Modeling and Simulation of a Swarm of Robots for Box-pushing Task
Yangmin Li and Xin Chen
Department of Electromechanical Engineering, Faculty of Science and Technology
University of Macau, Av.Padre Tomás Pereira S.J., Taipa, Macau SAR, P.R.China,
Email: {ymli@umac.mo | ya27407@umac.mo}
Abstract
This paper investigates the application of swarm
intelligence principles for box-pushing task, and proposes
the mathematical model of the system. The paper analyses
the system in two scopes: microscope and macroscope.
Firstly, the structure of individual robots is described
briefly. Secondly, in macroscope, based on the dynamic
process theory, the paper proposes the dynamic equations
of the system. The solution of the equations reveals the
mechanism of cooperation among robots. And based on
the solution and simulations, the paper discusses how to
realize obstacle avoidance during the process of
box-pushing.
1. Introduction
There has been increasing research interest in swarm
behavior. Two researching fields are regarded as the
sources of swarm behavior: social insects in mathematical
biology, and groups of interacting autonomous robots in
engineering. Different from traditional multi-agent
paradigm, which is based on deliberative agents and
central control, swarm paradigm need no central
controller to direct the behavior of the system. In other
words, swarm systems are self-organizing
Since it is social insects that inspire the research on
swarm behavior, “understanding the nature of
coordination in groups of simple agents is a first step
toward implementing useful multi-robot system.”[1]
Many researchers concentrate upon exploring the
underlying mechanism of social insects [2]. Based on
researching of biology systems, the community introduces
swarm behavior into engineering, and has achieved many
useful conclusions. In general, it’s believed that swarm
systems are more flexible, more robust, and, more
economical than traditional multi-robot systems. These
advantages result from characteristics of swam systems:
1) Decentralization; 2) Homogeneity; 3) self-organization;
4) simplicity.
Now swarm intelligence has become an alternative
approach to classical artificial intelligence. The paradigm
of swarm systems is complete distributed control. There is
no central controller directing the behaviors of swarm
systems. So, the group behavior ‘emerges’ from
individual behaviors of agents, or, robots in Robotics. It’s
called as ‘self-organization’. Normally, agents in swarm
systems need no explicit communication. Instead of it,
stigmergy communication is an alternative way. That
means all agents sense each other through the
environment.
Box-pushing has been an important benchmark for
testing swarm architectures in swarm community.
describes An ant-like transport system was proposed [3],
in which robots cooperate with each other with local
information. A transportation model was proposed on the
basis of social insects, which can handle the uncertainty in
transportation [4]. We think that the essential question of
swarm design is to find out a mechanism that relates
individual characteristics to the collective behavior of the
entire system. So, it’s required to analyze swarm systems
by mathematic model. It has been a challenge in swarm
community. Microscopic and macroscopic methodologies
were presented in [5], which based on Markov models for
predicting the dynamics of swarm system. The paper also
presented a physical system and simulations to illustrate
their viewpoints. In addition, A mathematical model was
constructed for asynchronous swarm with a fixed
communication topology [6]. A new methodology was
introduced to model coalition formation of electronic
markets [7].
In this paper, the goal is to analyze Markov properties
of the system and figure out the mathematic model of the
box-pushing task by swarm paradigm, and discuss the
superiority of swarm intelligence.
2. Behavior-based Robots
Fig. 1 shows the simulation interface of the
box-pushing. The center disk represents the box. And the
small disks represent the robots. It’s assumed that few
robots can not move the box. So, the aim of the task is to
push the box to the goal area marked as a sphere, shown
on the right bottom corner.
Figure 2.
Figure 1. Simulation interface.
The swarm is composed of a group of homogeneous
robots. All robots are controlled by behavior-based
control method [8]. The behavior layers include:
1) Wandering behavior: It is the lowest layer that
makes robot wander randomly in the arena. In every
certain period, it will send random speed command to
two motors.
