## 4 Theoretical Works
\begin{array}{c}v\cos\theta\\ v\sin\theta\\ \omega\end{array} \right] \tag{11}\]  
The relationship among interacting forces can be performed  
\[\left\{ \begin{array}{c}J\dot{\omega}=u_{1}+d_{1}(t)\\ m\dot{v}=u_{2}+d_{2}(t)\end{array} \right. \tag{12}\]  
where  
\[u=\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]=\frac{1}{r}\left[ \begin{array}{c}\tau_{1}-\tau_{2}\\ \tau_{1}+\tau_{2}\end{array} \right],\ d(t)=\left[ \begin{array}{c}d_{1}(t)\\ d_{2}(t)\end{array} \right]\]  
For physical meaning, \(d_{i}(t)\) denotes external noises or uncertain factors. \(u_{i}(t)\) implies the control signal for two wheels. To address the tracking control troubles, the reference signal is firstly determined so that system errors converge to zero values in finite time. \[\dot{q}_{r}=\left[ \begin{array}{c}\dot{x}_{r}\\ \dot{y}_{r}\\ \dot{\theta}_{r}\end{array} \right]=\left[ \begin{array}{c}v_{r}\cos\theta_{r }\\ v_{r}\sin\theta_{r}\\ \omega_{r}\end{array} \right] \tag{13}\]  
\(\left[x_{r}(t),y_{r}(t),\theta_{r}(t)\right]^{T}\in\Re^{3}\), \(\left[v_{r}(t),\omega_{r}(t)\right]^{T}\in\Re^{2}\). Then, the tracking error vector is defined as  
\[\vec{e}=\left[ \begin{array}{c}e_{x}\\ e_{y}\\ e_{\theta}\end{array} \right]=\left[ \begin{array}{ccc}\cos\theta&\sin\theta &0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}x-x_{r}\\ y-y_{r}\\ \theta-\theta_{r}\end{array} \right] \tag{14}\]  
Hence,  
\[\vec{e}=\left[ \begin{array}{c}\dot{e}_{x}\\ \dot{e}_{y}\\ \dot{e}_{\theta}\end{array} \right]=\left[ \begin{array}{c}v-v_{r}\cos e_{ \theta}+\omega e_{y}\\ v_{r}\sin e_{\theta}-\omega e_{x}\\ \omega-\omega_{r}\end{array} \right] \tag{15}\]  
In the real world, some constraints related to hardware platform or physical phenomenon still exist inside the system. With no loss of generality, we give the following assumptions. **Assumption 4:** The existing positive gains which are tuned could be the upper bound and lower bound of linear velocity and angular velocity. \[\omega_{r}^{lower}\leq\left|\omega_{r}\right|\leq\omega_{r}^{upper} \tag{16}\]  
\[\left|\dot{\omega}_{r}\right|\leq\dot{\omega}_{r}^{upper} \tag{17}\]  
\[\left|v_{r}\right|\leq v_{r}^{upper} \tag{18}\]  
\[\left|\dot{v}_{r}\right|\leq\dot{v}_{r}^{upper} \tag{19}\]  
Although the external disturbances or uncertainties are unknown, its positive upper limitation is always remaining so that \(\left|d_{i}(t)\right|\leq d^{upper}\)  
Obviously, the entire design idea is to categorize into angular control problem and positioning control problem as following. Even though the Multi-Input-Multi-Output (MIMO) controller design can be employed in this situation, a separated solution for second-order subsystem and third-order subsystem assists to lessen a burden computation. \[\left\{ \begin{array}{c}\dot{e}_{\theta}=\omega-\omega_{r}\\ J\dot{\omega}=u_{1}+d_{1}(t)\end{array} \right. \tag{20}\]  
\[\left\{\begin{array}{c}\dot{e}  
_with \(\varphi_{1}>0\). Substituting Equation (38) into Equation (35)_  
\[j_{1}=-\beta_{1}s_{1}+\widetilde{Y}_{1}-\varphi_{1}\text{sign}(s_{1})|s_{1}|^{ \frac{1}{2}} \tag{39}\]  
As the proposed control law (38) guarantees that the tracking angular error converges to the small region in finite time, we could obtain \(sin(e_{\theta})\to 0\) and \(cos(e_{\theta})\to 1\). Later, the subsystem (21) can be simplified to  
\[\left\{ \begin{array}{c}\dot{x}_{e}=v-v_{r}+\omega_{r}y_{e}\\ \dot{y}_{e}=-\omega_{r}x_{e}\\ m\dot{v}=u_{2}+d_{2}(t)\end{array} \right. \tag{40}\]  
Now, we can elect the transitional variable as \(e_{\omega}=x_{e}-sign(\omega_{r})y_{e}\) and \(\dot{e}_{\omega}=v-v_{r}+\omega_{r}y_{e}+\big{|}\omega_{r}\big{|}x_{e}\). Then, a sliding mode surface is constructed as  
\[s_{2}=\dot{e}_{\omega}+k_{1}e_{\omega}+k_{2}\Gamma_{e_{\omega}} \tag{41}\]  
\[\Gamma_{e_{\omega}}=\left\{