scholarly journals Formation mechanism of a basin of attraction for passive dynamic walking induced by intrinsic hyperbolicity

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160028 ◽  
Author(s):  
Ippei Obayashi ◽  
Shinya Aoi ◽  
Kazuo Tsuchiya ◽  
Hiroshi Kokubu
Keyword(s):  
Basin Of Attraction ◽  
Underlying Mechanism ◽  
The Body ◽  
Bipedal Walking ◽  
Dynamic Walking ◽  
Passive Dynamic Walking ◽  
Saddle Type ◽  
The Stability ◽  
Stable Passive ◽  
The Basin Of Attraction

Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The basin of attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the basin of attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the basin of attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

Passive Dynamic Biped Walking—Part II: Stability Analysis of the Passive Dynamic Gait

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Derek Koop ◽  
Christine Q. Wu
Keyword(s):  
Angular Momentum ◽  
Stability Analysis ◽  
Lyapunov Exponents ◽  
Basin Of Attraction ◽  
Dynamic Walking ◽  
Novel Method ◽  
The Stability ◽  
Dynamic Gait ◽  
Excellent Tool ◽  
The Basin Of Attraction

Passive dynamic walking is an excellent tool for evaluating biped stability measures, due to its simplicity, but an understanding of the stability, in the classical definition, is required. The focus of this paper is on analyzing the stability of the passive dynamic gait. The stability of the passive walking model, validated in Part I, was analyzed with Lyapunov exponents, and the geometry of the basin of attraction was determined. A novel method was created to determine the 2D projection of the basin of attraction of the model. Using the insights gained from the stability analysis, the relation between the angular momentum and the stability of gait was examined. The angular momentum of the passive walker was not found to correlate to the stability of the gait.

Novel Periodic and Turning Motions in the Simplest Passive Walking Model

2014 ◽  
Author(s):  
M. R. Sabaapour ◽  
M. R. Hairi Yazdi ◽  
B. Beigzadeh
Keyword(s):  
Periodic Motion ◽  
Essential Feature ◽  
Basin Of Attraction ◽  
Three Dimensional ◽  
Surface Profile ◽  
Passive Walking ◽  
The Stability ◽  
Special Case ◽  
Stable Passive ◽  
Passive Turning

The ability to move along curved paths is an essential feature for biped walkers to move around obstacles. This study is aimed at extending passive walking concept for curved walking and turning to generate more natural and effective motion. Hence three-dimensional (3D) motion of a rimless spoked-wheel, as the simplest walking model, about a general vertical fixed coordinate system has been derived. Then, two kinds of a stable passive turning, i.e. limited and circular continuous have been considered and discussed. The first kind is actually transferring from a 2D periodic motion to another, and can be implemented on a straight slope surface. While, it was shown that the second kind is just related to novel 3D periodic motions and can be recognized on a special surface profile namely “helical slope” introduced here. The latter are interpreted as 3D fixed points of a Poincare return map again. So, their stability was evaluated numerically by a Jacobian analysis and demonstrated through several simulations. Results show asymptotical stability of such motions and their considerable basin of attraction with respect to initial states. In addition, the characteristic of passive turning is shown to be strictly connected with the value of the initial perturbed condition, for instance, to the initial inclination of the wheel. It is then surprising to note that the stability of a 3D passive periodic motion (turning) is higher than 2D one (straight walking) which is actually a special case just with an infinite radius of turn.

Basin of Attraction of the Simplest Walking Model

2001 ◽  
Author(s):  
A. L. Schwab ◽  
M. Wisse
Keyword(s):  
Equations Of Motion ◽  
Basin Of Attraction ◽  
Mapping Method ◽  
Walking Robots ◽  
Natural Energy ◽  
Walking Motion ◽  
The Stability ◽  
The Basin Of Attraction ◽  
General Method ◽  
Small Disturbances

Abstract Passive dynamic walking is an important development for walking robots, supplying natural, energy-efficient motions. In practice, the cyclic gait of passive dynamic prototypes appears to be stable, only for small disturbances. Therefore, in this paper we research the basin of attraction of the cyclic walking motion for the simplest walking model. Furthermore, we present a general method for deriving the equations of motion and impact equations for the analysis of multibody systems, as in walking models. Application of the cell mapping method shows the basin of attraction to be a small, thin area. It is shown that the basin of attraction is not directly related to the stability of the cyclic motion.

