准坐标形式的非完整刚体系统动力学方程及数值模拟

Dynamic Equation and Numerical Simulation of Nonholonomic Multibody System in Quasi-Coordinate Form

摘要: 本文的目标是研究针对非完整多体系统动力学与控制问题的更简单规范的动力学建模方法。采用欧拉–拉格朗日形式的高斯原理推导了适于非完整多体系统的准坐标形式的动力学运动微分方程。与目前常用的以动力学状态函数表达的针对非完整系统的动力学微分方程相比，文中的推导方法简单直接，得到的方程维数与自由度一致，且方程以准速度的显式矩阵表达，具有较强的通用性及规范性，适用于多体系统的程式化建模及计算的要求。针对例子中的典型非完整系统，建立了准速度表达的一阶运动微分方程并进行了数值模拟以及相应的动力学特性分析，与通常用于该类问题的带乘子的Routh方程相比，文中的方法在计算速度上占优，同时在不进行额外的违约修正的前提下，可较好地解决违约及位置奇异问题。

Abstract: The goal is to study a simpler and more canonical dynamic modeling method for nonholonomic multibody system dynamics and control problems. By using the Euler-Lagrangian form of Gauss principle, the differential equations of dynamic motion suitable for nonholonomic multibody sys-tems in quasi-velocity form are derived. Compared with the dynamic differential equations for nonholonomic systems, which are usually expressed in terms of energy functions and generalized coordinates, the method of derivation presented is simple and direct, and the obtained equations are expressed in explicit matrices of quasi-velocity consistent with the degrees of freedom. It has strong generality and standardization, and is suitable for programming modeling and calculation of multibody system requirements. For the example of nonholonomic system, the differential equa-tion of motion expressed by quasi-velocity is established, and the numerical simulation and the corresponding dynamic characteristics are carried out. Compared with the Routh equation with multipliers, the proposed method is superior in computational speed and can be used to solve the constrained default and location singular problems.