## 3 Iterative Learning Construction and Analysis Based on RBF Neural Network  
### Controller Design  
For the passive gait rehabilitation training phase of lower limb exoskeleton robots, we developed an RBF neural network-based iterative learning control method incorporating a single hidden layer feedback neural network. The control architecture adopts a conventional PD feedback structure, with the RBF neural network serving to minimize interference model errors and accelerate convergence. Figure 2 presents the corresponding control block diagram. Figure 2: Iterative learning control block diagram for RBF neural networks. According to the characteristics of the dynamics model of the lower limb exoskeleton rehabilitation robot system, the iterative learning control law based on the RBF neural network is designed as follows:  
\[T_{k}(t)=K_{P_{k}}e_{k}(t)+K_{D_{k}}\dot{e}_{k}(t)+T_{k-1}(t)+\eta\text{sgn}( \delta\varphi_{k-1}) \tag{2}\]  
where the switching rules for \(K_{P}\) and \(K_{D}\) can be described as \(K_{P_{k}}=\alpha(k)K_{P_{k}}\), \(K_{D_{k}}=\alpha(k)K_{D_{k}}\), \(\alpha(k+1)>\alpha(k)\), \(k=1,2,3\ldots n\). Then, let \(T_{0}=0\)  
\[\left\{ \begin{array}{l}\dot{e}_{k}(t)=\dot{q}_{d}(t)-\dot{q}_{k}(t)\\ e_{k}(t)=q_{d}(t)-q_{k}(t)\end{array} \right. \tag{3}\]  
\(K_{P_{0}}\) and \(K_{D_{0}}\) are positive definite gain matrices, \(\alpha\) is a conversion factor for gain, and the following inequality is satisfied:  
\[\left\{ \begin{array}{l}\left\|d_{I_{k+1}}(t)-d_{I_{k}}(t)\right\|=\left\| \delta d_{I_{k}}(t)\right\|\\ \left\|\delta d_{I_{k}}(t)\right\|\leq\eta\\ \alpha(k)>1\end{array} \right. \tag{4}\]  
Linearization of the lower extremity exoskeleton dynamics model for gait rehabilitation training trajectories is transformed by [PERSON] formula expansion:  
\[M(t)\ddot{e}+(C+C_{I})\dot{e}+F(t)e+\mu\left[\ddot{e},\dot{e},e,t\right]=D- \left[M(t)\ddot{q}+C(t)\dot{q}+G\right] \tag{5}\]  
where, because of the existence of higher order variables in the variable \(\mu\left[\ddot{e},\dot{e},e,t\right]\), is organized by simplification as:  
\[\left\{ \begin{array}{l}D(t)=M(t)\ddot{q}_{d}+C(t)\dot{q}_{d}+G\\ C_{I}=\frac{\partial C}{\partial\dot{q}}\left|_{I_{d}(t)\dot{q}_{d}(t)}\ddot{q }_{d}(t)+\frac{\partial C}{\partial\dot{q}}\right|_{q_{d}(t)\dot{q}_{d}(t)}\\ F(t)=\frac{\partial M}{\partial t}\left|_{q_{d}(t)}\ddot{q}_{d}(t)+\frac{ \partial C}{\partial\dot{q}}\right|_{q_{d}(t)\dot{q}_{d}(t)}\dot{q}_{d}(t)+ \frac{\partial C}{\partial\dot{q}}\Big{|}_{q_{d}(t)}\end{array} \right. \tag{6}\]  
This is obtained by organizing (1) with (5):  
\[D(t)-T_{k}(t)=M(t)\dot{e}_{k}(t)+[C(t)+C_{I}(t)]\dot{e}_{k}(t)+F(t)e_{k}(t)+ \mu\left[\ddot{e}_{k},\dot{e}_{k},e_{k},t\right]-d_{k}(t) \tag{7}\]  
When \(d_{k}(t)=d_{I_{k}}(t)+\mu\left[\ddot{e},\dot{e},e,t\right]\), at the kth and k + 1 st iteration, Equation (7) can be reduced to:  
\[\left\{ \begin{array}{l}D(t)-T_{k}(t)=M(t)\dot{e}_{k}(t)+[C(t)+C_{I}(t)]\dot{ e}_{k}(t)+F(t)e_{k}(t)-d_{I_{k}}(t)\\ D(t)-T_{k+1}(t)=M(t)\ddot{e}_{k+1}(t)+[C(t)+C_{I}(t)]\dot{e}_{k+1}(t)+F(t)e_{k+ 1}(t)-d_{I_{k+1}}(t)\end{array} \right. \tag{8}\]  
Define \(K_{P_{0}}=\lambda K_{D_{0}}\) and \(\varphi_{k}(t)=\dot{e}_{k}(t)+\lambda e_{k}(t)\), the iterative learning control gain based on the RBF neural network needs to satisfy the following conditions:  
\[\left\{ \begin{array}{l}L_{i}=\lambda_{\min}\big{(}K_{D_{0}}+2C_{I}-2\lambda M \big{)}>0\\ L_{o}=\lambda_{\min}\Big{(}K_{D_{0}}+2C+\frac{2F}{\lambda}-\frac{2C_{I}}{ \lambda}\Big{)}>0\\ L_{i}L_{o}\geq Max\Big{|}\frac{F}{\lambda}-\big{(}C+C_{I}-\lambda M\big{)} \Big{|}^{2}\end{array} \right. \tag{9}\]  
This study employs a 12-10-6 RBF neural network architecture to achieve accelerated convergence of control state errors in the iterative learning controller.