Research Papers

Computing the Global Visibility Map Using Slice Geometry for Setup Planning

[+] Author and Article Information
Guangyu Hou

Department of Industrial and Manufacturing
System Engineering,
Iowa State University,
3023 Black Engineering,
Ames, IA 50011
e-mail: houes@iastate.edu

Matthew C. Frank

Department of Industrial and Manufacturing
System Engineering,
Iowa State University,
3023 Black Engineering,
Ames, IA 50011
e-mail: mfrank@iastate.edu

1Corresponding author.

Manuscript received August 11, 2016; final manuscript received March 21, 2017; published online May 8, 2017. Assoc. Editor: Xiaoping Qian.

J. Manuf. Sci. Eng 139(8), 081006 (May 08, 2017) (11 pages) Paper No: MANU-16-1426; doi: 10.1115/1.4036423 History: Received August 11, 2016; Revised March 21, 2017

This paper introduces a new method that uses slice geometry to compute the global visibility map (GVM). Global visibility mapping is a fundamentally important process that extracts geometric information about an object, which can be used to solve hard problems, for example, setup and process planning in computer numerical control (CNC) machining. In this work, we present a method for creating the GVM from slice data of polyhedron models, and then show how it can help determine around which axis of rotation a part can be machined. There have been various methods of calculating the GVM to date, tracing back to the well-known seminal methods that use Gaussian mapping. Compared to the considerable amount of work in this field, the proposed method has an advantage of starting from feature-free models like stereolithography (STL) files and has adjustable resolution. Moreover, since it is built upon slicing the model, the method is embarrassingly parallelizable in nature, thus suitable for high-performance computing. Using the GVM obtained by this method, we generate an axis of rotation map to facilitate the setup planning for four-axis CNC milling machines as one implementation example.

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Fig. 1

An example showing the visibility cone of a facet on the pocket of the cube

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Fig. 2

The discretization of the visibility sphere

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Fig. 3

An example of a 3D visibility cone built up by a set of 2D visibility arcs: (a) An actual visibility cone is approximated by a set of arcs/points, (b) a visibility arc is represented by a set of visibility points, and (c) a sweep of the visibility arcs yields the visibility cone. In this paper, the continuous visibility arcs are showed by a set of points.

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Fig. 4

The distribution of sampling longitude lines where visibility arcs are evaluated: (a) Top view showing evenly spaced sampling longitude lines captures multiple visibility cones and (b) isometric view showing five visibility arcs of a visibility cone discovered by the densely angular spaced sampling longitude lines

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Fig. 5

The slice-based visibility computation showed by an example: (a) an example part, (b) the slice geometry, (c) one slice for demonstration, (d) visibility computation for a segment on the slice chain, (e) and (f) example where sliced segments changes with the slicing direction, and (g) example showing the visibility arc resides on the plane perpendicular to the slicing direction

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Fig. 6

The definition of facet visibility: (a) an example showing segment's visibility is constant for a distance due to approximation and (b)–(d) the formation of the facet visibility by intersecting visibility ranges of its containing segments

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Fig. 7

An example of improved visibility after mesh refinement: (a) visibility loss ratio of the original facet is 1.0 and (b) visibility loss ratio of the equivalent facets after refinement is 0.36. Assuming visibility range θ is constant for all visible segments: (a) original facet and (b) result facets after remesh.

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Fig. 8

Examples showing the union of visibility arcs from different slicing planes: (a) the orientations of slicing vectors, (b)–(d) visibility arcs generated from different slicing planes, and (e) the union of visibility arcs from 180 sampling planes results in a complete visibility cone

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Fig. 9

The use of two sets of slices (perpendicular to each other) to solve the missing parallel planes problem

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Fig. 10

An example of parallel plane visibility computation using the three edges of a triangle: (a) a demonstration part with its parallel planes marked, (b) a triangular facet on a parallel plane showing its vertices are shared by its neighboring triangles, and (c) obstacle range obtained by finding the maximum angle difference among six bounding rays. Rays coming from the same vertex are annotated with the same number near the arrow.

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Fig. 11

The possible position relations of a triangle facet and its obstacle chains together with the solution to numerical error. The triangle is hollow. The obstacle chain has solid interior. The triangle (a) is out of the obstacle chain, (b) has one vertex on the obstacle chain, (c) has two vertices on the obstacle chain, (d) has three vertices on the obstacle chain, and (e) is contained by the obstacle chain. (f)–(h) Due to numerical error, at least one vertex is contained in the obstacle chain, (i) the rays of min and max accumulated angle give the correct bounding rays if vertex is on the exterior, (j) the rays of min and max accumulated angle give the wrong bounding rays if vertex is in the interior, (k) offset the triangle's false interior vertex to the exterior by a small value ϵ2, and (l) offset the triangle inward to its centroid by a small value. The vertex is deemed false interior if Δ<ϵ1.

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Fig. 12

The correspondence between a visibility point and feasible axes of rotation: (a) A real four-axis CNC machine setup and (b) the feasible axes of rotation are any axes perpendicular to the visibility vector, assuming the component is fixed while two indexers rotate to align with different axes

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Fig. 13

The procedure of finding feasible axes of rotation from the GVM: (a) Find corresponding axes of rotation region of a visibility cone and (b) an example showing intersecting axes' regions of two facets to obtain the share axes of rotation region

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Fig. 14

The formation of the axes of rotation map. (a) Facets to axes map and its reverse map (for each axis, the reverse mapping collects every facet that maps to it in the original map) and (b) an example of axes of rotation map where axes are indicated by dots and their visibility value (normalized visible surface area) is represented by dots' distance to the origin ([0-1] range).

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Fig. 15

An overview of the computational procedure: (a) the flowchart showing entire computational flow and (b) the corresponding detailed illustration showing five critical steps in the flowchart

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Fig. 16

The resulting visibility cones for various models

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Fig. 17

The time on computing global visibility map for various models. Sample parts 5–8 are from the source in Refs. [3235].

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Fig. 18

The statistics of computational time on a various number of slices and facets for an example model. The example model is embedded in the line chart where the number of facets is used as horizontal axis (data from Table 1).

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Fig. 19

The method to obtain total area of the visibility cone from discrete visibility arcs: (a) example part and visibility cone, (b) total area of visibility cone is computed as the area summation of spherical triangles determined by corresponding visibility arcs, and (c) example showing the spherical triangle determined by the visibility arc could either overestimate or underestimate the portion of the cone area

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Fig. 20

The results showing areas of visibility cone for (a) errors of areas of visibility cone versus the number of slices and (b) errors of areas of visibility cone versus the number of slicing directions (data from Table 2)

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Fig. 21

Examples of the computed feasible axes of rotation and the axis of rotation map. (a) A cube with one pocket has feasible axes on a great circle, (b) a cube with two orthogonal pockets has only one feasible axis, (c) a cube with three orthogonal pockets has no feasible axis, (d) the axis of rotation map for a cube with three orthogonal pockets, and (e) a cube with a cylindrical through-hole has a feasible axes region of a ring shape.

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Fig. 22

The axes of rotation region which reveals more than 99% of component's surface area. Axes are indicated by dots which are pointed out by arrows. (a) An industrial bracket with two cylindrical through-holes and (b) an industrial linkage.

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Fig. 23

An example of an industrial bracket machined by a four-axis CNC machine where the axis of rotation is chosen as the part's Y-axis. (a) The bracket with supports on after CNC-RP machining and (b) The finished bracket with all supports removed.




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