多体系统Euler-Lagrange方程的最小二乘法与违约修正

LEAST SQUARE ALGORITHMS AND CONSTRAINTSTABILIZATION FOR EULER-LAGRANGE EQUATIONS OFMULTIBODY SYSTEM DYNAMICS

针对受完整约束的多体系统动力学Euler-Lagrange方程,在其传统的直接增广算法和零空间方法基础上提出了当系统存在冗余约束情形下的最小二乘法.同时,对应于最小二乘法提出了改进的约束违约修正方法.本文还针对Euler-Lagrange方程的计算过程给出了相应Jacobi矩阵的QR分解和零空间连续正交基的算法.最后,以平行五连杆机构给出了数值结果并与部分现有方法进行比较.

Numerical algorithms for Euler-Lagrange equations system M(x)x+ФxT(x,t)y= F(x,x,t),Ф(x,t)=0 are discussed and following results are obtained in this paper: 1) On the basis of direct expansion and null space technique, two least square algorithms for the problems with redundant constraints are raised. 2) For reducing constraint violations arisen in the process of numerical computation, two ways of constraint stabilization are presented, which are extensions of the works of Haug and Blajer. 3) Since some numerical methods raised in this paper require computation of a continuous differentiable base in the null-space of Фx which can not be achieved by using the algorithms found in Numerical Algebra Courses, a special Schmidt orthogonalize process is designed to solve the problem.Finally, numerical simulations of a planar five-bar parallel system are given as examples, and comparisons of the results with some known ones are made.