CN101672637B - Digitizing detection method of complicated curved face - Google Patents

Abstract

Translated fromChinese

本发明涉及一种复杂曲面的数字化检测方法，该方法包括以下步骤：(1)提取复杂曲面CAD模型的几何数据，建立复杂曲面的NURBS理论模型，再利用三坐标测量机测量曲面零件实物，将零件表面数字化，得到测量点参数；(2)通过在CAD曲面上搜索与测量点最近的两组等参数线构造等参数线区域，建立测量点到曲面等参数线区域最小距离的二元非线性数学模型；(3)运用遗传算法来计算复杂曲面形状误差，实现复杂曲面的误差评定。与现有技术相比，本发明建立了精确的曲面理论模型，并且提高了搜索效率、能够确保获得全局最优解并减轻测量强度。

The invention relates to a digital detection method for complex curved surfaces. The method comprises the following steps: (1) extracting the geometric data of a CAD model of the complex curved surface, establishing a NURBS theoretical model of the complex curved surface, and then measuring the actual object of the curved surface parts by using a three-dimensional coordinate measuring machine. Digitize the surface of the part to obtain the parameters of the measuring point; (2) Construct the isoparametric line area by searching two sets of isoparametric lines closest to the measuring point on the CAD surface, and establish the binary nonlinearity of the minimum distance from the measuring point to the surface isoparametric line area Mathematical model; (3) Use genetic algorithm to calculate the shape error of complex surface, and realize the error evaluation of complex surface. Compared with the prior art, the present invention establishes an accurate surface theory model, improves search efficiency, can ensure global optimal solution and reduces measurement intensity.

Description

A kind of complex-curved digitizing detection method

Technical field

The present invention relates to Machine Design and detection technique field, especially relate to a kind of complex-curved digitizing detection method.

Background technology

Increasingly mature along with the develop rapidly of modern manufacturing industry and the design of CAD/CAM mean camber and process technology; The demand of complex curved surface parts such as hi-tech, high performance mechanized equipment such as thruster, screw pump, compressor, extruder, steam turbine, fan blower is increasing; They have in a plurality of essential industrys field of national economy such as military affairs, aviation, the energy, machinery, chemical industry, new material widely uses, and its manufacturing accuracy directly affects the serviceability and the quality of equipment.

Complex-curved shape error digitizing detects and is meant the measuring equipment measuring complex curved surface part that utilizes the advanced person, obtains curved surface part surface discrete data point, this data point is compared with designing a model of part, thereby detect mismachining tolerance.Digitizing detect provide an analysis to design a model and its final products between the method for the margin of error, can find whether each characteristic of part and size have the deviation flaw with respect to designing a model.Before part puts into production, the batch part is done initial workpiece detect, can check whether product has carried out the preparation of putting into production, in detection information, revise the defective that exists the processing.Because utilized the data of precision measurement and desirable profile to compare, can realize high precision, high efficiency detection, so complex-curved digitizing detection is had the important engineering meaning.

At present complex-curved digitizing is detected and remain traditional detection and modern detection method and deposit, the problem of existence mainly shows as:

(1) traditional complex-curved detection is to adopt model and actual curved surface to carry out the method for artificial comparison; This method need be made many models in kind; The quality that detects to a certain degree depends on individual knowledge level and experience; Labour intensity is big, detection time is long, accuracy of detection is low, is difficult to carry out information interchange with automatic control system and quality control system; In addition, existing dedicated test utensil can only detect certain some position of a certain product or product, does not have versatility.

(2) because the complicacy of complex-curved theoretical model; The research of a large amount of shape error evaluations concentrates in the research of ruled surface; To measuring the computational problem of distance from point to curved surface in the complex-curved shape error evaluation; The method that adopts the plane to approach is mostly carried out approximate processing, and calculated amount is big and precision is not high.

(3) in the existing wrap-around error computing method, the mathematical model that has does not meet minimal condition; Although what have meets minimal condition, in computation process, there is the approximate error of model, influence computational accuracy; The distance that directly will measure the point to curved surface discrete point that has is as wrap-around error, and do not calculate the normal direction distance of point to curved surface, and the high-precision requirement of this and Error Calculation has big gap.

