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.     

Complete swept volume generation, Part I: Swept volume of a piecewise C-continuous cutter at five-axis milling via Gauss map

In this paper, we present a methodology to generate swept volume of prevailing cutting tools undergoing multi-axis motion and it is proved to be robust and amenable for practical purposes with the help of a series of tests. The exact and complete SV, which is closed from the tool bottom to the top of the shaft, is generated by stitching up envelope profiles calculated by Gauss map.The novel approach finds the swept volume boundary for five-axis milling by extending the basic idea behind Gauss map. It takes piecewise C¹-continuous tool shape into account. At first, the tool shape is transformed from Euclidean space into Tool map (T-Map) on the unit sphere and the velocity vector of a cutter is transformed into Contact map (C-Map) using Gauss map. Then, closed intersection curve is found between T-Map and C-Map on the Gaussian sphere. At last, the inverse Gauss map is exploited to get envelope profile in Euclidean space from the closed curve in the range. To demonstrate its validity, a cutting simulation kernel for five-axis machining has been implemented and applied to mold and die machining.     

Global optimization of tool path for five-axis flank milling with a conical cutter

In this paper, optimum positioning of the conical cutter for five-axis flank milling of slender surfaces is addressed from the perspective of approximating the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. Based on the observation that a conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter is represented as a sphere-swept surface. Then, an approach is presented to efficiently compute the signed distance between a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface is derived. By using the distance function, tool path optimizations for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework, and a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are given to confirm the validity and efficiency of the proposed approach. Comparing with the existing approaches, the present one improves the machining accuracy greatly. The rationale developed applies to general rotary cutters.     

Continuity-preserving tool path generation for minimizing machining errors in five-axis CNC flank milling of ruled surfaces

Five-axis CNC flank machining has been commonly used in the industry for shaping complex geometries. Geometrical errors typically occur in five-axis flank finishing of non-developable surfaces using a cylindrical cutter. Most existing tool path planning methods adjust discrete cutter locations to reduce these errors. An excessive change in the cutter center or axis between consecutive cutter locations may deteriorate the machined surface quality. This study developed a tool path generation method for minimizing geometrical errors on finished surfaces while preserving high-order continuity in the cutter motion. A tool path is described using the moving trajectory of the cutter center and changes in two rotational angles in compact curve representations. An optimization scheme is proposed to search for optimal curve control points and the resulting tool path. A curve subdivision mechanism progressively increases the control points during the search process. Simulation results confirm that the proposed method not only enhances the computational efficiency of tool path generation but also improves the machined surface finish. This study provides a computational approach for precision tool path planning in five-axis CNC flank finishing of ruled surfaces.     

Computing of the actual shape of removed material for five-axis flat-end milling

This paper describes an algorithm that predicts the shape of material removed by a flat-end milling tool, and this may be used to compute machining strip width and scallop height at different positions of the tool path track. The algorithm computes swept sections, profiles which are swept by a moving tool bottom by passing through given planes. The technique is applicable for finish and semi-finish multi-axis milling strategies that use flat-end tools. For these strategies, the algorithm complexity can be reduced from computation of the 3D envelope of swept volumes to computation of plane-circle intersections. A new adaptive derivative-free method to sample tool motion provides robust means to generate intermediate tool positions. The step length is constrained by and dependent on different geometrical measures. At each point of a tool path, in the plane perpendicular to the cutting direction, the bottom profile of the swept section is an estimate of the profile of material left. By calculating the distance between part geometry and the computed profile of removed material, machining strip width and a scallop profile can be derived. These results can be used by tool path generation and validation routines to accurately determine the step-over between tool path tracks and surface quality.     

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.     

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.     

Tool selection for five-axis curvature matched machining

This paper presents an automatic cutting tool selection methodology for five-axis finish surface machining based on the techniques of curvature matched machining. The criterion for cutter selection is to minimize the machine errors and to maximize material removal rate using an optimal filleted end mill selected from a standard cutting tool library. Tool parameters investigated include cutter radius, cutter corner radius and cutter length. The maximum swept silhouette of the inclined tool is proposed and implemented as tool radii selection protocols for matching the change in surface curvature. Algorithms for detection and correction of local tool gouging and global tool interference are presented. The local distance between the cutter bottom and the surface is used to detect and correct local tool gouging. Global tool interference detection and correction is solved by studying the shortest distance between the part surface and the cutter body axis. A faceted approach is used to accelerate the distance calculations. The solution to the local and global gouging problems leads to the shortest, most rigid, tool in the library. These methods of automatic tool selection have been implemented in ROBLINE using the C-language on the system. ROBLINE is a precursor to CODE (Cimetrix Open Development Environment) which is a complete commercial off-line/on-line machine modeling, development and control package. Machined examples confirm the effectiveness of these methods.     

