## 2 Working Principle and Modeling
### Mathematical Model  
The force state of the HFAM is shown in Figure 6. It is assumed that both rubber and liquid are incompressible. The reduction in the RD volume chamber is equal to the increase in the CD volume chamber. When the VCM propels shaft 1 to the right, the liquid pressure in the sleeve chamber increases. The heightened pressure propels the CD and shaft 2 in the same direction. Throughout the motion, there are static friction forces and viscous friction forces on the shafts, and the diaphragm deformations produce elastic forces. The axial force balance expressions of the RD and the CD are  
\[F_{\rm U}-F_{\rm IV}-F_{\rm 1k}-F_{\rm c1}-F_{\rm f1}=m_{\rm 1}\tilde{x}_{\rm 1} \tag{1}\]  
\[F_{\rm 2V}-F_{\rm 2k}-F_{\rm c2}-F_{\rm f2}=m_{\rm 2}\tilde{x}_{\rm 2} \tag{2}\]  
where F\({}_{\rm U}\) is the driving force of the VCM, F\({}_{\rm IV}\) and F\({}_{\rm 2V}\) are, respectively, the forces applied to the RD and the CD by the liquid, F\({}_{\rm 1k}\) and F\({}_{\rm 2k}\) are, respectively, the elastic forces arising from the deformation of the RD and the CD, F\({}_{\rm c1}\) and F\({}_{\rm c2}\) are, respectively, the viscous friction forces of shafts 1 and 2, F\({}_{\rm f1}\) and F\({}_{\rm f2}\) are, respectively, the static friction forces of shafts 1 and 2, x\({}_{\rm 1}\) and x\({}_{\rm 2}\) are, respectively, the displacements of the RD and the CD, and  
Figure 5: Working principle of the HFAM. Figure 6: Force state diagram of HFAM. m\({}_{1}\) and m\({}_{2}\) are, respectively, the masses of the shafts, sensors, and other objects connected to the RD and the CD. Considering that the mass of the fluid in the sleeve chamber is small and the precise motion state is difficult to calculate, the influence of the fluid mass is ignored in the system model. The output force of the VCM is related to the electric current acting on the VCM, and the expression is  
\[F_{\text{U}}=BIL_{\text{U}} \tag{3}\]  
where B is the magnetic induction, I is the electric current acting on the VCM, L\({}_{\text{U}}\) is the length of the coil actuator, and BL\({}_{\text{U}}\) is the force constant of the VCM, expressed by K\({}_{\text{f}}\). The electric current of the VCM depends on the voltage, and the expression is  
\[U=IR+L\frac{dI}{dt} \tag{4}\]  
where U is the voltage acting on the VCM, R is the resistance of the VCM, and L is the inductance of the VCM. The force exerted by the liquid on the RD and the CD depends on their effective areas in the axial direction, and the expression is  
\[\frac{F_{\text{1V}}}{F_{\text{2V}}}=\frac{A_{1}}{A_{2}}=k_{\text{p2}} \tag{5}\]  
where A\({}_{1}\) and A\({}_{2}\) are, respectively, the effective area in the axial direction of the RD and the CD, and k\({}_{\text{p2}}\) is the area proportional coefficient. The elastic forces of the RD and the CD are related to the displacements, and their expressions are  
\[F_{\text{1k}}=k_{1}x_{1} \tag{6}\]  
\[F_{\text{2k}}=k_{2}x_{2} \tag{7}\]  
where k\({}_{1}\) and k\({}_{2}\) are the elastic slopes of the RD and the CD. They must be determined through finite element simulation. The viscous friction forces associated with the RD and the CD depend on the speed, and the expressions are  
\[F_{\rm U}-F_{\rm IV}-F_{\rm 1k}-F_{\rm c1}-F_{\rm f1}=m_{\rm 1}\tilde{x}_{\rm 1} \tag{1}\]0  
\[F_{\rm U}-F_{\rm IV}-F_{\rm 1k}-F_{\rm c1}-F_{\rm f1}=m_{\rm 1}\tilde{x}_{\rm 1} \tag{1}\]1  
where c\({}_{\text{v1}}\) and c\({}_{\text{v2}}\) are, respectively, the viscous friction coefficients of shafts 1 and 2. To make the static friction forces of the objects at rest equal to the external forces, the expressions of F\({}_{\text{f1}}\) and F\({}_{\text{f2}}\) are set to  
\[F_{\text{f1}}=