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毕业设计（论文）任务书 机电工程学院 学院 机械设计及其自动化 专业设计（论文）题目 双立柱式巷道堆垛机的设计 学 生 姓 名 朱云香 班 级 起 止 日 期 指 导 教 师 毛瑞卿 教研室主任 张元越 发任务书日期 2008 年 6 月 25 日1.毕业设计的背景： 自动化立体仓库是物流中的重要组成部分，它是在不直接进行人工干预的情况下自动地存储和取出物流的系统。它是现代工业社会发展的高科技产物，对提高生产率、降低成本有着重要意义。近年来，随着企业生产与管理的不断提高，越来越多的企业认识到物流系统的改善与合理性对企业的发展非常重要。堆垛机是自动化立体仓库中最重要的起重堆垛设备，它能够在自动化立体的巷道中来回穿梭运行，将位于巷道口的货物存入货格；或者相反取出货格内的货物运送到巷道口。世界主要工业国家都把着眼点放在开发性能可靠的新产品和采用高新技术上，更加注重实用性和安全性。在堆垛机方面，我们应当看到和世界发达国家的差距，总结经验，找出不足，打破传统思路，推出新的外形和更高性能的堆垛机。2.毕业设计(论文)的内容和要求： 内容:堆垛机是整个自动化立体仓库的核心设备,通过手动操作、半自动操作或全自动操作实现把货物从一处搬到另一处。它由机架、水平行走机构、提升机构、载货台、货叉及电器控制系统构成。确定堆垛机的形式、确定堆垛机的速度、其他参数及配置。要求:总装图1张(0#)行走机构 升降机构 货叉的零件图 设计说明书(20000字) 译文5000字3.主要参考文献： 1吉国宏.自动化仓库堆垛机设计M：北京：中国铁道出版社，1979. 2杨长暌.起重机械J：北京：机械工业出版社，1982. 3濮良贵.机械零件J：北京：高等教育出版社，1982. 4刘鸿文.材料力学J:北京：高等教育出版社，1991.5.王丽洁、吴佩年.画法几何及机械制图J:哈尔滨：哈尔滨工业大学出版社,1998.4.毕业设计(论文)进度计划(以周为单位)：起 止 日 期工 作 内 容备 注第1、2周第3、4周第5、6周第7、8周第9、10周第11、12周第13、14周第15、16周老师布置课题,查找相关资料确定设计方案,根据设计方案和设计要求进行计算选择堆垛机的形式、速度,确定参数和配置绘制堆垛机的总装图绘制堆垛机的零件图整理论文资料,撰写论文初稿修改论文初稿完成论文,整理图纸,装订成册教研室审查意见： 室主任 年 月 日学院审查意见： 教学院长 年 月 日附录英文原文The Pre-Processing of Data Points for Curve Fitting in Reverse EngineeringReverse engineering has become an important tool for CAD model construction from the data points, measured by a coordinate measuring machine (CMM), of an existing part. A major problem in reverse engineering is that the measured points having an irregular format and unequal distribution are difficult to fit into a B-spline curve or surface. The paper presents a method for pre-processing data points for curve fitting in reverse engineering. The proposed method has been developed to process the measured data points before fitting into a B-spline form. The format of the new data points regenerated by the proposed method is suitable for the requirements for fitting into a smooth B-spline curve with a good shape. The entire procedure of this method involves filtering, curvature analysis, segmentation, regressing, and regenerating steps. The method is implemented and used for a practical application in reverse engineering. The result of the reconstruction proves the viability of the proposed method for integration with current commercial CAD systems.IntroductionWith the progress in the development of computer hardware and software technology, the concept of computer-aided technology for product development has become more widely accepted by industry. The gap between design and manufacturing is now being gradually narrowed through the development of new CAD technology. In a normal automated manufacturing environment, the operation sequence usually starts from product design via geometric models created in CAD systems, and ends with the generation of machining instructions required to convert raw material into a finished product, based on the geometric model. To realize the advantages of modern computer-aided technology in the product development and manufacturing process, a geometric model of the part to be created in the CAD system is required. However, there are some situations in product development in which a physical model or sample is produced before creating the CAD model:1. Where a clay model, for example, in designing automobile body panels, is made by the designer or artist based on conceptual sketches of what the panel should look like.2. Where a sample exists without the original drawing or documentation definition.3. Where the CAD model representing the part has to be revised owing to design change during manufacturing.In all of these situations, the physical model or sample must be reverse engineered to create or refine the CAD model. In contrast to this conventional manufacturing sequence reverse engineering typically starts with measuring an existing physical object so that a CAD model can be deduced in order to exploit the advantages of CAD technologies. The process of reverse engineering can usually be subdivided into three stages, i.e. data capture, data segmentation