A multi-robot task allocation algorithm based on universal gravity rules

Abstract

In this paper, a new multi-robot task allocation (MRTA) algorithm inspired by the Newtonian law of gravity is proposed. In the proposed method, targets and robots are considered as fixed objects and movable objects, respectively. For each target, a constant mass is assigned, which corresponds to its quality. The fixed objects (which refer to targets) apply a gravitational force to the movable objects (which are considered as robots) and change their positions in the feasible search space and therefore, the best target allocation of robots is determined by employing the law of gravity. In the proposed scenario, task allocation consists of assigning the robots to the found targets in a 2-D feasible area. The expected distribution is obtained from the targets’ qualities that are represented as scalar values. Decision-making is a distributed mechanism and robots choose their assignments, taking into account targets’ qualities and distances. Moreover, a control parameter is planned to make a remarkable balance between exploration and exploitation ability of the proposed algorithm. A self-adaptive mechanism is proposed to adjust the value of the exploration parameter automatically, aiming to maintain the balance between exploration and exploitation ability of robots. Furthermore, in order to decrease the time of reaching the target and accelerate computation, a selection memory is designed. In the experiments, we examine the scalability of the proposed method in terms of the number of robots and the number of targets and speed of algorithm to deliver robots to the desired targets with comparison to other competitors. The simulation results show the scalability of the algorithm, comparing the existing methods. Moreover, some non-parametric statistical tests are utilized to compare the results obtained in experiments. The statistical comparisons confirm the superiority of the proposed method compared over the existing methods.

Appendices

Appendix A

1.1 Performance evaluation of two proposed gravitational parameters

In order to find the best definition of ${G}_{t}$ parameter two main approaches are proposed that are discussed in detail:

(A) Fractional definition

To strive for a balance between exploration and exploitation in the optimization process of the proposed method, we propose a self-adaptive mechanism which can get feedback from the current population to control the value of ${G}_{t}$. ${G}_{t}$ is the control parameter that allows us to bias the decision-making mechanism toward the quality of the solution or its cost, respectively.

To evaluate adaptive, firstly, $c_{{\text{t}}} { }$ as a percent of the selected target $t \in { }\left\{ {1,{ }.{ }.{ }.{ },{ }T} \right\}$ by robots is calculated as follows:

$$c_{t} = \frac{{n_{t} }}{R}\quad t{ } \in { }\left\{ {1,{ }.{ }.{ }.{ },{ }T} \right\}$$

(10)

where $n_{{\text{t}}}$ present the number of robots which select the target $t$ and $R$ is the total number of robots.

It should be explained that based on MRTA problems, MAE is one of the most crucial criterion. In order to decrease the MAE value, the difference between $c_{t}$(percent of selected target or the resulting robots’ distribution) and ${\text{m}}_{{\text{t}}}$ (expected percent of selected target or expected robots’ distribution) should be decreased.

To reach this aim in our proposed algorithm, the gravitational parameter $G_{t}$ is considered as an adaptive controlling parameter which is evaluated as follows:

$$G_{t} = \left( {\frac{{m_{t} }}{{c_{{\text{t}}} + \varepsilon }}} \right)^{\alpha }$$

(11)

where $\varepsilon$ is a small positive number to avoid dividing by zero and $\alpha$ is a parameter of the algorithm which tunes the effect of $G_{t}$ on the calculation of the force.

(B) Exponential definition

$$G_{t} = \beta^{{c_{t} - m_{t} }}$$

(12)

In the following, the performance of two proposed gravitational parameters are evaluated. To reach this aim, Figs. 2 and 3 illustrate the MAE versus variable value of alpha for gravitational constant 1 and 2 when the number of robot is 100.

Fig. 9

MAE versus varying value of alpha for fractional formulation of gravitational parameter

Fig. 10

MAE versus varying value of beta for power formulation of gravitational parameter

It can be concluded from Figs. 9 and 10 that the best values of $\alpha$ and $\beta$ occur in 1.35 and 0.74, respectively. Moreover, the MAE value of fractional formulation is less than power formulation. To have a more precise investigation, the average and the maximum values of MAE obtained for setup 1 (number of targets is two with similar qualities) and setup 2 (four targets with same qualities) from the experiments are presented in Table. Based on Table 4, the performance of $G_{t}$ based on fractional formulation is better than power formulation.

Table 4 Comparison of two proposed gravitational parameters

As a result power formulation $G_{t} = \left( {\frac{{m_{t} }}{{c_{t} + \varepsilon }}} \right)^{1.35}$ is considered as the gravitational parameter of algorithm.

Appendix B

4.1 Discussion of algorithms for comparison

4.1.1 Distributed bees algorithm (DBA)

In Jevtic et al. (2011), the task allocation problem is solved with a swarm-based meta-heuristic technique named DBA algorithm which is inspired by the intelligent behavior of the “bees” to optimize their search for “food” resources. In this algorithm, each robot is represented as a ‘bee’, and task utility, $p_{ik}$, is defined as a probability that the task $k$ is allocated to the task $i$ and depends on both target’s quality and the distance of the task from the robot:

$$p_{ik} = \frac{{q_{i}^{\alpha } \left( {\frac{1}{{D_{ik} }}} \right)^{\beta } }}{{\mathop \sum \nolimits_{j = 1}^{M} q_{j}^{\alpha } \left( {\frac{1}{{D_{jk} }}} \right)^{\beta } }}$$

(13)

where $q_{i}$ and $D_{ik} { }$ shows the quality or priority of target $i$ and distance between target $i$ and robot $k$, respectively. Moreover, $\alpha$ and $\beta$ are control parameters that bias importance of the priority and distance, respectively, ($\alpha$, $\beta > 0$; $\alpha$, $\beta \in R$).