2) Chasing behavior: When robot wanders around the
arena, the chasing behavior perceives the arena by
using three photoelectric sensors. The one is for
perceiving the light that denotes the goal area, and the
other two are for perceiving the box. If the robot is
behind the box and can push the box to the goal area,
the behavior will suppress the output of the
wandering behavior, and makes robot move to the
box directly.
3) Pushing behavior: The behavior always checks the
outputs of two bump switchs installed in the front of
the robot. If the robot touches something and judges
the object is the box through optical sensors, the
robot will adjust its pose and push the box.
4) Stagnancy behavior: The behavior check robot’s
movement. If robot is stagnant, the behavior will
count the stagnancy time. Once the stagnant duration
is greater than a certain threshold, the behavior will
compel the robot to obey wandering behavior for a
period. It should be emphasized that pushing is also
regarded as a kind of stagnancy. That means even a
robot is pushing the box, it will quit the pushing
action after certain time and wander randomly for
another period, this important for the system.
In the control architecture, the higher layer behaviors
can suppress the output the lower layer behaviors. Then
the whole control architecture is shown in Fig.2.
The behavior-base architecture
Obviously, if there are not enough robots pushing the
box together, the box could not be moved. So, robots need
to cooperate with each other. But from the description of
four-layer microscopic construction of the system, there is
no direct behavior for cooperation. Even there is no
communication behavior to coordinate their actions. In
the following analysis of macroscopic model of system,
we will explain that the model of cooperation is the
property of system’s structure, and need no explicit
cooperation indention at all.
3. Macroscopic model of the system
Normally, the swarm behaviors can be regarded as a
kind of dynamic process. So, a mathematical model is an
idealized representation of a process.
A mathematical model can describe a swarm system at
two levels. One is microscopic model, which describes
the agent’s interaction with other agents and the
environment. The description of individual architecture
refers to describe the microscopic model. The other is
macroscopic model, which describes the collective group
behavior of a multi-agent system.
The swarm robotic system is composed of a group of
homogeneous robots. Individual’s behavior is determined
by itself. And among the robots, there is no explicit
communication. Now, we explain how coordination
behavior ‘emerges’ from interaction among robots.
To simplify the analysis, we assume that obstacle
avoidance can be ignored while wandering. And pushing
behavior can count the pushing duration. Once the count
exceeds a threshold, robots quit pushing, and wander
randomly.Then the behavior of stagnancy can be canceled.
The behaviors can be reduced to three ones: Wandering,
Pushing, and Chasing.
Firstly, we define the states of the system S i as:
S1 : The average number of robots in state of
Pushing;
S 2 : The average number of robots in state of
Chasing;
S 3 : The average number of robots in state of
Wandering.
φ is defined as the total number of robots, then:
φ=
∑S .
3
(1)
i
i =1
S=
∑ S qˆ
3
i =1
i
i
,
(2)
where vector q̂i is a unit vector to represent each
state.
Figure 3.
State diagram.
Fig. 3 shows the system’s states transition. In fact, it’s
also individual robot’s state transition. Briefly, the
transition can be described as: when a wandering robot
finds the box, it changes from wandering state into
chasing state to move to the box. And once it touches the
box, it adjusts its pose to change into pushing state. But
instead of pushing the box until reaching the goal area, it
just pushes the box for a certain of period. And then it will
give up pushing and wander again. In the latter section,
we will explain why we adopt such pushing duration.
Because of such individual state transition, the number of
robots in each state also increases or decreases 1.
From the description, it can be concluded that the
system obeys Markov property. More accurately, the
mathematic model analyzed in the paper is a
states-discrete, parameters-continuous Markov chains.
That means the future motion of the system is determined
by the present states, not by the past. And the dynamics of
system is time continuous.
If the probability of state S
is defined as P (S , t ) , then,
P (S , t + ∆t ) =
∑ P(S , t + ∆t S ′, t)P(S ′, t) .
