A Physarum-Inspired Decision-Making Strategy for Multisource Task Searching of Mobile Robots

In the real world, there are many different kinds of sources, such as light, sound, and gas, distributed randomly over an area. Source search can be carried out by robotic system in applications. However, for a single robot, the multisource search has been receiving relatively little attention compared to single-source search. For multisource task searching, a single robot has a high travel cost and is easy to trap a source which has been located before. In order to overcome these shortages, twomultisource search algorithms inspired by the foraging behavior of Physarum polycephalum are proposed in this paper. First, a Physarum-inspired Strategy (PS) is designed based on the gradient climbing characteristic of Physarum polycephalum during foraging. .e PS is simple and effective to let a mobile robot traverse all sources. .en, an extension algorithm named Physarum-inspired Decisionmaking Strategy (PDS) is proposed based on PS. .erein the synthetical field gradient model is established by introducing decision-making factor to obtain more accurate gradient information estimation. .e PDS also introduces an obstacle avoidance model. Various simulation results obtained in the multisource environments show that the performance of PDS is better than other algorithms.


Introduction
Source search is one of the most popular behaviors in the biological world, such as foraging, mate-finding, and cell migration. Strategies behind these behaviors have encouraged the development of mobile robots that can search various sources. Many applications can be found in locating chemical sources leakage [1][2][3], searching and rescuing targets [4,5], and searching for a gas/odor source [6][7][8].
Using robotic system to search for a single source has been broadly researched. However, in the real world, multiple sources are often distributed over an area and need to be discovered [9,10]. We may consider a realistic scenario, such as survivor rescue after an earthquake. e people waiting to be rescued are scattered in different areas, and we can use many autonomous mobile robots equipped with life signal detectors to find all people as soon as possible. In general, multisource search task has distinct challenges for robots [9]: (1) how to continue after each source is found; (2) how to avoid obstacles; and (3) how to find all sources in minimal time.
e methods for source search task can be divided into two categories: single-robot based methods and multirobot based methods [11]. Generally, most of the source search problems have a common characteristic that no matter what kind of source is, the closer the source is, the stronger source signal is. us, the robot can perceive the gradient information of the surroundings to search for the potential source, and a number of gradient-based source search methods for a single robot have been published. e gradient-based source search strategies can be classified into two categories: chemotaxis and biased random walk. e simplest chemotaxis-based algorithm was proposed for a search task [1]. By comparing concentration difference between two chemical sensors or different positions, the robot moves to the direction of higher intensity value at a constant speed. e limitation of this model is that fewer sensors will decrease the accuracy of the chosen direction. A chemotaxis-based source search algorithm takes the wind direction as an important factor [2]. e motion vector of robot is a linear combination of the concentration gradient direction and wind direction. Similar strategies were also presented in [6,12]. Although these methods have achieved in single-source search, they have not been extended to solve multi-source search. Biased random walk is also called bacterial chemotaxis, which is inspired by the foraging behavior of bacteria on the liquid surface. ere is evidence that the gradient-induced search strategy inspired by bacterial chemotaxis can guide a robot to locate the sources [13]. In this approach, the robot makes a decision on either moving straight or turning a random angle by comparing current measured concentrations with previous ones. For the environment that is not fully covered by the source signals, a Yuragi-Based adaptive biased random walk strategy has been proposed to guide a mobile robot to search for a sound source [14]. is method improves success rate and efficiency of source search by combining Bacterial Chemotaxis and Levy walk. Another related study has investigated a bacteria-inspired mobile robot with I-DOF and a single sensor, revealing the relation between the navigation behavior and environmental noise [15]. However, the related algorithms based on biased random walk cannot guarantee the repeatability of path.
