## 3 Proposed Control Methodology
### Adaptive Backstepping Sliding Mode Algorithm
\tag{11}\]  
with \(\mathbf{\lambda}_{1}\triangleq diag\Big{(} \begin{array}{cccccc}\lambda_{11} &\lambda_{12}&\ldots&\lambda_{1i}&\ldots&\lambda_{1n}\end{array} \Big{)}\) as the positive definite matrix. A new equivalent variable is defined as  
\[\left\{ \begin{array}{l}\mathbf{y}_{2s}=\mathbf{y}_{2}-\mathbf{s}_{1}=\mathbf{ y}_{2d}-\mathbf{\lambda}_{1}\mathbf{e}_{1}\in R^{n}\\ \dot{\mathbf{y}}_{2s}=\dot{\mathbf{y}}_{2}-\dot{\mathbf{s}}_{1}=\dot{\mathbf{ y}}_{2d}-\mathbf{\lambda}_{1}\mathbf{e}_{2}\in R^{n}\end{array} \right. \tag{12}\]  
New estimated error between the actual and estimated lumped uncertainties is defined as \(\widetilde{\Delta}=\Delta-\hat{\Delta}\in R^{n}\) in which the estimated lumped uncertainties satisfies:  
\[\dot{\hat{\Delta}}=-\mathbf{s}_{1} \tag{13}\]  
**Assumption 2**. : _The lumped uncertainties are supposed to vary slowly relative to the system dynamics, i.e., \(\dot{\Delta}_{i}\approx 0\)._  
A Lyapunov candidate \(V_{1}\) is chosen as  
\[V_{1}=\frac{1}{2}\mathbf{s}_{1}{}^{T}\mathbf{M}\mathbf{s}_{1}+\frac{1}{2} \tilde{\Delta}^{T}\tilde{\Delta} \tag{14}\]  
Taking derivative the Lyapunov candidate \(V_{1}\) with Property 1 and substituting Equations (11) and (12) yields:  
\[\dot{V}_{1}=\mathbf{s}_{1}^{T}\Big{(}\mathbf{y}_{3}-\Big{(}\tilde{\Delta}+ \tilde{\Delta}\Big{)}-\mathbf{M}(\mathbf{y}_{1})\dot{\mathbf{y}}_{2s}-\mathbf{ C}(\mathbf{y}_{1},\mathbf{y}_{2})\mathbf{y}_{2s}-\mathbf{G}(\mathbf{y}_{1})- \mathbf{J}(\mathbf{y}_{1}){}^{T}\mathbf{F}_{ext}\Big{)}+\dot{\tilde{\Delta}}^{ T}\tilde{\Delta} \tag{15}\]  
An error between an actual virtual torque \(\mathbf{y}_{3}\) and desired torque \(\mathbf{y}_{3d}\) as \(\mathbf{e}_{3}=\mathbf{y}_{3}-\mathbf{y}_{3d}\) is defined. Then the desired torque is designed as  
\[\mathbf{y}_{3d}=\mathbf{M}(\mathbf{y}_{1})\dot{\mathbf{y}}_{2s}+\mathbf{C}( \mathbf{y}_{1},\mathbf{y}_{2})\mathbf{y}_{2s}+\mathbf{G}(\mathbf{y}_{1})+ \mathbf{J}(\mathbf{y}_{1}){}^{T}\mathbf{F}_{ext}+\hat{\Delta}-\mathbf{K}_{1} \mathbf{s}_{1}-\mathbf{\eta}_{1}\text{tanh}\Bigg{(}\frac{\mathbf{s}_{1}}{ \sigma_{1}}\Bigg{)} \tag{16}\]  
where \(\mathbf{K}_{1}=diag\Big{(} \begin{array}{cccccc}K_{11}&K_{12}&\ldots&K_{1i}& \ldots&K_{1n}\end{array} \Big{)}\) and \(\mathbf{\eta}_{1}=diag\Big{(} \begin{array}{cccccc}\eta_{11}&\eta_{12}&\ldots &\eta_{1i}&\ldots&\eta_{1n}\end{array} \Big{)}\) are positive definite matrices. \(\mathbf{\sigma}_{1}\) is the width vector. The term \(-\mathbf{K}_{1}\mathbf{s}_{1}\) is added to enhance the convergence rate when \(\mathbf{s}_{1}\) is large. Define \(\mathbf{e}_{3}=\mathbf{y}_{3}-\mathbf{y}_{3d}\in R^{n}\), then substituting this and Equation (16) into Equation (15) yields:  
\[\dot{V}_{1}=\mathbf{s}_{1}^{T}\Big{(}\mathbf{J}_{a}^{T}\mathbf{e}_{3}-\mathbf{ K}_{1}\mathbf{s}_{1}-\mathbf{\eta}_{1}\text{tanh}\Bigg{(}\frac{\mathbf{s}_{1}}{ \sigma_{1}}\Bigg{)}-\tilde{\Delta}\Big{)}+\dot{\tilde{\Delta}}^{T}\tilde{\Delta} \tag{17}\]  
Substituting the adaptive law in Equation (13) yields:  
\[\dot{V}_{1}=\mathbf{s}_{1}^{T}\Big{(}\mathbf{J}_{a}^{T}\mathbf{e}_{3}-\mathbf{ K}_{1}\mathbf{s}_{1}-\mathbf{\eta}_{1}\text{tanh}\Bigg{(}\frac{\mathbf{s}_{1}}{ \sigma_{1}}\Bigg{)}\Big{)} \tag{18}\]  
**Remark 2**. : _The aim of using the adaptive law in Equation (13) is to cancel out influences of the lumped uncertainties to force the model as same as an ideal model. In practical simulation, to enhance the convergence rate of estimation, this equation is modified as_  
\[\dot{\hat{\Delta}}=-\mathbf{K}_{\Delta}\mathbf{s}_{1}-\mathbf{\eta}_{\Delta} \text{sign}(\mathbf{s}_{1}) \tag{19}\]  
_where \(\mathbf{K}_{\Delta},\mathbf{\eta}_{\Delta}\) are diagonal positive definite matrices to increase the convergence rate of the estimated lumped uncertainties._  
**Remark 3**. : _The lumped uncertainty \(\mathbf{\Delta}\) is unknown but bounded by \(\|\Delta\|\leq\overline{\Delta}\).