Mathematical Modeling and Analysis of Multirobot Cooperative Hunting Behaviors

This paper presents a mathematical model of multirobot cooperative hunting behavior. Multiple robots try to search for and surround a prey. When a robot detects a prey it forms a following team.When another “searching” robot detects the same prey, the robots form a new following team. Until four robots have detected the same prey, the prey disappears from the simulation and the robots return to searching for other prey. If a following team fails to be joined by another robot within a certain time limit the team is disbanded and the robots return to searching state. The mathematical model is formulated by a set of rate equations. The evolution of robot collective hunting behaviors represents the transition between different states of robots. The complex collective hunting behavior emerges through local interaction.The paper presents numerical solutions to normalized versions of the model equations and provides both a steady state and a collaboration ratio analysis. The value of the delay time is shown through mathematical modeling to be a strong factor in the performance of the system as well as the relative numbers of the searching robots and the prey.


Introduction
It is critical to model multirobot cooperative behavior when analyzing the multiagent system [1,2].The first model is a mathematical model which will provide us with computer simulations of the robot's essential properties, while the second is a model which will retain the important aspects of the system operation and provide an essential tool for the analysis of multirobot system.Up till now, most researches focus on the implementations of the real systems.The control strategies are improved ceaselessly by simulations or experiments in a specific context until the robot system can accomplish the specific task successfully [3].Generally, the methods mentioned above can prove that the robot systems are meeting the specific tasks.But there is no guarantee that the control strategies can be implemented well on different robot platforms or under the different environmental conditions.The mechanisms of interactions between robots and environment remain enigmatic based on the real systems.Likewise, the behavior of robot systems cannot be modeled using mathematic methods, and it is difficult to analyze the behavior quantitatively [4].
The mathematical models of robot cooperative behaviors are the abstract of multirobot systems in the real world [5].The interaction between robots and environments can be better understood through the research of mathematical models [6].Many works can be found that attempt to model robot cooperative behaviors.The mathematical methods can help us to find the major factor for the system behaviors, and to enhance the understanding of emergent properties of robot collective behaviors.Probabilistic robotics is concerned with perception and control in the face of uncertainty [7].Thrun [8] describes a methodology for programming robots known as probabilistic robotics.The central conjecture is that the probabilistic approach to robotics scales complex real-world applications better than approaches that ignore a robot's uncertainty.Burgard et al. [9] presented a probabilistic approach for the coordination of multiple robots.The proposed method simultaneously takes into account the costs of reaching a target point and the utility of target points.Fox et al. [10] presented a statistical algorithm for collaborative mobile robot localization.The approach was capable of localizing mobile robots in an any-time fashion by using a sample-based version of Markov localization.
In robotics, there is a small variety in mature models to support the algorithm design [11].The behaviors of individual robot in a swarm are triggered by many complex influences.From a micro perspective, the evolution of individual behavior is a stochastic process for the limits of sensors, noise, and unpredictability of environment [12].The evolution of robot collective behaviors can be regarded as a conversion process under different states of multirobot system.A model can be described by some variables, which include many behaviors or actions of robots.In reactive or behavior-based robotics, the future state of systems depends only on its present state and is conditionally independent of the previous states.Therefore, the evolution of multirobot collective behaviors is a Markov process [13].The rate equation can describe a multirobot system at a macroscopic level, which includes an increase in the occupation number due to transitions to a state from other states and the loss due to the transitions to other states.The rate equation has been widely used to model dynamic processes of robot collective behaviors [14].It is more computationally efficient and theoretically analytical, because it uses fewer variables.Martinoli et al. [15] proposed a time-discrete, macroscopic model able to capture the dynamics of a robotic swarm system engaged in a collaborative manipulation task.Lerman et al. [16] presented a mathematical model of foraging in a homogeneous group of robots and analyzed the behavior of the system, focusing on quantitative characterization of the effects of interference on the performance of the group.
A mathematical model of multirobot cooperative hunting behaviors is proposed in this paper.The mathematical model is established at state level based on the macroscopic modeling.The model incorporates the concept of teams as a basis for cooperation.When a robot detects a prey it forms a 1-robot team.If another searching robot detects the same prey, the team changes to a 2-robot team.When a third robot detects the same prey, the team changes to a 3-robot team.If the fourth robot detects the same prey, the prey is removed, the 3-team is disbanded, and the robots return to searching.If the teams fail to cooperate with another robot within a certain time limit the team is disbanded and the robots return to search for another prey.Theoretical analyses show that the delay time and the ratio of predator to prey are the critical parameters, which influence robots collective behaviors performance.The optimal parameters of multirobot system are found via mathematical analysis.This will provide the essential theory basis for the design and analysis of robots collective behaviors.

