Rough cut tool path planning for B-spline surfaces using convex hull boxes

The objective of this paper is to present a non-uniform layered rough cut plan for B-spline surfaces using convex hull boxes. The tool path plan generated by this method will rapidly remove most redundant material from stock material without overcutting.First, a B-spline surface is decomposed into piecewise Bezier surfaces, of which the convex hull boxes form an approximate model for rough cutting. Then, according to the top planes of those convex hull boxes, the stock material is divided into layers of different thickness. The cavity contour for each layer is obtained using a simplified union Boolean operation on convex hull boxes. Finally, from the top down, each layer is processed like a 2D pocket die cavity.The algorithm is implemented on a personal computer. It is shown that the rough cut plan is very efficient since no computation for solving nonlinear equations is needed, and no overcutting occurs since B-spline surfaces are protected by the convex hull property of Bezier surfaces.     

一种基于插值拟合的复杂刀具轨迹在线平滑压缩算法

提出了一种基于插值拟合的在线复杂刀具轨迹平滑压缩算法,该算法依据主导点的选取策略,对原始数据点进行离线预处理,然后进行主导点的在线插值拟合以及非主导点的误差检测,进而生成一条满足拟合精度要求的B样条曲线。主导点依据离散数据点的曲率阈值、曲率极大值、曲线拐点,以及分段Bezier曲线逼近拟合后的误差最大值点进行选取。在具有C2连续性的分段Bezier曲线逼近拟合前,需要利用长度均分策略,提取长度突变点作为新增的主导点,以保证拟合的准确性。对主导点进行B样条插值拟合后,利用轮廓误差跟随法对非主导点到拟合曲线的误差进行检测。该方法与牛顿迭代法相比,其计算速度更快且能提高算法效率。仿真结果表明,提出的算法可对复杂刀具轨迹进行平滑压缩,且误差检测的精度能够满足要求。     

Multipatched B-Spline Surfaces and Automatic Rough Cut Path Generation

A method is proposed to generate rough cut machining tool paths for multipatched B-spline surfaces. For a group of B-spline surfaces, smoothly connected, separated or intersecting one another, this method can guarantee generating paths without overcutting any of the surfaces under consideration. The use of multipatched surfaces to construct complex surfaces can greatly increase the controllability and friendliness in the design of surfaces. A method for rough cut planning by convex hull boxing is proposed to generate paths guaranteeing no over-cut. Since no computation for solving nonlinear equation is involved, the rough cut plan is robust and efficient.     

Adaptive tool-path generation on point-sampled surfaces

In this paper, we present a new approach to generate tool paths for machining point sampled surfaces using a direct projection algorithm, which is based on generating tool paths along planar intersection curves. In our implementation, a guide surface, with simple geometry like planes or cylinder surfaces, is first created according to the bound volume of the point cloud and initial tool paths are planned on it in terms of the motion pattern of the cutters. For each point of the initial tool paths, then, the corresponding cutter contact point (CC) of the point set surface is located by projecting the point onto the point cloud using the direct projection algorithm. In order to obtain adaptive cutter location points (CL), a least squares-based curve fitting method is applied to approximate the CC points using piecewise cubic Bezier, and a numerical method derived to estimate the length of the curve is used to adjust the position of the points along the curve, and make them evenly spaced on the curve with equal arc lengths. In addition, considering that offset curves or surfaces are necessary for locating CL points in many applications, such as machining using ball end milling cutter, torus ended milling cutters, an offset strategy for cubic Bezier curves is also studied. By testing the proposed method on several point clouds, it has been demonstrated to be promising.     

轴流风机叶片的自由曲线建模研究

介绍了利用自由曲线构造轴流式风机叶型的方法，基于叶栅参数给出了二次Bezier型、三次Bezier型和三次B样条型叶片中线的参数方程，通过实例分析了它们的几何特性以及作功时加压强度沿孤长方向的变化规律，并与传统的双圆孤型中线进行了对比。分析表明，几何特性直接影响其工作性能，采用自由曲线构造叶型可以使轴流风机叶片做功强度的分布更合理，尤其是三次B样条型叶型可以保证加压强度从零开始逐渐增大至最大值。然后又逐渐减小至零。     

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.     

基于角度分割的截面线数据匹配点对建立算法

研究了从截面线数据到B样条曲面拟合的过程中,建立最佳匹配数据点对的问题。提出了一种基于角度分割建立截面线数据匹配点对的算法,该算法利用各层截面线的公共几何中心,计算各点到中心的斜率,再利用基于斜率的插值获取给定角度在原参数曲线上的参数值。最后运用德布尔点的递推公式得到参数对应的坐标。当处理完所有的截面线数据,就得到了一组采样个数统一的匹配点对。该算法在进行截面线数据重采样的同时就完成了对数据点的对正处理,不仅适用于对凸轮廓的处理,而且也能处理层间形状变化较大的截面线数据。     

NURBS output based tool path generation for freeform pockets

A robust method is proposed to generate tool paths for NURBS-based machining of arbitrarily shaped freeform pockets with islands. Although the input and output are all of higher-degree NURBS curves, only one simple category of geometric entities, i.e., line segments, is required for initial offsetting and for detecting and removing self-intersecting loops. Furthermore, using those linear non-self-intersecting offsets as the legs of NURBS control polygons, NURBS-format tool paths can be smoothly reconstructed with G¹-continuity, no overcutting, no cusps, and global error control. Since all operations involved in computing tool path curves are linear geometric calculations, the method is robust and simple. Examples with integrated rough and finish cutting tool paths of pockets demonstrate the usefulness and effectiveness of this method.     

