Lie Group Integration Methods in Multibody Dynamics
Summary
Multibody dynamics concerns the simulation of interconnected rigid or flexible bodies undergoing large spatial rotations and translations. Traditional time‐integration schemes often struggle with singularities, drift from constraint manifolds and poor long‐term energy behaviour. Lie group integration methods embed the configuration of each body in a smooth manifold structure, typically the special orthogonal group or a semi-direct product incorporating translations. By formulating updates via exponential and Cayley maps, or by using Runge–Kutta–Munthe–Kaas techniques, these integrators ensure that numerical solutions evolve on the exact group manifold, preserving orthogonality of rotation matrices and respecting holonomic constraints without ad hoc reprojection. Variational and symplectic Lie group methods further conserve discrete analogues of momentum and energy properties, delivering enhanced stability over long simulations. Such structure-preserving algorithms have found application in robotics, aerospace attitude control and flexible beam modelling, offering improved accuracy, constraint handling and computational robustness when simulating complex multibody assemblies.
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Technical terms
Lie group: A differentiable manifold endowed with a group operation, enabling smooth composition of rotations and translations.
Lie algebra: The tangent space at the identity of a Lie group, providing a linearised representation of infinitesimal motions.
Variational integrator: A numerical scheme derived from a discrete action principle, conserving discrete analogues of momentum and symplectic form.
Symplectic integrator: An algorithm that preserves the symplectic two-form of Hamiltonian systems, yielding favourable long-term energy behaviour.
Cosserat beam: A continuum model representing slender structures with both translational and rotational degrees of freedom at each cross-section.
Holonomic constraint: A restriction expressible as an algebraic relation among configuration variables, such as fixed distances or orthogonality.
References
- A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model. Multibody System Dynamics (2021).
- Local coordinates on Lie groups for half-explicit time integration of Cosserat-rod models with constraints. Multibody System Dynamics (2024).
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