Screw-theoretic PDE-based synthesis for unified representation of multibody flexible manipulator dynamics

This paper addresses the open problem of partial differential equation (PDE)-based dynamic modeling for flexible multibody robotic systems by presenting a screw-theoretic synthesis methodology, developed within a unified Lie-algebraic framework, for serial flexible manipulators with an arbitrary number of links in three-dimensional motion. The proposed approach expresses all dynamic states (rigid-body motion, elastic deformation, and inter-link interaction forces) uniformly within the screw theoretic structure as body-fixed twists and their dual wrenches, treating all physical components consistently within the same geometric framework. Hence, the PDE structure of the deformation field is retained exactly, while the $$\mathfrak {se}(3)$$ se ( 3 ) representation admits the linear-algebraic formulation required for multibody assembly and formal well-posedness analysis. Building on a previously developed single-link screw-theoretic PDE model, joint constraints are enforced as screw-compatibility equations connecting the twist and wrench fields of adjacent links at their connection points, and the per-link models are assembled via a closed-form linear-algebraic stacking procedure. The resulting system matrix exhibits near-tridiagonal block structure, interaction wrenches appear explicitly as algebraic variables, and adding a link requires only appending block rows, making the synthesis automatable for arbitrary n . The assembled system is formulated as a semi-explicit index-1 differential-algebraic equation, and well-posedness is established by recasting it in abstract Cauchy form through modal projection. Presented solutions are validated experimentally on a two-link flexible manipulator in three-dimensional motion, and numerically for a three-link flexible manipulator as a scalability demonstration, confirming implementability and physical consistency.