On the effective tool path planning algorithms for sculptured surface manufacture

This paper describes the cutter path planning and cutter interference (gouging) analysis algorithms developed to generate optimal tool path for manufacturing sculptured surfaces on three axes CNC machine tools. Cutter path planning algorithm approximates the parametric curves on three dimensional surfaces by a sequence of straight line segments and generates optimal tool paths by minimizing the number of interpolation points while keeping the path deviations within the specified tolerances. Cutter interference analysis algorithm checks for the self intersection of an offset surface and determines the self-intersection curve. The tool path is then planned over the cutter contact (CC) surface after removing the CC data that lies inside the self-intersection curve. Finally, the effectiveness of these algorithms is demonstrated by implementing them in CAD/CAM system.     

Constant scallop-height tool path generation for three-axis sculptured surface machining

This paper presents a new approach for the determination of efficient tool paths in the machining of sculptured surfaces using 3-axis ball-end milling. The objective is to keep the scallop height constant across the machined surface such that redundant tool paths are minimized. Unlike most previous studies on constant scallop-height machining, the present work determines the tool paths without resorting to the approximated 2D representations of the 3D cutting geometry. Two offset surfaces of the design surface, the scallop surface and the tool center surface, are employed to successively establish scallop curves on the scallop surface and cutter location tool paths for the design surface. The effectiveness of the present approach is demonstrated through the machining of a typical sculptured surface. The results indicate that constant scallop-height machining achieves the specified machining accuracy with fewer and shorter tool paths than the existing tool path generation approaches.     

Smooth closed surfaces with discrete triangular interpolants

Discrete interpolants which involve cross boundary derivatives in an attempt to form C¹ surfaces have the following major problem: Requiring C¹ joins between patches makes sense only if the patch domains are adjacent in the domain space. This makes it impossible to form C¹ closed surfaces, or indeed any surface which contains more connections than can be achieved in the domain.

This paper develops a method of forming smooth closed (or otherwise complexly connected) surfaces from a discrete triangular interpolant by relaxing the C¹ property of an interpolant to ‘Visually C¹”.

The only constraint on the scheme is that the data to be interpolated define a unique tangent plane at each vertex where several triangles meet. Then each patch can be calculated independently of its neighbors, using only data defined at its vertices, and the domain for each triangular patch can be chosen without regarding the connectivity of the patch with others. This last feature could be of great interest to a designer of a surface since one can choose the domain of each patch to be an equilateral triangle, and give it no further thought.     

ADAPTIVE STRATEGIES FORNC MACHINING COMPOUND FREE-FORM SURFACES

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A mapping-based approach to eliminating self-intersection of offset paths on mesh surfaces for CNC machining

Geometrically, a tool path can be generated by successively offsetting its adjacent path on the surface with a given path interval, which preferably starts from one of the surface boundaries or a primary curve. The key issues involved in offset path planning are the generation of raw offset paths and the elimination of the self-intersection of raw offset paths. Most researches available in this area are focused on how to generate the raw offset paths, however, the latter, especially how to eliminate the self-intersection of the offset paths on mesh surfaces, has not been sufficiently addressed. In this paper, a mapping-based approach to eliminating the self-intersection of offset paths is proposed for the CNC machining of mesh surfaces. The method first flattens the mesh surface onto a predefined plane by using a mesh mapping technique, and then taking the mapping as a guide, the offset paths are also naturally mapped onto the plane, from which those invalid self-intersection loops can be effectively identified and eliminated. To handle the issue of self-intersection for all types of offset path, a notion of local loop is introduced to detect and eliminate the invalid self-intersection loops. After that the planar paths are inversely mapped into the physical space and the final tool paths used for the machining of mesh surface are obtained. Meanwhile, in order to improve the kinematic and dynamic performance of the machine tool when machining along the generated offset paths, a method for rounding the sharp corners of tool paths, which result from the process of eliminating the self-intersection of raw offset paths, is also preliminarily investigated. Finally, the proposed method is validated by the results of simulations and machining experiments.     

On describing complex surface shapes

The fractal surface model¹ is extended to provide a formalism that is competent to describe complex, natural three-dimensional surfaces in either a quantitative or qualitative manner and which, in addition, closely mimics human perceptual judgments of surface structure (eg ‘peaks’, ‘ridges’ or ‘valleys’) and threedimensional texture. This representation is then used to develop a statistical version of scale-space filtering that is applicable to one-, two- or three-dimensional data.     