and CAD modeling and/or updating .A physical mock-up or prototype is first measured by a coordinate measuring machine or a laser scanner to acquire the geometric information in the form of 3D points. The measured results are then segmented into topological regions for further processing. Each region represents a single geometric feature that can be represented mathematically by a simple surface in the case of model reconstruction. CAD modeling reconstructs the surface of a region and combines these surfaces into a complete model representing the measured Part or prototype.In practical measuring cases, however, there are many situations where the geometric information of a physical prototype or sample cannot be measured completely and accurately to reconstruct a good CAD model. Some data points of the measured surface may be irregular, have measurement errors, or cannot be acquired. As shown in Fig. 1, the main surface of measured object may have features such as holes, islands, or roughness caused by manufacturing inaccuracy, consequently the CMM probe cannot capture the complete set of data points to reconstruct the entire surface.Fig. 1. The general problems in a practical measuring caseMeasurement of an existing object surface in reverse engineering can be achieved by using either contact probing or non-contact sensing probing techniques. Whatever technique is applied, there are many practical problems with acquiring data points, for examples, noise, and incomplete data. Without extensive processing to adjust the data points, these problems will cause the CAD model to be reconstructed with an undesired shape. In order to rebuild the CAD model correctly and satisfactorily, this paper presents a useful and effective method to pre-process the data points for curve fitting. Using the proposed method, the data points are regenerated in a specified format, which is suitable for fitting into a curve represented in B-spline form without the problems previously mentioned. After fitting all of the curves, the surface model can be completed by connecting the curves.The Theory of B-splineMost of the surface-based CAD systems express shapes required for modeling by parametric equations, such as in Bezier or B-spline forms. The most used is the B-spline form. B-splines are the standard for representing freeform curves and surfaces in current commercial CAD systems. B-spline curves and Bezier curves have many advantages in common. Control points influence the curve shape in a predictable natural way, making them good candidates for use in an interactive environment. Both types of curve are variation diminishing, axis independent, and multivalued, and both exhibit the convex hull property. However, it is the local control of curve shape which is possible with B-splines that gives the technique an advantage over the Bezier technique, as does the ability to add control points without increasing the degree of the curve. Considering the real-world applications requirement, the B-spline technique is used to represent curves and surfaces in this research.A B-spline curve is a set of basis functions which combines the effects of n+1 control points. A parametric B-spline curve is given byp(u)= （1）pi= control pointsn+1= number of control pointsNi,k(u) = the B-spline basis functions u = parameterFor B-spline curves, the degree of these polynomials is controlled by a parameter k and is usually independent of the number of control points, and the B-spline basis functions are defined by the following expression: (2) and (3)Where k controls the degree (k-1) of the resulting polynomials in u and thus also controls the continuity of the curve.A B-spline surface is defined in a similar way to a tensor product in a B-spline curve. It is also possible to define a B-spline surface having different degrees in the u- and v-directions: (4)Curve FittingGiven a set of data points measured from existing object, curve fitting is required to pass through the data points. The least-squares fitting technique is the most used algorithm which aims at approximating, based on an iterative method, a set of data points to form a B-spline.Given a set of data points Qk, k = 0,1,2,. . .,n, that lie on an unknown curve P for certain parameter values uk, k = 0,1,2,. . .,n; it is necessary to determine an exact interpolation or best fitting curve, P.To solve this problem, the parameter values (uk) for each of the data points must be assumed. The knot vector and the degree of the curve are also determined. The degree in practical applications is generally 3 (order = 4). The parameter values can be determined by the chord length method: (5) (6)Given the parameter values, a knot vector that reflects the distribution of these parameters has the following form: (7)Fig.2. Curve fitting with unequal distribution of data points.It can be proved that the coefficient matrix is totally positive and banded with a bandwidth of less than p, therefore, the linear system can be solved safely by Gaussian elimination without pivoting.Equation (5) can be written in a matrix form: (8)where Q is an (m + 1) 1 matrix, N is an (m + 1)*(n + 1) matrix, and P is an (n + 1)*1 matrix. Since m . n, this equation is over-determined. The solution is (9)The Requirement for Fitting a Set of Data into a B-Spline CurveIn order to produce a B-spline curve with a “good shape”,some characteristics are required to fit the data point set into a curve presented in B-spline form. First, the data points must be in a well-ordered sequence. When applying the program to fit a set of data points into a B-spline curve, the data points must be read one by one in a specified order. If the data points are not in order, this will cause an undesired twist or an out-of-control shape of the B-spline curve.Secondly, an even dispersion of the data points is better for curve fitting. In the measuring procedure, some factors, such as the vibration of the machine, the noise in the system, and the roughness of the surface of the measured object will influence the result of the measurement. All of these phenomena will cause local shakes in the curve which passes through the problem points. Therefore, a smooth gradation of the location of the data points is necessary for generating a “high quality” B-spline curve.Having the data points equally distributed is important for improving the result of parameter for fitting a B-spline curve. As the mathematical presentation shows in Eq. (9), the control points matrix P is determined by the basis functions N and data points Q, where the basis functions N are determined by the parameters ui which are correspond to the distribution of the data points. If the data points are distributed unequally, the control points will also be distributed unequally and will cause a lack of smoothness of the fitting curve. As mentioned above, in practical measuring cases, the main surface of a physical sample often has some features such asholes, islands, and radius fillets, which prevent the CMM probe from capturing data points with equal distribution. If a curve is rebuilt by fitting data points with an unequal distribution, as shown in Fig. 2, the generated curve may differ from the real shape of the measured object. Figure 3 illustrates that a smoother and more accurate reconstruction may be obtained by fitting an equally spaced set of data points.The Pre-Processing of Data PointsTo achieve the requirements for fitting a set of data points into a B-spline curve as mentioned above, it is very important and necessary that the data points must be pre-processed before curveFig.3. Curve fitting with unequal distribution of data points.Fig.4.The procedure of data points pre-processingfitting. In the following description, a useful and effective method for pre-processing the data points for curve fitting is presented. The concept of this method is to regress a set of measuring data points into a non-parametric equation in implicit or explicit form, and this equation must also satisfy the continuity of the curvature. For a plane curve, the explicit nonparametric equation takes the general form: y = f (x). Figure 4 illumination an overview of the procedure to pre-process the data points for reverse engineering.Fig.5. Curvature is calculated by three discrete points on a circle.Data point filtering is the first step in displacing the unwanted points and the noisy points. The original data points measured from a physical prototype or an existing sample by a CMM are in discrete format. When the measured points are plotted in a diagram, the noisy points which obviously deviate from the original curve can be selected and removed by a visual search by the designer for extensive processing. In addition the distinct discontinuous points which apparently relate to a sharp change in shape may also be separated easily for further processing.Many approaches have been developed for generating a CAD model from measured points in reverse engineering. A complex model is usually constructed by subdividing the complete model into individual simple surfaces. Each of the individual surfaces defines a single feature in a CAD system and a complete CAD model is obtained by further trimming, blending and filleting, or using other surface-processing tools. When the designer is given a set of unorganized data points measured from an existing object, data point segmentation is required to reconstruct a complete model by defining individual simple surfaces. Therefore, curvature analysis for the data points is used for subdividing the data points into individual group.In order to extract the profile curves for CAD model reconstruction, in this step, data points are divided into different groups depending upon the result of curvature calculation and analysis of the data points. For each 2D curve, y = f(x), the curvature is defined as: (10)If the data is expressed in discrete form, for any three consecutive points in the same plane (X1,Y1) (X2,Y2) (X3,Y3), the three points form a circle and the centre (X0, Y0) can be calculated as (see Fig. 5):a = (X1 + X2) (X2 - X1) (Y3 - Y2)b = (X2 + X3) (X3 - X2) (Y2 - Y1)c = (Y1 - Y3) (Y2 - Y1) (Y3 - Y2)d = 2(X2 - X1) (Y3 - Y2) -(X3 - X2) (Y2 - Y1)e = (Y1 + Y2) (Y2 - Y1) (X3 - X2)f = (Y2 + Y3) (Y3 - Y2) (X2 - X1)g = (X1 - X3) (X2 - X1) (X3 - X2)Fig.6. The fillet of the modelFig.7.The curvature change of the filletAnd,the curvature k of (X2,Y2) can be defined as: (11)Figure 6 illustrates an example in which the curvatures of a plane curve consisting of a data point set are calculated using the previous method. The curvature of the curve determined by the data point set changes from 0 to 0.0333, as shown in Fig. 7. This indicates that there is a fillet feature with a radius 30 in the data points set. Thus, these points can be isolated from the original data points, and form a single feature. By curvature analysis, the total array of data points is divided into several groups. Each of these groups is a segmented form of the original data points which is devoid of any sharp change in shape.After segmentation, individual groups of data points are separately regressed into explicit non-parametric equations, and then the data points can be regenerated from the regression equation in a well-ordered sequence, with appropriate spacing and an equal distribution so that better fitting can be achieved. The format of the new data point set is valid for fitting into a single simple B-spline curve without inner constraints, which can be applied for further editing and modifying, such as trimming and extending. By combining individual curves to construct all of the surfaces, designers may effortlessly achieve a complete CAD model conforming to the design intent.Additionally, some regression errors are introduced by the regression operation between the measured points and the regression equation. In the following example, the order of the