Characterizing envelopes of moving rotational cones and applications in CNC machining

https://doi.org/10.1016/j.cagd.2020.101944Get rights and content

Highlights

  • Higher order contact between rotational cones and curved surfaces is studied.

  • The isotropic model of Laguerre geometry simplifies the analysis of the contact.

  • Surfaces enveloped by a one-parametric family of congruent cones are characterized.

  • Ruled surfaces and their offsets are characterized by a partial differential equation.

  • Applications towards 5-axis flank CNC machining are indicated.

Abstract

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.

Introduction

Various manufacturing technologies, such as hot wire cutting, electrical discharge machining or computer numerically controlled (CNC) machining are based on a moving tool, the active part of which can be a curve or a surface. They generate surfaces which are swept by a simple curve, e.g. a straight line segment or a circular arc, or are enveloped by a simple surface. The latter case mostly refers to CNC machining where the moving tool is part of a rotational surface (sphere, rotational cylinder, rotational cone, torus). In order to produce a given shape with such a manufacturing process, one has to approximate the target shape by surfaces which are generated by a moving tool of the available type. Depending on the application, such an approximation has to be highly accurate and, for example in the case of CNC machining may have to meet a numerical tolerance of a few micrometers for objects of the size of tens of centimeters. Such high precision pushes demands on the path-planning algorithms which greatly benefit from a higher order analysis of the contact between the reference surface and the surface generated by the moving tool.

A moving tool, conceptualized as a truncated cone, in the context of 5-axis flank CNC machining is shown in Fig. 1. Its motion is visualized by the motion of its axis (tracing a ruled surface). Ideally, the tool is supposed to touch the desired reference surface not only at a single point, but along a whole curve, known as characteristic. The characteristic is the intersection of the current position of the cone with the position at an “infinitesimally close” moment, therefore is an algebraic curve of degree four. For some special instantaneous motions, e.g. translation, this characteristic can degenerate to a pair of straight lines passing through the vertex of the cone, however, for a generic screw motion, the characteristic is a spatial curve lying on the cone. Therefore solving the flank-milling problem by approximating the given surface by developable patches, as widely done in engineering literature, is not a correct approach as it restricts the space of solutions. In this work we look for good initial positions of cones that admit higher order contact with the surface.

With the flank CNC machining application in mind, we present such an analysis for envelopes of rotational cones. We emphasize here that we strictly focus on cones of revolution (aka rotational cones). A rotational cone is formed by all lines passing through a fixed point (vertex) and a fixed circle such that the orthogonal projection of the vertex to the plane of the circle is its center (but the vertex does not coincide with the center). In order to obtain contact of order n between an envelope of a moving rotational cone and a design surface Φ, it is not necessary that each position of the cone has contact of order n with Φ, when viewing the surfaces as point sets. This is obvious anyway, since 2nd order contact between a cone and a surface Φ would already imply vanishing Gaussian curvature of Φ, i.e., a developable surface Φ. One needs contact of order n between the cone and the surface, viewed in the space of planes. It is related to the fact that a cone possesses just a one-parameter family of tangent planes. This indicates the advantage of using a geometry, in which the (oriented) planes in Euclidean space are the basic elements. Therefore, we use Laguerre geometry and work in a point model of the set of oriented planes, known as the isotropic model of Laguerre geometry. There, a cone appears as a curve (an isotropic circle) and not as a surface. That is, the analysis of cone-surface contact is transferred to the study of a curve-surface contact, which is conceptually simpler.

When we speak of higher order contact between a surface generated by a conical milling tool and a reference surface, it is important to note the following: Second order contact, also referred to as osculation, means that the surfaces locally penetrate tangentially. Thus, this case is not directly suitable for CNC machining, but may still be useful for initial estimates of good tool positions. However, third order contact, so-called hyperosculation, is locally penetration-free in the very neighborhood of the contact point and therefore very well-suited for CNC machining, in particular for initialization of optimization algorithms which aim at high-precision machining.

Our main contribution is a careful analysis of plane-based higher order contact between cones of revolution and a given reference surface. This leads to a nonlinear PDE which characterizes exact envelopes of congruent rotational cones (see Theorem 13). From a practical perspective, this means that we can detect the (rare) cases in which a surface can be milled exactly in a single path by flank milling with an appropriate conical tool, provided that this tool motion is collision free and accessible. Probably more importantly, a computational approach to locally well fitting tool positions is very helpful for the initialization of numerical optimization algorithms for high-precision tool motion planning. On our way towards the characterization of envelopes of moving rotational cones, we discuss other special types of surfaces as well.

