NURBS curve and surface fitting for reverse engineering

Reverse engineering is an approach for constructing a computer-aided design (CAD) model from a physical part through dimensional measurement and surface modelling. This paper presents alternatives for reverse engineering of free-form parts using Non-Uniform Rational B-Spline (NURBS) surfaces. A two-step linear approach is employed for fitting NURBS curves and surfaces using the measured points. During the first step, the weights of the control points are identified from a homogeneous system using symmetric eigenvalue decomposition. The control points are further processed in a way similar to B-spline curve and surface fitting. Some examples are presented to illustrate both the steps of reverse engineering and the process of NURBS curve and surface fitting.     

大型旋转曲面表面涂层仿形加工方法

针对航天领域中某大型易变形薄壁工件旋转曲面表面的涂层仿形加工问题,提出一种高效的实时加工方法。将距离传感器安装于刀具前方一定角度处,加工时,传感器先检测金属曲面外形数据,以测量的离散数据点计算刀具进给轨迹进而控制刀具进给,从而实现高效率的涂层加工,解决易变形薄壁工件表面涂层的仿形加工难题。在实时加工点到达测量点的时间段内,系统完成数据采集、数据处理和刀具轨迹计算。为提高运算实时性,提出对NURBS曲线进行近似拟合的轨迹快速生成方法,以拟合误差为依据将轨迹分成NURBS曲线段和折线段,在误差大的拟合点以直线段连接。最后,通过加工实验证明该加工方法高效、实用。     

NURBS skinning surface for ship hull design based on new parameterization method

Surface reconstruction from sets of cross-sectional data is important in a variety of applications. The problem of generating a ship hull surface from non-regular cross-sectional curves is addressed. Generating non-uniform rational B-splines (NURBS) surfaces that represent cross-sectional curves is a challenge, since the number of control points is growing due to the non-avoidable process of having compatible cross-sectional curves. A new NURBS parameterization method that yields a minimum number of control points, and is adequate in generating a smooth and fair NURBS surface for ship hulls is proposed. This method allows for multiple knots and close domain knots. The results of applying different parameterization methods on the forward perpendicular (FP) region of a ship hull (organized in eight sections) shows that the proposed method reduces the number of control points and generates a smooth and fair NURBS surface, without sacrificing the original object shape of the FP region.     

基于螺旋样条的旋翼无人机区域轨迹规划

针对旋翼无人机在区域作业中的应用需求，提出一种螺旋样条区域轨迹生成方法，设计了给定区域的螺旋控制点生成方法，基于非均匀有理B样条拟合了轨迹。采用线性规划方法求解了以飞行时间最短为目标的规划问题，利用二阶泰勒展开法插值生成用于控制的轨迹序列。通过轨迹追踪实验可知，与传统折线区域轨迹规划方法对比，在相同的动力学约束及作业要求的前提下，该方法有效缩短了作业时间，提高了作业效率。     

NURBS skinning surface for ship hull design based on new parameterization method

Surface reconstruction from sets of cross-sectional data is important in a variety of applications. The problem of generating a ship hull surface from non-regular cross-sectional curves is addressed. Generating non-uniform rational B-splines (NURBS) surfaces that represent cross-sectional curves is a challenge, since the number of control points is growing due to the non-avoidable process of having compatible cross-sectional curves. A new NURBS parameterization method that yields a minimum number of control points, and is adequate in generating a smooth and fair NURBS surface for ship hulls is proposed. This method allows for multiple knots and close domain knots. The results of applying different parameterization methods on the forward perpendicular (FP) region of a ship hull (organized in eight sections) shows that the proposed method reduces the number of control points and generates a smooth and fair NURBS surface, without sacrificing the original object shape of the FP region.     

Efficient analysis-suitable T-spline fitting for freeform surface reconstruction and intelligent sampling

Optical scanning instruments require sampling and reconstruction with high accuracy and low computational cost. T-splines have recently been developed that allow significant reductions in the number of control parameters by overcoming some of the topological constraints of B-splines and NURBS. As a subset of T-splines, analysis-suitable T-splines (ASTS) show promise due to the linear independence and partition of unity of their basis functions. In this paper, a computationally efficient ASTS fitting algorithm for freeform surface reconstruction is proposed. This algorithm starts with adaptive construction of an initial analysis-suitable T-mesh according to the distribution of high-curvature feature points. A local refinement and local optimisation algorithm of the analysis-suitable T-mesh is then iteratively performed until a preset accuracy condition is satisfied. Our experimental results show that the proposed ASTS fitting can produce over 50% root-mean-square reconstruction error reduction compared to NURBS fitting, with the same number of control parameters. The computing efficiency of the proposed algorithm is equivalent to or higher than that for simple T-spline fitting. Fast derivative analysis of the ASTS has also been carried out, where two automatic intelligent sampling design methods have been developed, namely element area sampling and element curvature sampling. Up to 50% reconstruction error reduction is observed when compared to uniform and statistically optimised sampling designs. With the novel reconstruction and compatible intelligent sampling design techniques, freeform surface measurement accuracy and efficiency could be effectively improved using coordinate measuring machines.     