Nowadays, multi-robot based algorithms have been used in the multi-source search tasks. An agent-based attractionrejection model has been proposed for searching for multiple sources [16]. However, the model is centralized, which means that the robots know relative positions to each other in real time. A novel 3D pseudolinear Kalman filter (PLKF) and gradient-descent path optimization algorithm are proposed for angle-of-arrival (AOA) target tracking using multiple own-ships [17]. As an extension study, a distributed 3D angle-of-arrival (AOA) target searching and tracking method in three-dimensional (3D) space has been proposed for multiple UAVs [18]. is method consists of a distributed estimator and path optimization algorithm. Particle Swarm Optimization (PSO) as a distributed algorithm has been used in robotic multi-source search task. e related algorithms inspired by PSO consider each robot as a particle, and the communication and cooperation between robots are similar to that of particles in PSO [19][20][21][22][23]. e MPSO is developed that chemotaxis enables the robots to follow a local gradient of the chemical concentration, while anemotaxis-driven PSO measures the direction of wind and guild robot to locate the odor source [19]. e subgroup-based PSO algorithms were proposed to divide swarm robots into subgroups, in which there is a main robot [20,21]. e main robot will attract the members within its team and reject other main robots to ensure different subgroups search for different sources. e Adaptive Robotic PSO for target search has been introduced, considering the obstacle avoidance and local optimal escape [22]. e inertia weight of each robot is adaptively adjusted by evolutionary speed factor and aggregation degree at each iteration. A modified guaranteed convergence particle swarm optimizer (MGCPSO) was presented [23], in which the low fitness robot will give up the role in the current group and to assist the globally best robot.
is algorithm is superior to other algorithms in terms of success rate and iteration time. As the number of robots increases, the multirobot algorithms have the following problems, such as collision, high load of the communication network, and increased computational complexity. Researchers should make effort to find a better way of completing the multisource search task by fewer robots. In addition, there are some cases where the number of sources is much larger than that of robots. In these cases, each robot in swarm robotic system gets desire to maximize the chance of finding at least one source, the chance of finding as many sources as possible within a limited time. erefore, it is essential to have an effective multi-source search strategy for a single robot, in which the robot plans its movement whilst considering environment status and obstacle avoidance. is paper deals with the multi-source search using a mobile robot, considering two aspects: First, robot avoids getting trapped around a source and traverses all the sources. Second, the robot balances the gradient information of each source and makes an autonomous decision action during the search process, which results in reducing the probability of generating the redundant path. e proposed algorithms are inspired by the foraging behavior of Physarum polycephalum in this paper. Physarum polycephalus has been shown to rely on reactive navigation to explore the environment and search for multiple food. A number of foraging models have been proposed [24][25][26][27][28][29]. Related researches illustrate that Physarum has an ability to solve maze navigation [30] and graph theory problems [31][32][33][34]. e contributions of this work are as follows: (1) we propose a novel multi-source search strategy named PS by modifying the grown rule of the Physarum foraging model in [28]; and (2) an extension algorithm named PDS is developed, considering the decision-making of each source and obstacle avoidance. In simulations, the PDS has the best performance compared to other three algorithms: Sweep, Yuragi-based strategy, and PS. e rest of this paper is organized as follows. Section 2 describes problem description. Section 3 presents the PS and PDS, respectively. e simulation results and discussions are presented in Section 4. Finally, Section 5 concludes this paper and discusses future work.

Problem Description
In this paper, the shape of each source is represented as an ellipse. e center of ellipse has the maximum field intensity. Each source diffuses particles from the center to the surroundings layer by layer. e synthetical field intensity at one point is inversely proportional to the distance from the sources. is paper establishes a Four-tuple 〈R, E, S, L〉 to describe the multi-source search task for a robot: (1) where R � 〈f, δ, r〉 represent the properties of robot: where f is the field intensity perceived by robot at the current position. δ is the speed of robot, and r (x, y) is the current coordinate of robot. All the parameters in R are known to robot. (2) where E ⊂ R 2 denote a closed environment. e synthetical field intensity is obtained by equations (1)-(3) [28]. If there is only one source in the environment E, the field intensity at the point n (x, y) is defined as follows: where λ i is the central field intensity of s i , and e is the natural logarithm. d no can be calculated based on equation (2) to denote distance information between the center of s i and the point n (x, y), where a and b are the semimajor axis and semiminor axis of (2) When multiple sources are distributed in the environment, concentration field is the superposition of them. e synthetical field intensity at the n (x, y) is calculated by equation (3), where m is the number of sources:

