Fast and accurate NURBS fitting for reverse engineering

Fast and accurate fitting of non-uniform rational B-spline (NURBS) curves and surfaces through large sets of measured data is an important problem in applications such as reverse engineering and geometric modelling. This paper presents a method for realising significant improvements in the computational efficiency of this task. The basic idea is that the sparsity structures of the relevant matrices that are specific to the problem of NURBS fitting can be precisely defined and that full exploitation of these structures leads to significant savings in both computational and storage requirements. These savings allow for a large number of control points to be used in order to define the surface and consequently to improve the accuracy of shape representation. The achieved computational complexity is linear in both the number of measured points and the number of control points while the storage requirements of the algorithm are linear with the number of control points only. The complexity analysis, as well as the analysis of actual running times is presented. The results demonstrate that, using this approach, highly complex shapes may be modelled accurately with a single NURBS surface.     

An Approximate Lofting Approach for B-Spline Surface Fitting to Functional Surfaces

Owing to their specific functionality, surfaces are often given as bivariate functions including non-polynomial or higher-orde polynomial terms. It is necessary to represent them in standard formats such as non-uniform rational B-splines (NURBS) for approximation. As most such surfaces require fine quality of surfacing and high precision of manufacturing, accuracy should be guaranteed in their representations. As high accuracy is likely to result in bulky and redundant representations, it is also important to make them more com-pact. For NURBS surfaces, control points must be reduced without sacrificing accuracy. This paper presents an approximate lofting method for B-spline surface fitting to a functional surface within a specified accuracy. It adopts adaptive sampling and multiple B-spline curve fitting. The method is very effective when the surface shape is longish or the cross-sectional curves vary regularly in shape along a specific direction. Some experimental results demonstrate its usefulness and quality.This revised version with a corrected online cover date was published online in April 2004.     

采用截面特征点云的NURBS自由曲面构建

结合NURBS曲面的构建理论,以工程软件UG为平台,探讨了基于截面特征点云的NURBS自由曲面构建的工程实现方法.实现了由截面特征点云进行NURBS曲线拟合,再由拟合曲线进行自由曲面的构建过程,并从曲面的表面状态与误差两个方面对构建的自由曲面进行评估.实践表明:采用该方法构建的自由曲面能保证NURBS曲面拟合质量,且精度高,能有效地反映实物原型.     

Three-dimensional NURBS surface estimated by lofting method

For reverse engineering, nonuniform rational B-spline (NURBS) surfaces expressed by the tensor product are fitted to measured coordinates of points. To estimate the unknown control points, the lofting or skinning method by cross-sectional curve fits leads to efficient computations. Its numerical complexity for estimating k ² control points is O(k ³), while simultaneously estimating the control points possesses a complexity of O(k ⁶). Both methods give identical results. The lofting method is generalized here from a two-dimensional surface represented by the tensor product to a three-dimensional one. Such a surface is needed for a deformation analysis or for solving dynamical problems of reverse engineering, where surfaces change with time. It is shown that the numerical complexity to estimate k ³ control points for a three-dimensional surface is only O(k ⁴). It is also shown by an analytical proof and confirmed by a numerical example that the lofting method for estimating the control points and their simultaneous estimation give identical results. The numerical complexity increases from O(k ⁴) for the lofting method to O(k ⁹) for the simultaneous estimation of k ³ control points. Thus, the lofting method leads to an efficient way of estimating three-dimensional NURBS surfaces for time-depending problems.     

Reverse engineering from uni-directional CMM scan data

The design and manufacture of sculptured surfaces using the reverse engineering (RE) approach is popular in industry. RE requires the definition of surfaces, a process that is usually performed using a coordinate measuring machine (CMM). However, the present methods for surface model reproduction require a high density of data points, resulting in increased manufacturing times.This paper proposes a method to reduce the manufacturing times. In this method constrained least-square fitting techniques are employed. The splines are then blended to a surface model by a spline segment matching and subdividing algorithm. A model refinement method is proposed, that uses only a fraction of the data points. Experimental results and future developments are also included.     

The Pre-Processing of Data Points for Curve Fitting in Reverse Engineering

Reverse 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.     

