Five-axis tool path generation for a flat-end tool based on iso-conic partitioning

doi:10.1016/j.cad.2008.09.005

Computer-Aided Design

Volume 40, Issue 12, December 2008, Pages 1067-1079

https://doi.org/10.1016/j.cad.2008.09.005 Get rights and content

Abstract

Traditionally, for the flat-end tool, due to the intertwined dependence relationship between its axis and reference point, most 5-axis tool-path generation algorithms take a decoupled two-stage strategy: first, the so-called cutter contact (CC) curves are placed on the part surface; then, for each CC curve, tool orientations are decided that will accommodate local and/or global constraints such as minimum local gouging and global collision avoidance. For the former stage, usually simplistic “offset” methods are adopted to determine the cutter contact curves, such as the iso-parametric or iso-plane method; whereas for the latter, a common practice is to assign fixed tilt and yaw angle to the tool axis regardless the local curvature information and, in the case of considering global interference, the tool orientation is decided solely based on avoiding global collision but ignoring important local machining efficiency issues. This independence between the placement of CC curves and the determination of tool orientations, as well as the rigid way in which the tilt and yaw angle get assigned, incurs many undesired problems, such as the abrupt change of tool orientations, the reduced efficiency in machining, the reduced finishing surface quality, the unnecessary dynamic loading on the machine, etc. In this paper, we present a 5-axis tool-path generation algorithm that aims at alleviating these problems and thus improving the machining efficiency and accuracy. In our algorithm, the CC curves are contour lines on the part surface that satisfy the iso-conic property — the surface normal vectors on each CC curve fall on a right small circle on the Gaussian sphere, and the tool orientations associated to a CC curve are determined by the principle of minimum tilt (also sometimes called lead) angle that seeks fastest cutting rate without local gouging. Together with an elaborate scheme for determining the step-over distance between adjacent CC curves that seeks maximum material removal, the presented algorithm offers some plausible advantages over most existing 5-axis tool-path generation algorithms, particularly in terms of reducing the angular velocity and acceleration of the rotary axes of the machine. The simulation experiments of the proposed algorithm and their comparison with a leading commercial CAM software toolbox are also provided that demonstrate the claimed advantages.

Introduction

5-axis Numerical Controlled (NC) machining produces better quality surfaces and is more efficient than 3-axis NC machining. However, on the other hand, the two rotational axes also make its control and planning much more difficult. Over the past 15 years or so, more and more researchers have focused their study on 5-axis machining. Among a number of issues, the following are the most focused.

On local gouging detection and prevention, several techniques [1], [2], [3], [4], [5] have been developed that are based on comparing the effective cutting curvature of the tool’s swept surface with the normal curvature of the part surface at the contact point. And others, such as the Rolling Ball Method [6], the Arc-intersection Method [7], the Penetration–elimination Method [8] and the algorithm proposed by Xu [9], are area-based methods. Yoon [10] and Jensen [11] not only considered local gouging but also an improvement of the process efficiency by selecting the best tool orientations.

Global interference detection and avoidance has remained to be a major challenge. Lee [12] used quick feasibility checking and detailed feasibility checking to judge 2D and 3D collision between the cutter and the sculpture surface. Morishige [13] used the 2D configuration space to describe the relationship between postures of a ball-end tool and their collision with the environment. Vafaeesefa [14] projected infeasible domains to a unit sphere centered at the cutter contact point to find the feasible tool orientations. Ho [15] handled the real-time collision detection problem with haptic rendering. Ilushin [16] proposed an approach that uses space subdivision techniques and ray-tracing algorithms to derive a surface-tool intersection algorithm to find better results. To reduce the calculation time, Umehara [17] proposed the idea of expanding a group of interference-free tool postures to fill the machined surface before defining the cutter contact points. Besides these, Lauwers [18], [19] also considered collisions between the machine and part or the tool.

