## 3 Methods
### Design of Adaptive Sliding Mode Controller
The basic idea of the RBF neural network is to use the radial basis function as the basis of the hidden layer neurons to form the hidden layer space. The hidden layer transforms the input data of the low-dimensional mode into the high-dimensional space, which makes the linear indivisible problem in the low-dimensional space linearly separable in the high-dimensional space. The RBF neural network is simple in structure, simple in training, and fast in convergence, and can approach any nonlinear function; therefore, the RBF neural network is used to approximate the uncertain term \(\mathbf{f}\). Taking \(\mathbf{x}=\left[\mathbf{e}^{T},\dot{\mathbf{e}}^{T},\mathbf{q}_{\mathbf{d}} ^{T},\dot{\mathbf{q}}_{\mathbf{d}}^{T},\ddot{\mathbf{q}}_{\mathbf{d}}^{T}\right]\) as the input of the RBF neural network, and taking Gaussian function as a membership function, the RBF neural network algorithm can be denoted as follows:  
\[\mathbf{h}_{\mathbf{i}}=\exp(-\|\mathbf{x}-\mathbf{c}_{\mathbf{i}}\|^{2}/2 \mathbf{b}_{\mathbf{i}}{}^{2}), \tag{21}\]  
\[\mathbf{f}(\mathbf{x})=\mathbf{W}^{T}\mathbf{h}(\mathbf{x})+\varepsilon, \tag{22}\]  
where \(i=1,2,3,\cdots,m\), and \(m\) is the number of neurons in the hidden layer of the RBF neural network, \(\mathbf{c}_{\mathbf{i}}\) and \(\mathbf{b}_{\mathbf{i}}\) are the center value and standardization constant of Gaussian function of the i-th neuron in the hidden layer of the RBF neural network, respectively, \(\mathbf{W}\) is the connection weight matrix from the hidden layer to the output layer, \(\varepsilon\) is the approximation error. Due to the existence of approximation error, the actual output of the neural network can be set as:  
\[\mathbf{\hat{f}(x)}=\mathbf{\hat{W}}^{T}\mathbf{h}(\mathbf{x}), \tag{23}\]  
where \(\mathbf{\hat{W}}\) is the actual weight matrix of the RBF neural network when approaching \(\mathbf{f}\), \(\mathbf{h}(\mathbf{x})\) is the hidden layer basis vector of the RBF neural network. The sliding mode control law is designed as follows:  
\[\mathbf{u}=\mathbf{\hat{W}}^{T}\mathbf{h}(\mathbf{x})+\mathbf{J}_{\mathbf{e} \mathbf{h}}\mathbf{}^{T}\mathbf{F}_{\mathbf{h}}+\mathbf{J}_{\mathbf{e}\mathbf{h }}\mathbf{}^{T}\mathbf{F}_{\mathbf{k}}+\mathbf{K}\mathbf{r}+\mathbf{v}, \tag{24}\]  
where \(\mathbf{K}\) is the control parameter of the sliding mode controller, \(\mathbf{r}\) is the designed sliding surface, \(\mathbf{v}\) is the robust term of the controller to overcome system disturbance. Considering the approximation error \(\varepsilon\) of the neural network and the disturbance term \(\mathbf{\tau}_{\mathbf{d}}\) in the actual exoskeleton control, the robust term \(\mathbf{v}\) is designed as:  
\[\mathbf{v}=-(\varepsilon_{\mathbf{N}}+\mathbf{b}_{\mathbf{d}})\text{sgn}( \mathbf{r}), \tag{25}\]  
where \(\varepsilon_{\mathbf{N}}\geq\|\mathbf{f}-\mathbf{\hat{f}}\|\) is the maximum permissible error of RBF neural network approximation, \(\mathbf{b}_{\mathbf{d}}\geq\|\mathbf{J}_{\mathbf{e}\mathbf{h}}\mathbf{}^{T} \mathbf{F}_{\mathbf{h}}+\mathbf{J}_{\mathbf{e}\mathbf{h}}\mathbf{}^{T}\mathbf{F} _{\mathbf{k}}+\mathbf{\tau}_{\mathbf{d}}\|\) is the maximum disturbance of the system.