## 5 Conclusions  
In this study, we developed a novel control strategy by integrating Objective Sliding Mode Control (OSMC) and a Stochastic Configuration Network (SSCN) with a cyclic structure to address trajectory tracking in multi-joint manipulators under uncertain conditions and external disturbances. Utilizing the Lagrangian method, we constructed a dynamic model for a manipulator with six joints and drives, which served as the basis for the OSMC design. The SSCN was crafted with a two-stage node selection process within its hidden layer. Initially, during the'single time' phase, weights and biases were randomly set via a uniform distribution, allowing for extensive parameter exploration. This was followed by the 'double time' phase, where parameters were fine-tuned from a normal distribution centered on earlier selections, enhancing precision. This dual-phase approach not only enables robust adaptation to dynamic environments but also increases manipulator response accuracy. Online adaptive updating of the SSCN's output weights during the real-time phase greatly enhances its ability to simulate real-world dynamics. Employing Lyapunov theory, we confirmed the system's stability and convergence. Detailed dynamic model parameters obtained from a virtual prototype facilitated simulations that showcased superior position-tracking and disturbance resistance. This research highlights the benefits of incorporating \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Articulation q** & **OSMC** & **NOSMC** & **SMC** \\ \hline
1 & 0.2463 s & 0.6423 s & 0.4773 s \\
2 & 0.2463 s & 0.6423 s & 0.4773 s \\
3 & 0.2463 s & 0.6423 s & 0.4773 s \\
4 & 0.2463 s & 0.6423 s & 0.4773 s \\
5 & 0.2463 s & 0.6423 s & 0.4773 s \\
6 & 0.2463 s & 0.6423 s & 0.4773 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The position tracking error for six joints near the initial system position is below 0.01. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Articulation q** & **OSMC** & **NOSMC** & **SMC** \\ \hline
1 & 0.8152 s & 2.5317 s & 1.1322 s \\
2 & 0.8152 s & 2.5298 s & 1.1322 s \\
3 & 0.8152 s & 2.5244 s & 1.1322 s \\
4 & 0.8152 s & 2.5230 s & 1.1322 s \\
5 & 0.8152 s & 2.5224 s & 1.1322 s \\
6 & 0.8152 s & 2.5211 s & 1.1322 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The position tracking error for six joints far from the initial system position is below 0.01. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Articulation q** & **Compensation** & **No Compensation** \\ \hline
1 & 0.2463 s & 0.2572 s \\
2 & 0.2463 s & 0.2572 s \\
3 & 0.2463 s & 0.2572 s \\
4 & 0.2463 s & 0.2572 s \\
5 & 0.2463 s & 0.2572 s \\
6 & 0.2463 s & 0.2572 s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time required for position tracking error of six joints with and without compensation. an advanced SSCN with a dual-phase node selection strategy in robotic control systems, significantly boosting operational reliability and precision in complex scenarios. In the future, optimizing the SSCN parameters could benefit from the incorporation of advanced machine learning techniques, such as reinforcement learning or meta-learning, to further enhance its adaptive capabilities and performance in more complex environments. By leveraging these emerging techniques, the SSCN could dynamically adjust its network structure and parameters based on real-time feedback, potentially reducing computational costs and improving the accuracy of trajectory tracking under varying conditions. Additionally, the integration of these optimization methods could facilitate the deployment of the proposed control strategy in larger-scale industrial applications, where manipulators must operate under diverse and unpredictable conditions. At the same time, the exploration of more sophisticated parameter optimization strategies and scalability opens new avenues for applying this control method to real-world industrial environments, further enhancing its practicality and impact. Conceptualization, Y.T. ; Writing-original draft, Y.T.