Download Description of Attraction-Repulsion Forces by

Volume 5, Issue 4, 2015
ISSN: 2277 128X
International Journal of Advanced Research in
Computer Science and Software Engineering
Research Paper
Available online at: www.ijarcsse.com
Description of Attraction-Repulsion Forces by Probabilistic Fuzzy
Logic for Handling Uncertainty in Swarm Aggregations
1
Samane Sharif* 2Alireza Rowhanimanesh
Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP),
Ferdowsi University of Mashhad, Iran
2
Robotics Laboratory, Department of Electrical Engineering, University of Neyshabur, Iran
1
Abstract— Attraction-repulsion forces are widely used to describe the interactions among individuals in natural and
artificial swarm aggregations from macro to nano scale. In most of the previous works, it is assumed that these forces
are mathematically describable by deterministic functions. But there exist two crucial problems for mathematical
modeling of attraction-repulsion forces in real swarms especially in micro and nano scales. First, taking sampling
data by measuring the available forces among individuals is difficult and costly; therefore, system identification
techniques cannot be used for finding crisp mathematical models. Instead, experts can usually describe forces
linguistically based on the acquired knowledge from observations. Second, due to the presence of non-deterministic
uncertainty in addition to deterministic uncertainty, forces among individuals cannot be described deterministically;
thus, the stochastic behaviour of forces should be considered in mathematical modeling. In contrast to the prior
researches, this paper benefits from probabilistic fuzzy logic to address both of the above-mentioned problems.
Simulation results demonstrate that the proposed approach is an efficient method for handling uncertainty in swarm
aggregations.
Keywords— Swarm Aggregation - Attraction-Repulsion Forces – Deterministic Uncertainty –Non-deterministic
Uncertainty - Probabilistic Fuzzy Logic
I.
INTRODUCTION
In recent years, considerable researches have been performed on multi-agent systems and specially swarm intelligence.
This field of research is currently recognized as a promising applied paradigm in distributed artificial intelligence. Swarm
intelligence have a very wide range of applications from mathematics, physics and biology such as modeling the
dynamics of ants, bees, moving particles and micro-organisms to engineering problems like distributed computing,
wireless sensor networks, traffic control, and swarm robotics. More importantly, swarm intelligence can be considered as
a powerful approach for realizing collective artificial intelligence in swarms of very simple nano-scale agents that is a hot
research topic in today‟s nanorobotics ([1]-[5]).
Previous studies on the behaviour of natural swarms in physics and biology demonstrates that the cooperative
behaviour among particles or individuals in a swarm have very important benefits in reaching the final goal like finding
food, avoiding enemy, reaching nest, etc. Ant colony, bees, bacteria, fishes and birds are all well-known samples of
natural swarms. The central question in studying the behaviour of swarms is: How can a swarm reach its final goal
intelligently while its individuals are very simple and they individually have not any notable intelligence? Studies in
swarm intelligence have answered this question. Indeed, the mystery of success in a swarm lies in the interactions among
individuals. A famous sentence of “1+1=3” points to this fact that good local interactions among very simple individuals
in a swarm can lead to an emerging collective intelligence, called swarm intelligence, which can globally generate
complex self-organized patterns.
Inspired by the dynamics of particle swarms in physics and biology, in recent decades, special focus has been
performed on artificial attraction-repulsion forces to mathematically model the interactions among individuals in a swarm.
In this paradigm, the force that two individuals apply to each other is a function of their distance from each other.
Previous researches demonstrate that this approach can be widely used for modeling the dynamics of swarms from macro
to nano; however it is promisingly applicable for natural and artificial micro and nano scale swarms. Some of the
previous works are reviewed.
