# Robust coverage control of multiple USVs with time-varying disturbances

*Intell Robot*2023;3(3):242-56.

## Abstract

This paper investigates the problem of optimal coverage control for multiple unmanned surface vehicles (USVs) in the presence of time-varying disturbances. To solve this problem, the disturbance vector observer is designed to approximate the unknown time-varying disturbances. It is demonstrated that the estimated disturbance vector converges to the actual disturbance vector within a finite time. To achieve the optimal coverage effect of the task region, the control idea of layer-by-layer design is borrowed, and the desired velocities of the USV are designed. By following the desired velocities, the USV network can achieve the optimal coverage effect of the task region. Based on the estimated disturbances, a robust coverage controller is designed to achieve the tracking of desired velocities by the USV within a finite time, ultimately achieving optimal coverage effect of the task region by the USV network. Finally, corresponding simulation results are provided to validate the effectiveness of the proposed approach.

## Keywords

*,*disturbance observation

*,*multiple unmanned surface vehicles

*,*finite time control

## 1. INTRODUCTION

With advancements in technology and theoretical progress in multi-agent systems, multi-agent cooperative control has gained popularity in both military and civilian applications. One of the key research areas in this field is multi-agent system coverage control within a designated task region. The coverage control problem has a wide range of practical applications, including environmental monitoring, search and rescue missions, harbor patrolling, and area defense. As a result, researchers have dedicated considerable attention to addressing this problem in recent decades ^{[1–7]}.

The coverage control problem poses a common challenge in deploying an agent network within a task region to optimize task execution. One widely used approach for addressing this problem is based on Voronoi partition, which was first proposed in the work of Ref.^{[8]}. Since then, numerous scholars have conducted extensive research on the coverage control problem using Voronoi partition. For example, Ref.^{[9]} solved the coverage control problem concerning the deployment of mobile sensor networks in non-convex domains. Meanwhile, Ref.^{[5]} studied the coverage control problem for non-convex regions, taking into account the heterogeneity of sensing range in mobile robot networks. In the research presented by Ref.^{[10]}, a coverage control strategy for mobile sensor networks with limited communication range was proposed, where the trajectory of the robots is constrained to a circle. Additionally, Ref.^{[11]} proposed a region coverage control law for a team of first-order kinematic model mobile robots operating within a two-dimensional region with time-varying risk density.

In view of the presence of unknown information in the coverage control problem, several scholars have proposed adaptive coverage control methods to optimize the coverage efficiency^{[12–14]}. In the work of Ref.^{[12]}, the unknown density function is approximated using the feedforward neural network method, followed by a coverage control algorithm for the sensor network based on this approximation. An observer is introduced to estimate the unknown information in Ref.^{[13]}, and a controller is designed to achieve optimal coverage effects. Additionally, in Ref.^{[14]}, a multi-agent coverage control law with time-varying model uncertainty is proposed, utilizing function approximation techniques.

It is worth noting that the kinematic model of the agent used in the above coverage control studies is the first-order integral model ^{[5–14]} or the second-order model ^{[4]}. However, the agent typically exhibits underactuated characteristics in practical applications, and designing a controller for underactuated agents can present additional challenges. To address this issue, Ref.^{[15]} proposed gradient-descent coverage control algorithms for underactuated wheeled vehicles. Meanwhile, Ref.^{[16]} proposed an observer-based coverage control law for a unicycle multi-agent network with external disturbance in a dynamic environment. The swarm-based coverage control in Ref.^{[17]} considers two different types of agents: the unicycle agent and the single-integrator agent. However, in practical scenarios such as ocean environmental monitoring, marine scientific research, and marine security defense ^{[18–20]}, the underactuated unmanned surface vehicle (USV) is widely used, which has stronger underactuated characteristics and operates in complex working environments. The agent models considered in the aforementioned studies on area coverage control are relatively simple models, and their proposed control laws cannot be directly applied to the USV model. As a result, there is currently limited research on marine area coverage control of the USV. In the application of USV control, the movement of USVs is frequently affected by the marine environment, and disturbances generated by the marine environment may prevent the movement of the USV from achieving the desired performance. Therefore, it is necessary to take into account the impact of unknown disturbances. Considering the influence of model uncertainties and environmental disturbances, Ref.^{[21]} and Ref.^{[22]} proposed path following strategies for the USV based on robust neural damping adaptive methods and the fuzzy logic system, respectively. Addressing time-varying delay and uncertainty topology, Ref.^{[23]} studied the consensus problem among agents operating under Markov switching topology. With a focus on the network security and uncertainty, Ref.^{[24]} studied the elastic consensus problem of dynamic network agents based on the media consensus strategy. Considering bounded uncertainties and external disturbances, Ref.^{[25]} proposed an adaptive control strategy for the super-twisting controller to achieve trajectory tracking of USVs. In the presence of disturbances, a fixed time line of sight (LOS) guidance law and a fixed time heading controller based on the fixed time disturbance period are proposed in Ref.^{[26]} to drive the USV to track the expected path within a fixed time frame.

