Design of modified fractional-order PID controller for lower limb rehabilitation exoskeleton robot based on an improved elk herd hybridized with grey wolf and multi-verse optimization algorithms

Rehabilitation robots are widely recognized as vital for restoring motor function in patients with lower-limb impairments. A Modified Fractional-Order Proportional-Integral-Derivative (MFOPID) controller is proposed to improve trajectory tracking of a 2-DoF Lower Limb Rehabilitation Exoskeleton Robot (LLRER). The classical FOPID is augmented with a modified control formulation by which steady-state error is reduced and the transient response is sharpened. Controller gains and fractional orders were tuned offline using a hybrid metaheuristic Improved Elk Herd Optimization hybridized with Grey Wolf and Multi-Verse Optimization algorithms (IElk-GM) so that exploration and exploitation are balanced. Superiority over the classical FOPID was demonstrated in simulations under linear and nonlinear trajectories, with disturbances and parametric uncertainty: 0% overshoot was achieved at both hip and knee joints; settling time was reduced from 6.998 s to 0.430 s (hip) and from 7.150 s to 0.829 s (knee); ITAE was reduced from 23.39 to 2.694 (hip) and from 16.95 to 3.522 (knee); and the hip steady-state error decreased from 0.018 Rad to 0.0015 Rad, while the knee steady-state error remained within 0.011 Rad. Control torques remained bounded under linear tracking (345 N·m at the hip; 95 N·m at the knee) and under nonlinear cosine tracking (350 N·m at the hip; 100 N·m at the knee). These results indicate that safer, smoother, and more effective robot-assisted rehabilitation can be supported by the proposed controller.