Chebyshev fitting of complex surfaces for precision metrology

The form qualities of precision components are essential for their functionalities. The Peak-to-Valley parameters are widely adopted to assess the form accuracies of optical components. The commonly used least squares method is prone to over-estimation, thus the Chebyshev fitting should in turn be implemented. In this paper the original minimax optimisation problem is converted into an unconstrained differentiable minimisation problem by the exponential penalty functions. The fitting accuracy and numerical stability are balanced by employing an active-set strategy and adjusting the configuration parameters adaptively. Finally some benchmark data sets are applied to demonstrate the validity and efficiency of this method.     

Quadrilateral mesh generation on trimmed NURBS surfaces

An automatic mesh generation scheme with unstructured quadrilateral elements on trimmed NURBS surfaces has been developed. In this paper NURBS surface geometries in the IGES format have been employed to represent geometric models. For unstructured mesh generation with quadrilateral elements, a domain decomposition algorithm employing loop operators has been modified. As for the surface meshing, an indirect 2D approach is proposed in which both quasi-expanded planes and projection planes are employed. Sample meshes for complex models are presented to demonstrate the robustness of the algorithm.     

Optimal door fitting with systematic fixture adjustment

A systematic approach is presented to obtain the best door gap quality through optimal door fitting in automobile body manufacturing. First, three indexes of gap quality are defined; they are: (1) door gap width deviation relative to design nominal; (2) door gap parallelism; and (3) car-to-car gap consistency. Then the door-fitting problem is formulated into a general constrained optimization problem. The effects of optimal door fitting on the three quality indexes are evaluated through computer simulation. These results provide a lower bound on the design of nominal door gap by considering process capability. Finally, a computer-aided fixture adjustment scheme is developed to orient a door in a body side opening to achieve the optimal fitting. The amount of adjustment, with the desired orientation obtained from optimization, is calculated based on parametrically modeled local surface features of the fixture and the door. The adequacy of door feature modeling is verified through a door-fitting experiment.     

基于网格分割的自适应细分曲面算法研究

细分曲面中,随着对模型细分次数的增多,模型的面片数迅速增长,巨大的数据量加大了模型后处理的难度。为此,结合网格分割技术,提出了一种自适应细分曲面算法。该算法利用控制网格每个顶点与其对应的极限位置之间的欧氏距离不同、采用K-均值算法和区域合并技术对控制网格进行分割,随后对用户指定的网格区域或不满足精度的区域进行局部自适应细分。实验结果表明,该算法在给定精度范围内,有效地以相对较少的面片数获得了理想光滑的细分曲面,且大大提高了模型的后处理速度。     

三角网格曲面上的勾画曲线及其编辑

提出一种在三角网格模型上生成与编辑勾画曲线的方法.首先通过人机交互获得一个屏幕坐标系下的有序点集S,拟合点集s为一条3次非均匀B样条曲线C,对曲线进行重采样,得到一系列的采样点.把采样点由屏幕坐标系映射到三维世界坐标系,即获得了一个位于三角网格模型上的有序点集,用直线段将这些点依次顺序相连,即获得了三角网格模型上的勾画曲线,该曲线线性逼近于3次非均匀B样条曲线.同样地,通过位于屏幕空间的曲线的编辑来实现三角网格模型上勾画曲线的编辑修改.最后,给出了所介绍算法在帮样设计CAD系统中的应用实例,结果显示,算法实时交互性好,能够满足工程应用的要求.     

基于网格评判法的机械产品优化设计

指出利用非线性规划求解工程设计中多目标优化问题的一些局限性,提出用网格评判法处理机械产品的多目标优化设计,并以行星减速器为例印证了该方法的有效性。     

Optimal localization of complex surfaces in CAD-based inspection

Complex surface inspection requires the optimal localization of the measured surface related to the design surface so that the two surfaces can be compared in a common coordinate frame. This paper presents a new technique for solving the localization problem. The basic approach consists of two steps: 1) rough localization of the measured points to the design surface based on curvature features, which can produce a good initial estimate for the optimal localization; 2) fine localization based on the least-square principle so that the deviation between the measured surface and the design surface is minimized. To efficiently compute the closest points on the design surface of the measured points, a novel method is proposed. Since this approach does not involve an iterative process of solving non-linear equations for the closest points, it is more convenient and robust. The typical complex surface is used to test the developed algorithm. Analysis and comparison of experimental results demonstrate the validity and applicability of the algorithm.     

