Rigid-flexible coupling dynamics of pectoral fins for robotic manta rays based on rotating Mindlin plate theory

Bio-inspired underwater robots, such as bionic manta rays, have attracted increasing attention due to their efficient propulsion and superior maneuverability. Unlike conventional rigid-body robots, their bionic fins undergo rigid-body motion and elastic deformation simultaneously. To model the motion and deformation of bio-inspired fins from a solid mechanics perspective, a rigid-flexible coupling dynamic framework is proposed. A floating reference frame is used to represent large rigid-body motions, while a Mindlin plate model captures in-plane extension, bending, and first-order shear deformation. As this study focuses on solid mechanics, the fluid loads are simplified using the engineering approximation formulas common in bio-inspired underwater robots, whose validity has been partially verified experimentally. To obtain the time-dependent deformation of the plate under the adopted fluid loading approximation, the governing equations are discretized using a weak-form finite element formulation and solved via a Newmark time integration scheme combined with fixed-point iteration. The deformation predicted from the presented model with substantially reduced mesh size agrees well with that obtained from 3D solid finite element analysis under the same loadings, with computational cost being reduced by nearly two orders in comparison. Thus, this reduction brings the computational time scale into a range potentially useful for real-time robotic motion control. A further numerical study systematically explores the effects of pitch amplitude, roll amplitude, and actuation frequency on plate deformation. As such, for emerging bio-inspired underwater robots with flexible fins, the proposed model provides a structural mechanics framework for dynamic analysis, control, and structural optimization.