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 (D2PSO), 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 [7,8,9] 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 PSO-based realization.

The authors present in [11] PSO-based 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 collision-free 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 real-world 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 A-star algorithm. Nakisa proposes a method based on the multi-swarm 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 (ML-PSO)” applied in the exploration search space by adding a local search algorithm such as A-star 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 (IPSO-DV) 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 multi-objective particle swarm optimization algorithm (AMOPSO) for five robots to reach the shortest path. The algorithm PSO-NAV 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 maze-like 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.

Multiple robot path planning problem is formulated by considering the set of principles using the following assumptions:

  1. 1.

    For each robot, the current position (recent position) and goal position (target position) is known in a given reference coordinate system.

  2. 2.

    Each robot is performing its action in steps until all robots reached in their respective target positions.

The following principles have been taken care of for satisfying the given assumptions.

  1. 1.

    For determining the next position from its current position, the robot tries to align its heading direction toward the goal.

  2. 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. 3.

    If a robot can align itself with a goal without collision, then it will move to that position.

  4. 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 z-axis.

Obstacle avoidance approach

Plenty of algorithms for obstacle avoidance were mentioned in the robotic literature [23,24,25].

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).

To solve the latter problem, we propose a method which is able to detect collisions between the robot and an object (figure 1). Let us define a repulsive function \(F_{o}\) in a region \(Z\) around an obstacle \(O\). The region \(Z\) is defined as a circular disk centered at \(r_{o}\) with radius \(\rho_{O}\); the parameter \(\left( {\rho_{O} + \varepsilon } \right)\) is the minimum distance that the robot should keep with respect to the boundary of the obstacle \(O\). \(\varepsilon\) represents the minimum allowed distance between the robot and the obstacle. The repulsive function can then be defined out of Eq. (1), where \(\left( {x, y} \right)\) is the position of the robot and \(\left( {r_{ox} , r_{oy} } \right)\) is the obstacle position.

$$F_{o} = \sqrt {\left( {x - r_{ox} } \right)^{2} + \left( {y - r_{oy} } \right)^{2} }$$
(1)

Afterward, we define a Boolean function \(\delta_{ij}\) described in (2), where i refers to the ith robot and j refers to the jth obstacle in the environment.

$$\delta_{ij} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\text{if}}\; {F_{{o_{ij} }} \le \varepsilon } \hfill \\ 0 \hfill & {\text{Otherwise}} \hfill \\ \end{array} } \right.$$
(2)

Robots must be able to handle limited sensing range for the obstacles through considering the latter function value in order to check collision (Fig. 1).

Fig. 1
figure 1

Selection of next position (x next i , y next i ) of robot i from current position (x curr i , y curr i ) for avoiding collision with obstacle (r o , ρ O )

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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.

Each of these particles is provided with:

  • 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.

At every moment, each particle knows:

  • 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.

After finding the two best values, the particle updates its velocity and positions with following equations:

$$v_{n + 1} = wv_{n} + c_{1} *{\text{rand}}*({\text{pBest}}_{n} - p_{n} \, ) + c_{2} *{\text{rand}}*({\text{gBest}}_{n} - p_{n} ),$$
(3)
$$p_{n + 1} = p_{n} + v_{n + 1} .$$
(4)

where w is the inertia coefficient which slows velocity over time; v n is the particle velocity; p n is the current particle position in the search space; pBest n and gBest n are defined as the “personal” best and global best; rand is a random number between (0, 1); c 1 and c 2 are the acceleration coefficients. The stop condition is usually the maximum number of allowed iterations for PSO to execute or the minimum error requirement. As with the other parameters, the stop condition depends on the problem to be optimized.

D3GPSO: the dynamic distributed double guided particle swarm optimization algorithm

The D3GPSO 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 D3G2A [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 SpecieFVR. 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 min-conflict 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 D3GPSO, 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.

D2PSO: 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 D3GPSO described in [4, 27], we introduce two new parameters to the PSO: local optima detector for global best LODgBest and local optima detector for personal best LODpBest. 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 (D2PSO) 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 D2PSO would follow the PSO’s behavior for the rest of situations.

The flowchart for multiple robot path planning using D2PSO is presented in Fig. 2.

Fig. 2
figure 2

Flow chart of multiple robot path planning using D2PSO

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Fig. 3
figure 3

Obstacle avoidance of 10 robots moving in the same environment (rectangles = initial positions, stars = goal positions)

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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 sub-process is trapped in a local optimum, and so the LODpBest will increment, respectively, for gBest and LODgBest.

Dynamic concept

Our work consists in implementing a dynamic distributed version of PSO. It is a multi-agent 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].

