## 3 Kinematics of a Robot-Trailer System
### Constraint Equations of the Two-Wheeled Robot and Trailer with Conventional Wheels
 \tag{40}\]  
Integrating the equation for \(\alpha\), we obtain  
\[\alpha(t)=2\,\arctan\bigg{(}C\,\exp\Big{(}-\,\frac{R_{c}\omega}{b_{s}\sqrt{2} }\,t\Big{)}\bigg{)}\,-\,\frac{\pi}{4}\,. \tag{41}\]  
Here, the constant \(C\) is determined from the initial condition. The constant is zero \(C=0\), if \(\chi(0)\,=\,0\), \(\,\alpha(0)\,=\,\chi(0)\,-\,\psi(0)\,=\,-\,\pi/4\) at initialization. This means that \(\alpha(t)\,=\,-\,\pi/4\) during the entire time of the motion. Note that for any initial condition for the considered motion, the following limitations hold:  
\[ \begin{split}\lim_{t\to\infty}\alpha(t)\,=\,-\,\frac{\pi}{4}\,, \quad\lim_{t\to\infty}\dot{x}_{S}(t)\,=\,\frac{R_{C}\,\omega}{\sqrt{2}}\,, \quad\lim_{t\to\infty}\dot{y}_{S}(t)\,=\,0\,,\\ \lim_{t\to\infty}\dot{\theta}_{1}(t)\,=\,\lim_{t\to\infty}\dot{ \theta}_{2}(t)\,=\,\frac{R_{C}\,\omega}{R_{S}\sqrt{2}}\,.\end{split}  \tag{42}\]  
For simulation, the parameters of the robot-trailer system are defined as follows:  
\[ \begin{split} l_{C}\,=&\,0.15\,\text{m}\,,\quad l_ {S}\,=\,0.10\,\text{m}\,,\quad b_{c}\,=\,0.10\,\text{m}\,,\quad b_{S}\,=\,0.12 \,\text{m}\,,\\ R_{C}\,=\,0.05\,\text{m}\,,\quad R_{S}\,=\,0.03\,\text{m}\,,\quad \omega\,=\,10\,\text{rad}/\text{s}\,.\end{split} \]  
The angle \(\chi\) is initialized with \(\chi(0)\,=\,3\pi/4\) so that \(\alpha(0)\,=\,\chi(0)\,-\,\psi(0)\,=\,\pi/2\). Taking into account expression (41), we obtain \(C\,=\,\tan(3\pi/8)\,=\,\sqrt{2}+1\,\). Figure 4 presents the dependency of the angle \(\alpha\) on time \(t\) for these initial conditions. Figure 5 shows the time-domain results of the trailer's velocity components \(\dot{x}_{S}\) and \(\dot{y}_{S}\). The time-domain results of the trailer's coordinates \(x_{S}\) and \(y_{S}\) are presented in Figure 6. The robot's coordinates \(x_{C}\) and \(y_{C}\) are initialized at \(x_{C}(0)\,=\,y_{C}(0)\,=\,0\). Figure 7 shows the angular velocities \(\dot{\theta}_{1}\) and \(\dot{\theta}_{2}\) of the wheels of the trailer versus time. Figure 4: The angle \(\alpha\) vs. time \(t\). Figure 5: The velocities \(\dot{x}_{S}\) and \(\dot{y}_{S}\) of the trailer’s body vs. time \(t\). Figure 6: The coordinates \(x_{S}\) and \(y_{S}\) of the trailer’s body vs. time \(t\). Now, let us consider an example of a system consisting of a robot with four Mecanum wheels and a trailer with two conventional wheels (see Figure 1). The system has three degrees of freedom too. When studying the dynamics of the robot-trailer system, it is necessary to define the torques applied to the robot wheels. When analyzing the kinematic relations, the angular velocities of the wheels are given. The angular velocities of the wheels are related by the second expression (12). \[\omega_{4}\,=\,\omega_{1}+\omega_{2}-\omega_{3}\,,\qquad\omega_{i}\,=\,\dot{ \varphi_{i}}\,\,\,(i\,=\,1,\ldots,4)\,. \tag{43}\]  
At given angular velocities of the robot wheels, the remaining motion parameters are determined by integrating the system of equations of kinematic constraints (25)-(29). Let the angular velocities of the wheels be constant and equal:  
\[\omega_{1}\,=\,5\,\mathrm{rad}/\mathrm{s}\quad\omega_{2}\,=\,10\,\mathrm{rad}/ \mathrm{s}\quad\omega_{3}\,=\,15\,\mathrm{rad}/\mathrm{s}\,.\]  
The remaining quantities included in the equations of kinematic constraints will be given as follows:  
\[l_{C}\,= 0.15\,\mathrm{m}\,,\quad\rho\,=\,0.15\,\mathrm{m}\,,\quad l_{S} \,=\,0.10\,\mathrm{m}\,,\quad b_{c}\,=\,0.25\,\mathrm{m}\,,\] \[b_{S}\,=\,0.12\,\mathrm{m}\,,\quad R_{C}\,=\,0.05\,\mathrm{m}\,, \quad R_{S}\,=\,0.03\,\mathrm{m}\,.\]  
Figures 8-10 plot the computational result of integrating the system of differential Equations (25)-(29) for the following initial conditions:  
\[x_{C}(0)\,=\,0\,,\quad y_{C}(0)\,=\,0\,,\quad\psi(0)\,=\,\pi/4\,,\quad\alpha(0 )\,=\,\pi/2\,,\] \[x_{S}(0)\,=\,x_{C}(0)-b_{S}\cos\left(\alpha(0)+\psi(0)\right)-b_ {c}\cos\left(\psi(0)\right)\,,\] Figure 4:0  
Figure 8 shows the dependencies between the robot's body rotation angle \(\psi\) and angle \(\alpha\) vs. time \(t\).