## 6 Conclusions  
The authors of this paper proposed a new multi-degree-of-freedom seat damping suspension that was based on the cubic Stewart mechanism and an MRF damper. Furthermore, a virtual prototype model and test prototype model of the seat vibration-damping platform were established. Through theoretical analysis, experimental research, and numerical simulation, both passive and semi-active control of the seat suspension were studied. The main conclusions are as follows:  
1. The cubic Stewart mechanism was subject to kinematic analysis. The relative motion velocity of each leg of the Stewart mechanism was calculated from the center velocity of the upper and lower platforms, according to the reverse kinematics equation. Using this method, the measurement of parameters was simplified. When semi-active control was implemented, we used just two sensors to control six legs separately.
2. Forward and inverse dynamic models of the MRF damper were built. The experiment and simulation results showed that the forward dynamic model could accurately reflect the mechanical characteristics of the MRF damper; the inverse model developed  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Condition** & **Direction** & **\(z\)-Axis Translation** & **\(y\)-Axis Translation** & **Rotate Around \(x\)-Axis** \\  & & **(m/s\({}^{2}\))** & **(m/s\({}^{2}\))** & **(rad/s\({}^{2}\))** \\ \hline Lower platform & 0.78 & 0.17 & 2.02 \\  & Passive damping & 0.82 & 0.20 & 2.45 \\ Upper platform & Semi-active control & 0.71 & 0.16 & 1.83 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation results (RMS value) of the multi-DOF damping seat platform.  
Figure 28: Simulation results of the angular acceleration around the \(x\)-axis direction.  
by the BP neural network could accurately calculate the control current from the expected damping force.
3. Experimental research on the test prototype model showed that for high-frequency excitations, the seat damping suspension proposed in this paper could achieve vibration damping in the vertical, horizontal, and roll directions. Moreover, increasing the MRF damper's current for low-frequency excitations could effectively improve the low-frequency vibration response of the seat suspension in all directions.
4. A semi-active control method for multi-dimensional damping, based on the optimized fuzzy skyhook control method, was proposed. On this basis, the control algorithm was applied to the multi-dimensional seat vibration-damping platform, combined with the characteristics of the cubic Stewart mechanism; through the individual control of each leg, the vibration-damping performance of the platform on the seat was improved in all directions. The research results showed that using this method could simultaneously improve vibration-damping performance of the seat suspension in the vertical, horizontal, and roll directions, and relative to the lower platform of the seat suspension, the upper platform achieved vibration reduction in all three directions.  
In this study, a semi-active control method for multi-dimensional damping based on the optimized fuzzy skyhook control method was proposed. This method was based on the skyhook control method, and its control force output mode is consistent with the damping force output mode of a magnetorheological fluid damper, which is convenient to achieve. However, its control effects still have room for further improvement. Therefore, the control method should be further studied to obtain better control effects.  
Formal analysis, T.M.; Funding acquisition, G.J.; Methodology, T.L. and H.L.; Writing--original draft, T.M.; Writing--review & editing, F.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Informed consent was obtained from all subjects involved in the study.  
Not applicable.  
The authors declare no conflict of interest.