Elsevier

European Journal of Control

Volume 22, March 2015, Pages 30-45
European Journal of Control

Persistent coverage control for a team of agents with collision avoidance

https://doi.org/10.1016/j.ejcon.2014.12.001Get rights and content

Abstract

In this paper, the idea of persistent coverage to be accomplished by multiple agents while avoiding collisions is considered and developed. The persistent coverage problem is formulated by assuming that the coverage degrades over time. In this framework, our contribution is a new distributed control law which is capable of carrying out the persistent coverage without computing agents׳ paths explicitly. The proposed setup considers agents with nonholonomic motion constraints and it is based on the combination of local and global strategies to achieve efficient coverage while avoiding bottlenecks such as local minima. The local strategy is based on the gradient of the coverage error in the neighborhood of an agent whereas the global strategy leads the agents to uncovered areas of the domain. Furthermore, we present a new bounded potential repulsion law and a proof of safe navigation is provided for the case of unicycle vehicles. We also propose a modification of the tangent-bug algorithm to deal with multiple non-point agents which allows the team to navigate in environments with non-convex obstacles in a reactive manner. Simulation results illustrate the performance of the proposed control law.

Introduction

The problem of area coverage by a team of agents is of interest in a wide variety of applications such as cleaning [23], lawn mowing [1] and monitoring [32]. In general, the use of multiple agents to solve this problem enhances the coverage performance by, for example, decreasing the coverage time. However, multiple agents introduce additional issues like collision avoidance and coordination of agents.

In the case of static coverage the agents are assumed to be able to cover an area by being placed at particular positions [9]. However, if the agents are mobile, it is possible to deploy the resources and adapt their positions to a variety of environments. Deployment has been solved through different approaches: with Voronoi partitions [7], [15], by using potential fields [27], [4], [24], or with gradient methods [33]. In dynamic coverage problems agents are assumed to have a limited sensing range and can not cover the domain statically, that is, by any deployment. To accomplish the task, some approaches compute a path explicitly [6], [30], [17], whereas others solve the problem without computing a path [18], [13], [14]. If the environment is invariable, the problem is solved by covering all the points once [6], [18], [13], [14]. However, some tasks require to re-cover all the points over time because the environment evolves and the task is to monitor the area persistently [30], [17], [19]. Our work is focused on the latter scenario which we refer to as the persistent coverage.

A relevant issue that arises when dealing with multiple agents and environments with obstacles is the problem of collision avoidance. Obstacle avoidance for navigation purposes has been addressed by many different strategies: potentials [21], vector field histogram [2], dynamic window approach [11], elastic bands [28], nearness diagram navigation [25], and model predictive control [29], to name a few.

In the field of coverage, the work presented in [3] introduces the tangent-bug algorithm to avoid convex obstacles and develop deployment in multi-agent environment. However, they consider point agents and do not consider inter-agent avoidance. On the other hand inter-agent avoidance has been commonly treated with a repulsive force that grows relative to the proximity among the agents. In [5], [26] authors introduce forces that modify the angular velocity whereas Liu et al. (2006) [22] and Dimarogonas et al. (2006) [8] introduce a repulsive force that modifies both module and direction of the motion toward the main objective. In the field of dynamic coverage with sensors networks, Hussein and Stipanović [18] introduce a scheme based on the collision avoidance ideas from [31]. In those papers inter-agent avoidance is solved but obstacle avoidance is not addressed.

In this work, we focus on the problem of persistent coverage control by a team of nonholonomic agents in an environment with a coverage decay. We propose a coverage control law based on ideas introduced in [13], [14] where a local strategy and a global strategy are combined. We begin by proposing a new model for the evolution of the coverage, based on a differential equation that evolves between zero and a maximum coverage level. The behavior of the model can be tuned with two gains, the sensing gain and the decay gain, to adjust the values for different scenarios.

To develop persistent coverage we propose reactive strategies that do not compute a path explicitly. We use a strategy based on the gradient of the coverage error to find the best direction to move instantaneously. As gradient strategies may get trapped in local minima, we combine the local strategy with a global strategy that leads agents to uncovered areas. Both strategies are continuously weighted in such a way that agents obey their local control laws if the error in their neighborhoods is high, and they move to new areas obeying the global control law when there is no benefit in covering the nearby areas. To reach uncovered areas by avoiding obstacles and other agents in a reactive fashion we use the tangent-bug algorithm [20] with a modification which allows the algorithm to work in environments with multiple non-point agents. Once the coverage action is obtained from the local and global strategy, it is combined with a new bounded repulsion law. The coverage control law and the repulsion law are weighted depending on the danger of collision to obtain the desired motion.

