Tenkor, student engineer in cybernetics of the university of Stuttgart, carried out a thesis on the reversed balance of the pendulums. It for that produced a robot able to balance a double reversed pendulum.
The robot has only one type of movement, that of a translation along a rail. It is initially obliged to balance the pendulum from left to right to make it turn around an axis.
After some rotations, it is necessary for him then to stop the pendulum in high position and to stabilize it from left to right using small movements.
With a double pendulum, that becomes more perilous, but the robot arrives there after a phase a little longer of stability.
One can even see Him, operate the pendulum in balance, and the robot carries out negative feedbacks for always keeping a balance, with a return in median position of the rail.
That appair simple like that, but it is done only mathematics with some formulas…. here it video: