Modular Lie Algebraic PDE Control of Multibody Flexible Manipulators

2026-05-06

AI Summary

This study addresses the challenging problem of partial differential equation (PDE)-based control for spatially arbitrary-configured flexible multi-body robotic arms by proposing a modular, subsystem-based control framework. Leveraging screw theory within the se(3) Lie algebra, the approach unifies the modeling of rigid motion, elastic deformation, and internal forces, while introducing a strain-based PDE formulation to eliminate distributed elastic acceleration terms. An inverse kinematics solution compensating for deformation is integrated with task-space end-effector trajectory generation to design both nominal and adaptive controllers. Exploiting the frame invariance of power pairings on se(3) and Newton’s third law, the method achieves exact cancellation of inter-link interaction power, enabling the first proof of global exponential stability for flexible arms of arbitrary length. Theoretical analysis guarantees exponential convergence of twist and parameter estimation errors and bounded elastic deformation, with simulations confirming physical consistency and superior control performance.

Abstract

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.

Problem

• flexible multibody systems
• PDE control
• Lie algebraic modeling
• deformation compensation
• exponential stability

Innovation

• Modular control
• Lie algebraic PDE
• Screw theory
• Exponential convergence
• Flexible multibody manipulators