Lie Group Variational Integrator for Multi-Body System with Rotation Coupling in Space

Abstract

In multi-body system dynamics modeling, the body which rotate in space is complex and not easily to be expressed by Newton method, because the space rotation is realized by multi-joints rotation coupling with different direction. In this exploration, the Lie group variational integrator method is used to the dynamics modeling problem of multi-body system with three orthogonal direction joints coupling.  Firstly, a mechanism accords with Cardan rotation regular is designed which can represent 3 different directions coupling, the kinematics model is derived out by matrix operation without triangle function. With inertia matrix and mass matrix of the multi-body system according to the topology structure, and the Lagrange function is built, and the dynamics equation is derived out with Lie group variational integrator method. With Legendre transformation, the Hamilton dynamics model is obtained. The differential computation of the momentum part is avoided, the scale of the dynamics model is greatly reduced. The Hamilton dynamics model with two different kinematics part are compared in simulation. The simulation results indicates that the different kinematic expression can lead to different structure conservation characters under same numerical computation method. This exploration offers a benefit attempt of using geometry method to dynamics modeling problem of tree structure multi-body system with different structure rotation matrixes coupling.