Fractional potential field in path planning for mobile robot obstacle ...

Fractional potential field in path planning for mobile robot obstacle
avoidance in the feedback control of a diffusion process
Zhongmin Wang
Center for Self-Organizing and Intelligent Systems
Dept. of Electrical and Computer Engineering
Utah State University
April 19, 2005
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References
• Jorge Cortes, Sonia Martinez, Timur Karatas and Francesco Bullo,
“Coverage Control for Mobile Sensing Networks”, IEEE Trans on Robotics
and Automation, Vol.20, NO.2, April 2004.
• Lili Ju, Qiang Du and Max Gunzburger, “Probabilistic methods for centroidal
Voronoi tessellations and their parallel implementtaions”,Journal of Parallel
Computing,Volume 28 , Issue 10, October 2002, Pages: 1477
• S. S GE and Y. J. CUI, “Dyanmic motiong planning for mobile robots using
potentail field method”, Autonomous Robots,Volume 13, 2002,
pages:207–222.
• A.Poty, P. Melchior and A. Oustaloup, “Dynamic path planning for mobile
robots using fractional potential filed”, in Proceedings of First International
Symposium on Control, Communications and Signal Processing, 2004,
Page(s):557 - 561.
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Coulombian Potential Field
The potential field method is widely used for autonomous mobile robot path planning
due to its elegant mathematical analysis and simplicity [3]. The coulombian
electric field E(r) generated by a punctual charge in the vacuum is given by:
E(r) =
q
4πε0r 2er.
where ε0 is vacuum permittivity, r is the distance to charge q, and er a radial unit
vector. A single integration provides the coulombian potential from the electric
field of a punctual charge:
V (r) = q
4πε0r .
We can use two integration and more to set up the potential filed.
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Fractional Potential Field
The fractional potential definition using the Weyl fractional integral is given below:
Vn(r) = W r ∞
q (θ − r)
E(r) =
4πε0Γ(n)
n−1
r2 dθ.
where Γ(n) is the Gamma function defined by:
Γ(n) =
1
0
r
[ln( 1
t )]n−1 dt.
The fractional potential is then normalized between 0 and 1. It has a maximum
effective distance rmax. To avoid the singularity at r = 0, a minimum distance
rmin should also be chosen.
∀n ∈ [0, 2[, ∀r ∈ [rmin, rmax],
Un(r) = Vn(r) − Vn(rmax)
Vn(rmin) − Vn(rmax) = rn−2 − rn−2 max
rn−2 min − rn−2 .
max
n is the risk coefficients.
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FRACTIONAL POTENTIAL FIELD IN PATH PLANNING FOR MOBILE
AVOIDANCE OF DYNAMIC OBS
Normalized fractional potential
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
n=2: segment
n=1: ponctual charge
0
0 5 10 15 20 25 30
r =1 r
min Distance
=30
max
1: Influence • n = 1 gives distance the Coulombian of the normalized potential fieldfractional generated by potential an isolated based charge. on We
n (1 •≤ n n= ≤2 gives 5 ); the coulombian Coulombian potential field created generated by by a uniformly punctualdistributed charge (n=1
mly distributed charge along charge a straight-line on a segment segment. (n=2)
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n=4
n=3

