A new multiple robot path planning algorithm: dynamic distributed particle swarm optimization
 Asma Ayari^{1}Email authorView ORCID ID profile and
 Sadok Bouamama^{1}
Received: 21 September 2017
Accepted: 14 October 2017
Published: 2 November 2017
Abstract
Multiple robot systems have become a major study concern in the field of robotic research. Their control becomes unreliable and even infeasible if the number of robots increases. In this paper, a new dynamic distributed particle swarm optimization (D^{2}PSO) algorithm is proposed for trajectory path planning of multiple robots in order to find collisionfree optimal path for each robot in the environment. The proposed approach consists in calculating two local optima detectors, LOD_{pBest} and LOD_{gBest}. Particles which are unable to improve their personal best and global best for predefined number of successive iterations would be replaced with restructured ones. Stagnation and local optima problems would be avoided by adding diversity to the population, without losing the fast convergence characteristic of PSO. Experiments with multiple robots are provided and proved effectiveness of such approach compared with the distributed PSO.
Keywords
Background
The concept of multiple robot systems (MRS) began in the 1990s, in particular in works regrouping mobile robots, gathering objects [1] and robot colonies [2, 3]. Arai et al. [4] identified seven primary research themes in the MRS: biological inspirations, communication, architectures, location/cartography/exploration, transport and handling of objects, motion coordination and reconfigurable robots.
Multiple robot systems are well known by the synchronization process and having better spatial distribution capability as compared to a single robot. This coordination addresses the problem of how teams of autonomous mobile robots can share the same workspace while avoiding interference with each other, collision with static obstacles and/or while achieving group motion objectives.
There are two basic approaches to solve the problem of multiple robot path planning: centralized and distributed. In the case of the centralized approach, each robot is treated as a composite system, and planning is done in a composite configuration space, formed by combining the configuration spaces of the individual robots. While in the case of the distributed approach, paths are first generated for robots independently and then their interactions are considered. The advantage of centralized approaches is that they always find a solution when there is one. However, the practical difficulty is the temporal complexity which is exponential in composite configuration space. Distributed planners help generate robot trajectories independently before using different strategies to resolve potential conflicts. But, they are incomplete in nature (probabilities of various and varied configurations) and can therefore lead to blocking situations. This distributed approach can be applied to each robot taking into account the positions and orientations of all other robots at each point in time. Thus, the general problem is reduced to several versions of the planning problem for a single mobile robot in the presence of other robots which move in the presence of fixed obstacles. A trajectory of the robot is then found by the search of a path from the start to the arrival thanks to the spatial–temporal configuration.
The control of the MRS becomes unreliable and even infeasible if the number of robots increases. In addition, the multiple robot path planning problem becomes more and more complex. The latter has been extensively studied since the 1980s. Swarm behavior has proven its effectiveness in such problems thanks to interesting properties like robustness, flexibility and scalability. One of the successful optimization methods is particle swarm optimization (PSO) algorithm.
This paper proposed a novel approach to determine the optimal trajectory of the path for distributed multiple robots system using dynamic distributed particle swarm optimization (D^{2}PSO), where each robot is considered to be a mobile, autonomous and physically independent agent.
The remaining part of the paper is outlined as follows. “Literature survey of particle swarm optimization use in MRS path planning” section covers briefly the latest works done in the MRS path planning search domain using different PSO variants. Formulation of the problem for multiple robot path planning has been elaborated in “Problem formulation for multiple robot navigation” section. “Obstacle avoidance approach” section describes our obstacle avoidance approach. The classical particle swarm optimization, dynamic distributed double guided particle swarm optimization algorithm and dynamic distributed particle swarm optimization are described in the “Particle swarm optimization (PSO) for MRS path planning” section. “Conclusion” section demonstrates the computer simulation for path planning of multiple robots.
Literature survey of particle swarm optimization use in MRS path planning
Since the inception of PSO [5, 6], several variants have been proposed to improve the performance of original PSO. The first versions of PSO for MRS were proposed in [7–9] to find a target in a given environment, and studies have demonstrated that the PSO algorithm has acceptable performances in the searching task. In the study of Chakraborty et al. [10], behavioral cooperation of the robots was realized through selection of alternative local trajectories for collision avoidance among teammates. In fact, he compared the performances using differential evolution (DE) with a PSObased realization.
