Sharpening an axe is no different than sharpening a knife, just that it is more awkward usually. I sharpen them by holding the axe fixed in place and honing them to shape freehand. Then after the edge has been shaped and roughly sharpened I put a finish polish on with fine sandpaper (5 micron, SiC) on leather and finish on a loaded strop.
Most working axes have convex bevels which you should keep because of the combination of shallow edge durability, smooth cutting and low binding. To shape such an edge you either need a slack belt sander (or a large piece of slack sandpaper), or simply use a hone and work the convex bevel freehand, you need to arc the hone over the bevel and match the curvature. This is not as difficult as it may seem.
Remember in the early stages of sharpening, it is just a shaping process. You start out with a file if the edge is damaged or has been poorly ground to start. Once the edge has been formed you move on to a coarse stone. Generally I will go something like a bastard file, 220 grit SiC waterstone, 800 grit AO waterstone, 5 micron SiC sandpaper, 0.5 micron CrO loaded strop. This produces a fine shaving polish which is the optimum for cutting performance and durability when chopping.
This of course if a complete sharpening. After a period of extensive use I will sharpen by simply hitting the edge with the strop and when this fails (after 3-4 uses typically) I go back to the sandpaper and then back to the strop. After 3-4 times of doing this I go back to the coarse hone and reshape the edge. You should only have to file the axe once, unless you damage the edge.
Note as well that the benefits of each additional stage decrease rapidly. You would be hard pressed to tell the difference in penetration between a well shaped edge finished with a 800 grit waterstone and the one that has been brought all the way up to 0.5 micron CrO. Fact is that even a well shaped edge left with a file finish will chop very well, however the edge will degrade very quickly.
If the axe has flat bevels then you can sharpen pretty much on any sharpening system. In the Sharpmaker video, Sal Glesser sharpens an axe (Gerber I think), on the Sharpmaker. Most rigs however tend to have really obtuse angles, axes should have bevels that are from about 8-16 degrees per side depending on the type of wood being cut, assuming it is fresh, seasoned wood will require a slightly more obtuse edge.
-Cliff