Methods
In this section, two Physarum-inspired search strategies, PS and PDS, are proposed. e former modifies the growth rules of Physarum foraging model in [28], mainly serving as a benchmark. e robot is equipped with corresponding sensors according to the source types. During the process of searching, the robot perceives the concentration information of the current position in real time. For the latter, we introduce the decision-making factor of each source and obstacle avoidance model.

e Steps of the PS.
is part describes the steps of the PS. e dynamic process of source search based on PS is shown in Algorithm 1, where the initial coordinate of robot r ini (x ini , y ini ) is necessary to activate the model. e robot only determines the next direction of movement according to the field gradient of the current position. On initialization, we set maximum iterations I max and o i (x i , y i ). e synthetical field gradient G fs (n) and the normalized vector N fs (n) at the point n (x, y) can be given by equations (4) and (5), respectively (line 4-5): e robot moves along the synthetical field gradient. e position of robot at the next iteration is updated by equation (6) (line 6): where δ denotes the speed of robot and is dimensionless. If the condition meets the ‖r − oi‖ ≤ min a i , b i , it means that s i is accessed (line 8). e robot will mark the current position as a new start point for searching for the next source. Ignoring the exploitation time at s i , the robot immediately enters the exploration stage to search for other sources. At the same time, we set the central field intensity value of s i to 0 and the S eliminate s i (line 9). It results in updating the synthetical field intensity distribution. e robot relies on the updated synthetical field intensity distribution to search for the next source autonomously, which prevents the repeated search for the same source. If the robot cannot perceive any field intensity (S � ϕ) or t > I max , the algorithm will terminate (line 11). Finally, a trajectory is obtained while the robot traverses all sources within a limited time or the task fails. e trajectory generated by the robot is a curve containing a start point and an end point.

e
Steps of the PDS. In PS, this reactive search strategy lacks decision-making for each source and obstacle avoidance in the search process. During the food search process, Physarum exhibits two distinct capabilities: decision-making on food and repellent stimuli avoidance. Inspired by these foraging behavior characters of Physarum, the PDS based on PS is proposed. In the PDS, we remodel the synthetical field gradient by introducing decision-making factor for each source to improve the accuracy of gradient estimation and establish obstacle field gradient model to integrate into the synthetical field gradient.
At first, the original gradient component for each source at point n (x, y) is calculated by We defined that the field gradient for each source of the robot perception is the original gradient multiplied by its own decision-making factor. erefore, the field gradient for all sources is calculated by

Complexity 3
where function m is the number of sources. p i is the decisionmaking factor for each source and endows the robot with the ability to balance the quality information of each source. In other words, during the robot movements without sensing obstacles, the robot will consider the proportion of field gradient of each source in synthetical field gradient and decide to choose the most appropriate direction of next move. p i is defined as follows: To avoid obstacles, an obstacle field gradient model is given by is the center coordinate of obstacle j. As shown in Figure 1, in order to not change the movement trend of the robot, the influence range of obstacle is local. ρ is the shortest distance from robot to the obstacle outline. c is the redundancy parameter, which denotes the influence range of obstacle. λ j is the central field intensity of obstacle and is a negative value. p j is decision-making factor for the obstacle j, which is calculated by where ‖g f i (n)‖ and ‖g f j (n)‖ is modulus of original field gradient component of the source i and obstacle j, respectively. Finally, we remodel the synthetical field gradient as follows: In Figure 2, we show the flowchart of source search based on PDS, and the details are described in Algorithm 2. In each iteration, the robot performs source search and obstacle avoidance based only on the gradient of the current position (line 4-line 10). In the applications, once the robot enters the influence range of obstacle, the robot with distance measuring sensors (laser-ranging-sensor or camera, etc.) converts the distance information into the gradient information and integrate it into the synthetical field gradient. e robot can rely on the synthetical field gradient to move until the termination condition is triggered.