What (4) have adopts STL to represent cad model; The directed distance of putting triangle grid model with original measurement is that spatial point is represented shape error to the triangular plate distance; And stl file is the data file of being come the approximate representation cad model by a series of triangular nets; For the steep excessively model of changes in surface, approach poor, error is very big.

(5) the employing measurement point that has is found the solution to the gradient method of B-spline surface distance, and gradient method and aforesaid method of steepest descent solving result are absorbed in the locally optimal solution in the primary iteration vertex neighborhood easily.

(6) great majority research at present is curved surface to be dispersed change into little grid plan, calculates measurement point then and evaluates complex-curved error to the minor increment of all little grid surfaces.Adopting the triangular plate of STL form to approach curved surface profile, also is present more common method to nearest triangular plate distance as wrap-around error according to measurement point, and the problem that these methods exist is that computational accuracy is not high enough.

(7) have with the distance of contour design point and corresponding measurement point as the error evaluation amount, but in actual measurement, can not guarantee measurement data points just and the design point that the CAD curved surface provides overlap, can not guarantee precision.

Because the shape error mathematical model that complex-curved shape error calculates by the Minimum Area method is very complicated, the utilization conventional calculation is difficult to direct calculating, usually adopts certain approximation method such as least square method to carry out indirect calculation.

Complex-curved digitizing detects problem because the complicacy of its shape error mathematical model is used conventional calculation to be difficult to directly find the solution, and adopted approximate method to calculate, and the resulting conclusion of different measuring methods is inconsistent.

Summary of the invention

The object of the invention be exactly provide in order to overcome the defective that above-mentioned prior art exists a kind of model accurately, search the complex-curved digitizing detection method that plain efficient is high, detection accuracy is good.

The object of the invention can be realized through following technical scheme:

A kind of complex-curved digitizing detection method is characterized in that, this method may further comprise the steps:

(1) geometric data of the complex-curved cad model of extraction is set up complex-curved NURBS theoretical model, utilizes three coordinate measuring machine that part curved surface in kind is carried out three-dimensional measurement again, with the piece surface digitizing, obtains the measurement point parameter;

(2) through parameter lines such as search on the CAD curved surface and measurement points nearest two groups, the binary nonlinear mathematical model of the regional minor increments of parameter line such as measurement point to curved surface is set up in zone such as the structure parameter line of etc.ing;

(3) the utilization genetic algorithm is come calculation of complex curve form error, realizes complex-curved error evaluation.

In the described step (1) during the NURBS theoretical model of complex structure curved surface,, calculate the data point on the nurbs surface of u, v parameter correspondence according to U and the V knot vector on two parametric directions of cad model.

When setting up the binary nonlinear mathematical model in the described step (2); To each measurement point, select 2 nearest points in curved surface u parameter line direction, the point that on two v parameter lines at these 2 places, selects 2 nearest u parameters to equate then; Wait the parameter line zone by these of four some structures; The intersection point of surface normal and curved surface that guaranteed measurement point to four somes orderings, is found out less u in four points in these four determined rectangular areas of point_a, v_aParameter value and bigger u_b, v_bParameter value.

When using genetic algorithm in the described step (3), at variable range [u_a, u_b] and [v_a, v_b] interior search and the point of measurement point apart from minimum, the parameter value of this point is u^*, v^*, measurement point is the mismachining tolerance of part curve form to the distance of this point.

Compared with prior art, the present invention has the following advantages:

(1) accurately sets up curved face theoretical model: the theoretical model of CAD curved surface that utilized the NURBS construction of function; Simultaneously in the feasible zone of its knot vector parameter; Utilize u, v parameter as optimization variable, set up and calculated the two yuan nonlinear mathematical models of measurement point to CAD curved surface minor increment.

(2) improve search efficiency: propose the method for solving that division waits the parameter line zone; At first seek parameter line zones such as near the CAD curved surface u of measurement point, v; Through parameter line zones such as the nearest u of structure on the CAD curved surface and measurement point, v, as the hunting zone, simultaneously; Propose parameter line regional extents such as the segmentation node parameter dwindles, improved search efficiency.

