Modular Lie Algebraic PDE Control of Multibody Flexible Manipulators
For roboticists working on flexible manipulators, this work provides a scalable and modular control framework that guarantees stability for arbitrary-length chains, though it is an incremental extension of existing PDE control methods.
This paper presents a modular PDE-based control framework for flexible multibody manipulators, achieving global exponential convergence of tracking errors and bounded elastic deformation through a screw-theoretic Lie-algebraic model. Numerical simulations confirm the approach's physical consistency.
This paper addresses PDE-based control for flexible multibody robotic systems, presenting a subsystem-based framework for serial manipulators with arbitrary links in 3D space. The approach uses a screw-theoretic Lie-algebraic model where motion, deformation, and forces are expressed as body-fixed twists and wrenches in se(3). By substituting a strain-based deformation PDE into the dynamics, distributed elastic acceleration is eliminated, yielding a model governed by twist acceleration and the deformation field. Subsystem twist trajectories are generated from task-space endpoints via deflection-compensating inverse kinematics, providing real-time correction for tip deformation. A nominal controller for each link ensures exponential decay of twist errors via a Lyapunov function nu_i. An adaptive modification replaces physical parameters with online estimates, establishing exponential convergence of both twist and parameter errors. Summing over all links, composite Lyapunov functions V = sum(nu_i) and V^a = sum(nu_i^a) yield time derivatives where inter-link interaction power terms telescope to zero. This cancellation is ensured by Newton's third law and the frame invariance of the power pairing on se(3) x se*(3), establishing global exponential convergence of tracking errors. Bounded elastic deformation is guaranteed by an Euler-Bernoulli energy argument. The screw-theoretic structure renders interaction cancellation exact, making the stability certificate modular and scalable to chains of arbitrary length. Numerical simulations demonstrate the scheme's physical consistency.