## 2 Kinematic Analysis of a Dual-Parallelogram Passive Rocking Vibration Isolator
### Kinematic Analysis  
The kinematic relationship between the rods in a multiple-rod isolation mechanism is relatively complicated. The basic platform rocks around a turning center \(O\), with angular \(\theta\). Meanwhile, the vibration isolation platform exhibits coupled motion--translation and rotation around \(o_{1}\). \(\phi\) is the angle between the local coordinate system \(o_{1}-xyz\) and the absolute coordinate system \(O-XYZ\). Figure 2 shows the kinematic positions of the rods.  
The mass center (\(A_{1}A_{2}\), \(A_{3}A_{4}\), \(B_{1}B_{2}\), \(B_{3}B_{4}\), \(C_{1}C_{2}\)) is the respective geometric center of each rod. \(d_{2}\) is the distance from the mass center of each \"T\"-frame rod (\(A_{2}A_{3}C_{1}\), \(B_{2}B_{3}C_{2}\)) to \(C_{1}\) and \(C_{2}\), respectively. The load and rod masses are \(m_{0},m_{1},...m_{7}\). The mass center of each rod is defined as \(p_{i}=(x_{i},y_{i},\  
Figure 2: Motion diagram of the dual-parallelogram vibration isolation.  
\(i=1\), 2,..., 7. \(\theta\) and \(\phi\) are generalized coordinates. Symmetry yields the following kinematic relationship:  
\[\left\{\begin{array}{l}X=\left[\begin{array}{cccccccc}x_{0}&x_{1}&x_{2}&x_{3} &x_{4}&x_{5}&x_{6}&x_{7}\\ Y=\left[\begin{array}{cccccccc}y_{0}&y_{1}&y_{2}&y_{3}&y_{4}&y_{5}&y_{6}&y_{7} \\ \varphi=\left[\begin{array}{cccccccc}\varphi_{0}&\varphi_{1}&\varphi_{2}& \varphi_{3}&\varphi_{4}&\varphi_{5}&\varphi_{6}&\varphi_{7}\\ \end{array}\right]^{T}\end{array}\right]^{T} \tag{2}\]  
where \(X=AC\), \(Y=BC\), \(\varphi=\left[\begin{array}{cccccccc}\varphi&\theta-\phi&\theta&\theta- \phi&\phi&\phi-\theta&-\theta&\phi-\theta&\phi-\theta\end{array}\right]^{T}\), and  
\[A=\left[\begin{array}{cccccccc}-R&0&h&0\\ -H-\frac{\sqrt{2}}{2}l_{2}&-\frac{1}{2}-\frac{\sqrt{2}}{2}l_{2}&\frac{\sqrt {2}}{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ -R+\frac{\sqrt{2}}{2}d_{2}&-\frac{\sqrt{2}}{2}d_{2}&d_{1}&\frac{1}{2}\\ -H&-\frac{1}{2}&\frac{\sqrt{2}}{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ -R&0&d_{1}&0\\ -H-\frac{\sqrt{2}}{2}l_{2}&\frac{t}{2}+\frac{\sqrt{2}}{2}l_{2}&\frac{\sqrt{2} }{4}l_{1}&\frac{\sqrt{2}}{4}l_{1}\\ -R+\frac{\sqrt{2}}{2}d_{2}&\frac{\sqrt{2}}{2}d_{2}&d_{1}&\frac{1}{2}\\ -H&\frac{t}{2}&\frac{\sqrt{2}}{4}l_{1}&\frac{\sqrt{2}}{4}l_{1}\\ \end{array}\right],B=\left[\begin{array}{cccccccc}0&R&0&-h\\ -\frac{1}{2}-\frac{\sqrt{2}}{2}l_{2}&H+\frac{\sqrt{2}}{2}l_{2}&-\frac{\sqrt{2} }{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ -\frac{\sqrt{2}}{2}d_{2}&R-\frac{\sqrt{2}}{2}d_{2}&-\frac{1}{2}&-d_{1}\\ -\frac{1}{2}&H&-\frac{\sqrt{2}}{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ 0&R&0&-d_{1}\\ \frac{t}{2}+\frac{\sqrt{2}}{2}l_{2}&H+\frac{\sqrt{2}}{2}l_{2}&\frac{\sqrt{2} }{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ \frac{\sqrt{2}}{2}d_{2}&R-\frac{\sqrt{2}}{2}d_{2}&\frac{1}{4}&-d_{1}\\ \frac{t}{2}&H&\frac{\sqrt{2}}{4}l_{1}&-\frac{\sqrt{2}}{4}l_{1}\\ \end{array}\right]\]  
\[C=\left[\begin{array}{cccccccc}\sin\theta&\cos\theta&\sin\phi&\cos\phi\\ \end{array}\right]^{T}\]  
This kinematic relationship could be used to obtain the kinematic position of the center of mass and rotation angle of each rod. Table 1 shows the geometrical and mass parameters and moment of inertia.