The edge–torus tangency problem in multipoint machining of triangulated surface models

This paper presents a method for computing the tangency between an edge and a torus. This tangency is used in positioning a toroidal tool on an edge that is part of a triangulated surface. This method is easier to implement and faster to execute than earlier solutions to this problem. The method was tested on a triangulated surface modeling a pyramid and on a triangulated tensor product Bézier patch.     

Analytical envelope surface representation of a conical cutter undergoing rational motion

Flank milling with a taper cutter is widely used in industry. The analytical representation of the envelope surface generated by a conical cutter undergoing rational motion is derived by bringing together the theories of line geometry and kinematics. Based on the projective duality between a point and a plane in line geometry, a cone surface is represented as two pieces of rational quadratic Bézier developable surfaces in terms of the plane coordinates instead of the traditional point coordinates. It provides a way to describe and calculate the envelope surface exactly by analyzing the trajectory of a plane undergoing a two-parameter rational motion. The rotation around the axis of the cone is adopted to ensure that the characteristic curve is located on the same piece of rational quadratic Bézier developable surface of the cone. The degenerate cases that the characteristic curve does not exist are also discussed. Examples are provided, in which the envelope surfaces of a conical cutter undergoing rational Bézier and B-spline motions are computed. The results can be applied to tool-path planning and error analysis for five-axis flank milling machining.     

Tool path planning for five-axis flank milling with developable surface approximation

This paper presents a novel approach that automatically generates an interference-free tool path for five-axis flank milling of a ruled surface. A boundary curve of the machined surface is subdivided into curve segments. Each segment works as a guide curve in the design method for developable Bézier surface that controls a developable patch for approximating the surface with available degrees of freedom. Geometric algorithms are proposed for calculating consecutive patches with G¹ continuity across the patch boundary. A tapered tool can move along the rulings of these patches without inducing local tool interference as a result of their developability. The machining deviation is controlled by the surface approximation error. A machining test is conducted with the generated CL data and the result verifies the feasibility of the proposed approach. This work successfully transforms avoidance of tool interference into a geometric modeling problem and provides a simple solution. It thus demonstrates a good potential for the developable surface theory of five-axis flank machining .     

Cycloids for polishing along double-spiral toolpaths in configuration space

An automated polishing process of free-form surfaces requires a tool path that covers the entire surface equally and forms an overlapping pattern without visible artifacts. The recently presented double-spiral tool paths assure a coverage of the entire surface with a continuous, non-overlapping path and low variation in distance between adjacent traces of the path. We build upon this approach by constructing cycloids of flexible radii that fill the space between adjacent traces. The use of cycloids mimics the cyclic movement when polishing by hand. The approach operates in a precomputed configuration space (c-space) given in form of an adaptive quadrilateral heightfield mesh. Operating in c-space avoids having to deal with the issues of patch-boundary oscillations or long, stretched triangles in non-uniform rational b-spline surface or triangular mesh representations, respectively. Our algorithm computes appropriate spheroids that are intersected with the c-space to compute the cycloids. We derive a smooth representation of the cycloids using arcs in a rational Bézier formulation. We apply our approach to real-world examples to demonstrate its effectiveness in covering the entire surface with the desired polishing movements.     

参数曲线弧长的一种估算方法

研究了如何估算任意次多项式形式参数曲线的弧长曲线的问题。与目前已有类似方法相比,我们方法的优点是对曲线的次数没有限制。这里的关键问题是如何求出弧长曲线的拐点以及如何对曲线进行分割。利用Bézier曲线的变差缩减性用牛顿迭代法计算出了弧长曲线的全部拐点,进而可以得到原曲线的Bézier曲线形式的近似弧长曲线。数值实验表明,用该方法得到的弧长曲线是令人满意的。     

The selection of radius correction method in the case of coordinate measurements applicable for turbine blades

