-
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
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Bacterial chemotaxis
Immune cells
Swarm behaviour
Nonlocal movement
Human movement
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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.)
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 4 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Absent/sparse attractant ⇒ change in τ distribution:
Figure 1: Movement ofDictyostelium cells. From L. Li etal., PLoS
one (2008).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 4 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Bio-inspired strategy
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 4 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Bio-inspired strategy
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:
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 4 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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.
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 5 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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
).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 6 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Movement of individual robots
We assume that each individual moves in Rn according to
thefollowing rules:
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.
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 6 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Movement of individual robots
We assume that each individual moves in Rn according to
thefollowing rules:
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 .
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 6 /
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 7 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 8 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 9 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 9 /
19
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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:
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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.
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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.
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Kinetic equation for the 2-particle equation
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
Kinetic equation for the 2-particle equation
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) .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
Conclusions
General strategy
I Integrate out the microscopic quantity τ1,2.
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 = 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).
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 = 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.
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
Gissell Estrada-Rodríguez Bio-inspired strategies 27/11/2018 16
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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) .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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 .
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Introduction
Bio-inspiredstrategy
Anomalousdiffusion
Swarmroboticsystems
Results I
Result II
Individualrobotssimulations’comparison
MacroscopicPDEdescription
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).
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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.
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IntroductionBio-inspired strategyAnomalous diffusionSwarm
robotic systemsResults IResult IIIndividual robots simulations'
comparisonMacroscopic PDE descriptionConclusionsAppendix
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