## 3 Controller Design and Stability Analysis
### Stability Analysis  
Based on the robust Nussbaum gain-based approach in Lemma 2 and above formulations, the conclusion is summarized as follows.  
**Theorem 1**.: _For \(ACV\) mathematical models (16)-(18), the virtual control laws are chosen as (32) and (42), adaptive control laws of RBFNN are provided as (38) and (48),the position control laws are given by (35)-(37), and yaw control laws are (45)-(47). Then, we can obtain the following properties:_  
1. _The tracking errors of the_ \(ACV\) _can converge to small neighbourhoods around zero._
2. _The yaw rate_ \(r\) _is constrained to_ \(k_{b}\) _for_ \(t\in[0,+\infty]\)_All the signals in the closed-loop system are bounded._  
Proof.: To facilitate the analysis, the positive definite Lyapunov function is chosen as:  
\[\begin{split} V&=V_{2}+V_{4}\\ &=\frac{1}{2}(z_{e}-\delta_{\Delta})^{2}+\frac{1}{2}\xi_{u}^{2}+ \frac{1}{2}mu_{e}^{2}+\frac{1}{2}\tilde{W}_{u}^{2}+\frac{1}{2}\psi_{e}^{2}+ \frac{1}{2}\xi_{r}^{2}+\frac{1}{2}l_{z}\ln\frac{k_{b}^{2}}{k_{b}^{2}-r_{e}^{2}} +\frac{1}{2}\tilde{W}_{r}^{2}.\end{split} \tag{49}\]  
Differentiating (49) with the help of (34) and (44), we have:  
\[\dot{V}\leq -k_{1}(z_{e}-\sigma_{\Delta})^{2}-\frac{1}{2}[\xi_{u}-c_{u}(z_{e}- \sigma_{\Delta})]^{2}-\frac{1}{2}[\xi_{u}-b_{u}\Delta a_{u}]^{2}\] \[-(\alpha_{u}-1)\xi_{u}^{2}-k_{2}u_{e}^{2}-g_{u}(t)N_{u}(\zeta_{u} (t))\bar{u}-u_{e}\bar{u}+u_{e}\varepsilon_{u}(x)-\eta_{u}\tilde{W}_{u}\tilde{ W}_{u}\] \[-k_{3}\psi_{e}^{2}-\frac{1}{2}[\xi_{r}-c_{r}\psi_{e}]^{2}-\frac{ 1}{2}[\xi_{r}-b_{r}\Delta a_{r}]^{2}-(\alpha_{r}-1)\xi_{r}^{2}-k_{4}r_{e}^{2} +r_{e}\varepsilon_{r}(x)\] \[-g_{r}(t)N_{r}(\zeta_{r}(t))r-r_{e}\bar{r}-\eta_{r}\tilde{W}_{r} \tilde{W}_{r} \tag{50}\]  
Utilizing the Yang inequality, we obtain  
\[u_{e}\varepsilon_{u}(x)\leq\frac{1}{2}u_{e}^{2}+\frac{1}{2}\varepsilon_{u}^{2 }(x)\leq\frac{1}{2}u_{e}^{2}+\frac{1}{2}\bar{\varepsilon}^{2} \tag{51}\]  
\[r_{e}\varepsilon_{r}(x)\leq\frac{1}{2}r_{e}^{2}+\frac{1}{2}\varepsilon_{r}^{2} (x)\leq\frac{1}{2}r_{e}^{2}+\frac{1}{2}\bar{\varepsilon}^{2} \tag{52}\]  
\[-\eta_{u,r}\tilde{W}_{u,r}\tilde{W}_{u,r}\leq-\frac{\eta_{u,r}}{2}\tilde{W}_{ u,r}^{2}+\frac{\eta_{u,r}}{2}\tilde{W}_{u,r}^{2} \tag{53}\]  
Subsequently, substituting (49) and (51)-(53) into (50), and according the Lemma 2, we have:  
\[\dot{V}\leq-\Lambda V+\sum_{k=u,r}\frac{1}{\rho_{k}}[g_{k}(t)N_{k}(\zeta_{k}( t))+1]\zeta_{k}(t)+\Gamma \tag{54}\]  
where  
\[\Lambda=\min\{2k_{1},2(\alpha_{u}-1),\frac{2k_{2}-1}{m},\eta_{u,r},2k_{3},2( \alpha_{r}-1),\frac{2k_{4}}{l_{z}}\}\]  
\[\Gamma=\frac{\eta_{u,r}}{2}\hat{W}_{u,r}^{2}+\varepsilon^{2}.\]  
Taking the integration of (54) over the time \([0,t]\), we get:  
\[V\leq\sum_{k=u,r}\frac{e^{-\Lambda t}}{\rho_{k}}\int_{0}^{t}[g_{k}(t)N_{k}+1] \zeta_{k}(\lambda)e^{\Lambda\lambda}d\lambda-V(0)e^{-\Lambda t}+\frac{\Gamma }{\Lambda}. \tag{55}\]  
Applying Lemma 2 to (55), here \(k=u,r\), then we draw the conclusion that \(N_{k}(\zeta_{k}(t))\), \(V(t)\), \(e^{-\Lambda t}\int_{0}^{t}\zeta_{k}^{r}(\lambda)e^{\Lambda\lambda}d\lambda\), \(e^{-\Lambda t}\int_{0}^{t}\delta_{k}(t)N_{k}(\zeta_{k}(\lambda))\zeta_{k}( \lambda)e^{\Lambda\lambda}d\lambda\) are bounded over \([0,t]\).  
**Remark 4**.: _To improve the efficiency of selecting the parameters for design, the parameters \(k_{1},k_{2},k_{3},k_{4}\) in the controller are required to be positive definite. The designed parameter \(\eta_{u,r}\) in (38) and (48) are positive constants; and \(\rho_{u,r}\) in (37) and (47) is defined as positive definite. For the given parameters \(k_{1},k_{2},k_{3},k_{4},\eta_{u,r},\rho_{u,r}\), if the designed gain \(\eta_{u,r}\) is set smaller but \(k_{1},k_{2},k_{3},k_{4}\), and \(\rho_{u,r}\) are chosen as larger, then the tracking error can be small._