Screw theoretic nonlinear PDE dynamic model of a flexible link using body-fixed coordinates

This paper presents a screw-theoretic partial differential equation (PDE) dynamic model for a flexible link undergoing finite rigid-body motion in three-dimensional space. The flexible body is represented using three body-fixed twist quantities in se ( 3 ) : a twist describing motion of the body-fixed frame on the Lie group SE(3), a reference geometry descriptor for the undeformed configuration, and a continuous deformation twist field governing distributed elastic displacement. Kinematic quantities and their variations are derived via the Lie derivative on se ( 3 ) and its variational analogue, yielding a systematic treatment of rigid–flexible coupling. The dynamic model is obtained from the extended Hamilton principle in body-fixed coordinates, producing a symmetric positive definite inertia structure with nonlinear Coriolis–centrifugal terms emerging naturally from the adjoint action. The result is a finite-dimensional rigid-body equation coupled to an infinite-dimensional deformation PDE with boundary conditions derived from the variational principle. Well-posedness – local existence, uniqueness, and continuous dependence on initial conditions – is established by the Picard–Lindelöf theorem via reformulation as an abstract Cauchy problem through modal projection, which also yields a semi-analytical numerical solution procedure. The per-link formulation facilitates rigorous modal and ability analysis, with the se ( 3 ) representation retaining exact nonlinear coupling without coordinate overhead. Fidelity and versatility are demonstrated through numerical comparisons with the floating frame of reference and co-rotational finite element methods, and experimental validation on a flexible manipulator.