CN103268378B - A kind of globoid cam curve movement discrimination method based on multistage correlation analysis - Google Patents

A kind of globoid cam curve movement discrimination method based on multistage correlation analysis Download PDF

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CN103268378B
CN103268378B CN201310182841.4A CN201310182841A CN103268378B CN 103268378 B CN103268378 B CN 103268378B CN 201310182841 A CN201310182841 A CN 201310182841A CN 103268378 B CN103268378 B CN 103268378B
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要义勇
赵丽萍
王亚菲
赵虎
王龙轩
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Xian Jiaotong University
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Abstract

本发明公开了一种基于多阶相关分析的弧面凸轮运动曲线辨识方法：(1)弧面凸轮运动曲线辨识；(2)对获取的弧面凸轮运动采集数据进行相关分析并拟合出实际运动曲线，进而确定弧面凸轮实际运动曲线与理想的运动规律的相似性；(3)多阶相关分析，用于对获取的弧面凸轮运动采集数据进行多阶相关分析，进行弧面凸轮运动规律的一阶导数、二阶导数和多阶导数与理想的运动规律的相关分析，得到弧面凸轮实际运动曲线在弧面凸轮的位置、速度和加速度的相关系数及其相似形态类别，最终辨识出弧面凸轮实际运动曲线。本发明不仅有利于提高机械行业中机械运动机构的位置精度，而且有利于提高机械运动机构的速度和加速度精度，进而提高运动平稳性。

The invention discloses a cam motion curve identification method based on multi-order correlation analysis: (1) motion curve identification of cam cam; Motion curve, and then determine the similarity between the actual motion curve of the arc cam and the ideal motion law; (3) Multi-order correlation analysis, used for multi-order correlation analysis on the acquired arc cam motion data, and perform arc cam motion The correlation analysis of the first-order derivative, second-order derivative and multi-order derivative of the law and the ideal motion law can obtain the correlation coefficient of the actual motion curve of the arc-shaped cam in the position, velocity and acceleration of the arc-shaped cam and its similar shape category, and finally identify The actual motion curve of the arc-out cam. The invention not only helps to improve the position accuracy of the mechanical motion mechanism in the machinery industry, but also helps to improve the speed and acceleration accuracy of the mechanical motion mechanism, thereby improving the motion stability.

Description

一种基于多阶相关分析的弧面凸轮运动曲线辨识方法A Cam Motion Curve Identification Method Based on Multi-order Correlation Analysis

技术领域technical field

本发明属于加工过程弧面凸轮运动规律的设计领域，涉及一种基于多阶相关分析的弧面凸轮运动曲线辨识方法。The invention belongs to the design field of arc cam motion laws in the processing process, and relates to an arc cam motion curve identification method based on multi-order correlation analysis.

背景技术Background technique

在机械工程领域，弧面凸轮是实现机械运动的关键零件之一，其运动曲线是保证弧面凸轮运动精度和平稳性的基础，因此，各个弧面凸轮制造企业以及研究学者对此进行了大量的理论研究和试验。In the field of mechanical engineering, arc cam is one of the key parts to realize mechanical movement, and its motion curve is the basis for ensuring the accuracy and stability of arc cam movement. Therefore, various arc cam manufacturers and researchers have conducted a lot of research on this. theoretical research and experiments.

弧面凸轮运动曲线辨识主要是对弧面凸轮的运动规律进行反求，得到运动规律的数学表达。但是现有的数据拟合和曲线相关分析研究都集中于各种曲线的一次拟合或相关分析方面，现有辨识方法采用最小二乘方法直接进行曲线拟合，这样拟合得到的拟合曲线和多种运动曲线都非常相关，相关系数都接近1，无法进行区分。因此，为了避免冲击，面对弧面凸轮的设计，需要一种新的运动曲线的辨识方法，从而使得弧面凸轮结构运动地更加平稳。面对弧面凸轮的加工，需要一种运动曲线的辨识方法，来分析已有的弧面凸轮的运动规律，检验制造的弧面凸轮的运动误差。The motion curve identification of arc cam is mainly to invert the motion law of arc cam to obtain the mathematical expression of motion law. However, the existing data fitting and curve correlation analysis researches all focus on the one-time fitting or correlation analysis of various curves. The existing identification methods use the least squares method to directly perform curve fitting. It is very correlated with various motion curves, and the correlation coefficients are all close to 1, making it impossible to distinguish. Therefore, in order to avoid the impact, facing the design of the arc cam, a new method for identifying the motion curve is needed, so that the arc cam structure can move more smoothly. Facing the processing of arc cams, a motion curve identification method is needed to analyze the motion laws of existing arc cams and check the motion errors of manufactured arc cams.

发明内容Contents of the invention

本发明的目的在于提供一种基于多阶相关分析的弧面凸轮运动曲线辨识方法。The object of the present invention is to provide a cam motion curve identification method based on multi-order correlation analysis.

为达到上述目的，本发明采用了以下技术方案：To achieve the above object, the present invention adopts the following technical solutions:

(1)弧面凸轮运动曲线的最小二乘拟合：对采集到的弧面凸轮运动数据，采用最小二乘法进行曲线拟合，得到弧面凸轮的实际运动曲线；(1) Least squares fitting of the arc cam motion curve: for the collected arc cam motion data, the least square method is used for curve fitting to obtain the actual motion curve of the arc cam;

(2)相关分析：根据弧面凸轮的实际运动曲线与理想的弧面凸轮运动曲线，进行弧面凸轮运动规律的相关分析得到相关系数；(2) Correlation analysis: according to the actual motion curve of the arc cam and the ideal cam motion curve, the correlation analysis of the motion law of the arc cam is carried out to obtain the correlation coefficient;

(3)根据相关系数的差异完成弧面凸轮的实际运动曲线的识别过程。(3) Complete the identification process of the actual motion curve of the arc cam according to the difference of the correlation coefficient.

