An arrovian analysis on the multi-robot task allocation problem: Analyzing a behavior-based architecture

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Highlights

  • An axiomatic framework based on Social Choice Theory is proposed to the MRTA problem.

  • It establishes an analogy between the Social Choice Theory and the MRTA problem.

  • A behavior-based MRTA architecture is discussed under the ordinal preference domain.

  • The results show that the proposal can be used to analyze implemented architectures.

Abstract

Research in multi-robot systems is a rich field that has attracted much attention in recent decades. However, robot coordination and task allocation to a correct mission accomplishment are still challenging even with technological advances. Despite many proposals presented in the literature, the applications and theories about the task allocation problem are not yet exhausted. Thus, this work proposes an axiomatic framework based on Social Choice Theory to analyze the task allocation problem in intentional cooperation multi-robot systems. It uses Kenneth J. Arrow’s framework of his famous Impossibility Theorem. The conditions imposed by Arrow aim to create an ideal for preference aggregation mechanisms through axiomatic analysis. This paper aims to transport this analysis to the multi-robot domain. A behavior-based Multi-robot Task Allocation architecture is used to present simulation results and discuss two cases in the ordinal preference domain. The analysis results show that using the proposed framework to analyze, under the Arrovian perspective, implemented MRTA architectures is feasible.

Introduction

According to Kenneth J. Arrow [1], Social Choice theory starts with reasonable conditions relating choice outcomes to communications or interactions. The imposed conditions support feasible preference communication. Assuming that social choice depends on individual well-being, then the individual should communicate its preference. Therefore, the collective outcome is a function of individuals’ communications [1]. Gaertner [2] affirms that the theory intends to obtain a verdict (a collective homologation) based on society members’ opinions and values. In this way, social preference results from individual preference aggregation, and it should reflect the general opinion of society. This work focus on the Welfare Economics and Social Choice Theory to establish similar premises applied to Robotics, mainly to Multi-Robot Systems (MRS) and to the Task Allocation problem.

When working with a Multi-robot Task Allocation (MRTA) architecture with heterogeneous robots, robots may have different capabilities, e.g., different sensors. They may also have different metrics and utility functions, managing well-defined and individual variables, such as battery level, distance, and communication range. The paper’s contribution is the proposal of an ordinal preference comparison over the alternatives set for a robot society decision-making. Also, it proposes social choice mechanisms for task allocation that attend, even if relaxed, Arrovian conditions.

The primary objective of this article is to establish an analogy between the Social Choice Theory and the MRTA problem. The paper aims to address Arrow’s axiomatic framework to propose an analysis of social choice mechanisms on the MRTA domain. We first described the conditions and the Arrovian MRTA problem in [3] defining the Arrovian framework from a general multi-agent perspective. Also, it discusses the Arrovian conditions briefly from the standpoint of Single-task robots, Single-robot tasks, and Instantaneous task assignment (ST-SR-IA) and Single-task robots, Multi-robot tasks, and Time-extended task assignment (ST-MR-TA) multi-robot task allocation problems. The present paper revisits and extends the Arrovian conditions with more straightforward definitions in the problem domain of MRTA, further discussing succinctly other MRTA problem classifications. It exemplifies the proposed analysis using an MRTA architecture simulation based on the well-known ALLIANCE architecture, a behavior-based architecture proposed by Parker [4].

The work focuses on multi-robot architectures with intentional cooperation, heterogeneous robots, and decentralized decision-making. It focuses on the single-task robot problems, following Gerkey and Matarić’s [5] taxonomy definition, and briefly extends the discussion to the other cases. We define the required conditions for the problem domain to avoid the impossibility of the group of robots’ collective decisions. Also, we examine the allocation mechanisms based on social choice rules observing the conditions met and those that should be relaxed. Moreover, the goal is to establish an ordinal comparing structure of individual preferences rather than comparing scalar utility.

Arrow’s axiomatization [6] is not a structure of task allocation. Instead, it is a standard to state when a given structure meets the basic requirements of a democratic social choice that maximizes social welfare. Thus, the paper does not propose a new MRTA architecture but an analysis framework based on Social Choice Theory.

This paper follows the organization: Section 2 presents the related works on the MRTA problem; Section 3 introduces the Social Welfare and Social Choice in Multi-robot systems; Section 4 examines the MRTA problem from an Arrovian point of view; Section 5 analyzes a problem from Gerkey and Matarić’s MRTA problem taxonomy [5] under the Arrovian framework, while Section 6 presents simulation results of an MRTA architecture; Section 7 discusses the results from the proposed Arrovian framework view and other aspects of the proposal. Finally, Section 8 renders the conclusions.

