Review ArticleThe continuous-discrete PSO algorithm for shape formation problem of multiple agents in two and three dimensional space
Graphical abstract
Introduction
Recently, the formation problem is one of the interesting and important topics in the fields of robotics, evolutionary computation, computer science [1], control system and the consensus in the multi-agent systems [2], etc. The related theory of formation problem can be further applied to airplanes, automobiles, ships and agents in the civilian and military systems. Most of the existing results, by using the theory of the consensus in multi-agents system, mainly analyze and design the control law to achieve the desired shape of several agents (e.g. [[3], [4], [5], [6], [7], [8], [9], [10]]; see also [11] for a brief overview to robot formation). Ren and Beard [3] provide that the agent's status can be asymptotically achieved under dynamically topologies when the union of the directed graphs has a spanning tree in the T steps. Jadbabaie and Lin [4] provide the theoretical explanation on the coordination of agents by using the nearest neighbor rule. Olfati-Saber [5] gives two cases of flocking in free space and in the presence of the obstacles, in addition, the author presents three flocking algorithms and interaction rules to achieve the consensus of agents. Tanner and Jadbabaie [6] provide the coordination control rule in the fixed topology and time invariant, which is the combination of attractive/repulsive and alignment forces. Tanner and Jadbabaie [7] also provide the control law of the agents in the dynamic topology, mainly depending on the state of the neighbor agents. Olfati-Saber [8] discusses the control law on the flocking of multiple agents in the presence of several obstacles. Lafferriere and Williams [9] design the decentralized control law of vehicle formation. Olfati-Saber and Murray [10] provide the control law on the consensus of agents under the switching topology and time-delay. Chen and Wang [11] summary the current control issues and the corresponding strategies on the group UAV/robots formation, and provide several considerations on formation control. In summary, the above-mentioned works mainly study the formation problem from the optimal control viewpoint.
In addition, the existing results also concentrate on the trajectory planning from the initial point to the destination point. Chen and Li [12] search for the optimal trajectory between the initial point and the known destination in the presence of static and moving obstacles. Prasanna and Saikishan [13] design a co-ordination path planning method, which is based on particle swarm optimization method, can generate two cooperative trajectories from the initial points to the known destinations. Liu and Ma [14] provide the hybrid PSO algorithm with convex hull for overcoming optimal ships formation problem in the realistic application, where the number of ships or agents is smaller than 15 in the different shapes. Additionally, Liu and Ma [15] also utilize the continuous and discrete PSO algorithm to handle the optimal formation problem in the three-dimensional space. Yoshida and Fukushima [16] provide one control method for mobile agents formation based on local optimization. In terms of the above-mentioned research works, they mainly focus on the control law to achieve the desired shape of several agents or design the optimal trajectory of each robot from the initial position to the known destination.
To the best knowledge of the authors, few researchers from the efficiency viewpoint concentrate on the performance index of formation problem considering the time factor and the distance factor in the presence of the unknown desired positions of agents. On this background, this paper from the perspective of optimization focuses on a class of formation problems which are called the shape formation problem from the initial shape to the desired shape. To form the desired shape of agents, those positions in the desired shape should be yielded to several constraints, therefore, optimal shape formation problem of agents is essentially classified as the constraint optimization problem with some objectives. Because of difficulties in handling with constraints, a new method is utilized to become the continuous problem with several constraints into the continuous-discrete problem without any constraints. In order to handle the above-mentioned problem, the CDPSO algorithm is provided to seek the center point and the initial rotated angle of the desired shape, together with the matching pair between the points in the initial shape and those of the desired shape.
The PSO algorithm is originally developed by Kennedy and Eberhart in 1995 [[17], [18]] and originates from the behavior of finding the food of swarm animals, such as the birds, the fish, etc. In addition, the PSO algorithm has the following merits [[19], [20]]. Firstly, the common method of gradient algorithm greatly depends on the initial positions of all particles in the solution space, however, the effectiveness of PSO algorithm is not greatly dependent of the initial points of all particles. Secondly, it is not required that the gradient of the objective function is defined over the whole solution space. Thirdly, the code of PSO algorithm is relatively simple and the efficiency of searching for the suboptimal or global optimum is relatively high according to one previously reported result [20]. Because of the aforementioned advantages, PSO algorithm has been successfully and widely applied to a broad range of optimization problems, such as electric power system [[21], [22], [23], [24]], electromagnetic [25], locating and tracking [[26], [27], [28], [29]], intelligent control [30], neural network [[31], [32]], fault detection [[33], [34]], feature selection [[35], [36], [37]], path planning [38] and others [[39], [40]], etc.
