基于三维凸包的公差基准轴线拟合

Datum axis fitting for tolerances based on 3D convex hulls

针对采用坐标测量系统(CMM)进行基准轴线测量的特点,提出了一种基于凸包的基准轴线拟合算法。该算法首先根据计算几何的凸包理论,利用礼品包裹算法获得轴线采样点构建的三维凸包;考虑基准体系(Datum Reference Frames DRF)中基准的定义,确定基准轴线处于不同的基准优先级时被约束的自由度以及基准的评价方法。然后,利用轴线的数学方程将带约束的轴线分解成转动向量和平动向量并依次建立带约束的基准轴线转动和平动的数学模型。最后,利用三维凸包的相关定理以及基准的相关实体要求完成基准轴线的变动区域的求解。实验结果显示:采用该算法进行轴线基准体系建立得到的结果在处理无约束的轴线基准拟合时与CMM最小二乘算法得到的结果近似重叠,在处理存在约束的轴线基准拟合时两者的误差较大。不过,本文算法保证了受约束轴线相对位置关系,符合基准轴线的建立原则,满足工程需求。

According to the characteristics of datum axis measurement by using a Coordinate Measuring Machine（CMM）,a method based on 3Dconvex hulls was proposed to realize the datum plane fitting.Firstly,the 3Dconvex hulls of discrete sampling points were rapidly constructed by using ＂ gift encapsulated algorithm＂ based on the computational geometric theory.According to the definition of datum axis in Datum Reference Frames（DRF）,the constrained degrees of freedom as well as the method to establish the datum axis were determine by the priority of datum axis and the features of other datum.Then,the mathematical equation of datum axis was used to resolve the constrained axis into a rotational vector and a translational vector and then to establish mathematical models with constrained datum axis rotation and constrained datum axis translation.Finally,the variable domain of datum axis was obtained by using 3Dconvex hulls based on material requirements.The experiment demonstrate that the results of datum axis determined by proposed method is the same as that of the CMM least square method in an unconstrained axis processing,however,it has more different as copared with that in a constrained axis processing.Fortunately,the method ensures the relative positionrelationship between the constrained axes,and it accords with the principle of establishing datum axes and meets the requirements of engineering applications