Implementation of online NURBS curve fitting process on CNC machines

The study aims to develop a method for implementing an online non-uniform rational B-spline (NURBS) curve fitting process on CNC machines for improving the quality and efficiency of machining. Conventional CAD/CAM/CNC systems usually induce some machining difficulties and limit the machining results in real applications. Therefore, some researchers have proposed various methods for improving the machining results. An online NURBS curve fitting process consists of the fetching and fitting stages for CNC machines is developed in this paper. The fetching stage fetches a block of NC code and generates motion commands for obtaining data points. The fitting stage executes the NURBS curve fitting method along with a given set of data points. Moreover, the optimal search method is designed for obtaining good results in the NURBS curve fitting process. Simulations and machining tests carried out on a vertical machining center show that the proposed approach reduces the machining time to about 23% while maintaining the machining quality for the sample test.     

基于Foley参数法反算三次NURBS曲线的算法研究

非均匀有理B样条(NURBS)方法已成为定义工业产品几何形状的唯一数学描述方法.工程实践中,常需要通过曲线已知的一批型值点反算曲线.针对三次NURBS曲线反算问题,采用矩阵形式来表示三次NURBS曲线,由已知的型值点及其权因子,利用Foley参数法确定节点矢量,并推导出以抛物条件即贝塞尔条件作为边界条件构造附加方程,反算出三次NURBS曲线的控制点及其权因子.曲线拟合结果表明,该方法生成的插值曲线具有良好的光滑性,很好地适用于工程实践.     

利用ARX构造反求近似曲面

面对反求工程中仅仅具有少量测量点构造近似曲面的要求,通过利用Ob ject ARX的内置类库,提出了采用NURBS曲线构造近似曲面的方法。该方法简单有效,可以满足一般少量测量点近似曲面构造的要求。     

Accuracy improvement method for flank milling surface design

In this paper, a variation of the method of designing surfaces for flank milling proposed by Li et al. 2006 (Surface design for flank milling. Submitted to CAD, July) is presented. Li’s method is based on the premise that the surface flank milled by a cylindrical tool can be represented by a NURBS surface and can be used by designers to build efficient impellers, blades and other engineering parts. In the proposed method, a four control point curve is used to approximate the grazing curves and for subsequent generation of a polynomial surface. This eliminates the need of weights for the interior control points and still results in a good surface. The accuracy of the surface can be controlled by adding control points. Examples are given to demonstrate the proposed surface design method.     

CNC codes conversion from linear and circular paths to NURBS curves

A novel approach is proposed for converting NC files of G01/G02/G03 codes into NURBS curves. The NC points are first acquired from an NC file composed of lines and arcs. A segmentation algorithm based on the angles and lengths evaluated from adjacent points is then used to subdivide the entire group of points into segments. An error-bounded constrained curve fitting algorithm is implemented to fit each segment of points into a curve, maintaining the accuracy, smoothness and boundary continuity between the curves. The chord errors between the NC points and the fitted curves are controlled, which ensures the accurate recovery of NC paths. Also, a knot insertion algorithm is proposed to determine the appropriate number of control points automatically, allowing effective fitting of all segments. Several examples are presented to demonstrate the feasibility of the proposed approach.     

NURBS曲线、曲面造型过程的可视化设计

由于ＮＵＲＢＳ曲面、曲线构造的复杂性,描述ＮＵＲＢＳ曲线、曲面必然采用复杂的数据结构,这样给人—机交互带来很大的困难。本文利用ＶＣ＋＋编写的ＮＵＲＢＳ曲线、曲面造型系统可以实时观察曲线、曲面的控制点、节点信息,可以实时对控制点、节点进行诸如添加、修改、删除等操作,这对ＮＵＲＢＳ曲线曲面造型无疑会带来极大的方便。     

Reverse engineering of turbine blades based on design intent

The purpose of this paper is to describe a new approach to process the data points measured from turbine blade airfoils in order to make a valid shape via reverse engineering method. Currently, preliminary B-rep models can be created by fitting surfaces to point clouds using a 3D laser scanner. In case of a turbine blade, due to high shape complexity, the resulting model is often unsuitable in practice. A small change in blade geometry can lead to a large change in turbine performance. Therefore, control of the blade shape is critical to the design process. Authors believe that the only way to capture the valid shape of a blade airfoil out of the many manufacturing deviations is to incorporate design key-points during reverse engineering. Implementation of the new method using segmentation and constrained fitting algorithm (SCFA) on a heavy-duty industrial gas turbine blade has been reported and discussed.     

