## 2 Dynamical Model and Problem Statement  
The dynamic model of n-link robot manipulators is shown as the following:  
\[\mathbf{M}(q(t))\ddot{\mathbf{q}}(t)+\mathbf{C}\big{(}\mathbf{q}(t),\dot{\mathbf{q}}(t)\big{)}\dot{ \mathbf{q}}(t)+\mathbf{G}(\mathbf{q}(t))+\mathbf{F}\big{(}\dot{\mathbf{q}}(t)\big{)}=\mathbf{\tau}(t)+ \mathbf{\tau}_{d}(\mathbf{t}) \tag{1}\]  
where \(\mathbf{M}(\mathbf{q}(t))\in\mathbb{R}^{\mathrm{n}\times\mathrm{n}}\) is the inertia matrix, \(\mathbf{q}(t)=[q_{1}(t),q_{2}(t),..,q_{i}(t),..,q_{n}(t)]^{T}\!\in\mathbb{R}^{ \mathrm{n}}\) is the vector of angular positions of joints of manipulator. \(\mathbf{C}\big{(}\mathbf{q}(t),\dot{\mathbf{q}}(t)\big{)}\in\mathbb{R}^{\mathrm{n}\times \mathrm{n}}\) is the Coriolis and centrifugal matrix. \(\mathbf{G}(\mathbf{q}(t))\!\in\mathbb{R}^{\mathrm{n}}\) is the gravity vector. \(\mathbf{F}\big{(}\dot{\mathbf{q}}(t)\big{)}\!\in\mathbb{R}^{\mathrm{n}}\) is the vector of friction. \(\mathbf{\tau}(t)\in\mathbb{R}^{\mathrm{n}}\) is the vector of torques applied on the joints. \(\mathbf{\tau}_{d}(\mathbf{t})\in~{}R^{\mathrm{n}}\) is the external disturbance.  
The model (1) can be further written as (2) to indicate the system uncertainty.  
\[\begin{split}\ddot{\mathbf{q}}(\mathbf{t})=\mathbf{M}(\mathbf{q}(t))^{-1}& \big{[}-\mathbf{C}\big{(}\mathbf{q}(t),\dot{\mathbf{q}}(t)\big{)}\dot{\mathbf{q}}(t)-\mathbf{G}( \mathbf{q}(t))-\mathbf{F}\big{(}\dot{\mathbf{q}}(t)\big{)}+\mathbf{\tau}_{d}(\mathbf{t})\big{]}+ \big{[}\mathbf{M}(\mathbf{q}(t))^{-1}-\hat{\mathbf{M}}(\mathbf{q}(t))^{-1}\big{]}\mathbf{\tau}(t) \\ +&\hat{\mathbf{M}}(\mathbf{q}(t))^{-1}\mathbf{\tau}(t)=\mathbf{ \Gamma}(t)+\hat{\mathbf{M}}(\mathbf{q}(t))^{-1}\mathbf{\tau}(t)\end{split} \tag{2}\]  
\[\begin{split}&\text{where }\hat{\mathbf{M}}(\mathbf{q}(t))\text{ is the estimation of }\mathbf{M}(\mathbf{q}(t))\text{. }\mathbf{\Gamma}(\mathbf{t})=\mathbf{M}(\mathbf{q}(t))^{-1}\big{[}-\mathbf{C}\big{(}\mathbf{q}(t),\dot{ \mathbf{q}}(t)\big{)}\dot{\mathbf{q}}(t)-\mathbf{G}(\mathbf{q}(t))\\ -\mathbf{F}\big{(}\dot{\mathbf{q}}(t)\big{)}+\mathbf{\tau}_{d}(\mathbf{t})\big{]}+ \big{[}\mathbf{M}(\mathbf{q}(t))^{-1}-\hat{\mathbf{M}}(\mathbf{q}(t))^{-1}\big{]}\mathbf{\tau}(t) \text{ is the system uncertainty.}\end{split}\]  
The vector of error between the desired angular position and actual angular position is defined in (3):  
\[\mathbf{e}(t)=\mathbf{q}(t)-\mathbf{q}_{d} \tag{3}\]  
where \(\mathbf{q}_{d}=[q_{d1},q_{d2},\ldots,q_{dn}]^{T}\in R^{n}\) is the vector of desired angular position of joints. \(\mathbf{e}=[e_{1},e_{2},\ldots,e_{n}]^{T}\in R^{n}\) is the tracking error vector.  
The main control objective is to design the torque \(\mathbf{\tau}(\mathbf{t})\) that can drive all the joints of the robotic manipulator system (1) to approach their desired angular positions. Meanwhile, both the angular positions and angular velocities for each joint should be guaranteed to not exceed the given constraints. The control objective can be described by (4)-(6).  
\[||e(t)||\leq\sigma,\;\forall\;t\geq 0 \tag{4}\]  
\[|q_{i}(t)|<\epsilon_{i},\;\forall\;t\geq 0,\;i=1,2,..,n \tag{5}\]  
\[\big{|}\dot{q}_{i}(t)\big{|}<\Lambda_{i},\;\forall\;t\geq 0,\;i=1,2,..,n \tag{6}\]  
where \(\sigma^{*}>\sigma\geq 0\), and \(\sigma^{*}>0\) is a positive constant, \(\epsilon_{i}>0\) is a positive constant referring to the angular position constraint of the \(i^{th}\) joint, and \(\Lambda_{i}>0\) is a positive constant, meaning the angular velocity constraint of the \(i^{th}\) joint.  
The positive constant \(\sigma^{*}\) is related to the initial state of the system and parameters of the controller. \(\sigma\) reflects the tracking accuracy. Notably, \(\sigma(t\rightarrow\infty)=0\) means the asymptotic convergence of tracking errors.  
**Remark 2**: _In this paper, we consider the angular position tracking problem for each joint of the manipulator. Hence, \(q_{di}\) is a constant for \(i=1,2,..,n\), which means \(\dot{q}_{di}=0\). The angular trajectory tracking problem will be considered in our future work._