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.     

实体模型的三轴数控粗加工刀轨生成算法

提出一种适合实体模型的三轴数控粗加工刀位轨迹生成算法。首先根据加工行距作一组平行于刀轴的平面，与模型的待加工表面求交，得到一系列交线；再根据精度规划一组垂直于刀轴的分层平面，与上述交线求交，在每一分层平面上判断加工区域，规划出加工刀位轨迹，将每一分层平面上的刀位轨迹较适当的方式连接起来，就构成零件的整体加工轨迹。该算法避免了轮廓环等距、自交处理和布尔运算等复杂的计算过程；同时，对生成开型腔的加工刀位轨迹也是有效的。     

三角网格表面等残留高度刀轨生成算法

针对截平面法规划的三角网格表面的刀轨长度较长、加工表面残留高度不均匀的问题,提出一种基于改进截平面法的等残留高度刀轨生成算法.首先在估算刀触点轨迹线垂直方向曲率半径的基础上,计算刀触点轨迹投影线并对其进行修正,去除其中冗余的投影点;然后由修正后的刀触点轨迹投影线构造驱动表面,利用驱动表面和网格表面迭代计算刀触点轨迹线;最后由刀触点轨迹线计算无干涉刀轨.与截平面法生成的刀轨进行比较分析的结果表明,文中算法生成的刀轨长度较小且获得的残留高度保持均匀,适合于三角网格表面表示的复杂表面的精加工.     

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.     

A Tool Path Generation Strategy for Sculptured Surfaces Machining

This paper presents a strategy to generate interference-free tool paths for machining sculptured surfaces. The strategy proposed here is first to determine the tool path topology. The values of the step length and the path interval are then calculated based on the machining tolerance requirements. After detecting and eliminating the tool interference, the interference-free tool path is generated. The effectiveness of the developed algorithm is demonstrated through simulation and actual cutting tests.     

Sculptured surface machining using triangular mesh slicing

In this paper, an optimized procedure for tool path generation in regional milling is presented. The proposed procedure computes tool paths by slicing a CL-surface (Cutter Location surface), which is a triangular, mesh containing invalid triangles. Tool path generation consists of two steps: firstly, it obtains a set of line segments by slicing the triangular mesh with two-dimensional geometric elements (slicing elements), and, secondly, it extracts a valid tool path from the line segments by removing invalid portions. Two algorithms based on the slicing elements are presented: a ‘line projection’ algorithm based on the plane sweeping paradigm, which works efficiently by using the characteristics of a monotone chain; and a ‘curve projection’ algorithm for the projection of curves, which transforms the curve projection problem into a line projection problem by mapping the XYZ-space of the cylinder surface to the TZ-plane of the unfolded cylinder. The proposed procedure has been implemented and applied to tool path generation in regional milling. Performance tests show the efficiency of the proposed procedure.     

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.     

Iso-scallop tool path planning for triangular mesh surfaces in multi-axis machining

Triangular mesh enables the flexible construction of complex surface geometry and has become a general representation of 3D objects in computer graphics. However, the creation of a tool path with constant residual scallop height on triangular mesh surfaces in multi-axis machining is not a convenient task for current algorithms. In this study, an isoscallop tool path planning method for triangular mesh surfaces, in which the tool path is derived directly from the contours of a normalized geodesic distance field (GDF), without any post-processing is proposed. First, the GDF is built to determine the shortest geodesic distance from each vertex to the mesh boundary. Then, the normalizing process is performed on the GDF to ensure that its first contour meets the isoscallop height requirement considering the mesh curvature and effective cutter radius. To improve the computational efficiency, the GDF is only built in the mesh area related to the first contour by specifying a stop distance. Moreover, an adaptive refinement process is conducted on the mesh to improve the smoothness and accuracy of the tool path. Finally, the triangular mesh is trimmed along this first contour for a new round of tool path planning. The proposed method is organized recursively and terminated when no new paths are generated. Simulations and experiments are conducted to verify the effectiveness and superiority of the proposed tool path planning method.     

