## 3 Kinematics of a Robot-Trailer System  
The kinematic conditions for robot-trailer systems contain the following variables: \(x_{C}\), \(y_{C}\), \(\psi\), \(\varphi_{i}\), \(x_{S}\), \(y_{S}\), \(\chi\), \(\theta_{1}\), and \(\theta_{2}\). Thus, we have twelve (\(i=1,\dots,4\)) or ten (\(i=1,2\)) variables (generalized coordinates) that are related to the constraint equations. The number of independent variables (the number of degrees of freedom) depends on the configuration of the robot-trailer system. We introduce the angle \(\alpha\) of rotation of the rods relative to each other as shown in Figures 1 and 2:  
\[\alpha\,=\,\chi\,-\,\psi\,. \tag{24}\]  
Constraint Equations of a Four-Wheeled Robot with Mecanum Wheels and a Trailer with Conventional Wheels  
Combing Equations (12), (13), (17), and (23), we obtain a system of nine equations, including twelve generalized coordinates \(x_{C}\), \(y_{C}\), \(\psi\), \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{3}\), \(\varphi_{4}\), \(x_{S}\), \(y_{S}\), \(\alpha\), \(\theta_{1}\), and \(\theta_{2}\):  
\[ \begin{split}\dot{x}_{C}&=\,\frac{R_{C}}{\sqrt{2}} \cos{(\psi-\frac{\pi}{4})}\,\dot{\varphi}_{1}\,+\,\frac{R_{C}}{2}(\cos{\psi}\, \dot{\varphi}_{2}-\sin{\psi}\,\dot{\varphi}_{3})\,,\\ \dot{y}_{C}&=\,\frac{R_{C}}{\sqrt{2}}\sin{(\psi- \frac{\pi}{4})}\,\dot{\varphi}_{1}\,+\,\frac{R_{C}}{2}(\sin{\psi}\,\dot{\varphi} _{2}+\cos{\psi}\,\dot{\varphi}_{3})\,,\end{split}  \tag{25}\]\[\dot{\psi} =\frac{R_{C}}{2(\rho+l_{C})}\left(\phi_{2}\,-\,\phi_{3}\right), \tag{26}\] \[\dot{\phi}_{4} =\dot{\phi}_{1}+\dot{\phi}_{2}\,-\,\dot{\phi}_{3}\,.\]  
\[\dot{\alpha} =\,-\,\frac{R_{C}}{2b_{S}}\bigg{(}\sqrt{2}\,\sin\left(\alpha+\pi/4 \right)\dot{\phi}_{1}\,+\,\Big{(}\sin\alpha+\frac{b_{C}\cos\alpha+b_{S}}{\rho+ l_{C}}\Big{)}\dot{\phi}_{2}\,-\,\Big{(}\cos\alpha+\frac{b_{C}\cos\alpha+b_{S}}{\rho+ l_{C}}\Big{)}\dot{\phi}_{3}\bigg{)}\,, \tag{27}\] \[\dot{x}_{S} =\,\frac{R_{C}}{2}\bigg{(}\sqrt{2}\,\cos\left(\alpha+\frac{\pi}{ 4}\right)\dot{\phi}_{1}\,+\,\Big{(}\cos\alpha-\frac{b_{C}\sin\alpha}{\rho+l_{ C}}\Big{)}\dot{\phi}_{2}+\sin\alpha\Big{(}1+\frac{b_{C}}{\rho+l_{C}}\Big{)}\dot{ \phi}_{3}\bigg{)}\cos\left(\psi+\alpha\right),\] (28) \[\dot{y}_{S} =\,\frac{R_{C}}{2}\bigg{(}\sqrt{2}\,\cos\left(\alpha+\frac{\pi}{ 4}\right)\dot{\phi}_{1}\,+\,\Big{(}\cos\alpha-\frac{b_{C}\sin\alpha}{\rho+l_{ C}}\Big{)}\dot{\phi}_{2}+\sin\alpha\Big{(}1+\frac{b_{C}}{\rho+l_{C}}\Big{)}\dot{ \phi}_{3}\bigg{)}\sin\left(\psi+\alpha\right),\]  
\[\dot{\theta}_{1} =\,\frac{R_{C}}{2R_{S}}\bigg{(}\sqrt{2}\Big{(}\cos\left(\alpha+ \pi/4\right)\,+\,\frac{l_{S}}{b_{S}}\sin\left(\alpha+\pi/4\right)\Big{)}\dot{ \phi}_{1}+\Big{(}\big{(}1+\frac{l_{S}b_{C}}{b_{S}(\rho+l_{C})}\big{)}\cos \alpha+\big{(}\frac{l_{S}}{b_{S}}-\frac{b_{C}}{\rho+l_{C}}\big{)}\sin\alpha \Big{)}\dot{\phi}_{2}+ \tag{29}\] \[\Big{(}\big{(}1+\frac{b_{C}}{\rho+l_{C}}\big{)}\sin\alpha+\frac{l _{S}}{b_{S}}\big{(}1+\frac{b_{C}}{\rho+l_{C}}\big{)}\cos\alpha\Big{)}\dot{ \phi}_{3}\bigg{)}\,,\]  
The system of kinematic constraints (25)-(29) is a nonlinear system of first-order differential equations with nine equations and twelve unknowns. As can be seen, the mechanical system of a four-wheeled Mecanum robot coupled with a trailer featuring conventional wheels has three degrees of freedom. Non-holonomic constraints require different methods for deriving the dynamic equations of the mechanical system. [PERSON]'s equations which were used to describe the motion of a single robot in [23] are not applicable to the robot-trailer system discussed in our article. It follows from Equation (27) that it is not possible to provide the right-hand side as a function of only independent coordinates since the coefficients of the right-hand side depend on the generalized coordinate \(\alpha\).