Editing Inverted pendulum (section)

== Overview ==
A pendulum with its bob hanging directly below the support [[Lever|pivot]] is at a [[Mechanical equilibrium|stable equilibrium]] point, where it remains motionless because there is no torque on the pendulum. If displaced from this position, it experiences a restoring torque that returns it toward the equilibrium position.  A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an [[unstable equilibrium]] point.  At this point again there is no torque on the pendulum, but the slightest displacement away from this position causes a gravitation torque on the pendulum that accelerates it away from equilibrium, causing it to fall over.  
  
In order to stabilize a pendulum in this inverted position, a [[feedback controller|feedback control system]] can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced.   The inverted pendulum is a classic problem in [[dynamics (mechanics)|dynamics]] and [[control theory]] and is widely used as a benchmark for testing control algorithms ([[PID controller]]s, [[state-space representation]], [[Artificial neural network|neural networks]], [[fuzzy control]], [[genetic algorithm]]s, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability.<ref>{{cite web| url = https://www.grc.nasa.gov/WWW/k-12/VirtualAero/BottleRocket/airplane/rktstab.html| title = Model Rocket Stability}}</ref> The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing [[personal transporter]]s such as the [[Segway PT]], the [[self-balancing scooter|self-balancing hoverboard]] and the [[self-balancing unicycle]].

Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down.  This is called [[Kapitza's pendulum]].  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. If the driving point moves in [[simple harmonic motion]], the pendulum's motion is described by the [[Mathieu equation]].<ref>{{Cite web |last=Mitchell |first=Joe |title=Techniques for the Oscillated Pendulum and the Mathieu Equation |url=https://math.ou.edu/~npetrov/joe-report.pdf |access-date=2023-11-06 |website=math.ou.edu}}</ref>