A mapping-based approach to eliminating self-intersection of offset paths on mesh surfaces for CNC machining

Highlights

• A method of eliminating self-intersection of offset paths on mesh surfaces is proposed.

• The new orientation rule based on the local loop is developed and discussed in detail.

• The method of rounding the sharp corner is preliminarily investigated.

• Simulation and real machining are performed.

Abstract

Geometrically, a tool path can be generated by successively offsetting its adjacent path on the surface with a given path interval, which preferably starts from one of the surface boundaries or a primary curve. The key issues involved in offset path planning are the generation of raw offset paths and the elimination of the self-intersection of raw offset paths. Most researches available in this area are focused on how to generate the raw offset paths, however, the latter, especially how to eliminate the self-intersection of the offset paths on mesh surfaces, has not been sufficiently addressed. In this paper, a mapping-based approach to eliminating the self-intersection of offset paths is proposed for the CNC machining of mesh surfaces. The method first flattens the mesh surface onto a predefined plane by using a mesh mapping technique, and then taking the mapping as a guide, the offset paths are also naturally mapped onto the plane, from which those invalid self-intersection loops can be effectively identified and eliminated. To handle the issue of self-intersection for all types of offset path, a notion of local loop is introduced to detect and eliminate the invalid self-intersection loops. After that the planar paths are inversely mapped into the physical space and the final tool paths used for the machining of mesh surface are obtained. Meanwhile, in order to improve the kinematic and dynamic performance of the machine tool when machining along the generated offset paths, a method for rounding the sharp corners of tool paths, which result from the process of eliminating the self-intersection of raw offset paths, is also preliminarily investigated. Finally, the proposed method is validated by the results of simulations and machining experiments.

Introduction

Computer numerical control (CNC) machining has been commonly used in the aerospace, automobile, die/mold industries. Since CNC machine tools cut a part out of a blank by moving the cutter along a given tool path, generating an efficient tool path becomes very important for sculptured surface machining. At present, although some primary tool path generation methods such as iso-parametric method  [1], [2], iso-planar method  [3], [4], [5] and iso-scallop method  [6], [7], [8] have been proposed, most of these methods are only applicable to parametric surfaces. Straightforward extensions of these methods to other type of surfaces are not obvious to us. For example, few of them can be directly used for the machining of mesh surfaces. From the manufacturing point of view, parametric models utilized widely in design stage may not be a good choice for the purposes of CAM and process planning due to its complexity in data exchange between different CAD/CAM systems and geometric computations related to tool path generation  [9], [10], [11], [12].

Compared with parametric model, the mesh model saved by STL format only simply records the positions of vertices and the normal vectors of all facets, so that it can be handily translated between different CAD/CAM systems, and its geometric computation is much simpler than that of parametric model due to the linear model representation. As an alternative it is often confronted with the issue of generating tool paths from mesh surfaces and is now becoming the focus of interest. In most previous studies  [13], [14], [15], [16], [17], mesh surfaces are usually used to check the gouge and collision in CNC machining, and almost only one topology of iso-planar type is adopted for tool path generation. Recently, several different strategies have been proposed for planning tool paths based on mesh surfaces. For example, Kim and Yang  [18] first proposed the iso-scallop method for mesh surfaces by means of CL surface deformation and mesh slicing technique. Lee et al.  [19] extended the concepts of conventional drive surface and guide plane, and used a series of planes at regular intervals to slice the CL surface to generate iso-scallop tool paths. Sun et al.  [20] and Li et al.  [21] proposed the methods of generating contour-parallel tool paths for mesh surfaces. In their works, the offset paths are planned on the plane, but not directly on the original mesh surface. Latter, Xu et al.  [22] proposed another method of generating closed contour-parallel tool paths on mesh surfaces by means of mesh flattening technique. Geometrically, iso-scallop and contour-parallel tool paths can both be regarded as offsets of a target curve on mesh surfaces. Up to date, two types of offsets on 3D spatial surfaces, namely geodesic offsets and functional offsets, have been proposed for diverse applications  [23]. The former is based on the geodesic metric, which is essentially a non-Euclidean distance metric and needs the construction of Voronoi diagram as a premise. It is mainly used in pattern analysis. The latter is a dominant method applied in NC machining since it takes a classical Euclidean distance metric in computing the path interval. In CNC machining, the functional offsets of a tool path may often intersect in more complicated ways than that in planar domain. By far, the issue of self-intersection of offset paths on mesh surfaces, especially aiming at that of open offset paths with variable path intervals (e.g. iso-scallop tool path and also be called open non-uniform offset paths), has either not been addressed  [18], [19] or addressed insufficiently  [20], [21], [22], [24], [25] in the previous works.

Offset paths in machining are usually represented by directed line segments linked end to end on the mesh surface, and as a result it will bring the issue that some path segments should intersect theoretically at the self-intersection points, but actually not intersect each other due to the linear expression, so the direct calculation of intersection points and the removal of self-intersection interferences on the mesh surface is highly undesirable. In  [26], Lee calculated the self-intersection points of offset paths by projecting them onto $XY$ plane along $Z$ axis, however this method is only suitable for convex surfaces. When projecting the offset paths on non-convex surfaces, the corresponding planar paths on $XY$ plane may possibly produce such self-intersection points which do not exist in offset paths. Similarly, Hatna et al.  [27] and Tam et al.  [23] mapped the offset paths back onto the uv parameter plane via the equation of surface, on which the self-intersection interference is removed. The method works well for parametric surfaces, but it is infeasible for mesh surface because of its dependency on the parameter domain. Some other methods  [28], [29], [30], [31] were also proposed specifically for the planar offset path generation of pockets, and their extensions to offset paths on mesh surfaces are still not obvious and have not been reported yet. Moreover, most of these methods are limited to uniform offsets paths of a closed curve. Needless to say, the identification and removal of self-intersection of offset paths on the mesh surface is a complicated issue, and more researches are needed to be done.

Based on our previous work  [22], this paper develops a unified mapping-based method of eliminating the self-intersection of all types of offset paths on mesh surfaces, including uniform or non-uniform, open or closed offset paths. Mesh parameterization is introduced to realize the bidirectional mapping between the mesh surface and the plane, which enables that the identification and removal of complex self-intersection interferences are reduced from 3D surface to 2D plane. A principle of orientation rule is adopted and has been extended to deal with the issue of the self-intersection of more complex open or necked offset paths by means of the proposed notion of the local loop. An approach to rounding sharp corners on the generated offset paths is also investigated for improving the kinematic and dynamic performance of the machine tools. The rest of this paper starts with raw offset path generation and the detail of self-intersection elimination are discussed in Section  3. Sharp corner rounding is investigated in Section  4. Section  5 shows the results of simulations and experiments with discussion. Section  6 concludes the paper.