Special Issue on Dynamically and Biologically Inspired Legged Locomotion

2017 ◽  
Vol 29 (3) ◽  
pp. 455-455
Author(s):  
Tetsuya Kinugasa ◽  
◽  
Koh Hosoda ◽  
Masatsugu Iribe ◽  
Fumihiko Asano ◽  
...  
Keyword(s):  
Design Method ◽  
Joint Stiffness ◽  
Legged Locomotion ◽  
The Body ◽  
Biological Characteristics ◽  
Dynamic Walking ◽  
Special Issue ◽  
Biologically Inspired ◽  
Passive Dynamic Walking ◽  
Wide Range

Legged locomotion, including walking, running, turning, and jumping, strongly depends on the dynamics and biological characteristics of the body involved. Gait patterns and energy efficiency, for example, are known to be greatly affected by not only travel velocity and ground contact conditions but also by body configuration, such as joint stiffness and coordination, as well as foot sole shape. To understand legged locomotion principles, we must clarify how the body’s dynamic and biological characteristics affect locomotion. Effort must also be made to incorporate these characteristics inventively to improve locomotion performance, such as robustness, adaptability, and efficiency, which further refine the legged locomotion. This special issue on “Dynamically and Biologically Inspired Legged Locomotion,” studies on legged locomotion based on dynamic and biological characteristics, covers a wide range of themes, such as a rimless wheel, a design method for a biped based on passive dynamic walking, the analysis of biped locomotion based on passive dynamic walking and dynamically inspired walking, an analysis of gait generation for a triped robot, and quadruped locomotion with a flexible trunk. Since there are interesting papers on legged robots with different numbers of legs, we basically organized the papers based on the number of legs. Studies on “Dynamically and Biologically Inspired Legged Locomotion” are expected to not only realize and improve legged locomotion as engineering, but also to reveal the locomotion mechanism of various creatures as science.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

2020 ◽  
Vol 30 (05) ◽  
pp. 2050072 ◽  
Author(s):  
Yingjuan Yang ◽  
Guoyuan Qi ◽  
Jianbing Hu ◽  
Philippe Faradja
Keyword(s):  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The Self ◽  
Numerical Continuation ◽  
Chaotic Attractors ◽  
Phase Portrait Analysis ◽  
The Stability ◽  
Two Parameters ◽  
The Basin Of Attraction

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

scholarly journals Basins of Attraction for Various Steffensen-Type Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa ◽  
Stanford Shateyi
Keyword(s):  
Computer Algebra System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Numerical Tests ◽  
Derivative Free ◽  
Iteration Functions ◽  
The Stability ◽  
Programming Package ◽  
The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

Further Investigation of the Period-Three Route to Chaos in the Passive Compass-Gait Biped Model

2015 ◽  
pp. 279-300 ◽  
Author(s):  
Hassène Gritli ◽  
Nahla Khraief ◽  
Safya Belghith
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Limit Cycle ◽  
Basin Of Attraction ◽  
Biped Robot ◽  
Computation Method ◽  
Dynamic Walking ◽  
Route To Chaos ◽  
Inclined Surface ◽  
The Basin Of Attraction

This chapter presents further investigations into the period-three route to chaos exhibited in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. This discovered kind of route in the passive bipedal locomotion was found to coexist with the conventional period-one passive hybrid limit cycle. The further analysis on the period-three route chaos is realized by means of the Lyapunov exponents and the fractal Lyapunov dimension. Numerical computation method of these two tools is presented. The first return Poincaré map of the chaotic attractor and its basin of attraction are presented. Furthermore, the further study of the period-three passive gait is realized. The analysis of the period-three hybrid limit cycle is given. The balance between the potential energy and the kinetic energy of the biped robot is illustrated. In addition, the basin of attraction of the period-three passive gait is also presented.

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