Figure 11 Topology of basic Cartesian curvilinear grids.
Related Figures (38)
Figure 1. Five-axis milling machine with rotary axes on the table Moodleah and Makhanov 2015. it in the machine coordinates and does not necessarily minimise the machining time. However, optimisation in the work- piece coordinates is machine independent and computationally easy. That is why the strategy to minimise the toolpath is often based on a machine independent criterion such as maximisation of the machining strip (Chiou and Lee 2002) and minimisation of the scallops. However, the reduction of the kinematic error and the machining time are the ultimate goals of five-axis optimisation (Lee 1997; Yoon 2003; Anotaipaiboon, Makhanov, and Bohez 2006; Gao et al. 2006; Anotaipaiboon and Makhanov 2011). That is why in this paper, we consider general optimisation, which may include a machine dependent criterion. Figure 3. Five-axis milling machine with rotary axes on the table Moodleah and Makhanov 2015. Figure 4. MAHO 600E. We also consider a quality criterion measure represented by a vector function. At every point W, = (X;, Y,, Zp), we evaluate an optimal direction (vector), using one of the above scalar criterion C’. The collection of these vector constitutes the target vector field denoted by V. The ontimication nroblem ic then oiven hv parametrisation and the spiral toolpath is a feasible option which can be implemented by the ACT as well. Figure 6 compares the proposed algorithm with MeshFlatten (Accessed June 1, 2015 http://www.resurf3d.com/MeshFlatten_stan- dalone.htm). Figure 6. Flattening (a) 3D Model (b) and (c) MeshFlatten, (d) and (e) radial plane flattening. Figure 7. The reference point. where A = ee o(xpivpi+1 —Xp,itiyp,) is the area of the convex hull, and (xp; yp,) are the corresponding vertices numbered in order of their occurrence along the perimeter. Finally, O, is the intersection of the axis of the corresponding cylinder with the radius ,/A/z and the STL surface Figure 7 (e)). Ld Ee) EC es es, ee ee Ct, ) ee s coe ooo) ee Cer: Oe Se, ee ee, ey ee a ee ee ee i. ef a rn 6 ee ae ame eee igure 8. Estimation of the curvature tensor on the STL surface. The surface normal and the curvature required by the toolpath generation methods are evaluated using a modificatior of an algorithm (Rusinkiewicz 2004) for finding per-vertex characteristics by averaging adjacent per-face characteristics The per-face computation of the curvature is based on the second fundamental tensor given by Figure 9. Material removal rate. The actual rotation speed is given by Figure 10. Coordinate transformations and the curvilinear grids, A denotes the computational domain, K the parametric domain Recall that S(u, v) = (x(u, v), y(u, v), z(u, v)) is the part surface, where u and v are the parametric variables. Let us con- sider the CC points {(ui;, viz), 0<i< Ne O<j<N,} arranged as a curvilinear grid. In other words, the CC points are a discrete analogy of a mapping {u(é,7),v(€,1)} from the computational region A= {0<€<Nz,0<n<N,} onto a physical region K in the parametric coordinates u, v. In other words, the rectangular grid {(i,7), 0<i< Ne O<j<N,} being fed to {u(€,7),v(€,m)} becomes {u;;, vi;} (see Figure 10). Figure 12. Partition of the vector field V(u, v) into (a(u, v), B(u, v)). For instance, the ‘important points’ can be defined as points characterised by a high curvature or large rotation angles At the unimportant points F;=0, which simplifies the numerical solution and reduces the number of iterations. The numerical approximation is based on the discrete Laplacian and the central differences for the first derivatives. Figure 14. The grid slides along the boundary, where 7, is the original position of the boundary node, and nz is a new position Figure 15. BSFC correction in the case of frequent turns. Figure 16. Synthetic (ridge-cusp) dental surface. The conventional methods and the ACT have been applied with an appropriate set-up optimisation (Anotaipaiboon, Makhanov, and Bohez 2006). The accuracy of the machining has been evaluated in terms of the kinematic error, roughness and waviness. The improvement is defined as fore. 100. The grid alignment is based on cost functions representing the material removal rate and the toolpath length. We will denote these options by subscripts R and L, respectively, for instance, VFAGp and VFAG,. Table 1. Synthetic dental surface: ACT vs. the reference methods. 7.1 Example 1: Surface decomposition and ACT for a synthetic dental surface Table 2. Synthetic dental surface: CC points, kinematic error, ACT vs. ISO. Table 3. Synthetic dental surface: roughness and waviness, ACT vs. ISO. optimal directions corresponding to the material removal rate and machining strip width, respectively, in Figure 16(c) and (d), show that it is possible to construct a single grid such that one family of the grid lines is adapted to Vp and another to V;. Table 4. Human face surface: ACT vs. the reference methods. Table 5. Human face surface: CC points, kinematic error, ACT vs. ISO. Table 6. Human face surface, roughness and waviness, ACT vs. ISO. Figure 18. Machining the incisor tooth. Table 3 shows the quality of the produced surface for the rough and finishing cuts evaluated by stylus profiling, (Vorburger and Raja 1990; ). The surface profile obtained by a high-resolution probe is post processed by the Gaussian filter (Tabenkin 1998). Twenty sample profiles were measured with the standard cutoff of 0.8 mm (Tabenkin 1998). Roughness of the machined surface is within the acceptable range for surface milling operations (American National Standard 1985). The rough cut by the ACT has a better quality compared to ISO, that is, roughness: 7.8 vs. 8.1 um and waviness 17.8 vs. 31.9. We hypothesise that the major improvement in waviness is due to a reduced variation of the rotation angles, which are implicitly included in the evaluation of the material removal rate, since maximisation of the removal rate leads to the reduction in the angular distance between the CC points. Table 7. Incisor tooth: ACT vs. the reference methods. Table 8. Incisor tooth: CC points, kinematic error, ACT vs. ISO. Figure 19. H-grid, C-grid and O-grid (Siladic 1988). The toolpath is 43% shorter and 46% faster against the standard ISO. The advantage with regard to NX9, in terms of the length of the tool-path, is also very significant. For instance, when h = 0.01, the ACT, generates the tool-path of 570.9, whereas the best NX9 result is 1049.3. The advantage in the machining time on HAAS is 17%. It should be noted that the neither MasterCam nor NX9 can produce the five-axis tool-path for the STL surface directly. In order to compare their performance, we flattened the STL surface, generated a boundary-fitted curvilinear grid, and approximated the surface on that grid. Therefore, our MasterCam and NX9 toolpath generation is partly based on the proposed tech- niques. Finally, Table 8 shows that the efficiency of the ACT, measured by the number of the required CC points, increases, whereas the average kinematic error practically does not change. The STL-premolar and its flattened version are presented in Figure 20(a)-(d). The boundary-fitted initial grids in Figure 20(e) and (f) are the H-grid and the O-grid, respectively. In order to evaluate the suitability of the grid, we com- pute the alignment functional (3) as follows: Table 9. ACT vs. the reference methods, molar, premolar and canine tooth. Table 10. Molar, premolar and canine: CC points, kinematic error, ACT vs. ISO. Table 11. Performance of ACT.
