## 3 Controller Design and Stability Analysis
### Stability Analysis  
**Theorem 1**. : _If the initial angular position and velocity of all joints are within their preset constraints such that \(|q_{i}(0)|<\varepsilon_{i}\) and \(\big{|}\hat{q}_{i}(0)\big{|}<\Lambda_{i}\), and (8) in lemma 1 holds, the control law consisting of (7) and (9)-(14) can achieve the uniformly ultimately bounded tracking errors of robotic manipulator system (1). Meanwhile, the angular velocity and angular position of each joint of manipulator are within the preset constraints such that \(|q_{i}(t)|<\varepsilon_{i}\) and \(\big{|}\hat{q}_{i}(t)\big{|}<\Lambda_{i}\), \(\forall\;t>0\). Namely, the control target (4)-(6) is achieved._  
**Proof. ** By inserting (9) into (2) and using the fact of \(\hat{M}^{-1}\hat{M}=I\) with \(I=diag([1,1,..,1])\in R^{n\times n}\), the MIMO robotic manipulator system can be decoupled into \(n\) uncertain SISO subsystems (15). \[\ddot{q}_{i}(t)=u_{i}(t)+d_{i}(t)i=1,2,\ldots,n \tag{15}\]  
where \(q_{i}\in q\) is the angular position of the \(i^{th}\) joint. \(u_{i}\) is designed in (10). \(d_{i}(t)=\Gamma_{i}(t)-\hat{\Gamma}_{i}(t)\) is the TDE error. The following Lyapunov candidate is designed:  
\[V=\sum_{i=1}^{n}V_{i}=\sum_{i=1}^{n}[\frac{1}{2}\ddot{q}_{i}^{2}+\frac{1}{2}k_ {pi}e_{i}^{2}+\frac{1}{2\psi_{i}}\ddot{d}_{i}^{2}+k_{si}\frac{e_{i}^{2}z_{i}^{ 2}}{2(\varepsilon_{i}^{2}-q_{i}^{2})}+k_{yi}\frac{\dot{q}_{i}^{2}}{\Lambda_{i} ^{2}-\dot{q}_{i}^{2}}+\Lambda_{i}\,\Lambda_{i}\mathbb{G}_{i}\ln\left(\cosh\frac {e_{i}}{\dot{\mathbb{G}}_{i}}\right))\ ] \tag{16}\]  
where \(\ddot{d}_{i}=\Gamma_{i}^{*}-\dot{d}_{i}\). \(\Gamma_{i}^{*}\) is the upper bound TDE error defined in Lemma 1. **Remark 5**. : _Clearly, \(k_{si}\frac{e_{i}^{2}z_{i}^{2}}{2(\varepsilon_{i}^{2}-q_{i}^{2})}>0\) holds as long as \(|q_{i}|<\varepsilon_{i}\). \(k_{yi}\frac{\dot{q}_{i}^{2}}{\Lambda_{i}^{2}-\dot{q}_{i}^{2}}>0\) as long as \(\big{|}\dot{q}_{i}\big{|}<\Lambda_{i}\). Furthermore, \(\Lambda_{i}\mathbb{G}_{i}ln\left(cosh\left(\frac{e_{i}}{\dot{\mathbb{G}}_{i}} \right)\right)\ \geq 0\) holds because of the fact of \(cosh(\cdot)\geq 1\) and the fact of \(ln(x)\geq 0\) with \(x\geq 1\). Therefore, the Lyapunov candidate (16) is positive defined as long as \(|q_{i}|<\varepsilon_{i}\) and \(\big{|}\dot{q}_{i}\big{|}<\Lambda_{i}\)._  
Taking the derivative of the Lyapunov function (16) with respect to the time \(t\) and using (11), (12) and (15), we have:\[\dot{V}=\sum\
olimits_{i=1}^{n}[\hat{q}_{i}\ddot{q}_{i}+k_{pi}\dot{e}_{i}\dot{q}_ {i}-\frac{1}{\Psi_{i}}\ddot{\dot{d}_{i}}\dot{d}_{i}+\ k_{si}A_{i}\eta_{i}\dot{q}_ {i}+k_{si}B_{i}z_{i}\dot{z}_{i}+k_{yi}C_{i}\dot{q}_{i}\ddot{\ddot{q}}_{i}+\Lambda _{i}\Lambda_{i}tanh\bigg{(}\frac{\dot{e}_{i}}{\dot{\dot{\Phi}}_{i}}\bigg{)}\hat {q}_{i}] \tag{17}\] \[\qquad=\sum\limits_{i=1}^{n}\dot{q}_{i}\bigg{[}\big{(}1+k_{yi}C_{i }\big{)}(u_{i}+d_{i})+k_{pi}\dot{e}_{i}+\ k_{si}A_{i}\eta_{i}+\ k_{si}B_{i}z_{i }\ +\Lambda_{i}\Lambda_{i}tanh\bigg{(}\frac{\dot{e}_{i}}{\dot{\dot{\Phi}}_{i}} \bigg{)}\bigg{]}\] \[\qquad-\ \sum\
olimits_{i=1}^{n}\bigg{(}\frac{1}{\Psi_{i}}\ddot{d}_{i }\dot{d}_{i}+\ k_{si}B_{i}z_{i}\ \eta_{i}\bigg{)}\]  
Substituting (10) and into (17) with the fact of \(C_{i}>0\) as long as \(\big{|}\dot{q}_{i}\big{|}<\Lambda_{i}\):  
\[\dot{V}=\sum\limits_{i=1}^{n}\dot{q}_{i}\big{(}1+k_{pi}C_{i}\big{)} \big{[}-\dot{d}_{i}sgn\big{(}\dot{q}_{i}\big{)}+d_{i}\big{]}-\sum\
olimits_{i= 1}^{n}\frac{1}{\Psi_{i}}\dd_{i}\dot{d}_{i} \tag{18}\] \[\qquad\qquad-\sum\
olimits_{i=1}^{n}\ k_{si}B_{i}\beta_{i}z_{i}^ {2}-\sum\
olimits_{i=1}^{n}(\Lambda_{i}+k_{di})\dot{q}_{i}^{2}\] \[\qquad\leq\sum\limits_{i=1}^{n}\big{|}\dot{q}_{i}\big{|}\big{(}1+ k_{yi}C_{i}\big{)}\left(\Gamma_{i}^{*}-\hat{d}_{i}\right)-\sum\
olimits_{i= 1}^{n}\frac{1}{\Psi_{i}}\dd_{i}\dot{d}_{i}\] \[\qquad=\sum\limits_{i=1}^{n}\bigg{[}\big{|}\dot{q}_{i}\big{|} \big{(}1+k_{yi}C_{i}\big{)}-\frac{1}{\Psi_{i}}\dd_{i}\bigg{]}\bigg{(}\Gamma_ {i}^{*}-\hat{d}_{i}\bigg{)}\]  
Therefore, for each \(V_{i}\), we can derive (19) by combining (13) and (18):  
\[\dot{V}_{i}\leq\left\{