Local Optima Escape.
When there are multiple sources in the environment, local optima points may be distributed in the map. Local optima points can be caused by a combination of sources. e original synthetical field gradient G fs (n) obtained by the robot in the local optima point is (0, 0). erefore, to obtain a valid successful path and ensure that all sources are accessed, it is necessary to provide an escape strategy. In this situation with local optima points, we introduce a fluctuating noise ε to the robot which is similar to the biological fluctuation that exists in organisms. When the robot falls in the local optima point, ε becomes dominant and the robot transforms into a "stochastic state." In stochastic state, the robot changes the direction randomly. We can construct kinematic equation of robot to describe the escape strategy which is as follows: where _ α(t) represents the amplitude of change in the direction of movement. θ is the movement direction of a robot when it is trapped in a local optimal position. ε affects the rotation of robot angular velocity and is set as zero-mean Gaussian noise. e triggering condition of escape strategy is that the robot can perceive the synthetical field intensity but the original synthetical field gradient is (0, 0).

Simulation Results and Analysis
To verify effectiveness of the proposed algorithms, we present several simulations with respect to source types, different center field intensities of source, and environments with obstacles. ese simulations are presented in four parts. Part 1 studies the impact of the different source models by comparing the trajectory of PDS with PS. e source properties are shown in Table 1. Part 2 studies the performance comparison of PS and PDS under different central field intensities. Part 3 studies several simulations in environments with obstacles. Part 4 studies comparison of statistical efficiency value for PDS and other algorithms. Related parameters, we set I max to 2000 and δ to 0.1. Each source is programmed as an ellipse.
is paper does not consider the real physical shape of the robot. e robot is represented by a diamond just to make the simulation effect more intuitive. e environments are unknown and have a size of 20 * 20. We consider the terrain flat, without considering slopes, holes, or any other external factors.

e Comparisons under Different Source Models.
In this part, it will be shown that how a robot accomplishes source search task based on PS and PDS under different source models. In Figure 3(a), the field intensity distribution of source is Gaussian-like shape.
e field intensity at the center of the source is the maximum and decreases nonlinearly outward from the source center. In Figure 3(b), there are six sources in the environment, the synthetical field intensity model based on equation (3), wherein we set all sources to have the same central field intensity λ i � 1, ∀i ∈ {1, 2, . . ., 6}. e field intensity value is 1.5 > 1 at some points, because the synthetical field intensity distribution is the superposition of the field intensity distribution of each source. In Figures 3(a) and 3(b), the number and locations of sources can be directly observed.
For the multi-source with Gaussian-like shape field intensity distribution, a comparison is made between PS and PDS. e process of searching for sources based on PDS is shown in Figures 4(a)-4(g). In Figure 4(a), r ini (1, 1) is the initial position of the robot. e robot accesses the s 2 at t � 43. Ignoring the exploitation time, the robot starts to search for the next source as shown in Figure 4(b). During the search process of s 2 ⟶ s 6 , s 2 will not contribute field intensity to any point in the environment. is strategy will increase the probability that the robot accesses the remaining sources and prevent the robot from trapping a source. Figures 4(c)-4(g) are the segments of robot movement process from t � 52 to t � 333. Finally, the robot generates a trajectory that traverses all sources. e access sequence is s 2 ⟶ s 6 ⟶ s 3 ⟶ s 1 ⟶ s 4 ⟶ s 5 . Figure 4(h) represents the trajectory comparison between PS and PDS. e trajectories of r ini ⟶ s 2 ⟶ s 6 ⟶ s 3 (from t � 0 to t � 72) have little difference, because {s 2 , s 6 , s 3 } are aggregated, and the joint gradient contributed by them accounted for the largest proportion in the synthetical field gradient. During searching the first three sources, the robot based on the two algorithms makes not much difference in direction selection in each iteration. After accessing {s 2 , s 6 , s 3 }, the robot starts to search for the remaining sources which are scattered. e most obvious difference based on the two algorithms is the trajectories of s 3 ⟶ s 1 , and the trajectory generated by robot based on (1) Initialize input a Four-tuple 〈R, E, S, L〉, r ini (x ini , y ini ), I max and t � 0 (2) For t � 1: I max (3) Calculate f s (n) based on equation (3)  (4) Calculate g f i (n) based on equation (7)  (5) Calculate p i based on equation (9)  (6) Calculate p j based on equation (11)  (7) Calculate G p f s (n) based on equation (8)  (8) Calculate g p f j (n) based on equation (10)  (9) Calculate G p f sj (n) based on equation (12)  (10) Choose the new position of the mobile robot for next move And t � t + 1 (12) If ‖r − o i ‖ ≤ min a i , b i then (13) S eliminate the s i (14) End if (15) If S � φ‖I ≥ I max (16) Break (17) End if (18) End for ALGORITHM 2: e PDS-based source search. 6 Complexity PDS is straighter. We can conclude that when the robot traverses the same sequence, the performance of PDS is better than PS. e complexity of multi-source search mainly depends on the robot platform and source types. In the absence of background flow, the field intensity distribution of the source with chemical characteristics is a Gaussian-like shape distribution, while the light, RF, sound, etc. are modeled on an inverse square law distribution. e general search strategy is generally applicable to multi-source search tasks under different source types. If the source is light source, the field intensity model follows: Equation (7) is replaced by equation (15) which denotes the original gradient component: e synthetical field intensity distribution is formed by six sources, as shown in Figure 5(a). However, it is difficult to directly observe the number and locations of the sources from a surface diagram. In Figure 5(b), the trajectories for two algorithms are similar to the Figure 4(h). In other words, both algorithms are versatile for searching for the light sources.