所述步骤(1)包括如下具体流程：Described step (1) comprises following specific flow process:

①假设某弧面凸轮的实际运动曲线上包含n个采集点，则将该弧面凸轮的实际运动曲线表示为C={yt=fc(xt)}，t=1,2,3…,n；将理想的弧面凸轮运动曲线表示为Pk={ykt=fk(xt)}，t=1,2,3…,n；k=1,2,3…,K，K为理想的弧面凸轮运动曲线种类总数；① Assuming that the actual motion curve of a certain arc cam contains n collection points, then the actual motion curve of the arc cam is expressed as C={y t =f c (x t )}, t=1,2,3 …,n; Express the ideal arc cam motion curve as P k ={y kt =f k (x t )}, t=1,2,3…,n; k=1,2,3…,K , K is the total number of types of ideal arc cam motion curves;

②定义实际运动曲线的拟合多项式为：②The fitting polynomial defining the actual motion curve is:

fC(x)=α0g0(x)+α1g1(x)+α2g2(x)+...αmgm(x)(1)f C (x)=α 0 g 0 (x)+α 1 g 1 (x)+α 2 g 2 (x)+...α m g m (x)(1)

公式(1)中，gj(x)=xj，j=0,1,2,…,m；m表示拟合多项式的阶数；In formula (1), g j (x)=x j , j=0,1,2,...,m; m represents the order of the fitted polynomial;

③由弧面凸轮的实际运动曲线采集点和实际运动曲线的拟合多项式计算误差平方和：③ Calculate the sum of squared errors from the actual motion curve collection points of the arc cam and the fitting polynomial of the actual motion curve:

SS CC == ΣΣ tt == 11 nno (( ythe y tt -- ff cc (( xx tt )) )) 22 →&Right Arrow; minmin -- -- -- (( 22 ))

④然后求偏导：④ Then find the partial derivative:

∂∂ SS CC ∂∂ αα jj == 00 ,, jj == 0,1,2,30,1,2,3 ,, .. .. .. mm -- -- -- (( 33 ))

定义：definition:

AA ijij == ΣΣ tt == 11 nno xx tt ii ++ jj ,, ii == 0,1,20,1,2 ,, .. .. .. ,, mm ;; jj == 0,10,1 ,, .. .. .. mm ;; -- -- -- (( 44 ))

bb jj == ΣΣ tt == 11 nno ythe y tt ×× xx tt jj ,, jj == 0,1,20,1,2 ,, .. .. .. ,, mm -- -- -- (( 55 ))

所以有：So have:

AA 0000 AA 0101 .. .. .. AA 00 mm AA 1010 .. .. .. AA 11 mm .. .. .. .. .. .. .. .. .. .. .. .. AA mm 00 AA mm 11 .. .. .. AA mmmm αα 00 αα 11 .. .. .. αα mm == bb 00 bb 11 .. .. .. bb mm -- -- -- (( 66 ))

通过求解公式(6)得到拟合多项式的系数αj，将系数αj对应带入拟合多项式得弧面凸轮的实际运动曲线方程。The coefficient α j of the fitting polynomial is obtained by solving the formula (6), and the coefficient α j is correspondingly brought into the actual motion curve equation of the arc cam by fitting the polynomial.

所述弧面凸轮运动规律的相关分析包括弧面凸轮的实际运动曲线与理想的弧面凸轮运动曲线的多阶相关分析，所述多阶相关分析包括弧面凸轮的实际运动曲线与理想的弧面凸轮运动曲线的一阶导数或二阶导数的相关分析，所述相关系数为与相关分析对应的弧面凸轮速度的相关系数或弧面凸轮加速度的相关系数。The correlation analysis of the arc cam motion law includes the multi-order correlation analysis between the actual motion curve of the arc cam and the ideal arc cam motion curve, and the multi-order correlation analysis includes the actual motion curve of the arc cam and the ideal arc The correlation analysis of the first derivative or the second derivative of the surface cam motion curve, the correlation coefficient is the correlation coefficient of the arc cam speed or the correlation coefficient of the arc cam acceleration corresponding to the correlation analysis.

所述弧面凸轮运动规律的相关分析还包括以下步骤：在多阶相关分析前，将弧面凸轮的实际运动曲线与理想的弧面凸轮运动曲线进行相关分析，得到相关系数γxy。The correlation analysis of the motion law of the arc cam further includes the following steps: before the multi-order correlation analysis, correlating the actual motion curve of the arc cam with the ideal motion curve of the arc cam to obtain a correlation coefficient γ xy .

所述弧面凸轮的实际运动曲线与理想的弧面凸轮运动曲线进行相关分析，包括如下具体流程：The actual motion curve of the arc cam and the ideal arc cam motion curve are correlated and analyzed, including the following specific processes:

由实际运动曲线采集点和理想运动曲线点计算相关系数γxy：Calculate the correlation coefficient γ xy from the actual motion curve collection points and the ideal motion curve points:

γγ xyxy == nno ΣΣ ii == 11 nno ythe y cici ythe y kithe ki -- ΣΣ ii == 11 nno ythe y cici ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki nno ΣΣ ii == 11 nno ythe y cici 22 -- (( ΣΣ ii == 11 nno ythe y cici )) 22 ·&Center Dot; nno ΣΣ ii == 11 nno ythe y kithe ki 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki )) 22 -- -- -- (( 88 ))

其中，yci表示拟合的实际运动曲线上对应xi的值，yki表示理想的弧面凸轮运动曲线k上对应xi的值，n表示采集点个数。Among them, y ci represents the value corresponding to xi on the fitted actual motion curve, y ki represents the value corresponding to xi on the ideal arc cam motion curve k, and n represents the number of collection points.

