# A High-Precision Active Vibration Isolation Control System: Experimental Study
## 2 System Modeling  
The schematic illustration of the controlled 6-DoF AVIS with its coordinates is introduced in Figure 1. The system consists of two platforms: the top platform and the base platform. The top and base platforms' coordinates are defined as right-handed systems, in which the \(z\)-axis points upward, the \(x\)-axis is directed forward, and the positive \(y\)-axis is pointing right. Define \(O_{t}\) as the origin of the top platform, located at the center of mass of the top platform. Hence, \(x_{t}\), \(y_{t}\), and \(z_{t}\) are the three axes in the top platform. Similarly, \(O_{b}\) is the origin and \(x_{b}\), \(y_{b}\), \(z_{b}\) are the three axes in the base platform. The top platform is supported by four vertical coil springs at the four corners, connecting the top and base platform. The vibration from the working environment will be transferred through the base platform and springs to the top platform. The springs act as passive isolators, isolating the vibration affecting the top platform. However, due to the characteristics of the system, this passive vibration isolation performs well in the high-frequency region but does not have any effect in the low-frequency region. Hence, the actuators are inserted between the two platforms to deal with this problem. The system uses four vertical VCMs, \(Z_{1},~{}Z_{2},~{}Z_{3},~{}Z_{4}\), to generate the forces \(F_{vz1}\), \(F_{vz2}\), \(F_{vz3}\) and \(F_{vz4}\) along the \(z_{t}\)-axis. The two horizontal VCMs \(X_{1},~{}X_{2}\) generate the forces \(F_{vx1}\), \(F_{vx2}\) along the \(x_{t}\)-axis, and the two horizontal VCMs \(Y_{1},~{}Y_{2}\) operate along the \(y_{t}\)-axis. Each VCM consists of two halves, one half is attached to the underside of the top platform, while the other half is attached to the base platform's upper surface, and they move in conjunction with each other. The motion of the top platform is limited by the plate guides, hence, protecting the VCMs from collision in the case of non-control. The motion of the AVIS is collected by 11 geophones, in which eight sensors are used to collect the velocity of the top platform in three directions (\(x_{t}\), \(y_{t}\), and \(z_{t}\)). The remaining three sensors are aimed at collecting the translational velocity from the base in three directions (\(x_{b}\), \(y_{b}\), and \(z_{b}\)). In the top platform, four geophone sensors are installed vertically along the \(z_{t}\)-axis, two geophone sensors are installed horizontally along the \(x_{t}\)-axis, and the other two geophone sensors are along the \(y_{t}\)-axis, as shown in Figure 1.  
The state-space form of the system dynamics is obtained as follows [18]:  
\[\dot{\mathbf{X}}=\mathbf{AX}+\mathbf{BU_{\mathrm{p}}}+\mathbf{DX_{\mathrm{b}}}+\mathbf{F_{\mathrm{t}}}, \tag{1}\] \[\mathbf{Y}=\mathbf{CX}\]  
in which  
* \(\mathbf{X}\in\mathbf{R}^{12}\) is the state vector containing the translational and rotational displacements and velocities of the top platform;
* \(\mathbf{U_{\mathrm{p}}}\in\mathbf{R}^{6}\) is the control input vector that contains the total of forces and torques in the corresponding 6-DoF;
* \(\mathbf{Y}\in\mathbf{R}^{6}\) is the output vector of the control system, which consists of the 6-DoF velocities;
* \(\mathbf{X_{\mathrm{b}}}\in\mathbf{R}^{12}\) is the vibration, i.e., all displacements and velocities, from the base platform;
* \(\mathbf{F_{\mathrm{t}}}\in\mathbf{R}^{6}\) contains the other disturbances acting directly on the top platform;
* The coefficient matrices \(\mathbf{A},\;\mathbf{B},\;\mathbf{C}\) and \(\mathbf{D}\) are written as in [18]. Another point is that without the external vibrations and disturbances, the relationship between the transfer matrix and the state space of the system is described as [19]  
\[\mathbf{G}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B} \tag{2}\]  
Figure 1: Configuration of the 6-DoF AVIS.