准双曲面齿轮齿面的NURBS曲面拟合

概述了NURBS曲面的表达方法及其在准双曲面齿轮齿面拟合中的应用 ,重点分析了用NURBS方法重构齿面时影响拟合精度的因素 ,如型值点的选取、型值点的数量、节点矢量的计算、边界条件的确定等 ,最后探讨了如何快速准确地得到齿面的拟合误差。     

NEURAL NETWORK APPROACH TO SURFACE BLENDING BASED ON DI GITIZED POINTS

0 INTRODUCTIONIn practical application of industrial area, thefreefOrm surface is often comprised of a lot of surfacepatches (pieces) and the generation of only a singlesurface is difficult to meet design requirements.Hence, we can say, the surface patches are the basicelements from which the surfaces are constructed,and they can be represented by mathematicalequations. Their form and accuracy depend on thefOllowing three factors: ro The wire-frame geometryform which the surfaces are con…     

A Novel Method of Efficient Machining Error Compensation Based on NURBS Surface Control Points Reconstruction

A novel method to improve the efficiency of error compensation in free-form surface machining based on the Non-Uniform Rational B-Splines (NURBS) surface control points reconstruction is proposed in this article. With the presented method, a relatively small number of inspection points are needed to be measured for error compensation. The machined surface is obtained by reconstructing the control points of the designed surface based on the on-machine measurement data. The machining error of the surface is obtained by calculating the difference between the machined surface and the designed one. Then a compensate surface is achieved using the mirror symmetry model and surface modification method to compensate the machining error. Experimental validation for the milling of a NURBS surface shows that the machining accuracy of the surface is improved by 62.57% through use of the proposed method.     

Accurate evaluation of free-form surface profile error based on quasi particle swarm optimization algorithm and surface subdivision

Although significant progress has been made in precision machining of free-form surfaces recently, inspection of such surfaces remains a difficult problem. In order to solve the problem that no specific standards for the verification of free-form surface profile are available, the profile parameters of free-form surface are proposed by referring to ISO standards regarding form tolerances and considering its complexity and non-rotational symmetry. Non-uniform rational basis spline(NURBS) for describing free-form surface is formulated. Crucial issues in surface inspection and profile error verification are localization between the design coordinate system(DCS) and measurement coordinate system(MCS) for searching the closest points on the design model corresponding to measured points. A quasi particle swarm optimization(QPSO) is proposed to search the transformation parameters to implement localization between DCS and MCS. Surface subdivide method which does the searching in a recursively reduced range of the parameters u and v of the NURBS design model is developed to find the closest points. In order to verify the effectiveness of the proposed methods, the design model is generated by NURBS and the measurement data of simulation example are generated by transforming the design model to arbitrary position and orientation, and the parts are machined based on the design model and are measured on CMM. The profile errors of simulation example and actual parts are calculated by the proposed method. The results verify that the evaluation precision of freeform surface profile error by the proposed method is higher 10%-22% than that by CMM software. The proposed method deals with the hard problem that it has a lower precision in profile error evaluation of free-form surface.     

Identity of simultaneous estimates of control points and of their estimates by the lofting method for NURBS surface fitting

For reverse engineering, nonuniform rational B-spline surfaces are fitted to measured coordinates of points. Two methods are often applied. The first one consists of simultaneously estimating the unknown control points of the surface. The second method is a special case of the lofting or skinning method where cross-sectional curves are not interpolated but fitted to the measurements, thus obtaining the control points. This approach is considered to give an approximate solution of the first one. However, it is shown here by an analytical proof and confirmed by a numerical example that both methods give identical results. Since the computational complexity of the first approach is the square of the second one, the simultaneous estimation of a large number of control points should be avoided.     