Second order approximation of tool envelope surface for 5-axis machining with single point contact

For 5-axis machining with single point contact, this paper proposes a method to calculate second order approximation of the tool envelope surface by using only one tool position. As we known, the true machining errors are deviations between designed surface and tool envelope surface. But it is impossible to determine the whole shape of the tool envelope surface before all tool positions are obtained. Hence, it is difficult to position the tool individually and consider true errors at the same time. Basic Curvature Equations of Locally Tool Positioning (BCELTP) are presented to solve this problem in some degree. By using them under some special conditions, given one tool position, the local shape (second order approximation) of the tool envelope surface can be calculated precisely at the corresponding cutter contact point. These equations make it convenient to adjust the tool position individually until true errors are reduced in some degree. So, they are useful for optimizing tool positions locally. Finally, some examples are given to verify the correctness and practicability of theory.     

Complete swept volume generation — Part II: NC simulation of self-penetration via comprehensive analysis of envelope profiles

In this paper the swept volume with self-penetration (or self-intersection) of the cutter is presented. The complete swept volume (SV), which describes the side and bottom shape of a milling cutter undergoing self-penetration, is generated by using the Gauss map method proposed in the authors’ previous paper [Lee SW, Nestler A. Complete swept volume generation—part I: swept volume of a piecewise C¹-continuous cutter at five-axis milling via Gauss map. Computer-Aided Design 2011; 43(4): 427–41]. Based on the Gauss map method, the comprehensive analysis of envelope profiles of the tool is accomplished. Through the analysis the necessary condition of the self-penetration of the cutter at five-axis movement is identified. After having classified movement types of the milling cutter in an in-depth manner, the topologically consistent boundary of SV is generated by trimming the invalid facets interior to the SV. To demonstrate the validity of the proposed method, a cutting simulation kernel for five-axis machining has been implemented and applied to cavity machining examples such as intake ports of automobile engines and so forth where the self-penetration occurs. The proposed method is proved to be robust and amenable for the practical purpose of the NC simulation.     

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。     

Estimation of tool orientation contour errors for five-axismachining

Recently, there has been a growth of interest in high-precision machining in multi-axis feed drive systems, which are subject to problems such as friction, cutting force and incompatibility of individual driving axis dynamics. Tracking errors in an individual driving axis during five-axis machining result in tool tip contour error and tool orientation contour error. Based on the conventional definition of the tool orientation contour error, that is, it is the deviation in the normal direction from the desired orientation in spherical coordinates, even if the tool tip and tool orientation contour errors are very small, a mismatch between the tool tip position and tool orientation causes an overcut or undercut when these errors are treated independently. To address this problem of mismatch, this paper presents a new definition of the actual tool orientation contour error. This definition considers synchronization between the tool tip and tool orientation contour errors. In addition, we propose an estimation model for the tool orientation contour error. Experimental results demonstrate that the proposed model provides a better indication of the actual tool orientation contour error than the conventional definition.     

数控仿真技术的回顾与评述

本文在对当前国内外数控切削加工几何仿真和切削过程力学仿真的两个并行发展的研究领域的研究现状和成果的分析评述的基础上，指出了仿真的发展趋势，并提出一种几何仿真和力学仿真相结合的更精确的试切模型。     

Generating cutter swept envelopes in five-axis milling by two-parameter families of spheres

This paper presents an efficient parametric approach of determining the shape of the envelope surface by a generalized cutter that follows five-axis tool path during NC machining. In this approach the cutter is modeled as a canal surface. By considering the tool motions the cutter is decomposed into a set of characteristic and great circles which are generated by two-parameter families of spheres. The center of a sphere from these families is described by two parameters which represent the spine curve and the tool path, and the radius of the sphere is described by one parameter representing the spine curve. Considering the relationship between characteristic and great circles the grazing points on the tool surface are identified. Analytically it is proven for the NC cutter geometries that any point on the envelope surface is located at the intersection of the characteristic and great circles. Then based on the proofs a closed-form solution for computing the grazing points generated by a surface of revolution is presented. The presented methodology is reduced to a simpler parametric form when the NC cutters are described by pipe surfaces.     

Machine models and tool motions for simulating five-axis machining

This paper presents a method of determining the tool motion of a five-axis machine. The method is motivated by the imprint point method, where points on the machined surface are computed based on the tool position and tool motion. While simple linear motion can be used as a coarse approximation to this motion, this paper looks at more accurate models of tool motion based on machine kinematics that can be generalized and applied to any CNC machine with one tool head. A kinematic chain is created by decomposing the linear motion of a machine’s translational and rotational axes into parameterized affine transformations, and a hierarchical model of the machine combines the transformations to provide an accurate model of machine motion.     

Geometric parameter optimization in multi-axis machining

This paper presents a systematic method for the determination of optimal geometric machining parameters in multi-axis machining. Machining accuracy is considered to be determined by a set of geometric parameters: the design parameters of the cutter, the positioning of the cutter, the orientation of the cutter etc. First, we formulate the general nonlinear constrained optimization model of the machining process. The optimal machining result is expected to produce the least deviation between the designed surface and the actual surface. This objective is accomplished by minimizing the deviation between the designed surface and the actual surface during machining. The details of how to characterize and calculate the deviation is then discussed for both ruled surface milling and general free-form surface milling. The swept surface is developed based on robotic manipulation and is used to model the actual surface. A signed distance function is constructed to perform the comparison which returns the signed distance from each sampled point to the designed surface. The direct search algorithm (Nelder-Mead simplex algorithm and pattern search algorithm in this paper) is used to solve our optimization problems due to possible discontinuity of the objective function and large nonlinearity of the problem. Three numerical examples and necessary comparisons are given to demonstrate the effectiveness of our method. The first example shows the generation of the swept volume of a filled-end cutter. The second example employs the swept surface generation method to solve a parameter optimization problem. Sensitivity analysis is performed for the parameters critical to machining accuracy. The third example optimizes the cutter orientation relative to the part surface to minimize the kinematics error caused by kinematics transformation and interpolation of multi-axis machines.     

An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears

To transfer power, a pair of spiral bevel or hypoid gears engages. From beginning to end of two tooth surfaces engaging with each other: for their rigid property, they contact at different points; and for their plastic property, they contact at small ellipses around the points. On each surface, the contact line (or called as contact path) by connecting these points and the contact area by joining these ellipses are critical to driving performance. Therefore, to machine these surfaces, it is important to machine the contact line and area with higher accuracy than other areas. Five-axis flank milling is efficient and is widely used in industry. However, tool paths for flank milling the gears, which are generated with the existing methods, can cause overcuts on the contact area with large machining errors. To overcome this problem, an innovative approach to NC programming for accurate and efficient five-axis flank milling of spiral bevel or hypoid gears is proposed. First, the necessary conditions of the cutter envelope surface tangent with the designed surface along a designed line are derived to address the overcut problem of five-axis milling. Second, the tooth surface including the contact line and area are represented using their machining and meshing models. Third, according to the tooth surface model, an optimization method based on the necessary conditions is proposed to plan the cutter location and orientation for flank milling the tooth surface. By using these planned tool paths, the overcut problem is eliminated and the machining errors of contact area are reduced. The proposed approach can significantly promote flank milling in the gear manufacturing industry.     

Simultaneous optimization of tool path and shape for five-axis flank milling

By representing the swept envelope of a generic rotary cutter as a sphere-swept surface, our previous work on distance function based tool path optimization is extended to develop the model and algorithm for simultaneous optimization of the tool path and shape for five-axis flank milling. If the tool path is fixed, a novel tool shape optimization method is obtained. If the tool shape is fixed, a tool path optimization method applicable to any rotary cutter is obtained. The approach applies to non-ruled surfaces, and also finds applications in cutter dimension optimization and flank millable surface design. Numerical examples are given to confirm its validity.     

NURBS扫描体的逼近算法研究

提供了一种NURBS扫描体的逼近方法。该方法主要步骤：(1)通过系列平面切割，把NURBS曲面(实体)进行降维处理，变成平面曲线；(2)为曲线设置局部标架；(3)在局部标架下求出每一曲线在每一时刻的极值点后将其转换成原曲线的奇异点；(4)使用marching cubes算法剔除扫描体内部点，保留扫描体边界上的奇异点；(5)由所有保留点拟合成奇异曲面。本算法能较好地逼近NURBS扫描体，其逼近精度可通过控制切割精度和扫描过程中时间间隔的选取而得到有效控制。     

Tool path planning for 5-axis flank milling of ruled surfaces considering CNC linear interpolation

This paper investigates tool path planning for 5-axis flank milling of ruled surfaces in consideration of CNC linear interpolation. Simulation analyses for machining error show insights into the tool motion that generates a precision machined surface. Contradicting to previous thoughts, the resultant tool path does not necessarily produce minimal machining error when the cutter contacts the rulings of a developable surface. This effect becomes more significant as the distance between two cutter locations is increased. An optimizing approach that adjusts the tool position locally may not produce minimal error as far as the entire surface is concerned. The optimal tool path computed by a global search scheme based on dynamic programming supports this argument. A flank milling experiment and CMM measurement further validate the findings of this work.