## 7 Dynamic Simulation
Figure 21 presents a side-by-side comparison of the simulated model and real trunk motion over time. To replicate the natural curling movement, a time-dependent bending moment of \(M=30t\) was applied at a point located one-quarter of the trunk's length from the distal end, acting about the y-axis. The external force formulation described in Equation (16) was applied to the ANCF-based trunk model. Initially, both the simulated and real trunks remained nearly straight, indicating a minimal bending state. As time progressed, the trunk gradually bent, forming a pronounced curvature that closely resembled the real trunk's upward motion towards the mouth in an active manner. The comparison demonstrates a strong agreement between the simulated and real movements, particularly in the bending pattern and temporal evolution of the curvature. This confirms that the proposed dynamic model effectively captures the natural biomechanics of an elephant trunk. Additionally, another simulation was carried out where the system was subjected to double-acting forces to further explore the dynamic response under combined loading conditions; see Figure 22. The trunk rested in a horizontal position and was subjected to a bending moment about the \(y\)-axis, while the rigid head experienced an applied torque about the \(z\)-axis. Figure 21: Dynamic simulation of an elephant’s trunk motion along with the corresponding actual movement during feeding. Figure 20: Dynamic simulation of an elephant’s head–trunk when the trunk is subjected to a bending moment. Figures 23-26 illustrate the violations of the constraints (rigidity, distance, angle, and orthogonality) over time and demonstrate the effectiveness of the integration method and  
Figure 22: Dynamic simulation of elephant’s head–trunk coupling. the rigid-flexible interface. For rigidity constraints, the deviations remained within \(10^{-7}\), ensuring the rigid body's behavior was accurately maintained. The distance constraints showed minimal violations, confined to the order of \(10^{-8}\), confirming the precise preservation of the distance between nodes. Angle constraints exhibited negligible deviations, within \(10^{-16}\), indicating the relative orientation between vectors remained constant as expected. Orthogonality constraints also displayed small violations, within \(10^{-7}\), maintaining the orthogonal relationship between vectors throughout the simulation. These results collectively highlight the robustness and precision of the simulation framework, demonstrating that the system adheres to the constraints effectively over time. Figure 24: Violation of distance constraints, Equation (44). Figure 23: Violation of rigidity constraints, Equation (7). It is important to note that the exact inherent damping of an elephant trunk is not explicitly documented in the literature. However, studies on elephant trunk dynamics [1; 35] suggest that trunk movements rely on a combination of passive elasticity and active muscular control, both of which inherently contribute to energy dissipation. In previous implementations, the damping effect was not explicitly modeled; however, the results demonstrated realistic oscillatory behavior, ensuring consistency with observed trunk movements. The simulation approach inherently incorporates constraint stabilization techniques to manage the numerical behavior of the system. Specifically, damping in constraint dynamics is handled using the Baumgarte and Fuzzy Logic Control stabilization methods. These stabilization methods effectively regulate constraint violations, preventing unrealistic oscillations and ensuring smooth, physically meaningful motion. The effectiveness of this approach is evident in the dynamic results presented in Figures 17 and 19-21), where the simulated motion decay closely aligns with natural trunk behavior.