# Application of Improved Sliding Mode and Artificial Neural Networks in Robot Control
## 2 Application of Improved Sliding Surfaces and Artificial Neural Networks in Robotics
### Controlling Robots Using an Improved Slider Interface  
Schematic structure of the sliding controller based on an improved sliding mode control (ISMC), as shown in Figure 2.  
Figure 2 depicts a schematic structure for a sliding control system applied to a mobile robot, employing an enhanced sliding method. At the heart of the diagram is the ISMC controller, an acronym for \"Integral Sliding Mode Control\". This represents a robust control strategy aimed at mitigating the impact of uncertainties and ensuring system stability, despite changes in the robot's dynamics or configuration. The reference signal (\(\boldsymbol{\Gamma}_{d}\)) contains the desired parameters that the robot aims to achieve, such as position or orientation. This signal is input into a comparator, where it is subtracted from the actual feedback signal (not directly shown in the diagram) to generate the error signal (\(\boldsymbol{e}\)). The error signal represents the difference between the current state and the desired state of the robot, providing critical  
Figure 2: Schematic structure of the sliding controller based on an improved sliding mode control. information for the control process. The ISMC then processes the error signal to produce a control signal (**U**), with the primary goal of appropriately adjusting the robot's actions to reduce this error. The mobile robot, with input parameters including position \((x_{w},\ y_{w})\) and orientation (\(\phi\)), will act based on the control signal, **U**, received. Feedback from the robot post-action (**\(\Gamma\)**) becomes the input for the next control cycle, creating a closed-loop that allows for automatic adjustment and continuous improvement. This ensures that the robot not only reacts to the current controls, but also self-adjusts based on feedback, enhancing its ability to achieve goals accurately and efficiently. The schematic also suggests an improvement to the traditional sliding control mechanism, potentially related to enhanced noise handling or dynamic changes of the robot. The discrepancy between the actual output, \(\Gamma\), and the reference input, \(\Gamma_{d}\), is as Equation (2):  
\[\mathbf{e}=\Gamma-\Gamma_{d} \tag{2}\]  
where \(\Gamma= \begin{bmatrix}x_{w}&y_{w}&\phi\end{bmatrix} ^{T}\) represents the actual output, and \(\Gamma_{d}= \begin{bmatrix}x_{d}&y_{d}&\phi_{d}\end{bmatrix} ^{T}\) is the reference input of the robot. The first and second derivatives of Equation (2) yield Equations (3) and (4):  
\[\dot{\mathbf{e}}=\ddot{\Gamma}-\dot{\Gamma}_{d} \tag{3}\]  
\[\ddot{\mathbf{e}}=\ddot{\Gamma}-\ddot{\Gamma}_{d}= \begin{bmatrix}\ddot{x}_{w}\\ \ddot{y}_{w}\\ \dot{\phi}\end{bmatrix} - \begin{bmatrix}\ddot{x}_{d}\\ \ddot{y}_{d}\\ \dot{\phi}_{d}\end{bmatrix} =\dot{\mathbf{X}}-\ddot{\Gamma}_{d}=\mathbf{A}_{w}\mathbf{X}+ \mathbf{B}_{w}\mathbf{U}+\mathbf{D}_{f}-\ddot{\Gamma}_{d} \tag{4}\]  
The developed sliding surface is as Equation (5) [29]:  
\[\mathbf{s}=2\dot{\mathbf{e}}+(\mathbf{\lambda}+2\mathbf{\alpha})\mathbf{e}+\mathbf{\kappa}^{2}\int_{0} ^{t}\mathbf{e}(\tau)d\tau \tag{5}\]  
where \(\mathbf{\lambda}=diag(\lambda_{1},\lambda_{2},\lambda_{3})\), with each \(\lambda_{i}>0\), and \(\mathbf{\alpha}=diag(\alpha_{1},\alpha_{2},\alpha_{3})\), with each \(\alpha_{j}>0\).