## 3 Proposed Control Methodology
### Adaptive Backstepping Sliding Mode Algorithm
Hence, the derivative of the Lyapunov function \(V_{2}\) becomes:  
\[\dot{V}_{2}=-\mathbf{s}_{1}^{T}\mathbf{K}_{1}\mathbf{s}_{1}-\mathbf{s}_{1}^{T }\mathbf{\eta}_{1}\text{tanh}\bigg{(}\frac{\mathbf{s}_{1}}{\mathbf{\sigma}_{1 }}\bigg{)}-\mathbf{s}_{2}^{T}\mathbf{K}_{2}\mathbf{s}_{2}-\mathbf{s}_{2}^{T} \mathbf{\eta}_{2}\text{tanh}\bigg{(}\frac{\mathbf{s}_{2}}{\mathbf{\sigma}_{2}} \bigg{)}+\mathbf{e}_{3}^{T}\mathbf{\
abla}\mathbf{e}_{\text{l}ak}+\mathbf{e}_{ C_{\text{l}ak}}^{T}\dot{\mathbf{e}}_{C_{\text{l}ak}} \tag{25}\]To achieve system performance as an ideal model, the term \(\mathbf{e}_{3}^{\mathsf{T}}\mathbf{Fe}_{\text{leak}}+\mathbf{e}_{\text{C}_{\text{ leak}}}^{\mathsf{T}}\dot{\mathbf{c}}_{\text{L}_{\text{ leak}}}\) should be cancelled out. Therefore, the following adaptive law is satisfied:  
\[\dot{\mathbf{C}}_{\text{leak}}=\mathbf{\
abla}\mathbf{e}_{3} \tag{26}\]  
**Remark 4**. : _The aim of using the adaptive law in Equation (26) is to cancel out influences of the leakage to achieve the ideal model with no uncertainties or disturbances. In practical simulation, to enhance the convergence rate of estimation, this equation is modified as_  
\[\dot{\mathbf{C}}_{\text{leak}}=\mathbf{K}_{\text{C}_{\text{ leak}}}\mathbf{\
abla}\mathbf{e}_{3}+\mathbf{\eta}_{\text{C}_{\text{ leak}}}\mathbf{\
abla}sign(\mathbf{e}_{3}) \tag{27}\]  
_where \(\mathbf{K}_{\text{C}_{\text{ leak}}}\) and \(\mathbf{\eta}_{\text{C}_{\text{ leak}}}\) are a diagonal positive definite matrix to adjust the convergence rate of the estimation. A function \(sign(\bullet_{i})\) in both Equations (19) and (27) is defined as \(sign(\bullet_{k})=\left( \begin{array}{cccc}sign(\bullet_{k1})&sign(\bullet_ {k2})&\ldots&sign(\bullet_{kn})\end{array} \right)^{\mathsf{T}}\)._  
**Remark 5**. : _Despite using the adaptive law in Equation (26), an error of estimated leakage coefficient is still existing, but its estimated error is bounded and predetermined as \(\|\mathbf{e}_{\text{leak}}\|=\|\mathbf{C}_{\text{leak}}-\mathbf{\hat{C}}_{ \text{leak}}\|\leq\epsilon_{\text{leak}}\). Then \(\mathbf{\eta}_{2}\) is chosen to be greater than the error of estimated leakage coefficient, i.e., \(\eta_{2i}\geq\epsilon_{\text{leak},i}\). This means the controller gains are much smaller instead of designing with large values enough to cover the upper bound of uncertainties in conventional design._  
Consequently, the derivative of Lyapunov function \(V_{2}\) becomes:  
\[\dot{V}_{2}=-\mathbf{s}_{1}^{\mathsf{T}}\mathbf{K}_{1}\mathbf{s}_{1}-\mathbf{ s}_{1}^{\mathsf{T}}\mathbf{\eta}_{1}\text{tanh}\left(\frac{\mathbf{s}_{1}}{ \sigma_{1}}\right)-\mathbf{s}_{2}^{\mathsf{T}}\mathbf{K}_{2}\mathbf{s}_{2}- \mathbf{s}_{2}^{\mathsf{T}}\mathbf{\eta}_{2}\text{tanh}\left(\frac{\mathbf{s} _{2}}{\sigma_{2}}\right)+\epsilon_{\text{humped}}\leq 0 \tag{28}\]  
where \(\epsilon_{\text{humped}}\) denotes lumped error of estimated procedures. **Remark 6**. : _In a practical system, the position of the end-effector can be calculated from angular position of each joint. These values can be directly measured by using encoders. However, angular and acceleration are not available. Conventionally, these variables are obtained from taking derivative of joint angles and, thereby, measured noises are also amplified. To effectively estimate unmeasurable variables for designing the controller, a second-order exact differentiation (SOED) is utilized as_  
\[ \begin{array}{l}\dot{\mathbf{z}}_{0}=\mathbf{f}_{0}\\ \mathbf{f}_{0}=-\mathbf{\kappa}_{0}\|\mathbf{z}_{0}-\mathbf{q}\|^{2/3}sign( \mathbf{z}_{0}-\mathbf{q})+\mathbf{z}_{1}\\ \dot{\mathbf{z}}_{1}=\mathbf{f}_{1}\\ \mathbf{f}_{1}=-\mathbf{\kappa}_{1}|\mathbf{z}_{1}-\mathbf{f}_{0}|^{1/2}sign( \mathbf{z}_{1}-\mathbf{f}_{0})+\mathbf{z}_{2}\\ \dot{\mathbf{z}}_{2}=-\mathbf{\kappa}_{2}sign(\mathbf{z}_{2}-\mathbf{f}_{1}) \end{array}  \tag{29}\]  
_where \(\mathbf{z}_{0},\mathbf{z}_{1},\mathbf{z}_{2}\) are estimated values of \(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}\).