Different studies have been performed on the effect of interactions among individuals on the dynamics of natural and
artificial swarms ([6]-[9]). Gazy and Passino ([10]-[12]) mathematically modeled the emerging collective behaviour of
swarms by attraction-repulsion forces among individuals and introduced the fundamental concept of swarm stability as
an analytical tool for studying swarm dynamics. Moreover, Liu and Passino [13] considered swarm stability in the
presence of uncertainty, in which, noise takes effect on the interactions among individuals. Many researches have been
done on mathematical modeling of swarm aggregation by attraction-repulsion forces and different models have been
proposed ([14]-[25]). These models, as basic algorithms, can be broadly applied to simulate multi-agent and multi-robot
systems ([26]-[28]). For example in [29], attraction and repulsion forces are employed for target tracking and obstacle
© 2015, IJARCSSE All Rights Reserved
Page | 16
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
avoidance, respectively. In comparison to the earlier works, in [30], fuzzy logic is employed for modeling the dynamics
and stability of swarm aggregation, where force functions are described by fuzzy if-then rules.
In most of the previous works, it is assumed that these forces can be mathematically modeled by deterministic
functions, either crisp or fuzzy. But in this paper, we show that there exist two crucial problems for mathematical
modeling of attraction-repulsion forces in natural swarms especially in micro and nano scales which cannot be
simultaneously addressed by previous works. In contrast, this paper proposes a novel approach based on probabilistic
fuzzy logic to deal with both of these problems. Simulation results demonstrate that the proposed approach is an efficient
method for handling both types of deterministic and non-deterministic uncertainty in swarm aggregations.
II. PROPOSED APPROACH
A. Conventional Model of Swarm Aggregation
Fig.1 shows a swarm of N individuals, in which, interactions among individuals can be described by attractionrepulsion forces. The force between each two individuals is a function of the distance between them. The velocity of each
individual is highly affected by the forces that other individuals apply to it. According to the prior researches in the
literature ([10]-[13]), the governing equation of motion for each individual in the swarm of Fig.1 is:
𝑖
𝑗
𝑥𝑖 = 𝑁
𝑥 𝑖 − 𝑥 𝑗 , 𝑖 = 1, … , 𝑁
(1)
𝑗 =1 ,𝑗 ≠𝑖 𝐹 𝑥 − 𝑥
where𝑥 𝑖 = 𝑥1𝑖 , 𝑥2𝑖 , … , 𝑥𝑛𝑖 ∈ ℝ𝑛 is the position of individual𝑖in n-dimensional space,𝑥 𝑗 is the position of individual𝑗,
𝐹 . : ℝ → ℝis the force that individual𝑗applies to individual𝑖andit is a function of 𝑥 𝑖 − 𝑥 𝑗 ,and𝑥 𝑖 is the velocity of
individual𝑖.
Fig1. Inter-individual interactions in form of attraction-repulsion forces in the presence of both types of uncertainty.
Different functions have been already proposed forF . in the literature ([6]-[30]). Inspired by these works, in this
paper, we focus on the following force function:
𝐹 𝑑𝑖𝑗 = 𝑓𝑟 (𝑑𝑖𝑗 ) − 𝑓𝑎 (𝑑𝑖𝑗 ) + 𝑓𝑛 (𝑑𝑖𝑗 )
(2)
𝑖
𝑗
where𝑑𝑖𝑗 = 𝑥 − 𝑥 is the Euclidean distance between individuals𝑖and𝑗, 𝑓𝑟 and𝑓𝑎 are positive force functions which
describe repulsion and attraction, respectively.𝑓𝑛 (𝑑𝑖𝑗 )isthe stochastic term of force (noise) whose characteristics can be a
function of𝑑𝑖𝑗 . Usually, the mean of𝑓𝑛 (𝑑𝑖𝑗 )is zero for all distances and there exists an equilibrium distance between two
individuals, in which, attraction and repulsion forces are equal. If distance is less than this value, F is repulsive (𝑓𝑟 > 𝑓𝑎 ),
and repulsion is increased by decreasing the distance. If distance is larger than equilibrium distance, F is attractive(𝑓𝑟 <
𝑓𝑎 ), and attraction is increased by increasing the distance (untilreaching athreshold distance).
