两个正交矢量的点积为0.将矢量概念拓展，即将这两个矢量换成线性空间，会怎么样？例如，多体系统中的约束力空间和速度空间就是正交的两个空间。利用这一点，可将基于Newton-Euler方程多体系统动力学DAE形式变换为不含约束反力的动力学方程最小形式。下面文章进行了这方面的分析。

基于Newton-Euler方法的多体系统常微分方程

摘  要：常微分方程是多刚体系统动力学模型的一种重要形式。本文讨论了多体系统常微分形式的动力学建模方法。多体系统由相关广义坐标描述，广义坐标受约束方程限制。根据约束方程的Jacobi矩阵，可以将广义坐标分解为相关坐标和独立坐标。组合各构件的Newton-Euler方程可得到多体系统的动力学方程，方程由广义坐标描述并包含约束反力。将相关坐标表示为独立坐标的函数，并使用Jacobi矩阵表示约束反力，可消去方程中的相关坐标和约束反力，得到使用独立坐标刻画的、不含约束反力的常微分方程。

关键词：多体系统；Newton-Euler方法；Jacobi矩阵；常微分方程

中图分类号：TP242.2                 文献标识码：A

Formulation of Ordinary Differential Equations of Multi-body Systems Based on Newton-Euler Methods

Abstract: Ordinary differential equations (ODE) is an important dynamic model of multi-body systems. Formulation of ODE of multi-body systems is discussed in this paper. The configuration space of multi-body system is described by dependant generalized coordinations which are satisfied with constraint equations. Applied Jocobian matrix of constraint equations, generalized coordinations can be partitioned to dependant coordinates and independent ones. Combining Newton-Euler equations of each part, dynamic equations of multi-body system can be attained which are described with generalized coordinates and have constraint forces in it. Subsituting dependant coordinates with independant ones and describing contraint forces with Jacobiian matrix, dependant coordinates and contraint forces can be eliminated from dynamic equations of multi-body system. Therefore, the minimal form of dynamic equations, ordinary differential equations, of multi-body-system can be obtained.

Keywords: Multi-body systems; Newton-Euler Methods; Jacobian Matrix; Ordinary Differential Equations