## 3 Design and Realization
### Relative Displacement Feedback Strategy  
As shown in Figure 14, a relative displacement feedback strategy was adopted to generate negative stiffness. The driven current of the electromagnetic actuator was set proportionally to the displacement of the optical platform relative to the ground, which can be expressed as  
\[I=k_{i}[x_{1}(t)-x_{2}(t)], \tag{7}\]  
where \(k_{i}\) is the current feedback coefficient, \(x_{1}(t)\) is the absolute displacement of the optical platform, and \(x_{2}(t)\) is the absolute displacement of the ground.  
In this research, the displacement of the optical platform relative to the ground, i.e., \(x_{1}(t)\)-\(x_{2}(t)\), was measured by the proposed laser interferometry sensor. According to Equations (5) and (7), the dynamic equation of LFPEI, which is composed of the pneumatic spring and paralleled EANSG, is given by  
\[\left\{\begin{array}{l}m\ddot{x}_{1}(t)+c\dot{[}\dot{x}_{1}(t)-\dot{x}_{2}(t )\big{]}+k[x_{1}(t)-x_{2}(t)]=F(t)\\ F(t)=k_{s}[x_{1}(t)-x_{2}(t)]\end{array}\right., \tag{8}\]  
where \(m\) is the mass of the payload, \(c\) and \(k\) are the damping and stiffness of the pneumatic spring. \(F(t)\) is the force exerted by the electromagnetic actuator on the payload, \(k_{s}=NBLk_{i}\).  
Figure 14: Scheme of relative displacement feedback.  
Figure 13: Bandwidth test curve of the amplifier.  
After the Fourier transform and dimensionless processing, the displacement transmissibility of LFPEI can be expressed as  
\[\left|\frac{X_{1}}{X_{2}}\right|=\sqrt{\frac{\left(1-\alpha\right)^{2}+4\xi^{2} \beta^{2}}{\left[1-\alpha-\beta^{2}\right]^{2}+4\xi^{2}\beta^{2}}}, \tag{9}\]  
where \(\alpha=k_{\text{s}}/k\) is the feedback coefficient, \(w_{\text{n}}=\sqrt{k/m}\) is the natural frequency of the pneumatic spring, \(\xi=c/2 mw_{\text{n}}\) is the damping ratio, and \(\beta=w/w_{\text{n}}\) is the frequency ratio.  
According to Equation (9), the natural frequency \(w_{\text{n}}\)', damped natural frequency \(w\), and resonance peak \(T_{\text{max}}\) of LFPEI, respectively, can be expressed as  
\[w_{n}^{\prime}=w_{n}\sqrt{1-\alpha}, \tag{10}\]  
\[w=w_{n}^{\prime}\sqrt{1-\xi^{2}}, \tag{11}\]  
\[T_{\text{max}}=\left|\frac{X_{1}}{X_{2}}\right|_{\text{max}}=\sqrt{\frac{1- \alpha}{4\xi^{2}}+1}, \tag{12}\]  
Due to the low damping of pneumatic springs, its damped natural frequency \(w\) was usually taken as the natural frequency \(w_{\text{n}}\)'. Equation (10) shows that a positive \(\alpha\) makes the natural frequency \(w_{\text{n}}\)' of LFPEI lower than that of a pneumatic spring, which results from the negative stiffness characteristics generated by the relative displacement feedback strategy. Moreover the closer the feedback coefficient \(\alpha\) is to 1, the lower the natural frequency \(w_{\text{n}}\)' will be. To prevent the stiffness of LFPEI from being negative, which would result in instability, the range of the feedback coefficient \(\alpha\) was limited to (0, 1). Therefore, in this paper, \(\alpha=0.5\) was adopted to validate the effectiveness of the proposed method. The theoretical natural frequency of LFPEI \(w_{\text{n}}\)' is 71% of that of the pneumatic spring \(w_{\text{n}}\).  
The expression of resonance peak \(T_{\text{max}}\) in Equation (12) indicates that it is affected by the feedback coefficient \(\alpha\) in the same way as the natural frequency. Therefore, it can be concluded that the proposed method can enhance the low-frequency vibration isolation performance and simultaneously suppress the resonance peak of a pneumatic spring through adopting a relative displacement feedback strategy.  
The control block diagram of the electromagnetic actuator is shown in Figure 15. An analog control voltage \(V\) is the output according to the displacement of the optical platform relative to the ground, an amplifier converts the control voltage \(V\) into driven current \(I\), and drives the electromagnetic actuator to exert force \(F\)(t) on the optical platform.