# Nonlinear Dynamics and Machine Learning for Robotic Control Systems in IoT Applications
## 5 Discussion  
The results of this study demonstrate the efficacy of integrating nonlinear dynamics with machine learning (ML) in optimizing control systems for a 20 degrees of freedom (DOFs) robotic platform. By utilizing a hybrid control approach that combines traditional feedback methods with neural networks, the proposed system adapts to real-time changes in complex environments, such as those characterized by nonlinearity and time variance. \begin{table}
\begin{tabular}{l c} \hline \hline  & **Circular Motion** \\ \hline radius/amplitude (m) & 1 \\ time (s) & 3.5 \\ max acceleration (m/s\({}^{2}\)) & 6.445 \\ max speed (m/s) & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of circular motion. Figure 17: Position of the robot’s joints during the circular motion with a radius of 1 m and a time period of T = 3.5 s.  
This significantly enhances the accuracy and robustness of the robot's trajectory tracking capabilities, particularly in IoT-driven scenarios where unpredictable disturbances and diverse environmental inputs are common. In comparison to previous studies, this work advances the field of robotic control in several ways [43]. For instance, [PERSON] et al. [11; 44] successfully applied a deep learning-based Model Predictive Control (MPC) framework to a three DOFs biped robot  
Figure 19: A detailed visual representation of the simulation: references and actual trajectories during the robot’s circular motion. Figure 18: Tracking errors during the circular motion with a radius of 1 m and a time period of T = 3.5 s.  
leg, showing improvements in trajectory tracking. However, their focus was on a lower dimensional system and did not account for as much real-time adaptability in nonlinear environments. Similarly, [PERSON] et al. [24; 45; 46] applied auxiliary physics-informed neural networks to solve nonlinear integral differential equations, showing promise in adapting to complex environments, but with limited integration in real-time control systems for robotics. [PERSON] and [PERSON] [32] explored the use of multi-layer neural networks in trajectory tracking for industrial robots. Their results highlighted the potential for ML in improving control precision, yet their study did not integrate the additional complexity of nonlinear dynamics. Our hybrid control architecture addresses this gap by providing a more comprehensive solution that allows for faster and more accurate adjustments in dynamic environments, particularly with the inclusion of real-time sensor data from IoT platforms. This study also builds upon earlier works on robotic control using deep reinforcement learning [47; 48], where [PERSON] et al. reviewed real-world successes in the application of these techniques. While deep reinforcement learning offers significant benefits for robotic systems, our hybrid approach enhances the control system by combining machine learning models with traditional nonlinear control techniques. This hybrid method provides superior adaptability in real-time applications, particularly in IoT-driven settings. Furthermore, [PERSON] [49] explored deep and recurrent neural architectures for control tasks in high-dimensional robotic systems. Similarly, studies by [PERSON] et al. [33] and [PERSON] et al. [34] applied recurrent neural networks (RNNs) in trajectory tracking for high-dimensional robotic systems, underscoring the importance of adaptive learning models in nonlinear environments. While these works contributed valuable insights into ML applications with high-dimensional control, our study goes further by leveraging the iterative learning capabilities of neural networks within a feedback control loop, allowing for continuous system optimization in real-time scenarios. Additionally, [PERSON] and [PERSON] [50] demonstrated the application of MPC for trajectory tracking and control [51; 52] in mobile robots, addressing challenges in time-varying environments.