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TECHNICAL PAPERS

# Morphological Offset Computing for Contour Pocketing

[+] Author and Article Information
R. Molina-Carmona

Department of Computer Science and Artificial Intelligence, University of Alicante, Apdo. Correos 99, 03080 Alicante, Spainrmolina@dccia.ua.es

A. Jimeno

Department of Computer Science Technology and Computation, University of Alicante, Apdo. Correos 99, 03080 Alicante, Spainjimeno@dtic.ua.es

R. Rizo-Aldeguer

Department of Computer Science and Artificial Intelligence, University of Alicante, Apdo. Correos 99, 03080 Alicante, Spainrizo@dccia.ua.es

J. Manuf. Sci. Eng 129(2), 400-406 (Sep 28, 2006) (7 pages) doi:10.1115/1.2540741 History: Received February 02, 2006; Revised September 28, 2006

## Abstract

Background. Tool path generation problem is one of the most complexes in computer aided manufacturing. Although some efficient algorithms have been developed to solve it, their technological dependency makes them efficient in only a limited number of cases. Method of Approach. Our aim is to propose a model that will set apart the geometrical issues involved in the manufacturing process from the purely technology-dependent physical issues by means of a topological system. This system applies methods and concepts used in mathematical morphology paradigms. Thus, we will obtain a geometrical abstraction which will not only provide solutions to typically complex problems but also the possibility of applying these solutions to any machining environment regardless of the technology. Presented in the paper is a method for offsetting any kind of curve. Specifically, we use parametric cubic curves, which is one of the most general and popular models in computer aided design (CAD)/computer aided manufacturing (CAM) applications. Results. The resulting method avoids any constraint in object or tool shape and obtains valid and optimal trajectories, with a low temporal cost of $O(n∙m)$, which is corroborated by the experiments. It also avoids some precision errors that are present in the most popular commercial CAD/CAM libraries. Conclusions. The use of morphology as the base of the formulation avoids self-intersections and discontinuities and allows the system to machine free-form shapes using any tool without constraints. Most numerical and geometrical problems are also avoided. Obtaining a practical algorithm from the theoretical formulation is straightforward. The resulting procedure is simple and efficient.

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## Figures

Figure 1

(a) Raw offset of a curve. Local self-intersections in (a) and discontinuities in (b) will clash with part curve in shaded regions if it were machined. (b) Offset after trimming self-intersections and avoiding the discontinuity. If this offset is used as a tool compensated trajectory, machined surface will differ from the original in c due to curve curvature.

Figure 2

Geometric description of the instant basic operator: (a) initial position, (b) X transformation, and (c) distance computing

Figure 3

Classical morphologic operations on 2D images. On the left, a morphologic dilation. On the right, a morphologic erosion. The SE (a circle of 20pixels in radius) is shown at the top left corner.

Figure 4

Experiment in two dimensions. Shaded regions will not be machined. On the left, a morphological erosion for a toroidal tool (two radii implied). On the right, the classical offset for a spherical tool.

Figure 5

A torical tool with the rotating axe attacking parallel to the surface. In this case, the tool can be considered as having two radii.

Figure 6

Analysis of s1 and s2 segments of shape C. Shaded SE positions are discarded due to shape collision. Note that discontinuity at pd is solved by a vector swept generation.

Figure 7

Recursive algorithm for contour pocketing. The recursion tree can be easily simulated by a stack.

Figure 8

Offset of curves using the morphological approach. (a) Offset with a one radius tool (e.g., spherical tool). (b) Offset with a rectangular tool.

Figure 9

Contour pocketing using the morphological approach. (a) Pocketing with a one radius tool (e.g., spherical tool). (b) Pocketing with a rectangular tool.

Figure 10

Contour pocketing using a free form tool and a one radius tool. Shaded regions will not be machined. On the left, the shape is machined correctly. On the right, some regions are not machined because of the use of a one radius tool.

Figure 11

Contour pocketing using the morphological approach (left) and the commercial library (right). The library does not detect the two segments that are almost parallel.

Figure 12

Results for a sample of 50 experiments. Time of execution (t) compared with the product of n and m:t is linearly dependent of n×m.

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