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Sumona Mukhopadhyay (University of Calgary, Canada) and Henry Leung (University of Calgary, Canada)

Copyright: © 2015
|Pages: 12

DOI: 10.4018/978-1-4666-5888-2.ch651

Top## 2. Background

Robots are a complex dynamical system characterized through statistical measures and mathematical expressions describing the system dynamics. Cooperative robots accomplish the goal of workspace coverage faster than an atomic robot (Cao et al., 1997). Effective collision-free robot motion and coverage, planning (Shiller & Gwo, 1991; Klomkam & Sooraksa, 2004; Yang & Luo. 2004; Hazan, et. al, 2006; Willms & Yang, 2006; Fallahi & Leung, 2010; Volos & Kyprianidis, 2012), robustness of synchronous speeds against robot and communication failures (Zang et al., 2006) are important issues in robotics currently under research. In order to execute cooperative task assignment, control strategies must be formulated.

Complex System: An emerging discipline of science that involves the analysis of the interaction and behavior of sub parts of a system and its elementary constituents together with the collective system which govern in the emergence of patterns of behavior demonstrating non- linear dynamics. It aids in understanding how interactions of the components with its environment result in the rise of patterns by quantification of its variables and parameters. It encompasses disciplines of science and biological systems, engineering, and management. The emergent behavior demonstrated is neither completely ordered and predictable, nor completely random and unpredictable. These systems generate patterns and hierarchy of structures that emerge due to its attractor.

Shannon’s Entropy: Entropy implies the amount of “disorder” associated with a random variable. In information theory, this important metric gives the expected value of the information conveyed by a message. Shannon entropy measure calculates an estimate of the average minimum number of bits that is required to encode a string of symbols based on an alphabet size and the frequency of the occurrence of the symbols.

Kolmogorov-Sinai Entropy: The maximum capacity of information that can be generated by a dynamical system is governed by the Kolmogorov-Sinai (KS) complexity. A system with value 0 is fully deterministic while 1 represents randomness. Thus, it is a measure of the amount of order and randomness associated with a system. In general, a reduction of entropy would imply that the receiver is certain to receive the actual information.

Chaos Synchronization: It is a technique wherein chaotic systems are coupled to achieve identical dynamics asymptotically with time. Consider two coupled systems xn+1=f(xn) and yn+1=f(yn). They will be completely synchronized in the influence of a coupling parameter if the difference en= yn - xn asymptotically converges to zero, as n?8. The coupling may be unidirectional or bidirectional. Therefore, synchronization of two chaotic dynamical systems implies tracking the trajectories of the response system with that of its master.

Lyapunov Exponents: Gives the measure of the rate of divergence of trajectories of chaotic system. It tells us the sensitivity of the system to initial conditions. Chaotic systems with positive lyapunov exponents have bounded orbits. The number of exponents equals to the dimension of the system. When the largest lyapunov exponent becomes negative, then the system is synchronized.

Predator-Prey System: Is a vertical food-web ecological structure described by an ordinary differential equation whose variables represent vegetation grazed on by herbivores which in turn are fed on by predators. The parameters describe the respective growth or mortality rates of each trophic species under different conditions.

Symbolic Dynamics: Nonlinear dynamical systems are theoretically well described by a fundamental tool known as symbolic dynamics. It describes the time evolution of a chaotic dynamical system by representing its trajectory by sequences of symbols belonging to a finite alphabet set. The evolution of the trajectory with time is represented by the transition between symbols. Such continuous assemblies of symbols constitute a word which is a distinguishing feature of the system. The word describes the dynamics of the system. Thereby, a coarse grain partition of the phase space can be obtained. It helps to investigate chaotic behavior with finite precision and can be used combined with information theory.

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