It doesn't matter if there is a board underneath, there is a carpet. Most of us here use a table and tape our sets down, I don't think you did that, correct me if I am wrong.

Do you have a test you could upload, for us to see how shaky it is.

Your not taking frames through the camera and not your compute, right? Thats the problem, you see, even though you can't notice it, the camera moves a bit every time you take a picture manually. What is your camera? You might be able to hhok your computer up to it.

-Tejas VIM

You have the mode on manual, no?

-Tejas VIM

oh, and it doesn't have live preview in iStopmotion. But I'll still try.

  Check me out!  Please take a minute to watch my film! Lego Batman Joker Returns

Here it shows all cameras supported and unsupported by remote capture with iStopmotion
EDIT:Good, since you have the focus on try and make a lego case or of some type to keep on your setup, Nathan Wells has done the same and gotten good results

-Tejas VIM

I wouldn't zoom in if I were you, because then, the focus is messed up.

Zooming doesn't mess up the focus, it just makes the picture pixelated.

It's be more worth your while sticking your camera to your desk, instead of supporting it on a tripod. It makes it very steady and if you knock it you could easily mess up your shot.