## 3 Self-Adaptive-Coefficient Double-Power Sliding Mode Control Algorithm
### Design of Self-Adaptive-Coefficient-Double-Power Sliding Mode Control Method
 \tag{16}\]  
where \(\mathbf{Y}(\mathbf{\theta},\hat{\mathbf{\theta}},\hat{\mathbf{\theta}}_{r},\tilde{\mathbf{\theta}} _{r})\mathbf{\tilde{p}}\) is the linear expression of the model error; \(\mathbf{Y}(\mathbf{\theta},\hat{\mathbf{\theta}},\hat{\mathbf{\theta}}_{r},\tilde{\mathbf{\theta}} _{r})\) is a known regression matrix about the function of robot joint variables; \(\mathbf{Y}(\mathbf{\theta},\hat{\mathbf{\theta}},\hat{\mathbf{\theta}}_{r},\tilde{\mathbf{\theta}} _{r})=[\mathbf{Y}^{\prime}_{ij}]\Big{|}\mathbf{Y}^{\prime}_{ij}\Big{|}\leq\mathbf{\bar{Y}}^ {\prime}_{ij}\); and \(\mathbf{\tilde{p}}\) is a constant parameter vector that describes the quality characteristics of the robot, where \(\mathbf{\tilde{p}}=[\mathbf{\bar{\rho}}_{1},\cdots,\mathbf{\bar{\rho}}_{j}]^{\text{T}}, \big{|}\mathbf{\bar{\rho}}_{j}\big{|}\leq\varepsilon_{j}\), \(i=1,2\), and \(j=1,\cdots,l\). Therefore,  
\[ \begin{array}{ll}\mathbf{Y}(\hat{\mathbf{\theta}},\hat{\mathbf{\theta}},\hat{\mathbf{\theta}} _{r},\tilde{\mathbf{\theta}}_{r})\mathbf{\tilde{p}}&=[\mathbf{Y}^{\prime}_{ij}]\cdot[\mathbf{ \bar{\rho}}_{1}\cdots\mathbf{\bar{\rho}}_{j}]^{\text{T}}\\ &=\left[\begin{array}{ccc}\mathbf{Y}^{\prime}_{11}&\cdots&\mathbf{Y}^{\prime}_{1l} \\ \vdots&\mathbf{Y}^{\prime}_{ij}&\vdots\\ \mathbf{Y}^{\prime}_{ni}&\cdots&\mathbf{Y}^{\prime}_{ni}\end{array}\right]\cdot\left[ \begin{array}{ccc}\mathbf{\bar{p}}_{1}\\ \vdots\\ \mathbf{\bar{\rho}}_{j}\end{array}\right]\leq\left[\begin{array}{ccc}\mathbf{\bar{ \tau}}_{11}&\cdots&\mathbf{\bar{\tau}}_{1l}\\ \vdots&\mathbf{\bar{\tau}}^{\prime}_{ij}&\vdots\\ \mathbf{\bar{\tau}}^{\prime}_{ni}&\cdots&\mathbf{\bar{\tau}}^{\prime}_{ni}\end{array} \right]\cdot\left[\begin{array}{c}\mathbf{\epsilon}_{1}\\ \vdots\\ \mathbf{\epsilon}_{l}\end{array}\right]=\overset{-\tau}{Y_{ij}\cdot\varepsilon_{j}} \end{array}  \tag{17}\]  
From Equation (16), \(\mathbf{Y}(\mathbf{\theta},\hat{\mathbf{\theta}},\hat{\mathbf{\theta}}_{r},\hat{\mathbf{\theta}}_{r}) \mathbf{\bar{p}}\), \(\mathbf{\tau}_{c}\), and \(\mathbf{\tau}_{d}\) determine whether the control system of the lower limb rehabilitation exoskeleton robot is closed-loop stable. To ensure the stable convergence of the control system, it is necessary to be satisfied as follows. \[\dot{\mathbf{V}}(t)=\mathbf{s}^{\text{T}}\Big{[}\mathbf{Y}(\mathbf{\theta},\dot{\mathbf{\theta}}, \dot{\mathbf{\theta}}_{r},\ddot{\mathbf{\theta}}_{r})\mathbf{\bar{p}}-(\mathbf{\tau}_{c}-\mathbf{ \tau}_{\text{d}})\Big{]}\leq 0 \tag{18}\]  
To enhance the stable convergence of the control system and ensure that the compensation term follows the model error, adjusts adaptively, and improves the anti-interferenceability of the robot control system, a self-adaptive-coefficient \(\mathbf{\Lambda}\) was designed for the compensation term \(\mathbf{\tau}_{c}\) and \(\mathbf{\Lambda}\) as follows. \[\mathbf{\Lambda}=[\mathbf{\Lambda}_{1},\cdots\mathbf{\Lambda}_{n}]^{\mathrm{T}},\mathbf{\Lambda} _{i}=\left\{ \begin{array}{l}\sum\limits_{j=1}^{l}\overline{Y}_{ij}^{\prime} \mathbf{\varepsilon}_{j},\mathbf{\Lambda}_{i}\geq 1\\ 1,\mathbf{\Lambda}_{i}<1\end{array} \right. \tag{19}\]  
According to Equations (12) and (19), we rewrite Equation (16) as follows.