An algorithm for the interpolation of hybrid curves

Real time tool path generation consists of off-line design and real time interpolation of tool paths. An hybrid curve is the intersection of a parametric surface and an implicit surface. Previous work in tool path interpolation focused mainly in the interpolation of parametric curves. Tool paths designed by drive surface methods are hybrid curves which, in general, cannot be represented as parametric curves. An algorithm for the interpolation of hybrid curves is proposed in this paper. The algorithm is based on interpolation of the projection of the hybrid curve into the parametric domain. Each increment involves a second-order interpolation step augmented by iterative error reduction.Simulations of hybrid curve interpolation have been carried out. They are based on practical surfaces represented as NURB surfaces and implicit surfaces including a plane, a cylinder and a high order algebraic surface. They demonstrate that under typical machining conditions, interpolation error is well within the accuracy requirements of typical machining and that the use of one iteration error reduction can significantly reduce the path deviation. These show that the proposed algorithm is potentially useful for tool path interpolation for the machining of parametric surfaces.     

测量点数据等残留高度刀具路径规划

在介绍计算几何模型的基础上,提出一种刀具路径算法.首先分别以刀具半径值和残留高度值为等距距离,通过等距计算求出等距点集和残留高度点集;然后以前一行刀具路径为中心构建刀具包络面,并求出刀具包络面与残留高度点集的交点,即过渡点集;最后通过求以过渡点集为中心的刀具包络面与等距点集的交点,得出相邻行刀具轨迹,依次递推,求出所有的刀具路径行.针对计算过程中可能出现的欠切问题,给出了应用密切圆追踪的边界处理方法.通过实例验证了该算法的可行性.与等间距刀具路径生成方法进行比较表明,应用文中算法解决针对测量点数据的数控加工刀具路径生成问题,可缩短刀具路径长度,提高加工效率.     

Graphics-assisted Rolling Ball Method for 5-axis surface machining

In this paper, a graphics hardware-assisted approach to 5-axis surface machining is presented that builds upon a tool positioning strategy named the Rolling Ball Method presented in an earlier paper by the present authors [Comput. Aided Des. 35 (2003) 347]. The depth buffer of the computer's graphics card is used to compute the data needed for the Rolling Ball Method, which generates gouge-free 5-axis curvature-matched tool positions. With this approach, the tool path for a workpiece can be computed with triangulated data instead of parametric surface equations. It also permits the generation of tool paths for multiple surface patch workpieces that have only positional continuity. The method is easy to implement and it is robust since every tool position is computed with the same algorithm regardless of the type of surface. For illustration, tool paths were generated for a workpiece with two bi-cubic surface patches, connected with only position continuity. Simulations for gouge-checking and machining tests were performed. Workpiece cusp heights were measured using a coordinate measuring machine. The maximum undercutting measured in the machining examples was 0.07 and 0.05 mm, which is within the expected NC machine accuracy and measuring capabilities for surfaces.     

One-sided offset approximation of freeform curves for interference-free NURBS machining

Generating valid tool path curves in NURBS form is important in realizing an efficient NURBS machining. In this paper, a method for computing one-sided offset approximations of freeform curves with NURBS format as tool paths is presented. The approach first uses line segments to approximate the progenitor curve with one-sided deviations. Based on the obtained line approximating curve and its offsets, a unilateral tolerance zone (UTZ) is constructed subsequently. Finally, a C¹-continuous and completely interference-free NURBS offset curve is generated within the UTZ to satisfy the required tolerance globally. Since all of the geometric computations involved are linear, the proposed method is efficient and robust. Interference-free tool path generation thus can be achieved in NURBS based NC machining.     

A new approach to z-level contour machining of triangulated surface models using fillet endmills

Precision z-level contour machining is important for various computer-aided manufacturing (CAM) applications such as pocket machining and high-speed machining (HSM). This paper presents a new z-level contour tool-path generation algorithm for NC machining of triangulated surface models. Traditional approaches of z-level machining rely on the creation of accurate CL (cutter location) surfaces by surface offsetting or high-density z-map generation, which is computationally expensive and memory demanding. In contrast, this paper presents a novel approach to the generation of CL data directly from the section polygon of a triangulated surface model. For each polygon vertex of the contour, the offset direction is determined by the normal to the edge, while the offset distance is not fixed but is determined from the cutter shape and the part surface. An interference-free tool-path computation algorithm using fillet endmills is developed. Since there is no need to create a complete CL surface or high-density z-map grids, this proposed method is highly efficient and more flexible, and can be directly applied to triangulated surfaces either tessellated from CAD models, or reconstructed from 3D scanned data for reverse engineering (RE) applications.     

