## 4 Experimental Procedure  
As in previous works [6; 11; 12], we used a linear polishing machine with a single head, specifically built for polishing research (by PIRRA--Mauinas Ferramentas Lda, Estremoz, Portugal), with the following working ranges: belt speed (500-2000 mm/min), transverse speed (50-200 mm/s), rotational speed (500-1000 rpm), head pressure (1-8 bar), and water flow (10-40 L/min). Results are automatically sampled at a 10 Hz frequency and stored in a database.  
The polishing head of the machine was equipped with six Frankfurt tools. The diameter of the polishing head was \(d\) = 435 mm and the cross distance on a transverse movement was \(b\) = 240 mm. We used constant values, previously optimized, for the conveyor belt speed \(V_{L}\) = 600 mm/min, the head pressure \(P\) = 2 bar, and the water flow Q = 30 litter/min. These values were chosen to maximize quality and reproducibility of the results, and to emulate industrial data. For example, a minimum coefficient of variation (standard variation/mean value) of 6% in gloss measurements was achieved for a head pressure \(P\) = 2 bar. Six slabs of limestone, as equal as possible, were selected for these tests due to the high homogeneity of its surface. This is required to simplify the comparison between experimental and simulation data, as the simulator assumes a perfect, two dimensional, stone surface. After applying the sequence of abrasives, the surface gloss is measured and compared with the abrasion predicted in the same conditions by the polishing simulator (see Figures 7 and 8). To make the measurements more precise, we used a physical grid made of wire to precisely define the sections where gloss was measured for comparison with simulated values. Throughout this work, sections with homogeneous simulated abrasion values were identified as green cells (see Figure 7) while sections with simulated abrasion values too different were marked red and discarded in the subsequent analysis and comparison with experimental results. The gloss values reported represent average values over the various green cells (typically 10, depending of each experimental pattern). After each experiment, the slab surface was reset, using the more abrasive tool head (320 grit), thus slabs were only discarded after significant wear.  
Figure 7: Output image showing the simulated abrasion with homogeneous (green) cells and heterogeneous (red) cells.  
To obey condition 1 (\(l<d\)), with \(V_{L}\) = 600 mm/min as stated above, the transverse speed \(V_{T}\) must be \(V_{T}\) \(\geq\) 20 mm/s. To obey condition 2 (\(x<r\)), with \(V_{L}\) = 600 mm/min and \(V_{T}\) \(\geq\) 20 mm/s, the head rotation speed must be \(\omega\) \(\geq\) 10 rpm. Table 1 shows the various operational conditions tested in this work, whereas \(l\) values are determined by condition 1 and \(\omega\) values are determined by condition 2. Clearly, setting conditions 1 and 2 leads to an increase of the transverse velocity (to decrease \(l\)) and an increase of the rotational velocity (to decrease \(x\)). The simulated linear velocity (\(V\)) matches the vectorial combination of both cross and conveyor belt velocities from the polishing machine, according to the following equation:  
\[V=\sqrt{{V_{L}}^{2}+{V_{T}}^{2}}\,. \tag{3}\]