What you are looking for is called a four bar linkage. (see
) The vertical part of the jawbone is one bar. The servo horn is another bar. A piece of rod between the servo horn and the jaw is the third bar. The solid structure of your animatron's head what the servo and the jaw hinge are attached to is the fourth bar.
The easiest way to make a four bar linkage is to make the length of the jawbone bar the same length as the servo horn bar and have the center of the servo axis on a horizontal line directly behind the axis of the jaw bone. With the jaw closed, the servo horn should be pointing downward, parallel to the vertical part of the jaw. The connecting rod going from the servo horn to the jaw will be horizontal. To open the jaw, the servo is activated to move the bottom of the horn away from the jaw. Since the connecting rod can not stretch, the jaw has to rotate to keep the maintain the same distance between the points on the servo horn and the jaw.
One thing you have to be careful about when designing a linkage is that you don't make a structure that will bind - i.e. get into a configuration where it can not move without some part of the linkage bending. In the old days, before solid modeling, linkages were tried out using cardboard cutouts and a type of push pin where the shank was made of two pieces of brass that could be bent around (any of you old guys remember what these are called? I still have a box of them in my desk at work, but I won't be back there until next week). Try to make a model of your linkage in cardboard before you start drilling holes in your animatron. You can also try to learn how to use the linkage modeler available at: