Theory of inverted pendulum with follower force revisited

The effect of the type of springs on the equilibrium states of an inverted pendulum is examined. The angular and linear eccentricities of the follower force are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–137, June 2007.     

The equations of an inverted pendulum with an arbitrary number of links and an asymmetric follower force

This paper formulates a problem for and gives the differential equations of plane-parallel motion of an inverted n-link pendulum subject to an asymmetric follower force applied at the upper end via a spring. The physical nonlinearities of springs are taken into account. The possible mechanisms of energy dissipation are described __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 106–114, May 2007.     

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.     

Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum

The effect of the concurrent use of springs with characteristics of different types (hard, soft, or linear) on the equilibrium of an inverted simple pendulum is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 107–113, July 2007.     

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.     

Evolution of the equilibrium states of an inverted pendulum

The influence of the pendulum parameters and the follower force on the evolution of equilibrium states is analyzed using a generalized mathematical model of inverted pendulum. Equilibrium curves are plotted using the parameter continuation method. It is shown that the pendulum with certain values of the angular eccentricity has one or three nonvertical equilibrium positions __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 122–131, March 2007.     

Rotating a pendulum with an electromechanical excitation

The objective of this paper is showing investigation of pendulum rotations via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid, which is originally built for an electro-vibro-impact mechanism. Various non-linear phenomena of pendulum dynamics, namely period-1 rotation, period-1 oscillation and period-2 oscillation, have been observed experimentally from the proposed apparatus. A mathematical model has been developed for the experimental rig and the system parameters have also been identified for the mathematical model. Finally, numerical results have been generated using the developed mathematical model and identified parameters, and their correlations with experimental observations have been discussed.     

Improvement of traction performance and off-road mobility for a vehicle with four individual electric motors: Driving over icy road

To control speed and wheel slip for severe conditions of tire-surface interaction is a challenging task in the design of traction control system for electric vehicles with off-road capability. In this regard, the present paper focuses on a specific traction control for an electric vehicle with four individual in-wheel motors over icy road. The study demonstrates that a proper integration of the speed controller and wheel slip controller can essentially improve the mobility of the vehicle in the cases of acceleration and slope climbing. The paper discusses relevant case studies with particular attention given to the system architecture (sliding mode and PID control methods), extremum-seeking algorithm for maximum tire-road friction and corresponding slip value, and experimental validation of the tire model used in the controller with the help of the Terramechanics Rig in the Advanced Vehicle Dynamics Laboratory (AVDL) at Virginia Polytechnic Institute and State University.     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

An analogy between a gravity modulated porous layer heated from below and the inverted pendulum with an oscillating pivot point

The governing equations for the inverted pendulum is developed and shown to be analogous to the gravity modulated porous layer heated from below. In particular the temperature in a gravity modulated porous layer heated from below (R>0-unstable) is likened to the motion of an inverted pendulum (unstable). As gravity modulation stabilizes convection, Transport Porous Med. 57 (2004), 113, it is found that oscillating the pivot point of an unstable, inverted pendulum stabilizes the motion.     

Equilibrium states of a pendulum with an asymmetric follower force

The effect of the linear eccentricity of the follower force on the equilibrium states of an inverted pendulum is examined. Bifurcation points and catastrophes associated with changes in pendulum parameters and type of springs are analyzed. Phase flows are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 121–129, April 2007.     

Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.     

Stability domains of the vertical equilibrium state of a triple simple pendulum

The stability boundaries for the vertical equilibrium position of a triple simple pendulum subject to asymmetric follower force are analyzed. The effect of the upper spring and the magnitude of the follower force on the roots of the characteristic equation is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 122–134, October 2008.     

Controlling the inverted pendulum by means of a nested saturation function

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.     

Asymptotic stability boundaries for the equilibrium positions of a double pendulum with vibrating point of suspension

The mechanisms whereby a double pendulum with vibrating point of suspension loses stability in equilibrium positions are studied. Stability conditions for the equilibrium positions in critical cases are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 120–133, July 2008.     

Modeling of system dynamics of a slewing flexible beam with moving payload pendulum

When a tower crane is handling payload via rotation and moving the carriage simultaneously the jib structure and the payload can be modeled as a system consisting of a slewing flexible clamed-free beam with the spherical payload pendulum that moves along the beam. The present work completes the dynamic modeling of the system mentioned above. The clamed-free beam attached to a rotating hub is modeled by Euler–Bernoulli beam theory. The payload is modeled as a sphere pendulum of point mass attached to via massless inextensible cable the carriage moving on the rotating beam. Non-linear coupled equations of motion of the in- and out-of-plane of the beam and the payload pendulum are derived by means of the Hamilton principle. Some remarks are made on the equations of motion.     

Development of a vehicle to study the tractive performance of integrated steering-drive systems

This paper describes a test-bed vehicle for studying the integration of the steering system of a wheeled vehicle with the drive system. The vehicle was produced in order to determine whether such an integrated system is practical; to investigate tractive performance compared to other steering-drive systems; and to determine under which conditions such a system has better performance. The integrated steering-drive system of the test-bed vehicle uses a computer to co-ordinate the independently driven wheel speeds of the drive system (which is also the primary steering system) with the steer angles of the non-driven steerable wheels to produce a beneficial secondary steering effect. The secondary steering system assists the primary steering system when side forces act on the vehicle, while producing minimal conflict. This concept can be applied to agricultural vehicles such as tractors, harvesters, mowers, sprayers and self-propelled windrowers. The test-bed vehicle is able to be configured for the following steering-drive systems types: open differential drive with steerable wheels, independent drive wheels with castors, locked differential drive with steerable wheels and a computer integrated steering-drive system. The capacity of the test-bed vehicle to be configured as described is a significant advantage when measuring tractive performance, as the results obtained will be more valid due to the vehicle parameters being the same.     

On some problems of stability of systems with an infinite number of degrees of freedom

The problem of stability of equilibria of a physical pendulum with a nonstretchable thread attached to it is considered from the standpoint of the Lagrange theorem on stability and its inversion. Specific difficulties which one faces when studying an infinite dimensional mechanical system are discussed. A new approach to the study of stability with respect to two metrics is suggested. The influence of resonant phenomena on the motion of the shortened (linearized) system is considered.

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Sommario Si considera il problema di stabilità di punti di equilibrio di un pendolo fisico con un filo inestensibile appeso ad esso dal punto di vista del teorema di Lagrange sulla stabilità e sulla sua inversione. Difficoltà specifiche relative allo studio di un sistema meccanico di dimensione infinita sono discusse. Si suggerisce un nuovo metodo per lo studio della stabilità rispetto a due metriche. L'influenza di fenomeni di risonanza sul moto del sistema ridotto (linearizzato) è considerata.