Decomposition of the Vector Field of Preferred Directions for Optimization of Five-Axis Machining

A new algorithm to increase the production rate of a five-axis milling machine through improving the coordinates of the to-be-milled points transformed from the workpiece to the machine coordinates is presented and analysed. The method optimizes the cutting path by following a vector field (VF) of optimal directions maximizing the material removal rate (MRR). The algorithm includes grid generation, space filling curves (SFC) and a VF decomposition using rotation invariant complex moments. The case of a radial tool path requires a special treatment called Compact Radial Zigzag (CRZ). To reduce the redundancy, the CRZ is composed of layers with a varying step between the tracks. The combination of the proposed techniques generates tool paths which produce complex shaped Stereolithography (STL) surfaces faster than the conventional methods.


Introduction
A five-axis machine is a programmable mechanism fed with the CNC program (G-code) i.e. a sequence of commands composed of three Cartesian coordinates of the tool-tip in the machine coordinate system and two rotation angles required to establish the orientation of the tool (Fig 1).
Matching a five-axis toolpath with a VF of preferred directions (VFPD) has been analysed in many publications and is becoming popular in the five-axis machining industry. However, the majority of the algorithms obtain toolpaths by propagating an initial curve inside the parametric region. Optionally, the region is subdivided into clusters characterized by homogeneous VFs. Most of the algorithms have not been verified on industrial formats such as STY, IGES, or STEP.
The VFPD is machine-dependent and is evaluated using the kinematics transformations based on coordinate systems shown in figure 1. The idea to match the tool path and the VFPD based on the maximization of the machining strip was proposed by Chiou and Lee [1]. The first algorithm to cluster the VFPD was introduced in [2]. Some recent modifications are the tensor based approach [3] and segmentation into regions of similar pathlines [4]. A comprehensive survey of these techniques is given in [5] (see also [6], [7]).
When a VFPD-method offsets an initial path and propagates it inside the region, the path may substantially deviate from the VFPD. Then the algorithm re-initializes it.
This procedure often creates configurations which does not actually match the desired VFPD. The tool paths often become redundant i.e. the distance between the tracks is considerably smaller than the maximum allowed machining strip. Besides, these algorithms are sensitive to the choice of the initial path which itself is a computationally expensive, NP-hard problem. Finally, as mentioned above, only a few algorithms have been verified on real industrial formats such as STL, IGES or STEP [8].
In this paper a new VFPD-type toolpath generation method based on curvilinear grids [9]- [11] and Adaptive Space Filling Curves (ASFC) [12]- [13] has been proposed. The method has been applied to ICMENS 2020 IOP Conf. Series: Materials Science and Engineering 840 (2020) 012005 IOP Publishing doi:10.1088/1757-899X/840/1/012005 2 the most popular industrial format-STL. The resulting VF-aligned path combines the adaptivity of pathpropagating methods with a convenient, regular topology of the conventional patterns such as the zigzag or the spiral. The VF can be obtained using a variety of utility functions such as the machining time, toolpath length (machine independent), kinematic error, etc. In this paper the algorithm constructs a curvilinear path which partly or even entirely aligns with the VF based on the maximum material removal rate MMRR. The reference methods are the standard iso-parametric path (ISO), MasterCam and advanced toolpath generation algorithms of NX11 (former UG) e.g. Helical/Spiral (HS) and Follow Periphery (FP). The cutting experiments have been performed on a milling machine HAAS VF2TR. -Define the order of the machining using the shortest path strategy.

Vector field of optimal tool directions Consider a part surface ( , )
S u v . Let 1 W be a CC (cutter contact) point. Define a collection of points given by   W on a complex shaped part surface usually is not possible. On the one hand, the tool path has to follow the prescribed VF. On the other hand, the desirable geometric structure of such a path is the zigzag, spiral or the radial pattern. The latter requirement improves the smoothness of the machined part, reduces the possibility (although does not guarantee the avoidance) of gouging and global collisions.

Grid generation
We arrange the cutter location (CC) points { ,, u v u v using a modification of a standard grid generation approach [6]- [9]. The smoothness of the grid is represented by a functional given by where subscripts denote the partial derivatives. The numerical solution is based on approximating of the second-order elliptic PDEs and an iterative solution by the Newton-Raphson method.

Numerical and cutting experiments
This section presents a VF-grid alignment method combined with the ASFC [10], tested against ISO, "Follow Periphery" (FP) and "Helical or Spiral" (HS) routines of NX11. A five-axis toolpath cannot be generated by NX11 directly from the STL format. Therefore, we convert the STL to a solid model, using fit surface functions of NX11. The average Euclidean error is 0.05 mm, which ensures the accuracy of the approximated surface. The HS and FP toolpaths are generated from the solid model. An example of the test surface is shown in figure 3. The blank workpiece is 200x200x50 mm. The curvilinear grid adapted to the dual VF in figure 3 is shown in figure 3(a) and the corresponding tool path in figure 3 (b). The surfaces produced by the conventional ISO zigzag and by the proposed method are displayed in figure 4 (a) and (b) respectively.

Decomposition of the VF using flusser complex moments
Several algorithms have been proposed to decompose the VF of optimal directions for tool path generation of five-axis machines e.g. [3], [4]. However, most of the algorithms are not applicable to the STL surfaces since the corresponding VF is usually highly irregular and noisy. In this section we present a clustering approach based on complex moments proposed by Flusser [14] to circumvent the above drawbacks. We illustrate the algorithm by radial (star) pattern which requires a special treatment due to the high redundancy near the pole. The basic idea of the CRZ is minimization of the machining time by decomposing the region into several subregions characterized by an increasing stepover between the radial tracks [15]. A video demo of the algorithm is available at: https://drive.google.com/open?id=1OM_z4cAOUqGu2RPAzkZOIBcEnfptdTq7 . The model of the crown of the canine tooth produced by the proposed method is shown in figure 5.

Conclusions
A new method for generation of VF-aligned tool paths for five-axis machining has been presented and analysed. The new idea is numerical generation of a curvilinear tool path adapted to the VF of optimal directions. This is combined with clustering of the VF and generation of local optimal subgrids providing faster machining. By introducing internal radial layers, the proposed algorithms eliminates the redundant parts. The experiments show a substantial decrease in the machining time with regard to preceding methods. In our experiments the proposed method outperforms the conventional algorithms for every prescribed scallop height, moreover, the advantage increases as the scallop height decreases. The CRZ partition makes it possible to set up the maximum machining strip on every circular boundary, reducing