Iso-planar interpolation for the machining of implicit surfaces

The paper presents an approach for on-line path generation and interpolation for the machining of implicit surfaces. For a given implicit surface, once the cutting plane direction and cut-in points have been selected, iso-planar tool paths and interpolated points can be calculated on-line according to the feedrate and scallop height requirements. The approach enables the tool position and orientation to be correctly calculated at each interpolated point. Validation examples are provided for the interpolation of cyclide surfaces with planar and curved boundaries.     

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.     

Smooth tool path generation for 5-axis machining of triangular mesh surface with nonzero genus

NC machining of a nonzero genus triangular mesh surface is being more widely confronted than before in the manufacturing field. At present, due to the complexity of geometry computation related to tool path generation, only one path pattern of iso-planar type is adopted in real machining of such surface. To improve significantly 5-axis machining of the nonzero genus mesh surface, it is necessary to develop a more efficient and robust tool path generation method. In this paper, a new method of generating spiral or contour-parallel tool path is proposed, which is inspired by the cylindrical helix or circle which are a set of parallel lines on the rectangular region obtained by unwrapping the cylinder. According to this idea, the effective data structure and algorithm are first designed to transform a nonzero genus surface into a genus-0 surface such that the conformal map method can be used to build the bidirectional mapping between the genus-0 surface and the rectangular region. In this rectangular region, the issues of spiral or contour-parallel tool path generation fall into the category of simple straight path planning. Accordingly, the formula for calculating the parameter increment for the guide line is derived by the difference scheme on the mesh surface and an accuracy improvement method is proposed based on the edge curve interpolation for determining the cutter contact (CC) point. These guarantee that the generated tool path can meet nicely the machining requirement. To improve further the kinematic and dynamic performance of 5-axis machine tool, a method for optimizing tool orientation is also preliminarily investigated. Finally, the experiments are performed to demonstrate the proposed method and show that it can generate nicely the spiral tool path or contour-parallel tool path on the nonzero genus mesh surface and also can guarantee the smooth change of tool orientation.     

Iso-planar piecewise linear NC tool path generation from discrete measured data points

This article presents a method of generating iso-planar piecewise linear NC tool paths for three-axis surface machining using ball-end milling directly from discrete measured data points. Unlike the existing tool path generation methods for discrete points, both the machining error and the machined surface finish are explicitly considered and evaluated in the present work. The primary direction of the generated iso-planar tool paths is derived from the projected boundary of the discrete points. A projected cutter location net (CL-net) is then created, which groups the data points according to the intended machining error and surface finish requirements. The machining error of an individual data point is evaluated within its bounding CL-net cell from the adjacent tool swept surfaces of the ball-end mill. The positions of the CL-net nodes can thus be optimized and established sequentially by minimizing the machining error of each CL-net cell. Since the linear edges of adjacent CL-net cells are in general not perfectly aligned, weighted averages of the associated CL-net nodes are employed as the CL points for machining. As a final step, the redundant segments on the CL paths are trimmed to reduce machining time. The validity of the tool path generation method has been examined by using both simulated and experimentally measured data points.     

Adaptive iso-planar tool path generation for machining of free-form surfaces

The iso-planar (Cartesian) tool path generation method has been used for several decades. However, it suffers an inherent drawback: in the region where the direction of the surface normal is close to that of the parallel intersecting planes, the intersecting plane intervals have to be reduced because of the influence of surface slopes. This causes redundant tool paths in the associated flatter regions and results in lower machining efficiency. This paper presents an algorithm that overcomes the disadvantage of the iso-planar method while keeping its advantages of robustness and simplicity. In this algorithm, the concept of isophote is applied to partition the surface into different regions. In each region the tool path side steps are adaptive to the surface features. Therefore redundant tool paths are avoided. By applying the region-by-region or global-local machining strategy, the machining efficiency is increased.     