(3) guarantee to obtain globally optimal solution: the utilization genetic algorithm is calculated the minor increment of measurement point to the CAD curved surface, and this method ability global optimizing only need be utilized target function value, need not derivative operation, has simplified computation process greatly.Can overcome the deficiency of traditional numerical optimization, this method detection accuracy is good, precision is high, strong robustness, detects for complex-curved digitizing to have important use value.

(4) alleviate measured intensity: native system has alleviated measuring operation personnel's labour intensity, and the analysis to measure result has improved the measurement accuracy automatically.

Description of drawings

Fig. 1 is the process flow diagram of this method;

Fig. 2 is near the area schematic such as parameter line such as grade the measurement point.

1 is measurement point among the figure, and 2 is apart from the nearest point of measurement point on the curved surface.

Embodiment

Below in conjunction with accompanying drawing and specific embodiment the present invention is elaborated.

Embodiment

A kind of complex-curved digitizing detection method, its flow process is as shown in Figure 1, and this method may further comprise the steps:

(1) geometric data of the complex-curved cad model of extraction is set up complex-curved NURBS theoretical model, utilizes the three-dimensional measurement curved surface part in kind again, with the piece surface digitizing, obtains the measurement point parameter;

(2), set up the binary nonlinear mathematical model of zone such as the parameter line of measuring point to curved surface minor increments through parameter lines such as parameter line structure zones such as search on the CAD curved surface and measurement points nearest two groups;

(3) the utilization genetic algorithm is come calculation of complex curve form error, realizes complex-curved error evaluation.

Adopt the NURBS method to describe the theoretical model of CAD curved surface, two knot vector U of nurbs surface and V have the unit square field of definition 0≤u of standard usually, v≤1, and this field of definition is divided into (m-k+1) * (n-l+1) individual sub-rectangle by its internal segment dotted line.Nurbs surface is the reasonable parametric polynomial curved surface of a kind of burst of special shape, and wherein each subsurface sheet is defined in that certain has on the sub-rectangular domain of non-zero area in the unit square.

The step of calculation of complex curve form error is generally speaking:

(1) calculates the minor increment of all measurement points to the CAD curved surface

{dst_i|i＝0，1，...，n}

(2) calculate the maximal value of each minor increment

dst_max＝max{dst_i|i＝0，1，...，n}

(3) peaked 2 times of Minimum Areas that promptly contain the desirable profile equidistant surface of whole measuring points, this is the shape error of curved surface.

f＝2×dst_max (1)

Can know that by complex-curved shape error calculation procedure the key of complex-curved shape error evaluation is to calculate the distance of measurement point to the CAD curved surface.A measurement point m_i(x_i, y_i, z_i) (i=0,1 ..., n) on nurbs surface, must find a nearest some p_i^*(p_x(u_i^*, v_i^*), p_y(u_i^*, v_i^*), p_z(u_i^*, v_i^*)), this point was the surface normal of measurement point and the intersection point of curved surface.Therefore, it is exactly to each measurement point that complex-curved shape error calculates, one group of u of search on the CAD curved surface^*, v^*Value, make point on their pairing curved surfaces to the distance of corresponding measurement point for minimum, this is that measurement point arrives complex-curved distance.In view of the above, the objective function of measurement distance from point to curved surface is:

$d_i(u,v)=\min \left(\sqrt{{(x_i-p_x(u,v))}^2+{(y_i-p_y(u,v))}^2+{(z_i-p_z(u,v))}^2}\right)---\left(2\right)$

I=0 wherein, 1 ..., n, n are measure dot number.

The key of calculation of complex curve form error is certain measurement point of computer memory to the minor increment of its CAD curved surface, promptly will on the CAD curved surface, search for a bit, make its and this measurement point apart from minimum.

