Inverted pendulum
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Also called a cart and pole, an inverted pendulum is a classic problem in dynamics and control theory consisting of a pole attached at its bottom to a moving cart. Whereas a normal pendulum is stable when hanging downwards, a vertical inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright, typically by moving the cart horizontally as part of a nonlinear feedback system.
The inverted pendulum is a widely used benchmark for testing control algorithms (PID controllers, neural networks, Genetic algorithms, etc). Modern controllers are capable of even balancing multiple linked pendulums.
Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the support rapidly up and down. If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner.
In practice, the inverted pendulum is frequently made of an aluminium strip, mounted on a ball-bearing pivot; the oscillatory force is conveniently applied with a jigsaw.
If the driving point moves in simple harmonic motion, the pendulum's motion is described by the Mathieu equation.