Duarte Dias Fabio Stradelli

Zeynab Talebpour

Distributed Intelligent Systems

Multi-level modeling of a distributed robotic system

Outline   Motivation

  System modeling

  Experiments and results

  Conclusions

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Why Modeling

 Simplified System Descriptions

  Faster and more straightforward analyses

 Additional tools for design and optimization

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Why Modeling

Cockroaches motion pattern

Nshelter1(t)

Nshelter2(t)

shelter 1

shelter 2 Average population numbers in shelters

16 cockroaches with no robots

12 cockroaches + 4 robots

real model

Same shelter type

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Halloy et. al. (2007) LEURRE project

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darker and ligher shelters

16 cockroaches with no robots

12 cockroaches + 4 robots

Why Modeling

Cockroaches motion pattern

Nlighter(t)

Ndarker(t)

lighter shelter

darker shelter Average population numbers in shelters Halloy et. al. (2007)

LEURRE project

model real

Multi Level Modeling Sub-microscopic model

Detailed representation of the physical environment and the robots ‒  Include inter-robot interactions

Microscopic model Multi-agent model Relevant robots features only

Macroscopic model Average representation of the whole swarm

Mathematical model (ODE)

Definition of model structure

Faithfulness

Tractability

Abstraction

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Sub-microscopic model: α-Algorithm •  Task: Each robot must maintain a minimum of α connections

•  Colision avoidance: ‒  Detected using threshold on

proximity sensor intensity ‒  uses braintenberg controller ‒  Lasts a period of Ta steps

•  Connectivity check Robot counts neighbors (r < Rw) every Tc steps (di): ‒  di ≥ α : ‒  di < α : (U-turn) ‒  If di doesn’t decrease untill next period resume forward motion with a random turn ‒  Otherwise repeat U-turn maneuver

Winfield et. al. (2008)

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Microscopic model: PFSM

  Pl,i : probability to get from Fi to a state with less neighbours   Pg,i : probability to get from Fi to a state with more neighbours   Pla,i: probability to get from Ci to a state with less neighbours   Pr,i : probability to recover from Ci to Fi+1   Pf,i : probability to get from Ci to forward Fi  Invalid transitions

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Microscopic model: PFSM

  Forward and coherence states are independent of avoidance   Build a sub-PFSM to model avoidance for both   Pl,i : probability to enter avoidance from state Fi

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Probabilities Estimation

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Macroscopic Model   Target: describe how many robots are in a given state at time step k   Example of DE (Forward)

•  Example of DE (Avoidance substate)

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Steady state and validation

  Evaluation time: 15mins   Plot the submicroscopic

steady state against the model steady state

  Submicroscopic steady state computed using an average window

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Sample plot for validation

Experiment Setup

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  40 robots   Unbounded environemnt   Parameters adjusted to the e-puck dimensions:

 Communiction range Rw = 0.5m   Speed V = 6 cm /s  Tc = 5 and Ta =2 (small):

  Less invalid states   Not to go outside the communication range

  α parameter: 5, 10, 15  Algorithm is not as effective for the values close to the bounds

TC = 5 TA = 2

RW = 0.5

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Findings   Good prediction of the system   Lost robots

 Depends on parameters:   Example: Large Tc and small Rw lost connections between

connectivity updates

 Tc and Rw must be correctly chosen

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Conclusion

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  Model closely follows submicroscopic model   A correct choice of the parameters is needed for more

effective model   Simplifying assumptions lead to some innacuracies

Microscopic Model

 Define states of the problem  Fi: forward with i neighbours, i=0,…,Nrobots-1  Ci: coherence with j neighbours, j=0,…,alpha  FiA , CiA: forward/coherence with avoidance

 Build a Probabilistic Finite State Machine  Estimate transition probabilities from the

submicroscopic model

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Probability computation    

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Multi Level Modeling   Sub-microscopic model – faithful representation of the physical

environment and the robots (detailed description)

  Microscopic model – relevant robot features captured through a multi-agent model (here we define the structure and the probabilities)

  Macroscopic model – average representation of the whole swarm, mathematical model (ODE)

  To represent explicitly different design choices, trade off computational speed and faithfulness to reality, bridge mathematically tractable models and reality in na incremental way

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Multi Level Modeling

  Trade off computational speed and faithfulness to reality

  bridge mathematically tractable models and reality in an incremental way

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Results

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 Evaluate the steady state  Run the model for a long time

 Compare with simulation to validate the model   Investigate the effect of different parameters

Probabilities Estimation

  Average transitions over time for each robot: pi   Sum over time the number of corresponding transitions

between states for each robot: Sumi  Divide by total number of time steps robot i was in the required

starting state (Ti ): pi = Sumi / Ti

  Final probability : p  Average pi’s   Sum pi over robots: Sum  Divide by Nrobots: p = Sum / Nrobots

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Macroscopic Model

  Target: describe how many robots are in a given state at time step k

  Build a system of DE’s (Markov chain)   Exploit conservation laws  Use probabilities for transitions

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Probabilities Estimation

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darker and ligher shelters

16 cockroaches with no robots

12 cockroaches + 4 robots

Why Modeling

Cockroaches motion pattern

Nlighter(t)

Ndarker(t)

lighter shelter

darker shelter Average population numbers in shelters Halloy et. al. (2007)

LEURRE project

model real