e Comparisons under Different Central Field Intensities.
In this part, we study the performance of PS and PDS under different central field intensitieas on robotic source search task. All simulations were carried out using 20 sources. e central field intensity values of 60%, 20%, and 20% of the total number of sources are set to 1, 1.2, and 1.5, respectively. e locations of the sources are randomly generated, and there is no intersection between sources. In Figure 6, the results show that both algorithms are effective for different source types with different central field intensities. In Figures 6(a)-6(d), it can be concluded that the trajectories for PS have more intersections than the corresponding ones for PDS. In other words, the trajectories for PDS are more reasonable. e relevant statistical efficiency analysis will be discussed in Section 4.5.

Obstacle Avoidance Analysis.
is part presents several simulations in the environments with obstacles. e obstacle is represented by a circle (a � b), and the central field intensity value λ j is set to be opposite of that of the source. c is set to 1/4 length of the sensing radius of robot, where c � 0.25. Two set of comparison simulations were designed for different obstacle radiuses r j . Although the obstacle provides repulsive gradient, the robot based on PS still cannot plan a reasonable direction of movement when encountering obstacles. In Figures 7(a) and 7(c), the robot hovers back and forth between the previous location and the obstacle center, and cannot avoid obstacles. In Figures 7(b) and 7(d), the robot based on PDS with decision-making factor has the ability to avoid obstacles, and the trajectories are smooth. On the other hand, it demonstrates that the PDS can also cope with complicated environment with obstacles.

Local Optima Problem.
We construct a special case in which the robot is trapped in local optima point. For this simulation, the relevant parameters are shown in Table 2. In Figure 8, there is a point where the original synthetical field gradient is (0, 0) between two sources. In both PS and PDS, the position of robot is (9,9). However, the robot falls in the local optima trap as shown in Figure 9. e trajectories for PS and PDS with escape strategy (with ε) are shown by Figures 10(a) and 10(b), respectively. e robot successfully escapes the local optima point. e robot can retrieve a nonzero field gradient information and continue its search task. It is important to note that the robot judges whether all the sources have been accessed by perceiving the field intensity information rather than the field intensity gradient information.