二维轮廓嵌套加工刀轨生成算法

提出一种新的刀轨生成算法,成功解决了由通用等距方法的刀轨加工而成的凸台和凹槽在嵌套时产生干涉的问题。该算法只需要三次等距即可计算出凸台和凹槽加工刀轨,并且能够处理包含自由曲线的轮廓。最后给出的算例验证了该算法的有效性。     

采用Voronoi图生成任意形状凹槽的刀具轨迹

为了自动生成任意形状凹槽的刀具轨迹 ,采用Voronoi图为基础的曲线置偏方法 ,避免传统的置偏算法所遇到的拓扑问题 ,提供了一种以Voronoi图为基础的简便有效的算法 ,可用来处理带“岛屿”的任意形状凹槽的刀具轨迹生成。     

基于VB.NET2005在AutoCAD中操纵拟合曲线的研究

以参数化思想为指导,利用VB.NET2005为编程语言和开发工具,在研究各种拟合曲线原理基础上,对AutoCAD软件进行二次开发,实现了对CAD中二次、三次贝塞尔曲线和B样条曲线,以及相应的控制多边形的绘制。同时,实现了可以在CAD中自由取点,并在程序界面中随意修改参数点来操控拟合曲线的目的。     

Real-time variable feed rate NURBS curve interpolator for CNC machining

This paper presents a real-time control algorithm based on Taylor’s expansion for implementing variable feed rate non-uniform rational B-spline (NURBS) curve interpolators using a digital signal processor for precision CNC machining. To efficiently compute the NURBS curve and its derivatives in real-time, an effective method is proposed. The variable feed rate NURBS curve interpolator can be used to realise the ACC/DEC before feed rate interpolation in which the ACC/DEC (acceleration/deceleration) planning on the feed rate command executes before the interpolation takes place, so that the path command errors caused by conventional ACC/DEC planning using the post feed rate interpolation can be effectively eliminated. To demonstrate the performance of the proposed algorithm, an X-Y table driven by two servomotors is controlled to track command paths represented by multiple blocks of NURBS curves. Experimental results verify the effectiveness of the proposed method.     

Double-spiral tool path in configuration space

Using a spiral tool path is a common strategy in many NC machining applications. It can be used for high-speed machining of pockets or as a space-filling curve for finish applications. When using spirals for finishing application, a double spiral is desired to avoid having a starting or end-point at the center of the workpiece. In this paper, we present an algorithm to create a double spiral from offset curves in a precomputed configuration space (c-space). CAD/CAM systems that operate on NURBS surfaces or on triangular meshes have to deal with the issues of patch-boundary oscillations or long, stretched triangles, respectively. This can be avoided when operating in c-space. The c-space is given in the form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and has proven to be beneficial in CAD/CAM frameworks. Our algorithm creates a double spiral by blending adjacent offset curves. The center of the spiral is filled by a b-spline curve. When given offset curves split into multiple components, the algorithm creates multiple smaller spirals and connects them appropriately. The resulting tool path is one large intersection-free curve with starting and end-point on the boundary of the workpiece.     

网格曲面上测地B样条曲线交互操作与重用

针对现有曲面上自由曲线设计重用方法的不足,提出一种流形网格曲面上曲线几何变换方法,达到曲线重用与再设计的目的。网格曲面上的曲线用测地B样条表示,具有与欧氏空间中传统B样条相一致的明确数学模型;引入对数映射理论将给定的源曲线控制顶点映射到切空间,获得它们的法坐标,按照曲线迁移前后控制顶点法坐标保持不变的原则,建立曲线迁移前后控制顶点的对应关系,实现类似于欧氏空间中的平移、旋转和缩放等几何变换。以网格曲面上离散对数映射理论为基础,将欧氏空间中的对称定义拓展到曲面空间,提出曲面上曲线的广义镜像概念并给出具体的算法实现。法坐标很好地保持了控制顶点之间的测地距离和相对位置关系,因而也保证了曲线迁移重用过程中的形状保持性。试验结果表明,所介绍方法健壮、有效,能满足曲面上曲线的交互设计要求。     

An enhanced algorithm for estimation of a digitized curve length using B-splines

A digitized curve length estimation is a hot topic of research. All the methods tend to estimate the length of the original curve from which the digitized curve is obtained. When comparing different methods of estimation, the main performance criterium is the accuracy of this estimation. In this paper a novel approach is presented. It shows a significant improvement compared to all the other methods, regardless of the resolution and the type of curve. The method is based on B-splines. The initial number of control points and their location is determined by the reverse polygonization, which is one of the polygonal approximation techniques. The method then iteratively updates the location of the control points of the B-spline based on the specified cost function. Due to the fact that it shows the same level of accuracy for different B-spline orders it is also much more convenient to use than the other methods because it doesn’t require the selection of appropriate parameter(s). Beside that it doesn’t require nonlinear optimization techniques. The performance of the method is verified on various test curves including the ones with multiple paths within pixel at different grid resolutions.     

