I made the puppet in AI and the finished character does not stay still stage.

The puppet, for no reason I can determine, just has decided to hop around every few seconds.

I took this shot to try and help you see what I mean. This is AI made and does not respond to anything I do to the staid rig mode. I have added several pins to bottom and it doesn't help a thing. It pins bottom of the art but her head still keeps flailing around. I stuck a pin in her head and she continued to flail, just with her head pinned.