# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 2 System Modeling  
Figure 1 shows the schematic drawing of a 6-DoF AVIS and the allocation of the actuators and sensors is highlighted. The mechanical structure consists of a bottom plate fixed to the ground and a top plate carrying the machine that needs isolating from vibrations. There are four coil springs at the four corners connecting the two plates and supporting the mass of the top parts. To start with, let the coordinate systems be defined as follows:  
* The top plate fixed frame \(\mathbf{T}\) is a right-handed coordinate system whose origin is at the mass center of the plate; the Z-axis points upward, the X-axis points forward, the counterclockwise is the positive rotation direction. This coordinate system is depicted by red-colored vectors in Figure 1.
* The bottom plate fixed frame \(\mathbf{B}\) has its origin at the centroid of the bottom plate. Translation and rotation directions are similar to frame \(\mathbf{T}\). For the active isolation operation, eight VCMs generate linear forces between the two plates. Four of the VCMs generate translational movements perpendicular to the top plate, i.e., along the Z-axis. Two VCMs generate motions along the X-axis direction and the others along the Y-axis direction. Accordingly, eight vibration sensors have been arranged as shown in Figure 1.  
Assumption 1: The top plate is geometrically symmetric, and its mass is uniformly distributed. Assumption 2: Along an \(i\)-axis direction, each of the springs acts as a linear spring with the same stiffness \(k_{i}\) and the same damping factor \(c_{i}\). For convenience's sake, nomenclatures used in the remainder of the paper are listed as follows:  
* \(\mathbf{x}=\left[ \begin{array}{cc}x&y&z\end{array} \right]^{T}\) is the displacement of the origin \(\mathbf{O}\) in the \(x\)-, \(y\)-, and \(z\)-directions;
* \(\mathbf{\varphi}=\left[ \begin{array}{cc}\varphi_{x}&\varphi_{y}&\varphi_{z}\end{array}  \right]^{T}\) is the rotation about \(x\)-, \(y\)-, and \(z\)-axes of the upper plate;
* \(x_{b}=\left[ \begin{array}{cc}x_{b}&y_{b}&z_{b}\end{array} \right]^{T}\) and \(\mathbf{\varphi}_{b}=\left[ \begin{array}{cc}\varphi_{xb}&\varphi_{yb}&\varphi_{ zb}\end{array} \right]^{T}\) are the displacement and the angular position of the bottom plate, respectively;
* \(m\) is the mass of the sprung weight;
* \(\mathbf{J}=\left[ \begin{array}{ccc}J_{xx}&0&0\\ 0&J_{yy}&0\\ 0&0&J_{zz}\end{array} \right]\) is the inertia tensor of the sprung mass;
* \(f_{ji}\) is the force in the \(j\)-axis direction generated by the \(i^{th}\) actuator in the same direction;
* \(f_{kj}\) is the force in the \(j\)-axis direction generated by the four springs;
* \(L_{ij}\) is the distance in the \(j\)-axis direction from the center of mass to the sensor measuring the velocity in the \(i\)-axis direction (m);
* \(l_{ij}\) is the distance in the \(j\)-axis direction from the center of mass to the actuator that generates the force in the \(i\)-axis direction (m). The dynamics of the system are obtained from the translational and rotational motion equations. \[ \begin{array}{c}m\ddot{\mathbf{x}}=\sum\limits_{i}\mathbf{f}_{i}\\ \mathbf{f}\ddot{\mathbf{\varphi}}=\sum\limits_{i}\mathbf{\tau}_{i}\end{array}  \tag{1}\]  
where \(\mathbf{f}_{i}\) is the force vector. The total force includes the actuator's forces and spring's forces. It is worth noting that the force of a spring is proportional to its displacement and velocity by the spring stiffness and damping coefficient, respectively. Subsequently, \(\mathbf{\tau}_{i}\) is the torque vector. In addition, suppose there is a point at the top plate with the initial position given by a vector \(\mathbf{P}_{0}=\left[ \begin{array}{cc}x_{p0}&y_{p0}&z_{p0}&1\\ \end{array} \right]^{T}\) in \(\mathbf{T}\) (1 is the scale factor). After a differential  
Figure 1: Schematic drawing of the 6-DoFs AVIS and the allocation of actuators and sensors.