Ferdowsi University of Mashhad, Iran 2 Robotics Laboratory, Department of Electrical Engineering, University of Neyshabur, Iran

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1 Volume 5, Issue 4, 2015 ISSN: X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: Description of Attraction-Repulsion Forces by Probabilistic Fuzzy Logic for Handling Uncertainty in Swarm Aggregations 1 Samane Sharif * 2 Alireza Rowhanimanesh 1 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 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

2 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: x i N = j =1,j i F x i x j x i x j, i = 1,, N (1) wherex i = x i 1, x i i 2,, x n R n 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 d ij = f r (d ij ) f a (d ij ) + f n (d ij ) (2) whered ij = x i x j is the Euclidean distance between individualsiandj, f r andf a are positive force functions which describe repulsion and attraction, respectively.f n (d ij )isthe stochastic term of force (noise) whose characteristics can be a function ofd ij. Usually, the mean off n (d ij )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 (f r > f a ), and repulsion is increased by decreasing the distance. If distance is larger than equilibrium distance, F is attractive(f r < f a ), 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.f r andf a ) 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.f n ) 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

3 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 A l then F f is B l And the relation between input (d)andoutput (F f ) is: F f (d) = M l=1 μ A l (d) C B l M l=1 μ A l (d) where μ A l (. ) is the membership function (MF) of A l (the antecedent part of the l th rule), C B l is the center of B l (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 (f r (d) f a (d)), but cannot handle stochastic part (f n (d)). In contrast to fuzzy systems, in probabilistic fuzzy systems, the consequent part of a probabilistic fuzzy rule contains different fuzzy sets (B l 1, B l 2,..., B l K ) with different corresponding probabilities(p l 1, p l 2,..., p l K ): Rule l: if d is A l then F pf is l l B 1 with probability p 1 and l l B 2 with probability p 2 and l l B K with probability p K And the relation between input (d) and output (F pf ) is: F PF (d) = M l=1 μ A l (d) C Bk l M l=1 μ A l (d) l where p 1 l + p 2 l p K l = 1. wherec l Bk is the center ofb k that is selected amongb l 1, B l 2,..., B l K by a random selection mechanism according to the corresponding probability valuesp l 1, p l 2,..., p l K. 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. (3) (4) 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: x i N = j =1,j i F pf x i x j x i x j, i = 1,, N (5) 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

4 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 Eq.1 with the actual force of Eq.2, where f r d ij = 6 2, f d a d ij = 4, and f ij d n d ij is a zero-mean Gaussian noise ij whose standard deviation is proportional to d ij. 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 valuesc Bk l and the vertical axis illustrates their corresponding probability valuesp k l, wherek = 1,2,, , IJARCSSE All Rights Reserved Page 19

5 (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

6 (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 , [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 , [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, [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, [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 , , IJARCSSE All Rights Reserved Page 21

7 [6] 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, [7] E. Boabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, Oxford, [8] 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 [9] 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 , [10] V. Gazi and K. M. Passino, Stability Analysis of Swarms, IEEE Transactions on Automatic Control, vol. 48, no. 4, pp , [11] 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 , [12] V. Gazi and K. M. Passino, Stability Analysis of Social Foraging Swarms, IEEE Transactions On Systems, Man, And Cybernetics, vol. 34, no. 1, pp , [13] Y. Liu and K. M. Passino, Stable Social Foraging Swarms in a Noisy Environment, IEEE Transactions on Automatic Control, vol. 49, no. 1, pp , [14] 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 [15] 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 , [16] H. Shi and G. Xie, Collective Dynamics of Swarms with a New Attraction/Repulsion Function, Mathematical Problems in Engineering, Article ID , [17] Z. Xue, Z. Liu, C. Feng and J. Zeng, Stability Analysis of Exponential Type Stochastic Swarms with Time- Delay, International Journal of Innovative Management, Information & Production, vol. 2, no. 3, pp. 1-12, [18] 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 , [19] 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 [20] 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 [21] 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 [22] 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 [23] 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 [24] 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 , [25] 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 , [26] V. Gazi, Swarm aggregations using artificial potentials and sliding mode control, IEEE Transactions on Robotics, vol. 21, no. 6, pp , [27] 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, [28] 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 [29] 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, [30] 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 [31] H. Meghdadi and M. R. Akbarzadeh-T., Probabilistic Fuzzy Logic and Probabilistic Fuzzy Systems, in Proc. International Conference on Fuzzy Systems. Melbourne, 2001, pp [32] 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 , IJARCSSE All Rights Reserved Page 22

8 [33] 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 [34] 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 [35] S. Sharif and M-R. Akbarzadeh-T, Self-Organized Probabilistic Fuzzy Network for Handling Inconsistent Data, in Proc. Iranian Conference on Fuzzy Systems, [36] 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 , IJARCSSE All Rights Reserved Page 23