# Nonlinear Dynamics and Machine Learning for Robotic Control Systems in IoT Applications
## 2 Materials and Methods
### Adams-Bashforth-Moulton Method for Nonlinear Dynamics
The forward kinematics \(f\) maps the joint angles to the task space positions, while the inverse kinematics \(f^{-1}\) maps the task space positions to the necessary joint angles. The additional mappings \(f_{spec}\) and \(f_{spec}^{-1}\) connect the actuation commands with the configuration space, effectively linking the entire control system from the actuator inputs to the task-specific outputs. The system's foundation is built on mathematical equations representing inverse kinematics and the dynamic model of the robot. The equations below define the positional errors and control inputs: \(e_{x}=x_{d}-x\,e_{y}=y_{d}-y\,e_{\psi}=\psi_{d}-\psi\). The robot's control inputs \(u_{1}\) and \(u_{2}\) are derived based on these error measurements and are fed into the controller to generate the desired driving force \(\mathrm{F_{D}}\) and steering wheel commands \(\delta\). In Figure 8, the control system, which integrates both the traditional control methods and a neural network within a hybrid control architecture, is presented. This configuration is composed of two main loops: the tracking loop and the nonlinear inner loop with a neural network. The tracking loop generates the desired trajectory signals, including the reference positions \(q_{1d}\) and velocities \(\dot{q}_{1d}\), and computes the trajectory errors \(e\) and \(\dot{e}\). These errors are processed through the gain matrices \(\mathrm{L}\) and \(\mathrm{Kv}\) to produce a reference signal that guides the robust control term and the nonlinear inner loop. The summation operation, represented by the plus sign in the diagram (Figure 8), combines the reference signals and feedback states to compute the error signals. By minimizing these errors, the tracking loop ensures that the robot's movement remains as close as possible to the desired path. The nonlinear inner loop incorporates a neural network that provides an adaptive control term \(\hat{f}(x)\). This neural network estimates and compensates for the nonlinearities and uncertainties in the robot's dynamics, particularly those not addressed by the traditional control elements. The neural network receives inputs corresponding to the robot's states, including positions and velocities, and outputs an adaptive control signal \(\hat{f}(x)\). This signal is combined with the robust control term \(v(t)\), enhancing the system's adaptability and providing improved tracking accuracy by adjusting to dynamic changes in real time. The robust control term \(v(t)\) provides stability to the control system by handling model uncertainties and disturbances. It receives the reference input \(\tau\) from the tracking loop and works in conjunction with the adaptive term \(\hat{f}(x)\) generated by the neural network. Unity gain, represented by the number \"1\" in the diagram, indicates that certain control signals pass through unchanged as they proceed to different stages of the system. The robust control term, therefore, adds an additional layer of reliability, ensuring that the system remains resilient to unexpected changes and modeling inaccuracies. The robot system block represents the actual physical dynamics of the robot. The combined output from the robust control term and force control loop is fed into this block, which calculates the resulting joint positions \(\mathrm{q_{1}}\) and velocities \(\dot{\mathrm{q_{1}}}\). The feedback from these actual states is looped back into the tracking and force control loops, allowing the control  
\[y_{n+1}=y_{n}+\frac{h}{2}\left[\partial_{0}f(t_{n+1},y_{n+1}^{predict})+\sum_ {i=0}^{m-1}\partial_{i}f(t_{n-1},y_{n-1})\right] \tag{14}\]0 Hybrid control system integrating neural network and traditional control methods. system to continuously refine its actions based on the current robot state, ensuring accurate trajectory tracking and force application.