Precise gouging-free tool orientations for 5-axis CNC machining

Highlights

• We present an algorithm for generating optimized gouging-free tool path for 5-Axis CNC machining.

• We employ analysis of hyper-osculating circles that provides third order approximation of the surface.

• Double tangential contact between the tool and the target surface is employed to connect feasible hyper-osculating tool paths.

• A robust collision and gouging detection algorithm is provided.

• We introduce a global optimization algorithm that maximizes the geometric matching between the tool and the target surface.

Abstract

We present a precise approach to the generation of optimized collision-free and gouging-free tool paths for 5-axis CNC machining of freeform NURBS surfaces using flat-end and rounded-end (bull nose) tools having cylindrical shank. To achieve high approximation quality, we employ analysis of hyper-osculating circles (HOCs) (Wang et al., 1993a,b), that have third order contact with the target surface, and lead to a locally collision-free configuration between the tool and the target surface. At locations where an HOC is not possible, we aim at a double tangential contact among the tool and the target surface, and use it as a bridge between the feasible HOC tool paths. We formulate all such possible two-contact configurations as systems of algebraic constraints and solve them. For all feasible HOCs and two-contact configurations, we perform a global optimization to find the tool path that maximizes the approximation quality of the machining, while being gouge-free and possibly satisfying constraints on the tool tilt and the tool acceleration. We demonstrate the effectiveness of our approach via several experimental results.

Introduction

Free-form NURBS surfaces are the de facto industry standard representation for 3D modeling and have been widely used in a variety of applications. Consequently, CNC machining of NURBS surface has been studied extensively in the past few decades. 5-axis machining, which provides two additional degrees of freedom, compared to 3-axis machining, attracted much attention due to the flexibility of 5-axis tool path planning, in contrast with the difficulties that 5-axis tool path generation entails.

There are two major technical challenges in 5-axis machining. One is how to optimally orient the tool so that it can approximate the local target surface properly. The other challenge is how to avoid collisions and gouging into the target surface, $S$, and other check surfaces, or other surfaces of the object. Collision and gouging¹ avoidance in 5-axis machining is more challenging compared to other applications, because the tool tip is typically in a tangential contact with $S$ at the milling contact point, denoted $P_C$.

Consider 5-axis machining planning using a flat end cylindrical tool, $T$, having a bottom circle, $C_T$, as a cutting edge.² For each contact point $P_C$, two orientation degrees of freedom must be determined for $T$, while maintaining tangential contact between $C_T$ and $S$ at $P_C$. One natural approach to determine the orientation degrees of freedom uses a 2nd order approximation, to examine the osculating circle of a planar section of $S$ using the plane through $C_T$. From Meusnier’s theorem  [1], and using the two orientation degrees of freedom, one can attempt to match the curvature of $C_T$ and this planar section of $S$ at $P_C$, which is also known as curvature matched machining   [2], [3], [4], [5], [6], [7]. Curvature matched machining provides a 2nd order approximation and is a simple way to define the tool orientation, and it has been widely adapted in tool path computation. However and as a side effect of curvature matched machining, $T$ is likely to locally gouge into $S$, near $P_C$, because the planar section curve of $S$ typically has an increasing or decreasing curvature at $P_C$, while $C_T$ presents a constant curvature (see Fig. 1(a)).

Hyper-osculating circles (HOCs)  [8], [9] alleviate this difficulty and vastly reduce the possibility of $C_T$ gouging into $S$. Based on a 3rd order differential analysis, HOCs share the same position, tangent direction, curvature, and curvature derivative (which is zero for a circle), with $S$’s planar section at $P_C$. In other words, an HOC is an osculating circle that is located at a curvature extreme point of the planar section of $S$ and thus it resolves the local potential gouging of $T$ into $S$ (see Figs. 1(b) and  5). However, this 3rd order approximation also has a limitation. Since the HOCs should satisfy both curvature and curvature derivative constraints, typically only a limited number of candidate configurations exist. As a consequence, a hyper-osculating configuration is not always feasible, and even less so when various constraints in practical situations are imposed, e.g., angular orientation limits on the CNC machine, global collisions, etc.

To help overcome this limitation of the HOCs, we also consider cases of two-tangential contacts between $C_T$ and $S$ (see Fig. 1(c)). Henceforth and unless otherwise stated, ‘contact’ will denote a tangential contact. The key idea of our approach is based on the fact that there always exist two-contact configurations in the local neighborhood of hyper-osculating configurations.³ From this observation, we can augment the HOCs and switch to nearby two-contact configurations whenever necessary, while providing good approximation quality in terms of the curvature difference between the bottom circle and the surface section at $P_C$ and being gouging-free.

The rest of this paper is organized as follows. In Section  2, we briefly review previous related work. Section  3 introduces HOCs and establishes the algebraic conditions for two-contact configurations between $T$ and $S$. Then, we propose a global optimization algorithm for the tool path that maximizes the approximation quality of the HOCs and the two-contact configurations. Experimental results are reported in Section  4, and the paper is concluded in Section  5.