Limits...
Multiscale modelling and analysis of collective decision making in swarm robotics.

Vigelius M, Meyer B, Pascoe G - PLoS ONE (2014)

Bottom Line: Our approach encompasses robot swarm experiments, microscopic and probabilistic macroscopic-discrete simulations as well as an analytic mathematical model.Following up on previous work, we identify the symmetry parameter, a measure of the progress of the swarm towards a decision, as a fundamental integrated swarm property and formulate its time evolution as a continuous-time Markov process.Contrary to previous work, which justified this approach only empirically and a posteriori, we justify it from first principles and derive hard limits on the parameter regime in which it is applicable.

View Article: PubMed Central - PubMed

Affiliation: FIT Centre for Research in Intelligent Systems, Monash University, Melbourne, Australia.

ABSTRACT
We present a unified approach to describing certain types of collective decision making in swarm robotics that bridges from a microscopic individual-based description to aggregate properties. Our approach encompasses robot swarm experiments, microscopic and probabilistic macroscopic-discrete simulations as well as an analytic mathematical model. Following up on previous work, we identify the symmetry parameter, a measure of the progress of the swarm towards a decision, as a fundamental integrated swarm property and formulate its time evolution as a continuous-time Markov process. Contrary to previous work, which justified this approach only empirically and a posteriori, we justify it from first principles and derive hard limits on the parameter regime in which it is applicable.

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Jump probabilities for the toy model.Jump probabilites  (red curves and markers) and  (blue curves and markers) for the toy model [Eqs. (21)–(28)] obtained from the analytic approximation [Eq. (44), curves] and a macroscopic-discrete simulation (circles). For comparison, we include simulation results of the full macroscopic-discrete model (Sec. 2, "plus'' markers).
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pone-0111542-g012: Jump probabilities for the toy model.Jump probabilites (red curves and markers) and (blue curves and markers) for the toy model [Eqs. (21)–(28)] obtained from the analytic approximation [Eq. (44), curves] and a macroscopic-discrete simulation (circles). For comparison, we include simulation results of the full macroscopic-discrete model (Sec. 2, "plus'' markers).

Mentions: In Fig. 12, we compare the approximate solution (44) of the Master equation (curve) with a macroscopic-discrete simulation of our toy model (circles) and a macroscopic-discrete simulation of the full model (“plus'' markers). We first observe that the toy model systematically overestimates the tendency to agree on an emerging decision, i.e. are higher for the toy model. This is easy to explain, as the toy model only allows for two states, the ground state and the excited state, whereas the full model has a decision depth of . This means that, in the toy model, a robot can be more easily convinced to change its decision. In a similar manner, the approximate solution (curve in the Figure) tends to promote consensus more than the actual macroscopic-discrete system (plus markers). The reason for this is that the time scale for the fast variables to equilibrate, which is given by the eigenvalues of the linear system (Sec. 3) to . For the macroscopic-discrete toy system (circles), is larger than the typical time scale for a change in the symmetry parameter. The latter can be obtained from through [20]. Hence, the fast variables cannot generally attain equilibrium before the slow variable changes and the fast variable elimination procedure is not fully accurate. We would expect that an analytic solution that allows more than two internal states would cause a more distinct separation of time scales and hence allow better equilibration of the fast states. We speculate that the more detailed model would result in a more accurate approximation of the full Master equation of the toy model and the full macroscopic-discrete simulation. Ultimately, only solving the analytic model for a higher decision depth will provide a definite confirmation.


Multiscale modelling and analysis of collective decision making in swarm robotics.

Vigelius M, Meyer B, Pascoe G - PLoS ONE (2014)

Jump probabilities for the toy model.Jump probabilites  (red curves and markers) and  (blue curves and markers) for the toy model [Eqs. (21)–(28)] obtained from the analytic approximation [Eq. (44), curves] and a macroscopic-discrete simulation (circles). For comparison, we include simulation results of the full macroscopic-discrete model (Sec. 2, "plus'' markers).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4219724&req=5

pone-0111542-g012: Jump probabilities for the toy model.Jump probabilites (red curves and markers) and (blue curves and markers) for the toy model [Eqs. (21)–(28)] obtained from the analytic approximation [Eq. (44), curves] and a macroscopic-discrete simulation (circles). For comparison, we include simulation results of the full macroscopic-discrete model (Sec. 2, "plus'' markers).
Mentions: In Fig. 12, we compare the approximate solution (44) of the Master equation (curve) with a macroscopic-discrete simulation of our toy model (circles) and a macroscopic-discrete simulation of the full model (“plus'' markers). We first observe that the toy model systematically overestimates the tendency to agree on an emerging decision, i.e. are higher for the toy model. This is easy to explain, as the toy model only allows for two states, the ground state and the excited state, whereas the full model has a decision depth of . This means that, in the toy model, a robot can be more easily convinced to change its decision. In a similar manner, the approximate solution (curve in the Figure) tends to promote consensus more than the actual macroscopic-discrete system (plus markers). The reason for this is that the time scale for the fast variables to equilibrate, which is given by the eigenvalues of the linear system (Sec. 3) to . For the macroscopic-discrete toy system (circles), is larger than the typical time scale for a change in the symmetry parameter. The latter can be obtained from through [20]. Hence, the fast variables cannot generally attain equilibrium before the slow variable changes and the fast variable elimination procedure is not fully accurate. We would expect that an analytic solution that allows more than two internal states would cause a more distinct separation of time scales and hence allow better equilibration of the fast states. We speculate that the more detailed model would result in a more accurate approximation of the full Master equation of the toy model and the full macroscopic-discrete simulation. Ultimately, only solving the analytic model for a higher decision depth will provide a definite confirmation.

Bottom Line: Our approach encompasses robot swarm experiments, microscopic and probabilistic macroscopic-discrete simulations as well as an analytic mathematical model.Following up on previous work, we identify the symmetry parameter, a measure of the progress of the swarm towards a decision, as a fundamental integrated swarm property and formulate its time evolution as a continuous-time Markov process.Contrary to previous work, which justified this approach only empirically and a posteriori, we justify it from first principles and derive hard limits on the parameter regime in which it is applicable.

View Article: PubMed Central - PubMed

Affiliation: FIT Centre for Research in Intelligent Systems, Monash University, Melbourne, Australia.

ABSTRACT
We present a unified approach to describing certain types of collective decision making in swarm robotics that bridges from a microscopic individual-based description to aggregate properties. Our approach encompasses robot swarm experiments, microscopic and probabilistic macroscopic-discrete simulations as well as an analytic mathematical model. Following up on previous work, we identify the symmetry parameter, a measure of the progress of the swarm towards a decision, as a fundamental integrated swarm property and formulate its time evolution as a continuous-time Markov process. Contrary to previous work, which justified this approach only empirically and a posteriori, we justify it from first principles and derive hard limits on the parameter regime in which it is applicable.

Show MeSH