In DBA algorithm, roulette wheel selection method is chosen in which a target with a probabilistic procedure is selected by each robot. In other words, in DBA the selection probability of target is proportional to their fitness value.

The probabilities $p_{ik}$ are normalized, and it is easy to show that:

$$\mathop \sum \limits_{i = 1}^{M} p_{ik} = 1$$

(14)

4.1.2 Modified distributed bees algorithm (MDBA)

In the MDBA, the original DBA robot utility function is modified to take advantage of heterogeneous robots or targets with different performances aiming to improve system performance by correlating the robot’s utility with their performances. In order to apply MDBA to heterogeneous robots or targets, a control parameter was defined in Tkach et al. (2018) as a function of the robot’s performance on a target. In simple words, when a robot receives information about an available target, it calculates its performance for that task. The robot’s utility function is updated accordingly, and depends on the target quality, the distance from the task and the robot’s performance on that task:

$$p_{ik} = \frac{{q_{i}^{\alpha } \left( {\frac{1}{{D_{ik} }}} \right)^{\beta } V_{ik}^{\chi } }}{{\mathop \sum \nolimits_{j = 1}^{M} q_{j}^{\alpha } \left( {\frac{1}{{D_{jk} }}} \right)^{\beta } V_{jk}^{\chi } }}\quad if\,\Delta_{i} > 0$$

(15)

where ${\upchi }$ is a control parameter that biases the importance of the robots performance and $V_{ik}$ is the performance of robot $k$ on task $i$. Moreover, each task has a time limit, or a deadline which is evaluated as follows:

$$\Delta_{i} = \frac{1}{{q_{i} }} > 0,\quad q_{i} > 0$$

(16)

The MDBA decision-making mechanism applies the same wheel-selection rule that is used in DBA to choose from a set of available tasks.

4.1.3 Linear ranking for distributed bees algorithm (LRDBA)

LRDBA shows a linear ranking selection for DBA. In order to do linear ranking selection, each target is defined by its fitness score which is named as "rank of target". In other words, the selection probability of each target ($Prob_{i}$) in linear ranking selection is evaluated as follows:

$$Prob_{i} = q - \left( {q - q_{0} } \right) \times \frac{{R_{i} - 1}}{T - 1}$$

(17)

where $q$ and $q_{0}$ are probability of selection of the best target and the worst target. Moreover $R_{i}$ is the rank of target $i$ and $T$ shows the total number of targets.

4.1.4 Exponential ranking for distributed bees algorithm (ERDBA)

In ERDBA an exponential ranking selection for DBA is utilized. This technique is different from linear ranking selection technique in a way that the probabilities of ERDBA are exponentially weighted as follows:

$$Prob_{i} = q\left( {1 - q} \right)^{{R_{i} - 1}}$$

(18)

4.1.5 Tournament based for distributed bees algorithm (TBDBA)

TBDBA is a tournament based selection strategy for DBA. Tournament Selection is a selection procedure used for selecting the fittest candidates from the current generation. These selected candidates are then passed on to the next generation. In a $K$-way tournament selection, $k$-individuals are selected and run a tournament among them. Only the fittest candidate among those selected candidates is chosen and is passed on to the next generation. In this way many such tournaments take place and the final selection of candidates who move on to the next generation are given.

4.1.6 Market-based algorithm (MBA)

A market-based algorithm used in Zlot et al. (2002) for distributed system was applied with application specific modifications. In this approach, the bid of robot $k$ to task $i$ is defined as (19):

$$Bid_{ik} = q_{ik} + \delta \left( {\frac{{V_{ik} }}{{D_{ik} }} - q_{ik} } \right)$$

(19)

where $q_{ik}$ serves as the reservation price of task $i$, and $\delta$ is a control parameter with values between 0 and 1. A task $i$ is selected by robot $k$ if it maximizes its bid value:

$$select = {\text{max}}(Bid_{ik} )$$

(20)

4.1.7 Greedy algorithm (GrA)

A greedy algorithm that was used previously for a multi target observation problem with broadcast messaging (Broadcast of Local Eligibility for Multi-Target Observation 2002) was modified to fit the described problem. The greedy algorithm was set to perform task allocation based on the best possible allocation of each individual robot to task that maximizes $\frac{{V_{ik} }}{{D_{ik} }}$, where $V_{ik}$ is the $k$-th robot’s performance on the $i$-th task and $D_{ik}$ is the Euclidean distance between the robot and the task:

$$task_{k} = {\text{max}}\left( {\frac{{V_{ik} }}{{D_{ik} }}} \right)\quad i \in Z$$

(21)

where $task_{k}$ is the task chosen by the $k$-th robot, $i$ shows the index of task, and $Z$ is the set of tasks within $k$-th robot range. Note that $Z$ is a subset of all $M$ available tasks.

4.1.8 Niching immune-based optimization algorithm based on softmax regression (sNIOA)

The sNIOA presents a niching immune-based optimization algorithm based on Softmax regression (sNIOA) to handle it (Huang et al. 2018). A pre-judgment of population is done before entering an evaluation process to reduce the evaluation time and to avoid unnecessary computation. Furthermore, a guiding mutation operator inspired by the base pair in theory of gene mutation is introduced into sNIOA to strengthen its search ability. It should be mentioned that in Huang et al. (2018), a discrete version of immune optimization algorithm is used in which each antibodies is utilized to present the allocated robot to defined tasks.