Lie-Group Modeling and Numerical Simulation of a Helicopter

Alessandro Tarsi,Simone Fiori

doi:10.3390/math9212682

Abstract

Helicopters are extraordinarily complex mechanisms. Such complexity makes it difficult to model, simulate and pilot a helicopter. The present paper proposes a mathematical model of a fantail helicopter type based on Lie-group theory. The present paper first recalls the Lagrange–d’Alembert–Pontryagin principle to describe the dynamics of a multi-part object, and subsequently applies such principle to describe the motion of a helicopter in space. A good part of the paper is devoted to the numerical simulation of the motion of a helicopter, which was obtained through a dedicated numerical method. Numerical simulation was based on a series of values for the many parameters involved in the mathematical model carefully inferred from the available technical literature.

Highlights

Conventional helicopters are built with two propellers that can be arranged as two coplanar rotors both providing upward thrust, but spinning in opposite directions in order to balance the torques exerted upon the body of the helicopter, or as one main rotor providing thrust and a smaller side rotor oriented laterally and counteracting the torque produced by the main rotor, as shown in the Figure 1
The aim of this paper was to devise a mathematical model of a fantail helicopter in the framework of Lie-group theory
The main theoretical instrument, besides of Lie-group theory itself, is the Lagrange–d’Alembert–Pontryagin principle, which generalizes the Lagrangian formulation of dynamics to curved manifolds and to dissipative systems

Summary

Introduction

Conventional helicopters are built with two propellers that can be arranged as two coplanar rotors both providing upward thrust, but spinning in opposite directions in order to balance the torques exerted upon the body of the helicopter, or as one main rotor providing thrust and a smaller side rotor oriented laterally and counteracting the torque produced by the main rotor, as shown in the Figure 1. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Conclusion