Optimal localization of complex surfaces in CAD-based inspection

Complex surface inspection requires the optimal localization of the measured surface related to the design surface so that the two surfaces can be compared in a common coordinate frame. This paper presents a new technique for solving the localization problem. The basic approach consists of two steps: 1) rough localization of the measured points to the design surface based on curvature features, which can produce a good initial estimate for the optimal localization; 2) fine localization based on the least-square principle so that the deviation between the measured surface and the design surface is minimized. To efficiently compute the closest points on the design surface of the measured points, a novel method is proposed. Since this approach does not involve an iterative process of solving non-linear equations for the closest points, it is more convenient and robust. The typical complex surface is used to test the developed algorithm. Analysis and comparison of experimental results demonstrate the validity and applicability of the algorithm.     

Optimal method for auto-body closure panel fitting using Hausdorff distance criteria

Auto-body closure panel-fitting positions at the final stage of body-in-white (BIW)’s hierarchical assembly. It is such a critical process that contributes greatly to automotive dimensional integrity and quality. Poor fitting results in unsatisfied gap width and flushness that would cause functional problems and make adverse effect on vehicle quality perceived by customers. Although this problem has been studied by several authors in the past, two aspects still require further research. One is the improvement of fit quality measure suitable for complicated contours of closure panel and body opening’s profile. The other is the effect of weld-induced distortion during fitting process. In this paper, a novel computer-aided approach is proposed to address these above two problems. Firstly, panel fitting is mathematically modeled as a relative position optimization between panel and body opening which adopts translation and transposition parameters in homogeneous transformation as independent variables. Hausdorff distance between two continuous closed panel curves is firstly employed as the measure to evaluate the fitting quality of panel-opening profile contour. Secondly, since hinge axis plays important role for panel-opening relative position, its deviation from nominal design intent could not be neglected. After obtaining weld-induced hinge distortion using numerical analysis for these subassembly process, deviation of closure panel profile contour from design stage is calculated by homogeneous transformation during optimization and factors resulting from this physical thermal effects on gap width and flushness are taken into account and compensated. For the reason that the panel-fitting optimal function contains more than three variables, it is not suitable for applying algorithm with gradient descent or direct search. Genetic algorithm is used to search optimal fitting variables in a global and parallel way. Case study of certain car door fitting shows that the proposed method can effectively improve panel-fitting quality and accuracy.     

Optimal flattening of freeform surfaces based on energy model

Flattening of freeform surfaces (i.e., non-developable surfaces) is a difficult problem in engineering application. According to flattening distortions, a new development based on energy model is put forward in this paper. Given a doubly curved surface and using a triangle mesh subdivision, some triangles will be distorted when the surface is flattened. Generally, the developed plane will possess many splits. There are two kinds of developments: unconstrained flattening and constrained flattening, which are both analysed. In order to obtain a better pattern, the strain energy required to force a 3-D shape adopt the plane pattern should be minimised. At last a flattening algorithm is developed on the basis of theoretical analyses. Compared with the others, the proposed method can control the local precision easily, and is an effective method for flattening doubly curved surfaces.     

Optimal combinations of contact surfaces in coaxial frictional pairs

Various combinations of surfaces in the frictional pairs of primary diesel-engine bearings are investigated by computer calculation. The parameters of the gap formed by the frictional surface are calculated, as well as the lubricant volume in the gap.     

采样点分布对基于面形斜率径向基模型的自由曲面拟合精度的影响

鉴于自由曲面模型的面形拟合精度在自由曲面表征以及面形初始结构选取等研究中的重要性,本文针对基于面形斜率的高斯径向基表征模型,研究了不同的采样点分布类型对该模型面形拟合精度的影响。采用不同采样点分布拟合离轴二次曲面和带凸起的抛物面,结果表明采用均匀随机分布的采样点有利于实现高精度的面形拟合,且达到一定的拟合精度后,采样点的数目对拟合精度的影响有限。以离轴三反系统为设计实例,对比了由不同采样方式生成初始面形后系统的像质优化结果。结果显示,采用均匀随机型采样方式得到的初始面形进行系统优化,最终全视场平均调制传递函数(MTF)可以达到0.72以上,远高于由边缘集中采样方式生成初始面形后系统像质的优化结果,从而印证了理论研究结果。     

A robust surface fitting and reconstruction algorithm for form characterization of ultra-precision freeform surfaces

Ultra-precision freeform surfaces are non-rotational symmetric surfaces possessing sub-micrometer form accuracy and nanometric surface finish. Although they can be fabricated accurately by ultra-precision machining technology, their surface quality is difficult to be characterized. Surface reconstruction is a vital task in the form characterization of ultra-precision freeform surfaces. This paper presents a robust surface fitting algorithm to reconstruct a high fidelity surface from measured discrete points while the surface smoothness can be ensured as well. A fitting threshold named confidence interval of fitting error is proposed to strike the balance between fitting accuracy and surface smoothness in the fitting process. The fitting algorithm is in two steps. In the first step, bidirectional sampling method is developed to extract a curve network from measured points cloud to construct an initial surface. In the second step, the fitting error of the initial surface is minimized to meet the prescribed fitting error threshold. A series of experimental work has been conducted and the results show that the proposed algorithm is able to provide effective means for increasing the accuracy in the form characterization of ultra-precision freeform surfaces.     

