About the possibility of self-oscillations formation of a two-link inverted pendulum with a movable fulcrum

Аuthors

Alexandrov V. V.^*, Sidorenko G. U.^**

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

*e-mail: vladimiralexandrov366@hotmail.com
**e-mail: gingrul@yandex.ru

Abstract

In this paper we consider a movement of the inverted two-link pendulum on a cart system with restricted resources of perturbation. The pendulum system has two ideal motors. One of the motors is located in the node of the pendulum, and with the aid of the second motor pendulum is mounted on the cart. The control of the ideal motors is constructed as a feedback of potentiometers measurements. This provides oscillatory behavior of the system and ensures asymptotic stability of the zero equilibrium point. By means of the changing coordinates the system represents in a new form. The new system is considered under assumption of smallness of coefficients ligation and multiplicity of eigenfrequencies. The selection of the cart system acceleration as piecewise constant function of the angular velocity is based on the analysis of maximum deviation of pendulum’s center of gravity using Pontryagin`s maximum principle. The orbitally-stable limit cycle for fourth-order system has been constructed by integrating the system with selected deviation and constructing the successor function on a secant plane in four-dimensional space. Self-oscillations correspond to the limit cycle. The achievable line was founded on the secant plane. The achievable line means that in the phase space trajectories of the system that begin from this line necessarily come back to the line infinite times. Obtained system’s self-oscillations can be a maximum, i.e. limit cycle can be a boundary of the achievable set for the pendulum center of gravity. The resulting algorithm of the cart movement can be used as a test motion for the imitation stand for the vestibular prosthesis verification.

Keywords:

inverted pendulum, autooscillation, limit cycle, worst disturbance, vestibular prosthesis

References

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