The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

Uncertain inference control for balancing an inverted pendulum

Fuzzy inference control uses fuzzy sets to describe the antecedents and consequents of If-Then rules. However, most surveys show the antecedents and consequents are uncertain sets rather than fuzzy sets. This fact provides a motivation to invent an uncertain inference control method. This paper gives an introduction to the design procedures of uncertain inference controller. As an example, an uncertain inference controller for balancing an inverted pendulum system is successfully designed. The computer simulation shows the developed uncertain inference controller is of good robustness.     

The non-linear oscillations of a pendulum of variable length on a vibrating base

A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone.     

Chaotic responses of a harmonically excited spring pendulum moving in circular path

In this paper, the chaotic response of a harmonically excited spring pendulum that is moving in circular path is studied. This system is a nonlinear multi-degrees-of-freedom system and is a good example for several engineering applications such as ship motion. Using the multiple scales (MS) method [A. Nayfeh, Perturbation Methods, Wiley-Interscience, New York, 1973], the original non-autonomous system is reduced to a third-order approximate autonomous system. The approximate system is shown to have bifurcation leading to chaos.     

Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation

A class of periodic motions of an inverted pendulum with rigid lateral barriers is analysed under the hypothesis that the system is forced by impulsed periodic excitation. Due to the piece-wise linear nature of the problem, the existence and the stability of the cycles are determined analytically. It is found that they depend on both classical (saddle-node and period-doubling) and non-classical bifurcations, the latter involving a ‘synchronization' between impulses and impacts which leads to the sudden disappearing of the orbits. Attention is paid to the physical interpretation of these bifurcations, and to the determination of analytical criteria for their occurrence. We study how the relative position (with respect to the excitation amplitude) of the local bifurcations determines the system response and the bifurcation scenario. Symmetric and unsymmetric excitations are considered and the regions of stability of the periodic solutions are analytically determined. Finally, a comparison with the case of harmonic excitation is presented showing both analogies and differences, and highlighting how the impulsed excitation allows to obtain stable periodic responses at higher values of the excitation amplitude.     

Expansion of solution of an inverted pendulum system with time delay

In this paper, the solution expansion for an inverted pendulum system with time delay is studied. The linearized model of this nonlinear system near its equilibrium is derived on the assumption that a unique equilibrium exists in it. Then the asymptotic expressions of its eigenvalues and the eigenvalues’ corresponding eigenvectors are obtained. Moreover, although the set of these eigenvectors does not form a Schauder basis for the state space, the solution of this model still can be expressed by these eigenvectors in the form of infinite series under certain conditions. Finally, a simulation is provided to support these results.     

Control of the spatial motion of a multilink inverted pendulum using a torque applied to the first link

An approach to constructing a control for non-linear mechanical systems in which the number of degrees of freedom exceeds the dimension of the generalized control forces is developed. An n-link pendulum with two-degree-of-freedom joints, controlled by a torque applied to the first link, is considered as an example. Such a pendulum has 2ⁿ different equilibrium positions. A feedback control with an absolute value constraint, which transfers the pendulum from the neighbourhood of an arbitrary equilibrium position to this equilibrium position in a finite time, is constructed. For this purpose, the equations of motion of the pendulum are linearized in the neighbourhood of the equilibrium position under consideration, complete controllability of the linear model is established, and a control is constructed for it using the linear matrix inequality technique. The applicability of the control law obtained to the solution of the problem of controlling a non-linear multilink pendulum is verified. ©2014     

On the existence of periodic motions of the excited inverted pendulum by elementary methods

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form

$$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$

where l, g are constants and

$$a(t) := \begin{cases} A &\text{if } 2kT\leq t < (2k+1)T,\\ -A &\text{if } (2k+1)T\leq t < (2k+2)T,\end{cases}\quad (k=0,1,\dots);$$

A, T are positive constants. The exact stability zones for the upper equilibrium are presented.

     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.     

The stability of an inverted pendulum when there are rapid random oscillations of the suspension point

The stability of the upper equilibrium position of a pendulum when the suspension point makes rapid random oscillations of small amplitude, is investigated. A class of random oscillations that make the system stable with unit probability for small friction is indicated. It is shown that, if there is no friction, there is no stability, which, as is well known, is not the case for harmonic oscillations of the suspension point. Some general results concerning the impossibility of stochastic stabilization of Hamiltonian systems are proved.     

An inverse problem for a system with a moving boundary

The problem of estimating the right-hand side of a nonlinear parabolic equation is considered. A finite-step algorithm based on the model positional control method and the finite element method is proposed. The algorithm is robust to informational noise and computational errors. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 23–33.     

On the number of maximal dependencies in a data base relation of fixed order

The paper gives asymptotic bounds for the maximum number N_n of non-trivial maximal elements in a data base relation of given order. The result shows that there exist relations which are very rich in maximal elements.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

Control and stability of the complex inverted pendulum models in application to postural sway analysis of the vertical human stance

Mechanism of balance control is investigated on computerized posturography data on the vertical two–legged and one–legged stances measured on healthy subjects and the patients with osteohondrosis and coxarthrosis. Oscillations of the center of mass over 30 s of keeping the quiet stance and the trajectories for a step forward off the force platform have been computed. It was shown that during the one–legged stance the balance control strategy includes ‘scanning’ of the larger area with bigger sway amplitudes in the vicinity of the stable state as compared to the two–legged stance. A generalized model of the body as a multi–link system is proposed. Mass and inertia of each body segment and torques in joints are taken into consideration. The calculated own and forced frequencies of the model correspond to the spectral analysis of the posturography data. One–legged stance is proposed as a proper test for revealing the balance disorders. It is shown that investigation of the oscillations and trajectories of the center of mass for the step forward off the force platform is perspective for medical diagnostics to distinguish between the spine and joint pathologies. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum

We present a continuation method to obtain a family of T-periodic solutions for a family of T-periodic systems.In particular, we present some sufficient analytical conditions, which have the advantage of being an easy application to some systems of interest in physics or engineering. We apply these conditions to the exact tracking problem for the inverted spherical pendulum.     

An example concerning fixed points

An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2) ⁿ does not converge.     

An intuitionistic fixed point theory

In this article we prove that a certain intuitionistic version of the well-known fixed point theory is conservative over for almost negative formulas. Received October 24, 1995