Fractional Second Order Reaching Law‐based Sliding Mode Control for Assistive Robotics

ABSTRACT Lower‐limb rehabilitation exoskeletons must deliver accurate gait assistance while operating under nonlinear dynamics, bounded disturbances, and time‐varying human‐robot interaction that can amplify transient torque peaks and degrade comfort. This paper proposes a Fractional Second‐Order Reaching Law (FSO‐RL) sliding mode control strategy that explicitly shapes the reaching phase to obtain fast and well‐damped convergence to the sliding manifold while mitigating high‐frequency torque oscillations. The method couples a fractional‐order sliding variable with a second‐order reaching dynamics, providing additional degrees of freedom to regulate transient behavior without sacrificing sliding‐mode robustness. A Lyapunov‐based analysis is developed to establish reachability and closed‐loop convergence in the presence of bounded disturbance torques. The approach is evaluated in simulation on a three‐degree‐of‐freedom lower‐limb exoskeleton model using clinically relevant pediatric gait trajectories. Comparative results against a first‐order fractional reaching‐law baseline show that the proposed FSO‐RL controller preserves tracking accuracy while significantly improving torque smoothness and reducing control effort, yielding approximately a 30% reduction in torque‐based effort measures over the gait cycle. These results indicate that second‐order reaching dynamics combined with fractional‐order effects provide an effective mechanism for comfort‐oriented exoskeleton assistance under uncertainty.