Algorithm Evaluation.
In this section, we present the statistical efficiency values for PDS compared with that for other algorithms in a closed environment without obstacles. First, a simple evaluation function is introduced. en, the comparison results of statistical efficiency value for various search strategies are given.
For the same source type, even the sources are distributed at different locations, the nature of all sources is similar. A simple and general statistical efficiency evaluation function is given in [10,35] as follows:   where N is the number of sources found. L is the trajectory length. Since the robot with constant speed δ, equation (16) can be expressed as η � 1/(δ · 〈t se 〉), 〈t se 〉 is the average time between successive encounters. is efficiency function does not depend on the number of sources but on the distribution of sources. erefore, the evaluation function can reflect the efficiency of the algorithms, and the higher η value means the better performance. e base reference is the Sweep strategy which requires a robot move from left to right and back and visit every point only once. e robot uses the Sweep strategy to carry out local environmental coverage and search for sources. e robot stops when traverses all sources. We conducted related simulations for the comparison of statistical efficiency for PDS, the Sweep strategy, Yuragi-based strategy [14], and PS. For the Yuragi-based strategy, the noise parameter obeys ε ω (t)∼N (0, 1) and Levy does not occur because the source particles completely cover the map.
For a scenario for specific number of sources, each algorithm is executed 30 times to reduce the influence of stochastic factors on the performance. e source locations are randomly generated each time. η represents the mean statistical efficiency value. For all sources with central field intensity set to 1, 960 simulations are conducted: number of    Complexity algorithms (4) × number of the source number schemes (4) × number of source types (2) × number of simulation executions for a specific source number scheme (30). e comparison results of η are shown in Figure 11. For different source types, we can conclude that the η values of PDS are higher than that of the Sweep strategy, Yuragi-based strategy, and PS, respectively. is means that the robot with decisionmaking ability performs the best. In Figures 11(a) and 11(b) is a common trend that the η value increases with the number of sources. at is because η value depends on the location distribution of sources. In a closed environment, all the source locations are randomly generated and have no intersection area with each other. e greater number of sources along with the shorter mean time between successive encounters results in a larger η value. For the sources with different central field intensities, the central field intensity values of 60%, 20%, and 20% of the total number of sources are set to 1, 1.2 and 1.5, respectively. 480 simulations are carried out: number of algorithms (4) × number of the source number schemes (2) × number of source types (2) × number of simulation executions for a specific source number scheme (30). e comparison results of η are shown in Figure 12. Because all center field intensities are not completely uniform, we consider two schemes with 10 and 20 sources for simplicity. In Figures 12(a) and 12(b), the change trend and comparison results of η are similar to that described above. e PDS improves the search efficiency of the robot in various conditions by introducing decision-making factor in the synthetical field gradient. It should be noted that the evaluation function is no longer applicable when the central field intensity of the source changes with different magnitudes over time, which is beyond the scope of this paper.

Conclusions
In this paper, inspired by the foraging behavior of Physarum polycephalus, we establish two bio-inspired strategies for multi-source search task. e PS is proposed as a benchmark, which is efficient and easy to implement in an obstacle-free environment. e PDS based on PS has taken decision-making for each source and obstacle avoidance into account. e advantage of the two proposed algorithms are that the robot does not need to know the location of all sources, but only perceives the field intensity at the current position and estimates the gradient to determine the next action. e proposed algorithms prevent the robot from relocating the same source which has been located before. Simulations under different source types and central field intensity show that PS and PDS have certain versatility. Compared with PS, PDS can reduce redundant paths and endow the robot with obstacle avoidance capacity, improving greatly the search security. Furthermore, the related simulation results show that PDS has the best statistical efficiency value compared with other three methods.
Also, as future work, one research direction is multisource searching task in such condition, and there is no source signal in some areas of the environment. In addition, this paper focuses on a single robot, and the algorithms can be extended to swarm robotics system by increasing the cooperation between them.
Data Availability e simulation data and codes used to support this paper have not been made public since related information is related to personal privacy of academic achievements of each author.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.