(3)
S′
For the continuous system, the transition rate is defined
as the limit of transition probability:
W (S S ′; t ) = lim
P( S , t + ∆t S ′, t )
∆t → 0
∆t
.
(4)
So, the normal dynamic equations of the system is [9]:
∂S i
=
∂t
So, the configuration of system can be expressed as:
∑w
j
ji
(S )S j + S i
∑ w (S ) .
ij
(5)
j
To get the dynamic equation of the system, the
following assumptions are proposed:
1) Once a robot perceived the box, it can adjust its
direction to the box instantly. That means, the time for
pose adjustment can be ignored. And robot moves to the
box at a constant speed of v .
2) Comparing with the size of box, the size of robots can
be ignored. That means there’s no density limit. And the
quantity of robots is large enough. Then, we need not
consider the limit of pushing robots.
3) Because the box’s shape is cylinder, the force acting
on the box is through its center of mass, which is also
the geometry center of box, and parallel with horizon.
4) The origin of reference frame is on the center of the
box. And it’s assumed that, comparing with the velocity
of robots, the speed of the box can be ignored. So, no
matter the box is static or dynamic, robots will move to
the box at a speed of v .
Secondly, there are some definitions:
1) R : The radius of the box.
2) T P : The average pushing time. i.e., the period in
which the robot pushes the box.
3) L : The boundary of a valid perceiving area.
Every robot has identical perceiving ability, i.e. the
individual perceives bound. Consequently, from the view
of the box, there exists an area around it. If robot is in this
area, it can perceive the box and move to the box directly.
This area is called as the box’s valid perceiving area. And
its boundary is described as L . The max distance
between the boundary and center of box is L0 . Fig.4
shows a kind of simple perceiving area, which is the
sector marked by dashed line. And the maximum of the
boundary is 50 units.
4) H : The template of the attraction.
The action of moving to the box can be regarded as a
kind of attraction, therefore we need a template to
describe such attraction field.
According to the assumption 2), robot chases the box at
constant velocity of v . It can be regarded as the product
of certain constant and gradient of the attraction potential
field. Because the velocity is constant, the gradient of the
template should point to the center of the box, and the
intensity of the gradient should be 1. So, the template is:
H = L0 − x 2 + y 2 , v = A ⋅ ∇ R H = 1 ,
(6)
where R represents the position of the robot.
In Fig. 4, circle lines denote the template. Of course,
the template’s effect range should be equal to or larger
than the valid perceiving area. Obviously, outside of the
valid perceiving area, this attraction is of no effect.
Figure 4. The template and the
valid perceiving area of the box.
The radius of the box is 10cm .
L0 = 50cm .
(8)
where S1 = C1 dθ , S = C dxdy .
2
∫
∫∫ 2
R
The dynamic equations can be obtained according to
general expression Eq.(5). But Eq.(5) only reflects
temporal change of the system, we should also find out
special distribution of robots to reveal how to push the
box. For example, if all pushing robots distribute around
the box averagely, the composition of forces acting on the
box is zero. And the box would not be moved. So, what
we should concern is the density change of robots around
the box.
Of course, the density is differential of states on space,
S i . Then, based on these assumptions and definitions, the
dynamic equation of the system can be expressed as
following:
∂C1
1
(7)
= − C1 + Rv C 2 R ,
∂t
TP
∂C2
= f (φ − S1 − S2 ) − v∇ ⋅ (C2∇H ) ,
∂t
robots attach to the box is T P . Averagely, there is 1 TP
robots depart from the box per unit of time. At the same
time, vC2 |B represents the rate of chasing robots that
touches the box and pushes it. For unit coordination,
vC2 |R multiplies with the radius of the box, R . It must be
figured out that we can prove Eq. (7) is valid only when
the template is in the form of Eq. (6).
The second equation describes the dynamics of robots
moving to the box, or the arrow from wandering state to
chasing state. f (φ − S1 − S 2 ) represents the distribution
density of robots that changes from wandering behavior
into chasing behavior per unit of time. v∇ ⋅ (C2∇H )
describes the interaction between robots and the template,
i.e. the attractiveness of the template gradient. Obviously,
we take the action of chasing as a kind of diffusion. There
exists a trend that robots move along the gradient of
attraction field.
