Analytical estimation of error in flank milling of ruled surfaces

This article introduces a method to estimate geometric error during flank milling of a ruled surface. The various positioning schemes developed by researchers are intended to reduce this geometric error in order to mill with larger sized milling cutters while respecting the tolerance interval. There are two trends in positioning: either positioning is simple and right from the start it is easy to determine design of the maximum allowed milling cutter radius, or positioning is complex and determination of the maximum milling cutter dimensions can only be conducted after digital calculations of the error. It will then be necessary to choose another milling cutter radius and recommence the positioning procedure and error calculation in order to validate the tool. In the present study, a method to estimate error in the scope of complex positioning is presented. The aim is to be capable of choosing a maximum cutting tool radius that respects the tolerance interval.     

Arc-intersect method for 5-axis tool positioning

A new method for 5-axis CNC tool positioning is presented in this paper that improves upon a previous tool positioning strategy named the rolling ball method (RBM), which was developed by the present authors [Gray P, Bedi F, Ismail S. Rolling ball method for 5-axis surface machining. Comput Aided Des 2003;35(4):347-57]. The special property of the RBM is that it computes tool positions by considering the area beneath the tool that the tool will be positioned to cut instead of using surface curvatures computed at a single point on the surface. This enables the RBM to generate gouge-free tool positions without secondary iterative gouge-check and correction algorithms. However, the RBM generates conservative tilt angles in order to guarantee gouge-free tool positions. The new arc-intersect method (AIM) presented in this paper improves upon the RBM by directly positioning the tool to contact the surface and thereby eliminates the conservative nature of the RBM to give optimal tool positions. Like the RBM, the AIM is an area-based method that generates gouge-free tool positions without the use of iterative gouge-check and correction algorithms. The implementation described in this paper uses triangulated surfaces and the computer's graphics hardware to assist in the tool position calculations. However, the method can be applied to any surface representation since it only uses surface coordinates and surface normals for computation. A section of a stamping die was machined to demonstrate the AIM and to show the improvement over the RBM and for comparison with 3-axis ballnose machining. The results showed that the AIM was 1.33 times faster than the RBM and that the AIM, with single direction parallel tool passes, was 1.62 times faster than a zig-zag pattern 3-axis ballnose tool path for the same feed rate, cusp height and tool diameter. The workpieces were measured with a CMM and the data were compared to the CAD model to show no gouging occurred and to check the cusp heights.     

Improved positioning of cylindrical cutter for flank milling ruled surfaces

An optimized positioning procedure for flank milling ruled surfaces with cylindrical cutter is described in the paper. The tool axis trajectory surface is a ruled surface, which is generated by moving the tool axis. The proposition that the envelope surface of cylindrical cutter is the offset surface of tool axis trajectory surface is proved using kinematics approach. It is a complement of Bedi's [Bedi S, Mann S, Menzel C. Flank milling with flat end cutter. Comput Aided Des 2003; 35:293-300] analysis about the envelope surface of cylindrical cutter. Subsequently, we get another proposition that the deviation at extremum point between the designed surface and the envelope surface of cylindrical cutter is equal to that between the offset surface of designed surface and the tool axis trajectory surface. Based on this proposition, we propose three points offset (TPO) strategy to approximate the offset surface. In order to reduce errors further, a simple least square approximation scheme is established to make the tool axis trajectory surface fit the offset surface of designed surface as much as possible. By solving the linear system of equations, the tool axis trajectory surface is deformed. Simultaneously, the corresponding envelope surface is deformed to approximate the designed surface better. Two examples are given to verify the effectiveness of the developed 5-axis flank milling technique.     

