Potentially you could even make two "arms" one with potentiometers to measure the angles at different points on the linkage and a second with servos set to reproduce those angles. So a person can trace out an object or drawing on the first one, then the second one shadows/reproduces it. It would simplify any code substantially and would probably be more impressive to the judges

It might be a good idea to look at a pantograph for something like this, you could probably build a simple linkage out of popsicle sticks and use a couple of servos to control it. It would take some trigonometry to get it right but it's probably the easiest to build with those materials