含柔性连杆的空间闭链机器人动力学数值分析

Numerical Solution of Dynamic Equations for Multi-flexible Closed Chain Spatial Robots

以系统中既存在柔性构件,又存在刚性构件的空间闭链机器人为例,分析了模型降阶法在求解柔性多体系统指标-3微分代数方程组中的求解策略。首先,将加速度约束方程并入指标-3的DAEs,动力学指标降为1,通过等价方程组,直接求出难于初值估算的拉格朗日乘子λ,则DAEs变为纯微分方程组;其次,采用逐步积分Newmarkβ法求各时刻的广义位移和速度坐标;最后,运用Baumgarte约束违约修正法消除位置和速度违约,再通过泰勒展开式和Newton-Raphson公式迭代出位移坐标和λ的修正值。由于数值求解过程中没有引入新的未知向量,避免了因模型扩大而导致的新的求解问题,并减少了对加速度坐标的违约修正过程,为求解柔性多体系统动力学方程组提供了一种新方法。

Taking the spatial closed-chain robot with flexible and rigid links in the system as an example,the solution strategy of the model reduction method for solving the differential-algebraic equations(DAEs)of the flexible multi-body system is analyzed.First,the acceleration constraint equation is incorporated into the DAEs of index-3,the index is reduced to 1.Through the equivalent equations,the Lagrangian multiplierλ,which is difficult to estimate from the initial value,is directly obtained,and the dynamic equations become a pure differential algebraic equation.Secondly,the generalized displacement and velocity coordinates can be obtained by using the Newmarkβmethod.Finally,the correction method based on Baumgarte stabilization method is used to eliminate positional and velocity violation,and then iterate out the modified value of displacement coordinates and Lagrangian multiplierλby utilizing the Taylor expansion and Newton-Raphson formulas.New unknown variables are not introduced in the numerical solution process,the new solving problem caused by the model expansion is avoided,and the violation correction process is reduced,which provides a new way to solve the dynamic equations of the flexible multibody systems.