The authors present in [11] PSObased technique for determining the optimal set of parameters for a second PSO for collective robotic search. Particle swarm optimization technique was used to optimize the velocity parameters of robots in [9], to arrive at the shortest collisionfree trajectory, satisfying dynamic constraints. A hybrid technique for the control of swarms of robots, based on particle swarm optimization (PSO) and consensus algorithms, is presented in [12]. A MOPSO algorithm is utilized in [13] to generate trajectories for mobile robots that are working on the environments that the robots are working on and may be included danger sources. Darvishzadeh and Bhanu [14] present a framework to use a modified PSO (MPSO) algorithm in a multiple robot system for search task in realworld environments. Nakisa et al. [15] also proposes a new method (APSO) to create an efficient balance between exploration and exploitation by hybridizing basic PSO algorithm with Astar algorithm. Nakisa proposes a method based on the multiswarm particle swarm optimization (PSO) with local search on the multiple robot search system to find a given target in a complex environment that contains static obstacles [16]. Rastgoo et al. [17] proposed an algorithm named the “modified PSO with local search (MLPSO)” applied in the exploration search space by adding a local search algorithm such as Astar to guarantee global convergence with a reduction in the search time. Allawi and Abdalla [18] used PSO combined with reciprocal velocity obstacles (RVO) method, in order to choose the best paths for robots without collision and to get to their goals faster. Das [19] proposed a new methodology to determine the optimal trajectory of the path for multiple robot in a clutter environment using hybridization of improved particle swarm optimization (IPSO) with an improved gravitational search algorithm (IGSA).
A hybridization of improved particle swarm optimization (IPSO) with differentially perturbed velocity (DV) algorithm (IPSODV) was also proposed by Das et al. [20] for trajectory path planning of multiple robots in a static environment. Abbas et al. discusses in [21] an optimal path planning algorithm based on an adaptive multiobjective particle swarm optimization algorithm (AMOPSO) for five robots to reach the shortest path. The algorithm PSONAV presented in Raffaele Grandi’s work [22] focuses on the possibility to drive a group of very simple robots from a starting zone to a final one inside a mazelike environment unknown a priori.
Problem formulation for multiple robot navigation
The problem formulation for multiple robot path planning is provided in this section. We consider a group of mobile robots to navigate by maintaining predefined geometric shapes (line, column, triangle, etc.), controlling the location of each robot relative to the others. The geometric formation is established from predetermined initial positions, or even from random positions, and is maintained during the movement of the group. This navigation must ensure the avoidance of obstacles in the environment. This kind of navigation is useful in many cooperation tasks such as moving a sports field, transporting or manipulating objects involving several mobile robots.
 1.
For each robot, the current position (recent position) and goal position (target position) is known in a given reference coordinate system.
 2.
Each robot is performing its action in steps until all robots reached in their respective target positions.
 1.
For determining the next position from its current position, the robot tries to align its heading direction toward the goal.
 2.
The alignment may cause a collision with the robots/obstacles (which are static in nature) in the environment. Hence, the robot has to turn its heading direction left or right by a prescribed angle to determine its next position.
 3.
If a robot can align itself with a goal without collision, then it will move to that position.
 4.
If the heading direction is rotated to the left or right, then it is required for the robot to rotate the same angle about its zaxis.
Obstacle avoidance approach
Plenty of algorithms for obstacle avoidance were mentioned in the robotic literature [23–25].
The obstacle avoidance approaches in MRS studies aim to find a path from an initial position S of a robot to a desired goal position G, with respect to positions and shapes of known obstacles O. The penalty function to be minimized by the planning algorithm consists of two parts. While the first one evaluates a length of the trajectory (or time needed to execute the trajectory), the second part ensures safety of the path (i.e., distance to obstacles).
Particle swarm optimization (PSO) for MRS path planning
Classic PSO
Particle swarm optimization (PSO) is a stochastic optimization method for nonlinear functions based on the reproduction of social behavior developed by Berhart and Kennedy [5, 6] in 1995.
The origin of this method comes from the observations made during computer simulations on grouped flights of birds and fish [26]. These simulations highlighted the ability of individuals in a moving group to maintain an optimal distance between each other’s and to follow a global movement in relation to neighbors one.
To apply the PSO, we have to define a particle search space and an objective function to optimize. The principle of the algorithm is to move these particles so that they find the optimum.

