Fractional-Order Modelling of Pneumatic Transmission Dynamics in Soft Robotic Actuation

Pneumatic transmission lines play a critical role in the dynamic performance of soft robotic actuation systems, yet their behaviour is difficult to capture using conventional integer-order (IO) models. In long, slender pipelines, compressibility, viscothermal losses, and wave propagation give rise to distributed damping and non-exponential relaxation dynamics that are not well represented by finite-dimensional models. This paper presents a control-oriented, experimentally validated fractional-order (FO) modelling framework for pneumatic pipeline dynamics under closed-end boundary conditions. Models are calibrated using measured step-response data from a 13.2 m pipeline, with all parameters—including the fractional order—identified through a unified optimisation procedure. In addition to global fitting accuracy, model performance is evaluated using control-relevant metrics, including effective delay, initial slope and early transient behaviour, and early-time error. The results show that FO models provide a more compact and structurally consistent representation of long-memory dynamics while improving the accuracy of control-relevant features compared to their IO counterparts. These findings demonstrate that fractional dynamics offer a physically meaningful and practically useful framework for modelling pneumatic transmission lines, with direct implications for high-performance control design in soft robotic systems.