regression equation is discussed, because it bears a close relationship to the regression errors. Given a set of existing data points, the set is regressed using a different order of the regression (order = 2,3,4,5). Figure 8 illustrates the relationship between the order of the regression equations and the regressed errors calculated by the root-mean-square (r.m.s.) method. This figure shows that increasing the equation order causes a decrease of the r.m.s. error. However, in most cases, when the 5th-order of the regression equation is used, the coefficient of the 5th-order item becomes zero. i.e. the rams. error of the 4th-order equation is equal to the 5th-order equation. This means that the designer only has to regress the data points into a 4th-order equation. In practice, a 4th-order equation has already satisfied the demand for curvature continuity in CAD model construction for industrial applications.Fig.8.The relationship between the order and the r.m.s. error.ImplementationIn order to prove the effectiveness and feasibility of the proposed method the pre-processing of data points for curve fitting, an implemented case is developed following the steps of the flowchart (Fig. 9). A Mitutoyo BN706 coordinate measuring machine equipped with a Reni Shaw PH9 touch probe and SAS statistics software is used as a tool for system implementation. The measurement of the part surface is performed via standard CMM control and measurement software (Geopak 2800). To ensure that the proposed method is useful for practical applications, a commercial CAD system, Pro/Engineer, is integrated in the implementation. The overall configuration of the system components is shown in Fig. 10.First, the cross-sectional curves describing the shape of the implemented sample are measured by the CMM. The physical object which is typically of symmetric geometry, as shown Fig.9. The procedure of implemnationin Fig. 11, is used in the implemented case. The CAD model of a symmetric object can easily be constructed by mirroring the symmetric features about the centerline. Therefore, some cross-sectional curves which are symmetric require only data for half the curve and then the other half can be mirrored to generate the complete curve. The result of the measurement is shown in Fig. 12.When the measurement is completed, the individual data point sets representing different cross-sectional curves are processed separately. In this implemented case, the central cross-sectional curve is processed as an instance to demonstrate the procedure for pre-processingFig.10.Configuration of system components for implementation.Fig.11.The physical model implementation Fig.12.The result of measurement.the data points, where 144 points are obtained in this curve, as shown in Fig. 13(a). In the data points filtering step, the noisy points and distinct discontinuous points, which obviously deviate from the group of data points, are removed directly for pre-processing. After filtering, the residual data consist of 132 points, as shown in Fig. 13(b). In order to segment the data points, the curvatures of the curve representing the residual data points are calculated and plotted in Fig. 14. As the surface of the implemented physical object is unrefined, the curvature determined by these measured points may greatly deviate from the original curve so that it is difficult to achieve curve segmentation. To obtain the apparent curvature variation, the measured points must be smoothed by the median method before curvature calculation. Figure 15 describes the algorithm of the median method in which point x1, the new coordinate of point x1, is the average of point x0, x1 and x2, x1 = (x0 + x1 + x2)/3. The result of the curvature calculation of the new points, shown in Fig. 16, may be used to segment the curve roughly. Observing the change of curvature and considering the scheme of surface construction, these filtered points are divided into several groups which represent