###### Abstract  
The goal of this study was to construct a mathematical and computational model that accurately represents the complex, flexible movements and mechanics of an elephant's trunk. Rather than serving as a biological study, the elephant trunk model was used as an application to demonstrate the effectiveness of a proposed rigid-flexible coupling framework. This model has broader applications beyond understanding the mechanics of an elephant trunk, including its potential use in designing flexible robotic systems and prosthetics, as well as contributions to the fields of biomechanics and animal locomotion. An elephant's trunk, a highly flexible and muscular organ without bones, is best modeled using continuum mechanics to capture the dynamic behavior of its motion. Given the rigid body nature of an elephant's head movement and the highly flexible nature of the trunk, a robust geometric framework for the rigid-flexible interface is crucial to accurately capture the complex interactions, force transmission, and dynamic behavior arising from their distinct motion characteristics and differing coordinate representations. Under the umbrella of flexible multibody dynamics, this study introduced a hybrid coordinate system, integrating the Natural Coordinates Formulation (NCF) and the Absolute Nodal Coordinates Formulation (ANCF), to establish the geometric constraints governing the interaction between the rigid body (the head) and the highly flexible body (the trunk). Moreover, the model illustrates how forces and moments are transmitted between these components in both direct and inverse scenarios. Various finite elements were evaluated to identify suitable elements for modeling the elephant's trunk. The model's accuracy was validated through simulations of bending, twisting, compression, and other characteristic trunk movements. The solution method is presented alongside the simulation analysis for various motion scenarios, providing a comprehensive framework for understanding and replicating the trunk's complex dynamics. n 2025 8, 42 complex, flexible movements and mechanics of an elephant's trunk. The study began with a thorough study of an elephant's trunk anatomy and movement; see Figure 1. The trunk is a highly flexible and muscular organ with no bones, comprising an intricate network of muscles, tendons, and skin. It can perform a wide range of motions and functions, from lifting heavy objects to delicate manipulations. Such a model could have applications beyond understanding the mechanics of the elephant trunk, including the design of flexible robotic systems and prosthetics, as well as contributing to the fields of biomechanics and animal locomotion. Developing this model would be an interdisciplinary endeavor that requires expertise in mechanical engineering, computational modeling, biomechanics, and zoology. It would also likely involve iterative testing and refinement to accurately capture the complex, multifaceted movements and functions of an elephant's trunk. This research highlights the advantages of continuum robots over traditional rigid-link robots, particularly in their ability to navigate complex trajectories and confined spaces, making them ideal for various advanced engineering applications. An elephant's trunk serves as an exemplary biological model, offering inspiration for designing flexible robotic systems capable of unprecedented versatility and adaptability. A novel approach to continuum robot backbone design, inspired by the flexibility and versatility of an elephant's trunk, was introduced and validated in [2]. The proposed design incorporated helical compression springs with varying stiffness, arranged in a multi-section structure, allowing the robot to achieve smooth elongation, bending, and shortening. The paper focused on the effects of variable stiffness on its motion control and stability.