B. Inter-individual Interactions in Presence of Uncertainty
It is very important to note that in most of the previous researches, it is assumed that the deterministic terms of forcein
Eq.2 (i.e.𝑓𝑟 and𝑓𝑎 ) can be mathematically modeled by deterministic functions including crisp functions (in mostcases) and
fuzzy modeling (in rare cases). Also, the non-deterministic term of force (i.e.𝑓𝑛 ) is supposed to bemodeled via one of the
familiar probability distribution functions (PDFs). To obtain accurate model for inter-individual interactions, the model
of forces should be very close to actual forces. But there exist two crucial problems for mathematical modeling of forces
in natural swarms especially in micro and nano scales. First, taking sampling data by measuring the available forces
among individuals is difficult and costly; therefore, system identification techniques cannot be used for finding crisp
mathematical models. Instead, usually experts can linguistically describe forces based on the acquired knowledge from
observations. Second, due to the presence of stochastic uncertainty, forces among individuals cannot be described
deterministically; thus, the stochastic behaviour of forces should be considered in mathematical modeling.
As mentioned in Section 1, only a few of previous works have used fuzzy logic to address the first problem, but
theirproposed approaches are not powerful enough to deal with the second problem. It should be mentioned that fuzzy
logic is a very good technique for handling deterministic uncertainty. But real world is full of the both types of
uncertainty including deterministic (possibilistic) and non-deterministic (probabilistic or stochastic). So, fuzzy logic
cannot individually deal with non-deterministic uncertainty. On the other hand, probability theory is a well-known
classical method for working on this type of uncertainty. But as a fundamental feature of probability theory, it is based on
conventional mathematics which uses crisp functions for describing probability distribution functions. Therefore, finding
accurate approximation for the crisp functions of PDFs still has all difficulties of the first problem mentioned above, and
probability theory is not individually desirable and applicable for our purpose. As an alternative, we have been inspired
© 2015, IJARCSSE All Rights Reserved
Page | 17
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
by the sentence of Prof. Zadeh, the father of fuzzy logic and computing with words, which says the fuzzy logic and
probability theory are complementary rather than competitive. So, we must look for a synergic combination of fuzzy
logic and probability theory. In this paper, we benefit from Probabilistic Fuzzy Logic, as a powerful method in soft
computing ([31]-[36]), to simultaneously handle deterministic and non-deterministic uncertainty in mathematical
modeling of forces. In the proposed method, force is modeled by a probabilistic fuzzy system. In contrast to the previous
works, the proposed approach could efficiently address both of the above-mentioned problems.
C. Probabilistic Fuzzy Approach
Regarding the important property of universal function approximation, fuzzy systems are very efficient approach for
handling deterministic uncertainty which plays a central role in approximating functions using expert's linguistic
knowledge. In fuzzy systems, the consequent part of a fuzzy if-then rule consists of only one fuzzy set:
𝑅𝑢𝑙𝑒 𝑙:
𝑖𝑓 𝑑 𝑖𝑠 𝐴𝑙 𝑡ℎ𝑒𝑛 𝐹𝑓 𝑖𝑠 𝐵𝑙
And the relation between input (𝑑)andoutput (𝐹𝑓 ) is:
𝑀
𝑙=1 𝜇 𝐴 𝑙 (𝑑) 𝐶𝐵 𝑙
𝑀 𝜇 (𝑑)
𝑙=1 𝐴 𝑙
𝐹𝑓 (𝑑) =
(3)
𝑙
𝑡ℎ
where 𝜇𝐴𝑙 (. ) is the membership function (MF) of 𝐴 (the antecedent part of the 𝑙 rule), 𝐶𝐵 𝑙 is the center of 𝐵𝑙 (the
consequent part of the𝑙 𝑡ℎ rule), and𝑀is the number of rules in fuzzy rule base. Comparing the above equation with Eq.2
demonstrates that a fuzzy system is able to approximate the deterministic part of force (𝑓𝑟 (𝑑) − 𝑓𝑎 (𝑑)), but cannot handle
stochastic part (𝑓𝑛 (𝑑)). In contrast to fuzzy systems, in probabilistic fuzzy systems, the consequent part of a probabilistic
fuzzy rule contains different fuzzy sets (𝐵1𝑙 , 𝐵2𝑙 , . . . , 𝐵𝐾𝑙 ) with different corresponding probabilities(𝑝1𝑙 , 𝑝2𝑙 , . . . , 𝑝𝐾𝑙 ):
𝐵1𝑙 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑝1𝑙 𝑎𝑛𝑑
𝑅𝑢𝑙𝑒 𝑙: 𝑖𝑓 𝑑 𝑖𝑠 𝐴𝑙 𝑡ℎ𝑒𝑛 𝐹𝑝𝑓 𝑖𝑠
𝐵2𝑙 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑝2𝑙 𝑎𝑛𝑑
⋮
𝐵𝐾𝑙 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑝𝐾𝑙
where 𝑝1𝑙 + 𝑝2𝑙 + . . . + 𝑝𝐾𝑙 = 1.
