Create a basic platform for the inverted pendulum. You can worry about the balancing mechanism for the pendulum later, so the platform you make should be something that you can adapt to your balancing mechanism. The balancing mechanism can be fairly complicated, as evidenced by some of the robotic structures created by university engineering departments, like the one at Cornell University. Your platform can be something as simple as a board or a small wooden platform on wheels. Having a platform with wheels will make it easier to perform balancing movements with less friction.
Attach the base of the pivot point to the top of the base or platform. Determining what to use for the base of your pendulum can also be a matter of creativity. Attach a small L-bracket with screw holes in both ends of the "L" to the center of the platform by drilling wood screws through the holes on one side. Doing this should leave the "L" sitting in an upright position with at least one hole on the upper side.
Drill a 3/8-inch hole in one end of a hollow aluminum tube. Ensure that the hole goes all the way through the tube. The tube should be about 36 inches long.
Insert a 3/8-inch bolt through the holes in the end of the tube. Insert the end of the bolt through the hole in the top of the L-bracket also and secure the rod to the bracket using a washer and nut. Tighten the bolt by hand, but do not over-tighten. Ensure that the aluminum rod can swing from side to side with little to no friction. If there is significant friction on your aluminum rod, remove the bolt and drill out the hole to make it slightly bigger than the bolt.
Attach a weight to the end of the aluminum tube farthest away from the pivot point. The weight you use needs to be uniform and cannot put more weight on one side of the pendulum than another. A small weighted steel ball placed on the end of the aluminum rod should suffice for this part of the pendulum. Secure the ball in place by wrapping it in duct tape. Ensure that you wrap the ball evenly so that the tape's weight does not cause uneven weight distribution.