Obviously, the box can move only under the condition
that the composition of forces acting on the box is big
than the static friction. If we assumed that every robot can
provide the same pushing force, D f , the condition is
expressed as:
S
C1 represents the density of robots for pushing the box.
Because we ignore robots’ size, C1 is one dimensional
distribution of number of pushing robots around the box.
C 2 represents the density of robots in the valid
perceiving area, i.e. the density of robots moving to the
box. It’s two dimensional distribution of number of
chasing robots.
The first equation describes the dynamics of robotic
density around the box, or, the arrow from state ‘chasing’
to ‘pushing’ in Fig. 3. Because the average time that
2π
F > Fstatic , where F = D f ∫ e iθ C1 dθ ,
0
(9)
and the direction of F points to the goal area.
So, the sufficient condition that the system will push
the box to the goal area is that Eq. (9) is satisfied when
∂C1 ∂t = 0 , and ∂C2 ∂t = 0 , or,
−
1
C1 + RvC 2
TP
R
= 0,
(10)
(11)
f (φ − S1 − S 2 ) − v∇ ⋅ (C2∇H ) = 0 .
Solving Eq.(10) and Eq.(11), and expressing the
solution in polar coordinates:
T f (φ − S 1 − S 2 ) 2
(12)
(L − R 2 ) ,
C1 = P
2
C2 =
.
f (φ − S1 − S 2 )  L2


 r − r
2v


(13)
In fact, from the state paradigm and Eq.(7) and Eq.(8),
we know that the states of the system form a limit event
set. And the Markov chain is an irreducible chain. If we
assume that f (φ − S1 − S 2 ) is time-invariant. Say, it
only relates to the number of robots in three states. And in
the beginning, the box is away from the goal area far
enough, there must be a stationary distribution of Markov
chain during the process of moving the box. Obviously,
this stationary distribution is expressed as Eq.(12) and
Eq.(13).
4. Analysis on solutions
From Eq.(12) and Eq.(13), it can be observed that if the
distribution density of robots changing into chasing
behavior is known, the density of robots pushing the box
is affected by T P , and L . So, proper design of average
pushing time and perceiving area of a single robot can
make robots work together! The cooperation among
robots is a property of the system’s structure.
An example is illustrated with following assumptions:
After robots depart from the box, firstly they will wander
randomly for a period without perceiving the arena. The
average duration is TW . Then robots perceive the arena
for a short time. If nothing found, they will change into
random wandering state again. Or, they will move to the
box. TW is long enough that position at which a robot
suspends its perceiving ability has no relationship with the
position at which the robot resumes perceiving ability.
Consequently,
γ
(14)
(φ − S1 − S 2 ) ,
f (φ − S1 − S 2 ) =
TW S
where γ represents the probability that robot is in the
valid perceiving area when it resumes perceiving ability.
S represents the area of the valid perceiving area. So,
Eq.(14) means that the robots which are changing from
wandering into chasing behavior ordinarily distribute in
the valid perceiving area.
Rewriting the expression of C1 as
Tγ
(15)
C1 = P (φ − S1 − S 2 )(L2 − R 2 ).
2TW S
If it is assumed that, T P = 3S , TW = 3S , γ = 0.4 ,
v = 40cm / S , the template and the valid perceiving area
are as the same as Fig.3 shown, when Eq.(10) and Eq.(11)
are satisfied, the average density of robots pushing the
box is C1 = 6.27rad −1 . If D f = 10 N , the average
composition of forces acting on the box is
FMax = 108.6 N . And the force points to negative Y-axis.
So, if FMax > Fstatic , the box will be moved.
The pushing time T P plays a very important role in
the system. Normally, all robots are required to get
together to push the box in proper direction. Once robots
attached to the box, they should not depart from it until
the box arrives at goal area. Why do robots depart from
the box after T P ?