所述理想的弧面凸轮运动曲线包括余弦曲线、正弦曲线、五次多项式曲线、修正等速曲线、修正梯形曲线和修正正弦曲线，所述相关系数为弧面凸轮加速度的相关系数。The ideal arc cam motion curve includes cosine curve, sine curve, quintic polynomial curve, modified constant velocity curve, modified trapezoidal curve and modified sine curve, and the correlation coefficient is the correlation coefficient of arc cam acceleration.

所述步骤(2)包括如下具体流程：Described step (2) comprises following specific flow process:

当K种理想的弧面凸轮运动曲线和弧面凸轮的实际运动曲线的相关系数γxy≈1，说明弧面凸轮的实际运动曲线和K种理想的弧面凸轮运动曲线在位置曲线的相似程度非常高，需要进行一阶导数或二阶导数相关分析；When the correlation coefficient γ xy ≈ 1 between K kinds of ideal arc cam motion curves and the actual motion curves of arc cams, it shows the similarity between the actual motion curves of arc cams and K ideal arc cam motion curves in the position curve Very high, need to conduct correlation analysis of first order derivative or second order derivative;

由实际运动曲线采集点和理想运动曲线点计算相关系数，如果实际运动速度曲线和理想运动速度曲线的相关系数γ′xy还不能区分弧面凸轮的实际运动曲线的速度属性，则进行二阶求导和计算加速度的相关系数γ″xy，直到辨识出来弧面凸轮的实际运动规律：The correlation coefficient is calculated from the actual motion curve collection points and the ideal motion curve points, if the correlation coefficient γ′ xy of the actual motion speed curve and the ideal motion speed curve cannot distinguish the speed attribute of the actual motion curve of the arc cam, the second-order calculation is performed Derivation and calculation of the correlation coefficient γ″ xy of the acceleration until the actual law of motion of the arc cam is identified:

γγ xyxy ′′ == nno ΣΣ ii == 11 nno ythe y cici ′′ ythe y kithe ki ′′ -- ΣΣ ii == 11 nno ythe y cici ′′ ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki ′′ nno ΣΣ ii == 11 nno ythe y cici ′′ 22 -- (( ΣΣ ii == 11 nno ythe y cici ′′ )) 22 ·&Center Dot; nno ΣΣ ii == 11 nno ythe y kithe ki ′′ 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki ′′ )) 22 -- -- -- (( 1111 ))

γγ xyxy ′′ ′′ == nno ΣΣ ii == 11 nno ythe y cici ′′ ′′ ythe y kithe ki ′′ ′′ -- ΣΣ ii == 11 nno ythe y cici ′′ ′′ ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ nno ΣΣ ii == 11 nno ythe y cici ′′ ′′ 22 -- (( ΣΣ ii == 11 nno ythe y cici ′′ ′′ )) 22 ·&Center Dot; nno ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ )) 22 -- -- -- (( 1212 ))

其中，y′ki表示理想的弧面凸轮运动曲线k一阶导数上对应xi的值，y′ci表示拟合的实际运动曲线一阶导数上对应xi的值，y″ci表示拟合的实际运动曲线二阶导数上对应xi的值，y″ki表示理想的弧面凸轮运动曲线k二阶导数上对应xi的值，n表示采集点个数。Among them, y′ ki represents the value corresponding to xi on the first derivative of the ideal arc cam motion curve k, y′ ci represents the value corresponding to xi on the first derivative of the fitted actual motion curve, and y″ ci represents the fitting The value corresponding to x i on the second order derivative of the actual motion curve, y " ki represents the value corresponding to x i on the second order derivative of the ideal arc cam motion curve k, and n represents the number of collection points.

本发明与现有技术相比，其优点在于：Compared with the prior art, the present invention has the advantages of:

1)本发明为弧面凸轮运动曲线辨识提供了完整的参考解决方案以及清晰的控制流程。1) The present invention provides a complete reference solution and a clear control process for the identification of arcuate cam motion curves.

2)本发明提出了一种基于多阶相关分析的弧面凸轮运动曲线辨识方法，并首次将其应用于弧面凸轮运动曲线辨识。弧面凸轮实际运动曲线辨识包括：①弧面凸轮实际运动曲线的数据采集；②弧面凸轮实际运动曲线数据的相关分析；③弧面凸轮实际运动曲线在一阶导数、二阶导数和更高阶导数的相关分析。2) The present invention proposes a cam motion curve identification method based on multi-order correlation analysis, and applies it to cam motion curve identification for the first time. The identification of the actual motion curve of the arc-shaped cam includes: ① data collection of the actual motion curve of the arc-shaped cam; ② correlation analysis of the actual motion curve data of the arc-shaped cam; Correlation Analysis of Order Derivatives.

3)本发明不仅有利于提高机械行业中机械运动机构的位置精度，而且有利于提高机械运动机构的速度和加速度精度，进而提高机械运动的平稳性。3) The present invention not only helps to improve the positional accuracy of the mechanical motion mechanism in the machinery industry, but also helps to improve the speed and acceleration accuracy of the mechanical motion mechanism, thereby improving the stability of the mechanical motion.

