## 1 Introduction
On the other hand, the Lagrangian technique is applied for developing flexible manipulators' dynamic models, including payload and hub inertia, using finite elements and assumed mode methods [39; 40]. Advanced control strategies, such as fuzzy set theory and passivity-based controllers, have also been proposed to achieve precise control and energy dissipation in these systems. In summary, variable stiffness manipulators have been designed and developed using various approaches and materials that enable soft robots to perform different applications and tasks. These manipulators have the potential to revolutionize several industries, including manufacturing and healthcare. These modeling methods provide insights into the flexible link manipulators' dynamics and can aid in the development and control of robotic systems. This manipulator's dynamic model helps to predict and control the robot's motion and the forces acting on it. An innovative idea for the development of a collaborative 3 DOF robot is presented by [PERSON] et al. using VSL. Using a hybrid of silicone rubber construction, polypropylene plastic mesh, and fabric materials, they have created links whose stiffness is controllable, and which are pneumatically actuated. These links can be adjusted to various stiffness levels by changing the pressure value within their structure, allowing the robot's performance to be tailored to different applications [11; 41]. The pressure sensor is used to sense collisions between the robot's body and a human. When a collision is detected, the pressure in the links is released and the links become soft, so it does not cause any harm to the human worker. The performance of the VSL is evaluated when it is integrated into a robotic manipulator. They analyzed the different loads' effects and pressures on the manipulator's workspace and evaluated the effectiveness detection of the collision control system and hardware [11; 41]. But the mathematical model is not derived for this 3D VSL manipulator. The lack of a derived mathematical model is a significant obstacle to the implementation of model-based control systems, a fundamental requirement for robust control methods. Dynamic mathematical models play a significant role in the manipulators with flexible links development It enables accurate prediction of the complex behavior of the system, including link deflection, vibrations, and coupling effects. Accurate models form the foundation for designing effective control strategies and optimizing performance. Theyfacilitate parameter estimation, system identification, and the development of vibration suppression techniques. Therefore, it is essential to address this critical gap in our current understanding to unlock the full capabilities of model-based control strategies and improve overall system performance. This research makes an important contribution to the modeling of VSL robotic manipulators presented in [11; 41]. In particular, a 3 DOF VSL prototype comprehensive kinematic and dynamic models are derived. The dynamic model exclusively uses the Euler-Lagrange formulation with lumped parameters for representing the system efficiently yet accurately. The drawbacks of the other modeling methods and the advantages of this method are discussed in the Dynamics section. The complete experimental validation proves the proposed techniques effectiveness in capturing system behavior. By providing a precise mathematical representation of VSL dynamics, this work lays the foundation for optimizing performance, stability, and human-robot collaboration through model-based control strategies tailored to these VSL manipulators. As VSL robots are increasingly used in safety-critical tasks, the rigorous modeling methodology presented will support efforts to analyze, predict, and regulate the behavior of similar systems. This paper contains Seven sections covering various aspects of the research. Section 2 deals with the mathematical model derivation. Section 3 deals with the simulation of the derived mathematical model.