macroscopic description of bio-inspired strategies for swarm …hg94/labtalk.pdf · 2019-02-20 ·...

Macroscopic description of bio-inspiredstrategies for swarm robotic systems

Gissell Estrada-Rodríguez 1

Together with H. Gimperlein 1, K. J. Painter 1, J. Štoček 1 andEdinburgh Centre for Robotics

1Maxwell Institute and Heriot-Watt University (Edinburgh)

Math Biology Group Meeting

Introduction

Bio-inspiredstrategy

Anomalousdiffusion

Swarmroboticsystems

Results I

Result II

Individualrobotssimulations’comparison

MacroscopicPDEdescription

Conclusions

Bacterial chemotaxis

Immune cells

Swarm behaviour

Nonlocal movement

Human movement

Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 2 / 19

Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Results of this talk

I We derived macroscopic fractional PDE descriptions fromthe movement strategies of the individual robots:

I Hyperbolic limit: Alignment dominates.I Parabolic limit: Competition between long trajectories

and alignment.

I Showed that the system allows efficient parameterstudies for search and targeting problems.

(Inspired by Lévy walks of E. coli and immune cells.)


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Bio-inspired strategy

Classical case of chemotaxis: the individual runs for sometime τ , it stops at (x, t) and chooses a new direction at random.τ follows a Poisson process. Patlak-Keller-Segel equations.

∂tu = ∇ · (C (u, ρ)∇u − χ(u, ρ)∇ρ),∂tρ = Dρ∆ρ+ f (u, ρ).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions


Classical case of chemotaxis: the individual runs for sometime τ , it stops at (x, t) and chooses a new direction at random.τ follows a Poisson process. Patlak-Keller-Segel equations.

∂tu = ∇ · (C (u, ρ)∇u − χ(u, ρ)∇ρ),∂tρ = Dρ∆ρ+ f (u, ρ).

Absent/sparse attractant ⇒ change in τ distribution:

Figure 1: Movement ofDictyostelium cells. From L. Li etal., PLoS one (2008).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions


In this talk: τ with long tail fractional diffusion orchemotactic equations that involve non-local, fractionaldifferential operators.

Evidence: (1) E. Korobkova et al., Nature (2004), (2) L. Li etal., PLoS one (2008), (3) T. Harris et al., Nature (2012).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions


In this talk: τ with long tail fractional diffusion orchemotactic equations that involve non-local, fractionaldifferential operators.

Evidence: (1) E. Korobkova et al., Nature (2004), (2) L. Li etal., PLoS one (2008), (3) T. Harris et al., Nature (2012).

Biology:

Robotic systems:


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Nonlocal diffusion = Lévy walk

Diffusion (Brownianmotion):

〈x2〉 ∝ t

Nonlocal diffusion (Lévymotion):

〈x2〉 ∝ t2/α, 1 ≤ α ≤ 2

Note: The systems we study don’t directly follow a Lévyprocess in space.


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Movement of individual robots

We assume that each individual moves in Rn according to thefollowing rules:

I Running probability: ψi (xi , τi ) =(

ς0ς0+τi

)α, α ∈ (1, 2) .

I Collision avoidance (elastic reflection): the new direction isθ′i = θi − 2(θi · ν)ν, with normal ν =

xi−xj|xi−xj | .

I The stopping frequency during a run phase is βi (xi , τi ).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions



I With probability ζ ∈ [0, 1] it chooses a new direction θ∗iaccording to the turn angle distribution

k(·, θi ; θ∗i ) = k̃(·, |θ∗i − θi |) where∫S k(·, θ; θ

∗)dθ = 1.


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions



I With probability (1− ζ) the robot aligns with the directionof the neighbors according to Φ(Λi · θi ) where

Λi (xi , t) =J (xi , t)|J (xi , t)|

, J (xi , t) = nonlocal flux .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Long time dynamics: long range movement +alignment

Theorem: As ε→ 0, the first two moments of the solution tothe kinetic equation describing the previous movement satisfythe following fractional diffusion equation1

∂tu +∇ · w = 0 , w − `G (u)

F (u)Λw = − 1

F (u)Cα∇α−1u ,

where Λw is the mean direction, ` is the strength of thealignment, F (u) is the collision term, G (u) depends on ζ andCα is the diffusion coefficient.

1Estrada-Gimperlein. “Interacting particles with Lévy strategies: Limitsof transport equations for swarming robotic systems”. In: Submitted,arXiv:1807.10124v3 (2018).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Short time dynamics: alignment dominates

Theorem: As ε→ 0, the solution p to the kinetic equationdescribing the previous movement admits an expansion

p(x, t, θ) = Φζ(θ)u(x, t) + ε(µ+1)(α−1)p1 +O(ε2(µ+1)(α−1))

with Φζ(θ) = (1− ζ)Φε(Λ · θ) + ζ. The functions u and Λsatisfy the following system of equations2

∂tu + zc0(1− ζ)∇ · (uΛ) = 0 ,u(C0∂tΛ + C1Λ · ∇Λ) + C2P⊥∇u = 0 .