附图说明Description of drawings

图1是修正正弦运动规律曲线；其中，（a）为修正正弦运动规律的位移曲线，（b）为修正正弦运动规律的速度曲线，（c）为修正正弦运动规律的加速度曲线，（d）为修正正弦运动规律的加速度变化曲线；Figure 1 is the curve of the modified sinusoidal motion law; among them, (a) is the displacement curve of the modified sinusoidal motion law, (b) is the velocity curve of the modified sinusoidal motion law, (c) is the acceleration curve of the modified sinusoidal motion law, (d) Acceleration change curve for correcting the law of sinusoidal motion;

图2是加速度理论曲线与拟合曲线图；其中，实线为实际加速度曲线，（a）中虚线为修正等速加速度理论曲线，（b）中虚线为修正梯形加速度理论曲线，（c）中虚线为修正正弦加速度理论曲线，（d）中虚线为余弦加速度理论曲线，（e）中虚线为五次多项式加速度理论曲线，（f）中虚线为正弦加速度理论曲线；Figure 2 is a diagram of the theoretical acceleration curve and the fitting curve; among them, the solid line is the actual acceleration curve, the dotted line in (a) is the corrected constant acceleration theoretical curve, the dotted line in (b) is the corrected trapezoidal acceleration theoretical curve, and (c) The dotted line is the theoretical curve of modified sine acceleration, the dotted line in (d) is the theoretical curve of cosine acceleration, the dotted line in (e) is the theoretical curve of quintic polynomial acceleration, and the dotted line in (f) is the theoretical curve of sine acceleration;

图3是位移曲线辨识程序流程框图。Figure 3 is a block diagram of the displacement curve identification program.

具体实施方式detailed description

下面结合附图对本发明做进一步说明：The present invention will be further described below in conjunction with accompanying drawing:

一种基于多阶相关分析的弧面凸轮运动曲线辨识方法，包括三个步骤：弧面凸轮运动曲线的最小二乘拟合、弧面凸轮实际运动曲线数据的相关分析和多阶相关分析。A cam motion curve identification method based on multi-order correlation analysis includes three steps: least square fitting of cam cam motion curve, correlation analysis of actual motion curve data of cam cam and multi-order correlation analysis.

步骤（1）弧面凸轮运动曲线的最小二乘拟合Step (1) Least squares fitting of arc cam motion curve

为了能够识别与辨识弧面凸轮的实际运动规律，进一步提高弧面凸轮的加工精度，通过对弧面凸轮的实际运动曲线进行数据采集，在此基础上，进行最小二乘曲线拟合，得到弧面凸轮的实际运动规律的数学表达。因此，在弧面凸轮运动曲线辨识过程中，当采集到弧面凸轮运动数据后，采用最小二乘法进行曲线拟合，得到实际运动曲线；具体包括以下步骤：In order to be able to identify and identify the actual motion law of the arc cam and further improve the machining accuracy of the arc cam, the data of the actual motion curve of the arc cam is collected, and on this basis, the least squares curve fitting is carried out to obtain the arc cam. The mathematical expression of the actual motion law of the surface cam. Therefore, during the identification process of the arc cam motion curve, after the arc cam motion data is collected, the least square method is used for curve fitting to obtain the actual motion curve; specifically, the following steps are included:

①假设某弧面凸轮实际运动曲线上包含n个采集点，则将该弧面凸轮实际运动曲线表示为C={yt=fc(xt)}(t=1,2,3…,n)；理想的弧面凸轮运动曲线表示为Pk={ykt=fk(xt)}(t=1,2,3…,n；k=1,2,3…,K)，其中：K为弧面凸轮可能出现的理想运动曲线种类总数，有Pk={P1,P2...Pk}种；Pk={ykt=fk(xt)}表示第k种理想运动曲线上的第t个点；① Assuming that the actual motion curve of a certain arc cam contains n collection points, then the actual motion curve of the arc cam is expressed as C={y t =f c (x t )}(t=1,2,3..., n); the ideal arc cam motion curve is expressed as P k ={y kt =f k (x t )}(t=1,2,3...,n; k=1,2,3...,K), Among them: K is the total number of possible ideal motion curves of arc cam, there are P k ={P 1 ,P 2 ...P k } types; P k ={y kt =f k (x t )} means the first The tth point on k ideal motion curves;

②定义实际运动曲线的拟合多项式为：②The fitting polynomial defining the actual motion curve is:

fC(x)=α0g0(x)+α1g1(x)+α2g2(x)+...αmgm(x)(1)f C (x)=α 0 g 0 (x)+α 1 g 1 (x)+α 2 g 2 (x)+...α m g m (x)(1)

实际中，为了计算简单，一般定义：gj(x)=xj为常用幂函数。j=0,1,2,…,m；m表示拟合多项式的阶数，阶数越高，精度越高，对于机械运动来讲，经常用5～10。In practice, for the sake of simple calculation, the general definition: g j (x)=x j is a common power function. j=0,1,2,...,m; m represents the order of the fitting polynomial, the higher the order, the higher the precision, for mechanical movement, 5~10 is often used.

③由实际运动曲线采集点和拟合多项式计算误差平方和：③Calculate the sum of squared errors from the actual motion curve collection points and the fitting polynomial:

SS CC == ΣΣ tt == 11 nno (( ythe y tt -- ff cc (( xx tt )) )) 22 →&Right Arrow; minmin -- -- -- (( 22 ))

该误差平方和代表了实际运动曲线和拟合多项式的偏离程度。The error sum of squares represents the degree of deviation between the actual motion curve and the fitted polynomial.

