## 5 Conclusions  
In this paper, a robust adaptive backstepping control algorithm based on the PPF is proposed to enhance the riding comfort of heavy-duty vehicles. The development of the control algorithm addresses common system parameter uncertainties in detail. A PPF is employed to prescribe the dynamic behavior of the controlled system. Three estimators are introduced to derive the control signal from the unknown system parameters, and a novel adaptive law is designed to update these estimators, compensating for the effects of estimation errors. Both ride comfort requirements and safety constraints are taken into account. The CL stability is theoretically proven, and the performance of the proposed controller is evaluated through numerical simulations and experimental validation. The simulation and experimental results demonstrate that the proposed control algorithm effectively suppresses vertical vibration in the driver-seat system, even in the presence of parametric uncertainties. The advantage of the proposed control strategy lies in its ability to maintain the control target within a preset boundary range under system uncertainty. Through parameter estimation design, the proposed control can adapt to parameter variations within a certain range, and the uncertainties of each hardware in experiments also validate this argument. It is also noted that, from the definition (23) of the transformed error, if the system is subjected to a severe disturbance in a short period, the normalized error \(\varepsilon\) may exceed 1, making the control signal in such cases incalculable. To improve the robustness of the control strategy, PPF can be appropriately relaxed to ensure that the normalized error is always less than 1, alternatively, a buffer boundary \(\varepsilon_{th}<1\) can be employed to replace the normalized error \(\varepsilon\) when \(\varepsilon\) exceeds \(\varepsilon_{th}\), thereby ensuring that the control signal remains well-defined. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Suspension Type**} & **Displacement** & **Acceleration** & **Suspension Deformation** \\  & **(m)** & **(m/s2)** & **(m)** \\ \hline Original & 0.0163 & 1.2004 & 0.0145 \\ Passive & 0.0116 & 1.1674 & 0.0039 \\ Active & 0.0076 & 0.8020 & 0.0052 \\  & (\(\downarrow\)34.5\%) & (\(\downarrow\)31.3\%) & (\(\uparrow\)33.3\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMS values of the system response. The numbers marked with upward and downward arrows indicate the relative values compared to the original suspension. Figure 12: Actuating force in scenario 2. **Author Contributions:** Conceptualization, J.C. and Y.W. ; methodology, J.C. and Y.W. ; software, Y.W. ; validation, J.C.; formal analysis, W.H. ; investigation, Y.W. and D.X. ; resources, Y.W. ; data curation, Y.W. and D.X. ; writing--original draft preparation, Y.W. ; writing--review and editing, J.C., D.X. and W.H. ; visualization, W.H. ; supervision, Y.W. and W.H. ; project administration, Y.W. ; funding acquisition, J.C. and Y.W. All authors have read and agreed to the published version of the manuscript. **Funding:** This work is supported by the Chinese National Natural Science Foundation (52202498), the Open Fund of Intelligent Crane Engineering Research Center of Hunan Province (SYC202501), and partially supported by the Youth Fund of the National Natural Science Foundation of China (52305080) and the Natural Science Foundation for Youths of Hunan Province, China (2023 JJ40720). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Data are available upon reasonable request. **Acknowledgments:** We would like to express our sincere gratitude to [PERSON] and [PERSON] for their invaluable contributions to this research. Both [PERSON] and [PERSON] made significant contributions in writing, reviewing, and editing the manuscript, as well as in the visualization process. Their efforts and insights have greatly enhanced the quality and clarity of this study.