The objective of the research presented in the paper is the selection of suitable probe radius correction method in the case of coordinate measurements which can be applied for turbine blades. The investigations are based on theoretical analysis of geometric data and on further computer simulation of measurements and data processing. In the paper two methods for computing coordinates of corrected measured points are verified. Those so-called local methods of probe radius correction are based on Bézier curves. They are dedicated first of all to coordinate measurements of free-form surfaces which are characterized by big values of curvature, e.g. those surrounding the leading and trailing edges of a turbine blade. Numerical simulations are done for several models of the cross-sections of turbine blades with diversified magnitudes of radii of curvature. There are considered both manufacturing deviations and coordinate measurement uncertainty of each examined profile of a turbine blade. Numerical investigations based on the developed analytical models show the advantage of the algorithm which uses the second degree Bézier curves in probe radius correction. Moreover, in the paper there is explained the implementation of the developed algorithms for probe radius correction into the standard CMM software which amplifies the usability of the algorithms.     

A global collision-free path planning using parametric parabola through Geometry Mapping of obstacles in robot work space

In this paper, a new algorithm for palnning collision-free path connecting from start to target point is developed using Bézier curve of order two. The control point, i. e. the mid-point of quadratic Bézier curve, determines the shape of parabola and constitutes the Control Point Space, and this process is difined a Geometry Mapping. After Geometry Mapping of all obstacles, the clear area of CPS, an area not occupied by obstacle images, identifies collision-free path. The path planning algorithm, heance, transform path planning problem in Euclidian Space to point selection problem in CPS. The calculations involved in the algorithm do not require iterative procedures and all the formulas of the solution are derived in closed form. A CPS completely filled with obstacle images indicates that path planning based on parabola is not possible and requires higher order curve with more than one control point.     

Modeling of twist drills in terms of 3D angles

Traditional nomenclatures of specifying cutting tool geometries are two-dimensional (2D) in nature. The present work presents a paradigm to model the geometries of a variety of twist drills in terms of three-dimensional (3D) parameters. The work outlines the construction of a detailed computer-aided design (CAD) model for a fluted twist drill and establishes a new 3D definition for the geometry of drill in terms of biparametric surface patches. The flutes of the drill are modeled as helicoidal surfaces. For this, sectional geometry of tip-to-tip profile is developed and then swept. The geometric model of the shank is developed separately. The transitional surfaces are modeled as bicubic Bèzier surfaces. With this methodology, we propose a new 3D nomenclature for drill geometries in terms of 3D rotational angles. The relations necessary to map the proposed three-dimensional angles to two-dimensional conventional angles, known as forward mapping and their reverse relations (inverse mapping) are also developed. The new paradigm offers immense technological advantages in terms of numerous downstream applications.     

A novel approach for computing C2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C²-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C²-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bézier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C² continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve.     

Degree reduction of NURBS curves

Higher degree curves are used in applications because they are easier to manipulate interactively but require heavy computation. Most of the equations for curves used popularly in CAD software are of degree 2 and 3, because two curves of degree 3 can guarantee 2nd derivative continuity at the connection point. This study proposes a different but simpler method than any put forward before to deal with degree reduction of free-form curves. The reduced curves use the simplest knot vector type, i.e., the open uniform knot vector. Unlike other methods, this study does not modify or refine the knot vectors but perturb the control points globally. After obtaining an initial condition, a radiating web-like search algorithm is applied to detect the optimum positions. These NURBS curve formats reach basic industrial standards for CAD/CAM/CNC applications. By defining a global bound error function, this algorithm can achieve an optimum solution not only for NURBS curves but also Bézier/B-spline curves.     

Calculation of the unit normal vector using the cross-curve moving mask method for probe radius compensation of a freeform surface measurement

This paper proposes a cross-curve moving mask method to calculate the unit normal vector based on 5 or 9 data points of a freeform surface measurement for probe radius compensation. This is done in order to simplify the calculation and thus, produce a more efficient and time-saving process. Passing through the middle point and 4 or 8 neighboring points, two crossed curves – longitude and latitude can be constructed. The unit normal vector at the middle point can be determined by calculating the cross product of two tangent vectors along these two crossed curves. Different curve fitting methods for the curves passing through 5 or 9 data points, such as Bézier and B-spline methodologies, have been investigated. Three kinds of surfaces, namely, a spherical surface, a cosine-like surface and a shoe-shaped surface (hereby termed “shoe last”) are selected for evaluating the accuracy of the calculated unit normal vectors.     