④求拟合多项式的偏导并解方程，得到多项式的系数：④ Find the partial derivative of the fitted polynomial and solve the equation to obtain the coefficients of the polynomial:

∂∂ SS CC ∂∂ αα jj == 00 ,, jj == 0,1,2,30,1,2,3 ,, .. .. .. mm -- -- -- (( 33 ))

其中，定义：Among them, define:

AA ijij == ΣΣ tt == 11 nno xx tt ii ++ jj ,, ii == 0,1,20,1,2 ,, .. .. .. ,, mm ;; jj == 0,10,1 ,, .. .. .. mm ;; -- -- -- (( 44 ))

bb jj == ΣΣ tt == 11 nno ythe y tt ×× xx tt jj ,, jj == 0,1,20,1,2 ,, .. .. .. ,, mm -- -- -- (( 55 ))

所以有：So have:

AA 0000 AA 0101 .. .. .. AA 00 mm AA 1010 .. .. .. AA 11 mm .. .. .. .. .. .. .. .. .. .. .. .. AA mm 00 AA mm 11 .. .. .. AA mmmm αα 00 αα 11 .. .. .. αα mm == bb 00 bb 11 .. .. .. bb mm -- -- -- (( 66 ))

所以，通过求解这个方程就可以得到弧面凸轮的实际运动曲线方程，进而可以进行弧面凸轮的实际运动曲线相关分析。Therefore, by solving this equation, the actual motion curve equation of the arc cam can be obtained, and then the correlation analysis of the actual motion curve of the arc cam can be carried out.

步骤（2）相关分析Step (2) correlation analysis

通过计算弧面凸轮的实际运动曲线和理想的运动曲线的相关系数，进行弧面凸轮的实际运动曲线和理想的运动曲线的相关分析，具体包括：By calculating the correlation coefficient between the actual motion curve and the ideal motion curve of the arc cam, the correlation analysis between the actual motion curve and the ideal motion curve of the arc cam, including:

①由实际运动曲线采集点和理想运动曲线点计算误差平方和：①Calculate the sum of squared errors from the actual motion curve collection points and the ideal motion curve points:

SS kk == ΣΣ ii == 11 nno (( ythe y cici -- ythe y kithe ki )) 22 ,, kk == 1,2,31,2,3 .. .. .. ,, KK -- -- -- (( 77 ))

上式中误差平方和是辨别相关运动曲线的依据之一，该误差平方和代表了实际运动曲线和第k种理想运动曲线的偏离程度，可以与归一化的相关系数相配合使用。In the above formula, the error sum of squares is one of the basis for distinguishing the relevant motion curve. The error sum of squares represents the degree of deviation between the actual motion curve and the kth ideal motion curve, and can be used in conjunction with the normalized correlation coefficient.

②由实际运动曲线采集点和理想运动曲线点计算相关系数：② Calculate the correlation coefficient from the actual motion curve collection points and the ideal motion curve points:

γγ xyxy == nno ΣΣ ii == 11 nno ythe y cici ythe y kithe ki -- ΣΣ ii == 11 nno ythe y cici ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki nno ΣΣ ii == 11 nno ythe y cici 22 -- (( ΣΣ ii == 11 nno ythe y cici )) 22 ·&Center Dot; nno ΣΣ ii == 11 nno ythe y kithe ki 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki )) 22 -- -- -- (( 88 ))

该相关系数代表了实际运动曲线和第k种理想运动曲线的相似程度。其中，yci表示拟合的实际运动曲线上对应xi的值，yki表示理想运动曲线k上对应xi的值，n表示采集点个数。The correlation coefficient represents the similarity between the actual motion curve and the kth ideal motion curve. Among them, y ci represents the value corresponding to xi on the fitted actual motion curve, y ki represents the value corresponding to xi on the ideal motion curve k, and n represents the number of collection points.

步骤（3）多阶相关分析Step (3) Multi-order correlation analysis

在弧面凸轮实际运动曲线与理想的运动规律的密切相关基础上，进行弧面凸轮运动规律的一阶导数、二阶导数和更高阶导数的相关分析，得到弧面凸轮实际运动曲线在弧面凸轮速度、加速度等的相关系数及其相似形态类别，完成弧面凸轮实际运动曲线的识别过程，具体包括：On the basis of the close correlation between the actual motion curve of the arc cam and the ideal motion law, the correlation analysis of the first order derivative, the second order derivative and the higher order derivative of the arc cam motion law is carried out, and the actual motion curve of the arc cam is obtained in the arc The correlation coefficients of surface cam speed, acceleration, etc. and their similar shape categories complete the identification process of the actual motion curve of the curved surface cam, including:

①当K种理想运动曲线和实际运动曲线的相关系数γCP≈1，说明实际运动曲线和K种理想运动曲线在位置曲线的相似程度非常高，需要进行一阶导数、二阶导数和更高阶导数的相关分析。①When the correlation coefficient γ CP ≈ 1 between the K ideal motion curves and the actual motion curves, it means that the actual motion curves and the K ideal motion curves have a very high similarity in the position curve, and the first-order derivative, second-order derivative and higher Correlation Analysis of Order Derivatives.