A position, that is, its coordinates in the definition set.

A speed that allows the particle to move. In this way, during the iterations, each particle changes its position. It evolves according to its best neighbor, its best position and its previous position. This evolution makes it possible to fall on an optimal particle.

A neighborhood, that is, a set of particles that interact directly with the particle, especially the one with the best criteria.

Its best position visited. The value of the calculated criterion and its coordinates are essentially retained.

The position of the best neighbor of the swarm that corresponds to the optimal scheduling.

The value of the objective functions because it is necessary to compare the value of the criterion given by the current particle with the optimal value.
PSO is initialized with a group of random particles (solutions) and then searches for optima by updating generations. In every iteration, each particle is updated by following two “best” values. The first one is the best solution (fitness) the particle has achieved so far. This value is called pBest. Another “best” value that is tracked by the particle swarm optimizer is the best value obtained so far by any particle in the population. This best value is a global best and called gBest.
D^{3}GPSO: the dynamic distributed double guided particle swarm optimization algorithm
The D^{3}GPSO introduced by Bouamama in [4, 27] is a distributed PSO. It is a group of agents dynamically created and cooperating in order to solve a problem. Each agent performs locally its own PSO algorithm.
Inspired by works in [1, 2, 4, 28], this algorithm uses the same principle as the D^{3}G^{2}A [2], and it consists on dividing the initial population into subpopulations and affecting each one to an agent. Each agent is also called specie agent and is responsible of a set of particles having their fitness values in the same range. This range is called FVR (for fitness value range), and it is the specificity of the specie agent Specie_{FVR}. The species agents cooperate all by exchanging solutions to reach the optimal one. In fact, each one executes its own double guided PSO algorithm. The latter is double guided by the concept of template and the minconflict heuristic. It is enhanced by new parameters: guidance probability P _{guid}; a local optimum detector LOD and a weight ε (used by species agents to calculate their own PSO parameters).
For the D^{3}GPSO, we distinguish also a mediator agent to manage the communication between the species agents. This agent, called interface agent, can also create new species agents if necessary.
The local optimum detector LOD is an operator that we use in the PSO process. It represents the number of iterations in which the neighboring does not give improvement. If the best solution found by a specie agent remains unchanged for LOD generations, we can conclude that the particles are blocked in a local optimum. So, the best particle having this fitness value will be penalized. This variable is given by the user at the beginning of the optimization process, but it is changed by each specie agent according to the attained fitness value.
D^{2}PSO: dynamic distributed PSO for MRS path planning
In 2006, Hereford [9] has introduced a version of the PSO that “distributes” the motion processing among several, simple, compact, mobile robots, called distributed PSO (dPSO). Calculations were done “locally,” that is on each local robot. Simulation results showed that the dPSO appears to be a very good way of coordinating simple robots for a searching task operation. One of the most important advantages was that the algorithm appears to be scalable to large numbers of robots since the communication requirements do not increase as the number of robots is increased.
Although swarm intelligence approaches are attractive methods for robotic target searching problems, these strategies have two important disadvantages: First, they may get stuck on local optima. Second, they have slow progress in terms of fitness function in some situations (slow speed to converge to the target locations).
Inspiring from the D^{3}GPSO described in [4, 27], we introduce two new parameters to the PSO: local optima detector for global best LOD_{gBest} and local optima detector for personal best LOD_{pBest}. The purpose of the latest parameter is to count the number of successive iterations for which personal best and global best do not give improvement. Since these particles are unable to improve their pBest, they are no more contributing in finding the global optimal solution. This indicates that particles are saturated and require external thrust to boost their power. Dynamic distributed PSO (D^{2}PSO) provides thrust by heading particles toward potentially better unexplored regions which also add diversity to the search space. At the same time, when global best gBest is not improving for predefined number of successive iteration, it may be trapped in local optima and mislead other particles by attracting toward it. This also requires some external push that send trapped particle outside local optima position and mitigate its consequences. By this way, the stagnation and local optima problems would be avoided without losing the fast convergence characteristic of PSO since the D^{2}PSO would follow the PSO’s behavior for the rest of situations.
Local optimum detector
LOD is a parameter to the whole optimization process, and it will be locally and dynamically updated by each robot [1, 2]. If the personal best of the ith particle increases for a specific number of successive generations, we can conclude that the particle optimization subprocess is trapped in a local optimum, and so the LOD_{pBest} will increment, respectively, for gBest and LOD_{gBest}.
Dynamic concept
Our work consists in implementing a dynamic distributed version of PSO. It is a multiagent approach. It acquires its dynamic aspect from the agents that it uses. Indeed, they are capable to calculate their own parameters on the basis of the parameters given by the user (S _{ p }, S _{ g }, ε). ε ϵ [0, 1].
Template concept
As mentioned in [27], we define in (8) and (9) the parameters α and β which are function of template_{pBest} and template_{gBest}. Their worth is that they articulate the probability for a particle to propagate its knowledge. This confirms the fact that the best particles have more chance to be pursued by others.
Implementation and experiments
Experiment configuration
The multiple robot path planning algorithm is implemented in a simulated environment. The simulation is conducted through programming in C language on a Pentium microprocessor, and robot is represented with similar softbots of rectangular shape with different path color code. The robot is selfcontained in terms of power. It is mobile; it may be limited in terms of steering radius and speed, but it is mobile.
Predefined initial location and goal location for all the robots are assigned. The experiments were conducted with three different radius obstacles and assigned same velocities for each robot at the time of the program run.
Initial and target points used in simulation
Initial point  x  y  Target point  x  y 