individual feature curves, including the top curve, the side curve, and the fillet curve, as shown in Fig. 13(c) (refer to Fig. 16).Fig.13.The steps of pre-processing the data points of the central cross-sectional curve. Fig.14.Curvature variation of the central cross-sectional curve determined by original points.Fig.15.Smoothing the distribution of points by the media method.Fig.16.Curvature variation of the central cross-sectional curve determined by new pointsFig.17.The entir procedure of CAD model reconstructionAfter the segmentation step, individual groups of data points are separately regressed into explicit non-parametric equations. To eliminate the regression error caused by rough segmentation, remove several points at the start and end of each point group before regression. For example, the segmented points for the top curve are the 28th to 118th point, and the equation, regressing the 31st to the 115th point, can be obtained as (12)Depending on Eq. (12), the data points of the top curve can be regenerated with a well-ordered sequence, pre-determined spacing and equal distribution, as shown in Fig. 13(d). The result of pre-processing the original data point measured by the CMM allows smooth curves to be fitted to the regenerated data points. Points on a curve where the curvature is equal to zero are called inflection points. In some situations, there is more than one inflection point on a curve feature which can be applied to construct complex sculptured surfaces. For processing the data points fitting a single curve segment with multiple inflection points, a higher-order regression equation must be used to regress the data points in order to generate the shape of the curve. In applications of CAD, a curve-based modeling technique is widely applied in industry. The part is customarily divided into several cross-sections along a predetermined direction. Spatial curves for individual features are first fitted through the cross-sectional data points. By blending feature curves, the various surfaces can be constructed with the desired shape, using the different categories of surface construction schemes such as ruled surfaces, lofted surfaces, and Coons surfaces. A complex composite surface model is then constructed by combining these surfaces.When the entire pre-processing procedure is completed, the individual sets of regenerated data points can be transferred to a commercial CAD system (Pro/Engineer is applied here) via the IGES format. All of the feature curves on the measured object can be completely created by fitting different data points sets, which are represented in B-spline form, as shown in Fig. 13(e,f). Interpolating the feature curves, the various surfaces can be constructed with the desired shape. Finally, the complete CAD model, as shown in Fig. 17, is achieved by combining the various surfaces, for the further design operation or modification.ConclusionGeometric modeling is a technology that is already used extensively in industrial applications for developing new products. Reverse engineering has become an important tool for CAD model construction for an existing part from the measuring data. A major difficulty in reverse engineering techniques is to fit the irregular data points of an unequal distribution into a B-spline curve. The procedure of the pre-processing of data points for curve fitting in reverse engineering is described in this paper. The method proposed has been developed to process the data points measured from an existing object before curve fitting, and then new data points are regenerated which are suitable for the requirement for fitting into a smooth Bspline curve with a good shape. The entire procedure of this method involves filtering, curvature analysis, segmentation, regressing, and regenerating steps. The proposed method is implemented for practical applications in reverse engineering, and is an effective tool for integrating with current commercial CAD systems for reconstructing the geometric models of physical parts.A broader interpretation of the term “reverse engineering” might perhaps involve deducing the intent of the original designer to some degree. An ideal system of reverse engineering would be able to not only construct a complete geometric model of the source object but also catch the initial design intent. By applying the method proposed above, designers may regroup the data points in order to produce the individual feature curves for reconstructing a complete CAD model of the source object to achieve the original design intent.