And the relation between input (𝑑) and output (𝐹𝑝𝑓 ) is:
𝐹𝑃𝐹 (𝑑) =
𝑀
𝑙=1 𝜇 𝐴 𝑙 (𝑑) 𝐶𝐵 𝑙
𝑘∗
𝑀 𝜇 (𝑑)
𝑙=1 𝐴 𝑙
(4)
where𝐶𝐵 𝑙 ∗ is the center of𝐵𝑘𝑙 ∗ that is selected among𝐵1𝑙 , 𝐵2𝑙 , . . . , 𝐵𝐾𝑙 by a random selection mechanism according to the
𝑘
corresponding probability values𝑝1𝑙 , 𝑝2𝑙 , . . . , 𝑝𝐾𝑙 . It should be mentioned that in this study, we use Mamdani inference
engine, centers average defuzzifier and roulette wheel selection mechanism. Fig.2 schematically shows the roulette wheel
selection mechanism as well as the probability vector that have been used in the simulation study of the present paper.
More details about probabilistic fuzzy systems are available in ([31]-[36]). Comparing Eq.4 with Eq.2 shows that
probabilistic fuzzy system can simultaneously approximate both deterministic and non-deterministic parts of force.
Fig.2. The roulette wheel selection mechanism and the probability vector used in the simulation study.
It is very important to note that the proposed approach is not restricted to familiar PDFs such as uniform and normal
distributions, when any arbitrary probability distribution can be easily approximated by probabilistic fuzzy logic.
Therefore, probabilistic fuzzy systems can be considered as efficient approach for mathematical modeling of the force of
Eq.2. Finally, after approximating the actual force of Eq.2 by the probabilistic fuzzy system of Eq.4, the governing
equation of swarm in Eq.1 is converted to:
𝑥𝑖 = 𝑁
𝑥𝑖 − 𝑥 𝑗
𝑥 𝑖 − 𝑥 𝑗 , 𝑖 = 1, … , 𝑁
(5)
𝑗 =1 ,𝑗 ≠𝑖 𝐹𝑝𝑓
Simulation results in the next section demonstrate that the proposed approach can approximate the dynamics of the
given actual swarm accurately in the presence of both types of uncertainty.
© 2015, IJARCSSE All Rights Reserved
Page | 18
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
Fig.3. Actual force (the average of force is displayed as a solid blue plot).
III. SIMULATION RESULTS
In this section, we aim to demonstrate the efficiency of the proposed approach in approximating the dynamics of
actual swarms in comparison with fuzzy approach. Without loss of generality, swarms have been simulated in twodimensional space in MATLAB. For this purpose, we have simulated three different swarms. First, the actual swarm of
6
4
Eq.1 with the actual force of Eq.2, where 𝑓𝑟 𝑑𝑖𝑗 = 2 , 𝑓𝑎 𝑑𝑖𝑗 =
, and 𝑓𝑛 𝑑𝑖𝑗 is a zero-mean Gaussian noise
𝑑 𝑖𝑗
𝑑 𝑖𝑗
whose standard deviation is proportional to 𝑑𝑖𝑗 . Fig.3 represents this force. Second, the approximated model of Eq.5 with
fuzzy force of Eq.3, where the fuzzy system has 11 rules. Fig.4 shows the antecedent MFs and the centers of the
consequent MFs of the fuzzy system. Third, the approximated model of Eq.5 with probabilistic-fuzzy force of Eq.4,
where the probabilistic fuzzy system has 11 rules and its antecedent MFs are as same as the fuzzy system of Fig.4. The
consequent part of each probabilistic fuzzy rule contains 14 MFs and 14 probability values which are displayed in Fig.5.
