## 5 Stability Analysis  
To ensure that the system tracking error converges to the zero neighborhood, we design the following adaptive law:  
\[\dot{\hat{k}}_{p}=[\delta k_{p}(x_{d}-\hat{z}_{1})\varphi]/\widetilde{k}_{p}, \tag{39}\]  
\[\dot{\hat{k}}_{d}=[\delta k_{d}(\dot{x}_{d}-\hat{z}_{2})\zeta]/\widetilde{k}_{d}, \tag{40}\]  
\[\hat{M}_{g}=\kappa\dot{\chi}_{z}k_{f}\dot{z}. \tag{41}\]  
**Remark 5**.: _To avoid \(\widetilde{k}_{p}\) and \(\widetilde{k}_{d}\) being zero, \(\dot{\hat{k}}_{p}\) and \(\dot{\hat{k}}_{d}\) are integrated. The above formula can be rewritten as_  
\[\dot{k}_{p}=\int_{0}^{t}[\delta k_{p}(x_{d}-\hat{z}_{1})\varphi]/\widetilde{k }_{p}d\tau+k_{p0}, \tag{42}\]  
\[\dot{\hat{k}}_{d}=\int_{0}^{t}[\delta k_{d}(\dot{x}_{d}-\hat{z}_{2})\zeta]/ \widetilde{k}_{d}d\tau+k_{d0}. \tag{43}\]  
Appropriate constants \(k_{p0}\) and \(k_{d0}\) are chosen to make \(\widetilde{k}_{p}=k_{p}-\hat{k}_{p}\) and \(\widetilde{k}_{d}=k_{d}-\hat{k}_{d}\) non-zero.  
Proof of Theorem 1.: A positive definite Lyapunov function is designed as follows.  
\[V=\frac{1}{2}\sigma^{2}+\frac{1}{2}\chi_{z}^{2}+\frac{1}{2}\widetilde{k}_{p} \varphi\widetilde{k}_{p}+\frac{1}{2}\widetilde{k}_{d}\zeta\widetilde{k}_{d}+ \frac{1}{2}\widetilde{M}_{g}\kappa\widetilde{M}_{g}. \tag{44}\]  
Taking the derivative of Equation (46),  
\[\dot{V}=\sigma\dot{\sigma}+\chi_{z}\dot{\chi}_{z}+\widetilde{k}_{p}\varphi^{- 1}\dot{\widetilde{k}}_{p}+\widetilde{k}_{d}\zeta^{-1}\dot{\widetilde{k}}_{d} +\widetilde{M}_{g}\kappa^{-1}\dot{\widetilde{M}}_{g}. \tag{45}\]  
Bringing (31) and (40) into (47) can obtain  
\[\begin{split}\dot{V}&=\sigma\Big{[}-\mu\sigma+\hat {k}_{p}(v_{d}-\hat{z}_{1})+\hat{k}_{d}(\dot{v}_{d}-\hat{z}_{2})\Big{]}+\chi_{z} \Big{[}-\alpha\chi_{z}+k_{f}\dot{z}\hat{M}_{g}-k_{f}\dot{z}M_{g}\Big{]}\\ &+\widetilde{k}_{p}\varphi^{-1}\dot{\widetilde{k}}_{p}+ \widetilde{k}_{d}\zeta^{-1}\dot{\widetilde{k}}_{d}+\widetilde{M}_{g}\kappa^{- 1}\dot{\widetilde{M}}_{g}\\ &=-\mu\sigma^{2}+\hat{k}_{p}(v_{d}-\hat{z}_{1})\sigma-\alpha\chi_ {z}^{2}+k_{f}\dot{z}\hat{M}_{g}\chi_{z}+\hat{k}_{d}(\dot{v}_{d}-\hat{z}_{2}) \sigma-k_{f}\dot{z}M_{g}\chi_{z}\\ &+\widetilde{k}_{p}\varphi^{-1}\dot{\widetilde{k}}_{p}+ \widetilde{k}_{d}\zeta^{-1}\dot{\widetilde{k}}_{d}+\widetilde{M}_{g}\kappa^{- 1}\dot{\widetilde{M}}_{g}\\ &\leq-\mu\sigma^{2}+\hat{k}_{p}(v_{d}-\hat{z}_{1})\sigma-\alpha \chi_{z}^{2}+k_{f}\dot{z}\hat{M}_{g}\chi_{z}+\hat{k}_{d}(\dot{v}_{d}-\hat{z}_{ 2})\sigma-k_{f}\dot{z}M_{g}\chi_{z}\\ &-\widetilde{k}_{p}\varphi^{-1}\dot{\widetilde{k}}_{p}+ \widetilde{k}_{d}\zeta^{-1}\dot{\widetilde{k}}_{d}+\widetilde{M}_{g}\kappa^{- 1}\dot{\widetilde{M}}_{g}\end{split} \tag{46}\]  
Bringing (41), (42) and (43) into (48) can obtain  
\[\begin{split}\dot{V}&\leq-\mu\sigma^{2}+\hat{k}_{p} (v_{d}-\hat{z}_{1})\sigma-\alpha\chi_{z}^{2}+k_{f}\dot{z}\hat{M}_{g}\chi_{z}+ \hat{k}_{d}(\dot{v}_{d}-\hat{z}_{2})\sigma-k_{f}\dot{z}M_{g}\chi_{z}\\ &-\widetilde{k}_{p}\varphi^{-1}[\delta\hat{k}_{p}(v_{d}-\hat{z} _{1})\varphi]/\widetilde{k}_{p}+\widetilde{k}_{d}\zeta^{-1}[\delta k_{d}(\dot{ v}_{d}-\hat{z}_{2})\zeta]/\widetilde{k}_{d}+\widetilde{M}_{g}\kappa^{-1}\kappa\dot{ \chi}_{z}k_{f}\dot{z}\end{split} \tag{47}\]Formula (49) is simplified to obtain  
\[\dot{V}\leq-\mu\sigma^{2}-\alpha\chi_{\mathrm{z}}^{2}, \tag{48}\]  
where \(\mu\) and \(\alpha\) are positive parameters. Simplifying Equation (49) can obtain  
\[\dot{V}\leq 0. \tag{49}\]  
Through the above analysis, according to the Lyapunov stability theory, it can be proved that the system is uniformly stable.  
According to [PERSON] theorem, we can obtain  
\[\lim_{t\to\infty}\chi_{\mathrm{z}}(t)=0,\lim_{t\to\infty}\delta(t)=0. \tag{50}\]  
Therefore, when \(t\to\infty\), \(\delta(t)\to 0\); this can prove that the quadrotor system is asymptotically stable.