基于单开链有序求解的机构正向运动学建模原理及其两种求解方法

A Novel Forward Position Kinematics Modelling Principle for Mechanisms Based on Ordered Solutions for SOC Units and Its Two Solving Methods

提出了一种基于单开链有序求解的机构正向运动学建模原理。将机构分解为一系列具有不同约束度值的单开链单元,再根据约束度总和为零的原则,将一系列单开链单元划分为若干个自由度为零、耦合度为κi的基本运动链(BKCi),逐一按BKCi建立含最少虚拟变量数目的机构位置方程;给出了具体的数值法和封闭法两种方法。由于数值法较简单,故用κ维搜索法直接求解机构位置方程;封闭法求解时先用Mathematica进行符号处理,从含变量数为κ的机构位置方程中导出一个一元高次的非线性位置正解封闭方程,再求解该一元高次方程。分别给出4个实例予以详细说明与验证。所提原理及求解方法思路清晰,可使机构正向位置方程中的变量和计算量大大减少,适用于求解任意复杂平面机构、空间并联机构的位置正解。

A novel modeling principle for solving the position forward solution of any mechanism was presented based on ordered SOC units.Firstly,the mechanisms were decomposed into a series of SOC units with different constraint degrees based on the principle of mechanism composition.Then,according to the principle that the sum of all constraint degrees was equal to zero,a series of SOC units were grouped into several basic kinematic chains(BKCi)with zero degree-of-freedom and the value of coupling degree wasκi.Position equations for each BKCi with the minimum number of variables were obtained.In addition,two methods,i.e.,numerical solution and closed-form solution were given.The numerical method was relatively simple,and the position equation of the mechanisms was solved directly byκ-dimensional search method.However,the closed-form solution,nonlinear position equations with higher order were firstly derived from the position equation of the mechanisms withκvirtual variables by using the symbolic processing of Mathematica,and then the higher order equation was solved by using the conventional method.Four examples were illustrated in details.The solving processes of the kinematics modeling principle that considers the ordered topological decomposition and SOC unit solution and its two methods are clear,and the solving method for forward solution of parallel mechanisms may be simplified and its calculation amounts are greatly reduced.The kinematics modelling principle is suitable for solving the position forward solution of any complex planar and spatial parallel mechanisms.