In this simulation, each swarm has 9 individuals.
Fig.6 depicts the trajectory of individuals from three random initial positions for the above-mentioned three swarms.
This figure compares the generated spatial patterns by the actual swarm with fuzzy model and the proposed probabilistic
fuzzy model. Also, Fig.7 illustrates the temporal changes in the position of one of the individuals (for example) for each
test of Fig.6. The spatial and temporal patterns of Figures 6 and 7 clearly demonstrate that the generated pattern by fuzzy
model is deterministic and could not approximate the stochastic behaviour of the actual swarm. In contrast, the proposed
probabilistic fuzzy model could efficiently approximate the deterministic and non-deterministic dynamics of the actual
swarm and generate similar patterns. These results show that the proposed approach can be considered as a powerful
modeling tool for approximating the global spatial-temporal patterns generated by actual swarms, especially in micro and
nano scales which are full of deterministic and non-deterministic uncertainty.
(a)
(b)
Fig.4. Fuzzy approximation of force: a) Antecedent MFs, b) Centers of the consequent MFs.
Fig.5. Probabilistic fuzzy approximation of force: The𝑙 𝑡ℎ plot is related to the consequent part of the𝑙 𝑡ℎ rule, in which, the
horizontal axis shows the center values𝐶𝐵 𝑙 and the vertical axis illustrates their corresponding probability values𝑝𝑘𝑙 ,
𝑘
where𝑘 = 1,2, … ,14.
© 2015, IJARCSSE All Rights Reserved
Page | 19
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
(a)
(b)
(c)
Fig.6. Trajectory of individuals from 3 random initial positions (denoted by cross „x‟). This figure compares the actual
swarm with fuzzy model and the proposed probabilistic fuzzy model.
(a)
© 2015, IJARCSSE All Rights Reserved
Page | 20
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
(b)
(c)
Fig.7. The temporal changes in the position of one of the individuals for Tests a-c of Fig.6. In each case, the top, middle
and bottom plots are related to actual, fuzzy and probabilistic fuzzy swarms, respectively.
IV. CONCLUSIONS
In many natural and artificial swarms, especially in micro and nano scales, interactions among individuals can be
described by attraction-repulsion forces. In most of the previous works, these forces are mathematically modeled by
deterministic functions, either crisp or fuzzy, that cannot efficiently handle non-deterministic uncertainty in modeling. In
this paper, we benefits from probabilistic fuzzy logic to simultaneously handle deterministic and non-deterministic
uncertainty in mathematical modeling of forces. The performance of the proposed approach was compared with fuzzy
method in approximating the dynamics of a given actual swarm thorough computer simulation. The results demonstrated
that the proposed probabilistic fuzzy model could effectively model both deterministic and non-deterministic dynamics
of the actual swarm and generate similar patterns, while the generated pattern by fuzzy model was deterministic and
could not approximate the stochastic behavior of the actual swarm. Generally, the new approach is a promising modeling
technique for accurate approximation of the global spatial-temporal patterns generated by actual swarms, especially in
micro and nano scales, where the dynamics of swarms are highly affected by both deterministic and non-deterministic
uncertainty.
REFERENCES
[1]
A. Rowhanimanesh and M-R. Akbarzadeh-T, “Control of Low-Density Lipoprotein Concentration in the
Arterial Wall by Proportional Drug-Encapsulated Nanoparticles,” IEEE Transactions on Nanobioscience, vol.
11, no. 4, pp. 394-401, 2012.
[2]
A. Rowhanimanesh and M-R. Akbarzadeh-T, “Autonomous Drug-encapsulated Nanoparticles: Towards a Novel
Non-invasive Approach to Prevention of Atherosclerosis,”Iranian Journal of Medical Physics, vol. 10, no. 2, pp.
118-132, 2013.
