## 1 Introduction
This study addresses gait trajectory tracking control in lower-limb rehabilitation robots, focusing on nonlinear perturbations from patients' lower-limb muscle groups and exoskeleton gait compliance. We propose a novel hybrid control method integrating iterative learning control (ILC) and RBF neural network-based sliding mode control (SMC). An RBF neural network-based iterative learning controller is designed, utilizing a single hidden-layer feedforward neural network structure. This architecture replaces traditional multi-weight networks with a single-parameter approximation, enhancing real-time performance to meet the stringent requirements of exoskeleton rehabilitation training. MATLAB simulations demonstrate superior trajectory tracking performance of the proposed SMC method compared to conventional approaches. The proposed method eliminates chattering effects inherent in traditional SMC, enabling rapid and precise trajectory tracking in the exoskeleton system. Furthermore, the RBF-based iterative learning controller enhances tracking performance, with the global asymptotic stability of the closed-loop system rigorously proven via [PERSON]'s direct method. To validate training efficacy and safety, a 3 Dmotion capture system constructs a lower-limb musculoskeletal model for spasticity-stage patients, enabling personalized gait trajectory planning. The main contributions of this study can be summarized as follows:  
(1) We propose a novel method integrating iterative learning and sliding mode control using RBF neural networks. An RBF neural network-based iterative learning controller is designed, employing a single hidden-layer pre-feedback structure. By replacing the neural network weights with a single parameter, the system achieves strong real-time performance, fulfilling the training requirements of lower-limb exoskeleton rehabilitation robots. The trajectory tracking performance of the proposed sliding mode control method is compared with conventional approaches via MATLAB simulations. The chattering issue inherent in conventional sliding mode control is mitigated, enabling the lower-limb exoskeleton rehabilitation robot to track the desired trajectory rapidly and precisely. (2) To enhance system stability, we design an RBF neural network-based iterative learning controller and rigorously prove the global asymptotic stability of the closed-loop system using [PERSON]'s direct method. We establish a lower-limb skeletal muscle model for patients with leg spasticity and integrate a 3D motion capture system with sensor signals to validate rehabilitation training efficacy and safety while planning optimal gait trajectories. This approach significantly enhances patient outcomes in walking rehabilitation training. 2 The Overall Structure Description and Dynamic Model of Lower Limb Exoskeleton Rehabilitation Robot  
As illustrated in Figure 1, the lower limb rehabilitation exoskeleton system features a thirteen-degree-of-freedom configuration comprising: (a) a single-degree-of-freedom lumbar module; (b) a three-degree-of-freedom hip joint with actively actuated flexion/extension, passively compliant adduction/abduction, and passively compliant internal/external rotation; (c) a two-degree-of-freedom ankle joint with passively compliant plantar flexion/dorsiflexion and inversion/eversion; and (d) a single-degree-of-freedom knee joint with actively actuated flexion/extension. The ankle exoskeleton incorporates bioinspired elastic elements to mimic the natural foot rebound dynamics during plantarflexion/dorsiflexion movements [16]. The lower limb exoskeleton rehabilitation robot is a comprehensive medical rehabilitation system. To ensure wearing comfort, it incorporates a bionic and anthropomorphic joint design that aligns the exoskeleton structure with natural human joint movements. The system features a modular joint design, enabling independent motion control of the hip, knee, and ankle exoskeletons, thereby facilitating more efficient and adaptable rehabilitation training.