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.     

基于三次Q-Bézier曲线的车灯造型设计

用一条三次Q-Bézier曲线描述了汽车车灯造型轮廓;提出首先由设计师在平面视图中绘制并编辑汽车车灯造型的二维三次Q-Bézierr曲线轮廓图,然后将其投影到汽车车身上,最后通过对曲面进行修剪获得车灯造型轮廓的方法。该方法基于三次Q-Bézier曲线生成及拼接原理,可以迅速从全局或局部调整车灯轮廓线条,获得光顺的曲线效果,从而快速获得多个车灯轮廓的造型方案。     

Deviation modeling of manufactured surfaces from a perspective of manufacturing errors

Assembly analysis, as an important step in the product design stage, provides verifications of product design with specific requirements. The manufactured surfaces that participate in the analysis result from different manufacturing methods and show inconsistency with the designed surfaces due to the effects of numerous manufacturing errors. These deviations have a vital impact on the assembly analysis results and make a big difference on the final product design process. Therefore, consideration of such manufactured surface deviations in assembly analysis is critical in order to achieve an effective product design. In this paper, a general surface deviation modeling and representation approach based on curvilinear coordinate system is proposed. Different curvilinear coordinate systems for various manufacturing methods are defined on nominal product surfaces. By introducing the deviation dimension in the orthogonal direction of the curvilinear coordinates, deviation coordinate systems are formed. The deviation coordinate system could be used to describe deviation surfaces resulting from numerous manufacturing errors with deviation functions. As an example, non-ideal planar surfaces in the face milling process are modelled with the proposed method. With the generated non-ideal planar surfaces, the sealing performance analysis of an assembly is implemented. The validity of the proposed method and its applicability in simulation are verified in the end.     

自由型面的五轴激光淬火轨迹规划算法研究

为保证复杂型面的激光淬火质量,针对自由型面的激光淬火路径规划直接影响着淬火后自由型面表面硬度的均匀性这一特点,在分析激光淬火加工工艺特点的基础上,提出了等弧长的激光淬火轨迹算法来确定自由型面上的激光淬火轨迹。该方法通过激光淬火试验,确定了最佳激光淬火光道距离并进行了相应的计算。利用三次B样条参数曲面进行了实例计算,验证了所提出算法的可行性。     

Fundamental study on releasability of molded rubber from mold tool surface

In compression or injection molding of rubber products, small pieces of residual rubber often remain on the metal mold surface after releasing the product. This is caused by excessive localized adhesion to the mold surface. Therefore, cleaning of metal mold surface must be often performed, which results in longer molding cycle time, and the lifetime of metal mold is often reduced. In this study, the separation forces between molded rubber and metal mold surfaces are measured with a tensile tester to evaluate the releasability of molded rubber from the metal mold surface. Mold surfaces treated by various surface coatings and surface modification methods including EB polishing were tested and compared. Experimental results show that the separation force between molded rubber and metal mold surfaces depends on the true contact area between them and the chemical composition of the metal mold surface. The separation force decreases with a decrease in contact area. The chromium content at the metal mold surface significantly reduces the separation force. EB polishing is one of the most effective surface treatments for metal molds since the real contact area can be decreased while also decreasing the surface roughness of the tool surface in a short time. Electron beam melting is also shown to be an effective method of distributing chromium uniformly on the metal mold surface.     

