## 2 Principle of Active Vibration Isolation
### FB and FF Control Theory
Therefore, the platform residual disturbance from ground disturbance, together with \(f_{\text{noise}}(s)\), will be suppressed by the force component \(f_{\text{ffc}}(s)\). Thus, before the closed-loop control is applied, the motor output force \(f_{\text{M}}(s)\) is calculated as follows:  
\[f_{\text{M}}(s)=f_{\text{ffc}}(s)+f_{\text{ffc}}(s) \tag{9}\]  
According to the above formula, the relationship between the motor driving force \(f_{\text{M}}(s)\) and ground vibration \(v_{\text{b}}(s)\), together with surface disturbance \(f_{\text{noise}}(s)\), can be obtained as below:  
\[ \begin{array}{lcl}f_{\text{M}}(s)&=H_{\text{ffc}}(s)\frac{s}{ms^{2}+cs+k} \bigg{\{}\frac{cs+k}{s}+\frac{ms^{2}+|c-H_{\text{ffc}}(s)|+k}{H_{\text{ffc}}(s) s}H_{\text{ffc}}(s)\bigg{\}}v_{\text{b}}(s)\\ &+H_{\text{ffc}}(s)\frac{s}{ms^{2}+cs+k}f_{\text{noise}}(s)\end{array}  \tag{10}\]  
(2) In the closed-loop system, the motor force acts on the system at the value \(f_{\text{M}}(s)\) shown in Equation (10), and the table vibration is rapidly attenuated due to the effect of \(f_{\text{M}}(s)\). As the output of the controller remains the same before being updated, the dynamic process of the system from the perspective of computer control can be analyzed. Whenignoring the response delay of the system, assuming that at some point after the control loop is closed, the platform velocity under the combined effect of the motor driving force \(f_{\text{M}}{}^{\prime}(s)\), \(v_{\text{b}}(s)\), and \(f_{\text{noise}}(s)\) is \(v_{p}{}^{\prime}(s)\). This means that dynamic Equation (1) still holds. Then, the following equations can be derived:  
\[\left\{ \begin{array}{l}{m\text{s}{v_{p}}^{\prime}(s)+c\big{[}{v_{\text{p}}}^{ \prime}(s)-v_{\text{b}}(s)\big{]}+\frac{k}{s}\big{[}{v_{\text{p}}}^{\prime}(s)- v_{\text{b}}(s)\big{]}=f_{\text{noise}}(s)-{f_{\text{M}}}^{\prime}(s)}\\ {f_{\text{M}}}^{\prime}(s)=H_{\text{ffc}}(s)v_{\text{b}}(s)+H_{\text{flc}}(s){ v_{\text{p}}}^{\prime}(s)\end{array} \right. \tag{11}\]  
Simplifying the above equations, \(v_{p}{}^{\prime}(s)\) can be obtained in Equation (12):  
\[{v_{p}}^{\prime}(s)=\frac{f_{\text{noise}}(s)+[(c+k/s)-H_{\text{ffc}}(s)]v_{ \text{b}}(s)}{ms+c+k/s+H_{\text{flc}}(s)} \tag{12}\]  
Theoretically, if the system is ideal, the model of FF can be identified as:  
\[H_{\text{ffc}}(s)=\frac{cs+k}{s} \tag{13}\]  
From the derived formula, it is observed that FFC can eliminate the impact of ground vibration on the system. Therefore, rather than deteriorating the system's stability, proper FFC will improve the robust performance of the system. To better reflect the influence of FF and FB parameters on the damping performance of AVRS, the vertical vibration transmissibility curve, which is used to analyze the relationship between the ground disturbance \(v_{\text{b}}(s)\) and the platform vibration \(v_{\text{p}}{}^{\prime}(s)\) while ignoring \(f_{\text{noise}}(s)\), can be rearranged from the Equation (12) as:  
\[\frac{{v_{\text{p}}}^{\prime}(s)}{{v_{\text{b}}}(s)}=\frac{(c+k/s)-H_{\text{ ffc}}(s)}{ms+c+k/s+H_{\text{flc}}(s)} \tag{14}\]  
Compared with the passive curve shown by the red line in Figure 2, the pure FF proportional (P) control effect in the dotted red line is not ideal because it amplifies the amplitude at the resonant frequency point while suppressing the vibration at high frequencies. The optimal coefficient value of P is a trade-off between the high- and low-frequency damping performance. Therefore, merely P in FFC is not sufficient, especially when considering the high-performance requirements of ultra-precision equipment. Figure 2: The velocity transmissibility of one DOF AVRS in the vertical direction in different situations: Passive, FF P control FFC\({}_{\text{P}}\), FF I control FFC\({}_{\text{L}}\), FF PI control FFC\({}_{\text{PI}}\), FB P control based on FF PI control FBC\({}_{\text{P}}\) + FFC\({}_{\text{PI}}\), FB big P gain control based on FF PI control FBC\({}_{\text{BPI}}\) + FFC\({}_{\text{PI}}\).