Inverse pendulum

from Wikipedia, the free encyclopedia
Scheme of an inverse pendulum on a carriage

The inverse pendulum is a pendulum with the center of gravity above the axis. At its highest point, the pendulum is in an unstable position of rest . The inverted pendulum is one of the standard functions of the control technique for stabilizing an unstable control loop .

A standard example of an inverse pendulum is a cart with an inverse pendulum mounted on it. The pendulum movement can be influenced by the horizontal movement of the carriage.

Designs

The standard task of a linear inverse pendulum can be expanded in its dimensions. A possible extension is a second degree of freedom of the axis, another extension is an inverse pendulum with several axes and coupled pendulums.

Spatial inverse pendulum

Spatial inverse pendulum. The pendulum has two degrees of freedom, the base of the pendulum is movable in the plane along two directions (green and blue arrow).

In the case of a spatial inverse pendulum, the axis, for example the carriage, does not move along a straight line, but freely in the plane. The axis of the inverse pendulum is movable in two directions, so the pendulum can tilt in all directions.

The axis must now be moved in the plane analogous to the linear inverse pendulum in such a way that it does not tilt. Technical structures could solve this task either with a robot arm with at least two degrees of freedom or a carriage that can move freely in a plane.

The task that an artist performs when balancing a stick on the hand is clear here.

Inverse pendulum as a control task

The inverse pendulum is at its highest point in an unstable rest position . The inverse pendulum, viewed as a system, always strives for the state of the lowest total energy. This state is oscillating around the stable rest position or in this. Small deflections from the unstable rest position cause the pendulum rod to leave this rest position. The necessary energy then has to be constantly supplied to the system in order to remain in its unstable rest position.

The variable to be controlled with an inverse pendulum is the angle between the pendulum and the vertical axis. The angle is continuously measured and compared with a reference variable. The reference variable is what corresponds to the pendulum in its upright position. If there is a discrepancy between the measured variable, the actual value, and the reference variable, the setpoint, a control system takes place to bring the actual and setpoint back into line. The inverse pendulum in the above designs is controlled by moving the base point in order to move it below the center of gravity of the inverse pendulum.

The angle is usually measured either via a position sensor or a rotary encoder in the axis of the pendulum. The difference between the setpoint and actual value is translated into a manipulated variable for the actuator by a controller .

use cases

Segway driver

The inverse pendulum is used in various forms in everyday life. One example is the balancing of people and objects, as it is practiced by artists in acrobatics . A person standing upright can also be viewed as an inverse pendulum: The center of gravity is above the standing surface and body movements throw the pendulum out of balance, so that a countermovement is required to restore or maintain it.

Riding a unicycle can be interpreted as riding an inverse pendulum . The Segway Personal Transporter , a single-axle electric scooter, takes over the balancing for the driver by means of electronic control.

The Grasshopper launcher from the US company SpaceX uses the principle to be able to take off and land again upright.

literature

  • Fritz Tröster: Control and regulation technology for engineers . 2nd Edition. Oldenbourg Verlag, 2005, ISBN 3-486-57681-X .

Web links