How to plan a 5-axis tool path with both local and global gouging considered has remained the most difficult problem. The tool path planning strategies can be classified into five groups: the iso-parametric [5], the iso-offset [20], the iso-planar [21], [22], [23], [24], the iso-scallop [25], [26], [27] and the iso-phote [28]. Most tool paths are generated in the iso-parametric direction due to its straightforwardness. The iso-planar method is often preferred for three-axis machining and is capable of dealing with trimmed surfaces, despite its heavy computing cost due to the need of surface intersection. In the iso-scallop height approach, the initial tool path is the center line or the boundary curve of the part, and the following paths are offsetting lines of the current tool path. The iso-phote method and its variances are only suitable for three-axis machining, in which some inclination curves are defined as the initial paths, and other tool paths are generated by offsetting the inclination curves.

Otherwise, there have been some new developments in 5-axis tool path generation, apart from the aforementioned. They are the principal axis method [3], the fractal method [29], the guide surface method [30], the maximum kinematics performance based method [31], the contour-parallel offset method [32], [33], [34], the boundary conforming method [35], the spiral cutting method [36], [37], etc. In [38], Chiou and Lee introduced the machining potential field method that tries to define a series of contact points that should obtain the largest feasible machining strip width so as to reduce the machining time and improve the surface quality.

Traditionally, tool orientation determination was studied mainly concerning global collision avoidance but without considering other due factors such as kinematic or dynamic performance. Morishige [39] improved his original method in [13] and extended it to 3D configuration space, which is able to precisely describe the relationship between ball-end tool postures and the existence of collision. Jun et al. [40] proposed a method, which is also C-space based, to find better tool orientations with both local gouging and global interference considered. In their method, the minimum cusp height is used as the objective function for determining tool orientations, in addition to the interference-free constraint. Chiou [41] proposed a swept envelope approach to finding optimal tool orientations with the considerations of gouge and collisions. Balasubramaniam [42] used the concept of visibility to generate globally collision-free 5-axis tool postures. Gian [43] used open regions and vector fields to generate the tool orientation in 5-axis NC machining of cavity regions. If the orientation of the tool is predefined, the CL surface deformation approach introduced by Kim [44] could be used. Hsueh [45] proposed a method to generate the tool orientation automatically. First, the tilting collision-free angle range in the plane normal to the tool path is determined. Then, the corresponding collision-free yaw angle range is formed by intersecting the neighboring surfaces and the cone generated by the tilt collision-free angle range.

There is a number of limitations in the above mentioned methods for the determination of tool orientations. First, the tool is almost always assumed to be of the ball-end type. Second, and more critically, a series of tool postures computed by these algorithms often requires a drastic change in tool’s orientation between neighboring tool contact points. Such an extreme change in orientation can never be feasible in real machining due to the physical limits of the angular velocity and acceleration of the rotary motions of the machine tool. Meanwhile, the dynamics of machining tool will increase surface deviation (cf. [46]). As a remedy, a smoothing technique is usually adopted to lessen the angular change (cf. [39], [40], [47]). Recently, we presented a method [48] that uses VMap and FMap to determine the tool orientations that meet the angular velocity constraints. Nevertheless, surface properties are never considered in these remedy solutions.

When planning the 5-axis tool paths for a flat-end tool, which is preferred over ball-end type tools but whose computation is much more difficult than that of the ball-end tool due to the dependency of the reference point of the tool on its orientation, a particular tool posture is represented by the cutter contact (CC) point and the orientation of the tool. Except for very few works (cf. [38], [50]), the determination of the CC points has universally been in the offset-fashion: an initial CC curve is first decided (usually the boundary of the part surface), and subsequent CC curves are just offset curves of the previous ones with a constant step-over distance measured either in the parametric domain or geodesically on the part surface. Moreover, the determination of the CC points and tool orientations are decoupled — a CC point is established without concerning the eventual orientation of the tool at the point.

In this paper, we present an implemented work in automatic generation of 5-axis tool paths for a flat-end tool that seeks better correlation between CC points and tool orientations and also the overall planning/ordering of the CC points on the part surface. We introduce a concept called iso-conic that retrains the tool orientations along a CC curve to a right small circle on the Gaussian sphere. This constraint, together with the minimum inclination angle of the tool axis defined at a CC point, uniquely defines the tool axis at the CC point. The resultant tool postures along the CC curve enjoy the good characteristic of small change in tool orientations while at the same time achieving a faster material removal rate. And, by orchestrating a carefully designed propagation scheme for neighboring CC curves that seeks maximum machining strips, our algorithm outputs 5-axis tool paths that are believed to perform better in terms of the dynamics of the machine (i.e., the slower angular velocity and acceleration of the machine rotary axes, though not the overshoot and other issues due to potential large geodesic curvatures on the CC curves) and the machining efficiency (i.e., faster material removal rate), as well as the surface quality of the finished part.

