Grinding benches or barrels are necessary to hold the tool in place during both grinding and polishing.
On the left is a picture of the type of bench they use to make the mirrors at the Springfield Telescope Makers workshops, it pivots on a screw fastened to the bottom board. You need to be able to rotate the tool so that the mirror and tool have even contact. On the right is my grinding barrel just a fifty-five gallon drum that you walk around instead of rotating