Length-optimal tool path planning for freeform surfaces with preferred feed directions based on Poisson formulation☆
Introduction
Computer-aided design and manufacturing (CAD/CAM) systems have seen applications in many fields, including automotive, shipbuilding, and aerospace industries. One of the essential elements in a CAD/CAM system is tool path planning, which bridges part geometries designed in CAD with cutting processes controlled in CAM [1]. The generated tool paths govern how a three- or five-axis machine tool moves its cutter relative to the part geometry during machining. Therefore, the quality of tool paths directly impacts machining accuracy and efficiency.
Generating high-quality tool paths is, however, no trivial matter due to two major challenges. First, scallops are produced between adjacent tool paths, posing the machining accuracy problem [2]. Second, the machining strip width at a cutter contact point varies with the feed direction and the cutter orientation at that point, necessitating machining efficiency optimization [3]. Ideally, the most efficient (shortest) tool path would have a constant scallop height between adjacent tool paths to avoid redundant machining, and meanwhile have the maximum strip width along individual tool paths to attain maximum material removal. Kumazawa et al. [4] have, however, shown that tool paths having constant scallop height deviate considerably from those following feed directions of maximum strip width. That is, there is an incompatibility between the constant scallop height and the optimal feed directions, and this poses the challenging problem of finding the best trade-off between them.
Most studies related to the above problem focused on one of its two sub-problems, either tool paths having a constant scallop height or tool paths following exactly preferred feed directions. (In the literature, it is customary to call optimal feed directions as preferred feed directions; they are thus used interchangeably in this work.) Recently, some work to integrate the two lines of research has been reported [4]. Nevertheless, preferred feed directions were primarily used to assist the generation of constant scallop height tool paths, e.g., help choose the initial tool path, where little can be said about global optimality. In this paper, a new approach is to be presented to address this global tool path optimization problem.
The proposed method (Sections 3 Methodology, 4 Implementation) expresses the problem stated above in terms of the solution to a Poisson equation (a second-order linear partial differential equation). Unlike many existing methods, we approach the problem using an implicit tool path representation framework: a scalar function defined over the design surface is to be computed, and then tool paths are obtained by extracting appropriate iso-level curves (Fig. 1). Based on this representation, it is found that there is an integral relationship between the optimal tool paths and a vector field collectively representing the two requirements of constant scallop height and preferred feed directions. This allows us to express the tool path optimization problem in terms of a vector field matching problem, which eventually leads to a standard Poisson equation [5]. To the best of the author’s knowledge, this idea of converting tool path planning into Poisson problems has not been previously reported.
Formulating tool path optimization as a Poisson problem offers a number of advantages. Most notably, it generates the globally optimal tool paths in terms of constant scallop height and maximum strip width, and consequently the overall path length can be minimized. It is also conceptually simple and easy to implement as the solution reduces to solving a well-conditioned sparse linear system. In addition, as tool paths are represented as iso-level curves of a scalar function, there is no particular order among them, and thus there is no need to deal with the complex task of determining the initial tool path (a long-standing problem in the tool path planning domain). Another noteworthy benefit is that the use of the implicit representation scheme can automatically handle singularities and self-intersections in tool paths without any tedious, error-prone topological operations [6].
Section snippets
Related work
Tool path planning is an extensively studied problem in CAD/CAM, and many methods have been reported [7]. Among those methods, the categories of interest to this work are the constant scallop height (or iso-scallop) paradigm and the preferred feed direction paradigm. The iso-scallop paradigm was proposed as an improvement to the previous iso-parametric and iso-planar paradigms such that redundant machining observed in those two paradigms can be avoided. This paradigm was initialized by Suresh
Methodology
As already noted, tool paths would have a minimized overall length if they have a constant scallop height and follow preferred feed directions, but in general these two objectives cannot be satisfied concurrently. Minimizing the overall path length thus lies in finding the closest satisfaction of the two objectives, i.e., the optimal trade-off between the constant scallop height and the preferred feed direction field. Leaning towards either side, as in existing methods, could only give
Implementation
This section provides implementation details of the method outlined in the previous section, concerning numerical solutions to the Poisson equation, the consistency issue of feed directions, the singular region issue where preferred feed directions are ill-defined, and the issue of non-smooth preferred feed directions.
Results and discussion
Three case studies, based on a C++ implementation and a 2.4 GHz Intel Core i5 with 8G memory, are to be presented to demonstrate the effectiveness of the proposed method. Case study 1 considered a simple situation where no segmentation is needed; Case study 2 analyzed a comprehensive situation where the surface has to be segmented into multiple patches; Case study 3 involved a standard saddle surface, which was used to carry out error analysis. Case studies 1 and 2 also present comparisons with
Conclusion
A new method has been presented in this paper to generate length-optimal tool paths for freeform surface machining. The main features of this method include the minimum overall length of the generated tool paths and simplicity of the formulation. These features are essentially achieved by (1) formulating the problem of minimizing tool path length as the problem of finding the closest satisfaction of the conditions of constant scallop height and preferred feed directions, and (2) casting the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was partially supported by the Start-Up Grant from ZJU. The author is grateful to Profs. Hsi-Yung Feng and Charlie C.L. Wang for their valuable suggestions, especially at the revision stage.
Qiang Zou is currently an assistant professor of Computer Science at Zhejiang University, China. He got his Ph.D. from University of British Columbia, Canada, in 2019. His research interests lie in design modeling and manufacturing simulation.
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Cited by (0)
Qiang Zou is currently an assistant professor of Computer Science at Zhejiang University, China. He got his Ph.D. from University of British Columbia, Canada, in 2019. His research interests lie in design modeling and manufacturing simulation.
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This paper has been recommended for acceptance by Kai Tang.