# Designing and Simulating the Control Process of a Feed Pusher Robot Used on a Dairy Farm
## 3 Results  
### Results of an Artificial Experiment (Kinematic Model and Dynamic Model of the Robot)  
At the stage of selecting the element base of the pusher robot, kinematic and dynamic analysis was carried out, taking into account the robot's mass, the indicators of the acting forces in the process of motion, according to the analysis scheme presented in Figure 4.  
The robot is a mobile platform with two driving wheels and one driven one. The driving wheels are located on the same axle. The center of each whe  
Figure 4: Design scheme describing the robot motion kinematics. 1, mobile platform; 2, drive wheel; 3, drive wheel; 4, support wheels; 5, pushing auger. \(A_{1}\)—the fulcrum of the left driving wheel, \(A_{2}\)—the fulcrum of the right driving wheel, \(A_{3}\)—the center of mass of the robot, \(A_{4}\)—the point of support of the driven wheel.  
Figure 3: Binocular vision system.  
letter \(A_{1}\) and \(A_{2}\), respectively, with \(|A_{1}A_{3}|=|A_{3}A_{2}|=L\). The robot moves on a surface that is located in the Oxy plane.  
The robot's gravity center is presumably located at \(A_{3}\) point, belonging to the \(A_{3}x_{1}y_{1}\) coordinate system. Attachments are mounted on the robot body, which are designed to push the distributed feed.  
The \(A_{3}x_{1}\) axis is rigidly fixed to the robot and is the vertical of the movable base. The rotation angle of the base vertical line about the Ox axis is denoted by \(\alpha\), and the angular velocity of the platform is \(\dot{\alpha}\).  
The position of the mobile robot relative to the Oxy world reference system is indicated by \(A_{3}\) point with coordinates \((x,y)\).  
The motion speed of a point along the Ox and Oy coordinates is described by Equations (1) and (2), respectively:  
\[\dot{x}=\frac{dx}{dt} \tag{1}\]  
\[\dot{y}=\frac{dy}{dt} \tag{2}\]  
The following equality holds true: \(\dot{x}\cdot\sin(\alpha)=\dot{y}\cdot\cos(\alpha)\).  
The speed of \(A_{3}\) point is determined by Equation (3):  
\[\
u=\dot{x}\cdot\cos(\alpha)+\dot{y}\cdot\sin(\alpha) \tag{3}\]  
The rotation angles of the driving wheels (centered at points \(A_{1}\) and \(A_{2}\), respectively) relative to the horizontal planes are \(\beta_{1}\), and \(\beta_{2}\), then the angular velocity will equal \(\dot{\beta}_{1}\) and \(\dot{\beta}_{2}\), respectively.  
The motion speed of \(A_{3}\) point is determined by Equation (4)  
\[\
u=-L\cdot\dot{\alpha}+R\cdot\dot{\beta}_{1}=L\cdot\dot{\alpha}+R\cdot\dot{ \beta}_{2} \tag{4}\]  
where R--the radius of the driving wheels.  
The system of motion equations of the robotic installation will take the form:  
\[\left\{\begin{array}{l}\dot{x}\cdot\sin(\alpha)=\dot{y}\cdot\cos(\alpha)\\ \dot{x}\cdot\cos(\alpha)+\dot{y}\cdot\sin(\alpha)=-L\cdot\dot{\alpha}+R\cdot \dot{\beta}_{1}\\ \dot{x}\cdot\sin(\alpha)+\dot{y}\cdot\cos(\alpha)=L\cdot\dot{\alpha}+R\cdot \dot{\beta}_{2}\end{array}\right. \tag{5}\]  
Equations for solving inverse kinematics:  
\[\left\{\begin{array}{l}\dot{\beta}_{1}=\frac{\dot{x}\cdot\cos(\alpha)+\dot {y}\cdot\sin(\alpha)+L\cdot\dot{\alpha}}{R}\\ \dot{\beta}_{2}=\frac{\dot{x}\cdot\sin(\alpha)+\dot{y}\cdot\cos(\alpha)-L \cdot\dot{\alpha}}{R}\end{array}\right. \tag{6}\]  
To describe the dynamics of a mobile robot, we use the Appel equation, which can be presented in a matrix form as follows:  
\[\left(\frac{\partial S}{\partial\dot{\pi}}\right)^{T}=\Pi \tag{7}\]  
Here, the operation of differentiating the scalar Appel function \(S\) taking into account the vector of pseudo-accelerations \(\dot{\pi}=\left(\ddot{x}\,\ddot{\alpha}\right)\) is performed; \(\Pi=\left(\Pi_{\
u}\,\Pi_{\bar{\alpha}}\right)\) is the vector of pseudo-generalized forces. In this case, the components of the vector of pseudo-generalized forces are determined with the Equations:  
\[\Pi_{\
u}=\frac{1}{R}\cdot(M_{mot1}+M_{mot2});\,\Pi_{\bar{\alpha}}=\frac{1}{R }\cdot(M_{mot1}-M_{mot2}) \tag{8}\]  
where \(M_{mot1},\,M_{mot2}\) are the moments developed by the stepper electric drives of the wheels.