Best way is to sketch it out and then mock it up using whatever materials you have.
For the moving box puppets for Avenue Q, I used a trigger mechanism that pulled down on an L-bracket that was bent out to 135 degrees. The trigger pulled down on a rocker that pulled UP on the back of the L bracket. The parts were prototyped with foam core, and then modeled in Google Sketchup and sent to Shapeways for 3D printing. I used ABS the first version, but it broke easily. I later went with nylon, which was less expensive and was much more solid.
Moving Box MouthIf you sketch out what you're trying to do and shoot me an email, I'd be glad to help.
- Sean