For the parameter value among CAD curved surface knot vector U and the V; Calculate each parameter corresponding to the data point coordinate on the curved surface; Certain measurement point is compared with these data points, seeks out four the net surfaces lattice points nearest with measurement point, and guarantee these four net points u lines such as two groups with etc. on the v line; Make them corresponding to two groups of parameter values, i.e. [u among u and the v_a, u_b] and [v_a, v_b]; These two groups of parameter determining one sub regions of curved surface; Native system such as is referred to as at the parameter line zone, must be positioned at this zone, search of utilization genetic algorithm and the closest approach [u of measurement point apart from minimum in these parameter line regional extents then with point on the nearest curved surface of measurement point^*, v^*]; The distance of the two is promptly measured the distance of point to curved surface; As shown in Figure 2, wait the parameter line zone for the dash area among the figure, wait the parameter line dividing region to dwindle the hunting zone of genetic algorithm; Help improving computing velocity, be based on the calculation procedure that waits the regional measurement point of parameter line to CAD curved surface minor increment:

(1) reads in the geological information of cad model curved surface, construct its nurbs surface theoretical model,, calculate the data point on the corresponding nurbs surface of u, v parameter according to U and the V knot vector on its two parametric directions;

(2) to each measurement point; At 2 nearest points of curved surface u parameter line direction finding; On two v parameter lines at these 2 places, seek the point that 2 nearest u parameters equate then; Wait the parameter line zone by this of four somes structure, the intersection point of surface normal and curved surface that guaranteed measurement point is in these 4 determined rectangular areas of point;

(3) to 4 some orderings, find out less u in 4 points_a, v_aParameter value and bigger u_b, v_bParameter value;

(4) use genetic algorithm, at variable range [u_a, u_b] and [v_a, v_b] the interior searching and the minimum point 2 of measurement point 1 distance, its parameter value is u^*, v^*

Measurement point is the mismachining tolerance of part curve form to the distance of this point.

Claims (1)