Finally, with the target motion we design a control law to govern the nonholonomic agents. The angular velocity input is proportional to the angular error, and the linear velocity input takes into account the maximal speed of the agent, the local coverage error, the angular error, the distance to global goals and the danger of collision. If the local coverage error is high, the speed is decreased to provide a better coverage of the neighborhood. If the local coverage error is low, the speed is increased for the agent to leave the covered area. The speed is also decreased as the angular error gets larger to avoid high linear speeds while turning, and as agents approach global targets and obstacles. In this paper, the coverage information and the global strategy are centralized but the motion is agent-based. This is done to reduce the communication costs and increase the flexibility to changing environments while keeping a good level of efficiency. In fact, each agent can compute the coverage map, and the global goal of each agent is achieved with only position information.

Specifically this paper provides the following items as the main contributions:

  • An algorithm that develops persistent coverage without computing explicitly agents׳ paths.

  • An adaptation of the tangent-bug algorithm to multi-agent environments, to allow multiple non-point agents to reach their global goals in unconnected environments or with non-convex obstacles.

  • A new bounded potential repulsion law for agents that allows safe navigation for unicycles. Furthermore, proofs of collision avoidance in multi-agent environments with obstacles are provided.

The paper is organized as follows: Section 2 introduces the problem formulation and the model of the evolution of the coverage. Section 3 presents the coverage strategy and the repulsion law. Section 4 describes how coverage and collision avoidance objectives are combined, presents the nonholonomic motion control law, and provides collision avoidance proofs. Section 5 provides simulation results of the proposed algorithm in different environments. Finally, Section 6 presents the conclusions of the paper.

Section snippets

Problem formulation

In this section we introduce the problem formulation and a new evolution coverage model for a team of agents performing dynamic coverage tasks. We abuse notation by including the dependencies of the variables only when they are defined. One of the main objectives is to reach a desired coverage level Λ(x)R+ for all the points xDx over a bounded domain DxR2. We assume that mobile agents behave as differential drives, that is, each agent Ai of the team A={A1,,AN} of N agents is governed by

Dynamic coverage control laws

The crucial objective of our proposed distributed control law is to keep decreasing the error eDx. Let us now divide the domain into the points which have positive lack of coverage Dx+(Υ)={xDx|Υ>0}, and the rest Dx0(Υ)={xDx|Υ=0}. Notice that both domains are complementary and their union is the whole coverage domain Dx. The domains depend on time since the sign of Υ changes as the domain is covered by the agents or become uncovered due to the decay. We want to minimize the variation of the

Safe coverage

In this section we show how to combine the control laws related to coverage and obstacle avoidance in order to accomplish both objectives. We also propose a control law for the nonholonomic model considered. First, we introduce coverage gain as the complementary of the collision gain kicov(t)=1kicol and then we compute the desired motion action ui(t)R2 asui=kicovuicov+uicol=kicovuicov+kicolu^icol.Then, we can extract the desired orientation θdi(t)(π,π] for the unicycle, from the components

Simulation results

In this section we show simulation results of the proposed control laws. Firstly we introduce the coverage function of agent i:αi(r)={αMR(r2R2)2,rR0,r>Rwhere αM is the maximum level of coverage and R is the range of the agent. Note that other coverage functions could be chosen.

Using this coverage action we present a coverage simulation for 2000 units of time over a square domain Dx of 100×100 units by a team of four agents. The parameters of the coverage function are as follows: Ks=1/250, Kd=

Conclusion

In this paper we presented the first control algorithm that develops persistent coverage with reactive avoidance control laws. This is based on a new model for the evolution of the coverage level with decay. Assuming the unicycle model for the dynamics of the agents, we provide a controller which combines local and global control laws guaranteeing full coverage of the domain if there is no decay, even when there are non-convex obstacles or unconnected domains. The controller also provides a

Acknowledgements

This work was supported by projects DPI2012-32100, IPT-2011-1158-920000, RTC-2014-1847-6 from Ministerio de Economía y Competitividad, by FEDER funds, and by Grant B139/2010 by DGA.

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