## 3 Self-Adaptive-Coefficient Double-Power Sliding Mode Control Algorithm
### Design of Self-Adaptive-Coefficient-Double-Power Sliding Mode Control Method
\[\dot{\mathbf{V}}(t) =s^{\mathrm{T}}\Big{[}Y(\theta,\dot{\theta},\dot{\theta},\dot{ \theta},\ddot{\mathbf{\sigma}})\ddot{\mathbf{p}}-(\mathbf{\Lambda}\mathbf{\tau}_{c}-\mathbf{\tau}_{ d})\Big{]} \tag{20}\] \[=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}s_{i}Y_{ij}^{\prime} \ddot{p}_{j}-\sum\limits_{i=1}^{n}s_{i}\big{[}\mathbf{\Lambda}_{k}|s_{i}|^{8} \mathrm{sgn}(s_{i})-\ \mathbf{\Lambda}_{i}\mathbf{k}_{\beta i}|s_{i}|^{8}\mathrm{sgn}(s_{i})-(\mathbf{\Lambda}_ {f}\mathrm{sgn}(s_{i})-\mathbf{\tau}_{d})\big{]}\] \[=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l}Y_{ij}^{\prime} \ddot{p}_{j}s_{i}-\mathbf{k}_{\alpha i}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l} \overline{Y}_{ij}^{\prime}\mathbf{\varepsilon}|s_{i}|^{\alpha+1}-k_{\beta i}\sum \limits_{i=1}^{n}\sum\limits_{j=1}^{l}\overline{Y}_{ij}^{\prime}\mathbf{ \varepsilon}|s_{i}|^{\beta+1}-\sum\limits_{i=1}^{n}(\mathbf{\Lambda}_{f}|s_{i}|- \mathbf{\tau}_{d}\mathrm{is}_{i})\] \[\leq-k_{\beta i}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{l} \overline{Y}_{ij}^{\prime}\mathbf{\varepsilon}|s_{i}|^{\beta+1}=-k_{\beta i}\sum \limits_{i=1}^{n}\mathbf{\Lambda}_{i}|s_{i}|^{\beta+1}\leq-\sum\limits_{i=1}^{n} \mathbf{\Lambda}_{i}|s_{i}|^{\beta+1}\leq 0\]  
From Equation (20), we can see that \(\dot{\mathbf{V}}(t)\leq 0\). According to [PERSON]'s theorem [36], when \(\dot{\mathbf{V}}\equiv 0\), \(\mathbf{s}\equiv 0\). Thus \(\dot{\mathbf{\varepsilon}}\equiv 0\), \(\mathbf{e}\equiv 0\). Therefore, the introduction of the self-adaptive-coefficient \(\mathbf{\Lambda}\) can make \(\dot{\mathbf{V}}(t)\leq 0\) and ensure that the control system of the lower limb rehabilitation exoskeleton robot is closed-loop stable. In addition, the self-adaptive-coefficient \(\mathbf{\Lambda}\) is determined by the model error \(\mathbf{Y}(\mathbf{\theta},\dot{\mathbf{\theta}},\dot{\mathbf{\theta}}_{r},\ddot{\mathbf{\theta}} _{r})\ddot{\mathbf{p}}\). Therefore, \(\mathbf{\Lambda}\mathbf{\tau}_{c}\) can follow the model error and self-adaptive adjustment, thereby enhancing the anti-interference ability of the control system. In summary, the control law of the self-adaptive-coefficient double-power sliding mode control method proposed in this paper is shown in Equation (21) where \(\mathbf{u}\) is the actual control torque for each joint of a lower limb rehabilitation exoskeleton robot. \[\mathbf{\tau} =\mathbf{u}=\mathbf{u}+\mathbf{\Lambda}\mathbf{\tau}_{c}=\mathbf{M}(\mathbf{\theta}) \ddot{\mathbf{\theta}}_{r}+\mathbf{C}(\mathbf{\theta},\dot{\mathbf{\theta}})\dot{\mathbf{\theta}}_ {r}+\mathbf{G}(\mathbf{\theta})+\mathbf{\Lambda}\mathbf{\tau}_{c} \tag{21}\] \[=\left[ \begin{array}{cc}\overline{M}_{11}&\overline{M}_{12}\\ \overline{M}_{21}&\overline{M}_{22}\end{array} \right]\ddot{\mathbf{\theta}}_{r}+ \left[ \begin{array}{cc}\overline{C}_{11}&\overline{C}_{12}\\ \overline{C}_{21}&\overline{C}_{22}\end{array} \right]\dot{\mathbf{\theta}}_{r}+ \left[ \begin{array}{c}\overline{G}_{11}\\ \overline{G}_{21}\end{array} \right]+\] \[\left[ \begin{array}{c}\Lambda_{1}\Big{(}k_{\beta l}|\mathbf{s}_{1 }|^{\beta}\mathrm{sgn}(s_{1})+\overline{k}_{\beta l}|\mathbf{s}_{1}|^{\beta} \mathrm{sgn}(s_{1})\Big{)}\\ \Lambda_{2}\Big{(}k_{\beta l}|\mathbf{s}_{2}|^{\beta}\mathrm{sgn}(s_{2})+ \overline{k}_{\beta l}|\mathbf{s}_{2}|^{\beta}\mathrm{sgn}(s_{2})\Big{)}\end{array}  \right]\]