Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

2026-05-07

AI Summary

This study addresses the challenge of efficiently computing high-order dynamics for floating-base tree-structured robots. By leveraging the Lie group SE(3) and spatial vector algebra, the authors formulate, for the first time, unified high-order Newton–Euler, articulated-body inertia, and hybrid recursive algorithms within a consistent Lie group framework, and derive closed-form equations of motion. Key theoretical contributions include proving that the articulated inertia tensor remains invariant under arbitrary-order time derivatives and identifying a Coriolis matrix structure that satisfies passivity. The method is applied to a 12-degree-of-freedom aerial manipulator, enabling analytical computation of forward and inverse dynamics along with their first derivatives, and efficient numerical evaluation up to fifth-order derivatives. The computational complexity grows only quadratically with the derivative order, offering a significant advantage over the exponential complexity of automatic differentiation.

Abstract

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.

Problem

• Lie group
• recursive dynamics
• higher-order derivatives
• floating-base robots
• geometric dynamics

Innovation

• Lie group dynamics
• higher-order derivatives
• floating-base robots
• recursive algorithms
• computational efficiency