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I have been studying Acrobot and Cart-Pole. Both of these systems I can both work out the equations of motion and linear manipulator equations for how to maintain the poles vertically.
The problem is both of these examples are revolute joints. And I'm interested in developing a control system using a spherical joint.
What is a typical example used to explain how to create a control system for holding a rod vertically with a spherical joint?
The best example I can think of would be the spherical pendulum. Albeit, this is not exactly the easiest thing to regulate as you have two degrees of freedom and a frictionless contact (well depending on your design)
Here is a youtube of such an example of a method to control it.
You'll often read that the Equation of motion will be described through Hamilton, but don't let anyone kid you, this can also easily be derived via Lagrange (if thats your fancy). Whereby the important thing to keep in mind is that the spherical pendulum will only move within a single plane unless there is an initial velocity included in your initial conditions.
Some things to read:
not a direct answer, but I would recommend you the following lessons by Peter Corke: robotacademy: joint control
