# Nonlinear Dynamics and Machine Learning for Robotic Control Systems in IoT Applications
## 2 Materials and Methods
### Adams-Bashforth-Moulton Method for Nonlinear Dynamics  
The dynamics of the 20 DOFs robot are simulated using the ode113 solver, which implements the Adams-Bashforth-Moulton method. This method is particularly well suited for solving stiff ordinary differential equations (ODEs), which are common in robotic systems due to the presence of multiple interacting components. The Adams-Bashforth-Moulton method is a predictor-corrector method used to solve ODEs of the form \(\dot{y}=f(t,y)\). The predictor step estimates the solution at the next time step using the previous values:  
\[y_{n+1}^{predict}=y_{n}+h\sum_{i=0}^{m-1}\beta f(t_{n-1},y_{n-1}) \tag{13}\]  
The corrector step refines the estimate to enhance the accuracy:  
\[y_{n+1}=y_{n}+\frac{h}{2}\left[\partial_{0}f(t_{n+1},y_{n+1}^{predict})+\sum_ {i=0}^{m-1}\partial_{i}f(t_{n-1},y_{n-1})\right] \tag{14}\]  
where h is the step size and \(\beta_{\mathrm{i}}\) and \(\alpha_{\mathrm{i}}\) are the coefficients that depend on the order of the method. This method is particularly effective for ensuring that the solution remains stable and accurate over long simulation periods, making it ideal for scenarios that integrate both the outputs from numerical methods and adjustments from machine learning to generate the optimal control inputs for the robotic system. The hybrid control strategy, combining traditional control (PID and inverse kinematics) with machine learning for model-free adaptation, can be implemented within the controller block. The transformation to robot coordinates and feedforward components would be primarily handled by traditional control methods, while the controller can adapt using machine learning to fine-tune responses based on sensor feedback. The provided diagram (Figure 7) is consistent with the mathematical and control methodologies discussed in this paper. It effectively visualizes the hybrid control architecture where traditional control principles, numerical methods, and machine learning work together to optimize the robotic motion control. The control structure can be broken down into the following stages:  
Reference trajectory generator: This block generates the desired reference trajectory defined by \(x_{cd}\), which is the desired position in the x-direction; \(y_{cd}\), which is the desired position in the y-direction; \(\psi_{cd}\), which is the desired heading angle; and \(V_{d}\), which is the desired velocity. The feedforward block compensates for known disturbances and path deviations. In the methodology discussed earlier, this is supported by the traditional control methods, where the feedforward signal works as a primary reference, while the feedback corrects errors. Coordinate transformation and guidance errors: This block transforms the global reference trajectory into the robot's local coordinate frame. It computes the errors between the desired trajectory (from the reference generator) and the current trajectory of the robot. The outputs include the following: \(e_{\psi}\), which is the error in the heading angle and \(e_{x},\;\;e_{y}\), which are the position errors in the x- and y-directions. The desired heading and velocity values \(y_{cd},\;V_{d}\) are passed to the controller. The controller processes the errors and generates control signals \(u_{1}\) and \(u_{2}\), which correspond to the commands for the steering wheel and driving force, respectively. The controller seeks to minimize the errors \(e_{\psi}\) and \(e_{x},\;e_{y}\) by adjusting the steering and speed commands. This is analogous to the inverse kinematics calculation \(q=f^{-1}(p)\). Controller: The controller block is responsible for generating the actuation commands \(u_{1}\) (steering) and \(u_{2}\) (driving force). In the context of the earlier methodology, this would involve both traditional control (PID and adaptive control) and machine learning components for fine-tuning the actuation.