Inverted pendulum facts for kids
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger.
A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position.
Overview
A pendulum with its bob hanging directly below the support pivot is at a 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 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 and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, state-space representation, neural networks, fuzzy control, genetic algorithms, 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. 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 transporters such as the Segway PT, the 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.
Examples
Arguably the most prevalent example of a stabilized inverted pendulum is a human being. A person standing upright acts as an inverted pendulum with their feet as the pivot, and without constant small muscular adjustments would fall over. The human nervous system contains an unconscious feedback control system, the sense of balance or righting reflex, that uses proprioceptive input from the eyes, muscles and joints, and orientation input from the vestibular system consisting of the three semicircular canals in the inner ear, and two otolith organs, to make continual small adjustments to the skeletal muscles to keep us standing upright. Walking, running, or balancing on one leg puts additional demands on this system. Certain diseases can interfere with this reflex, causing dizziness and disequilibration, an inability to stand upright. A field sobriety test used by police to test drivers for the influence of alcohol, tests this reflex for impairment.
Some simple examples include balancing brooms or meter sticks by hand.
The inverted pendulum has been employed in various devices and trying to balance an inverted pendulum presents a unique engineering problem for researchers. The inverted pendulum was a central component in the design of several early seismometers due to its inherent instability resulting in a measurable response to any disturbance.
The inverted pendulum model has been used in some recent personal transporters, such as the two-wheeled self-balancing scooters and single-wheeled electric unicycles. These devices are kinematically unstable and use an electronic feedback servo system to keep them upright.
Swinging a pendulum on a cart into its inverted pendulum state is considered a traditional optimal control toy problem/benchmark.
See also
- Double inverted pendulum
- Inertia wheel pendulum
- Furuta pendulum
- iBOT
- Humanoid robot
- Ballbot
