Geometric Control Theory on Lie Groups
Summary
Geometric Control Theory on Lie Groups blends control theory with differential geometry to analyse and design controllers for systems evolving on smooth manifolds that possess a group structure. Unlike classical control approaches that operate in vector spaces, this framework recognises that many real-world systems—such as rigid-body attitudes in aerospace, robotic manipulators, and quantum spins—naturally evolve on Lie groups like SO(3) or SE(3). By exploiting group operations and inherent symmetries, practitioners derive coordinate-free formulations, intrinsic stability conditions and singularity-free control laws. Core elements include the use of the Lie algebra for infinitesimal motions, the exponential map for integration, and invariant Riemannian metrics that respect group symmetries. This theory underpins advances in precision attitude tracking, energy-efficient manoeuvring in robotics, synchronisation of multi-agent formations and robust quantum state control.
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Geometric Control Theory on Lie Groups publication trend
The graph below shows the total number of articles in geometric control theory on lie groups across all publications each year (not limited to Nature Index journals).
Technical terms
Lie group: A smooth manifold with a compatible group operation, enabling continuous transformations and multiplication of elements.
Lie algebra: The tangent space at the identity element of a Lie group, endowed with a commutator bracket that encodes infinitesimal symmetries.
Euler–Poincaré equations: The reduced form of Euler–Lagrange equations on a Lie group, expressing dynamics via algebraic relations aligned with group symmetries.
Sliding-mode control: A robust strategy that drives system trajectories onto and along a predefined manifold within the state space in finite time.
Exponential map: A mapping from the Lie algebra to the Lie group, converting infinitesimal generators into finite group motions.
References
- Geometric sliding mode control of mechanical systems on Lie groups. Automatica (2025).
- Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors. Mathematics (2019).
- Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems. Symmetry (2021).
- Manifold Calculus in System Theory and Control—Second Order Structures and Systems. Symmetry (2022).
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