Translated fromChinese

1.一种复杂曲面的数字化检测方法，其特征在于，该方法包括以下步骤：1. A digital detection method for complex curved surfaces, characterized in that the method may further comprise the steps:(1)提取复杂曲面CAD模型的几何数据，建立复杂曲面的NURBS理论模型，再利用三坐标测量机对零件实物曲面进行三维测量，将零件表面数字化，得到测量点参数；(1) Extract the geometric data of the CAD model of the complex surface, establish the NURBS theoretical model of the complex surface, and then use the three-dimensional coordinate measuring machine to perform three-dimensional measurement on the actual surface of the part, digitize the surface of the part, and obtain the measurement point parameters;(2)通过在CAD曲面上搜索与测量点最近的两组等参数线，构造等参数线区域，建立测量点到曲面等参数线区域最小距离的二元非线性数学模型；(2) By searching two groups of isoparametric lines closest to the measuring point on the CAD surface, constructing the isoparametric line area, and establishing the binary nonlinear mathematical model of the minimum distance from the measuring point to the surface isoparametric line area;(3)运用遗传算法来计算复杂曲面形状误差，实现复杂曲面的误差评定；(3) Use the genetic algorithm to calculate the shape error of the complex surface, and realize the error evaluation of the complex surface;所述的步骤(1)中构造复杂曲面的NURBS理论模型时，根据CAD模型两个参数方向上的U和V节点矢量，计算u、v参数对应的NURBS曲面上的型值点；所述的步骤(2)中建立二元非线性数学模型时，对每个测量点，在曲面u参数线方向选最近的2个点，然后在这两点所在的两条v参数线上选最近的2个u参数相等的点，由这四个点构造一个等参数线区域，保证过测量点的曲面法线与曲面的交点在这四个点所确定的矩形区域内，对四个点排序，找出四个点中较小的u_a、v_a参数值和较大的u_b、v_b参数值；所述的步骤(3)中使用遗传算法时，在变量范围[u_a，u_b]和[v_a，v_b]内搜索与测量点距离最小的点，该点的参数值为u^*、v^*，测量点到该点的距离即零件曲面形状的加工误差；When constructing the NURBS theoretical model of complex surface in the described step (1), according to the U and V node vectors on the two parameter directions of the CAD model, calculate the type value point on the NURBS surface corresponding to the u and v parameters; When establishing a binary nonlinear mathematical model in step (2), for each measurement point, select the nearest 2 points in the direction of the surface u parameter line, and then select the nearest 2 points on the two v parameter lines where these two points are located. A point with the same u parameter, construct an isoparametric line area from these four points, ensure that the intersection point between the surface normal and the surface at the measurement point is in the rectangular area determined by these four points, sort the four points, and find Out of the four points, the smaller u_a , v_a parameter values and the larger u_b , v_b parameter values; when using the genetic algorithm in the step (3), in the variable range [u_a , u_b ] and [v_a , v_b ] search for the point with the smallest distance to the measurement point, the parameter values of this point are u^* , v^* , the distance from the measurement point to this point is the machining error of the surface shape of the part;采用NURBS方法描述CAD曲面的理论模型，NURBS曲面的两个节点矢量U和V通常有规范的单位正方形定义域0≤u，v≤1，该定义域被其内节点线划分成(m-k+1)×(n-l+1)个子矩形，NURBS曲面是一种特殊形式的分片有理参数多项式曲面，其中每一子曲面片定义在单位正方形中某个具有非零面积的子矩形域上；Using the NURBS method to describe the theoretical model of the CAD surface, the two node vectors U and V of the NURBS surface usually have a canonical unit square definition domain 0≤u, v≤1, which is divided into (m-k +1)×(n-l+1) sub-rectangles, NURBS surface is a special form of piecewise rational parametric polynomial surface, where each sub-surface patch is defined in a sub-rectangular domain with non-zero area in the unit square superior;一般情况下计算复杂曲面形状误差的步骤为：In general, the steps to calculate the shape error of complex surfaces are:(1)计算所有测量点到CAD曲面的最小距离(1) Calculate the minimum distance from all measurement points to the CAD surface{dst_i|i＝0，1，...，n}{dst_i |i=0,1,...,n}(2)计算各个最小距离的最大值(2) Calculate the maximum value of each minimum distancedst_max＝max{dst_i|i＝0，1，...，n}dst_max = max{dst_i |i=0,1,...,n}(3)最大值的2倍即包容全部测点的理想轮廓等距面的最小区域，此即曲面的形状误差；(3) Twice the maximum value is the minimum area of the ideal contour equidistant surface containing all measuring points, which is the shape error of the curved surface;f＝2×dst_max            (1)f=2×dst_max (1)由复杂曲面形状误差计算步骤可知，复杂曲面形状误差评定的关键是计算测量点到CAD曲面的距离，一个测量点m_i(x_i，y_i，z_i)(i＝0，1，...，n)在NURBS曲面上必定能找到一个距离最近的点此点即为过测量点的曲面法线与曲面的交点，因此，复杂曲面的形状误差计算就是针对每一个测量点，在CAD曲面上搜索一组u^*、v^*值，使它们所对应的曲面上的点到相应测量点的距离为最小，此即测量点到复杂曲面的距离，据此，测量点到曲面距离的目标函数为：From the calculation steps_of complex surface shape error, it can_be seen that the key to_{theevaluation} of complex surface shape error is to calculate the distance from the measurement point to the CAD surface. ., n) must be able to find a nearest point on the NURBS surface This point is the intersection point of the surface normal and the surface passing through the measurement point. Therefore, the calculation of the shape error of the complex surface is to search for a set of u^* and v^* values on the CAD surface for each measurement point, so that their corresponding The distance from the point on the surface to the corresponding measurement point is the minimum, which is the distance from the measurement point to the complex surface. Accordingly, the objective