Modeling and simulating a mathematical tool for multi-robot pattern transformation

The work reported in this paper is motivated by the need to investigate general methods for pattern transformation. A formal definition for pattern transformation is provided and four special cases namely, elementary and geometric transformation based on repositioning all and some agents in the pattern are introduced. The need for a mathematical tool and simulations for visualizing the behavior of a transformation method is highlighted. A mathematical method based on the Moebius transformation is proposed. The transformation method involves discretization of events for planning paths of individual robots in a pattern. Simulations on a particle physics simulator are used to validate the feasibility of the proposed method.

Modeling and simulating a mathematical tool for multi-robot pattern transformation

The work reported in this paper is motivated by the need to investigate general methods for pattern transformation. A formal definition for pattern transformation is provided and four special cases namely, elementary and geometric transformation based on repositioning all and some agents in the pattern are introduced. The need for a mathematical tool and simulations for visualizing the behavior of a transformation method is highlighted. A mathematical method based on the Moebius transformation is proposed. The transformation method involves discretization of events for planning paths of individual robots in a pattern. Simulations on a particle physics simulator are used to validate the feasibility of the proposed method.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.