## 3 Methodology
### Adaptive Controller Design with Input Saturation
This indicates that \(v_{1}\) and \(v_{2}\) converge to \(\Omega_{1}:=\{v_{i}\in R,i=1,2||v_{i}|\leq B_{1}\}\). Step 2: Then, consider a Lyapunov function as  
\[V_{2}=\frac{1}{2}(\gamma_{e}^{2}+u_{e}^{2}+z_{1}^{2}). \tag{43}\]  
The derivative of \(V_{2}\) is  
\[ \begin{array}{l}\dot{V}_{2}=\gamma_{e}\dot{\gamma}_{e}+u_{e}\dot{u}_{e}+z_ {1}\dot{z}_{1}\\ \leq-k_{r}\gamma_{e}^{2}-(k_{1}-M_{1})u_{e}^{2}-(k_{2}-1.5M_{1})z_{1}^{2}+0.5 M_{1}(|l_{1}\phi|^{2}+\overline{d}_{1}^{2}+B_{1}^{2})\\ \leq-n_{2}V_{2}+\delta_{2}\end{array} , \tag{44}\]  
where \(n_{2}=2\text{min}\{k_{r},k_{1}-M_{1},k_{2}-1.5M_{1}\}\), \(k_{r}>0\), \(k_{1}>M_{1}\), \(k_{2}>1.5M_{1}\), and \(\delta_{2}=0.5M_{1}(|l_{1}\phi|^{2}+\overline{d}_{1}^{2}+B_{1}^{2})\). Multiply both sides of (44) by \(e^{n_{2}t}\), and then integrate both sides to obtain the following:  
\[V_{2}(t)\leq V_{2}(0)e^{-n_{2}t}+\delta_{2}(1-e^{-n_{2}t})/n_{2}\leq V_{2}(0)+ \delta_{2}/n_{2}. \tag{45}\]Additionally, we can determine that \(|\gamma_{e}|\leq B_{2}\), \(|u_{e}|\leq B_{2}\), \(|z_{1}|\leq B_{2}\), and \(B_{2}=\sqrt{2V_{2}(0)+2\delta_{2}/n_{2}}\). This indicates that \(\gamma_{e}\), \(u_{e}\), and \(z_{1}\) converge to a compact set: \(\Omega_{2}:=\{u_{e},z_{1}\in R\), \(||\gamma_{e}|\leq B_{2},|u_{e}|\leq B_{2},|z_{1}|\leq B_{2}\}\). \(\square\)  
Part 2: Controller design for the subsystem (28)  
To determine \(\tau_{r}\), according to the relationship \(v=v_{c}+v_{s}\), \(r=r_{c}+r_{s}\), we further design  
\[v_{s}=-\frac{a_{13}ur_{s}}{Y_{v}}. \tag{46}\]  
This accounts for the fact that the vehicle model herein is under-actuated. Accordingly, subsystem (28) is taken as:  
\[ \begin{cases}\dot{v}_{c}=g_{v}v_{c}+g_{r}r_{c}\\ \dot{r}_{c}=f_{r}+M_{3}u_{2}+M_{3}d_{2}\\ \dot{u}_{2}=-c_{2}u_{2}+\text{sat}(\tau_{2})\end{cases} , \tag{47}\]  
where \(f_{r}=M_{3}(N_{r}(r_{c}+r_{s})+a_{12}uv)\), \(g_{v}=M_{2}Y_{v}\), and \(g_{r}=M_{2}a_{13}u\). The error variables of the system are introduced as follows:  
\[ \begin{cases}v_{c}=v-v_{s}\\ z_{2}=r_{c}-\alpha_{2}^{c}\\ z_{3}=u_{2}-\alpha_{3}^{c}\end{cases} , \tag{48}\]  
where \(\alpha_{2}^{c}\) is the filtered signal of the virtual control quantity \(\alpha_{2}^{co}\) for \(r_{c}\). \(\alpha_{3}^{c}\) is the filtered signal of the virtual control quantity \(\alpha_{3}^{co}\) for \(u_{2}\). Define the compensation signal of the filter error:  
\[v_{3}=v_{c}-\xi_{3}, \tag{49}\]  
\[v_{4}=z_{2}-\xi_{4}, \tag{50}\]  
\[v_{5}=z_{3}-\xi_{5}. \tag{51}\]  
Construct the residual compensations of the filtering error and the saturation control quantity as follows:  
\[\dot{\xi}_{3}=-k_{3}\xi_{3}+[(\alpha_{2}^{c}-\alpha_{2}^{co})+\xi_{4}], \tag{52}\]  
\[\dot{\xi}_{4}=-k_{4}\xi_{4}+M_{3}(\alpha_{3}^{c}-\alpha_{3}^{co})+M_{3}\xi_{5}, \tag{53}\]  
\[\dot{\xi}_{5}=-k_{5}\xi_{5}+D_{2}, \tag{54}\]  
where, \(k_{3}>0\), \(k_{4}>0\), \(k_{5}>0\);\(\xi_{3}(0)=0\), \(\xi_{4}(0)=0\), \(\xi_{5}(0)=0\);\(D_{2}=\text{sat}(\tau_{2})-\tau_{2}+M_{3}\vec{d}_{2}\), and \(\vec{d}_{2}=d_{2}-\vec{d}_{2}\). Additionally, the control signals are designed as follows:  
\[\alpha_{2}^{co}=-k_{3}v_{c}/g_{v}+g_{r}r_{c}, \tag{55}\]  
\[\alpha_{3}^{co}=(-k_{4}z_{2}-f_{r}-M_{3}\vec{d}_{2}+\hat{\alpha}_{2}^{c}-v_{ 3})/M_{3}, \tag{56}\]  
\[\tau_{2}=c_{2}u_{2}-k_{5}z_{3}+\hat{\alpha}_{3}^{c}-M_{3}v_{4}+(M_{3}-1)D_{2}. \tag{57}\]  
**Theorem 3**. : _Considering the subsystem (28), select the filter as shown in (23), construct a compensation system as shown in (52)-(54), and the control signals are designed as (55)-(57); the filter compensation errors (49)-(51) are asymptotically stable, and the tracking errors (48) are bound and convergent._  
**Proof. ** Taking the derivative of (48), we could obtain the following:  
\[ \begin{cases}\dot{v}_{c}=\left[\dot{\alpha}_{2}^{co}+(\dot{\alpha}_{2}^{c}- \dot{\alpha}_{2}^{co})+z_{2}\right]-gr_{c}\\ \dot{z}_{2}=f_{r}+M_{3}[\dot{\alpha}_{3}^{co}+(\dot{\alpha}_{3}^{c}-\dot{ \alpha}_{3}^{co})+z_{3}]+M_{3}d_{2}-\dot{\alpha}_{2}^{c}\\ \dot{z}_{3}=-c_{2}u_{2}+\tau_{2}+D_{2}-\dot{\alpha}_{3}^{c}\end{cases} .