Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems

Hsu, Charles Ching–Hsiang; Tongue, Benson H.

doi:10.1115/1.3125869

Cited by 197 publications

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“…A periodic cycle of n cells forms a periodic group, with periodicity n (or shortly an n-P group). Each cell within the PG is a periodic cell with period n, or shortly n-P cell [1]. (For example, the sink cell is a 1-P cell…”

Section: Definitions and Abbreviationsmentioning

confidence: 99%

“…Hsu [1], in order to make the quick and thorough global analysis of nonlinear systems possible. CM methods discretize a region of the state space, thus creating the so called cell state space.…”

mentioning

confidence: 99%

“…The basic idea of the SCM method is that each cell has a single image, which is usually determined using the Centre Point Method [1], namely, a single trajectory from the centre of the cell domain is examined. In other words, all states within a cell are mapped to a single cell.…”

mentioning

confidence: 99%

“…Upon the joining procedure, new state space objects residing on the boundary of the two cell state spaces are also discovered. This paper introduces this extension, particularly for the Simple Cell Mapping method, because it is the simplest adequate method to discover all objects in the state space [1]. The method of joining separate SCM solutions to a cluster of SCM solutions is referred to as Clustered SCM method.…”

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confidence: 99%

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Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Gyebrószki

Csernák²

2017

Communications in Nonlinear Science and Numerical Simulation

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When a dynamical system has a complex structure of fixed points, periodic cycles or even chaotic attractors, Cell Mapping methods are excellent tools to discover and thoroughly analyse all features in the state space. These methods discretize a region of the state space into cells and examine the dynamics in the cell state space. By determining one or more image cells for each cell, the global behaviour within the region can be quickly determined. In the simplest case -Simple Cell Mapping (SCM) method -only one image corresponds to a cell and usually a rectangular grid of cells is used. In typical applications the grid of cells is refined at specific locations. This paper, however, introduces a different approach, which is useful to expand the analysed state space region to include all features which properly characterize the global dynamics of the system. Instead of refining the initial cell state space, we start with a small initial state space region, analyse other interesting regions of the state space and incorporate them into a cluster of cell mapping solutions. By this approach, trajectories escaping the original state space region can be followed automatically and additional objects in the state space can be discovered. To illustrate the benefits of the method, we present the exploration of the phase-space of the micro-chaos map -a simple model of digitally controlled systems.Keywords: discretized state space, adaptive global analysis, algorithm, micro-chaos Highlights:• Extension of the Simple Cell Mapping method by adaptively expanding cell state space.• New state space regions can be attached to the cluster of SCM solutions.• Clustering makes parallel computation trivial.

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Section: Definitions and Abbreviationsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Gyebrószki

Csernák²

2017

Communications in Nonlinear Science and Numerical Simulation

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“…Cell-to-cell mapping is applied as below [13]. The region of feasible initial conditions is subdivided into a large number(N) of small cells.…”

Section: Cell-to-cell Mappingmentioning

confidence: 99%

A Study on Stability of Limit Cycle Walking Model with Feet: Parameter Study

Jeon

Park

2013

International Journal of Advanced Robotic Systems

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In this paper, two kinds of feet, namely, curved and flat feet, are added to limit cycle walking model to investigate its stability properties. Although both models are already proposed and are investigated, most previous works are focused on efficiency and gait behaviors. Only the stability properties of the simplest walking model conceived Garcia et al. are well defined. Therefore, there is still a need for a precise research on the effect of feet, especially in the view of local stability, bifurcation route to chaos, global stability, falling boundary and energy balance line. Therefore, this article revisits the stability analysis of limit cycle walking model with various foot shape. To analyze the effects of feet, we re-derive the equation of motion of modified models by adding one more parameter of foot shape than the simplest walking model. Also, the falling boundary and energy balance line of modified models are derived to get proper initial conditions for stable walking and to explain global stability. Simulation results show us that the curved feet can enlarge both stable walking range and area of basin of attraction while the case of flat feet depends on foot shape parameter.

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Gait Generation and Stabilization for Nearly Passive Dynamic Walking Using Auto‐distributed Impulses

Moon

Stipanović

Spong

2015

Asian Journal of Control

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We propose a state feedback control design via linearization for flexible walking on flat ground. First, we generate nearly passive limit cycles, being stable or not, using impulsive toe‐off actuations. The term ‘nearly passive’ means that the dynamics is completely passive almost everywhere except at the toe‐off moment. A feature of our gait generation method is that walking gaits are characterized only by amounts of supplied energy, and we observe that other variables, including input torques, are auto‐balanced via our method. After gait generation, we design a feedback controller considering robustness and input saturation. As a result, each limit cycle can be matched with its respective controller classified only by energy levels. We have verified that walking speeds monotonically increase by adding more energy, and the ankle joint plays a significant role in compass‐gait walking. Finally, instead of applying impulsive torques, we discuss a practical issue regarding realistic control inputs that ensure stable gait transitions as energy levels are elevated.

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Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems

Cited by 197 publications

References 0 publications

Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

A Study on Stability of Limit Cycle Walking Model with Feet: Parameter Study

Gait Generation and Stabilization for Nearly Passive Dynamic Walking Using Auto‐distributed Impulses

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