1. Introduction
Pocketing cavities, which is common from an industrial point of view (e.g., in aeronautical structure parts or aerospace electronic crankcase), is a time-consuming operation.
As shown by [
1], manufacturing costs are time dependent and typically represent, for dies and molds,
of the total price of a piece [
2].
In this context, many studies aim to reduce the machining time. The main lines for optimization are enhancing the understanding of cutting phenomena [
3,
4,
5], improving tool performance [
6], selecting the best tools, defining the optimal milling parameters [
7,
8,
9], and, finally, enhancing the toolpath geometry [
10,
11,
12].
From this last perspective, the most common approach is to work on the tool trajectory to increase its speed. Multiple studies (e.g., [
13,
14]) have demonstrated that tangency discontinuities along the trajectory result in increased machining time. These discontinuities necessitate changes in the velocity vector (both in direction and magnitude) along the trajectory, which challenges the machine tool’s acceleration and jerk capabilities. As these capacities are limited, the numerical control system of the machine imposes restrictions on these variations, so the tool requires additional time and distance to achieve the programmed feed rate after negotiating corners or halts along the toolpath. Therefore, it is essential to minimize such discontinuities, for which two methods exist:
Local adjustments on classical toolpath (Zig-Zag or Contour Parallel) are employed to eliminate or to smooth out the discontinuities [
15,
16,
17];
Global strategy to build a spiral-like trajectory (or curvilinear spiral). Macroscopically, this kind of toolpath begins at a center point of the pocket and expands outward without “stop and go”. The toolpath morphs progressively to the shape of the pocket boundary. Natively, curvilinear spirals (or spiral-like toolpath) have a higher degree of continuity [
18,
19,
20,
21].
Primarily, two approaches are used to build curvilinear spirals:
Geometrical approaches such as mapping [
22,
23], morphing [
24], or the medial axis method [
19,
20];
Differential approaches using an elliptical partial differential equation (
PDE) [
21] or a level-set propagation model [
18]. In those cases, curvilinear trajectories are generated through interpolation between selected structure curves, which are the solutions of the used PDE.
To determine a point on a curvilinear path, an interpolation method is needed. Bieterman et al. [
21] use the intersection of a Fresnel vector (winding angle) and two discrete structure curves (
Figure 1). The trajectory point is established through a linear angular interpolation between the two intersecting points (Equation (
1)):
with
, discrete polar equation of a spiral-like turn;
, discrete polar equation of the external structure curve;
, discrete polar equation of the internal structure curve.
This method is effective when the cavity is sufficiently convex (i.e., without significant protrusions; mathematically, when the polar function of the pocket border is explicit). However, for pockets with more complex geometries, a partitioning task is required as proposed by [
25] and illustrated in
Figure 2.
Moreover, the structure curves are constructed through interpolations between a series of points calculated using finite element method software. In the context of G1 linear interpolation, the trajectory consists of numerous small segments. Although the trajectory may exhibit characteristics adapted to high feed rates on a macroscopic scale, the tangency discontinuities inherent to this method significantly increase the machining time on a smaller scale. Since ’polynomial’ interpolators are not yet widely adopted in the industry, efforts are needed to smooth the toolpath and improve its degree of continuity.
The simplest method to achieve this objective is the use of a Finite Impulse Response filter (
FIR); [
21] suggests a moving average technique. However, the use of such filters may result in a loss of tool position control, potentially leading to a machining in inappropriate zones. Other various studies [
22,
26,
27] achieve three of five-axis trajectory smoothing, via B-splines or Bézier curves.
Table 1 shows a summary of the methods proposed in the literature summarizing their main focuses.
This article proposes a method to reparameterize the structure curves of the curvilinear spiral with a set of Hermit quartic spline patches connected (continuity in tangency and curvature). Analytically, the spiral-like toolpath, formed through linear combination of these curves, maintains the continuity, ensuring a smaller machining time.
Moreover, the proposed method eliminates the need of Fresnel’s vector to establish the spiral parametrization, so the toolpath can be directly created without partitioning the pocket. It allows a generalization of Bieterman’s procedure for any pocket geometries, even with non-convex geometry.
3. Toolpath Quality Evaluation
As shown in many research works, notably in [
13,
14], discontinuities in the tangency and curvature of the toolpath geometry challenge the limited acceleration and jerk of each axis of the numerically controlled machine tool. Therefore, the average speed rate is always lower than the programmed one, and the milling time is longer than expected.
Here, a double objective is pursued: increasing the speed along the toolpath (which can reduce milling time) and mastering the position of the tool along the milling path. For the tool speed evaluation, two methods are employed:
VPOp (Velocity Profile Optimization—version 2.6) is licensed software developed by X. Beudaert and et al. [
33]. Based on kinematic data and the CNC parameters of a given machine tool, it generates optimized feedrate planning that respects the capabilities of the milling center and thus predicts the actual tool speed and slowdowns for the tool. So, the modelized milling time is close to the real one. VPOp is used locally to analyze the velocity profile along two turns of the curvilinear spiral.
To analyze the entire trajectory, histograms of the toolpath absolute curvature value ( in mm−1) are used as well as and quantiles of named Cut95 and Cut99. These last indicators can quickly give, unlike VPOp, the geometrical quality of a toolpath and, consequently, the possibility of NC to respect the programmed feed rate.
A comparison will be made between the “no fitted” toolpath (called RAW) and the fitted one with HQS method (called HQS). For another comparison point for milling time, a filtered toolpath is also computed, called FIR, and corresponding to a moving average on three points applied twenty times (this combination gives results close to the HQS method). Note that the more the filter is applied, the smoother the trajectory but the more the position control is lost. So, the FIR toolpath is absolutely not suitable to satisfy our goal.