②将实际运动曲线和K种理想运动曲线分别进行求导，计算实际运动曲线和K种理想运动曲线的一阶和二阶的导数误差平方和：②Deriving the actual motion curve and K kinds of ideal motion curves separately, and calculating the sum of squares of the first-order and second-order derivative errors of the actual motion curve and K kinds of ideal motion curves:

SS kk ′′ == ΣΣ ii == 11 nno (( ythe y cici ′′ -- ythe y kithe ki ′′ )) 22 ,, -- -- -- (( 99 ))

SS kk ′′ ′′ == ΣΣ ii == 11 nno (( ythe y cici ′′ ′′ -- ythe y kithe ki ′′ ′′ )) 22 ,, -- -- -- (( 1010 ))

k=1,2,3…,K；k=1,2,3...,K;

上式中误差平方和是辨别相关运动曲线的依据之一，代表实际曲线和理想曲线偏离的程度，与归一化的相关系数相配合使用。对于机械运动机构来讲，该一阶导数的误差平方和代表了实际运动曲线和理想运动曲线的速度偏离程度，二阶导数的误差平方和代表了弧面凸轮的加速度与理想运动的加速度偏离程度。The sum of squared errors in the above formula is one of the basis for distinguishing the relevant motion curve, which represents the degree of deviation between the actual curve and the ideal curve, and is used in conjunction with the normalized correlation coefficient. For the mechanical motion mechanism, the error sum of the squares of the first derivative represents the deviation degree of the speed between the actual motion curve and the ideal motion curve, and the error sum of the squares of the second derivative represents the deviation degree of the acceleration of the arc cam and the acceleration of the ideal motion. .

③由实际运动曲线采集点和理想运动曲线点计算相关系数，如果实际运动速度曲线和理想运动速度曲线的相关系数还不能区分弧面凸轮的实际运动曲线的速度属性，则重复进行二阶求导和计算加速度的相关系数，直到辨识出来弧面凸轮的实际运动规律：③ Calculate the correlation coefficient from the actual motion curve collection points and the ideal motion curve points. If the correlation coefficient between the actual motion speed curve and the ideal motion speed curve cannot distinguish the speed attribute of the actual motion curve of the arc cam, repeat the second-order derivation and calculate the correlation coefficient of the acceleration until the actual law of motion of the arcuate cam is identified:

γγ xyxy ′′ == nno ΣΣ ii == 11 nno ythe y cici ′′ ythe y kithe ki ′′ -- ΣΣ ii == 11 nno ythe y cici ′′ ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki ′′ nno ΣΣ ii == 11 nno ythe y cici ′′ 22 -- (( ΣΣ ii == 11 nno ythe y cici ′′ )) 22 ·· nno ΣΣ ii == 11 nno ythe y kithe ki ′′ 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki ′′ )) 22 -- -- -- (( 1111 ))

γγ xyxy ′′ ′′ == nno ΣΣ ii == 11 nno ythe y cici ′′ ′′ ythe y kithe ki ′′ ′′ -- ΣΣ ii == 11 nno ythe y cici ′′ ′′ ·&Center Dot; ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ nno ΣΣ ii == 11 nno ythe y cici ′′ ′′ 22 -- (( ΣΣ ii == 11 nno ythe y cici ′′ ′′ )) 22 ·· nno ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ 22 -- (( ΣΣ ii == 11 nno ythe y kithe ki ′′ ′′ )) 22 -- -- -- (( 1212 ))

其中，y′ki表示理想运动曲线k一阶导数上对应xi的值，y′ci表示拟合的实际运动曲线一阶导数上对应xi的值，y″ci表示拟合的实际运动曲线二阶导数上对应xi的值，y″ki表示理想运动曲线k二阶导数上对应xi的值。Among them, y' ki represents the value corresponding to xi on the first derivative of the ideal motion curve k, y' ci represents the value corresponding to xi on the first derivative of the fitted actual motion curve, and y″ ci represents the fitted actual motion curve The value corresponding to x i on the second order derivative, y " ki represents the value corresponding to x i on the second order derivative of the ideal motion curve k.

基于多阶相关分析的弧面凸轮运动曲线辨识方法应用实例Application example of arc cam motion curve identification method based on multi-order correlation analysis

常用弧面凸轮运动曲线有：余弦加速度曲线、正弦加速度曲线、五次多项式曲线、修正等速曲线、修正梯形曲线和修正正弦曲线等六种结构形式。其中，如图1所示为修正正弦曲线的位移曲线（0阶）、速度曲线（1阶）、加速度曲线（2阶）和加速度变化曲线（3阶）。Common arc cam motion curves include: cosine acceleration curve, sine acceleration curve, quintic polynomial curve, modified constant velocity curve, modified trapezoidal curve and modified sine curve. Among them, as shown in Figure 1, the displacement curve (0th order), velocity curve (1st order), acceleration curve (2nd order) and acceleration change curve (3rd order) of the modified sinusoidal curve are shown.

这些运动曲线的辨识主要是对弧面凸轮的运动规律进行反求，得到运动规律的数学表达。因此，为了验证本发明的正确性，进行实际弧面凸轮的数据采集，得到一组弧面凸轮的数据。The identification of these motion curves is mainly to obtain the mathematical expression of the motion law by reversing the motion law of the arc cam. Therefore, in order to verify the correctness of the present invention, the actual arc cam data collection is carried out to obtain a set of arc cam data.