EQUIVALENT NORMAL CURVATURE APPROACH MILLING MODEL OF MACHINING FREEFORM SURFACES

A new milling methodology with the equivalent normal curvature milling model machining freeform surfaces is proposed based on the normal curvature theorems on differential geometry. Moreover, a specialized whirlwind milling tool and a 5-axis CNC horizontal milling machine are introduced. This new milling model can efficiently enlarge the material removal volume at the tip of the whirlwind milling tool and improve the producing capacity. The machining strategy of this model is to regulate the orientation of the whirlwind milling tool relatively to the principal directions of the workpiece surface at the point of contact, so as to create a full match with collision avoidance between the workpiece surface and the symmetric rotational surface of the milling tool. The practical results show that this new milling model is an effective method in machining complex three- dimensional surfaces. This model has a good improvement on finishing machining time and scallop height in machining the freeform surfaces over other milling processes. Some actual examples for manufacturing the freeform surfaces with this new model are given.     

Influence of roughness parameters and surface texture on friction during sliding of pure lead over 080 M40 steel

In the present investigation, tests were conducted on a tribological couple made of cylindrical lead pin with spherical tip against 080 M40 steel plates of different textures with varying roughness under both dry and lubricated conditions using an inclined pin-on-plate sliding tester. Surface roughness parameters of the steel plates were measured using optical profilometer. The morphologies of the worn surfaces of the pins and the formation of transfer layer on the counter surfaces were observed using a scanning electron microscope. It was observed that the coefficient of friction and the formation of transfer layer depend primarily on the surface texture of hard surfaces. A newly formulated non-dimensional hybrid roughness parameter called ‘ξ’ (a product of number of peaks and maximum profile peak height) of the tool surface plays an important role in determining the frictional behaviour of the surfaces studied. The effect of surfaces texture on coefficient of friction was attributed to the variation of plowing component of friction, which in turn depends on the roughness parameter ‘ξ’.     

Integration of industrial photogrammetry and neuro-fuzzy system for measuring and modeling deformation of industrial product surfaces under external forces

Deformation of surfaces under external loadings greatly is function of physical laws. But according to the impossibility of identifying all effective factors and modeling their interactions parametrically, analytical methods do not provide good performance generally. So, it is necessary to apply practical methods by carrying out field tests and measuring deformations directly. To achieve this purpose it is required to integrate capabilities of an accurate measurement technique and a flexible modeling method. The capabilities such as: high accuracy and speed in measuring 3D coordinates of desired points, ability to perform measurement in a continuous space and no need to contact with the surface of objects at the time of measurement make close range photogrammetry a reliable tool for measuring geometric parameters of an object before and after deformation. The ability to measure geometric parameters of an object before and after deformation in one hand and following the deformation from physical laws on the other hand make neuro-fuzzy system the first choice for modeling the deformation of objects using outputs of close range photogrammetry.In this paper, a new method has been presented for measuring and modeling deformation of industrial product surfaces under external forces using close range photogrammetry (as an image-based measurement tool) and neuro-fuzzy network (as a behavior modeling tool).     

Research on surface normal measurement and adjustment in aircraft assembly

Drilling and riveting are commonly used in aircraft panel assembly process. Due to the fixture positioning error and the deformation of workpiece, the real position and orientation of the workpiece as well as its 3D geometry at the drilling position varies from the nominal CAD model, which would cause an unfavorable impact on assembly quality. Therefore, surface normal measurement and adjustment at the drilling position is of great importance. In this paper, a fast and effective non-contact measurement method for normal vector and height of moderately curved surfaces is accomplished by four laser displacement sensors, and a dedicated NC machine tool is also developed for normal adjustment. Firstly, a novel sensor calibration method based on laser tracker is introduced, which can acquire the sensors’ position and orientation in Tool Coordinate System (TCS) at the same time. The normal vector at hole position is calculated by cross product of any two non-parallel vectors constructed by the four laser projection points on the panel surface. Secondly, the kinematic model of the machine tool is established to calculate the adjustment of each axis of the machine tool with the Homogeneous Transformation Matrix (HTM). Besides, an innovative method to identify the distance of two rotary centers based on two laser interferometers is proposed. Finally, a series of experiments are conducted to validate the feasibility of the proposed method. The results show that the angle deviation can be reduced to less than 0.5° after adjustment, while the accuracy of the surface height is ±0.04 mm.