scholarly journals Delay-induced bifurcations in collocated position control of an elastic arm

2021 ◽  
Author(s):  
Bence Szaksz ◽  
Gabor Stepan
Keyword(s):  
Time Delays ◽  
Nonlinear Vibrations ◽  
Vibration Mode ◽  
Position Control ◽  
Control Force ◽  
Infinite Dimensional ◽  
Practical Procedure ◽  
Centre Manifold Reduction ◽  
Manifold Reduction ◽  
Simplified Mechanical Model

AbstractThe interference of the elasticity of a single robotic arm and the unavoidable time delay of its position control is analysed from nonlinear vibrations viewpoint. The simplified mechanical model of two blocks and a connecting spring considers the first vibration mode of the arm, while the collocated proportional-derivative (PD) control uses the state of the first block only and actuates also there. It is assumed that the relevant nonlinearity is the saturation of the delayed control force. The linear stability analysis proves that stabilizable and non-stabilizable parameter regions follow each other periodically even for large spring stiffnesses and for tiny time delays. Hopf bifurcation calculation is carried out after an infinite-dimensional centre manifold reduction, and closed-form algebraic expressions are given for the amplitudes of the emerging oscillations. These results support the experimental tuning of the control gains since the parameters of the arising and often unexpected self-excited vibrations can serve as a guide for this practical procedure.

scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Reduction of residual vibrations in a rotating flexible beam with a moving payload mass

1997 ◽  
Vol 211 (1) ◽  
pp. 25-34 ◽  
Author(s):  
S Choura
Keyword(s):  
Rigid Body ◽  
Position Control ◽  
Rate Of Change ◽  
Torque Control ◽  
Flexible Beam ◽  
Control Force ◽  
Axial Motion ◽  
Body Rotation ◽  
Rigid Body Rotation ◽  
Payload Mass

The reduction of residual vibrations for the position control of a flexible rotating beam carrying a payload mass is investigated. The common practice used to find the position control of a flexible multi-link arm is to assign a torque actuator to each joint while the payload mass is kept fixed relative to the end-link during the time of manoeuvre. This paper examines the stability of the system if either the payload is freed accidentally to move along the beam during the time of manoeuvre or is allowed to span the beam in a desired path for control purposes. A candidate Lyapunov function is constructed and its time rate of change is examined. It is shown that the use of a PD (proportional plus derivative) torque control yields a convergence of residual vibration to zero, an attainment of the rigid-body rotation to a prespecified desired angle of manoeuvre and a constant velocity of the payload mass as it moves relative to the beam. For manipulation purposes, an additional control force is added to the moving actuator in order to regulate its axial motion. It is shown that allowing the axial motion of the payload mass in a prescribed manner leads to a considerable reduction of its residual vibrations as compared to the case where the payload mass is fixed to the beam tip during the time of manoeuvre. Stability is also verified through simulations of rigid-body rotation and payload axial motion track prespecified reference trajectories.

Computing dynamics of copper-based SMA via centre manifold reduction of 3D models

2000 ◽  
Vol 18 (3-4) ◽  
pp. 255-268 ◽  
Author(s):  
R.V.N. Melnik ◽  
A.J. Roberts ◽  
K.A. Thomas
Keyword(s):  
3D Models ◽  
Centre Manifold ◽  
Centre Manifold Reduction ◽  
Manifold Reduction

Control of Two-Link Flexible Structures

Volume 1 ◽  
2004 ◽  
Author(s):  
Clarice Wagner-Nachshoni ◽  
Yoram Halevi
Keyword(s):  
Transfer Function ◽  
Time Delays ◽  
Wave Motion ◽  
Controller Design ◽  
Flexible Structures ◽  
Infinite Dimensional ◽  
Second Stage ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Two Stages

A method of noncollocated controller design for non-uniform flexible structures, governed by the wave equation, is proposed. An exact, infinite dimensional, transfer function, relating the actuation and measurement points, with general boundary conditions, is derived for the multi-link case. Three modeling methods are presented and discussed. A key element of the model is the existence of time delays, due to the wave motion, which play a major role in the controller design. The design consists of two stages. First an inner rate loop is closed in order to improve the system dynamic behavior. It leads to a finite dimensional plus delay inner closed loop, which is the equivalent plant for the outer loop. In the second stage an outer noncollocated position loop is closed. It has the structure of an observer–predictor control scheme to compensate for the response delay. The resulting overall transfer function is second order, with arbitrarily assigned dynamics, plus delay.

Nonlinear Bifurcation Analysis of a Single-DoF Model of a Robotic Arm Subject to Digital Position Control

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Giuseppe Habib ◽  
Giuseppe Rega ◽  
Gabor Stepan
Keyword(s):  
Center Manifold ◽  
Position Control ◽  
High Accuracy ◽  
Industrial Applications ◽  
Critical Parameters ◽  
Robotic Arm ◽  
Control Force ◽  
Local Center ◽  
The Stability ◽  
Stability Limits

Precision and stability in position control of robots are critical parameters in many industrial applications where high accuracy is needed. It is well known that digital effect is destabilizing and can cause instabilities. In this paper, we analyze a single DoF model of a robotic arm and we present the stability limits in the parameter space of the control gains. Furthermore we introduce a nonlinearity relative to the saturation of the control force in the model, reduce the dynamics of the nonlinear map to its local center manifold, study the bifurcation along the stability border and identify conditions under which a supercritical or subcritical bifurcation occurs. The obtained results explain some of the typical instabilities occurring in industrial applications. We verify the obtained results through numerical simulations.

scholarly journals Grain boundaries in the Swift–Hohenberg equation

2012 ◽  
Vol 23 (6) ◽  
pp. 737-759 ◽  
Author(s):  
MARIANA HARAGUS ◽  
ARND SCHEEL
Keyword(s):  
Grain Boundaries ◽  
Order Approximation ◽  
Spatial Dynamics ◽  
Heteroclinic Orbits ◽  
Existence Problem ◽  
Leading Order ◽  
Centre Manifold ◽  
Centre Manifold Reduction ◽  
Manifold Reduction ◽  
Wavenumber Selection

We study the existence of grain boundaries in the Swift–Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ordinary differential equations in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

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