Particles whose pBest is not improved for a predefined threshold, i.e., LODpBest = S p , would be restructured as pBest_temp given in (3). Respectively, for gBest, i.e., LODgBest = S g , would be restructured as pBest_temp given in (4).

$${\text{pBest}}\_{\text{temp}}_{i} = {\text{pBest}}_{{i_{1} }} + \frac{{i_{1} }}{{i_{1} + i_{2} }}*({\text{gBest}}\_{\text{hist}}_{{i_{2} }} - {\text{pBest}}_{i} ),$$
(5)
$${\text{gBest}}\_{\text{temp}}_{i} = \hbox{min} ({\text{pBest}}\_{\text{temp}}_{i} ,{\text{gBest}}_{i} ),$$
(6)

where i 1 = random (1, M), M is the population size, i 2 = random (1, size(gBest_hist), and gBest_hist presents the historical values of gBest.

Template concept

The concept of template was introduced by Tsang [29]. In our approach, inspired from D3GPSO [27], we will attach particles whose pBest is not improved for successive generations to a template called templatepBest and respectively particles whose gBest is not improved for a predefined threshold to a template called templategBest. Hence, the new robot path will be influenced by the best reached fitness since each robot would perform its own PSO algorithm guided by template concept [29].

$$\alpha = {\text{template}}_{\text{pBest}} /\left( {{\text{template}}_{\text{pBest}} + {\text{template}}_{\text{gBest}} } \right)$$
(7)
$$\beta = {\text{template}}_{\text{gBest}} /\left( {{\text{template}}_{\text{pBest}} + {\text{template}}_{\text{gBest}} } \right)$$
(8)

As mentioned in [27], we define in (8) and (9) the parameters α and β which are function of templatepBest and templategBest. 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 soft-bots of rectangular shape with different path color code. The robot is self-contained 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.

The objective function used in the simulation studies was a spherical function given by:

$$f = (x_{i} - x_{\text{target}} )^{2} + (y_{i} - y_{\text{target}} )^{2} ,$$
(9)

where (x i , y i ) is the position of the both and (x target, y target) is the position of the target point. The latter is given in Table 1. The spherical function is chosen because it approximates the expected dissipation pattern of chemicals, heat, etc., that would be emitted by real-world targets [9].

Table 1 Initial and target points used in simulation
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We have applied the distributed PSO and the proposed D2PSO 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.

We did the simulations in an environment with variant number of obstacles. Table 2 gives their positions.

Table 2 Description of obstacles present in Fig. 3
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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 D2PSO, ensuring of course the non-collision with the static predefined obstacles (Figure 3).

The overall effectiveness of the proposed algorithm is shown in Table 3. It confirms that it outperforms the dPSO with respect to the performance metric, i.e., the path length, for different number of robots (figures 4 and 5).

Table 3 Path lengths for M = 150
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Table 4 represents the same previous simulations for M = 300 particles. It is clear that we obtain shorter paths for each robot (Figs. 4, 5).

Table 4 Path lengths for M = 300
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Fig. 4
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Optimal path of 3 robots using D2PSO

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Fig. 5
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Optimal path of 5 robots using D2PSO

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Figures 6 and 7 show that the convergence of the objective function to the best value is faster with D2PSO approach than dPSO one.

Fig. 6
figure 6

Best cost variation for each robot when applying D2PSO (#robots = 5)

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Fig. 7
figure 7

Best cost variation for each robot when applying dPSO (#robots = 5)

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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.

After that, we have added more obstacles and made some changes in initial and target states. We obtained Figs. 8 and 9 for fixed number of robots.

Fig. 8
figure 8

Optimal paths of 7 robots using D2PSO with five obstacles

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Fig. 9
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Optimal paths of 7 robots using D2PSO with three obstacles

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Moreover, the result of the experiments performed is presented in Table 5 in the term of another performance metric which is the executed time to reach the best solution, i.e., the robot’s shortest path. We run the program with two different values of the population size M (Table 6). This time, the execution time would necessary be bigger as the number of particles increases. Table 5 and Fig. 10 confirm that D2PSO outperforms the remaining algorithm with respect to the cited metric for different robots.

Table 5 Execution time for M = 150
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Table 6 Execution time for M = 300
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Fig. 10
figure 10

Comparison of time taken to reach best solution in dPSO and D2PSO

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In addition, we have observed, for each algorithm, the number of iterations each robot takes to find its shortest path. Since the execution time is lower for D2PSO than dPSO, number of iterations would consequently be less. Table 7 confirms the latter observation.

Table 7 Comparison of number of iterations taken to obtain the shortest path by the different algorithms
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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. D2PSO 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 D2PSO 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 collision-free 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 inter-robot collision should be considered in future experiments. The task planning process for MRS would be also studied in order to ensure best coordination.