Section snippets

Preliminaries

Let our part surface $S$ be a $C^{2}$ surface with a parameter $u – v$ domain. A flat-end cutter could be fixed at any CC point on S with surface normal $N_{p}$ and tool axis $T$ , as shown in Fig. 1. Assuming that $T$ is not parallel with $N_{p}$ , the Cutter Location (CL) point could be calculated using Eq. (1), where $r$ is the radius of the cutter. $CL = CC + r \frac{N_{p} - (T \cdot N_{p}) N_{p}}{‖ N_{p} - (T \cdot N_{p}) N_{p} ‖} .$ There are two spherical coordinate systems shown in Fig. 1(a) and Fig. 1(b) — the Machine Coordinate System (MCS) and the Local Coordinate

Tool posture determination

For the time being, suppose that a list of CC points, { ${CC}_{1}$ , ${CC}_{2}, \dots, {CC}_{n}$ } that comprise a tool path, is already available. (How it will be generated will be discussed in the next section.) We are to determine the corresponding tilt angles ${α_{1} α_{2}, \dots, α_{n}}$ and yaw angles ${β_{1} β_{2}, \dots, β_{n}}$ .

At a CC point ${CC}_{i}$ , if the tilt angle $α_{i}$ is larger than some minimal angle $α_{min}$ , local gouging will not occur. We define the local gouging constraint cone (LGCC) at ${CC}_{i}$ to be the cone with its apex on ${CC}_{i}$ , its symmetry axis

Tool path determination

In the previous section, there is a pre-assumption that the surface normal vectors at the tool path ${{CC}_{1}, {CC}_{2}, \dots, {CC}_{n}}$ all fall on some cone (the iso-conic property). This requirement is actually the central point in our overall planning of tool paths. Let the part surface be given as a parametric surface, i.e., $S = (X (u, v), Y (u, v), Z (u, v)), (u, v) \in Φ \in {[0, 1]}^{2}$ where $Φ$ is some connected region in the $u – v$ plane. To facilitate our task, we define a scalar function $ϕ (u, v)$ , called the inclination function, in

Overall partitioning algorithm

Based on the single cluster generation procedure EXPANSION, the overall algorithm for generating all the clusters (the overall partitioning algorithm) is readily available. First of all, however, it is beneficial to note this observation: if a cluster does not contain an elliptic singular point, it must end on the boundary of the $u – v$ domain. For example, Fig. 11 depicts the partitioning of the part surface shown in Fig. 6(a) (also referring to Fig. 6(c)), in which all the clusters end at the

Consideration of global collision avoidance

So far in our work, global interference has not been considered. In the presence of global obstacles (and the part itself), the iso-conic condition imposed on a path some times might not hold. In practice, usually local machining quality and global interference are considered independently. This in large part is due to the fact that for many kinds of parts in which local quality is the utmost requirement, such as optical lenses, high quality watch frames, high precision instruments, etc., there

Experimental results and discussions

We have implemented our algorithm and applied to several examples, and compared the simulated cutting results to that of a leading commercial CAM software package.

The first test and comparison are shown in Fig. 13, in which the part surface is the one shown in Fig. 6(a). Fig. 13(a1) depicts the tool paths (including the tool retractions) generated by our algorithm, (a2) is the simulated machined surface, and (a3) shows the tool orientations on some tool paths. Fig. 13(b1)–(b3) show the

Conclusions

In this paper, we have presented a 5-axis tool-path generation algorithm for a flat-end tool. The primary goal of this work is to address some lingering and serious problems associated with most existing 5-axis tool-path generation algorithms. Specifically, the proposed algorithm aims at finding a better placement of CC curves on the part surface such that the tilt and yaw angle of the tool change adaptively according to the local geometry of the CC point so as to avoid or minimize the local

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