复杂散乱点重构方法的研究与实现

复杂散乱点重构技术历来是实物逆向工程中的关键技术.在深入分析现有经典的重构方法基础上,提出了一种基于三角域上改进的NURBS曲面重构方法.在复杂散乱点上进行基于Delaunay的三角刨分并重构出品质优良的Bezier曲面,然后在它的约束下完成NURBS曲面的二次重构.开发了复杂散乱点重构模块FastForm,并以IGES作为数据通讯接口搭建了一个实现反求全过程的反求平台,通过电话听筒实例验证了该反求系统的实效性.     

燃气轮机透平叶片截面形状重构技术研究

基于逆向工程,采用NURBS(非均匀有理B样条)理论对透平叶片截面数据点阵进行整体插值,实现对透平叶片截面形状的重构;然后采用NURBS小波分解的光顺算法,对重构的自由曲线进行整体光顺。实践表明,经过重构和小波光顺后,有效地构造出透平叶片截面线,并去除了曲线上的坏点,改善了截面线的品质,达到了预定目的。     

弧齿锥齿轮测量齿面的NURBS重构

利用三坐标测量机获得了弧齿锥齿轮齿面上采样点的坐标值,然后采用双三次NURBS插值方法拟合出了数值齿面,能够为弧齿锥齿轮的数字化啮合分析提供参数化的齿面模型。最后,通过一个实例说明了该方法的可行性。     

Global Continuity Adjustment and Local Shape Optimization Technique for Complex Trimmed Surface Model

Smoothly stitching multiple surfaces mainly represented by B-spline or NURBS together is an extremely important issue in complex surfaces modeling and reverse engineering.In recent years,a lot of progress has been made in smooth join of non-trimmed surface patches,while there has been seldom research on smoothly stitching trimmed surface patches together.This paper studies the problem of global continuity adjustment,damaged hole repair and local shape optimization for complex trimmed surface model,and presents a uniform scheme to deal with continuity adjustment of trimmed surfaces and geometric repair of local broken region.Constrained B-spline surface refitting technique and trim calculation are first utilized to achieve global G1 continuity,and then local shape optimization functional is adopted to reduce fitting error and improve local quality of refitted surface patch.The proposed approach is applied to a discontinuity ship hull surface model with an irregular hole,and the result demonstrates the validation of our method.Furthermore,on the premise of global continuity,the proposed locally repairing damaged surface model provides a better foundation for following research work,such as topology recovery technique for complex surface model after geometric repair.     

基于加权双向递推模型的自由曲面反向工程方法

提出了一种基于加权双向递推模型的自由曲面反向工程方法。根据这一方法，可由已采集到的自由曲面上一定数量的任意分布型值点来反向工程出自由曲面。加权双向递推模型的基本思想是，型值点影响并决定着其控制区域内的曲面形状，这种影响的大小与到型值点的距离及权指数相关。给出了将区域划分为ｍ×ｎ的矩形网格下沿ｘ、ｙ方向的加权递推模型。讨论了该方法中网格划分和权指数对反向工程曲面形状及精度的影响。     

Reverse engineering of compound surfaces using boundary detection method

This paper proposes an efficient reverse engineering technique for compound surfaces using a boundary detection method. This approach consists in extracting geometric edge information using a vision system, which can be used in order to drastically reduce geometric errors in the vicinity of compound surface boundaries. Through the image-processing technique and the interpolation process, boundaries are reconstructed by either analytic curves (e. g. circle, ellipse, line) or parametric curves (B-spline curve). In other regions, except boundaries, geometric data are acquired on CMM as points inspected using a touch type probe, and then they are interpolated on several surfaces using a B-spline skinning method. Finally, the boundary edge and the skinned surfaces are combined to reconstruct the final compound surface. Through simulations and experimental works, the effectiveness of the proposed method is confirmed.     

基于精度控制的刀具轨迹自适应拟合算法

为解决传统数控系统在进行曲线曲面离散连续微小线段刀具轨迹加工中频繁加减速、加工速度缓慢、加工质量不高等问题,提出了一种基于精度控制的刀具轨迹自适应NURBS曲线拟合算法,该算法通过提取连续微小线段刀具轨迹的主要特征点,进行基于精度控制的自适应添加特征点的迭代拟合。仿真测试结果表明,该算法可以在保证拟合精度的条件下有效提高计算效率,压缩数据量。