Elsevier

Graphical Models

Volume 103, May 2019, 101022
Graphical Models

Using curvature bounds towards collision free 5-axis tool-paths

https://doi.org/10.1016/j.gmod.2019.101022Get rights and content

Abstract

This paper presents how freeform surface properties can be estimated through bounding, rather than sampling. Bounding surface positions using the control mesh, and bounding surface normals using normal cones, are both well known procedures. In this paper, we add to these, a procedure to bound the normal curvature values of a surface.

We then show how collision free 5-axis tool-paths for CNC machining, using any convex shaped tool, can be generated using the normal curvature bounds of a surface, without calculating surface offsets. As the tool-paths are generated using conservative bounds on the surface, and not sampled points, the tool-paths can be both globally optimized and globally guaranteed to be collision free. Simulation results validating our approach are also presented.

Introduction

Computer numerical control (CNC) machining, is a widely used form of subtractive manufacturing. In CNC machining, a target model, usually expressed as a set of boundary surfaces, is manufactured from some initial model (stock) by removing excess material using a machining tool. In this paper, we present methods that can be adapted to any convex machining tool, and use two types of tools as examples:

Definition 1.1

A ball-end tool has a cylindrical shank, that ends in a tip: a hemisphere of the same radius. A flat-end tool is made of a cylindrical shank, with the bottom disc of the cylinder being the tip.

An important aspect of CNC machining, is the generation of collision free (valid) tool-paths. Collision (or gouging) free tool-paths are those in which the tool does not remove material that should remain as part of the target model. Generating collision free tool-paths is especially challenging in 5-axis machining, where both the tool position and orientation change along the tool-path. Collisions can be divided into two main types:

  • 1.

    Local, in which the tip of the tool gouges the target model (or the CNC machine, etc).

  • 2.

    Global, when the shank of the tool gouges the target model (or the CNC machine, etc).

In this paper, we show how bounds, computed for the normal curvatures of a given model surface, can be used to generate collision free 5-axis tool-paths for convex tools. The main contributions of this paper are:

  • 1.

    A method to generate tight bounds for the normal curvatures of a whole surface.

  • 2.

    A method to generate globally verified valid 5-axis tool-paths for convex tools.

  • 3.

    A collision avoidance strategy that enables global optimizations of the tool-path, without compromising its validity.

The rest of the paper is organized as follows: in Section 2, we discuss some of the relevant previous work. We lay down the theoretical background for bounding the normal curvature values of a surface in Section 3. Section 4 presents how we apply these normal curvature bounds in an algorithm that generates a globally collision free tool-path, for a flat-end or a ball-end tool. We present our experimental results, which include simulations validating our approach in Section 5. We mention some avenues for future research in Section 6, and conclude in Section 7.

Section snippets

Previous work

The concepts introduced in this work relate to local collisions in CNC machining, and so in this section we focus on other research efforts that deal with local collision avoidance. A more thorough (but not very recent) review of CNC machining research, in general, can be found in [1].

Perhaps the greatest challenge in local collision avoidance stems from the fact that the tool has to make contact with the target model as part of the machining process. This means collision avoidance algorithms

Calculation of normal curvature bounds

Given a polynomial or rational regular C2 surface S(u,v)R3, its normal surface n¯(u,v)=S(u,v)u×S(u,v)v, and its unit normal surface n^(u,v)=n¯(u,v)n¯(u,v), we recall the following terms of the first and second fundamental forms (following [11] or similar):E=(Su)2,F=SuSv,G=(Sv)2,L=2Su2n^,M=2Suvn^,N=2Sv2n^.

The principal curvatures at a point p=S(u0,v0) would be the roots of the following quadratic equation for the variable κ, evaluated at (u0, v0):(EGF2)κ2(GL+EN2FM)κ(LNM

Collision avoidance in 5-axis machining

Let M be our our target model, with a closed boundary that is expressed as a set of regular C2 surfaces, O. Further, let S={S1,,Sn}, be a subdivision of the boundary surfaces of M into smaller sub-surfaces. We wish to generate a collision free 5-axis tool-path for moving a given tool T along a given cutter contact (CC) curve C(t), t ∈ [t0, t1], on S.

To generate a locally as well as globally collision free 5-axis tool-path we use a configuration space (or C-space) based method. For more

Experimental results

We implemented the algorithm described in Section 4 as a C/C++ single threaded program. All tests ran on an Intel i7-4770 3.4 GHz windows 7 machine. To validate our results, using CNC simulations, we use the machining Verifier Application by ModuleWorks (https://www.moduleworks.com).

Throughout the following experiments, we use a tool with a unit (1) radius, regardless of type. For the normal curvature bounds calculation the maximum number of recursions is set to 8, and the accuracy value is set

Future work

One additional feature that can be added to the calculation presented in Section 3, for bounds on principal curvature values, is the computation of bounds on the principal curvature directions as well. Once the intervals for the principal curvatures are known, bounds on the principal curvature directions can also be calculated using interval arithmetic. The bounds for the two principal curvature directions will take the form of two cones, with perpendicular axes, that will encompass all

Conclusion

Bounding surface positions using the control mesh, and surface normals using normal cones, are both well known procedures. In this paper, we add to these, a procedure to bound the (normal) curvature values of a surface. In the context of machining, normal curvature bounds allow the second order behavior of a surface to be conservatively approximated by that of the tool, without resorting to point sampling. The methods presented in this work show that collision free tool-paths can be generated

Acknowledgements

This research was supported in part with funding from the Defense Advanced Research Projects Agency (DARPA), under contract HR0011-17-2-0028. The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

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