A tool path generation method for freeform surface machining by introducing the tensor property of machining strip width

Due to the complexity of geometry, the feed direction with maximal machining strip width usually varies among different regions over a freeform surface or a shell of surfaces. However, in most traditional tool path generation methods, the surface is treated as one machining region thus only local optimisation might be achieved. This paper presents a new region-based tool path generation method. To achieve the full effect of the optimal feed direction, a surface is divided into several sub-surface regions before tool path computation. Different from the scalar field representation of the machining strip width, a rank-two tensor field is derived to evaluate the machining strip width using ball end mill. The continuous tensor field is able to represent the machining strip widths in all feed directions at each cutter contact point, except at the boundaries between sub-regions. Critical points where the tensor field is discontinuous are defined and classified. By applying critical points in the freeform surface as the start for constructing inside boundaries, the surface could be accurately divided to such that each region contain continuous distribution of feed directions with maximal machining strip width. As a result, tool paths are generated in each sub-surface separately to achieve better machining efficiency. The proposed method was tested using two freeform surfaces and the comparison to several leading existing tool path generation methods is also provided.     

基于重组优化的轮式移动机器人路径处理方法

针对轮式移动机器人采集GNSS（全球卫星导航系统）路径过程中容易出现无效路段的问题,提出了基于自适应分段重组的无效路径剔除方法和多目标优化的路径平滑方法。该方法首先按照航向将采集的GNSS路径划分为DRD(drive-reverse-drive)形式的路段组合,通过设定剔除规则来剔除其中的无效路段,并对剔除无效路段后的离散路段按照邻域路段长度进行第2次分段,以实现有效路段快速重组。其次以重组后的路径点集为决策变量,建立多目标优化函数、端点渐近约束和矩形区域约束,转化为二次规划型进行最优化求解,在保证路径平滑的同时减小与原有路径的位置差。实验结果表明,该方法能够有效处理不同道路形态下采集的无效GNSS路径,优化结果与重组路径平均位置偏差小于0.2 m,平均处理时间为8.8 ms,处理后的路径可用于无人车轨迹跟随。     

基于边折叠和质点-弹簧模型的网格简化优化算法

通过边折叠实现网格曲面简化,提出了保持曲面特征的边折叠基本规则,引入边折叠顺序控制因子λ,给出了折叠点坐标获取方法,简化过程中网格边长度趋于均匀．在曲面简化基础上,利用质点-弹簧模型优化网格形状．将网格顶点邻域参数化到二维域上,在质点-弹簧模型中引入约束弹簧,约束调整网格顶点,并逆映射到三维原始曲面上,局部优化网格顶点的相邻网格;调整曲面上所有网格顶点,在全局上优化网格形状．在曲面简化优化过程中,建立原始模型曲面和简化优化后曲面之间的双向映射关系;曲面的网格顶点始终在原始模型表面上滑动,并以双向Hausdorff距离衡量、控制曲面间的形状误差．应用实例表明：文中算法稳定、高效,适合于任意复杂的二维流形网格．     

常用曲面的变尺寸三角形网格划分

针对常用的旋转面（包括圆锥面、圆柱面、圆台面、圆环面等）、柱状平扫面的特点,在参数平面内采用统一的数据结构表示,将曲面网格划分问题归结为对参数平面的平面网格划分问题。在对参数平面进行网格划分时,根据网格尺寸要求生成适当的内部结点,然后用Ｄｄｌａｕｎａｙ三角化方法生成变尺寸光滑过渡的高质量形,满足有限元高精度分析计算的需要。     

Trimming local and global self-intersections in offset curves/surfaces using distance maps

A robust and efficient algorithm for trimming both local and global self-intersections in offset curves and surfaces is presented. Our scheme is based on the derivation of a rational distance map between the original curve or surface and its offset. By solving a bivariate polynomial equation for an offset curve or a system of three polynomial equations for an offset surface, all local and global self-intersection regions in offset curves or surfaces can be detected. The zero-set of polynomial equation(s) corresponds to the self-intersection regions. These regions are trimmed by projecting the zero-set into an appropriate parameter space. The projection operation simplifies the analysis of the zero-set, which makes the proposed algorithm numerically stable and efficient. Furthermore, in a post-processing step, a numerical marching method is employed, which provides a highly precise scheme for self-intersection elimination in both offset curves and surfaces. The effectiveness of our approach is demonstrated using several experimental results.