[3]
A. Rowhanimanesh, “Swarm Control Systems for Nanomedicine and Its Application to the Prevention of
Atherosclerosis,” Ph.D. Dissertation of Control Engineering, Faculty Advisor: Prof. M-R. Akbarzadeh-T,
Ferdowsi University of Mashhad, 2013.
[4]
F. Razmi, R. Kardehi Moghadam and A. Rowhanimanesh, “Control of Cancer Growth Using Two Input
Autonomous Fuzzy Nanoparticles,”NANO: Brief Reports and Reviews, vol.10, no.4, 2015.
[5]
F. Razmi, R. Kardehi Moghadam and A. Rowhanimanesh, “Control of Cancer Growth Using Single Input
Autonomous Fuzzy Nanoparticles,”Journal of Fuzzy Set Valued Analysis, no.1, pp. 86-96, 2015.
© 2015, IJARCSSE All Rights Reserved
Page | 21
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
Iain D. Couzin and Jens Krause, “Self-Organization and Collective Behavior in Vertebrates,” Journal of
Advances in the Study of Behavior, vol. 32, pp. 1-75, 2003.
E. Boabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford
University Press, Oxford, 1999.
Shi, L.Wang, T. Chu and F. Xiao, “Self-organization of general multi-agent systems with complex interactions,”
in Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006, pp. 3203–3208.
S. W. Ekanayake and P. N. Pathirana, “Formations of Robotic Swarm: An Artificial Force Based Approach,”
International Journal of Advanced Robotic Systems, vol. 7, no. 3, pp. 173-190, 2010.
V. Gazi and K. M. Passino, “Stability Analysis of Swarms,” IEEE Transactions on Automatic Control, vol. 48,
no. 4, pp. 692 - 697, 2003.
V. Gazi and K. M. Passino, “A class of attractions/repulsion functions for stable swarm aggregations,”
International Journal of Control, vol. 77, no. 18, pp. 1567–1579, 2004.
V. Gazi and K. M. Passino, “Stability Analysis of Social Foraging Swarms,” IEEE Transactions On Systems,
Man, And Cybernetics, vol. 34, no. 1, pp. 539–557, 2004.
Y. Liu and K. M. Passino, “Stable Social Foraging Swarms in a Noisy Environment,” IEEE Transactions on
Automatic Control, vol. 49, no. 1, pp. 30–44, 2004.
D. H. Kim, H. O. Wang, G. Ye and S. Shin, “Decentralized Control of Autonomous Swarm Systems Using
Artificial Potential Functions: Analytical Design Guidelines,”in Proc. IEEE Conference on Decision and
Control, Atlantis,2004, pp. 159-164.
A. J. Leverentz, C. M. Topaz, and A. J. Bernoff, “Asymptotic Dynamics of Attractive-Repulsive Swarms,”
SIAM Journal on Applied Dynamical Systems,vol. 8, no. 3, pp. 880–908, 2009.
H. Shi and G. Xie, “Collective Dynamics of Swarms with a New Attraction/Repulsion Function,” Mathematical
Problems in Engineering, Article ID 735248, 2011.
Z. Xue, Z. Liu, C. Feng and J. Zeng, “Stability Analysis of Exponential Type Stochastic Swarms with TimeDelay,” International Journal of Innovative Management, Information & Production, vol. 2, no. 3, pp. 1-12,
2011.
P. Romanczuk and L. Schimansky-Geier, “ Swarming and pattern formation due to selective attraction and
repulsion,” Interface focus: a theme supplement of Journal of the Royal Society interface, vol. 2, no. 6, pp. 746–
756, 2012.
H. Hashimoto, S. Aso, S. Yokota, A. Sasaki, Y. Ohyama and H. Kobayashi, “Stability analysis of social
foraging swarm with general nonlinear attraction and repulsion forces and interaction time delays,” in Proc.
IEEE International Symposium on Industrial Electronics, Cambridge,2008, pp.1341 - 1346.
J. Hu, J. Yao and Lei Wang, “Cohesiveness analysis of hybrid swarm systems based on artificial potential
functions,” in Proc. The International Conference on Modelling, Identification and Control, Okayama,2010, pp.
639 – 644.