Two algorithms for volume-preserving approximations of surfaces of revolution

The paper deals with volume-preserving approximations of surfaces of revolution. The approximating surfaces are generated only by line segments and circular arcs of a constant radius r. Further, for r > 0, the approximating surfaces are visually C¹ surfaces. For r = 0, developable C⁰ surfaces are obtained (consisting of either congruent cylinders or frustums of cones of revolution). Two algorithms are discussed. The first algorithm preserves the volume enclosed by a surface of revolution and the planes of every two latitude circles; the approximating surface is, however, no longer a surface of revolution. The second algorithm applies an approximating surface of revolution; however, the volume preservation no longer holds globally.     

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves.     

An optimal approach to multiple tool selection and their numerical control path generation for aggressive rough machining of pockets with free-form boundaries

To machine pockets, especially ones with closed free-form boundary curves, roughing is crucial to part productivity, for this operation alone could take more than 60% of the total machining time. At present, there is a high demand from industry for a new machining technique that can efficiently cut pockets. Aggressive rough machining, in which the largest possible cutters are always employed and are fully immersed in workpieces, can be a solution. Although aggressive roughing is by far the most efficient machining strategy, compared to prior pocketing methods, no computer numerical control (CNC) programming technique has been developed to support it, resulting in few applications in machine shops. To address this urgent industrial need, based on the medial axis transform of a pocket, this work proposes an optimal approach to multiple tool selection and their numerical control (NC) path generation for aggressive roughing of the pocket. First, the NC paths of a specific tool are quickly generated using the pocket’s medial axis transform. Thanks to the unique characteristic of the medial axis transform, the paths can ensure the tool the largest accessible space for pocketing. At the same time, they can guarantee the tool to be free of gouging and interference. Then, an optimization model of selecting multiple cutters and generating their NC paths is built in order to achieve the highest efficiency of the aggressive rough machining. To demonstrate the advantages of this innovative approach, two examples are rendered, and their results are compared to those obtained by the existing methods. This approach can be directly implemented into current CAD/CAM software to promote aggressive rough machining of pockets in industry.     

Spatially referenced methods of processing raster and vector data

The authors consider a general method of constructing addressing and arithmetic systems for two-dimensional image data using the hierarchy of ‘molecular’ tilings based on an original isohedral ‘atomic’ tiling. (Each molecular title at level k is formed from a constant number of tiles at level k−1; this is termed the ‘aperture’ property of the hierarchy.) In addition they present 11 objective criteria (which are of significance in cartographic image processing), by which these hierarchies and tilings may be described and compared.

Of the 11 topologically distinct types of isohedral tiling, three ([3⁶], [4⁴] and [6³]) are composed of regular polygons, and two of these ([3⁶] and[4⁴]) satisfy the condition that all tiles have the same ‘orientation’. In general, although each level in a hierarchy is topologically equivalent, the tiles may differ in shape at different levels and only [6³], [4⁴], [4.8²] and [4.6.12] are capable of giving rise to hierarchies in which the tiles at all levels are the same shape. The possible apertures of hierarchies obeying this condition are n² (for any n > 1)in the cases of [6³] and [4⁴]; n² or 2n² in the case of [4.8²]; and n² or 3n² in the case of [4.6.12].

In contrast the only tiling exhibiting the uniform ‘adjacency’ criterion is[3⁶]. However, hierarchies based on this atomic tiling generate molecular tiles with different shapes at every level. If these disadvantages are accepted, hierarchies based on first-level molecular tiles referred to as the 4-shape, 4′-shape, 7-shape and 9-shape are generated. Of these the 4-shape and the 9-shape appear to satisfy many of the cartographically desirable properties in addition to having an atomic tiling which exhibits uniform adjacency.