1)弧面凸轮运动曲线的最小二乘拟合1) Least square fitting of arc cam motion curve

按照上述步骤（1）进行最小二乘拟合，得到弧面凸轮的实际运动曲线多项式为：According to the above step (1), the least squares fitting is carried out, and the actual motion curve polynomial of the arc cam is obtained as:

p(x)=91.4645×x10-448.3554×x9+934.9087×x8-1.078×103×x7 p(x)=91.4645×x 10 -448.3554×x 9 +934.9087×x 8 -1.078×10 3 ×x 7

+746.8583×x6-312.223×x5+70.0229×x4-5.7668×x3+2.156×x2 +746.8583×x 6 -312.223×x 5 +70.0229×x 4 -5.7668×x 3 +2.156×x 2

-0.1052×x1+0.0011-0.1052×x 1 +0.0011

2)相关分析2) Correlation Analysis

按照上述步骤（2）进行相关分析，得到弧面凸轮的实际运动曲线和理想运动曲线的相关系数如表1所示：Correlation analysis is carried out according to the above step (2), and the correlation coefficient between the actual motion curve and the ideal motion curve of the arc cam is shown in Table 1:

表1matlab程序运行得到的位移曲线六种误差及相关系数Table 1 Six errors and correlation coefficients of the displacement curve obtained by running the matlab program

理论曲线theoretical curve 余弦cosine 正弦sine 五次多项式quintic polynomial 修正等速corrected isokinetic 修正梯形corrected trapezoid 修正正弦Modified sine 位移误差displacement error 2.151×10-7 2.151×10 -7 8.687×10-8 8.687×10 -8 8.1775×10-8 8.1775×10 -8 5.9926×10-6 5.9926×10 -6 3.0269×10-6 3.0269×10 -6 4.8481×10-9 4.8481×10 -9 相关系数correlation coefficient 11 11 11 11 11 11

3)多阶相关分析3) Multi-order correlation analysis

以上表1所示，由于弧面凸轮的实际运动曲线和理想运动曲线的相关系数都等于1，说明这个弧面凸轮运动曲线和六种理想运动曲线都强相关，因此，需要进一步进行多阶的相关分析，按照上述步骤（3）进行多阶相关分析，得到弧面凸轮的实际运动曲线和理想运动曲线的二阶导数的相关系数如表2所示：As shown in Table 1 above, since the correlation coefficients between the actual motion curve and the ideal motion curve of the arc cam are equal to 1, it shows that the motion curve of the arc cam is strongly correlated with the six ideal motion curves. Therefore, further multi-level research is required. For correlation analysis, perform multi-order correlation analysis according to the above step (3), and obtain the correlation coefficient of the second-order derivative of the actual motion curve and the ideal motion curve of the arc cam, as shown in Table 2:

表2matlab程序运行得到的加速度曲线六种误差及相关系数Table 2 Six errors and correlation coefficients of the acceleration curve obtained by running the matlab program

4)应用效果4) Application effect

辨识效果如图2所示，实际凸轮的加速度曲线与六种理论加速度曲线进行比对的结果，其中实线为实际加速度曲线，虚线为各种理论加速度曲线，可以看出，实际加速度曲线只和图2c中加速度曲线相互重叠，其余不重叠。可见，在对弧面凸轮的运动规律进行辨识时，由于各种凸轮的位移规律极其相似，而加速度规律却有很大差别，而用加速度曲线对弧面凸轮运动曲线进行辨识更为直观，因此，经过相关分析和图形显示，实际的弧面凸轮曲线为修正凸轮曲线，其程序流程图如图3所示。The identification effect is shown in Figure 2. The actual acceleration curve of the cam is compared with six theoretical acceleration curves. The solid line is the actual acceleration curve, and the dotted line is various theoretical acceleration curves. It can be seen that the actual acceleration curve is only consistent with The acceleration curves in Figure 2c overlap each other, while the rest do not overlap. It can be seen that when identifying the motion law of the arc cam, since the displacement laws of various cams are very similar, but the acceleration laws are very different, it is more intuitive to use the acceleration curve to identify the motion curve of the arc cam, so , after relevant analysis and graphic display, the actual arc cam curve is a modified cam curve, and its program flow chart is shown in Figure 3.

修正正弦曲线是对余弦曲线的改进而来，它既克服了余弦曲线不连续的缺点，又保留了最大速度和最大加速度较小的优点，该曲线的平衡性极好，在负荷性质不清楚的情况下使用最没有危险，并具有较好的吸振性，可实现平稳运动。修正正弦曲线应用于高速运动以外的重载，具有特别的优越性。The modified sine curve is an improvement of the cosine curve. It not only overcomes the discontinuity of the cosine curve, but also retains the advantages of small maximum speed and maximum acceleration. The balance of the curve is excellent. It is the least dangerous to use under the circumstances, and has good shock absorption, which can realize smooth movement. The modified sinusoidal curve is applied to heavy loads other than high-speed motion, which has special advantages.

针对弧面凸轮运动曲线来讲，现有技术仅仅进行位置运动曲线的拟合，而没有考虑曲线对应的速度和加速度等高阶导数对应的曲线相关分析，针对运动曲线对应的机械运动机构来讲，这样没有考虑速度连续性和平滑性的相关分析结果，会给机械带来运动的冲击。上述应用实例的结果证明了所提出的基于多阶相关分析的弧面凸轮运动曲线辨识方法的准确性，也证明了基于多阶相关分析的弧面凸轮运动曲线辨识方法在弧面凸轮加工过程中实施的可行性和必要性。For the arc cam motion curve, the existing technology only fits the position motion curve, but does not consider the curve correlation analysis corresponding to the high-order derivatives such as velocity and acceleration corresponding to the curve. For the mechanical motion mechanism corresponding to the motion curve , which does not consider the relevant analysis results of velocity continuity and smoothness, which will bring movement impact to the machine. The results of the above application examples prove the accuracy of the proposed arc cam motion curve identification method based on multi-order correlation analysis, and also prove that the arc cam motion curve identification method based on multi-order correlation Feasibility and necessity of implementation.

