Limit Cycle Analysis and Control of the Dissipative Chaplygin Sleigh

2017 ◽  
Author(s):  
Vitaliy Fedonyuk ◽  
Phanindra Tallapragada ◽  
Yongqiang Wang
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Horizontal Plane ◽  
Translational Velocity ◽  
Periodic Forcing ◽  
Fluid System ◽  
Chaplygin Sleigh ◽  
Analysis Techniques ◽  
And Control ◽  
Gaining Insight

There are many types of systems in both nature and technology that exhibit limit cycles under periodic forcing. Sometimes, especially in swimming robots, such forcing is used to propel a body forward in a plane. Due to the complexity in studying a fluid system it is often useful to investigate the dynamics of an analogous land model. Such analysis can then be useful in gaining insight about and controlling the original fluid system. In this paper we investigate the behavior of the Chaplygin sleigh under the effect of viscous dissipation and sinusoidal forcing. This is shown to behave in a similar manner as certain robotic fish models. We then apply limit cycle analysis techniques to predict the behavior and control the net translational velocity of the sleigh in a horizontal plane.

Limit Cycle Behavior and Model Reduction of an Oscillating Fish-Like Robot

2018 ◽  
Author(s):  
Beau Pollard ◽  
Vitaliy Fedonyuk ◽  
Phanindra Tallapragada
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Nonholonomic Constraint ◽  
Dimensional Model ◽  
Underwater Robots ◽  
Chaplygin Sleigh ◽  
Shaped Body ◽  
Periodic Input ◽  
Low Dimensional ◽  
And Control

The design and control of underwater robots has to contend with the coupled robot-hydrodynamic interactions. A key aspect of this coupled dynamics is the interaction of the robot with the fluid via the vorticity that is created by the robot’s motion. In this paper we develop a simplified and very low dimensional model of this interaction. This is done recognizing that the vortex shedding is a nonholonomic constraint. We apply the harmonic balance approach to analyze and compare the limit cycle in the dynamics of the fish-shaped body propelled by a periodic input with that of a Chaplygin sleigh, a well known nonholonomic system. The dynamics on the limit cycles lead to a very low dimensional model of the swimming of the fish-shaped body that could be very useful from the perspective of controlling a swimming robot.

Advanced Fringe Analysis Techniques in Circuit Edit

2006 ◽  
Author(s):  
R.K. Jain ◽  
T. Malik ◽  
T.R. Lundquist ◽  
C.-C. Tsao ◽  
W.J. Walecki
Keyword(s):  
Imaging System ◽  
Success Rates ◽  
Endpoint Detection ◽  
Fringe Analysis ◽  
Analysis Techniques ◽  
Aspect Ratios ◽  
Slope Correction ◽  
Trench Area ◽  
And Control ◽  
Thickness Measurements

Abstract Novel Fabry Perot [1] fringe analysis techniques for monitoring the etching process with a coaxial photon-ion column [2] in the Credence OptiFIB are reported. Presently the primary application of these techniques in circuit edit is in trenching either from the front side or from the backside of a device. Optical fringes are observed in reflection geometry through the imaging system when the trench floor is thin and semi-transparent. The observed fringes result from optical interference in the etalon formed between the trench floor (Si in the case of backside trenching) and the circuitry layer beyond the trench floor. In-situ real-time thickness measurements and slope correction techniques are proposed that improve endpoint detection and control planarity of the trench floor. For successful through silicon edits, reliable endpoint detection and co-planarity of a local trench is important. Reliable endpoint detection prevents milling through bulk silicon and damaging active circuitry. Uneven trench floor thickness results in premature endpoint detection with sufficient thickness remaining in only part of the trench area. Good co-planarity of the trench floor also minimizes variability in the aspect ratios of the edit holes, hence increasing success rates in circuit edit.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

Determination of the Limit Cycle by He’s Parameter-Expansion for Oscillators in a u3 / (1 + u2) Potential

2007 ◽  
Vol 62 (7-8) ◽  
pp. 396-398 ◽  
Author(s):  
Li-Na Zhang ◽  
Lan Xu
Keyword(s):  
Exact Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Expansion Method ◽  
Nonlinear Oscillators ◽  
Mathematical Tool ◽  
Parameter Expansion ◽  
Parameter Expansion Method ◽  
Powerful Mathematical Tool

This paper applies He’s parameter-expansion method to determine the limit cycle of oscillators in a u3/(1+u2) potential. The results are compared with the exact solutions. This shows that the method is a convenient and powerful mathematical tool for the search of limit cycles of nonlinear oscillators.

scholarly journals Height control for a two-mass hopping robot based on stable limit cycle

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141665774
Author(s):  
Taihui Zhang ◽  
Honglei An ◽  
Qing Wei ◽  
Wenqi Hou ◽  
Hongxu Ma
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Inverted Pendulum Model ◽  
Hopping Robot ◽  
Control Objective ◽  
Upper Level ◽  
Mass Spring ◽  
And Control

Differing from the commonly used spring loaded inverted pendulum model, this paper makes use of a two-mass spring model considering impact between the foot and ground which is closer to the real hopping robot. The height of upper mass which includes the upper leg and body is the main control objective. Then we develop a new kind of control algorithm acting on two levels: The upper level aims to achieve the desired velocity of the upper mass based on a stable limit cycle, where three different controllers are used to regulate the limit cycle; the target of the lower level is to drive the system to converge to the desired state and control the contact force between the foot and ground within an appropriate range based on the inner force control at the same time. Simulation results presented in this paper confirm the efficiency of this control algorithm.

On Motion of Chaplygin Sleigh on a Horizontal Plane with Dry Friction

2019 ◽  
Vol 54 (5) ◽  
pp. 632-637
Author(s):  
A. V. Karapetyan ◽  
A. Yu. Shamin
Keyword(s):  
Dry Friction ◽  
Horizontal Plane ◽  
Chaplygin Sleigh

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

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