## 2 The Prototype and Modeling of the Proposed Isolation Platform
### Modeling
\begin{array}{l}l_{a}^{2}+\left(\left\|\mathbf{A}_{0}\mathbf{A^{\prime}}_{ 1}\right\|/n\right)^{2}=4l^{2}\\ l_{b}^{2}+\left(\left\|\mathbf{B}_{0}\mathbf{B^{\prime}}_{1}\right\|/n\right)^{2} =4l^{2}\\ l_{c}^{2}+\left(\left\|\mathbf{C}_{0}\mathbf{C^{\prime}}_{1}\right\|/n\right)^{2} =4l^{2}\\ l_{d}^{2}+\left(\left\|\mathbf{D}_{0}\mathbf{D^{\prime}}_{1}\right\|/n\right)^{2} =4l^{2}\end{array} \right. \tag{6}\]  
The potential energy \(V\) of the system is consisted of the elastic energy of the four springs in SLSs, which can be expressed as:  
\[ \begin{array}{ll}V&=\frac{1}{2}k(l_{a}-2l\cos\theta)^{2}+\frac{1}{2}k(l_{b} -2l\cos\theta)^{2}+\frac{1}{2}k(l_{c}-2l\cos\theta)^{2}+\frac{1}{2}k(l_{d}-2l \cos\theta)^{2}\\ &=\frac{1}{2}k\Bigg{(}\sqrt{4l^{2}-\frac{\left\|\mathbf{A}_{0}\mathbf{A^{ \prime}}_{1}\right\|^{2}}{n^{2}}}-2l\cos\theta\Bigg{)}^{2}+\frac{1}{2}k\Bigg{(} \sqrt{4l^{2}-\frac{\left\|\mathbf{B}_{0}\mathbf{B^{\prime}}_{1}\right\|^{2}} {n^{2}}}-2l\cos\theta\Bigg{)}^{2}\\ &+\frac{1}{2}k\Bigg{(}\sqrt{4l^{2}-\frac{\left\|\mathbf{C}_{0}\mathbf{C^{ \prime}}_{1}\right\|^{2}}{n^{2}}}-2l\cos\theta\Bigg{)}^{2}+\frac{1}{2}k\Bigg{(} \sqrt{4l^{2}-\frac{\left\|\mathbf{D}_{0}\mathbf{D^{\prime}}_{1}\right\|^{2}} {n^{2}}}-2l\cos\theta\Bigg{)}^{2}\end{array}  \tag{7}\]  
Considering the air damping of the system, the dynamic equation can be obtained by the Lagrange principle as:  
\[\frac{d}{dt}\left[\frac{\partial(T-V)}{\partial\dot{\mathbf{x}}}\right]-\frac {\partial(T-V)}{\partial\dot{\mathbf{x}}}=-\mathbf{\mathit{C}}\cdot\left( \dot{\mathbf{x}}-\dot{\mathbf{x}}_{e}\right) \tag{8}\]  
where \(T\) is the kinetic energy and \(V\) is the potential energy shown as Equations (3) and (7), the mass matrix \(\boldsymbol{M}\) of the system is:  
\[\boldsymbol{M}= \left[ \begin{array}{cccc}M_{1}+M_{2}&0&0\\ 0&M_{1}+M_{2}&0\\ 0&0&M_{1}+M_{2}\\ 0&-M_{2}L_{3}&M_{2}L_{2}\\ M_{2}L_{3}&0&-M_{2}L_{1}\\ -M_{2}L_{2}&M_{2}L_{1}&0\\ 0&M_{2}L_{3}&-M_{2}L_{2}\\ -M_{2}L_{3}&0&M_{2}L_{1}\\ M_{2}L_{2}&-M_{2}L_{1}&0\\ J_{xc}+J_{xm}+M_{2}\big{(}L_{2}^{2}+L_{3}^{2}\big{)}&-M_{2}L_{1}L_{2}&-M_{2}L_{ 1}L_{3}\\ -M_{2}L_{1}L_{2}&J_{yc}+J_{ym}+M_{2}\big{(}L_{1}^{2}+L_{3}^{2}\big{)}&-M_{2}L_{ 2}L_{3}\\ -M_{2}L_{1}L_{3}&-M_{2}L_{2}L_{3}&J_{zc}+J_{zm}+M_{2}\big{(}L_{1}^{2}+L_{2}^{2} \big{)}\end{array} \right] \tag{9}\]  
After first-order Taylor series expansion, the stiffness matrix \(\boldsymbol{K}\) is:  
\[\boldsymbol{K}=\left[ \begin{array}{cccc}\frac{k(l_{2}-l_{1})^{2}}{2l^{2}n^{4 }\cos^{2}\theta}&0&0&\frac{kl_{1}(l_{1}-l_{2})H}{2l^{2}n^{4}\cos^{2}\theta}&0 \\ 0&\frac{k(l_{2}-l_{1})^{2}}{2l^{2}n^{4}\cos^{2}\theta}&0&\frac{kl_{1}(l_{2}-l_{ 1})H}{2l^{2}n^{4}\cos^{2}\theta}&0&0\\ 0&0&\frac{kl^{2}}{2n^{4}\cos^{2}\theta}&0&0&0\\ 0&\frac{kl_{1}(l_{2}-l_{1})H}{2l^{2}n^{4}\cos^{2}\theta}&0&\frac{kl^{2}_{1}H^{ 2}}{2l^{2}n^{4}\cos^{2}\theta}&0\\ \frac{kl_{1}(l_{1}-l_{2})H}{2l^{2}n^{4}\cos^{2}\theta}&0&0&\frac{kl^{2}_{1}H^{ 2}}{2l^{2}n^{4}\cos^{2}\theta}&0\\ 0&0&0&0&0&0\end{array} \right] \tag{10}\]  
where \(H=\sqrt{4n^{2}l^{2}\sin^{2}\theta-\left(l_{2}-l_{1}\right)^{2}}\), and matrix \(\boldsymbol{C}\) is the damping matrix of the system:\[C=\left[ \begin{array}{ccccc}c_{1}&&&\cdots&0\\ &c_{2}&&&\\ &&c_{3}&&&\vdots\\ \vdots&&&c_{4}&&\\ 0&&\cdots&&&c_{6}\\ \end{array} \right] \tag{11}\]  
where \(c_{i}\) (\(i\) = 1,..., 6) are damping coefficients in the six coordinates. Due to the assumptions that the amplitudes of motions are sufficiently small, each term in the dynamic equation can be expanded by a Taylor series and higher-order terms can be neglected. Therefore, the dynamic equation of the system is:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{K}\mathbf{x}+\mathbf{C}\dot{\mathbf{x}}= \mathbf{K}\mathbf{x_{e}}+\mathbf{C}\dot{\mathbf{x_{e}}} \tag{12}\]  
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{K}\) is the stiffness matrix, and \(C\) is the damping matrix, respectively, and \(\mathbf{x}\) is the motion vector and \(\mathbf{x_{e}}\) is the base excitation vector.