## 3 Self-Adaptive-Coefficient Double-Power Sliding Mode Control Algorithm
### Design of Self-Adaptive-Coefficient-Double-Power Sliding Mode Control Method  
According to Equations (10) and (12), combining the estimated dynamic model of the lower limb rehabilitation exoskeleton robot and the compensation term based on the double-power reaching law, the control law of the double-power sliding mode control method can be expressed as follows. \[ \begin{array}{ll}\mathbf{\tau}&=\mathbf{u}=\mathbf{u}=\mathbf{\tau}_{c}=\mathbf{\bar{M}}(\mathbf{ \theta})\hat{\mathbf{\theta}}_{r}+\overline{C}(\mathbf{\theta},\hat{\mathbf{\theta}})\hat{ \mathbf{\theta}}_{r}+\overline{G}(\mathbf{\theta})+\mathbf{\tau}_{c}\\ &=\left[\begin{array}{cc}\overline{M}_{11}&\overline{M}_{12}\\ \overline{M}_{21}&\overline{M}_{22}\end{array}\right]\hat{\mathbf{\theta}}_{r}+ \left[\begin{array}{cc}\overline{C}_{11}&\overline{C}_{12}\\ \overline{C}_{21}&\overline{C}_{22}\end{array}\right]\hat{\mathbf{\theta}}_{r}+ \left[\begin{array}{c}\overline{G}_{11}\\ \overline{G}_{21}\end{array}\right]+\mathbf{k}_{\mathbf{\kappa}}|\mathbf{s}|^{\mathbf{\kappa}} \text{sgn}(\mathbf{s})+\overline{K}_{\mathbf{\beta}}|\mathbf{s}|^{\mathbf{\beta}}\text{sgn}( \mathbf{s})\end{array}  \tag{13}\]  
We defined the Lyapunov function as follows. \[\mathbf{V}(t)=0.5\mathbf{s}^{\text{T}}\mathbf{M}(\mathbf{\theta})\mathbf{s} \tag{14}\]  
According to Equations (3), (7) and (13), the derivative of Equation (14) with respect to time \(t\) is expressed as follows:  
\[ \begin{array}{ll}\dot{\mathbf{V}}(t)&=\mathbf{s}^{\text{T}}\mathbf{M}(\mathbf{\theta})\dot{ \mathbf{s}}+\frac{1}{2}\mathbf{s}^{\text{T}}\Big{(}2C(\mathbf{\theta},\hat{\mathbf{\theta}}) \Big{)}\mathbf{s}=\mathbf{s}^{\text{T}}\Big{[}\mathbf{M}(\mathbf{\theta})(\tilde{\mathbf{\theta}} _{r}-\tilde{\mathbf{\theta}})+C(\mathbf{\theta},\hat{\mathbf{\theta}})(\dot{\mathbf{\theta}}_{ r}-\dot{\mathbf{\theta}})\Big{]}\\ &=\mathbf{s}^{\text{T}}\Big{[}\mathbf{M}(\mathbf{\theta})\tilde{\mathbf{\theta}}_{r}+C(\mathbf{ \theta},\hat{\mathbf{\theta}})\dot{\mathbf{\theta}}_{r}+G(\mathbf{\theta})+\mathbf{\tau}_{d}- \mathbf{u}\Big{]}\\ &=\mathbf{s}^{\text{T}}\bigg{[}\Big{(}\mathbf{M}(\mathbf{\theta})-\mathbf{M}(\mathbf{\theta}) \Big{)}\tilde{\mathbf{\theta}}_{r}+\Big{(}C(\mathbf{\theta},\hat{\mathbf{\theta}})-C(\bm {\theta},\hat{\mathbf{\theta}})\Big{)}\dot{\mathbf{\theta}}_{r}+\Big{(}G(\mathbf{\theta})- G(\mathbf{\theta})\Big{)}+\mathbf{\tau}_{d}-\mathbf{\tau}_{c}\bigg{]}\\ &=\mathbf{s}^{\text{T}}\bigg{[}\tilde{\mathbf{M}}(\mathbf{\theta})\tilde{\mathbf{\theta}}_{r}+ \tilde{C}(\mathbf{\theta},\hat{\mathbf{\theta}})\dot{\mathbf{\theta}}_{r}+\tilde{G}(\mathbf{ \theta})+\mathbf{\tau}_{d}-\mathbf{\tau}_{c}\bigg{]}\end{array}  \tag{15}\]  
where \(\tilde{\mathbf{M}}(\mathbf{\theta})\) is the inertial matrix error and \(\tilde{\mathbf{M}}(\mathbf{\theta})=\mathbf{M}(\mathbf{\theta})-\tilde{\mathbf{M}}(\mathbf{\theta})\). \(\tilde{C}(\mathbf{\theta},\hat{\mathbf{\theta}})\) is the centrifugal and Coriolis forces error, and \(\tilde{\mathbf{C}}(\mathbf{\theta},\hat{\mathbf{\theta}})=C(\mathbf{\theta},\hat{\mathbf{\theta}} )-\tilde{C}(\mathbf{\theta},\hat{\mathbf{\theta}})\). Here, \(\tilde{G}(\mathbf{\theta})\) is the gravity term error, where \(\tilde{\mathbf{G}}(\mathbf{\theta})=\mathbf{G}(\mathbf{\theta})-\tilde{G}(\mathbf{\theta})\). Equations (4) and (15) can be expressed as follows. \[ \begin{array}{ll}\dot{\mathbf{V}}(t)&=\mathbf{s}^{\text{T}}\bigg{[}\tilde{\mathbf{M}}( \mathbf{\theta})\tilde{\mathbf{\theta}}_{r}+\tilde{C}(\mathbf{\theta},\hat{\mathbf{\theta}}) \dot{\mathbf{\theta}}_{r}+\tilde{G}(\mathbf{\theta})+\mathbf{\tau}_{d}-\mathbf{\tau}_{c}\bigg{]} \\ &=\mathbf{s}^{\text{T}}\Big{[}\mathbf{Y}(\mathbf{\theta},\hat{\mathbf{\theta}},\hat{\mathbf{ \theta}}_{r},\tilde{\mathbf{\theta}}_{r})\mathbf{\tilde{p}}+\mathbf{\tau}_{d}-\mathbf{\tau}_{c} \Big{]}\end{array}