## 8 Conclusions  
This paper successfully integrated the Natural Coordinates Formulation (NCF) and the Absolute Nodal Coordinates Formulation (ANCF) to model an elephant's head-trunk system as a rigid-flexible multibody system. The NCF was employed to model the rigid head, while the ANCF-based modeling of the trunk's tubular structure was carried out. The NCF/ANCF model constructed captured biomechanical properties while obeying the constraints of a rigid-flexible interface and demonstrated versatility and applicability. The proposed model provides an accurate and efficient framework for capturing the dynamics of large deformations of the trunk and large rotations of the head. The Fuzzy Logic Control (FLC) constraint stabilization method was employed to ensure the robust handling of constraint stabilization, effectively eliminating drift issues in long-term simulations. Dynamic analysis showed the realistic response of the system under various loading conditions, including bending moments, applied torques, and combined forces, highlighting the intricate interplay of external forces and internal constraints. Across all constraint types, the maximum violation values were exceedingly small, which means that the rigid-flexible interface proposed in this paper, as well as the integration and stabilization methods implemented, is effective. The system adheres to the constraints over time, proving the robustness and precision of the simulation framework. These results not only enhance the understanding of rigid-flexible dynamics but also offer valuable insights for applications in robotics, biomechanics, and continuum manipulator design, paving the way for future advancements in complex system modeling. While this study focused on modeling rigid-flexible coupling dynamics, the developed framework provides a solid foundation for future enhancements. Potential extensions include the integration of active control strategies, where model-based control methods could be explored to regulate trunk movement by utilizing the system's dynamic properties for improved adaptability and precision. Additionally, future work could incorporate soft tissue elasticity through compliant elements or reduced-order deformation models to further refine the head-trunk interaction, ensuring a more comprehensive representation of the system's biomechanics.  
Conceptualization, A.N. and A.E.-A.; methodology, A.N. and A.G.; software, A.G. and M.O.H.; validation, M.O.H. and A.G.; formal analysis, A.N. and A.E.-A.; writing--original draft preparation, A.G. and M.O.H.; writing--review and editing, A.N. and A.E.-A.; supervision, A.N. and A.E.-A.; data curation, M.O.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors would like to acknowledge E-JUST University for providing licensed MATLAB software version R2024, developed by MathWorks, and their cooperation and supervision, provided through Professor [PERSON].  
The authors declare that they have no conflicts of interest.  
The following abbreviations are used in this manuscript:  
ANCF Absolute Nodal Coordinates Formulation  
NCF Natural Coordinates Formulation  
FLC Fuzzy Logic Control