# Application of Improved Sliding Mode and Artificial Neural Networks in Robot Control
## 2 Application of Improved Sliding Surfaces and Artificial Neural Networks in Robotics  
### Mathematical Equations for Robotics  
This segment details the dynamic modeling of a tri-wheeled omnidirectional robot (Figure 1), which is based on the reference provided in [23; 24; 25; 26; 27; 28]. Formulating the robot's mathematical model is pivotal for the development of its control architecture.  
In this study, we examine a three-wheeled omnidirectional mobile robot positioned on a solid plane. The robot, as depicted in the image, has its own coordinate system anchored at its center of gravity, along with a corresponding real-world coordinate system. The arrangement of its three-wheel assemblies endows the robot with flexible mobility, with the driving forces of each wheel denoted as \(D_{1},\ D_{2}\), and \(D_{3}\). These forces collaborate to enable the robot to move in any direction across the plane. The angle, \(\Psi\), indicates the robot's orientation relative to the real-world coordinate system, while the rotational angles associated with each wheel indicate their specific orientations in relation to the robot's coordinate framework. The orientation and the forces combined provide a foundation for the robot to move accurately and respond to control directives, which requires precise coordination among the wheels to achieve the desired path.  
Figure 1: The robot model.  
\[\left[\begin{array}{c}\tilde{x}_{\text{W}}\\ \dot{y}_{\text{W}}\\ \dot{\phi}\end{array}\right] =\left[\begin{array}{cc}a_{1}&-a_{2}\dot{\phi}&0\\ a_{2}\dot{\phi}&a_{1}&0\\ 0&0&a_{3}\end{array}\right]\left[\begin{array}{c}\dot{x}_{\text{W}}\\ \dot{y}_{\text{W}}\\ \dot{\phi}\end{array}\right]+\left[\begin{array}{cc}b_{1}\gamma_{1}&b_{1} \gamma_{2}&2b_{1}\text{cos}\phi\\ b_{1}\gamma_{3}&b_{1}\gamma_{4}&2b_{1}\text{sin}\phi\\ b_{2}&b_{2}&b_{2}\end{array}\right]\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]+\left[\begin{array}{c}D_{f_{x}}\\ D_{f_{y}}\\ D_{f_{\phi}}\end{array}\right] \tag{1}\] \[=\boldsymbol{A}_{\text{W}}\beta+\boldsymbol{B}_{\text{W}} \boldsymbol{U}+\boldsymbol{D}_{f}\]  
With \(\boldsymbol{D}_{f}=\left[D_{f_{x}}\quad D_{f_{y}}\quad D_{f_{\phi}}\right]\) is the unknown system noise.  
where:  
\[\boldsymbol{A}_{\text{W}}=\left[\begin{array}{ccc}a_{1}&-a_{2}\dot{\phi}&0\\ a_{2}\dot{\phi}&a_{1}&0\\ 0&0&a_{3}\end{array}\right]\]  
\[\boldsymbol{B}_{\text{W}}=\left[\begin{array}{ccc}b_{1}\gamma_{1}&b_{1} \gamma_{2}&2b_{1}\text{cos}\phi\\ b_{1}\gamma_{3}&b_{1}\gamma_{4}&2b_{1}\text{sin}\phi\\ b_{2}&b_{2}&b_{2}\end{array}\right]\]  
\[\boldsymbol{U}=\left[\begin{array}{ccc}u_{1}&u_{2}&u_{3}\end{array}\right]^ {T}\]  
\[a_{1}=\frac{-3c}{(3I_{\text{W}}+2 Mr^{2})},a_{2}^{\prime}=\frac{2 Mr^{2}}{(3I_{ \text{W}}+2 Mr^{2})},a_{3}=\frac{-3 cL^{2}}{(3I_{\text{W}}L^{2}+l_{\text{F}}r^{ 2})}\]  
\[b_{1}=\frac{kr}{(3I_{\text{W}}+2 Mr^{2})},b_{2}=\frac{krL}{(3I_{\text{W}}+l_{ \text{F}}r^{2})}\]  
\[a_{2}=1-a_{2}^{\prime}=\frac{3I_{\text{W}}}{(3I_{\text{W}}+2 Mr^{2})}\]  
\[\gamma_{1}=-\sqrt{3}\text{sin}\phi-\text{cos}\phi,\;\gamma_{2}=\sqrt{3}\text {sin}\phi-\text{cos}\phi;\gamma_{3}=\sqrt{3}\text{cos}\phi-\text{sin}\phi; \gamma_{4}=-\sqrt{3}\text{cos}\phi-\text{sin}\phi\]  
where \(L\) denotes the distance from any wheel to the robot's center of mass; \(k\) is the drive gain factor; \(D_{i}\) represents the driving force of each robot wheel; \(r\) is the radius of each wheel; \(c\) stands for the viscous resistance factor of the wheel; \(\omega_{i}\) indicates the angular velocity of the robot; \(I_{\text{R}}\) is the moment of inertia of each wheel around its driving shaft; and \(u_{i}\) is the input torque for driving.