Optimize tool paths of flank milling with generic cutters based on approximation using the tool envelope surface

This paper presents a global optimization method to generate a tool path for flank milling free-form surfaces with a generic cutter based on approximation using the tool envelope surface. It is an extension of our previous work [Gong Hu, Cao Li-Xin, Liu Jian. Improved positioning of cylindrical cutter for flank milling ruled surfaces. Computer Aided Design 2005; 37:1205–13]. First, given initial tool path or tool axis trajectory surface, the grazing points of the tool envelope surface can be calculated. Second, the errors between the tool envelope surface and the designed surface along the normal direction of the tool envelope surface are calculated. Based on this new definition of error, an optimization model is established to get the global optimized tool axis trajectory surface. In order to simplify the calculation, two variants of this method based on the least square criterion are proposed to solve this model. Since this method is really based on the tool envelope surface, it can reduce the initial machining errors effectively. The proposed method can be used not only for cylindrical cutters and conical cutters, but also for generic cutters with a surface of revolution. In addition to ruled surfaces, it also can be used for machining non-ruled surfaces. Finally, several examples are given to prove its effectiveness and accuracy. The generated tool paths and calculated grazing points for test are available in supplementary files for the readers’ convenience in verifying this work in different CAD/CAM systems.     

Generation of reciprocating tool motion in 5-axis flank milling based on particle swarm optimization

This paper proposes a novel method for generation of optimized tool path in 5-axis flank milling of ruled surfaces based on Particle Swarm Optimization (PSO). The 3D geometric problem, tool path generation, is transformed into a mathematical programming task with the machined surface error as the objective function in the optimization. This approach overcomes the limitation of greedy planning methods employed by most previous studies. By allowing the cutter to move backforward, reciprocating tool path produces smaller machining error compared with the traditional one consisting of only forward cutter movement. A cutting experiment is conducted with different tool paths and the CMM measurement verifies the effectiveness of the proposed method.     

5-axis flank milling free-form surfaces considering constraints

A new tool path generation method of flank milling considering constraints is proposed for ball-end cutters in this paper. It will not only reduce the machining error range but also meet the following two constraints: (a) The ball end of the milling tool is tangential to the constraint surface; (b) There is no overcut and the minimum error is zero, which is called nonnegative-error constraint. The two constraints are very useful in some situations of engineering applications, such as flank milling impeller blades. Based on the proposed method, two types of cutter will be used to generate tool paths for the same designed surface and constraint surface. The effectiveness and accuracy of the proposed method will be finally proved with some examples.     

Accurate tool position for five-axis ruled surface machining by swept envelope approach

This paper presents a swept envelope approach to determining tool position for five-axis ruled surface machining. The initial tool position is traditionally located to contact with two directrices of a ruled surface. The swept profile of the tool is then determined based on the tool motion. By comparing the swept profile with the ruled surface, the tool position is corrected to avoid machining errors. The cutter's swept envelope is further constructed by integrating the intermediate swept profiles, and applied to NC simulation and verification. This paper presents the explicit solution for the swept profile of a taper-end cutter in five-axis ruled surface machining. The relation of the ruled surface geometry, the tool motion and the machining errors is developed. Therefore, the error sources can be detected early and prevented during tool path planning. The explicit swept envelope indicates that the machined surface is not a ruled surface in five-axis ruled surface machining. Manufacturing industries should take extra care in high precision ruled surface machining. Computer illustrations and example demonstrations are shown in this paper. The results reveal that the developed method can accurately position tool location and reduce machining errors for five-axis ruled surface machining.     

插值曲率线的直纹面可展设计

目的 曲率线在微分几何中起着非常重要的作用,它在曲面分析中是一个很有用的工具。可展曲面是曲面造型中最简单也最常用的一类曲面,目前大部分工作都是研究在给定曲面上寻找或者计算曲率线,而其反问题研究工作较少,为此,提出一种插值曲率线的可展曲面构造方法,并进一步将它应用到曲面造型中。方法 利用Frenet标架表示直纹面的母线,根据曲线为曲面曲率线以及曲面可展的充要条件,得到直纹面的母线需要满足的关系式。并引入控制函数控制曲面的形状。结果 给出了以给定曲线为曲率线的直纹面可展的具体表达式,根据可展曲面分类分析了设计曲面为柱面、锥面和空间曲线切线面的充要条件,并给出了两个代表性的实例验证该方法的有效性,实例结果表明,该方法不仅适用于一般参数曲线,对分段参数曲线也是有效的。结论 利用构造性的方法给出了插值曲率线的可展曲面的具体表达形式,并通过具体实例验证了该方法的有效性。     