In this paper, the shape formation problem, which can be applied to the helicopter and ship formation and the large-scale performance in the future can be roughly classified by two main cases. The first case is to discuss the shape formation problem of agents by the Lagrangian multiplier method, when the number of agents is equal to 3 and the number of constraints is equal to 1. While the second case is to solve the shape formation problem by the continuous and discrete PSO algorithm, when the number of agents is strictly larger than 3 and the number of constraints is larger than 2. Specifically speaking, the Lagrangian multiplier method can provide the optimal positions in the shape formation problem with three agents and one constraint, and the Lagrangian multiplier method only provides the necessary condition of the equilateral triangle formation problem. However, the Lagrangian multiplier method cannot address the shape formation problem when the number of agents is strictly larger than 3 and the number of constraints is larger than 2. Therefore, in order to overcome the complicated and challenging problem, the CDPSO algorithm is utilized to handle the shape formation problem of several agents. From the perspective of the optimization, the optimal shape formation problem can be classified as the continuous-discrete optimization problem, and the CDPSO algorithm has two main subtasks. One main subtask of CDPSO algorithm is to search for the center point and the rotated angle of the desired shape, such as the equilateral triangle, regular polygon, ellipse shape and square shape, etc. Another main subtask of CDPSO algorithm is to handle the marriage problem, which matches between points in the initial shape and points in the desired shape. Therefore, the CDPSO algorithm needs to optimize the continuous parameters including the center point and the rotated angle, together with the marriage pair between points in the initial shape and points in the desired shape. More importantly, the convergence analysis of three CDPSO algorithms is discussed from the viewpoint of Lyapunov theory. Finally, the CDPSO algorithm can be also applied to the conversion from the disorder shape to the order shape in the two and three dimensional space, together with shape conversion between two typical shapes.
The rest of this paper is organized as follows. Section 2 introduces the brief description of some assumptions and basic concepts in the case of the shape formation problem. By the Lagrangian multiplier method, two theorems and some important remarks on the shape formation problem with three agents are given in Section 3. The 2D/3D mathematical model of typical shape and the corresponding objective function are introduced in Section 4. It is important to describe the CDPSO algorithm and its convergence analysis of CDPSO algorithm with random variables in Section 5. To show the effectiveness and high performance of the CDPSO algorithm, simulation results focus on the shape formation problem in the two and three dimensional space in Section 6. The improved CDPSO algorithm solves the formation problem in the two dimensional space when the number of agents is equal to 100, 200, 500 and 1000 in Section 7. The computational time and the optimized results are compared among several different CDPSO algorithms in Section 8. Finally, Section 9 summarizes the interesting results and the future works with respect to shape formation problem of multiple agents.
Section snippets
Basic concepts of optimal shape formation problem
Shape formation problem can be viewed as a class of formation problems from the initial shape including agents to the desired shape including agents, and the corresponding objective function is to minimize the whole moving distance. The task of the optimal shape formation problem is to search for the continuous parameters including the center and the rotated angle, together with the discrete parameter including the matching pair between points in the initial shape and points in the desired
Shape formation problem with three agents by Lagrangian multiplier method
The shape formation problem is one of the challenging problems from the perspective of optimization. When the number of agents is equal to 3 and the number of constraints is smaller than 2, the shape formation problem can be solved by the Lagrangian multiplier method to obtain the corresponding analytic solution. However, the above-mentioned problem cannot overcome by the Lagrangian multiplier method, when the number of agents is larger than 3 and the number of constraints is larger than 2. In
The mathematical model of several typical shapes with agents in two or three dimensional space
The objective of this section is to introduce the mathematical model on some typical shapes in the two dimensional space and other shapes in the three dimensional space, such as the regular shape, the triangle shape, the square shape, the pyramid shape, etc. Additionally, the mathematical model of shape is composed of those parameters including the center point (xc, yc) and the initial angle ζ.
The CDPSO algorithm and its convergence analysis
As can be seen from the aforementioned results, the desired coordinates of three agents can be calculated by the Lagrangian multiplier method when the number of constraints and agents is equal to 3. However, it is very hard to calculate the desired coordinates of agents when the number of agents is strictly larger than 4 and some constraints are closely related to nonlinear functions with high order. In order to handle this problem, this section is mainly to provide a heuristic swarm
Simulation results
To demonstrate the effectiveness of the CDPSO algorithm, this section is to mainly study the shape formation problem in the two and three dimensional space. A detailed evolutionary process, which is closely related to objective function and the convergence speed, is described to introduce the CDPSO algorithm. In addition, the shape conversion between two typical shapes in the two and three dimensional space is solved by the CDPSO algorithm.
Shape conversion problem of multiple agents
The shape conversion problem is to discuss that the initial shape with multiple agents changes to another shape with multiple agents, leading to the smallest sum square distance. When the objective function is selected to be Eq. (6), the center of the initial shape is equal to the center of the desired shape. The objective of this section mainly to analyze and optimize the shape conversion problem, when the number of agents is larger than 100 and the desired shape is set to the regular shape.
Comparison with other existing CDPSO methods
The shape conversion problem is the hybrid optimization problem, which includes the continuous parameter and the discrete parameter. Therefore, this subsection is to optimize the continuous parameters and the discrete parameters by several above-mentioned algorithms, which are composed of CDPSO, CDDE, CDGA and the improved CDPSO algorithms. The number of particles in different formation methods is set to 20, and the maximum number of iterations is set to 1000, additionally, the generation
Conclusion
This paper mainly utilizes the CDPSO algorithm to handle the shape formation problems of agents in the two and three dimensional space. It not only introduces the definitions and the assumptions of the optimal shape formation problem to better describe formation problem of agents. From the viewpoint of theoretical analysis, this paper provides two theorems on the shape formation problem with three agents by the Lagrangian multiplier method and three theorems in the case of convergence analysis
Acknowledgements
This research is partially supported by Natural Science Foundation of Beijing under Grant No. 17D30118 and the National Natural Science Foundation of China under Grants No. 61433003, No. 61473038 and No. 91648117. Additionally, this work is also supported by Project funded by China Postdoctoral Science Foundation (2014M550988), China Academy of Railway Sciences (2015YJ080) and China Railway Corporation (2017X001-D).
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