Section snippets

Related works

Multi-robot Systems have several advantages if compared to a single-robot. To a monolithic robot, i.e., with various tools to allow the application in several different tasks, can be very complex to complete a series of tasks in a specific time interval, leading to reduced system performance [7]. For dynamic environment applications and where the priority of an emergency task may eventually appear, such as a hospital environment, Das et al. [8] recommended an MRS composed of heterogeneous

Social welfare and social choice in multi-robot systems

According to [38], investigations in social choice are expanding to the Artificial Intelligence community of multi-agent systems (MAS), particularly regarding preference and mathematical models. Many works deal with voting problems [39], [40]. Fischer [41] investigated the efficient use of cardinal and ordinal preferences in random assignment situations. MRS differs from MAS because of the embodiment and performance in a real environment of the former. However, MAS developments can be applied

The multi-robot task allocation problem under the Arrovian view

The preference ordering has advantages for the analysis of the collective decision in the task allocation problem. According to Fleurbaey and Maniquet [49], the ordinalism concept follows from the fact that any utility function can be replaced by any strictly increasing transformation of it, so only the ranking of bundles matters. The intensity will no longer take place in a decision-making process but rather the preference order.

Each robot keeps its internal utility function in an MRS to

Arrovian analysis of MRTA problem taxonomy

The analysis of task allocation problems that Gerkey and Matarić’s taxonomy [5] defines will be done individually, under each characteristic axis, i.e., robot, task, and allocation types. An initial structure was proposed by dos Reis and Bastos [3], which this paper expands. Some assumptions are considered for this analysis:

  • i.

    All the robots tell the truth and are not envious.

  • ii.

    The number of tasks is greater than or equal to the number of robots, so nm and n3.

  • iii.

    All mission tasks are known in advance

Behavior-based MRTA architecture simulation

The section presents an MRTA architecture simulation based on Alliance implemented in Robot Operational System (ROS) to illustrate the proposed ordinal axiomatic analysis of MRS. It analyzes two cases: (a) when the individuals have the same preference profile and (b) when individuals have a different preference profile. In the last case, the simulations aim to show that a social ordering emerges despite the individual preference profile and individual robot motivation.

The Alliance architecture

Discussion

This paper’s allocation mechanisms do not consider certain practical aspects of task allocation architectures in MRS. From the hardware point of view, some of these aspects are communication means, computational power, actuators, and sensors’ limitations. The simulation discussion shows the MRTA problem can be analyzed from the Arrovian point of view even in a task allocation architecture implemented in another domain.

The consideration of independent tasks simplifies the propositions of this

Conclusion

This paper proposes a Social Choice-based Arrovian structure to evaluate task allocation mechanisms in multi-robot systems. Moreover, it addresses the use of individual ordinal preferences and preference aggregation mechanisms for social decision-making. The proposed Arrovian view is another perspective to address the MRTA problem analysis. Applied to MRTA problems, it contributes to the theoretical and practical study of such problem solutions. Like Arrow’s axiomatic method, the paper proposes

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Wallace Pereira Neves dos Reis: Graduated in Electrical Engineering with an emphasis on Electronics, Specialization in New Educational Technologies, and Master in Electrical Engineering, the last from the Federal University of Itajubá - UNIFEI. He is currently a doctoral student in the Computer Science Program at the Federal University of São Carlos - UFSCar, research line Artificial Intelligence, in Industrial Automation. He is a Professor of Basic Technical and Technological Education at the

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    Wallace Pereira Neves dos Reis: Graduated in Electrical Engineering with an emphasis on Electronics, Specialization in New Educational Technologies, and Master in Electrical Engineering, the last from the Federal University of Itajubá - UNIFEI. He is currently a doctoral student in the Computer Science Program at the Federal University of São Carlos - UFSCar, research line Artificial Intelligence, in Industrial Automation. He is a Professor of Basic Technical and Technological Education at the Federal Institute of Education, Science, and Technology of Rio de Janeiro, IFRJ - Campus Volta Redonda, since 2014, working mainly on technical courses in Electrotechnics and Industrial Automation. He served as Director of Administration from March 2016 to June 2018 at the same campus. He has experience in Control Systems, Multirobot Systems, Social Robotics, Computational Social Choice, Embedded Electronics, Image Processing, Educational Robotics. His current research interests are AGV Dispatching and AGV Position Control Design.

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