Analysis of improved positioning in five-axis ruled surface milling using envelope surface

This article covers side milling of ruled surfaces using a milling cutter. Flank milling is useful for machining objects such as impellers, turbine blades, fan vanes and all workpieces defined by non-developable, ruled surfaces. In the present article, we first introduce two types of positioning on ruled surfaces developed within the Toulouse Mechanical Engineering Laboratory. The positioning studied is taken from the geometric situation not taking the instantaneous speed of rotation of the milling cutter into account. The swept profile of the tool is then determined based on the tool motion. Having defined the envelope surface, we seek to analyse improved and standard positioning errors comparing envelope surfaces with the ruled surface. We then introduce an example to illustrate positioning developed through a first theoretical study before experimentation including machining and measurement of the test piece. Finally, we give our conclusions as to the validity of improved positioning without taking the instantaneous speed of rotation of the milling cutter into account.     

一种曲线路径剪裁中刀具形变及其误差分析

动态移动切削阻力载荷对高速数控裁床加工过程中刀具形变及其剪裁误差具有的重要影响,提出了一种适用多层布料/皮革曲线剪裁路径的刀具形变及其误差计算方法;建立了动态负载条件下可伸缩刀具的挠度与转角方程,进而推导出高频振动裁刀剪裁误差及其随切削深度变化规律;计算结果表明,数控布料/皮革剪裁刀的动态载荷、高频振动参数、切削深度对剪裁误差具有重要影响,深入剖析高层数控裁床的加工机理,动态参数数据分析,对于提高机床加工效率,降低加工误差,提高刀具使用寿命具有一定的工程应用价值。     

Rolling ball method for 5-axis surface machining

Curvature matching for 5-axis surface machining has been plagued by the complexity of the task. As a result the current tool positioning strategies are likewise computationally complicated. Gouging the surface has been the main concern and has presented the greatest difficulty in the algorithms. Some of the methods perform exhaustive searches of the surface to avoid gouging while others incrementally adjust the tool orientation until gouges are no longer detected. In this paper a new positioning strategy is presented that is simple to implement and is not difficult to compute. The rolling ball method rolls a variable radius ball along the tool path and positions the cutting tool to cut the rolling ball. A small region of the ball's surface is used to approximate a small region of the surface being machined. The radius of each ball is computed by checking a grid of points in the area of the surface that the tool casts a shadow for each tool position. A pseudo-radius is computed for each grid point and the most appropriate radius is selected to be the rolling ball's radius. The selection process follows a hierarchy of surface profiles ranging from convex to concave. Convex, concave, and saddle (mixed) surface regions are all computed in a similar fashion and there are no special cases for which the positioning strategy must be changed to compute a tool position. Local gouge checking is automatically built-in to the positioning computations so that the typical iterative strategy of checking for gouging, then incrementally tilting the tool until no gouges are detected is eliminated. The method is robust and simple to implement and it only requires surface coordinates and surface normals. A simulation of the method and a cutting test were performed and are presented in this document.     

网格模型中直纹面的提取

提出一种基于直母线族提取与拟合的网格模型直纹面提取方法.首先通过集合误差权排序方法从模型中选择一个可信直母线种子,然后通过局部标架引导搜索邻接直母线,移动标架重复上述搜索过程,直到跨出网格边界或者开始循环搜索.利用"投影"光顺法对齐直母线段族首末端点,再通过定义欧氏6空间下的距离函数,将欧氏3空间下的直线族逼近直纹面问题转换成欧氏6空间下B样条曲线最小二乘拟合问题.为了使逼近的曲面光顺,在曲线拟合过程中引入了能量函数.与其他算法相比,文中方法获得了较强的直母线族的鲁棒性和精确性,并能有效、合理地拟合出光顺直纹面.     

