One approach would be to use an aircraft cable at each corner vertically, and cris-cross them in upper (sliding) bed to the diagonally opposite corner using pulleys/rollers.
Here's a very quick, rough diagram of one of the four cables; repeat at all the corners for the others. Some care in cable routing needs to be done at the middle so they don't all rub and chafe too much; either offsetting them by clever pulley angles/cable positions or routing them through a tube of some sort would probably be practical.
Similar arrangements are (were) used on manual drafting boards with built-in traveling straightedges, and probably in many other applications. You would want some sort of turnbuckle or other means for adjusting the tension in the cables and, indirectly, the level of the platform. It's not necessary to have them under tremendous tension.
I'm sure there are many other solutions that would also work. For power operation, a scheme of interconnected lead screws seems to me fairly straightforward and practical.