Claims (4)

1., based on a globoid cam curve movement discrimination method for multistage correlation analysis, it is characterized in that, this discrimination method comprises the following steps:
(1) least square fitting of globoid cam curve movement: to the globoid cam exercise data collected, adopts least square method to carry out curve fitting, obtains the actual motion curve of globoid cam;
(2) correlation analysis: according to actual motion curve and the desirable globoid cam curve movement of globoid cam, the correlation analysis carrying out the globoid cam characteristics of motion obtains related coefficient;
Actual motion curve and the desirable globoid cam curve movement of globoid cam carry out correlation analysis, comprise following idiographic flow:
Related coefficient γ is calculated by actual motion curve collection point and ideal movements curve point xy:
γ x y = n Σ i = 1 n y c i y k i - Σ i = 1 n y c i · Σ i = 1 n y k i n Σ i = 1 n y c i 2 - ( Σ i = 1 n y c i ) 2 · n Σ i = 1 n y k i 2 - ( Σ i = 1 n y k i ) 2 - - - ( 8 )
Wherein, y cirepresent corresponding x on actual motion curve ivalue, y kirepresent corresponding x on desirable globoid cam curve movement k ivalue, n represents collection point number;
As the related coefficient γ of the actual motion curve of the desirable globoid cam curve movement of K kind and globoid cam xy≈ 1, illustrate the actual motion curve of globoid cam and the desirable globoid cam curve movement of K kind very high at the similarity degree of position curve, need to carry out first order derivative or second derivative correlation analysis;
Related coefficient is calculated, if the related coefficient γ ' of actual motion rate curve and ideal movements rate curve by actual motion curve collection point and ideal movements curve point xythe Speed attribute of the actual motion curve of globoid cam can't be distinguished, then carry out the related coefficient γ of second order differentiate and calculating acceleration " xy, until the actual movement rule of identification out globoid cam:
γ x y ′ = n Σ i = 1 n y c i ′ y k i ′ - Σ i = 1 n y c i ′ · Σ i = 1 n y k i ′ n Σ i = 1 n y c i ′ 2 - ( Σ i = 1 n y c i ′ ) 2 · n Σ i = 1 n y k i ′ 2 - ( Σ i = 1 n y k i ′ ) 2 - - - ( 11 )
γ x y ′ ′ = n Σ i = 1 n y c i ′ ′ y k i ′ ′ - Σ i = 1 n y c i ′ ′ · Σ i = 1 n y k i ′ ′ n Σ i = 1 n y c i ′ ′ 2 - ( Σ i = 1 n y c i ′ ′ ) 2 · n Σ i = 1 n y k i ′ ′ 2 - ( Σ i = 1 n y k i ′ ′ ) 2 - - - ( 12 )
Wherein, y ' kirepresent corresponding x in desirable globoid cam curve movement k first order derivative ivalue, y ' cirepresent corresponding x in actual motion curve first order derivative ivalue, y " cirepresent corresponding x in actual motion curve second derivative ivalue, y " kirepresent corresponding x in desirable globoid cam curve movement k second derivative ivalue, n represents collection point number;
(3) identifying of the actual motion curve of globoid cam is completed according to the difference of related coefficient.
2. a kind of globoid cam curve movement discrimination method based on multistage correlation analysis according to claim 1, is characterized in that: described step (1) comprises following idiographic flow:
1. the actual motion curve supposing certain globoid cam comprises n collection point, then the actual motion curve of this globoid cam is expressed as C={y t=f c(x t), t=1,2,3 ..., n; Desirable globoid cam curve movement is expressed as P k={ y kt=f k(x t), t=1,2,3 ..., n; K=1,2,3 ..., K, K are desirable globoid cam curve movement kind sum;
2. the polynomial fitting defining actual motion curve is:
f C(x)＝α 0g 0(x)+α 1g 1(x)+α 2g 2(x)+...α mg m(x)(1)
In formula (1), g j(x)=x j, j=0,1,2 ..., m; M represents the exponent number of polynomial fitting;
3. by the actual motion curve collection point of globoid cam and the polynomial fitting error of calculation quadratic sum of actual motion curve:
S C = Σ t = 1 n ( y t - f c ( x t ) ) 2 → m i n - - - ( 2 )
4. then local derviation is asked:
∂ S C ∂ α j = 0 , j = 0 , 1 , 2 , 3 , ... m - - - ( 3 )
Definition:
A i j = Σ t = 1 n x t i + j , i = 0 , 1 , 2 , ... , m ; j = 0 , 1 , ... m ; - - - ( 4 )
b j = Σ t = 1 n y t × x t j , j = 0 , 1 , 2 , ... , m - - - ( 5 )
So have:
A 00 A 01 ... A 0 m A 10 ... A 1 m ... ... ... ... A m 0 A m 1 ... A m m α 0 α 1 ... α m = b 0 b 1 ... b m - - - ( 6 )
The factor alpha of polynomial fitting is obtained by solution formula (6) j, by factor alpha jcorrespondence brings the actual motion curvilinear equation that polynomial fitting obtains globoid cam into.
3. a kind of globoid cam curve movement discrimination method based on multistage correlation analysis according to claim 1, it is characterized in that: described multistage correlation analysis comprises the actual motion curve of globoid cam and the desirable first order derivative of globoid cam curve movement or the correlation analysis of second derivative, and described related coefficient is the related coefficient of the globoid cam speed corresponding with correlation analysis or the related coefficient of globoid cam acceleration.
4. a kind of globoid cam curve movement discrimination method based on multistage correlation analysis according to claim 1, it is characterized in that: described desirable globoid cam curve movement comprises cosine curve, sinusoidal curve, quintic algebra curve curve, revises curve of equal velocity, modified trapezoid curve and modified sinusoid, and described related coefficient is the related coefficient of globoid cam acceleration.
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