Here P⊥ = 1− Λ⊗ Λ and z , C0, C1 and C2 depend on ζ andΦε(Λ · θ).

2Estrada-Gimperlein. “Interacting particles with Lévy strategies: Limitsof transport equations for swarming robotic systems”. In: Submitted,arXiv:1807.10124v3 (2018).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Macroscopic PDE description

The macroscopic model for the movement of interactingrobots is given, in the limit, by the following theorem.

Theorem:a As ε→ 0, the macroscopic density u(x, t) satisfythe following fractional diffusion equation:

∂tu = nc0∇ ·(

1F (u)

(Cα∇α−1u

))where F (u) comes from the collisions and Cα is the diffusionconstant that only depends on microscopic parameters.

aDragone-Duncan-Estrada-Gimperlein-Štoček et al. “Efficient quantitativeassessment of robot swarms: coverage and targeting Levy strategies”. In:Preprint (2019).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

PDE description vs. individual robots simulations

Macroscopic PDE system with Neumann b.c.,

∂tu −∇ · (Cα∇α−1u) = 0 in Ω× [0,T ).

Since Ω = [220cm× 180cm], % = 7.5cm and c = 3cm/s we getε = 0.005, cn = 3, γ = 1/2, c0 = 3 · 0.005γ .Initial condition: u0(x) = max

(0, 1.2 exp −4N|x |

2

0.075 − 0.2).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

PDE description vs. individual robots simulations

Macroscopic PDE system with Neumann b.c.,

∂tu −∇ · (Cα∇α−1u) = 0 in Ω× [0,T ).

Since Ω = [220cm× 180cm], % = 7.5cm and c = 3cm/s we getε = 0.005, cn = 3, γ = 1/2, c0 = 3 · 0.005γ .Initial condition: u0(x) = max

(0, 1.2 exp −4N|x |

2

0.075 − 0.2).

Robots simulations: run distance r is generated from a Lévyprocess. The new positions are computed, for θ = π ∗ rand, as

xnew − xcurrent = r cos(θ),ynew − ycurrent = r sin(θ) .

Initial robots’ positions:


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Area coverage

We investigate area cov-erage for different valuesof α and N.

Macroscopic:

1t

∫ t0

∫Ω

min(u,1|Ω|

)dxds.

Robotic simulations:

# cells visitedtotal # of cells

Similar work: by A. Bertozzi’sgroup (2018), arXiv:1806.02488and IEEE Transactions. 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

Covera

ge a

t T

= 1

200 s

Macroscopic simulation

Average of webots simulations


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Hitting times (from robots)

From the robots simulations3:

3Dragone-Duncan-Estrada-Gimperlein-Štoček et al. “Efficientquantitative assessment of robot swarms: coverage and targeting Levystrategies”. In: Preprint (2019).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Hitting times (from PDE)

We seek the first time at which the density of the solution inthe target position T , reaches a certain threshold δ, i.e., weseek t0 such that

δ =

∫T

∫Ωu(x− y, t0)u0(y)dydx.

The robots simulation are compared to:

I Analytic hitting time4: Considers Ω = Rn and the initialconditions are given by delta functions.

I Numerical hitting time: We consider the numericalsolution when Ω = �.

4Estrada-Gimperlein-Painter-Štoček. “Space-time fractional diffusion incell movement models with delay”. In: M3AS 29 (2019), pp. 65–88.


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Hitting times (from PDE)Define

t0 'δπ

2αCα vol(T )∑

i |x0 − xi |−α−2 .

Here x0 is the centre of the target T , and xi corresponds to theinitial positions of the robots.5

1 1.2 1.4 1.6 1.8300

400

500

600

700

800

900

1000

1100

1200

Avera

ge h

itting tim

e

Tile 1 centered at (-0.45,-0.55)

Tile 2 centered at (-0.55,0.55)

Macroscopic simulation for Tile 1

Analytic estimate

5Dragone-Duncan-Estrada-Gimperlein-Štoček et al. “Efficientquantitative assessment of robot swarms: coverage and targeting Levystrategies”. In: Preprint (2019).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Kinetic equation for the 2-particle equation

For the N-individual system the density σ = σ(xi , t, θi , τi )evolves according to a kinetic equation

∂tσ + cN∑i=1

(∂τi + θi · ∇xi )σ = −N∑i=1

βiσ .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions


For the 2-individual system the densityσ = σ(x1, x2, t, θ1, θ2, τ1, τ2) evolves according to a kineticequation

∂tσ + c2∑

i=1

(∂τi + θi · ∇xi )σ = −2∑

i=1

βiσ .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions


For the 2-individual system the densityσ = σ(x1, x2, t, θ1, θ2, τ1, τ2) evolves according to a kineticequation

∂tσ + c2∑

i=1

(∂τi + θi · ∇xi )σ = −2∑

i=1

βiσ .

σ̃τ1∣∣τ1=0

=

∫SQ(θ1, θ

∗1)

∫ t0β1σ̃τ1(x1, x2, t, θ

∗1, θ2, τ1)dτ1dθ

∗1 ,

where

Q(θ1, θ∗1) = ζk(x1, t, θ

∗1; θ1) + (1− ζ)Φ(Λ1 · θ1) .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy

I Integrate out the microscopic quantity τ1,2.


