## 3 Design and Realization
### Electromagnetic Actuator  
To precisely generate negative stiffness within the micro-vibration range of an optical platform, an electromagnetic actuator with high linearity and high precision was proposed, as shown in Figure 7. A pair of cubic permanent magnets in a repulsive configuration was used to produce uniform magnetic flux density in the air gap. Two outer yokes and one inner yoke were symmetrically placed around the permanent magnets to form a closed double magnetic circuit. The cubic permanent magnet is made of NdFe35, the material of outer yokes and inner yoke are DT4C. A coil was located in the air gap and moves along the inner yoke, the magnetic force applied on the coil can be expressed as  
\[F=nB\Pi, \tag{5}\]  
where \(n\) and \(l\) are the turns and length of the coil, \(B\) is the magnetic flux density in the air gap, \(I\) is the driven current; \(F\) is the magnetic force, and its direction is perpendicular to \(I\) and \(B\). Figure 5: Resolution of the laser interferometry sensor. Figure 6: Stability of the laser interferometry sensor. The nonlinearity of the electromagnetic actuator was defined as the non-uniformity of the magnetic flux density in the air gap, which is expressed as  
\[B_{u}=\frac{\int_{l}^{m}\left|B(x)-B_{are}\right|\!dx}{B_{are}L_{lm}}\times 100 \%, \tag{6}\]  
where \(B_{are}=1/L_{lm}\int_{l}^{m}B(x)dx\) is the average magnetic flux density along the air gap, \(B(x)\) is the magnetic flux density along the air gap, and \(L_{lm}\) is the length of the air gap. As shown in Figure 8, an electromagnetic actuator with geometric parameters listed in Table 1 was developed. The uniformity of the magnetic flux density in the air gap is a key index that influences the force linearity of the electromagnetic actuator, so an experimental setup shown in Figure 9 was set on an optical platform to measure the magnetic flux density. The electromagnetic actuator was placed in parallel with an electronically controlled stage, and the stage was driven by a stepper motor  
Figure 7: Principle of the proposed electromagnetic actuator: (**a**) scheme of electromagnetic actuator; and (**b**) the simplified two-dimensional structure. to generate a standard step of 0.5 mm. One end of a Gaussian meter hall probe with a resolution of 0.1 mT was fixed with the stage, while the other was inserted into the air gap of the electromagnetic actuator. Experimental results in Figure 10 show that along the 60 mm stroke of the electromagnetic actuator, the maximum and minimum magnetic flux densities are 294.6 mT and 293.2 mT, respectively, and the nonlinearity of magnetic flux density calculated by Equation (6) is less than 0.48%. The small jump in the magnetic flux density curve was due to the resolution of the Gaussian meter. The _magnetic force-drive current_ curved surface in Figure 11 well shows the static characteristics of the electromagnetic actuator. The electromagnetic active-negative-stiffness method uses the magnetic force of the electromagnetic to realize feedback control. The magnetic force was generated instantaneously, so its dynamic performance mainly depends on the dynamic performance of an amplifier. It can be seen from the step response curve in Figure 12, the rise time of the amplifier is less than 10 us, and the settling time is less than 20 us. The bandwidth test curve in Figure 13 shows that there is no distortion and attenuation to the sweep signal from 1 to 5 kHz, which indicates that the bandwidth of the amplifier is higher than 5 kHz. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(I\) **(mm)** & \(\mathbf{h_{m}}\) **(mm)** & \(\mathbf{T_{m}}\) **(mm)** & \(\mathbf{h}\) **(mm)** & \(\mathbf{l_{se}}\) **(mm)** & \(\mathbf{T_{s}}\) **(mm)** & \(\mathbf{T_{y}}\) **(mm)** & \(\mathbf{l_{g}}\) **(mm)** & \(\mathbf{n}\) \\ \hline
40 & 15 & 30 & 7 & 17 & 3 & 8 & 70 & 220 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of an electromagnetic actuator. Figure 8: Photo of the electromagnetic actuator prototype.