The Behavioral Framework of Robot Cooperative Behaviors
2.1.The Hunting Behaviors of Multirobot.In this paper a large number of homogeneous mobile robots interact implicitly with each other, working towards a shared goal.The multirobot system consists of a group of identical reactive robots.The predators will cooperate with each other to search for, follow, and surround the prey in the hunting task.The cooperative pursuit is a very challenging task of multirobot system.The performance of the task has significance for multirobot collaborative behaviors.
The predators in the task will search for the prey by the limited perception and local interaction among the robots.The goal of predators is to follow and surround the prey (see Figure 1).In multirobot pursuit system sixteen predators and four prey are randomly distributed in a 10 × 10 area.The asterisk and dot denote the predators and prey, respectively.The task of the predators is to follow and surround the prey.The predators may perceive the ambient condition and choose appropriate actions.

The Structure of Hunting Behaviors.
The framework of the general hunting behaviors consists of many different kinds of basic behaviors: searching, following, and surrounding.The following teams include three types: 1-robot, 2-robot, and 3-robot teams.The basic behaviors of robots can be described as follows.
Searching.The predators execute a random walk at a fixed speed on the playing field until a prey is found with a group of sensors.
Following.The predators and their sensors should be capable of tracking a prey with a high degree of reliability.If the prey came into the detecting area, the predators will follow the prey based on one of tracking algorithms.
Surrounding.The predators will rapidly move to the ambient region of the target whenever possible.Until the prey is hunted, the surrounding behavior has been completed.
In the initial stage, all robots are evenly distributed in the whole playing field without any prior knowledge about the environment.For simplifying the model, the hunting behavior will be modeled in the case of not considering the conditions that prey are surrounded in the boundaries or collisions with obstacles.The transition between different states of multirobot system will be performed in a short time.Finally the predators will enclose the prey by forming a round-like formation of predators.With the capture of the prey, they will disappear from the playing field.The predators will search for prey randomly again.In the operation of hunting behaviors every predator will search for the prey randomly and avoid obstacles.Once a predator detects a prey, it will follow the prey in the next time period  and wait for the other predators to cooperate with it.If there are no other predators detecting the 1-robot team within the given time, the predator will give up the current target and randomly search for the prey again.When another "searching" robot detects the same prey, the team will change to a 2-robot team.The first predator to discover the prey will reset the waiting time and two predators will follow the prey synchronously.Similarly, when the third robot detects the same prey within the given time, the team will change to a 3-robot team.The first two predators to discover the prey will reset the waiting time.Otherwise, the first two predators to discover the prey will search for other prey again.If the fourth predator detects the same prey within the given time, the prey is removed, the 3-team is disbanded and the robots return to search for another prey.This means the capture of the prey by the hunting predators.Otherwise, the first three hunting predators will give up the track of the prey and randomly search for other prey again.
In the operation of hunting behaviors the robots will make decisions based on the environmental states.Generally robots just need the sensor information at the present time to design the succeeding behaviors based on the controller.The flowchart of controller can be shown in Figure 2 for cooperative hunting robot.