1  − 2.41  − 3.08  1  5.22  6.22 
2  − 1.72  − 0.47  2  6.10  6.70 
3  − 1.88  − 3.48  3  4.75  6.59 
4  − 2.53  − 0.95  4  6.09  6.05 
5  − 1.19  − 2.34  5  5.28  7.18 
6  − 2.81  − 2.39  6  5.23  5.78 
7  − 1.44  − 1.03  7  6.17  6.99 
8  − 2.09  − 3.21  8  4.61  6.41 
9  − 2.40  − 0.92  9  6.30  6.13 
10  − 0.21  − 2.59  10  5.03  7.21 
We have applied the distributed PSO and the proposed D^{2}PSO to the same environment. Population size M was fixed to 150 and then to 300.
We used the following parameters: inertia weight w = 1.0, damping ratio w _{damp} = 0.99, personal learning coefficient c _{1} = 1.5, global learning coefficient c _{2} = 1.5, S _{ p } = 3, S _{ g } = 3.
Experiment results and analysis
We made several simulation runs. We have run the program for 3, 5, 7 and 10 robots with different initial and target points, for 200 iterations. We evaluated the effectiveness of the algorithm by comparing the path lengths obtained from the basic distributed PSO and the D^{2}PSO, ensuring of course the noncollision with the static predefined obstacles (Figure 3).
Path lengths for M = 150
Path length M = 150  

Distributed PSO  D^{2}PSO  
Robot1  12.2489  12.1666 
Robot2  11.0501  10.6690 
Robot3  12.461  12.1351 
Robot4  11.1966  7.8620 
Robot5  11.6496  8.1610 
Robot6  12.3214  9.1203 
Robot7  9.4512  7.6291 
Robot8  10.9221  9.7347 
Robot9  11.3312  8.5025 
Robot10  12.3411  11.7196 
Path lengths for M = 300
Path length M = 300  

Distributed PSO  D^{2}PSO  
Robot1  12.2421  12.1664 
Robot2  11.0449  10.6679 
Robot3  12.4518  12.134 
Robot4  11.1951  7.8591 
Robot5  11.6412  8.1257 
Robot6  12.2114  9.1098 
Robot7  9.4232  7.6011 
Robot8  10.9119  8.6738 
Robot9  11.1931  8.4381 
Robot10  12.3391  10.6956 
Since restructuring both pBest and gBest depends on the number of robots and the population size, we deduce that each time we increase the number of bots, we obtain better values of the objective function, and respectively when increasing the population size M.
Execution time for M = 150
Time (s) M = 150  

Distributed PSO  D^{2}PSO  
Robot1  40.569334  39.676484 
Robot2  48.224949  30.275881 
Robot3  43.248937  32.953980 
Robot4  45.283051  30.535676 
Robot5  45.653371  35.419047 
Robot6  48.689387  30.173998 
Robot7  49.968918  39.833665 
Robot8  40.314716  30.654208 
Robot9  40.546997  30.701435 
Robot10  40.272204  31.146030 
Execution time for M = 300
Time (s) M = 300  

