## 3 Kinematics of a Robot-Trailer System
### Constraint Equations of the Two-Wheeled Robot and Trailer with Conventional Wheels  
In order to compare the kinematic capabilities of the Mecanum-based robot-trailer system, a pure setup with conventional wheels is considered as well. The conventional wheel setup consists of a two-wheeled robot and a trailer, both with conventional wheels. Taking into account Equations (17), (18) and (23), we obtain a system with kinematic constraints described by ten generalized coordinates \(x_{C}\), \(y_{C}\), \(\psi\), \(\varphi_{1}\), \(\varphi_{2}\), \(x_{S}\), \(y_{S}\), \(\alpha\), \(\theta_{1}\), and \(\theta_{2}\):  
\[\dot{x}_{C} = \frac{R_{C}}{2}(\dot{\varphi}_{1}+\dot{\varphi}_{2})\cos\psi\,, \tag{34}\] \[\dot{y}_{C} = \frac{R_{C}}{2}(\dot{\varphi}_{1}+\dot{\varphi}_{2})\sin\psi\,,\] (35) \[\dot{\psi} = \frac{R_{C}}{2l_{C}}(\dot{\varphi}_{2}-\dot{\varphi}_{1})\,,\] (36) \[\dot{x}_{S} = \frac{R_{C}}{2}\bigg{(}(\cos\alpha+\frac{b_{C}}{l_{C}}\sin \alpha)\dot{\varphi}_{1}+\left(\cos\alpha-\frac{b_{C}}{l_{C}}\sin\alpha \right)\dot{\varphi}_{2}\bigg{)}\cos(\psi+\alpha)\,,\] (37) \[\dot{y}_{S} = \frac{R_{C}}{2}\bigg{(}(\cos\alpha+\frac{b_{C}}{l_{C}}\sin \alpha)\dot{\varphi}_{1}+\left(\cos\alpha-\frac{b_{C}}{l_{C}}\sin\alpha \right)\dot{\varphi}_{2}\bigg{)}\sin(\psi+\alpha)\,,\]  
\[\dot{\theta}_{1} = \frac{R_{C}}{2R_{S}}\bigg{(}\left((1-\frac{l_{S}b_{C}}{l_{C}b_{S}} )\cos\alpha\right.+\left.\left(\frac{b_{C}}{l_{C}}+\frac{l_{S}}{b_{S}}\right) \sin\alpha\right)\dot{\varphi}_{1}\,+\,\left((1+\frac{l_{S}b_{C}}{l_{C}b_{S}} )\cos\alpha\right.-\left.\left(\frac{b_{C}}{l_{C}}-\frac{l_{S}}{b_{S}}\right) \sin\alpha\right)\dot{\varphi}_{2}\bigg{)}\,, \tag{38}\] \[\dot{\theta}_{2} = \frac{R_{C}}{2R_{S}}\bigg{(}\left((1+\frac{l_{S}b_{C}}{l_{C}b_{S}} )\cos\alpha\right.+\left.\left(\frac{b_{C}}{l_{C}}-\frac{l_{S}}{b_{S}}\right) \sin\alpha\right)\dot{\varphi}_{1}\,+\,\left((1-\frac{l_{S}b_{C}}{l_{C}b_{S}} )\cos\alpha\right.-\left.\left(\frac{b_{C}}{l_{C}}+\frac{l_{S}}{b_{S}}\right) \sin\alpha\right)\dot{\varphi}_{2}\bigg{)}\,.\]  
The system of kinematic constraints (34)-(38) has eight equations and ten variables and the mechanical system of the two-wheeled robot and the trailer with two conventional wheels has only two degrees of freedom. The mechanical system is again not a Chaplygin system due to Equation (36). A mechanical system with linear non-holonomic constraints is a Chaplygin system if the expression of dependent generalized velocities contains only independent coordinates [17]. For all the robot-trailer models considered in this article, this condition is not satisfied as the generalized velocity \(\dot{\alpha}\) depends on the generalized coordinate \(\alpha\). Examples for the Motion Behavior of a Robot with Mecanum Wheels and a Trailer with Conventional Wheels  
The kinematic characteristics of the robot motion are determined by solving the corresponding system of equations of kinematic constraints. Let us illustrate this with the example of a system consisting of a robot with two Mecanum wheels and a trailer with two conventional wheels. The system has three degrees of freedom. Let the robot move forward along the axis \(OX\) at a constant velocity (see Figure 2). Let us set the following conditions for the motion of the robot:  
\[\psi\,=\,\pi/4\,,\quad\dot{x}_{C}\,=\,\text{const}\,,\quad\dot{y}_{C}\,=\,0\,. \tag{39}\]  
Then, from the expressions we find  
\[ \begin{split}\phi_{1}\,=\,\text{const}&\,=\,\omega \,,\quad\phi_{2}\,=\,0\,,\quad\dot{\alpha}\,=\,-\,\frac{R_{C}}{b_{S}\sqrt{2}} \sin(\alpha+\pi/4)\,,\\ x_{S}\,=\,\frac{R_{C}\,\omega}{2\sqrt{2}}(1-\sin 2\alpha)\,, \quad\dot{\theta}_{1}\,=\,\frac{R_{C}\,\omega}{R_{S}\sqrt{2}}\bigg{(}\cos( \alpha+\pi/4)\,+\,\frac{l_{S}}{b_{S}}\sin(\alpha+\pi/4)\bigg{)}\,,\\ y_{S}\,=\,\frac{R_{C}\,\omega}{2\sqrt{2}}\cos 2\alpha\,, \quad\dot{\theta}_{2}\,=\,\frac{R_{C}\,\omega}{R_{S}\sqrt{2}} \bigg{(}\cos(\alpha+\pi/4)\,-\,\frac{l_{S}}{b_{S}}\sin(\alpha+\pi/4)\bigg{)}\,. \end{split}