In recent years the generalized balanced ternary addressing system has been developed to exploit the image processing power of the 7-shape. The authors have generalized and extended this system as ‘tesseral addressing and arithmetic’, showing how it can be used to render a 4-shape into a spatially correct linear quadtree.     

Approximate methods for simulation and verification of numerically controlled machining programs

Algorithms for simulation and verification of Numerically Controlled (NC) machining programs are presented. Compared to NC simulation based on conventional solid modeling systems, these models are designed to give approximate results, but with a substantial decrease in computer time. The surfaces of the part are discretized into a Surface Point Set (SPS) with a point spacing dependent on cutting tool size and shape local surface curvature and the desired accuracy of the approximate simulation. The surface-surface intersection calculations of the solid modeling approach are replaced by the intersection of the surface of the tool movement envelope with straight lines emanating from the surface points. The methods are applicable to both 3 and 5 axis machining. Samples test cases are presented, and implementation and efficiency issues are discussed.     

Offset of curves on tessellated surfaces

Geodesic offset of curves on surfaces is an important and useful tool of computer aided design for applications such as generation of tool paths for NC machining and simulation of fibre path on tool surfaces in composites manufacturing. For many industrial and graphic applications, tessellation representation is used for curves and surfaces because of its simplicity in representation and for simpler and faster geometric operations. The paper presents an algorithm for computing offset of curves on tessellated surfaces. A curve on tessellation (COT) is represented as a sequence of 3D points, with each line segment of every two consecutive points lying exactly on the tessellation. With an incremental approach of the algorithm to compute offset COT, the final offset curve position is obtained through several intermediate offset curve positions. Each offset curve position is obtained by offsetting all the points of COT along the tessellation in such a way that all the line segments gets offset exactly along the faces of tessellation in which the line segments are contained. The algorithm, based entirely on tessellation representation, completely eliminates the formation of local self-intersections. Global self-intersections if any, are detected and corrected explicitly. Offset of both open and closed tessellated curves, either in a plane or on a tessellated surface, can be generated using the proposed approach. The computation of offset COT is very accurate within the tessellation tolerance.     

A new curve-based approach to polyhedral machining

This paper presents a new approach to three-axis NC tool path generation for sculptured surfaces. In the proposed curve-based approach, the gouge-free tool paths are generated from a polyhedral model of the STL (stereolithography) format. The polyhedral model is offset by a local-offsetting scheme. Then, the offset elements such as triangular facets, trimmed cylinders, and trimmed spheres are sliced by a series of drive planes. The curve segments on a drive plane are sorted, trimmed and linked, while the concave gouge is removed during the trimming process. The method is implemented on a PC, and some illustrative examples are provided in this paper. The main advantage of the proposed method is that the tool path can be generated from a polyhedral model without any concave and convex gouge, especially on an NC machine that supports NURBS interpolation. Other advantages and disadvantages of the proposed model are also discussed.     

Quart-parametric interpolations for intersecting paths

The intersecting path is an important tool path generation method. This paper proposes an approach for the quart-parametric interpolation of intersecting paths. The objective of our approach is that the intersecting paths for surface machining can be directly interpolated within the computer numerical control (CNC) system. This enables the CNC interpolator to process the intersecting paths without geometric approximation as in existing approaches and take into consideration any specific feedrate profiles and further machining dynamical issues along the path.The interpolation of the intersection of two general parametric surfaces is transferred into interpolation of its projection curves and the time trajectories of four parameters along the intersecting curves are obtained. Our strategy is to carry out the quart-parametric interpolation based on the projection interpolation. The feedrate control method is developed, and then the interpolation algorithms for two projection curves are proposed. An error reduction scheme is presented to alleviate point deviation from the drive parametric surface. Simulations of quart-parametric interpolation have been carried out to verify the effectiveness of the proposed algorithm.     

直线与刀具扫描体求交算法及其应用研究

直线与刀具扫描体的交点问题是数控图形验证，加工过程仿真技术中的核心问题，本文根据五坐标联动机床的运动特点，把刀具扫描体表达成几组简单曲面的组合，利用直线与这几类曲交点来求解直线与刀具扫描体的交点。利用该算法，我们开发了一个用于检验五坐标铣削加工精度的计算机领导具软件。加工与仿真的实践证明，该算法计算正确正确，对进一步研制开发加工过程仿真系统有重要意义。