反求NURBS曲线及其插补算法的研究与应用

在模具设计和制造中,通过三坐标测量仪测量出复杂产品外形曲面上的型值点,通过NURBS反算构建NURBS模型。现代计算机数控系统中已经普遍使用NURBS插补,但大多数NURBS插补算法都致力于取得恒定的进给速度而不是轮廓精度。因此,提出限定弓高误差自适应进给速度的NURBS曲线插补算法,实现产品的精确造型与精密加工。     

曲面修改在逆向工程中的应用研究

针对汽车A—class特征曲线高精度、高光顺性的基本要求,提出了基于控制顶点规范的Bezier曲面迭代微调方法,用曲面反算变换为NURBS曲面,经过基于最小弯曲能曲面修改以后,对进行外形优化。该法核心是控制顶点的微调计算和基于最小弯曲能曲面修改,实现对空间型值点的光顺拟合,实现迭代拟合过程的快速收敛,同时实现了曲面微调和曲面修改的自动化,使基于逆向工程得到的复杂曲面不仅满足精度要求,更可以进行美学设计。结合某汽车的A级曲面的点云数据实例,应用迭代微调有效地提高了生成CAD数据的效率和准确性。     

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。     

Multivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures

Polynomial-basis response surface method has some shortcomings for truss structures in structural optimization,concluding the low fitting accuracy and the great computational effort. Based on the theory of approximation, a response surface method based on Multivariate Rational Function basis(MRRSM) is proposed. In order to further reduce the computational workload of MRRSM, focusing on the law between the cross-sectional area and the nodal displacements of truss structure, a conjecture that the determinant of the stiffness matrix and the corresponding elements of adjoint matrix involved in displacement determination are polynomials with the same order as their respective matrices, each term of which is the product of cross-sectional areas, is proposed. The conjecture is proved theoretically for statically determinate truss structure, and is shown corrected by a large number of statically indeterminate truss structures. The theoretical analysis and a large number of numerical examples show that MRRSM has a high fitting accuracy and less computational effort. Efficiency of the structural optimization of truss structures would be enhanced.     

A quick deviation zone fitting in coordinate metrology of NURBS surfaces using principle component analysis

Evaluation of the tolerance zone using discrete measured points plays a critical role in today’s manufacturing, metrology, and many industrial applications. The deviation zone is typically evaluated using a fitting method that locates an ideal desired geometry corresponding to a set of measured points while a function of the Euclidean distances of the measured points to the ideal surface becomes minimum. This paper presents a quick and reliable algorithm called Dynamic Principle Component Alignment (DPCA) for fitting complex surfaces to the coordinate metrology measured points using the information that is dynamically generated by Principal Component Analysis (PCA) of the measurement data and the corresponding fitted geometry. The developed algorithm efficiently eliminates the necessity for applying commonly used optimization methods for the fitting (localization) process, which decreases the computational cost and uncertainty of the evaluation process. Moreover, DPCA is very reliable and practical in coordinate metrology with large data sets in processes such as laser scanning and other optical methods. The results show that the proposed methodology more accurately finds fitting parameters in comparison with the other commonly used methods while the computational cost is considerably reduced.     

Adaptive Gregory Patch Approximation to Z-Map Data

A method is proposed for Gregory surface approximation to 3D array data points. Surface approximation is the process of constructing a compact representation to model an object surface based on a fairly large number of measured 3D data points. Based on an adaptive subdivision technique, the proposed method begins with a rough initialisation of the surface and progressively refines it in successive steps in the regions where the data is poorly approximated. The method has been implemented using piecewise bicubic Gregory patches with G 1 continuity. An advantage of this approach is that the refinement is essentially local, reducing the computational requirements that permit the processing of a large number of data points This method, combined with the inverse offsetting method, can be used to obtain an offset surface without self-interference. The offset surface can be used to generate gouging-free CL tool paths for machining compound surfaces on milling machines.     

五轴联动数控加工的刀具路径优化方法研究

为解决五轴联动加工复杂曲面过程中的刀具路径不连续问题,提出了五轴数控的刀具路径优化方法。通过五轴NC代码的坐标变换还原有效切削路径,对切削路径进行误差约束下的非均匀有理B样条(NURBS)曲线拟合。对旋转轴路径采用五次样条曲线进行插值,建立切削路径和旋转轴路径的参数映射关系,通过机床逆运动变换求解C2连续的平动轴路径。实验表明,经过该方法优化后,切削路径和各驱动轴运动路径具有良好的平滑性,显著提高了五轴加工曲面精度和表面质量。     

基于曲面广义偏置的EDM电极摇动补偿方法

针对EDM加工中电极摇动产生的加工误差,采用计算机辅助几何设计(CAGD)的思想,提出了基于曲面广义偏置的补偿方法。该方法通过直接对电极曲面几何信息的修正,有效、统一地解决了常用电极摇动所引起的加工误差问题,从而较大地提高了加工精度。从计算实例和试验加工结果看,该方法是正确有效的。     

NURBS体变形与NURBS曲面变形求解

主要研究了NURBS曲面与体实现自由变形时参数区间、曲面片或体元与控制顶点之间的关系。某一参数区间的曲面片或体元的变形只与有限个控制顶点有关。只对相关控制顶点进行自由变形求解,可以降低求解复杂度。