The paper is structured as follows: We discuss relevant previous work in Section 2. Section 3 derives a PDE that characterizes the graph of a bivariate function as a ruled surface (Theorem 1). To extend to envelopes of cones, in Section 4 we introduce the isotropic model of Laguerre geometry and discuss the contact order between a developable surface and a doubly curved surface, expressed in the space of planes. Section 5 characterizes envelopes of congruent rotational cones in the isotropic model and formulates conditions on second order and third order plane-based contact. This is the basis for a PDE characterization of envelopes of congruent rotational cones (Section 6, Theorem 13). In Section 7 we address the limit case of envelopes of congruent rotational cylinders (Corollary 21). Section 8 shows examples of hyperosculating cone positions and its application to flank CNC machining. Finally, Section 9 concludes the paper and indicates directions for future research. In Appendix A and Appendix B we prove Theorem 1, Theorem 10 respectively (the latter implying Theorem 13) and in the process give an “algorithm” to reconstruct rulings and special conics on a given surface (and hence cones enveloping a surface).

Section snippets

Geometry

Higher order contact between curves and/or surfaces has been well-studied in the past, see e.g. Colley and Kennedy (1994); Montaldi et al. (1986); Ye and Maekawa (1999). It appears, for example, in surface-surface intersection: Using marching methods is straightforward for transversal intersections, however, when the surfaces in question have higher order contact, the computation of the intersection curve is quite complex (Ye and Maekawa, 1999). Higher order contact between a circle and a

Ruled surfaces

Ruled surfaces are traced by a line moving in space. They appear as limits of surfaces enveloped by a one-parametric family of congruent rotational cones when the opening angle tends to zero whereas the vertices stay fixed. We first treat this well-known class of surfaces and in this way introduce to our approach at hand of a well-known case. A particular case of ruled surfaces are developable ones. The latter are enveloped by a one-parametric family of planes. For a developable surface, a

Surfaces enveloped by a family of rotational cones, using a point model of the space of planes

Now we come to the main topic of the paper: how to characterize surfaces enveloped by a one-parametric family of congruent cones? To minimize technicalities, we consider surfaces tangent to cones along curves rather than arbitrary envelopes of cones, and exclude certain positions of these curves. In this section we reduce the problem to the characterization of surfaces containing a special conic through each point, which is tractable by the methods already discussed.

Surfaces containing a special conic through each point

In this section we characterize the surfaces containing a conic satisfying condition (Θ) through each point (Theorem 10 below). This is required for the main result in Section 6. The characterization is similar to that of ruled surfaces in Section 3. We consider the graph of a smooth function f. The conics on the graph are parametrized by trigonometric functions. Differentiation with respect to the parameter gives a system of algebraic equations on the tangential direction to the top view of

Surfaces enveloped by a family of rotational cones: conclusion

Now we use the results of the previous two sections to complete the characterization of surfaces enveloped by a one-parametric family of congruent cones (Theorem 13 below). Then we show how to construct Φi from Φ and vice versa. Finally we show how to reconstruct the positions of cones in the enveloping family.

Envelopes of congruent rotational cylinders

Cylinders are a limit case of cones, but this limit is not straightforward. This is so, since the limit of cones with a constant opening angle are cones with vanishing opening angle, i.e., rotational cylinders. However, these cylinders need not be congruent. Hence, we now discuss envelopes of congruent rotational cylinders, i.e. offsets of ruled surfaces, which appear in flank CNC machining with a cylindrical tool.

The derivation of the PDE is analogous to Sections 4–6. Passing to the isotropic

Results and applications in CNC machining

In this section, we show how the proposed analysis of third order contact can be used in the context of 5-axis flank CNC machining with conical tools. First, we test our algorithm on an exact envelope, showing that we reconstruct the generators of the envelope.

Example 23

Reconstruction of an exact generator. Take a particular surface that is an exact envelope of one-parameter family of cones. In the isotropic space, consider the graph of the function f(x,y)=y2x2+y2. The graph contains a family of

Conclusion and future research

We have derived necessary and sufficient conditions on a surface to be an envelope of a one-parameter family of congruent rotational cones. Such a surface can be milled by flank CNC machining with an appropriate conical tool in a single trace (provided that the motion is collision free and technical constraints on available tool sizes, machine workspace etc. are fulfilled as well). This characterization comes in form of nonlinear PDEs. On our way towards that, we discussed similar PDEs for

CRediT authorship contribution statement

Mikhail Skopenkov: methodology, validation, formal analysis, investigation, writing- original draft preparation, writing-review and editing. Pengbo Bo: software, validation, investigation, data curation, visualization. Michael Barton: methodology, software, validation, investigation, data curation, writing-review and editing, visualization. Helmut Pottmann: conceptualization, methodology, formal analysis, investigation, writing- original draft preparation, writing- review and editing, project

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.

Acknowledgements

Three of the four coauthors are grateful to King Abdullah University of Science and Technology, where they met altogether and started this project. The authors are also grateful to R. Bryant, S. Ivanov, and A. Skopenkov for useful discussions. The first author has been supported within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (grant N18-01-0023) and by the Russian Academic Excellence Project5-100”. The second

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