Study On Measurement Points' Distribution Strategy And Distribution Indicator For Freeform Surface

Distribution of measurement points is one of the important steps for freeform surface machining quality testing.The method it used is called distribution strategy.Especially,the guiding factor among distribution is called distribution indicator.It is a basic feature of a curve.Usually,we use measurement method error to evaluate the pros and cons of a distribution strategy or a distribution indicator.Measurement method error means,on a theoretical curve,a number of measurement points were selected to interpolation with cubic spline curve,that the deviation of the interpolation of curve and the original theoretical curve.This paper researched on distribution strategy and distribution indicator.In distribution strategy,first of all,two requirements of distribution strategy were put forward.That is,being adaptive to distribution indicator and keeping spacing between points.A new class of distribution strategies with controllable curvature adaptability were proposed accordingly,which is called equal moment methods.It refers to that,in a curve segment composed of any adjacent three points,its left part and its right part have an equal mass moment to the intermediate point.The higher order the equal moment method has,the lower its curvature adaptability is.Finally,through two examples,the relationship between curvature adaptability and measurement method error was analyzed.The best curvature adaptability whit a minimum measurement method error was calculated.In distribution indicator,through the study of arc length and curvature the two existing distribution indicators,their two points of deficiency in using cubic spline curve as the measuring interpolation curve were analyzed.That is measurement point wastefulness and measurement point inaccuracy.In addition,two features a distribution indicator should possess were put forward.That is,1)satisfying the geometric invariance;2)no lack of value.Then according to the mathematical characteristics of cubic spline curve,this paper proposes an indicator based on cubic spline,and call it cubic spline rate.And then through cubic spline rate,a simulation experiment aiming at to find out piecewise points of cubic spline curve was conducted,and ideal results have been achieved.Then through a numerical example,compared respective measurement method error of curvature indicator and cubic spline rate indicator in a same distribution strategy.By comparison,it was concluded that cubic spline rate is a better distribution indicator than curvature.Finally,two key indicators for freeform surface measurement point distribution were studied,that is curvature and cubic spline rate.And their respective role in determining measuring point distribution was analyzed.By a normalization method,they were synthesized to a comprehensive distribution indicator.At the last,through comprehensive indicator,using equal moment method of distribution strategy,a simulated calculation of S specimens was conducted.