On the Fastest Small Movements of the Platform with Inverted Pendulums
Author(s):
Oleg Rashidovich Kayumov
Professor of the Department of Mathematics, Computer Science
and Vocational Training of the Branch of Omsk State
Pedagogical University in Tara
Oleg_Kayumov@mail.ru
Abstract:
We consider the problem of time-optimal movement of a platform moving translationally
along a horizontal straight line and carrying n inverted pendulums, and during
the movement the pendulums should not pass through the lower vertical positions.
The only control force is applied to the platform and is limited in magnitude,
there is no friction. The system is transferred from the one unstable state
of rest to a similar state at a given distance. It is assumed to be small
to the extent that linearized equations of motion can be used. The evolution
of optimal control functions depending on the amount of platform movement is studied.
A general approach to constructing a visual diagram reflecting such evolution is proposed.
It is shown that when the number of inverted pendulums is odd, the movement of the platform
begins from reverse, but when there is an even number, it does not. The optimal control
modes found for a linear system are applied to the problem of small displacement
of a platform with two inverted pendulums in a nonlinear formulation.
Keywords
- platform with inverted pendulum
- time-optimal problem
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