## 3 Controller Design and Stability Analysis
### Stability Analysis
\[\mathcal{V}_{i}(\infty)-\mathcal{V}_{i}(0)\leq-\Lambda_{i}\dot{\xi}_{i}\int_ {0}^{\infty}tanh^{2}\bigg{(}\frac{e_{i}}{\dot{\xi}_{i}}\bigg{)} \tag{27}\]  
In the light of [PERSON]'s Lemma, (27) implies \(tanh\bigg{(}\frac{e_{i}(\infty)}{\dot{\xi}_{i}}\bigg{)}=0\), which means \(e_{i}(\infty)=0\). Moreover, it is clear that \(|\dot{q}_{i}|=|\Lambda_{i}tanh\bigg{(}\frac{e_{i}}{\dot{\xi}_{i}}\bigg{)}|< \Lambda_{i}\) due to the fact of \(\tanh(\cdot)<1\). \(\square\)  
**Remark 6. **_The asymptotic convergence of tracking errors in Lemma 2 is stronger than the boundness of tracking errors that we achieved in Theorem 1, which means a better tracking accuracy. It is because asymptotic convergence means \(e_{i}\) is eventually going to zero, while boundedness only implies \(|e_{i}|\) bounded by a positive constant. Therefore, the proposed controller can be improved with a better tracking accuracy by finding the optimal parameters of the proposed controller that are able to minimize \(\bigg{|}\dot{q}_{i}+\Lambda_{i}tanh\bigg{(}\frac{e_{i}}{\dot{\xi}_{i}}\bigg{)} \bigg{|}\)._  
The converging behaviour of tracking errors of (24) is visualized in Figure 1.  
**Lemma 3**. : _For a moment \(t^{*}\), if the system states \(\big{(}q_{i}(t^{*}),\dot{q}_{i}(t^{*})\big{)}\) satisfy (5) and (6) as well as \(\dot{q}_{i}(t^{*})+\Lambda_{i}tanh\Big{(}\frac{e_{i}(t^{*})}{\overline{\xi}_{i} }\Big{)}\
eq 0\) holds, there always exists the optimal parameter \(\Lambda_{i}^{*}>0\) to decrease \((\dot{q}_{i}+\Lambda_{i}tanh\Big{(}\frac{e_{i}}{\overline{\xi}_{i}}\Big{)} \Big{)}^{2}\). Namely, (28) holds._  
\[\frac{d[\Big{(}\dot{q}_{i}+\Lambda_{i}tanh\Big{(}\frac{e_{i}}{\overline{\xi}_{i }}\Big{)}\Big{)}^{2}]}{dt}|_{t=t^{*}}<0 \tag{28}\]  
**Proof. ** Combining (10) and (15), we can obtain the 1 st derivative of \(\dot{q}_{i}+\Lambda_{i}tanh\Big{(}\frac{e_{i}}{\overline{\xi}_{i}}\Big{)}\) with respect to time, shown as (29). \[|e_{i}|\leq\sqrt{\frac{2}{k_{pi}}}\max\{V_{i}(0),\ (\overline{\Omega}_{i}+\frac{1}{2\psi_{i}} \big{(}2\Gamma_{i}^{*})^{2})i=1,2..,n \tag{21}\]0 \[|e_{i}|\leq\sqrt{\frac{2}{k_{pi}}}\max\{V_{i}(0),\ (\overline{\Omega}_{i}+\frac{1}{2\psi_{i}} \big{(}2\Gamma_{i}^{*})^{2})i=1,2..,n \tag{21}\]1  
\[|e_{i}|\leq\sqrt{\frac{2}{k_{pi}}}\max\{V_{i}(0),\ (\overline{\Omega}_{i}+\frac{1}{2\psi_{i}} \big{(}2\Gamma_{i}^{*})^{2})i=1,2..,n \tag{21}\]2 Converging behaviour of tracking error satisfying (24).The red line is the velocity constraint, and the black line with arrow is the converging trajectory. Then, we define a negative changing rate of at moment \(t^{*}\) in (30). \[|e_{i}|\leq\sqrt{\frac{2}{k_{pi}}}\max\{V_{i}(0),\ (\overline{\Omega}_{i}+\frac{1}{2\psi_{i}} \big{(}2\Gamma_{i}^{*})^{2})i=1,2..,n \tag{21}\]3  
Using (5), (6), (29) and (30), a solution of \(\Lambda_{i}^{*}\) satisfying (30) is obtained in (31). \[|e_{i}|\leq\sqrt{\frac{2}{k_{pi}}}\max\{V_{i}(0),\ (\overline{\Omega}_{i}+\frac{1}{2\psi_{i}} \big{(}2\Gamma_{i}^{*})^{2})i=1,2..,n \tag{21}\]4  
Notably, \(\dot{d}_{i}\), \(|e_{i}|\), \(|z_{i}|\), \(|A_{i}|\) and \(|B_{i}|\) are all bounded because of the bounded Lyapunov function (16). Therefore, a finite real solution of \(\Lambda_{i}^{*}\) satisfying (31) exists as long as  
\(\dot{q}_{i}+\Lambda_{i}tanh\left(\frac{e_{i}}{\dot{\mathbb{C}}_{i}}\right)\
eq 0\). **Remark 7**. : _According to Lemma 3 and Remark 6, the optimal parameters \(\Lambda_{i}^{*}\) leading to a decrease of \(|\dot{q}_{i}+\Lambda_{i}tanh\left(\frac{e_{i}}{\dot{\mathbb{C}}_{i}}\right)|\) at the moment \(t^{*}\) can improve the tracking accuracy. Although a large enough \(\Lambda_{i}\) can be selected to satisfy (31), an inappropriately large \(\Lambda_{i}\) could lead to a significant chattering of control torques and thereby compromise the tracking accuracy. Moreover, an optimal \(\Lambda_{i}\) is hard to be deterministically found due to the complexity of the system and the unknown TDE errors.