The Mathematical Model at State Level
The rate equation approach has been widely used to model the collective behaviors of multirobot systems obeying the Markov property [17,18].The rate equation is usually derived from the finite difference equation, which describes how the average number of robots in each state changes over some time interval.For a given multirobot system the rate equations are deterministic.Then the differential form of the rate equation can be obtained by taking the limit of time interval to become zero.Therefore, the rate equation describes the average quantities of the dynamics of a multirobot system.The precision of the model is inseparably linked with the size of the stochastic systems.Within a certain range, the model often represents the evolution of the collective behaviors better with the increasing size of system.
The Petri-net model is established to analyze the complex collective behaviors of multirobot systems.The Petri-net diagram connects all states by different transition rates between states.The mathematical model of multirobot system consists of a series of coupled rate equations, each describing how the dynamic variables change in time.Every state corresponds to a dynamic variable in the model.The arrows indicate the directions of the state transition in the Petri-net model (see Figure 3).The arrows labeled by the letter U indicate the state transition while a timeout occurs.The time delay is the maximum of the waiting time.The arrow labeled by the letter S indicates the state transition while the following team of three predators gets cooperation from their companions in the waiting time.And the hunting task is completed successfully.Other arrows indicate the state transition when a searching robot detects a following team and join the team.

Equations of the Model.
In the mathematical model each state indicates a dynamic variable.The dynamic variables correspond to the average number of robots in a certain state, which is an approximation of discrete number of robots.Therefore, the system variables can be treated as continuous variables.Let   () denote the number of searching robots at time ,  1 () denotes the number of following robots in the 1-robot teams,  2 () denotes the number of following robots in the 2-robot teams, and  3 () denotes the number of following robots in the 3-robot teams.
In the model of cooperative hunting behaviors all robots are evenly distributed in the workspace at the initial state.The mathematical model describing the evolution of dynamic variables of the system is shown as in the following equations: () =  1 () +  2 () +  3 () , where parameter  is the rate of detecting a single prey for predators,   is the rate of detecting a following team with  predators,  is the waiting time, and Γ  (; ) is the rate of giving up the hunting task of the following team with  predators at a given time .The first two terms in (1) account for a decrease in the number of searching predators which occurs when searching predators find a single prey or the other following teams.Then the searching predators will follow the target and wait for the cooperation from the companions.The third term accounts for the number of searching predators from a waiting state after a successful hunting task.The fourth term accounts for the number of searching predators which detect a single prey and fail to cooperate with other predators.The fifth and sixth terms describe the increase in the number of searching robots because the following teams of one or two predators fail to cooperate with other predators within the waiting time.The last four terms result in the increase of searching predators.The first term in (2) describes the number of searching predators which detect a single prey and form following teams with one predator.The second term describes the increase number of waiting predators which detect a following team of one predator and join them.The third term describes the decrease of the number of waiting predators because the following teams of one predator fail to cooperate with other predators, which are converted into searching state.The first term in (3) describes the increase of the following predators because the searching predators detect a following team of one predator and join them.The second term describes the decrease of the waiting predators of following teams of two predators because the searching predators detect a following team of two predators and join them.The third term describes the decrease of the waiting predators of following teams of two predators which fail to cooperate within the waiting time.The first term in (4) describes the decrease of the prey because the searching predators detect the following teams of three predators and accomplish a cooperative hunting task successfully.The variable   in ( 5) describes the number of the following teams of waiting predators.The parameter  0 in ( 6) is the number of searching predators in the initial state.
To obtain ratios instead of number of robots, let us introduce   () =   ()/ 0 ,  1 () =  1 ()/ 0 ,  2 () =  2 ()/ 0 , () = ()/ 0 ,  =  0 / 0 ,   =   /,   =   ,  = 1, 2, 3, and a dimensionless time  →  0 ,  →  0 ,   () is the ratio of searching predators to all predators,  1 (),  2 (), and  3 () are the ratio of the following teams of one, two, and three predators to all predators, respectively, and () is the ratio of the number of the prey at current moment to that in initial state.The mathematical model can be reformulated in dimensionless form as The variables of the model can be initialized as follows:   (0) = 1, (0) = 1,  1 (0) = 0, and  2 (0) = 0;  describes the probability of searching predators detecting the single prey and its values range from 0 to 1,   ( = 1, 2, 3) describes the probability of searching predators detecting following teams of  predators and its values range from 0 to 1, and  1 and  2 are the probabilities for a failed collaboration of teams with 1 and 2 searching robots during the time interval [ − , ].The numerical solutions of the differential equations by discrete simulation with computer are shown in Figure 4.
In the initial conditions all predators are searching prey,   (0) = 1, (0) = 1,  1 (0) = 0,  2 (0) = 0, and  3 (0) = 0.With the evolution of hunting behaviors more and more predators detect prey and form different following teams.The number of prey will decrease accordingly.The number of predators in different following teams will increase.As more and more prey are captured the number of prey will decrease continuously.Most of predators will detach the following teams and search for prey again.When the system reaches the steady state the numbers of following teams and prey all tend to zero.Eventually all prey are captured by predators, and all predators will search for prey randomly in workspace.
By comparing Figures 4(a) and 4(b), the steady state solutions are determined when waiting time takes different values.The oscillations of the solutions depend on the waiting time.Stronger oscillations will occur as waiting time is decreased.The reason is that many following teams have abandoned the hunting task before the other predators can detect them.The system is convergent after the period of transient oscillations.Figure 4(c) shows that the convergence speed of the system is accelerated obviously with the increasing number of the predators.In the course of evolution, the oscillations in the dynamic variables are not severe compared to Figure 4(b).The following teams can obtain the cooperation from other companions in a relatively short time with the increase of predators.To a certain extent it is easier for predators to form a new following team and finish the hunting tasks ultimately within the waiting time as the number of predators increases.
Above contents were concerned with the effects of the key parameters to the system performance.The evolution of the cooperative hunting behaviors was described qualitatively based on the mathematical model.The system modeling is intended to analyze the interaction among all parts of the system and the evolution of the system.Then the multirobot system can be described quantitatively and analyzed theoretically.