Distributed PSO  D^{2}PSO  
Robot1  45.103684  44.720177 
Robot2  55.595825  35.589437 
Robot3  48.550160  37.941275 
Robot4  51.812619  35.380362 
Robot5  49.960134  40.247571 
Robot6  50.862109  35.323138 
Robot7  55.566426  45.552622 
Robot8  45.396125  35.484535 
Robot9  46.271805  35.422918 
Robot10  44.100084  37.159384 
Comparison of number of iterations taken to obtain the shortest path by the different algorithms
# Iterations  

Distributed PSO  D^{2}PSO  
Robot1  93  77 
Robot2  122  54 
Robot3  102  66 
Robot4  112  56 
Robot5  119  70 
Robot6  127  52 
Robot7  135  79 
Robot8  80  58 
Robot9  91  59 
Robot10  82  61 
Conclusion
The problem of multiple robot motion planning focuses on computation of paths of different robots such that each robot has an optimal path, and so the overall path of all the robots combined is optimal. Many approaches have been proposed for solving multiple robot path planning problems. Particle swarm optimization algorithm is one of the successful optimization methods in this area. This paper has presented a successful improvement to the PSO algorithm. D^{2}PSO ensures diversity to stagnated particles in such a manner that they move to better and unexplored regions of search space. In addition, it does not disturb the fast convergence characteristics of PSO by keeping the basic concept of PSO unaffected. Experimental results show that our approach performs better in escaping local optimum and proves that applying D^{2}PSO to multiple robots path planning problem is practical and efficient for large number of robots in environments with variable obstacles.
Discussion and future work
The main contributions of our research are: (1) finding optimal paths of mobile robots moving together in the same workspace, (2) proposing to use the PSO evolutionary algorithm and (3) ensuring collisionfree trajectories.
However, there are still some issues and improvements to be addressed in our future work. First, dynamic obstacles, unknown environment, obstacles’ shapes and collision avoidance should be studied. In this paper, both the environment and obstacles are static relative to the robots, which is applicable in particular cases. In the future, work will be carried out using dynamic obstacles during the multiple robot path planning process. Second, the interrobot collision should be considered in future experiments. The task planning process for MRS would be also studied in order to ensure best coordination.
Declarations
Authors’ contributions
AA formulated, analyzed and implemented the D^{2}PSO and algorithms described in the paper inspiring from D^{3}GPSO. SB provided guidance and valuable suggestions for the improvement in the paper. Both authors read and approved the final manuscript.
Acknowledgements
I would like to thank the editor and the anonymous references for their helpful comments.
Competing interests
The authors declare that they have no competing interests.
Funding
Not applicable.
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Authors’ Affiliations
References
 Bouammama S, Ghedira K. Une nouvelle génération d’algorithmes génétiques guidés distribués pour la résolution des Max_CSPs. Revue des sciences et technologie de l’information, serie Technique et science informatique, TSI. 2008;27(1–2):109–40.Google Scholar
 Bouammama S, Ghedira K. A dynamic distributed double guided genetic algorithm for optimization and constraint reasoning. Int J Comput Intell Res. 2006;2(2):181–90.View ArticleGoogle Scholar
 Arai T, Pagello E, Parker LE. Editorial, “Advances in multirobot systems”. IEEE Trans Robot Autom. 2002;18(5):655–61.View ArticleGoogle Scholar
 Bouamama S, Ghedira K. A family of distributed double guided genetic algorithm for Max_CSPs. Int J Knowl Based Intell Eng Syst. 2006;10(5):363–76.View ArticleGoogle Scholar
 Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of the IEEE international conference neural networks, Perth, Australia, 1995, vol. 4, p. 1942–8.Google Scholar
 Eberhart RC, Kennedy J. A new optimizer using particle swarm theory. In: Proceeding of the 6th international symposium micromachine human science, Nagoya, Japan, 1995, p. 39–43.Google Scholar
 Doctor S, Venayagamoorthy G, Gudise V. Optimal PSO for collective robotic search applications. In: IEEE congress on evolutionary computation, Portland, OR, June 2004, p. 1390–5.Google Scholar
