# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## Appendix A  
From Equations (1) and (3), the motion equation of each of the 6-DoF AVIS is derived as follows:  
\[ \begin{array}{c}m\ddot{z}=\frac{4}{i-1}f_{xi}+\frac{4}{i-1}f_{xi}\\ m\ddot{x}=\sum\limits_{i=1}^{2}f_{xi}+\sum\limits_{i=1}^{4}f_{xi}\\ m\ddot{y}=\sum\limits_{i=1}^{2}f_{yi}+\sum\limits_{i=1}^{4}f_{yi}\\ \ddot{y}=\sum\limits_{i=1}^{2}f_{yi}+\sum\limits_{i=1}^{4}f_{yi}\\ l_{x}\ddot{\varphi}_{x}=l_{xx}(f_{yi}-f_{y2})+l_{yy}(f_{xi}-f_{y2})+b(f_{xixi}+f_ {xi}-f_{y2}-f_{y3})+a(f_{yi}+f_{y2}-f_{y3}-f_{y4})\\ l_{xx}\ddot{\varphi}_{x}=b(f_{x2}+f_{y3}-f_{x2}-f_{x1}-f_{x4})+l_{yy}(f_{z}+f_{ z3}-f_{z1}-f_{x4})+(c+l_{yz})(f_{yi}+f_{y2})+c(f_{yi}+f_{y2}+f_{y3}+f_{y4})\\ l_{yy}\ddot{\varphi}_{y}=a(f_{x2}+f_{x4}-f_{x1}-f_{x2})+l_{xx}(f_{z3}+f_{x4}-f_{ x1}-f_{x2})-(c+l_{xz})(f_{z1}+f_{x2})-c(f_{x1}+f_{x2}+f_{x3}+f_{x4})\end{array}  \tag{1}\]  
where the spring forces of the first spring are given by:  
\[ \begin{array}{c}\left[\begin{array}{c}f_{x1}\\ f_{y1}\\ f_{x2}\end{array}\right]=-\left[\begin{array}{ccc}k_{x}&0&0\\ 0&k_{y}&0\\ 0&0&k_{z}\end{array}\right]\left[\begin{array}{ccc}x_{A1}-x_{bA1}\\ y_{A1}-y_{bA1}\\ z_{A1}-z_{bA1}\end{array}\right]-\left[\begin{array}{ccc}c_{x}&0&0\\ 0&c_{y}&0\\ 0&0&c_{z}\end{array}\right]\left[\begin{array}{ccc}\dot{x}_{A1}-\dot{x}_{ bA1}\\ \dot{y}_{A1}-\dot{y}_{bA1}\\ z_{A1}-\dot{z}_{bA1}\end{array}\right]\\ =\left[\begin{array}{ccc}-k_{x}\Big{[}(x-x_{b})+b(\varphi_{z}-\varphi_{zb})-c \Big{(}\varphi_{y}-\varphi_{yb}\Big{)}\Big{]}-c_{x}\Big{[}(\dot{x}-\dot{x}_{ b})+b\Big{(}\ddot{\varphi}_{z}-\dot{\varphi}_{zb})-c\Big{(}\ddot{\varphi}_{y}- \dot{\varphi}_{yb}\Big{)}\Big{]}\\ -k_{y}\big{[}(y-y_{y})+a(\varphi_{z}-\dot{\varphi}_{zb})+c(\varphi_{x}-\varphi_ {xb})-c_{y}\Big{[}(\dot{y}-\dot{y}_{b})+a\Big{(}\ddot{\varphi}_{z}-\dot{\varphi }_{zb})+c\big{(}\dot{\varphi}_{x}-\dot{\varphi}_{xb}\big{)}\Big{]}\\ -k_{z}\Big{[}(z-z_{b})-a\Big{(}\varphi_{y}-\varphi_{yb}\Big{)}-b(\varphi_{x}- \varphi_{xb})\Big{]}-c_{z}\Big{[}(\dot{z}-\dot{z}_{b})-a\Big{(}\ddot{\varphi}_{ y}-\dot{\varphi}_{yb}\Big{)}-b\Big{(}\dot{\varphi}_{x}-\dot{\varphi}_{xb}\Big{)} \Big{]}\end{array}\right]\end{array}  \tag{2}\]  
The forces of the other springs are obtained in the same manner. Let \(\mathbf{X}_{1}=\left[ \begin{array}{cccccccc}x&y&z&\varphi_{x}&\varphi_{y}& \varphi_{z}\end{array} \right]^{T}\) and \(\mathbf{X}_{b1}=\left[ \begin{array}{cccccccc}x_{b}&y_{b}&z_{b}&\varphi_{bx}& \varphi_{by}&\varphi_{bz}\end{array} \right]^{T}\) be the displacements and rotations vectors of the upper and bottom plates of the AVIS, respectively. Substituting the spring forces into Equation (1) yields:  
\[\ddot{\mathbf{X}}_{1}=\mathbf{M}^{-1}\Big{(}\mathbf{A}_{1}\mathbf{X}_{1}+ \mathbf{A}_{2}\dot{\mathbf{X}}_{1}+\mathbf{B}_{\mathbf{a}}\mathbf{U}_{\mathbf{ a}}+\mathbf{D}_{1}\mathbf{X}_{\mathbf{b}1}+\mathbf{D}_{2}\dot{\mathbf{X}}_{ \mathbf{b}1}\Big{)} \tag{3}\]  
Thus, Equations (4) and (5) are derived. Applying Equation (3) on all the vibration sensors, the measured values are given by:  
\[ \begin{array}{c}\dot{x}_{1}=\dot{x}+L_{xy}  
+++ ==WARNING: Truncated because of repetitions==
\dot{\varphi}_{z}-L_{xz}\dot{ \varphi}_{y}\\ \dot{x}_{2}=\dot{x}-L_{xy}\dot{\varphi}_{z}-L_{xz}\dot{\varphi}_{y}\\ \dot{y}_{1}=\dot{y}+L_{zx}\dot{\varphi}_{z}+L_{xz}\dot{\varphi}_{x}\\ \dot{y}_{1}=\dot{y}-L_{zx}\dot{\varphi}_{z}+L_{xz}\dot{\varphi}_{x}\\ \dot{z}_{1}=\dot{z}-L_{zx}\dot{\varphi}_{y}-L_{zy}\dot{\varphi}_{x}\\ \dot{z}_{2}=\dot{z}-L_{zx}\dot{\varphi}_{y}+L_{zy}\dot{\varphi}_{x}\\ \dot{z}_{3}=\dot{z}+L_{zx}\dot{\varphi}_{y}+L_{zy}\dot{\varphi}_{x}\\ \dot{z}_{4}=\dot{z}+L_{zx}\dot{\varphi}_{y}-L_{zy}\dot{\varphi}_{x}\end{array} \] (4)  
Hence, Equation (6) is derived. ++  
* [PERSON] et al. (2006) [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Development of active six-degrees-of-freedom micro-vibration control system using hybrid actuators comprising air actuators and giant magnetostrictive actuators. _Smart Mater. Struct._**2006**, _15_, 1133-1142. [CrossRef]
* [PERSON] et al.