中文翻译在逆向工程中对适合曲线的数据点云的预处理逆向工程已经成为一种从现存物体通过CMM测量的数据点重建CAD模型的重要工具.在逆向工程中首要的问题是:测量到的点具有不规律形式和不对等分布很难用B-spline曲线拟合。这篇论文中介绍了一种在逆向工程中用预先处理数据点来拟合曲线的方法。适合B-spline形式之前来处理先前测量得到的数据点的方法已经得到了发展。通过这种方法产生的新的数据点形式，适合建立光滑精确B-spline曲线的要求。这种方法的整个的步骤包括：切片，弧度分析，分割，回归，和再生。在逆向工程中这种方法被实施和用于实践应用。重建的结果证实了此方法与目前流行的商业CAD系统的结合能力。随着计算机硬件的软件技术的发展，对促进产品发展的计算机辅助技术观念在工业领域已被广泛地接受通过新的CAD技术的发展,设计和制造之间的间隙已逐渐变得越来越密切。在正常的自动化制造环境下操作顺序经常是通过用CAD系统创建的几何模型的产品设计开始，在几何模型的基础上，产生机器制造指令将原材料转化成最终产品然后结束。由于意识到现代计算机辅助技术在产品发展和制造中的优势，因此在CAD系统着重要求创建物体的几何模型。然而，在创建CAD 模型之前，产品发展的物理模型和样本先被产生出来。例如，在设计汽车主体控制面板时，设计者和艺术家关于控制板的构想到底是在什么样的基础上制造黏土模型。没有最初的草图，确切的记录模型在哪里？在制造中由于设计的改变，CAD模型不得不重新修改的部分哪里？在所有这些情形中。物理模型或样本的建立是为了创建和建立CAD模型。与这些常规的制造顺序相反，典型的逆向工程从测量现存的物理实体开始，这样推断出来的CAD模型可以更好的利用CAD技术的优势。逆向工程经常可以细分为3个阶段：电子数据获取，数据分割，和用CAD模型构建一个物理模型。样本起先用CMM或激光扫描仪测量以得到以三维坐标形式存在的几何图案的信息。然后，为了更进一步的处理,测量结果被分割成拓扑状。就重建模型来说，每个小区域就代表一个简单的可以用数学方面知识描绘其简单外表的几何图案特征。CAD 模型重建区域的表面是把这些表面连接成完整的可以描述被测量部分或样本的模型。然而，在实际测量方案中，存在物理样本或模型的几何图案信息不能被完全测量和准确重建一个好的CAD 模型的情况。一些表面测量的数据可能是不规律的，还有一些测量误差或者表面是不要求的。如图1所示，测量物体的主要表面可能有这些特征：由于制造的不精确引起的凹坑，凸起，或噪声点，因此，CMM探针不能获取一套完全的数据点来重建整个物体的表面。图1.在一个实际测量情况中的一般的问题在逆向工程中，现存实体的测量，可以通过接触式测量或非接触式测量技术来实现。然而无论用什么技术，这里都有一系列获取数据的实际问题，例如，噪声和不完全数据。如果没有简单的程序去校对这些数据点。这些问题将引起令人不期望的CAD 模型的重建问题。为了正确和满意的重建CAD模型，这篇论文中介绍了一种先处理数据点去拟合曲线的有用和行之有效的方法，用这种方法，数据点被按指定的形式重新生成，并适合指定拟合B-spline曲线的形式，而没有先前提到的问题。在拟合了所有曲线之后，模型的表面才可能完全和曲线结合起来。 B-spline曲线理论通过含参数的方程，绝大多数外观基础上的CAD系统都表达了构造模型的要求， 如Bezier曲线或 B-spline曲线形式，最长用的是B-spline形式，在目前商业系统中，B-spline曲线是标准的代表自由曲线和外表的曲线。B-spline曲线和Bezier 曲线有许多共同的优势。用可预测的普通方法来移动控制点影响曲线形状，使它们两者成了构建曲面较好的曲线形式。这两种不同类型的曲线都具有控制点少，独立的对称轴和综合价值。都表现出了凸凹性。然而，在局部的控制曲线形状这方面，可能B-spline曲线表现出的优势超过了Bezier技术。如增加控制点而没有增加曲线的度数的能力。考虑到现实世界中应用的要求，在这篇论文中B-spline技术被用来代表曲线和曲面。一条B-spline曲线设定了连接n + 1个 控点。通过下面的列子给出了一条含参数的B-spline曲线：p(u)= （1）Pi=控制点n+1=控制点数 Ni,k(u)=B-spline 基本函数u=参数对于B-spline曲线，这些变量参数的度数经常通过参数K控制，它对应控制点的数量。一条B-spline曲线基本功能通过下面的表达式来定义： (2)和 (3) (4)拟合如果从现存的数据中测量一些数据点，拟合曲线不许经过数据点。最新的拟合技术，用接近的算法规则，在迭代方法的基础上，一系列数据点形成了B-spline曲线。假如一系列数据点，在一条不知道参数值的曲线P中，K从1到N决定一个准确加入位置或者是好的拟合曲线P是必要的。为了解决这个问题，每个数据点的参数值必须被假定出来。矢量的节点和曲线的度数也是要求的。在实际应用中度数一般都是3，参数值的确定可以通过下面的方法： (5) (6)如果给定参数值，反映这些参数分布的节点如下面的形式： (7)点如下面的形式： (8) (9) 图2.曲线与真实测量物体的形状不符.适合B-Spline曲线的数据要求 为了生成一条光滑准确的B-Spline曲线，还要求一系列数据点能适合呈现出的B-Spline形式的曲线特征。首先，数据必须有较好的排列顺序。当应用程序为了使一系列数据点能适合-Spline曲线，这些数据点必须以指定的顺序读入。如果数据点不是按顺序的，这将引起未预期的曲线或一条失去B-Spline曲线形状控制的曲线。其次，均匀分布数据点对拟合曲线来说是比较好的。在实际的测量中，一些因素如机器的颤抖，系统中的噪音，和被测量物体表面的粗糙，这都将影响测量的结果。所有这些现象都将引起在经过问题点的曲线的局部颤抖。因此，对于产生一个高质量的B-Spline曲线，光滑有序的点云数据是必须的。获得均匀分布的数据点，可以提高拟合B-Spline曲线参数的结果。就象在方程式（9）中数学方面所展示的那样，通过和数据点分布一致的参数UI决定的基本函数和数据点，确定了控制点。如果数据是不均匀的，这些控点也会分布不均匀还将引起拟合曲线的不平滑。正如上面所提及到的，在实际案例测量中一个物体模型经常有一些诸如空洞，内凹和小范围的切片，这些都将阻止CMM探针获得均匀分布的数据点。如果一条曲线不是用均匀分布的数据点拟合重建的，就像图2中所示，产生的曲线会和真实测量物体的形状不符。图3说明了更光滑和更准确的重建可以通过一系列均匀分布的空间数据点获得。图3.曲线与真实测量物体的形状相同图4.数据点预处理数据点预处理正如上面所述，为了达到使一系列数据点适合B-spline曲线的要求，在拟合曲线之前，对数据点进行预处理是非常重要和必须的。在下面的描述中，将介绍有种对拟合曲线有用而且有效的的数据预处理办法，这种办法的构想是：用绝对的或明确的形式将一系列测量结果设为不含参数的方程式，这些方程式必须满足曲率的连续性，对于一个飞机模型，一个明确的不含参数方程式的一般形式：图5.曲率是通过在圆里的三个离散的点来计算的图示说明，一个总的逆向工程中预处理数据点的程序。数据点的移动第一步是删除不需要和不规则的数据点。通过CMM从物理模型和现存模型测量的原始数据点是离散形式的，当这些测量的点用图表示出来时，明显偏离原始曲线的数据点，可通过设计者的一般处理和可见的搜寻能被有选择的剔除掉。此外，为进一步处理清晰的不连续的在形状上急转变化的点，可以很容易的把他们分开。逆向工程中，从测量点中产生一个CAD模型已经发展了很多种途径。一个复杂的模型经常要通过将完整的模型细分成单独的简单模型来构建。在一个CAD系统中，每一个单独的表面定义了一个简单的特性。一个完整的的CAD模型就可以通过更进一步的修整，融合，整合获得，或者用其他的表面处理工具。当一个设计者把从存在的物体中测量的一系列数据进行细分时，要求通过定义单独的简单表面来重新构建一个完整的模型。 因此，数据点的曲率分析被用来将细分的的数据点归成单独的小类。为了提炼出再建的CAD模型，在这一步中，依据曲率推算和数据点分析