Lie Group Variational Integrator Of Multi-rigid Body System

With the development of society,engineering application technology is becoming more and more nature,which promotes the research and development of multi-body system.With the deepening of multi-body problems research,the existing numerical modeling calculation methods can no longer meet the long time simulation needs of the system,so it is necessary to seek more accurate modeling methods and more efficient calculation methods.This thesis is written on the basis of some basic theories,including Lie group and Lie algebra.This article unites the discrete Lagrangian mechanics and Hamiltonian mechanics with Lie group method to get a method,namely Lie group variational integral method.This numerical integration method preserves not only the systemic energy and geometric property but also the Lie group structure.The main research contents are as described below:First,some basic concepts and related knowledge of Lie group and Lie algebra are introduced.According to Hamilton variational principle,we get the Euler-Lagrange equation on Lie Group G,and the dynamics equation of the system based on SO(3)and SE(3)are derived.Then,we get the general-format Lie group variational integral formula based on the discrete Hamilton variational principle,and the discrete Euler-Lagrange equations are drived.The discrete Legendre transformation is used to transform the discrete Lagrangian system into discrete Hamiltonian system,so the Lie group variational integral formula under the Hamiltonian system is obtained.The discrete Euler-Lagrange equations and the discrete Hamilton equations based on SO(3)are given.Finally,we carry out the rigid body system with the configuration space is SE(3)according to Lie group variational integral formula in general format.The discrete Euler-Lagrange equations and the discrete Hamilton equations of single rigid body and multi rigid body are derived.The 3D rigid body pendulum,spinning top and triple pendulum are calculated.The numerical simulation shows that the Lie group discrete variational method can preserve the energy and geometrical structure of the system.