Design and analysis of a fractional-order sliding mode controller for 3D overhead crane dynamics

This study proposes a fractional-order sliding mode control (FOSMC) framework for a three-dimensional underactuated overhead crane subject to modeling uncertainty and matched input disturbances. A novel fractional sliding variable that combines proportional error terms with a non-integer derivative Dβ is introduced, together with a composite reaching law that blends discontinuous and linear terms to balance robustness and smoothness. The controller is derived in state-space form for the 3D crane dynamics, and stability is established using a fractional-order Lyapunov analysis, leveraging the property DβD-β=I. Comprehensive case studies such as piecewise waypoints, sinusoidal tracking, and a coordinated move-and-hoist maneuver with time-varying rope length benchmark the proposed FOSMC against a PSO-optimized conventional SMC. Across all scenarios, the proposed scheme attains faster convergence, substantially improved sway suppression in both θx and θy, and markedly lower integral error indices (IAE, ISE, ITAE); notably, the hoisting channel achieves reductions on the order reported in this study. The results demonstrate that introducing fractional memory into sliding mode control yields a tunable trade-off between robustness and reduced chattering, enabling smoother multi-axis coordination and enhanced disturbance rejection in underactuated crane applications.