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.     

Tool path generation by offsetting curves on polyhedral surfaces based on mesh flattening

Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there are few tool path planning strategies for such surfaces but isoplanar method. This paper presents a contour offset approach to tool path generation for three-axis ball-end milling of polyhedral surfaces, based on a novel method for offsetting curves on polyhedral surfaces. One of its salient features is to reduce the task of removing complex interfering of offsets from 3D physical surfaces to 2D plane by flattening mesh surfaces and avoid costly 3D Boolean set operations and relatively expensive distance calculation. Moreover, in practical implement, the procedures of calculating offset points and removing interfering loops are merged and carried out simultaneously results in an efficient tool path generation method. Empirical examples illustrate the feasibility of the proposed method.     

An investigation of the robustness of the nonlinear least-squares sphere fitting method to small segment angle surfaces

This paper presents an investigation of the nonlinear least-squares sphere fitting algorithm (TLS_A). The work concentrates on investigating the reliability of the TLS_A algorithm when applied to a small segment angle of a sphere. The definition of small segment angle is discussed in the paper and taken to be below 1° (in both x and y directions) of the spherical surface. This application of the TLS_A method is important when it is used on data from optical scanning systems where the measurements are limited by the gauge range and the angular tolerance of the sensor. The TLS_A algorithm has been first compared with the TLS_D algorithm suggested by Forbes for this application. The results show that the TLS_A algorithm can be used in small surface segment angles. The main study is focused on testing the algorithm on a sphere superimposed with surface irregularities (sensor/measurement noise or roughness). Two properties of the TLS_A algorithm are covered: the bias and the uncertainty of the estimated radius. Both simulation and theoretical approaches have been attempted. A new algorithm to estimate the bias of the TLS_A algorithm has been derived in this paper based on Box's method. Together with uncertainty estimation, which can be produced by using either a conventional method or Zhang's error propagation function (EPF), a comprehensive understanding of the TLS_A algorithm in this application is thus achieved, and used to develop a number of recommendations for the precision metrology of spherical and near spherical surfaces.     

双圆弧及直线拟合技术及其在数控切割中的应用

提出双圆弧拟合和直线拟合混合应用的算法,解决数控切割中的拟合问题。获取切割对象轮廓的离散点后,使用逐点尝试回溯法,在满足精度和不破坏轮廓形状的前提下用一直线或者圆弧或者圆尽可能拟合更多的点,并且尽量保证拟合后的轮廓光滑性。根据离散点的切向变化和离散点前后连线的斜率来识别轮廓的拐点并且保留该拐点,对拐点的切向进行拟合前和拟合后的处理保证轮廓形状不变。结果表明,该算法性能良好,拟合后的矢量化结果在数控切割应用效果非常好。     

Optimal five-axis tool path generation algorithm based on double scalar fields for freeform surfaces

In order to generate efficient tool path with given precision requirements, scallop height should be kept under a given limit, while the tool path should be as short as possible to reduce machining time. Traditional methods generate CC curves one by one, which makes the final tool path far from being globally optimal. This paper presents an optimal tool path generation model for a ball-end tool which strives to globally optimize a tool path with various objectives and constraints. Two scalar functions are constructed over the part surface to represent the path intervals and the feedrate (with directions). Using the finite element method (FEM), the tool path length minimization model and the machining time minimization model are solved numerically. The proposed method is also suitable for tool path generation on mesh surfaces. Simulation results show that the generated tool path can be direction parallel or contour parallel with different boundary conditions. Compared to most of the conventional tool path generation methods, the proposed method is able to generate more effective tool paths due to the global optimization strategy.     

Influences of friction and mesh misalignment on time-varying mesh stiffness of helical gears

As one of the most important excitation sources of vibration, time-varying mesh stiffness of helical gear pairs need accurately calculated. Compared with spur gears, friction in helical gears is significant. This work for the first time presents an improved calculation method for the mesh stiffness of helical gears with effect of friction incorporated. Firstly, helical gear is sliced into number of pieces along its axis direction and each piece could be regarded as spur gear. Then forces applied to each piece including friction force are analyzed. Potential energy method is employed to develop time-varying mesh stiffness of each piece pair of both kinds of helical gears with different transverse and axial contact ratios. Furthermore, influences of various working conditions and misalignment on mesh stiffness are also investigated. Results indicate that effect brought by friction on total mesh stiffness should be not neglected. The reduction amount of stiffness increases with lower speed, heavier load and rougher surface. The stiffness difference between cases with and without friction is affected by gear geometry and mounting parameters like module, helix angle and mounting misalignment. This work provides an essential tool for comprehensive dynamics analysis with consideration of the relationship between stiffness and working conditions.