# Nonlinear Dynamics and Machine Learning for Robotic Control Systems in IoT Applications
## 2 Materials and Methods
### Integrating Nonlinear Methods with Neural Networks  
The diagram in Figure 4 outlines a hybrid control architecture that integrates traditional numerical calculations with neural network predictions to optimize robotic control. This architecture is designed to adaptively refine control strategies through iterative learning, making it highly effective for complex robotic applications. We proposed a hybrid control architecture that combines the strengths of nonlinear mathematical methods and neural networks. A neural network is used to optimize the robot's control strategy, such as path planning, joint control, or disturbance handling. The inputs to the neural network were as follows: the joint angles, joint velocities, external forces, etc. The outputs of the neural network were as follows: the predicted joint torques, optimized trajectories, or control signals. Machine learning models are integrated to learn from the robot's past trajectories and to dynamically adjust the control parameters. The inputs to the neural network are the variables that describe the current state of the robot and the environment. In this case, these include the following: the position of each joint in the robot, which can be represented as a vector Q (Equation (5)) of size n, where n is the number of joints \(q=\left[q_{1},\;q_{2},\;\ldots,\;q_{n}\;\right]\) the speed at which each joint is moving (vector \(dq=\dot{q}\)) of size n: \(dq=\dot{q}=\left[\dot{q}_{1},\dot{q}_{2},\dot{q}_{3},\;\ldots\;\dot{q}_{n}\;\right]\); and the external forces acting on the robot (including disturbances, gravity, or forces from contact with the environment). \(F=\left[F_{x},\;F_{y},\;F_{z}\;\right]\), where the components are forces in the three-dimensional space. The previous control signals or torques applied to the joints are used as inputs if the control strategy depends on past actions. Figure 3: Command and control process flowchart for robotic operations. Applying Equation (5), the outputs of the neural network are the variables that represent the control actions the robot should take (Figure 5). The joint torques \(\tau\) are applied to each joint to achieve the desired motion. With a vector \(\tau\) of size \(n\), \(y=\left[\tau_{1}\tau_{2}\tau_{3}\dots\tau_{n}\right]^{T}\), and the desired angles \(q_{n}\) (joint positions) and \(\dot{q}_{n}\) (joint velocities) for each joint, the robot should be guided by the neural network to a specific posture or trajectory. From the MATLAB robot code, the inputs are as follows: q(7:26), the angles of the robot's joints; dq(7:26), the velocities of the robot's joints; and FW, the external forces acting on the robot. Outputs: The torque values that should be applied to each joint to achieve that which is desired, such as from the main loop of the neural network, where the torques (currently with physics-based models) are calculated. Neural network architecture: The input layer has a size 2n + m, where n is the number of joints, and m is the number of force components. Hidden layers: the number of layers and neurons are chosen per layer based on the complexity of the control problem. A  
Figure 4: Hybrid control architecture workflow for robotic systems. Figure 5: Neural network-based dynamic control system for robotic actuation. common starting point is two hidden layers with 64 neurons each. The output layer has a size n, corresponding to the torques for each joint. Activation functions: typically, the ReLU (Rectified Linear Unit) is used for the hidden layers and a linear activation for the output layer. The control strategy for the robotic system involves using the torques predicted by a neural network to precisely actuate the robot's joints. This method incorporates real-time data on the joint positions, velocities, and external forces to calculate the necessary torques (Figure 6). These torques are then applied directly to the joints, enabling the robot to achieve the desired movements and postures efficiently.