Numerically Controlled Machining of Freeform Curves Using Biarc Approximation

To machine a shape incorporating freeform curves, the free-form curves must be approximated and represented in a form suitable for numerically controlled (NC) tool-path generation. Instead of the commonly used line segment approximation, a new method using biarc approximation is presented here. The purpose is to reduce sudden changes in tool movement and increase continuity and smoothness. A biarc can be described as two consecutive circular arcs with an identical tangent at the connecting point. Therefore, tangent continuity can be maintained. Furthermore, curvature continuity can be improved if the difference between the two curvatures is minimised. In this research, biarc segments are used to approximate the freeform curves by controlling the largest deviation distance between the curves and the biarcs. The test results show that the biarc approximation method generates fewer tool-path segments than the traditional line approximation. The definitions of the biarcs are used to generate NC tool paths. A new tool-path generation method is developed for processing rough cutting and finish cutting paths. Example parts modelled with B-spline curves are tested and machined.     

连续体结构综合优化设计

以连续体为对象的优化解法所求得的优化结果,经常只是反映最佳传力途径的具有锯齿状边界的某个区域。基于此,将拓扑优化、形状优化、有限元分析和计算机辅助几何设计有机地集成在一起,提出一种基于隐含边界描述的水平集连续体结构拓扑和形状渐进综合优化设计方法,将形状导数与拉格朗日乘子法引入到优化敏度分析中,控制水平集函数的动态运动,从而间接地实现结构边界的动态演化;用B样条曲线曲面逼近拓扑优化后的结构体边界,将前一优化过程所得到的反映传力途径的概念解上升为具有光顺边界,并被参数化了的物理解;在形状优化中,设计变量定义为B样条曲线或曲面的控制顶点的运动,建立边界节点移动速度场计算方法和边界形状调整方法,寻求较快的搜索方向,以合理速度分布,使结构变为最佳。通过一个典型算例证明所研究方法的有效性。     

Machining of complex-shaped parts with guidance curves

Nowadays, high-speed machining is usually used for production of hardened material parts with complex shapes such as dies and molds. In such parts, tool paths generated for bottom machining feature with the conventional parallel plane strategy induced many feed rate reductions, especially when boundaries of the feature have a lot of curvatures and are not parallel. Several machining experiments on hardened material lead to the conclusion that a tool path implying stable cutting conditions might guarantee a better part surface integrity. To ensure this stability, the shape machined must be decomposed when conventional strategies are not suitable. In this paper, an experimental approach based on high-speed performance simulation is conducted on a master bottom machining feature in order to highlight the influence of the curvatures towards a suitable decomposition of machining area. The decomposition is achieved through the construction of intermediate curves between the closed boundaries of the feature. These intermediate curves are used as guidance curve for the tool paths generation with an alternative machining strategy called “guidance curve strategy”. For the construction of intermediate curves, key parameters reflecting the influence of their proximity with each closed boundary and the influence of the curvatures of this latter are introduced. Based on the results, a method for defining guidance curves in four steps is proposed.     

开式整体叶盘通道侧铣粗加工技术的研究

整体叶盘从锻造毛坯到最终零件加工成型,需要切除大量的多余材料。本文提出了一种开式整体叶盘通道区域五坐标粗加工刀具轨迹规划方法。该方法通过生成整体叶盘薄壁叶片的直纹包络面,以确定叶盘通道粗加工区域的边界轮廓;基于直纹面五轴侧铣加工刀具轨迹双点偏置生成方式,利用投影法规划开式整体叶盘通道区域粗铣加工的刀心点轨迹与刀轴矢量方向。加工实践表明,开式整体叶盘的粗加工效率提高30%以上,同时,显著优化了后续半精加工和精加工工序的环境。     

Generation of Tool Paths for Machining Free-Form Pockets with Islands Using Distance Maps

In this paper, we present a new procedure for generating tool paths using discrete distance maps, especially for dealing with free-form shaped pockets with multiple islands. In this procedure, a discrete distance map is computed by shading a right circular cone having the height of a given offset distance, while moving its apex along the boundary curve segments. Using the discrete distance maps, the proposed algorithm effectively extracts the characteristic points or the valid self-intersection points of offset curve segments. For an offset distance, one or more offset profiles are constructed without the topological problems by offsetting the boundary curve segments within the parameter values specified by these chararteristic points, and connecting them into closed profiles using the topological information contained in the distance map. The gouging-free tool path is constructed by connecting these offset profiles, starting from the innermost offset profile. In the proposed method, we do not need any artificial bridges for a pocket with multiple islands to merge the pocket profile and the island profiles into a single boundary profile.