## 1 Introduction  
Soft robotics focuses on creating robots capable of working together safely with humans by adapting to their surroundings. The use of flexible link manipulators could bring several advantages: higher payload capacity, lower energy consumption, a more cost-effective design, faster movements, greater reach, and safer operation [1]. Variable stiffness (VS) is essential for this purpose, and numerous techniques have been proposed to achieve this, including soft actuators, shape-memory materials, additive manufacturing, and VS joints [2]. Researchers have developed various soft robotic systems and variable stiffness grippers, including a 3D-printed monolithic soft gripper and a fluidic actuation system [3; 4; 5; 6; 7; 8; 9; 10; 11]. Composite materials such as shape-memory alloys and magnetorheological elastomers (MREs) have also been used to modulate stiffness [12]. The modeling of flexible joints is becoming increasingly important in the development and control of robotic manipulators, as they need to move with high precision and speed. Various methods have been introduced for the manipulators' dynamic modeling with flexible links. The decoupling method, based on according to the decoupled natural orthogonal complement matrices and Euler-Lagrange equations, has been shown to provide computational efficiency and stability and has been extended to various systems, including serial, parallel, closed-loop, flexible, and hyper-degree-of-freedom systems [13; 14]. The equivalent rigid-link system approach is proposed to model spatial lightweight industrial robots, obtaining satisfactory alignment between the numerical outputs and the measured data points [15]. The formulation based on an Equivalent Rigid-Link System has been compared to other techniques for reducing flexible multibody dynamics model order [16]. An ERLS-based 3D dynamics formulation has also been developed for flexible articulated robots that can handle large displacements and small elastic deformations [17]. The Euler-Bernoulli method has been employed to develop single-link manipulators with cables dynamic models and payloads based on the extended Hamilton principle and validated by simulations and experiments [18; 19; 20]. The assumed mode method has also been applied to control flexible manipulators, investigating NN controllers with output feedback and full state and trajectory tracking fuzzy NN controllers and vibration suppression, with simulations and experiments showing their feasibility [21; 22; 23]. Other researches have examined the consequences of the elastic deflection boundary conditions and a flexible manipulator's torques [24]. A flexible manipulator model is formulated through Timoshenko beam theory. Divergences are quantified, and robust controllers are synthesized with l-synthesis, ensuring effective tip tracking and robustness to uncertainties and noise [25]. The popular technique finite element method (FEM) is a technique to model hyper-redundant or serpentine robots [26; 27; 28]. The Hamilton method is used for modeling single-link flexible beams with a tip mass [29]. The lumped parameter method is an accurate and efficient technique for spatially compliant and multi-link manipulators [30; 31; 32; 33]. The Newton-Euler method is used for modeling highly flexible 3D manipulators featuring links of arbitrary shape [34]. Finally, the pseudo-rigid body method is used for modeling compliant mechanisms, where the generalized PRBM PPRR model has better accuracy compared to other PRBM models [35; 36]. The Timoshenko beam theory method is used to investigate the frequency sensitivity and orthogonality of the modes of a flexible beam [37]. The sensitivity of flexible manipulators' vibration frequencies is analyzed to system parameters using variational methods, discussing both Euler-Bernoulli and Timoshenko models, and introduces a novel orthogonal relations derivation method [38].