L. Shu, Y. Zheng, H. Shao and W. Pan, “Aggregation stability of multiple agents with general nonlinear
attraction and repulsion forces,” in Proc. IEEE International Conference on Control and Automation,
Christchurch, 2009, pp. 303 – 308.
X. Li, J. Xiao and Z. Cai, “Stable swarming by mutual interactions of Attraction/Alignment/Repulsion,” in Proc.
IEEE Conference on Decision and Control, Cancun,2008, pp. 1036 – 1041.
W. Pan and Y. Zheng, "Aggregation stability of multiple agents with general nonlinear attraction/repulsion
forces and interaction time delays," in Proc. Conference on Control and Decision (CCDC), Guiyang, 2013, pp.
119 - 124.
R.C. Fetecau and Y. Huang, "Equilibria of biological aggregations with nonlocal repulsive–attractive
interactions," Journal of Physica D: Nonlinear Phenomena, vol. 260, no. 1, pp. 49-64, 2013.
J.A. Carrillo, Y. Huang and S. Martin, "Nonlinear stability of flock solutions in second-order swarming
models," Journal of Nonlinear Analysis: Real World Applications,vol. 17, no., pp. 332–343, 2014.
V. Gazi , “Swarm aggregations using artificial potentials and sliding mode control,” IEEE Transactions on
Robotics, vol. 21, no. 6, pp. 1208 - 1214, 2005.
R. P. Leland and R. P. Samples, “Coordination of systems of mobile robots with social potential functions,”
Ph.D. Dissertation, University of Alabama Tuscaloosa, AL, USA, 2009.
C. Depouhon, M. Masciotta,E. Garone and A. Gasparri, "Swarm aggregation with a multi-robot system
composed of three robotic units: A closed form analysis," in Proc. Conference of Control and Automation
(MED), Palermo,2014, pp. 1128 - 1133.
H. Hashimoto, S. Aso, S. Yokota, A. Sasaki, Y. Ohyama and H. Kobayashi, “Stability of swarm robot based on
local forces of local swarms,” in Proc. SICE Annual Conference, Tokyo, 2008, 1254 - 1257.
L. T. T. Nga and L. H. Lan, “Aggregation stability of multiple agents with fuzzy attraction and repulsion
forces,”in Proc. International Conference on Methods and Models in Automation and Robotics (MMAR),
Miedzyzdroje, 2013, pp. 81 - 85.
H. Meghdadi and M. –R. Akbarzadeh-T., “Probabilistic Fuzzy Logic and Probabilistic Fuzzy Systems,” in Proc.
International Conference on Fuzzy Systems. Melbourne, 2001, pp.1127-1130.
A. H. Meghdadi and M. –R. Akbarzadeh-T., “Uncertainty Modeling through Probabilistic Fuzzy Systems,” in
Proc. Int. Symposium on Uncertainty Modeling and Analysis, Maryland,2003, pp.56-61.
© 2015, IJARCSSE All Rights Reserved
Page | 22
[33]
[34]
[35]
[36]
Sharif et al., International Journal of Advanced Research in Computer Science and Software Engineering 5(4),
April- 2015, pp. 16-23
U. Kaymak and W.-M. v. d. Bergh, “A Fuzzy Additive Reasoning Scheme for Probabilistic Mamdani Fuzzy
Systems,” in Proc. International Conference on Fuzzy Systems, 2003, pp.331-336.
Z. LIU and H.-X. LI, “Probabilistic Fuzzy Logic System: a tool to process stochastic and imprecise information,”
in Proc. International Conference on Fuzzy Systems. Korea,2009, pp.849-853.
S. Sharif and M-R. Akbarzadeh-T, “Self-Organized Probabilistic Fuzzy Network for Handling Inconsistent
Data,” in Proc. Iranian Conference on Fuzzy Systems, 2013.
S. Sharif, “Distributed Decision Making Using Probabilistic Fuzzy Rules”, M. Eng. thesis, Ferdowsi University
of Mashhad, Mashhad, Iran,Faculty Advisor: Prof. M-R. Akbarzadeh-T, Sep. 2013.
© 2015, IJARCSSE All Rights Reserved
Page | 23