ALEXANDRE POTY, PIERRE MELCHIOR AND ALAIN OUSTALOUP
Figure 3: Fractional map 3D example - the height is the danger level
2.3 Fractional road
• The risk coefficients are fixed at 1.2 for the square, 1.5 for the chamfered
rectangle and 2.5 for the circle. rmax = 20 and rmin = 1.
The operator chooses a threshold of danger level beyond which it estimates that the risk
taken by the robot would be too important. An horizontal cross section of the fractional
card at the given threshold 4, provides some equipotential which separates zones where
the danger is too important, from zones of security to be used [27].
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Feedback control of a diffusion process - pollution neutralization
We consider the pollution neutralization problem. The scenario is described as
follows:
• A toxic diffusion source is releasing toxic material in 2D plane. The diffusion
process is modelled as a parabolic PDE system.
• Chemical concentration sensors are deployed to cover the polluted area and
collect data about the pollution.
• A few of mobile robots equipped with controllable dispensers of neutralizing
chemicals are sent to the polluted area with the mission to eliminate the
pollution by properly releasing the neutralizing chemicals.
Problem: How to choose the optimal positions for these robots and the trajectories
the robots will follow when the dynamic diffusion is evolving?
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Optimal Actuator location in feedback control of diffusion process
Let Ω be a convex polytope in R2 , including its interior. A concentration function
is a map ρ(x, y) : Ω → R+ that represents the pollutant concentration over Ω.
To simplify the presentation of our main idea, in our simulation experiment, we
assume ρ(x, y) is governed by the following PDE system:
2 ∂ρ ∂ ρ
= k
∂t ∂x2 + ∂2ρ ∂y2
+ fd(ρ, x, y, t), (1)
where k is a constant positive real system parameter; fd(ρ, x, y, t) represents the
source of the pollution. We assume that the diffusion process is evolving slowly.
Let P = (p1, · · · , pn) be the location of n actuators and let | · | denote the
Euclidean distance function. Every robot at pi will receive information of sensors
and release the neutralization chemical by some control law.
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The objectives are:
• Control the diffusion of the pollution to a confined area.
• Neutralize the pollution in a time optimal way while not making the area overdosed.
• Minimize the polluted area that is heavily affected.
n robots will partition Ω into a collection of n polytopes V = {V1, · · · , Vn},
pi ∈ Vi. It can be seen that to control the diffusion process and minimize the
heavily affected area, the robots should be close to those areas with high pollution
concentrations. To decide the positions of the robots, we consider the minimizing
of the following cost function.
K(P, V) =
n
i=1
Vi
ρ(q)|q − pi| 2 dq for q ∈ Ω. (2)
It is clear that to minimize K, the distance |q − pi| should be small when the
pollution concentration ρ(q) is big. A necessary condition for K to be minimized
is that {pi, Vi} k i=1 is a Centroidal Voronoi Tessellation of Ω
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Example:
z ∗ i =
Vi xρ(x)dx
We call the tessellation defined by (3) a Centroidal Voronoi Tessellation if and only if
Vi
ρ(x)dx for i = 1, · · · , k. (4)
zi = z ∗ i for i = 1, · · · , k.
So, the points zi that serves as the generators for the Voronoi regions Vi are themselves the mass centroids of those regions,
as shown in Fig. 1(b).
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
(a) The Voronoi regions constructed by 100 randomly selected
points in a square Ω = (−1, 1) 2 . The density function
is given by ρ(x, y) = e −8(x2 +y 2 ) . The dots are the Voronoi
generators and the circles are the mass centroids of the
corresponding Voronoi regions. The generators and the mass
centroids do not coincide.
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
(b) The centroidal Voronoi tessellations constructed by 100
points in the same Ω and same density function ρ as in
1(a). These points server both as the generators and the
mass centroids of the corresponding regions. More of them
aggregate to area where ρ(x, y) is bigger.
Fig. 1. The illustration of Voronoi diagram and Centroidal Voronoi Tessellations.
Centroidal Voronoi Tessellation has broad applications in many fields. It is the solution to optimal placement of resources,
but in general, CVT can only be approximately constructed. For algorithms to implement CVT, refer to [14]. In [12], an
example is given to show how CVT can be used to predict the cell divisions. It is shown that, after the cell division process,
the new cells’ shapes are very closely approximated10by- 15
Centroidal Voronoi Tessellations corresponding to the increased
number of generators. This is an example to show how CVT can be applied in a dynamically evolving environment.

Collision Avoidance
Now we consider the safety issue of the mobile robot. It is highly likely that the
robot will encounter moving obstacles that travel through the polluted area, for example,
the moving cargo vessels. In order to avoid collision between the robot and
the moving obstacle effectively, the relative position and velocity between robot
and obstacle should be considered. Here we introduce a potential field method
from reference 3 that is used for dynamic obstacle avoidance.
The repulsive potential generated by the obstacle is given by:
Urep(p, v) =
⎧
⎪⎨
⎪⎩
0 :
if ρs(p, p obs) − ρm ≥ ρ0 or vRO ≤ 0
η(Urep) :
if 0 < ρs(p, p obs) − ρm < ρm and vRO > 0
undefined :
if vRO > 0 and ρs(p, p obs) < ρm
The positive force between the robot and the obstacle is defined as the negative
gradient of the repulsive potential with respect to robot position and velocity.
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(3)

Let nRO⊥ be the unit vector perpendicular to nRO and n T RO⊥nRO⊥ = 1. The positive
force between the robot and the obstacle is defined as the negative gradient of
the repulsive potential with respect to the robot position and velocity:
⎧
where:
and
Frep(p, v) =
where vRO⊥ is given by
⎪⎨
⎪⎩
0 :
if ρs(p, p obs) − ρm ≥ ρ0 or vRO ≤ 0
Frep1 + Frep2 :
if 0 < ρs(p, p obs) − ρm < ρm and vRO > 0
undefined :
if vRO > 0 and ρs(p, p obs) < ρm
Frep1 = −ηU 2
rep(1 + vRO
)nRO.
amax
Frep2 = ηU 2 repvROvRO⊥
nRO⊥.
ρs(p, pobs)amax vRO⊥ = v(t) − vobs 2 −v 2 RO(t).
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(4)

The mobile robots are treated as virtual particles with unit mass and obey the
second-order dynamical equation:
where the control input Fi is given by
¨pi = Fi
Fi = fi − kv ˙pi + Frep
with fi the force input to control the motion of the robot by CVT. fi is given by a
proportional control law:
fi = −k(pi − ¯pi)
where ¯pi is the computed mass centroid of the current Voronoi cell.
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(5)

Simulation implementation
Diff-MAS2D is used as the simulation platform for our implementation. The
area concerned is given by Ω = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. The PDE
system with control input is modelled as
∂ρ(x, y, t)
∂t
= k( ∂2 ρ(x, y, t)
∂x 2
where k = 0.01 and the boundary condition is given by
+ ∂2 ρ(x, y, t)
∂y 2 ) + fc(x, y, t) + fd(x, y, t), (6)
∂u
∂n
= 0.
The stationary pollution source is modelled as a point disturbance fd to the the
PDE system with its position at (0.75, 0.35) and
fd(t) = 20e −t |(x=0.75,y=0.35).
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0.1. Question ?
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