Development of a new meso-scale machine tool for fabricating micro V-grooves

This work presents the development of a meso-scale machine tool with a nanometer resolution. The newly developed meso-scale machine tool consists of a pagoda structure for Z-axis, four HR8 ultrasonic motors, three linear encoders with a resolution of 2 nm, a coaxial counter-balance system, a XY coplanar positioning stage, a rotary stage, a Galil 4-axis motion control card, an industrial PC and a CCD camera system. The optimal geometrical dimensions of the pagoda structure have been determined by ANSYS software. The designed meso-scale machine tool is equipped with an X–Y coplanar positioning stage with nanometer resolution. The coplanar stage developed by National Taiwan University was integrated with two linear encoders, so that a two-axis closed-loop control was possible. A circular positioning test with the radius of 1 mm using the developed stage was tested, and the overall circular positioning error was about 83 nm based on the test results. The micro V-grooves and the micro pyramid cutting tests of the polished oxygen free copper using a single crystal diamond tool on the developed meso-scale machine tool have been performed. The cutting tests under various combination of the depth of cut and cutting speed have been carried out. It revealed that the cutting speed had no great influence on the cutting force. The measured cutting forces for the depth of cut of 5, 10, 15 μm were 1.2, 1.6 and 2.4 N, respectively. The results showed the meso-scale machining tool can be used in micro pyramid structures manufacturing.     

基于四点偏置法的非可展直纹面侧铣刀位计算

针对非可展直纹面五轴侧铣加工的问题,分析了非可展直纹面几何特点,根据等距 映射下的极差不变性,提出了一种计算非可展直纹面叶片五轴侧铣刀位数据的新方法。以刀具包 络面与设计曲面之间的整体误差为优化目标,建立了圆柱铣刀侧铣非可展直纹面的刀位计算方 法,运用四点偏置法确定初始刀位,采用最小二乘法对初始刀位进行优化,建立刀轴矢量偏转模 型进一步修正刀位以减小过切误差。通过实例计算分析,表明该方法可以在一定程度上减小加工 误差。     

Quasi-Developable B-Spline Surface Design with Control Rulings

We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible. Specifically, the terminal ruling lines are treated as hard constraints. Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control, our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely-designed rulings. We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications. The performance and the efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for path planning in 5-axis computer numerical control (CNC) flank milling.     

An accelerated K-means clustering algorithm using selection and erasure rules

The K-means method is a well-known clustering algorithm with an extensive range of applications,such as biological classification,disease analysis,data mining,and image compression.However,the plain K-means method is not fast when the number of clusters or the number of data points becomes large.A modified K-means algorithm was presented by Fahim et al.(2006).The modified algorithm produced clusters whose mean square error was very similar to that of the plain K-means,but the execution time was shorter.In this study,we try to further increase its speed.There are two rules in our method:a selection rule,used to acquire a good candidate as the initial center to be checked,and an erasure rule,used to delete one or many unqualified centers each time a specified condition is satisfied.Our clustering results are identical to those of Fahim et al.(2006).However,our method further cuts computation time when the number of clusters increases.The mathematical reasoning used in our design is included.     

