Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle

Simone Fiori; Simone Fiori

doi:10.3934/math.2024497

AIMS Mathematics

2024, Volume 9, Issue 4: 10157-10184. doi: 10.3934/math.2024497

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Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle

Simone Fiori ^{1,2
,
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1.
Department of Information Engineering, Marches Polytechnic University, Ancona, Italy
2.
Department of Mechanical Engineering, Keio University, Yokohama, Japan

Received: 07 December 2023 Revised: 12 February 2024 Accepted: 05 March 2024 Published: 14 March 2024
MSC : 70E60, 93C85

Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.
- Lie group theory,
- Lie algebra,
- Coordinate-free mathematical model,
- submersible vehicle,
- Lagrange-d'Alembert principle,
- Euler-Poincaré equation,
- Lie-group integration,
- numerical simulation
Citation: Simone Fiori. Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle[J]. AIMS Mathematics, 2024, 9(4): 10157-10184. doi: 10.3934/math.2024497

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Abstract

Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.

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