function of the distance from the measurement point to the surface is:${\mathrm{<span><span>d</span>d</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>(</span>(</span>\mathrm{<span><span>u</span>u</span>}<span><span>,</span>,</span>\mathrm{<span><span>v</span>v</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>min</span>min</span>}<span><span>(</span>(</span>\sqrt{{<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>x</span>x</span>}}<span><span>(</span>(</span>\mathrm{<span><span>u</span>u</span>}<span><span>,</span>,</span>\mathrm{<span><span>v</span>v</span>}<span><span>)</span>)</span><span><span>)</span>)</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{<span><span>(</span>(</span>{\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>y</span>the\; y</span>}}<span><span>(</span>(</span>\mathrm{<span><span>u</span>u</span>}<span><span>,</span>,</span>\mathrm{<span><span>v</span>v</span>}<span><span>)</span>)</span><span><span>)</span>)</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span>{<span><span>(</span>(</span>{\mathrm{<span><span>z</span>z</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>-</span>-</span>{\mathrm{<span><span>p</span>p</span>}}_{\mathrm{<span><span>z</span>z</span>}}<span><span>(</span>(</span>\mathrm{<span><span>u</span>u</span>}<span><span>,</span>,</span>\mathrm{<span><span>v</span>v</span>}<span><span>)</span>)</span><span><span>)</span>)</span>}^{\mathrm{<span><span>2</span>2</span>}}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span>$其中i＝0，1，...，n，n为测量点数；Wherein i=0, 1,..., n, n is the number of measurement points;计算复杂曲面形状误差的关键是计算空间某个测量点到其CAD曲面的最小距离，即要在CAD曲面上搜索一点，使其与该测量点的距离最小；The key to calculating the shape error of complex surfaces is to calculate the minimum distance from a measurement point in space to its CAD surface, that is, to search for a point on the CAD surface to minimize the distance from the measurement point;对于CAD曲面节点矢量U和V中的参数值，计算出各参数对应于曲面上的型值点坐标，将某个测量点与这些型值点相比较，寻找出与测量点最近的四个曲面网格点，并确保这四个网格点在两组等u线和等v线上，使它们对应于u和v中的两组参数值，即[u_a，u_b]和[v_a，v_b]，这两组参数决定了曲面的一个子区域，将其称为等参数线区域，与测量点距离最近的曲面上的点必定位于该区域内，然后在此等参数线区域范围内运用遗传算法搜索与测量点距离最小的最近点[u^*，v^*]，二者的距离即测量点到曲面的距离，等参数线区域的划分缩小了遗传算法的搜索范围，有利于提高计算速度，基于等参数线区域的测量点到CAD曲面最小距离的计算步骤为：For the parameter values in the CAD surface node vectors U and V, calculate the coordinates of each parameter corresponding to the value point on the surface, compare a certain measurement point with these value points, and find the four surfaces closest to the measurement point grid points, and make sure that these four grid points are on two sets of equal u lines and equal v lines, so that they correspond to two sets of parameter values in u and v, namely [u_a , u_b ] and [v_a , v_b ], these two groups of parameters determine a subarea of the surface, which is called the isoparametric line area, the point on the surface closest to the measurement point must be located in this area, and then the isoparametric line area range The genetic algorithm is used to search for the closest point [u^* , v^* ] with the smallest distance to the measurement point. The distance between the two is the distance from the measurement point to the surface. The division of the isoparametric line area reduces the search range of the genetic algorithm, which is beneficial to improve Calculate the speed, the calculation steps of the minimum distance from the measurement point to the CAD surface based on the isoparametric line area are:(1)读入CAD模型曲面的几何信息，构造其NURBS曲面理论模型，据其两个参数方向上的U和V节点矢量，计算u、v参数对应的NURBS曲面上的型值点；(1) Read in the geometric information of the CAD model surface, construct its NURBS surface theoretical model, and calculate the value points on the NURBS surface corresponding to the u and v parameters according to the U and V node vectors in its two parameter directions;(2)对每个测量点，在曲面u参数线方向寻找最近的2个点，然后在这两点所在的两条v参数线上寻找最近的2个u参数相等的点，由这四个点构造一个等参数线区域，保证过测量点的曲面法线与曲面的交点在这4个点所确定的矩形区域内；(2) For each measurement point, find the nearest 2 points in the direction of the surface u parameter line, and then find the nearest 2 points with equal u parameters on the two v parameter lines where these two points are located. Points to construct an isoparametric line area to ensure that the intersection of the surface normal and the surface through the measurement point is within the rectangular area determined by these 4 points;(3)对4个点排序，找出4个点中较小的u_a、v_a参数值和较大的u_b、v_b参数值；(3) Sort the 4 points, and find out the smaller u_a , v_a parameter values and the larger u_b , v_b parameter values among the 4 points;(4)应用遗传算法，在变量范围[u_a，u_b]和[v_a，v_b]内寻找与测量点距离最小的点，其参数值为u^*、v^*；(4) Apply genetic algorithm to find the point with the smallest distance to the measurement point within the variable range [u_a , u_b ] and [v_a , v_b ], and its parameter values are u^* , v^* ;测量点到该点的距离即零件曲面形状的加工误差。The distance from the measurement point to this point is the machining error of the surface shape of the part.

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