机器视觉识别袜品纹理与精准定位的测控装置

袜品生产属于劳动密集型行业,袜品的裁剪是非常重要的工序之一,目前主要依靠人工裁剪实现,存在工作强度大、效率低、人工成本高等问题,已经制约织袜行业的良性发展.设计和实现一种机器视觉识别袜品纹理与精准定位的测控装置,该装置利用光学成像技术和快速图像采集技术实时获取袜品纹理的动态数字图像,通过提出的Gabor能量积分算法和动态延时定位指令输出方法处理后,实现对袜品特定位置裁剪的精准快速识别定位.经过实际使用验证,该装置裁剪定位精度为±2mm,高于人工裁剪精度,可完全取代人工裁剪方式.一套装置可完成相当于3个工人的工作量,在大幅度提高生产效率的同时降低企业运营成本.     

The Bézier Tangential Surface System: a Robust Dual Representation of Tangent Space

This paper develops a robust dual representation for the tangent space of a rational surface. This dual representation of tangent space is a very useful tool for visibility analysis. Visibility constructs that are directly derivable from the dual representation of this paper include silhouettes, bitangent developables and kernels. It is known that the tangent space of a surface can be represented by a surface in dual space, which we call a tangential surface. Unfortunately, a tangential surface is usually infinite. Therefore, for robust computation, the points at infinity must be clipped from a tangential surface. This clipping requires two complementary refinements, the first to allow clipping and the second to do the clipping. First, three cooperating tangential surfaces are used to model the entire tangent space robustly, each defined within a box. Second, the points at infinity on each tangential surface are clipped away while preserving everything that lies within the box. This clipping only involves subdivision along isoparametric curves, a considerably simpler process than exact trimming to the box. The isoparametric values for this clipping are computed as local extrema from an analysis using Sederbergs piecewise algebraic curves. A construction of the tangential surface of a parametric surface is outlined, and it is shown how the tangential surface of a Bézier surface can be expressed as a rational Bézier surface.     

Inverse velocity analysis for line guidance five-axis robots

In this paper, inverse velocity problem for five-axis robots is investigated. The conventional method for a five-axis robot is to pseudo-inverse the 6×5 Jacobian matrix. The solution, primarily based on six freedoms inverse velocity analysis, is just an approximation with a least-square error. A five-axis robot can exactly guide an axis-symmetrical tool in 3-D space. Two exact solutions are provided for five-axis robots. One is based on the screw motion of the tool. The other is based on spherical angles of the tool to derive a 5×5 Jacobian matrix. A new type of singular configuration is discovered and is called the task singularity. The moving path of the line shaped tool is constructed as a ruled surface. Analysis of the angular acceleration shows the surface constructed based on the spherical angle representation has better characteristic. It is concluded that for five-axis robots, the tool position is better represented by five parameters rather than six parameters in order to get better solutions for inverse velocity as well as the motion planning.     

Coverage trajectory planning for a bush trimming robot arm

A novel motion planning algorithm for robotic bush trimming is presented. The algorithm is based on an optimal route search over a graph. Differently from other works in robotic surface coverage, it entails both accuracy in the surface sweeping task and smoothness in the motion of the robot arm. The proposed method requires the selection of a custom objective function in the joint space for optimal node traversal scheduling, as well as a kinematically constrained time interpolation. The algorithm was tested in simulation using a model of the Jaco arm and three target bush shapes. Analysis of the simulated motions showed how, differently from classical coverage techniques, the proposed algorithm is able to ensure high tool positioning accuracy while avoiding excessive arm motion jerkiness. It was reported that forbidding manipulation posture changes during the cutting phase of the motion is a key element for task accuracy, leading to a decrease of the tool positioning error up to 90%. Furthermore, the algorithm was validated in a real‐world trimming scenario with boxwood bushes. A target of 20 mm accuracy was proposed for a trimming result to be considered successful. Results showed that on average 82% of the bush surface was affected by trimming, and 51% of the trimmed surface was cut within the desired level of accuracy. Despite the fact that the trimming accuracy turned out to be lower than the stated requirements, it was found out this was mainly a consequence of the inaccurate, early stage vision system employed to compute the target trimming surface. By contrast, the trimming motion planning algorithm generated trajectories that smoothly followed their input target and allowed effective branch cutting.