## 3 Problem Formulation and Control Approaches
### Control Approach Based on the Bounded-Disturbance: First Control Approach  
In order to design a controller that fulfills our objective, we use the Lyapunov function candidate in Equation (6)  
\[V_{1}:=\frac{1}{2}\eta_{1}^{T}M(q)\eta_{1} \tag{6}\]  
where \(V_{1}\) is a scalar (i.e., \(V_{1}\in\Re\)). The time derivative of (6) gives  
\[\dot{V}_{1}=\frac{1}{2}{j{{\dot{\eta_{1}}}^{T}M(q)\eta_{1}}}+\frac{1}{2}\eta_{ 1}^{T}\dot{M}(q)\eta_{1}+\frac{1}{2}\eta_{1}^{T}M(q)\dot{\eta_{1}}=\frac{1}{2} \eta_{1}^{T}\dot{M}(q)\eta_{1}+\eta_{1}^{T}M(q)\dot{\eta_{1}}. \tag{7}\]Motivated by the term \(M(q)\dot{\eta}_{1}\) in (7), the following equation can be obtained by taking the derivative of (4) and then multiplying it by \(M(q)\):  
\[M(q)\dot{\eta}_{1}=M(q)\Delta\ddot{q}+\sigma_{1}M(q)\Delta\dot{q}. \tag{8}\]  
Equation (8) can be extended by substituting \(\Delta\ddot{q}=\ddot{q}_{d}-\ddot{q}\) to obtain  
\[M(q)\dot{\eta}_{1}=M(q)\ddot{q}_{d}-M(q)\ddot{q}+\sigma_{1}M(q)\Delta\dot{q}. \tag{9}\]  
Using \(M(q)\ddot{q}=\tau-\tau_{v}-\tau_{l}-V_{m}(q,\dot{q})\dot{q}-G(q)\) from (1), \(\dot{q}=\dot{q}_{d}-\Delta\dot{q}\), and \(\Delta\dot{q}=\eta_{1}-\sigma_{1}\Delta q\), we can write (9) as  
\[M(q)\dot{\eta}_{1}= M(q)\ddot{q}_{d}-\tau+\tau_{v}+\tau_{l}+V_{m}(q,\dot{q})\dot{q} +G(q)+\sigma_{1}M(q)\Delta q.\]  
Then  
\[M(q)\dot{\eta}_{1}= M(q)\ddot{q}_{d}+V_{m}(q,\dot{q})[\dot{q}_{d}-\eta_{1}+\sigma_{1 }\Delta q]+G(q)+\sigma_{1}M(q)\Delta q-\tau+\tau_{v}+\tau_{l}. \tag{10}\]  
**Assumption 4**. : _The external vibration and payload variation torques, and the gravity term in (1) are upper bounded as_  
\[\|\tau_{v}+\tau_{l}\|\leq v_{b}+l_{b},\quad\|G(q)\|\leq g_{b} \tag{11}\]  
_where \(v_{b}\), \(l_{b}\), and \(g_{b}\) indicate known, positive, and real constants._  
Depending on the above assumption, the context of Lyapunov stability [31], and the fact that it is possible to include the measurable and/or known terms in the input torque expression, we design \(\tau\) as  
\[\tau= K_{1}\eta_{1}+\frac{(v_{b}+l_{b})^{2}}{d}\eta_{1}+M(q)\ddot{q}_{d}+V_{m} (q,\dot{q})\dot{q}_{d}+V_{m}(q,\dot{q})\sigma_{1}\Delta q+G(q)+\sigma_{1}M(q) \Delta\dot{q} \tag{12}\]  
where \(K_{1}=K_{1}^{T}>0\) denotes the control gain diagonal matrix; \(d\) denotes a design parameter; \(v_{b}\) and \(l_{b}\) are introduced in Assumption 4. The term \(\frac{(v_{b}+l_{b})^{2}}{d}\eta_{1}\in\Re^{n}\) in (12) is considered as an auxiliary control signal. The validity of the designed \(\tau\) in (12) can be investigated by substituting (12) into (10) and then (10) into (7), which gives  
\[\dot{V}_{1}= \frac{1}{2}\eta_{1}^{T}\dot{M}(q)\eta_{1}-\eta_{1}^{T}V_{m}(q, \dot{q})\eta_{1}+\eta_{1}^{T}[\tau_{v}+\tau_{l}-\frac{(v_{b}+l_{b})^{2}}{d} \eta_{1}-K_{1}\eta_{1}]. \tag{13}\]  
Applying the skew-symmetric relationship in Property 2 and the inequalities in Assumption 4, (13) can be written as  
\[\dot{V}_{1}\leq(v_{b}+l_{b})\,\|\eta_{1}\|-\frac{(v_{b}+l_{b})^{2}}{d}\|\eta_{ 1}\|^{2}-\eta_{1}^{T}K_{1}\eta_{1}. \tag{14}\]  
Then, (14) yields  
\[\dot{V}_{1}=\frac{1}{2}{j{{\dot{\eta_{1}}}^{T}M(q)\eta_{1}}}+\frac{1}{2}\eta_{ 1}^{T}\dot{M}(q)\eta_{1}+\frac{1}{2}\eta_{1}^{T}M(q)\dot{\eta_{1}}=\frac{1}{2} \eta_{1}^{T}\dot{M}(q)\eta_{1}+\eta_{1}^{T}M(q)\dot{\eta_{1}}. \tag{7}\]0  
**Assumption 5**. : _Let \(\mu:=d/(v_{b}+l_{b})\); the norm value of the filtered signal \(\eta_{1}\) is considered to be_  
\[\dot{V}_{1}=\frac{1}{2}{j{{\dot{\eta_{1}}}^{T}M(q)\eta_{1}}}+\frac{1}{2}\eta_{ 1}^{T}\dot{M}(q)\eta_{1}+\frac{1}{2}\eta_{1}^{T}M(q)\dot{\eta_{1}}=\frac{1}{2} \eta_{1}^{T}\dot{M}(q)\eta_{1}+\eta_{1}^{T}M(q)\dot{\eta_{1}}. \tag{7}\]1  
In practice, the inequality in (16) is ascertained by initializing \(\eta_{1}\) in (4) suitably. This can be obtained through the initial value of the error. Simply put, since the desired joint displacement is known and the actual joint displacement \(q\) is measurable, the error \(\Delta q\) can be easily initiated to be greater than zero (i.e., \(\Delta q(0)>0\)).