T P describes the property, homeostasis, of the system.
It makes the system be more flexible. It’s also one of
properties of swarm systems. If the pushing duration is
infinite, all robots would attach to the box unavoidably.
Then the dynamic process falls into a static state, Eq. (7)
and Eq.(8) would be useless any more. Maybe collecting
all robots’ power is a high efficient way to accomplish the
task, but it will deduce the flexibility of the system. For
example, if there is an obstacle on the way of the box,
how do robots avoid it? Traditional methods require
robots abandon pushing and perceive the obstacle’s
position to reform the group. This will increase the cost of
decision-making. But if the system is homeostasis, there
are always a part of robots wandering around the box.
When they are near the obstacle, they can perceive the
obstacle. At the same time, if they also find that the box is
near them, they will move to it and work with other
pushing robots to drive the box away from the obstacle in
spite of whether they are in the valid perceiving area of
the box or not. Or, another reasonable explanation is that,
the obstacle changes the valid perceiving area of the box.
Fig. 5 shows a simple example of this change. The red
disk represents the box, the blue disk represents the
obstacle, and the gray area is the valid perceiving area of
the box. When the box is closed to the obstacle, just as
5(b) shows, the valid perceiving area is changed, because
some robots around the box will move to the box in spit
of the perceiving original area shown in 5(a). The arrow
indicates the direction change of the composition of
forces in two figures. Fig.6 is the simulation result. Fig.
6(a) shows the traces of the box in ten simulations. There
are 25 robots in simulation. The box’s radius is 10 cm .
The obstacle’s radius is 18cm . In each simulation, the
box was required to be pushed from the origin to the goal
area, (80, 80)cm. From the simulation, we can observe all
traces of the box avoiding the obstacle. Figure 6(b) shows
the relationship between average quantity of robots
pushing the box and distance form the obstacle to the box.
Then, without explicit indention for obstacle avoidance,
the system can form a trajectory that rounds the obstacle.
That means, without modifying anything about the
individual strategy or adding any complex coordination
methodology, the swarm can accomplish more complex
behavior.
(a)
(b)
Figure 5. The change of the valid
perceiving area of the box.
Now review the assumptions mentioned above.
Assumption 1) affects the rate of distribution density of
robots changing into chasing behavior f (φ − S1 − S 2 ) .
Assumption 2) means the density of robots around the box
should not be limited, if the robots’ size and quantity can
not be ignored, there would be a limit of C1 . Assumption
3) ensures that Eq.(9) is satisfied. The last assumption
ensures that the relative velocity of robot moving to the
box would be constant. Obviously, these assumptions have
no effects on the structure of dynamic equations Eq.(7)
and Eq.(8), hence the model is valid.
large scale of robots to work together. At this stage, we
have constructed a swarm of robots, and experimental
study will be carried out later.
6. Acknowledgements
This work was funded by the Research Committee of
University
of
Macau
under
grant
no.:
RG024/03-04S/LYM/FST.
80
70
60
7. References
50
[1] C. R. Kube and E. Bonabeau, “Cooperative Transport by
40
Obstacle
30
[2]
20
[3]
10
0
0
10
20
30
40
50
60
70
80
(a)
[4]
Averrage quantity of pushing robots
6
5.5
[5]
5
4.5
4
[6]
3.5
3
2.5
[7]
2
1.5
25
30
35
40
45
50
Distance from the obstacle to the box
(b)
Figure 6. Simulation results of obstacle avoidance.
5. Conclusion
Cooperation is a base property of multi-robot systems.
The paper provides a paradigm of swarm intelligence to
solve box-pushing issue and proposes the sufficient
condition for achieving the task. It can be concluded from
above analysis that cooperation can be achieved by proper
design of the system’s structure. The dynamic equilibrium
of system design brings about high flexibility. Because the
requirement for individual robot’s intelligence and
communication is very simple, it is feasible to construct a
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