## 2 Principle of Active Vibration Isolation
### FB and FF Control Theory  
Enlightened by the idea of independent channel control design that was based on the different disturbance transmission paths [29], and referring to the principle of force superposition of the FB and FF hybrid control system, the control force, generated by the voice-coil motor, can be decomposed into two parts: one provided by the FFC and the other by the FBC. It is worth noting that the driving forces of the FBC and FFC are not independent of each other in practical applications. It was found that the FFC will effectively prevent the ground disturbance that will transmit to the platform [11], and the corresponding FBC output force becomes smaller because the vibration amplitude of the payload decreases. Here, \(s\) represents the Laplace operator. Assuming that the initial speed of the platform is zero, and taking \(f_{\mathrm{noise}}\) and \(v_{\mathrm{b}}(s)\) as external disturbance sources, the value of FF force \(f_{\mathrm{ffc}}(s)\) is determined by \(v_{\mathrm{b}}(s)\) and has nothing to do with \(v_{\mathrm{p}}(s)\). Without considering the nonlinear factors of sensors, signal amplifier circuits, and actuators, the relationship between \(f_{\mathrm{ffc}}(s)\) and the ground detection signal \(v_{\mathrm{b}}(s)\) can be expressed as follows:  
\[f_{\mathrm{ffc}}(s)=H_{\mathrm{ffc}}(s)v_{\mathrm{b}}(s) \tag{2}\]  
Figure 1: The single DOF mass-spring damping system with active control. where \(H_{\text{ffc}}(s)\) denotes the velocity-force transfer function. Now, the dynamic characteristics of the AVRS in terms of two aspects during the feedback control are investigated in the following paragraphs. (1) When the AVRS is open-looped, according to the principle of speed superposition, the platform speed signal \(v_{\text{p}}(s)\) can be decomposed into two parts, \(v_{1}(s)\) and \(v_{2}(s)\). Here, \(v_{1}(s)\) is the component of platform velocity transmitted from ground disturbance \(v_{\text{b}}(s)\) under the control of \(f_{\text{ffc}}(s)\), while \(v_{2}(s)\) is caused by \(f_{\text{noise}}\). Then, the expression of the system's kinematic transfer functions described in Figure 1 is given by the Equation (3):  
\[\left\{ \begin{array}{l}msv_{1}(s)+c[v_{1}(s)-v(s)]+\frac{k}{s}[v_{1}(s)-v(s) ]=-f_{\text{ffc}}(s)\\ msv_{2}(s)+cv_{2}(s)+\frac{k}{s}v_{2}(s)=f_{\text{noise}}(s)\end{array} \right. \tag{3}\]  
The equations to calculate \(v_{1}(s)\) and \(v_{2}(s)\) can be presented as follows:  
\[\left\{ \begin{array}{l}v_{1}(s)=\frac{(cs+k)-sH_{\text{ffc}}(s)}{ms^{2}+cs+k }v_{\text{b}}(s)\\ v_{2}(s)=\frac{f_{\text{noise}}(s)+s}{ms^{2}+cs+k}\end{array} \right. \tag{4}\]  
Therefore, when the system is open-looped, the platform speed can be obtained by:  
\[v_{\text{p}}(s)=v_{1}(s)+v_{2}(s) \tag{5}\]  
By substituting Equation (4) into Equation (5), finally, \(v_{\text{p}}(s)\) can be expressed by Equation (6) as below:  
\[v_{\text{p}}(s)=\frac{s}{ms^{2}+cs+k}\bigg{[}\bigg{(}\frac{cs+k}{s}-H_{\text{ ffc}}(s)\bigg{)}v_{\text{b}}(s)+f_{\text{noise}}(s)\bigg{]} \tag{6}\]  
Considering the combined effects of \(v_{\text{b}}(s)\) and \(f_{\text{noise}}(s)\), the FB force \(f_{\text{ffc}}(s)\) can be calculated by \(H_{\text{ffc}}(s)\) multiplied by \(v_{\text{p}}(s)\), which is expressed in Equation (7) below:  
\[f_{\text{ffc}}(s)=H_{\text{ffc}}(s)v_{\text{p}}(s) \tag{7}\]  
where \(H_{\text{ffc}}(s)\) stands for the forward channel transfer function between the platform velocity \(v_{\text{p}}(s)\) and feedback force \(f_{\text{ffc}}(s)\). Then, substituting Equation (6) into Equation (7), yields:  
\[f_{\text{ffc}}(s)=H_{\text{ffc}}(s)\frac{s}{ms^{2}+cs+k}\bigg{\{}\bigg{[}\frac {cs+k}{s}-H_{\text{ffc}}(s)\bigg{]}v_{\text{b}}(s)+f_{\text{noise}}(s)\bigg{\}} \tag{8}\]  
From Equation (8), it can be seen that \(f_{\text{ffc}}(s)\) contains the term \(H_{\text{ffc}}(s)\), which indicates that the influence of ground disturbance has not been eliminated by \(f_{\text{ffc}}(s)\).