Abstract
The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.
Highlights
The dynamics of non-conservative systems may be formulated through the Lagrange–d’Alembert principle
As already done for the gyrostat satellite, we proceed on determining a model of quadcopter by a fully geometric approach, namely by means of the Euler–Poincaré equations written in compact form, without resorting to component-wise expressions or classical physics definitions
We focus on dynamical systems of the first order, namely initial value problems [23], since it is always possible to reformulate a dynamical system of higher order into a dynamical system of first order by augmenting the number of variables
Summary
The dynamics of non-conservative systems may be formulated through the Lagrange–d’Alembert principle. The present paper of archival nature aims at presenting in details, and in a self-contained manner, two interlaced topics, namely: 1) Formulation of Euler–Poincaré equations of motion on Lie groups: This part, covered by Sections 2 and 3, aims at recalling with some mathematical depth the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form. This part aims at presenting in a detailed way the derivation of the reduced Lagrangian function for two systems, namely a satellite gyrostat and a quadrotor drone.
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