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Published on January 4, 2008

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A Review of Modeling Methods for Swarm Robotic Systems:  A Review of Modeling Methods for Swarm Robotic Systems Kristina Lerman USC Information Sciences Institute Alcherio Martinoli Swarm-Intelligent Systems Group, EPFL Aram Galstyan USC Information Sciences Institute Chemical Reaction Rate Equations:  Chemical Reaction Rate Equations Robot Rate Equations:  Robot Rate Equations Robots are not molecules!:  Robots are not molecules! yes, but… Robot as a Stochastic Process:  Robot as a Stochastic Process Individual robot’s behavior subject to External forces may not be anticipated Noise fluctuations and random events in the environment Errors in sensors and actuators Other robots with complex trajectories Can’t predict which robots will interact Randomness programmed into controllers e.g., avoidance Individual robot’s actions are so unpredictable, they might as well be considered stochastic Controller as an FSA:  Controller as an FSA Robot is a stochastic Markov Process Reactive robots – act based on input from sensors Controller = Finite State Automaton (FSA) Robot controller for a simplified foraging scenario Box = robot’s state = action Arrows = transitions between states External stimuli Timer Stochastic Processes-based Modeling of Robot Swarms:  Stochastic Processes-based Modeling of Robot Swarms Definitions p(n,t) = probability robot is in state n at time t p(n,t) is also the fraction of robots in state n Markov property robot’s state at time t+Dt depends only on its state at time t Change in probability density The Rate Equation:  The Rate Equation Averaging both sides of Dp(n,t) over all robots gives the macroscopic Rate Equation Describes collective behavior with transition rates Rate Equation:  Rate Equation Describes dynamics of average quantities Compare with results averaged over many experiments No need to know exact probability distributions Or exact robot trajectories Used to study a variety of systems in natural sciences Usually phenomenological Can be written down simply by assessing what important characteristics of the problem are A “Recipe” for the Rate Equation:  A “Recipe” for the Rate Equation Initial conditions: Ns(t=0)=N, Nh(0)=0, Np(0)=0 A Word on Coarse-graining:  A Word on Coarse-graining search Detect object Avoid obstacle Coarse-graining reduces the complexity of the model Helps construct a minimal model that explains experiments Robot Swarm Applications :  Robot Swarm Applications Stick-Pulling Experiments in Robots:  Stick-Pulling Experiments in Robots Collaboration in a group of reactive robots Task completed only through collaboration Experiments with 2 – 6 Khepera robots Embodied simulations with up to 24 robots (Ijspeert et al. 2001) Flowchart of the Robot Controller:  Flowchart of the Robot Controller Experimental Results:  Experimental Results Key observations Different dynamics for different ratio of robots to sticks Optimal gripping time parameter Theoretical Results:  Theoretical Results Complete model+simulations Martinoli et al., 2004 Minimal 2-state model Lerman et al., 2001 Robot Foraging:  Robot Foraging Collect objects scattered in the arena and assemble them at a “home” location Single vs group of robots Benefits of a group robust to individual failure group can speed up collection Disadvantages of a group increased interference due to collision avoidance Goldberg & Matarić Foraging Efficiency vs Group Size:  Foraging Efficiency vs Group Size Comparison with embodied simulations Lerman & Galstyan, 2002 Collective Clustering:  Collective Clustering Aggregation 20 seeds scattered in a 80X80 cm working area (red zone) Goal: all the seed clustered in a single cluster, all the robots resting in the parking area (orange zone) Mean Cluster Size and Active Workers:  Mean Cluster Size and Active Workers Martinoli et al., 1999; Agassounon et al. 2004 Comparison with embodied simulations Conclusions:  Conclusions Macroscopic models of collective behavior based on theory of stochastic processes No need to know exact trajectories Allow quantitative analysis of collective behavior Results of mathematical models can be used in the design cycle to optimize robot controllers Caveats and simplifying assumptions to keep models tractable Models describe average swarm behavior Robots actions independent of one another Spatial uniformity Robot inhomogeneity not yet considered Future Directions:  Future Directions More realistic models Effects of noise Inhomogeneous robot characteristics Other systems Adaptation and learning Extended theory of stochastic processes to memory-based adaptation where robots change their behavior in response to series of observations of the other robots Other types of learning Reinforcement learning Pheromone-based stigmergy

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