Dynamical equations of multibody systems on Lie groups

Abstract

The Euler-Poinaré principle is a reduced Hamilton's principle under Lie group framework. In this article, it is applied to derive a hybrid set of dynamical equations of rigid multibody systems, which include four parts: the classical Euler-Lagrange equations of rigid bodies in their translational coordinates of mass center; Euler-Poinaré equations via orientation matrices and their related angular velocities; the constraint equations due to different joints in Cartesian coordinates and Lie groups; and the reconstruction equations between special orthogonal groups and their Lie algebras. The generalized mass matrices of dynamical equations are constant, which is computationally efficient. All the equations can be constructed systematically and can be solved easily. The construction equations can be used to design Lie group integrators of multibody system dynamics. The procedure presented in this article can be extended easily to flexible multibody systems, systems with non-holonomic constraints, and so on.

Lie-Group Modeling and Numerical Simulation of a Helicopter

“…The quantity M B denotes the total mass of the helicopter's fuselage. The matrix ĴB denotes a non-standard inertia tensor [21]. The standard inertia tensor of the helicopter's body is defined as…”

“…As it was already underlined while discussing equations (21), in the case of constant main rotor speed Ω m , the condition (72) will become S −1 ((D t u t − γu m )ξ z )S −1 = 0 that could be reduced to D t u t = γu m .…”

Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

“…and denotes a non-standard inertia tensor [14]. The standard inertia tensor of the platform is defined as…”

Modelling, control and simulation of a single rotor UAV with swashplateless torque modulation

“…First, the Euler-Poincaré equation was used to model the Equations of Motion of the drone. This is a special form of the Euler-Lagrange equation that can be used when the space of configurations is a member of a Lie group [22,23]. Using this equation allowed us to have a very complete theoretical model of the dynamics of the drone, using a minimal set of approximations.…”