基于四元数表示的一种改进的刚体动力学保辛积分

An Improved Symplectic Integration for Rigid Body Dynamics in Terms of Unit Quaternions

根据四元数刚体动力学基本理论,将四元数时间导数与角速度之间的恒等变换引入动能项,由此可以直接得到非奇异的四元数质量矩阵.将其与分析结构力学结合,可以得到4种形式的保辛积分算法.该算法以离散系统作用量变分原理代替四元数微分方程,单位长度约束以代数约束的方式在积分格点处满足.数值仿真结果表明该方法不仅避免了陀螺稳态进动数值仿真中严重的章动误差,并且对于一般情况也展现出很大的精度改善.

An identity transformation between the time derivative of quaternions and angular velocity was introduced into the kinetic energy term,according to the theory of quaternion-based rigid body dynamics.Mhis proposed approach yielded a non-singular mass matrix.Combined with the analytical structural mechanics,a newsymplectic integration scheme with 4 formulations,was proposed.In practice,the discrete variational principle of the action function was employed to replace the relevant quaternion differential equations for the proposed method.Correspondingly,the unit length constraint was met explicitly by means of the algebraic constraint at the integration grid points.The numerical results showthat the newscheme avoids the severe periodical nutation errors for the special cases of steady precession of a gyro top,which is a puzzling phenomenon in recent researches.In addition,the newscheme presents an impressive improvement of accuracy for the general cases as well.