## 5 Conclusions  
In conclusion, the development of VSL manipulators represents a significant advance in robotics, offering greater adaptability and safety in a range of applications. The VSL  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Time**} & \multicolumn{2}{c}{**Experimental (mm)**} & \multicolumn{4}{c}{**Theoretical (mm)**} & \multicolumn{4}{c}{**Error (mm)**} \\  & \multicolumn{2}{c}{**(sec)**} & \multicolumn{1}{c}{\(X\)} & \multicolumn{1}{c}{\(y\)} & \multicolumn{1}{c}{\(z\)} & \multicolumn{1}{c}{\(x\)} & \multicolumn{1}{c}{\(y\)} & \multicolumn{1}{c}{\(z\)} & \multicolumn{1}{c}{\(x\)} & \multicolumn{1}{c}{\(y\)} & \multicolumn{1}{c}{\(z\)} \\ \hline  
\hline \end{tabular}
\end{table}
Table 4: Sample of the data and dynamic errors along the \(x\)-, \(y\)-, and \(z\)-axes between the simulation values and those measured experimentally.  
manipulator's ability to adjust its stiffness in real time increases precision and efficiency, making it an ideal choice for tasks such as assembly line work. However, the non-linear dynamics of the VSL manipulator, resulting from the flexibility of its links, pose a challenge in modeling and control. This requires advanced techniques and leads to increased computational costs.  
The mathematical model presented for this 3 DOF VSL manipulator is a first step toward enhancing its performance in order to make the manipulator more feasible, increase safety, and reduce costs. The verification experiments have proven the model reliability and accuracy and underline the importance of mathematical modeling in engineering and scientific research. The kinematic model shows an average error percentage of 0.81% along the \(X\)-axis, 0.71% along the \(Y\)-axis, and 1.4% along the \(Z\)-axis for the end-tip position. The dynamic model has an average error of 0.63\({}^{\circ}\) for the first joint angular displacement of 1.06\({}^{\circ}\) for the second joint, and 1.27\({}^{\circ}\) for the third joint. The average percentage error in the position of the end tip along the X-, Y-, and Z-axes is 3.36%, 3.29%, and 2.49%, respectively.  
The results of the present study dynamic model of the present study exceed the results reported in [50]. In [50], a discrete Newton-Euler model is developed to analyze flexible manipulators, focusing on the fundamental dynamics of generic links. The rigid body motion is described using the Eulerian formulation, while elastic deformation is described using the Lagrangian formulation. The Jourdain principle guides the assumption of the Rayleigh-Ritz expansion for elastic variables. Manipulator dynamics are derived by assembling individual links, which leads to both articulated body (AB) and composite inertia (CI) methods. Only a single flexible link is used for validation. The maximum error reaches 44.6% at some points at the beginning of the fluctuation and 22% at some points towards the end [50], while the derived model in our study has a maximum error of about 18% at the beginning and an error of about 3% thereafter.  
The results and analysis of this study offer significant insights into the kinematics and dynamics of VSL manipulators. Future work will aim to implement a control algorithm tailored to the unique characteristics of this manipulator, enabling precise and adaptive manipulation in dynamic and uncertain environments.  
All authors contributed to the study's conception and design. Material preparation, data collection, and analysis were performed by [PERSON], [PERSON] and S.A.M. The first draft of the manuscript was written by [PERSON], and all authors commented on successive versions of the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available in the article.  
The authors have no conflicts of competing interests to declare.