 Pugh J, Segapelli L, Martinoli A. Applying aspects of multirobot search to particle swarm optimization. In: International workshop on ant colony optimization and swarm intelligence, Brussels, Belgium, 2006, p. 506–7.Google Scholar
 Hereford JM. A distributed particle swarm optimization algorithm for swarm robotic applications. In: IEEE congress on evolutionary computation, Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada, July 16–21, 2006.Google Scholar
 Chakraborty J, Konar A, Chakrabortyand UK, Jain LC. Distributed cooperative multirobot path planning using differential evolution. In: IEEE world congress on computational intelligence, 2008.Google Scholar
 Venayagamoorthy GK et al. Optimal PSO for collective robotic search applications. In: Proceedings of the congress on evolutionary computation, 2004. CEC2004, Institute of Electrical and Electronics Engineers (IEEE), Jan 2004.Google Scholar
 Grandi R, Falconi R, Melchiorri C. Coordination and control of autonomous mobile robot groups using a hybrid technique based on particle swarm optimization and consensus. In: ROBIO conference, 2013.Google Scholar
 Zhang Y, Gong DW, Zhang JH. Robot path planning in uncertain environment using multiobjective particle swarm optimization. Neurocomputing. 2013;103:172–85.View ArticleGoogle Scholar
 Darvishzadeh A, Bhanu B. Distributed multirobot search in the realworld using modified particle swarm optimization. In: GECCO’14, July 12–16, 2014, Vancouver, BC, Canada.Google Scholar
 Nakisa B, Rastgoo MN, Norodin MdJ. Balancing exploration and exploitation in particle swarm optimization on search tasking research. J Appl Sci Eng Technol. 2014;8:1429–34.Google Scholar
 Grandi R, Falconi R, Melchiorri C. A particle swarm optimizationbased multi robot navigation strategy. In: International workshop on bioinspired robots, 2011.Google Scholar
 Rastgoo MN, Nakisa B, Zakree M, Nazri A. A hybrid of modified PSO and local search on a multirobot search system”. Int J Adv Robot Syst. 2015;12:86.View ArticleGoogle Scholar
 Allawi ZT, Abdalla TY. A PSOoptimized reciprocal velocity obstacles algorithm for navigation of multiple mobile robots. Int J Robot Autom (IJRA). 2015;4(1):31–40.Google Scholar
 Das PK, Behera HS, Panigrahi BK. A hybridization of an improved particle swarm optimization and gravitational search algorithm for multirobot path planning. Swarm Evol Comput. 2016;28:14–28.View ArticleGoogle Scholar
 Das PK, Behera HS, Das S, Tripathy HK, Panigrahi BK, Pradhan SK. A hybrid improved PSODV algorithm for multi robot path planning in a clutter environment. Neurocomputing. 2016;207(C):735–53.View ArticleGoogle Scholar
 Abbas NH, Abdulsaheb JA. An adaptive multiobjective particle swarm optimization algorithm for multirobot path planning. J Eng. 2016;22(7):164–81.Google Scholar
 Nakisa B, Rastgoo MN, Nasrudin MF, Zakree M, Nazri A. A multiswarm particle swarm optimization with local search on multirobot search system. J Theoret Appl Inf Technol. 2015;71:129–36.Google Scholar
 Latombe JC. Robot motion planning. Norwell: Kluwer; 1991.View ArticleMATHGoogle Scholar
 Borenstein J, Koren Y. The vector field histogram: fast obstacle avoidance for mobile robots. IEEE J Robot Autom. 1991;7(3):278–88.View ArticleGoogle Scholar
 Meng H, Picton PD. A neural network for collisionfree path planning. Artif Neural Netw. 1992;2(1):591–4.Google Scholar
 Reynolds CW. A distributed behavioral model. In: Proceedings of the 14th annual conference on computer graphics and interactive techniques, 1987, vol. 21, issue 4, p. 25–34.Google Scholar
 Bouamama S. A new distributed particle swarm optimization algorithm for constraint reasoning. In: Proceedings of the 14th international conference on knowledgebased and intelligent information and engineering systems: part II, Sept 08–10, Cardiff, UK, 2010.Google Scholar
 Hu X, Eberhart R. Solving constrained nonlinear optimization problems with particle swarm optimization. In: 6th world multiconference on systemics, cybernetics and informatics, Orlando, Florida, USA, 2002.Google Scholar
 Deneubourg JL, Beckers R, Holland OE. From local actions to global tasks: stigmergy and collective robotics. In: Proceedings of the fourth international workshop on the synthesis and simulation of living systems, Cambridge, MA, USA, July 1994, p. 181–9.Google Scholar