism  
### Model for the Links from Actuator to Unison Ring  
The entire VGV link mechanism is divided into two parts and these were analyzed separately by reason of modeling convenience. First, link connections from the hydraulic actuator to the unison ring are shown in Figure 2. With the assumption that the unison ring rotates with respect to the center of the compressor plane, these link connections can be simplified as in-plane motion. With change in the length of the actuator stroke (represented as green line in Figure 2), the bell crank rotates on its pivot like a revolute joint having one degree of freedom, then it induces movement of the vertical link. The upper connection (consisting of links \(L_{x}\) and \(L_{e}\), and joints \(J0\), \(J1\), and \(J2\), represented as green and purple lines in Figure 2) can be summarized as three revolute joints and three parts, including a fixed part (ground). This results in zero DOF from a two-dimensional Grubler's equation (based on in-plane motion assumption). \[DOF = 3(N_{link}-1)-2N_{J1}-N_{J2} \tag{2}\]  
A lower connection (including links \(L_{u}\), \(L_{b}\), \(L_{c}\), and \(L_{d}\), and joints \(J2\), \(J3\), \(J4\), and \(J5\), represented as red lines in Figure 2) is a typical four-bar mechanism, and it has one DOF. However, the movement of the lower connection is coupled on the upper connection motion via the bell crank. The sole solution for positions of upper and lower connections thus can be obtained when the actuator stroke (\(L_{x}\)) is defined. The relation from the actuator stroke to the angle of the bell crank can be derived from the following simple transformation matrix. \[T^{0}_{J2} = T^{0}_{J0}T^{J0}_{J1}T^{J1}_{J2} \tag{3}\]  
where \(T^{P1}_{P2}\) represents a transformation matrix to change position with direction from \(P1\) to \(P2\). For example, \(T^{0}_{J2}\) indicates a transformation matrix to define the position and direction of joint J2, with respect to the left-end mount of the actuator. The transformation matrix contains a \(3\times 3\) directional \begin{table}
\begin{tabular}{c c c c} \hline  & **Description** & **Type** & **Numbers** \\ \hline \multirow{2}{*}{\(N_{J3}\)} & \multirow{2}{*}{Number of 3 DOF Joint} & Spherical Joint & \\  & & (2 x bell crank-vertical link) & 4 \\  & & (2 x clevis-vertical link) & \\ \hline \multirow{2}{*}{\(N_{link}\)} & \multirow{2}{*}{Number of Parts} & Links & \\  & & (2 vertical links, & 4 \\  & & 1 unison ring, ground) & \\ \hline \end{tabular}
\end{table} 
Table 1: Joints and links of the VGV’s link mechanisms below the bell cranks. transformation matrix based on Euler angles and the position information in the last column. In this case, Equation (3) can be represented in the form of the following detail matrix. \[\left[ \begin{array}{cccc}r_{11}&r_{12}&r_{13}&x_{1}\\ r_{21}&r_{22}&r_{23}&y_{1}\\ r_{31}&r_{32}&r_{33}&0\\ 0&0&0&1\end{array} \right]=\left[ \begin{array}{cccc}c\theta_{0}&-s\theta_{0} &0&0\\ s\theta_{0}&c\theta_{0}&0&0\\ 0&0&1&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}c\theta_{1}&-s\theta_{1} &0&L_{x}\\ s\theta_{1}&c\theta_{1}&0&0\\ 0&0&1&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&0&0&L_{x}\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{4}\]  
where \(c\theta\) and \(s\theta\) represent cosine and sine functions, respectively, and \(r_{xx}\) represents \(xx\)-directional values of the links (1 for \(x\)-axis, 2 for \(y\)-axis, 3 for \(z\)-axis). The unknown variables of Equation (4) are \(\theta_{0}\) and \(\theta_{1}\). They can be obtained using the actuator stroke, the bell crank angle, and the given joint J2 positions with Equation (4). Nonlinear simultaneous equations of Equation (4) are not directly computed, so simplification by multiplying the inverse matrix of \(T_{J0}^{0}\) to both sides of Equation (4) proceeds as shown in Equation (5). \[\left[ \begin{array}{cccc}c\theta_{0}&s\theta_{0}&0&0\\ -s\theta_{0}&c\theta_{0}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[