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy

I Integrate out the microscopic quantity τ1,2.I We now aim to derive an effective transport equation for

the one-particle density function

p(x1, t, θ1) =1|S |

∫ t0

∫ t0

∫Ω2

∫Sσdθ2dx2dτ1dτ2 .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy


the one-particle density function p = p(x1, t, θ1).I The density p satisfies the following kinetic equation

∂tp + cθ1 · ∇x1p = (1− ζ)Φ(Λ1 · θ1)(B(x1, t) ∗ u(x1, t)

)(t)︸︷︷︸

Alignment

−(1− ζT1)∫ t

0B(x1, t − s)p(x1 − cθ1(t − s), s, θ1)ds︸︷︷︸

Long-range movement

+ Collision term,

where u(x1, t) =∫S p(x1, t, θ1)dθ1, the operator B depends on

the power-law running probability ψ1(x1, τ1).


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy


the one-particle density function p = p(x1, t, θ1).I The density p satisfies the following kinetic equation

∂tp + cθ1 · ∇x1p = (1− ζ)Φ(Λ1 · θ1)(B(x1, t) ∗ u(x1, t)

)(t)

− (1− ζT1)∫ t

0B(x1, t − s)p(x1 − cθ1(t − s), s, θ1)ds

+ Collision term.

I Parabolic scaling (x, t, τ, c) 7→ (xns/ε, tn/ε, τn/εµ, c0/εγ)for µ, γ > 0. The diameter of each particle is small, % = εξ,while the number of particles N is large so that(N − 1)% = εξ−ϑ, with ξ − ϑ < 0.


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy

I Conservation equation

ε∂tu(x1, t) + εnc0∇ · w(x1, t) = 0 ,

where w(x1, t) =∫S θ1p(x1, t, θ1)dθ1 .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy


ε∂tu(x1, t) + εnc0∇ · w(x1, t) = 0 ,


I Use a quasi-static approximation for B̂ = L[B],

B̂ε(x1, ελ+ ε1−γc0θ1 · ∇) ' B̂ε(x1, ε1−γc0θ1 · ∇) .

I Molecular chaos assumption˜̃σ(x1, x1 ± εξν, t, θ1, θ2) = p(x1, t, θ1)p(x1, t, θ2) +O(εξ) .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

General strategy


ε∂tu(x1, t) + εnc0∇ · w(x1, t) = 0 ,


I Use a quasi-static approximation for B̂ = L[B],

B̂ε(x1, ελ+ ε1−γc0θ1 · ∇) ' B̂ε(x1, ε1−γc0θ1 · ∇) .

I Molecular chaos assumption˜̃σ(x1, x1 ± εξν, t, θ1, θ2) = p(x1, t, θ1)p(x1, t, θ2) +O(εξ) .

∂tu +∇ · w = 0 , w − `G (u)

F (u)Λw = − 1

F (u)Cα∇α−1u .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Short-time behaviour

I Hyperbolic scaling: xn = εx/s, tn = εt, τn = τεµ .

I The space and time variables are on the same scale

ε(∂tp + c0θ · ∇p) = (1− ζ)Φε(Λ · θ)∫ t

0Bε(x, t − s)u(x, s)ds

− (1− ζT )∫ t

0Bε(x, t − s)p(x− cθ(t − s), s, θ)ds .

So in this case no-quasistatic approximation.

I Obtain a generalized Chapman-Enskog expansion for p.

I Conservation equation: ∂tu + zc0(1− ζ)∇ · (uΛ) = 0 .


Introduction


Anomalousdiffusion

Swarmroboticsystems

Results I

Result II



Conclusions

Related work

I Networks. Superdiffusion in complex domains.

Estrada-Estrada-Gimperlein, arXiv:1812.11615, (2018).I Chemotaxis for E. coli and space-time fractional diffusion

models for immune cells:Estrada-Gimperlein-Painter, SIAP, (2018) andEstrada-Gimperlein-Painter-Stocek, M3AS, (2019).


References

Thanks! Any question?

www.macs.hw.ac.uk/∼ge5

Dragone-Duncan-Estrada-Gimperlein-Štoček et al.“Efficient quantitative assessment of robot swarms:coverage and targeting Levy strategies”. In: Preprint(2019).

Estrada-Gimperlein. “Interacting particles with Lévystrategies: Limits of transport equations for swarmingrobotic systems”. In: Submitted, arXiv:1807.10124v3(2018).

Estrada-Gimperlein-Painter-Štoček. “Space-time fractionaldiffusion in cell movement models with delay”. In: M3AS29 (2019), pp. 65–88.


IntroductionBio-inspired strategyAnomalous diffusionSwarm robotic systemsResults IResult IIIndividual robots simulations' comparisonMacroscopic PDE descriptionConclusionsAppendix

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macroscopic description of bio-inspired strategies for swarm …hg94/labtalk.pdf · 2019-02-20 ·...

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