## 3 Problem Formulation and Control Approaches
### Control Approach Based on the Bounded-Differentiable-Disturbance: Second Control Approach  
As shown above, the control approach drives the tracking error towards a small bound. Therefore, another approach will be followed in order to obtain an asymptotic result for the tracking error. A new Lyapunov function candidate is selected  
\[V_{2}:=\frac{1}{2}\eta_{2}^{T}M(q)\eta_{2}+\frac{1}{2}\eta_{1}^{T}\eta_{1}+ \frac{1}{2}\Delta q^{T}\Delta q+W \tag{20}\]  
where \(W\in\Re^{+}\) is an auxiliary function that will be formulated subsequently. The derivative of (20) gives  
\[\dot{V}_{2}=\frac{1}{2}\eta_{2}^{T}\dot{M}(q)\eta_{1}+\eta_{2}^{T}M(q)\dot{ \eta_{2}}+\eta_{1}^{T}\dot{\eta_{1}}+\Delta q^{T}\Delta q+\dot{W}. \tag{21}\]  
Analogous to the previous approach, (5) is multiplied by \(M(q)\) to obtain  
\[ \begin{split} M(q)\eta_{2}&=M(q)\dot{\eta_{1}}+M(q )\sigma_{2}\eta_{1}\\ &=M(q)(\Delta\dot{q}+\sigma_{1}\Delta q)+\sigma_{2}M(q)\eta_{1}\\ &=M(q)(\ddot{q}_{d}-\ddot{q}+\sigma_{1}\Delta q)+\sigma_{2}M(q) \eta_{1}.\end{split}  \tag{22}\]  
By substituting \(M(q)\ddot{q}\) from (1) into (22), \(M(q)\eta_{2}\) can be expressed as  
\[ \begin{split} M(q)\eta_{2}=& M(q)\ddot{q}_{d}+V_{m} (q,q)\dot{q}+G(q)+\tau_{b}+\tau_{l}-\tau+M(q)(\sigma_{1}\Delta q+\sigma_{2}\eta _{1}).\end{split}  \tag{23}\]To facilitate the analysis process, the terms in (23) that are linear in the parameters can be separated and written as [34]  
\[Y(q,\dot{q},\ddot{q})\phi=M(q)\ddot{q}_{d}+V_{m}(q,\dot{q})\dot{q}+G(q) \tag{24}\]  
where \(Y(q,\dot{q},\ddot{q})\in\Re^{n\times m}\) represents the matrix that is a function of \(q\), \(\dot{q}\) and \(\ddot{q}\) vectorsand \(\phi\in\Re^{m}\) represents the vector of constant parameters. The matrix \(Y\) can be expressed as a function of \(q_{d}\), \(\dot{q}_{d}\), and \(\ddot{q}_{d}\), which are considered to exist and be bounded. Therefore, (24) can be expressed as  
\[Y_{d}(q_{d},\dot{q}_{d},\ddot{q}_{d})\phi= M(q_{d})\ddot{q}_{d}+V_{m}(q_{d},\dot{q}_{d})q_{d}+G(q_{d}). \tag{25}\]  
By adding and subtracting (25) to (23), we can rewrite (23) as follows:  
\[M(q)\eta_{2}= (Y-Y_{d})\phi+M(q)(\sigma_{1}\Delta\dot{q}+\sigma_{2}\eta_{1})+Y_{ d}\phi+\tau_{v}+\tau_{l}-\tau. \tag{26}\]  
Now, we define a new auxiliary function as follows:  
\[Q:=(Y-Y_{d})\phi+M(q)(\sigma_{1}\Delta\dot{q}+\sigma_{2}\eta_{1}) \tag{27}\]  
where \(Q(q,\dot{q},q_{d},\dot{q}_{d},\ddot{q}_{d})\in\Re^{n}\) is a function of terms that can be upper bounded. **Assumption 6**. : _The disturbance torques due to the exogenous vibration and payload variation are continuous, bounded, and differentiable._  
Substituting (27) into (26) and then differentiating gives  
\[M(q)\eta_{2}+M(q)\eta_{2}=\dot{Q}+\dot{Y}_{d}\phi+\dot{\tau}_{v}+\dot{\tau}_{l }-\dot{\tau}. \tag{28}\]  
Based on the analysis thus far and motivated by the idea of proportional derivative control, the derivative of the torque input \(\dot{\tau}\) is designed as follows:  
\[\dot{\tau}=K_{2}\eta_{1}+\eta_{1}+K_{2}\eta_{2}+\eta_{2}+K_{3}sgn(\eta_{1}) \tag{29}\]  
where \(K_{2}\) and \(K_{3}\in\Re\) are positive gains. Since we need the torque input \(\tau\) in the dynamic system, \(\dot{\tau}\) in (29) is designed to be integrable. Substituting (5) into (29) and integrating both sides gives  
\[\tau= 2K_{2}\eta_{1}(t)+2\eta_{1}(t)+\int_{0}^{t}[K_{2}\sigma_{2}\eta_{1}( \epsilon)+\sigma_{2}\eta_{1}(\epsilon)+K_{3}sgn(\eta_{1}(\epsilon))]de. \tag{30}\]  
We know from (4) that \(\eta_{1}\) contains the joint displacement and velocity, which are measurable in this study, and thus, the term \(\int_{0}^{t}[K_{2}\sigma_{2}\eta_{1}(\epsilon)+\sigma_{2}\eta_{1}(\epsilon)+ K_{3}sgn(\eta_{1}(\epsilon))]de\) can be determined numerically.