Singularity-Free Lie Group Integration and Geometrically Consistent Evaluation of Multibody System Models described in terms of Standard Absolute Coordinates

2021-12-24

Journal of Computational and Nonlinear Dynamics

AI Summary

This work proposes a Lie group integration framework compatible with the standard absolute coordinate formulation for multibody system modeling. Traditional time integration methods struggle to preserve the geometric consistency of rigid body spatial motion due to their reliance on singularity-prone absolute coordinate parameterizations in vector spaces. The proposed approach overcomes this limitation by introducing a local–global transformation (LGT) mapping that seamlessly integrates pose representations on the Lie groups SO(3)×ℝ³ or SE(3) with classical vector-space integration techniques. For the first time, this method enables Lie group integrators to be directly embedded into existing absolute-coordinate-based multibody equations without requiring any reformulation of the simulation code. Consequently, the integration remains free of singularities while preserving the underlying geometric structure, significantly enhancing both accuracy and numerical stability in simulations.

Technology Category

• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Low Level & Physics-based Vision

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

A classical approach to the MBS modeling is to use absolute coordinates, i.e. a set of (possibly redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group integration methods, operating directly on the configuration space Lie group, are incompatible with standard formulations of the EOM, and cannot be implemented in existing MBS simulation codes without a major restructuring. A framework for interfacing Lie group integrators to standard EOM formulations is presented in this paper. It allows describing MBS in terms of various absolute coordinates and at the same using Lie group integration schemes. The direct product group SO(3)xR3; and the semidirect product group SE(3) are use for representing rigid body motions. The key element of this method is the local-global transitions (LGT) transition map, which facilitates the update of (global) absolute coordinates in terms of the (local) coordinates on the Lie group. This LGT map is specific to the absolute coordinates, the local coordinates on the Lie group, and the Lie group used to represent rigid body configurations. This embedding of Lie group integration methods allows for interfacing with standard vector space integration methods.

Problem

• multibody systems
• singularity-free integration
• absolute coordinates
• Lie group
• geometric consistency

Innovation

• Lie group integration
• singularity-free
• multibody systems
• geometric consistency
• local-global transition