## 3 System Development
### Mathematical Modeling of Telepresence Robot  
Mathematical models are used to model a system using symbolic computation. This approach helps analyze the dynamics of robots in real-life situations. During the modeling phase, four key components are considered: wheels, bodies, vertical joints, and intelligent devices. Figure 8 shows a schematic representation of the telepresence robot according to the coordinated reference system. The coordinate system for measuring the robot's position relative to the global reference system for each reference uses a Cartesian coordinate system. Independent coordinates are the vertical displacement x(t) and angular rotation \(\theta\)(t) of the wheel, as well as the stresses arising from sliding during horizontal and vertical displacements. To simplify this, we used a 2D model to look at the vertical displacements and rotations of the ramp. Lateral displacement is measured as irrelevant here. Overall lumped gravity central point is chosen in RF0 and RF3 in a side view of the diagram.  
Nonholonomic is likely the constraint defined in the system. For our telepresence robot, we consider that wheels rotate without slipping, which shows that both wheels have a similar angular rotation, as shown in Equation (1).  
\[\varphi=R\frac{d}{dt}\theta_{w}(t)=\frac{d}{dt}x(t) \tag{1}\]  
We know that \(\theta_{w}(t)\) is the wheel rotation angle, \(\varphi\) is the steering angle, x(t) is the longitudinal translation, and R is the wheel's radius.  
The system's kinetics is based on the Newton-Euler equation. The unknown variables are solved to derive the grip forces and regular forces. Longitudinal and rotational velocities with varying parameters are given by Equations (2) and (3), according to the symbolic equations.  
\[u(t)=\frac{d}{dt}x(t) \tag{2}\]  
\[u_{\theta}(t)=\frac{d}{dt}\theta(t) \tag{3}\]  
To determine the body weight caused by the upright structure causes the entire robotic platform system to topple concerning the Newton-Euler equations, the Euler equation for the telepresence robot body structure about RF0 and RF3 along with RF6 and RF9 are utilized. The modified equation is as follows in Equation (4).  
\[m_{tr}h\frac{d}{dt}u(t)+\frac{(2m_{tr}h^{2})}{L}\frac{d}{dt}u_{\theta}(t)-m_{ rr}h\theta(t)g=0 \tag{4}\]  
Figure 8: Schematic representation of telepresence robot.  
where \(\rm m_{tr}\) is the mass of the telepresence robot structure, h is the center of mass, L is the length of the telepresence robot platform (\(y\)-axis), as shown in Figure 4, \(\rm u_{\theta}(t)\) is the angular velocity of the robot wheel, and g is the gravitational force constant.  
The actual controlling equation of the telepresence robot system is defined in (5), where \(\rm Tc(t)\) is the torque, and \(\rm M_{tr}\) represents the equivalent total mass of the telepresence robot. The equivalent value of \(\rm\mu\) and \(\rm\psi\) is given by Equations (6) and (7), respectively.  
\[\rm M_{\rm{etr}}\frac{d}{dt}u(t)+\rm\mu\frac{d}{dt}u_{\theta}(t)-\frac{Tc(t)}{R }+\rm\psi\theta(t)=0 \tag{5}\]  
\[\rm\mu=\frac{(2m_{tr}(h^{2}+hR))}{RL} \tag{6}\]  
\[\rm\psi=\left(\frac{(m_{tr}hg)}{R}-\frac{\left(4m_{tr}h[u_{\theta}(t)]^{2} \right)}{L^{2}}\right) \tag{7}\]  
The angle of inclinations (9) is a maximum of 35% of the slope. Similarly, the height (h) of the robot support is one of the primary tasks affecting system instabilities.  
MATLAB simulates the system response and displays it concerning the time interval. With the initial values, the simulation runs for linearized feedback control. According to Figure 9, rotating and steering systems stabilize under 4 s when the angular motion is considered.