## 6 Conclusions  
Soft manipulators with different PAMs parallel to each other process asymmetric hysteresis, actuator couplings, and stiffness-dependent nonlinearities, leading to inaccuracy positioning. To compensate for system nonlinearities and to decouple the actuator coupling effects, a SDCHM model for the parallel PAMs-driven manipulator is proposed. The SDCHM presents a cascaded model architecture with an inherent hysteresis submodule, an actuator coupling submodule, and a stiffness-dependent submodule. Based on the established model, its compensator SDHC can be achieved easily without any simplification, allowing it to compensate and decouple for actuator hysteresis couplings at different stiffness states. The experimental results suggest that the SDCHM performs well and possesses good accuracy in predicting inputs-outputs of the soft manipulator in different working conditions. In addition, the compensation effect of the inverse compensator SDIHC is significantly better than that of the controller only with the inverse hysteresis compensator, successfully mitigating the hysteresis coupling phenomenon in the PAM-based soft manipulator. Although the proposed modeling and compensating methods are showcased on a manipulator driven by three parallel PAMs, they are believed to be easily extended to parallel driving systems based on more PAMs. However, the proposed model has not yet considered the impacts of high driving frequencies and external interferences, such as changes in external loads, on the hysteresis couplings of the PAM-based manipulator. Additionally, extending the proposed method to soft manipulators with multiple sections is worth further study, and a closed-loop strategy based on the proposed model will be the next key point for more robust control.  
Conceptualization, Y.Z.; methodology, Y.Z.; software, H.Q.; validation, Q.C.; formal analysis, Z.L.; investigation, Z.L.; resources, H.Q. and Q.C.; writing--original draft preparation, Y.Z.; writing--review and editing, H.Q.; supervision, L.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China [grant number 62103090], and Fundamental Research Funds for the Central Universities [grant numbers N2403020].  
Figure 13: Compensation control results with the SDHC compensator with \(p_{0}=0.6\)bar. (a) PAM I; (b) PAM II; (c) PAM III.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{4}{c}{**SDIHC Compensator**} & \multicolumn{2}{c}{**Individual Hysteresis Compensator**} & \multicolumn{2}{c}{**PID-Based Compensator**} \\ \hline
**Group** & \(\mathbf{e_{w}}\) **(\%)** & **RMSE (mm)** & \(\mathbf{e_{w}}\) **(\%)** & **RMSE (mm)** & **(\%)** & **RMSE (mm)** \\ \hline (d) & 0.83 & 1.06 & 7.02 & 9.12 & 3.92 & 4.01 \\ (e) & 0.62 & 0.34 & 3.58 & 3.55 & 1.75 & 3.36 \\ (f) & 0.75 & 1.02 & 6.88 & 11.10 & 3.42 & 3.89 \\ (g) & 1.86 & 1.33 & 10.23 & 13.86 & 6.21 & 5.87 \\ (h) & 2.33 & 1.56 & 8.89 & 10.21 & 3.89 & 5.12 \\ (i) & 0.61 & 0.29 & 1.89 & 2.03 & 2.05 & 1.96 \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.