Steady State Analysis.
The waiting time is the key parameter of the system and has a profound effect on the hunting strategy.Figure 4 shows that the ratios of searching robots to prey have steady state values after a period of transient oscillations.The steady state values simply depend on the time that searching robots spend in hunting prey.The analytic expressions of the ratios of searching robots can be obtained by setting the left-hand side of ( 7)-( 9) to zero.The solutions of steady states depending on waiting time satisfy the following equations: The evolution of  1 ,  2 ,   , and  3 ratios can be obtained by solving ( 13)-( 14).The solutions under the steady state are shown in Figure 5 when the waiting time takes different values.The hunting teams become increasingly stable as the waiting time increases with a fixed amount of predators.The following predators will complete the hunting task successfully through peer collaborative interaction.The number of following teams monotonically increases with waiting time.The searching predators will rapidly decrease owing to more and more predators following the prey.

Collaboration Ratio Analysis.
The collaboration ratio of robots represents the rate of successful hunting tasks for predators.The collaboration ratio of robots can be formulated in dimensionless form as in the following equation based on the values of   and  3 : The collaboration ratio of groups with different amounts of predators is shown in Figure 6.When there are fewer predators in the system, most of the predators can quickly detect the prey and follow it as the waiting time increases.The searching predators will increase rapidly.And the following teams are often difficult to obtain the cooperation from the companions.If there are sufficient predators in the system the hunting efficiency of the following teams will be improved as waiting time increases.The collaboration ratio will monotonically increase because there are more searching predators.In addition, for given number of predators the collaboration ratio will increase with the waiting time.When waiting time exceeds a certain threshold value the collaboration ratio will decrease with the waiting time.Therefore, the waiting time of the following teams has an optimal value.For given values  robot collective behaviors represents the transition between different states of robots.Each dynamic variable in the mathematical model corresponds to the number of robots in states.The dynamic characteristics of robot collective behaviors are investigated based on the macro mathematical model.This paper formulates the model as a set of rate equations and presents numerical solutions to normalized versions of these equations and provides both a steady state and a collaboration ratio analysis.According the presented modeling method the collective behavior of robots can be described mathematically and analyzed quantitatively.This paper is largely concerned with the cooperative hunting behaviors.The multirobot system consists of groups of reactive and homogeneous robots, which is viewed as a stochastic system with Markov property.The critical parameters that influence multirobot system performance are analyzed via mathematical model of robot collective hunting behavior.The optimal parameters of the system are found via mathematical analysis.Then the control strategy of the system will be greatly improved.This will provide the essential theory basis for the design and analysis of robots collective hunting behaviors.

Figure 1 :
Figure 1: The initial stage of the evolution of cooperative hunting behaviors.

Figure 5 :
Figure 5: Solutions under the steady state for  = 1.5.

Figure 6 :
Figure 6: The collaboration ratio of groups with different predators.