## 3 Problem Formulation and Control Approaches
### Control Approach Based on the Bounded-Differentiable-Disturbance: Second Control Approach
To investigate the stability of the system with the designed torque, we substitute (29) into (28) and apply some algebraic manipulations, which gives  
\[M(q)\eta_{2}= -\frac{1}{2}\dot{M}(q)\eta_{2}-\frac{1}{2}\dot{M}(q)\eta_{2}+\dot {Q}-(K_{2}+1)\eta_{1}+\tau_{v}+\dot{\tau}_{l} \tag{31}\] \[+\dot{Y}_{d}\phi-(K_{2}+1)\eta_{2}-K_{3}sgn(\eta_{1})-\eta_{1}\]  
where \(\eta_{1}\) is added and subtracted to facilitate the subsequent analysis. Using (31) and (4) and performing some simplifications, (21) can be re-expressed as  
\[\dot{V}= \eta_{2}^{T}[-\frac{1}{2}\dot{M}(q)\eta_{2}+\dot{Q}-(K_{2}+1)\eta _{1}+\tau_{v}+\dot{\tau}_{l}+\dot{Y}_{d}\phi-(K_{2}+1)\eta_{2}-K_{3}sgn(\eta_{ 1})] \tag{32}\] \[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]0Based on [PERSON]'s lemma [31], the condition that the auxiliary function \(W\) should be positive and its derivative will be used to cancel some terms in (32). \(W\) is formulated as follows:  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]1  
where \(i_{1}\) and \(i_{2}\in\Re^{n}\) are the values of \(\eta_{1}\) and \((\dot{\tau}_{v}+\dot{\tau}_{l}+\dot{Y}\phi)\) at the initial conditions, respectively. In order to guarantee that \(W\) is always positive, the gain \(K_{3}\) should be selected to be greater than \((\ddot{\tau}_{v}+\dot{\eta}_{l}+\dot{Y}\phi)\), which is possible since \(\ddot{\tau}_{v}\) and \(\dot{\tau}_{l}\) are bounded according to Assumption 6, the desired trajectories \(q_{d},\dot{q}_{d},\ddot{q}_{d}\) are known, which makes \(\dot{Y}_{d}\) known, and \(\phi\) is constant. Thus,  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]2  
where all terms in \(\dot{W}\) are uniformly continuous. Therefore, using (33) \(\dot{V}\) yields  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]3 \[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]4  
We segregate and upper bound some terms in the above expressions based on [17] as follows:  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]5  
where \(z\in\Re^{3n}\) is defined as  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]6  
and by using the triangle inequality, we can obtain the following inequality:  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]7  
We apply the above inequalities to upper bound \(\dot{V}\) as follows:  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]8  
By applying the nonlinear damping proof [35], we can obtain  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]9  
Therefore, we can simplify (35) as follows:  
\[M(q)\eta_{2}= (Y-Y_{d})\phi+M(q)(\sigma_{1}\Delta\dot{q}+\sigma_{2}\eta_{1})+Y_{ d}\phi+\tau_{v}+\tau_{l}-\tau. \tag{26}\]0  
To obtain a negative definite result from the above expression, \(K_{2}\) should be selected to be larger than \(\rho^{2}\). Since \(\rho\) is a function of the states that are bounded, we can obtain a semi-global stability result as the best depending on the initial conditions of the states. As a result we can invoke Lemma 2 in [36] to show that \(\Delta q\to 0\) as \(t\rightarrow\infty\).