## 3 Iterative Learning Construction and Analysis Based on RBF Neural Network
### Stability Analysis  
According to the characteristics of the lower limb exoskeleton rehabilitation robot dynamics model itself. An iterative learning controller based on the RBF neural network is proposed to realize the stability of the lower limb exoskeleton rehabilitation robot system. For the exoskeleton robot described above, the Lyapunov function is defined as:  
\[V_{k}=\int_{0}^{t}e^{\rho\tau}\varphi_{k}^{T}K_{D_{0}}\varphi_{k}d\tau(V\geq 0) \tag{12}\]  
where \(\rho\) is a positive number. According to \(\varphi_{k}(t)=\dot{e}_{k}(t)+\lambda e_{k}(t)\), and:  
\[\delta\varphi_{k}=\delta\dot{e}_{k}(t)+\delta\lambda e_{k}(t)=\varphi_{k+1}- \varphi_{k} \tag{13}\]  
Which is simplified by Formulas (8) and (13):  
\[ \begin{array}{l}\delta M\dot{\varphi}_{k}+K_{D_{0}}\varphi_{k}\\ =-\big{[}\big{(}C+C_{I}-\lambda M+K_{D_{k+1}}\big{)}\delta\varphi_{k}+\big{[}F- \lambda(C+C_{I}-\lambda M)\big{]}\delta e_{k}\big{]}\end{array}  \tag{14}\]  
Figure 3: RBF neural network model. According to the definition of the Lyapunov function, the \(k+1\)-th iterative expression is derived:  
\[V_{k+1}=\int_{0}^{t}e^{\rho\tau}\varphi_{k+1}^{T}K_{D_{0}}\varphi_{k+1}d\tau \quad\quad(V\geq 0) \tag{15}\]  
From \(\Delta V=V_{k+1}-V_{k}\), using Formula (14), we can obtain:  
\[ \begin{array}{rl}\Delta V&=\int_{0}^{t}e^{\rho\tau}\varphi_{k+1}^{T}K_{D_{0} }\varphi_{k+1}d\tau-\int_{0}^{t}e^{\rho\tau}\varphi_{k}^{T}K_{D_{0}}\varphi_{k} d\tau\\ &=\frac{1}{\alpha(k+1)}\Big{\{}\int_{0}^{t}e^{\rho\tau}\delta\varphi_{k+1}^{T}K _{D_{k+1}}\delta\varphi_{k+1}d\tau-2\int_{0}^{t}e^{\rho\tau}\delta\varphi_{k}^{ T}\delta M\dot{\varphi}_{k}d\tau\Big{\}}\\ &-2\int_{0}^{t}e^{\rho\tau}\delta\varphi_{k}^{T}\big{\{}\big{(}C+C_{I}-\lambda M +K_{D_{k+1}}\big{)}\delta\varphi_{k}+\big{[}F-\lambda(C+C_{I}-\lambda M) \big{]}\delta\varepsilon_{k}\big{\}}d\tau\end{array}  \tag{16}\]  
By calculating the integral of Formula (16) and combining it with Formula (14), the inequality is obtained by organizing as follows:  
\[ \begin{array}{rl}\Delta V&\leq-\frac{e^{\rho\tau}\delta\varphi_{k}^{T}\delta M \dot{\varphi}_{k}}{\alpha(k+1)}-\frac{1}{\alpha(k+1)}\Big{\{}\rho\int_{0}^{t}e ^{\rho\tau}\delta\varphi_{k}^{T}M\dot{\varphi}_{k}d\tau+\int_{0}^{t}e^{\rho\tau }\delta\varepsilon_{k}^{T}\big{(}K_{D_{0}}+2C_{I}-2\lambda M\big{)}\dot{ \delta\varepsilon}_{k}d\tau\\ &+2\lambda\int_{0}^{t}e^{\rho\tau}\delta\varphi_{k}^{T}\big{(}K_{D_{0}}+2C_{I }-2\lambda M\big{)}\delta\varepsilon_{k}d\tau+2\int_{0}^{t}e^{\rho\tau}\delta \varepsilon_{k}^{T}\big{[}F-\lambda(C+C_{I}-\lambda M)\big{]}\delta\varepsilon _{k}d\tau\\ &+\lambda^{2}\int_{0}^{t}e^{\rho\tau}\delta\varepsilon_{k}^{T}\big{(}K_{D_{0} }+2C_{I}-2\lambda M\big{)}\delta\varepsilon_{k}d\tau+2\lambda\int_{0}^{t}e^{ \rho\tau}\delta\varepsilon_{k}^{T}\big{[}F-\lambda(C+C_{I}-\lambda M)\big{]} \delta\varepsilon_{k}d\tau\Big{\}}\end{array}  \tag{17}\]  
Once again, Equation \(\int_{0}^{t}e^{\rho\tau}\delta\varepsilon_{k}^{T}\big{(}K_{D_{0}}+2C_{I}-2 \lambda M\big{)}\dot{\delta\varepsilon}_{k}d\tau\) is integrated, which is obtained by organizing Equation (9):  
\[ \begin{array}{rl}\Delta V&\leq-\frac{e^{\rho\tau}\delta\varphi_{k}^{T}M \delta\varphi_{k}}{\alpha(k+1)}-\frac{\lambda e^{\rho\tau}\delta\varepsilon_{ k}^{T}L_{i}\delta\varepsilon_{k}}{\alpha(k+1)}\\ &-\frac{1}{\alpha(k+1)}\Big{[}\rho\int_{0}^{t}e^{\rho\tau}\delta\varphi_{k}^{ T}M\delta\varphi_{k}d\tau+\lambda\rho\int_{0}^{t}e^{\rho\tau}\delta \varepsilon_{k}^{T}L_{i}\delta\varepsilon_{k}d\tau+\int_{0}^{t}e^{\rho\tau}Rd \tau\Big{]}\end{array}  \tag{18}\]  
where,  
\[ \begin{array}{rl}R&=\delta\varepsilon_{k}^{T}\big{(}K_{D_{0}}+2C_{I}-2 \lambda M\big{)}\dot{\delta\varepsilon}_{k}+2\lambda\delta\varepsilon_{k}^{T} \Big{[}\frac{F}{\lambda}-(C+C_{I}-\lambda M)\Big{]}\delta\varepsilon_{k}\\ &+\lambda^{2}\delta\varepsilon_{k}^{T}\bigg{(}K_{D_{0}}+2C+2\frac{F}{\lambda}-2 \frac{\dot{C}_{I}}{\lambda}\bigg{)}\delta\varepsilon_{k}\end{array}