New Lie Algebra Framework Controls Flexible Robot Arms with Exponential Precision

Researchers Sadeq Yaqubi and Jouni Mattila at Tampere University have developed a modular control framework for flexible multibody robotic manipulators using a Lie algebraic approach. The paper, published on arXiv, addresses the challenge of controlling serial robots with flexible links in 3D space. The framework uses a screw-theoretic model where motion, deformation, and forces are expressed as body-fixed twists and wrenches in the Lie algebra se(3). By substituting a strain-based deformation PDE into the dynamics, the distributed elastic acceleration is eliminated, resulting in 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.

The nominal controller for each link ensures exponential decay of twist errors via a Lyapunov function, while an adaptive modification replaces physical parameters with online estimates, establishing exponential convergence of both twist and parameter errors. Composite Lyapunov functions across all links yield time derivatives where inter-link interaction power terms telescope to zero, guaranteed by Newton's third law and frame invariance of the power pairing on se(3). This cancellation makes the stability certificate modular and scalable to chains of arbitrary length. Numerical simulations demonstrate the scheme's physical consistency and effectiveness. The work provides a rigorous foundation for real-time control of flexible robots, with potential applications in manufacturing, space robotics, and medical devices.