This paper investigates the coverage control problem for the USV network in the presence of unknown time-varying disturbances. To address this challenge, the disturbance vector observer is designed to estimate time-varying disturbances. Subsequently, the coverage controller is developed based on the observer to guide each USV to track desired velocities. Furthermore, the optimal location configuration of the USV network is implemented to optimize the coverage of the task region.

The paper is structured as follows. Section II presents the simplified kinematic and dynamic models of the USV, along with an overview of the coverage control problem. Next, in Section III, the design of expected velocities for USVs is discussed, and it is demonstrated that optimal coverage of the task region can be achieved by utilizing these velocities. Subsequently, Section IV details the design of a finite-time disturbance observer, which aims to estimate external input disturbances. Using the observer, a control law is then developed to drive the velocities of USVs toward the desired values within a finite time. Lastly, Sections V and VI present simulation results and conclusions, respectively.

## 2. PROBLEM FORMULATION

The USV set

where

where

and

The USV is equipped with an actuator module to perform the corresponding tasks. The performance of the actuator is optimal when the USV is located nearby, but it gradually weakens as the distance to be covered increases. The performance function ^{[27,28]},

where

The risk density function

where ^{[29,30]}.

The generalized Voronoi partition method is introduced to assign areas for each USV, as described in previous studies^{[8,9]}. The region

Next, the effect of USVs executing tasks within the task region ^{[31,32]}

where

The goal of achieving task region coverage with the USVs is to drive the location configuration of USVs to maximize the metric function

## 3. THE EXPECTED VELOCITIES DESIGN OF THE USVS

In this section, we will design the expected velocities of each USV to achieve optimal coverage of the task region and provide rigorous proofs to support our design.

**Assumption 1***Each USV can accurately measure its position and angle information *

From the Voronoi partition, there is

where

Take the derivative of the

where

Let

Next, the expected velocities of the

where constant coefficient

**Lemma 1**^{[33]} Consider the system

*where *

**Lemma 2**^{[33]} For the cascade system

*where *

**Theorem 1***Consider the kinematic model of the USV (0.1) with the performance function (0.4); the expected velocities (0.13) of USVs can maximize the coverage effect metric function (0.7), and the optimal coverage of the task region is achieved.*

**Proof.***Consider the following Lyapunov function*

*Taking the derivative of (0.16) yields,*

*Substituting the expected angular velocity *

*It can be inferred that the error *.

*Next, we define two quantities,*

Then, one has

*Taking the derivative of the quantity *

*From equation (0.12), using *

*According to equation (0.11), *

*Note that*

*For the system*

*one has*

*Since *

*The globally exponentially stable equilibrium point lies at the origin for the system *.

*As the system *

*Because *

**Remark 1***It can be shown that the quantity *.

## 4. THE ROBUST COVERAGE CONTROL LAW FOR USVS

This section presents the design of an observer that can estimate the unknown time-varying disturbances of the USV within a finite time. Subsequently, a controller is designed based on the observer to drive the velocities of the USV to track the desired velocities (0.13) within a finite time.

**Assumption 2***The first time derivative of the unknown time-varying disturbances *

**Lemma 3***( ^{[34]}) Consider the nonlinear system *

*where *

Considering the dynamic model (0.3), the control input

First, the observer is designed to approximate the unknown external disturbance

and the observer of the USV is designed as

where

**Theorem 2.***For the unknown time-varying disturbances *

**Proof.***Let*

*and differentiate with respect to time, considering variables *

*According to assumption (2), the derivative of unknown time-varying disturbance *.

Based on the above analysis and proof, it can be known that when the velocities of the USV are the desired velocities (0.13), the USV network can ultimately achieve the optimal coverage of the task region

Let

where

**Theorem 3.***Consider the USV dynamics described by (0.2) and (0.3), the designed control law (0.33) is capable of driving the velocities of USVs to track the desired velocities (0.13) within a finite time, and the optimal coverage effect of the task area *

**Proof.***Taking the derivative of time with respect to *

*According to Theorem 2, the observer's estimate of the disturbances converges to the actual value of the disturbances in a finite time, which is denoted by *

*When *

*So, it follows that the error *.

*It can be observed from theorem (2) that the designed observer (0.30) can estimate the external time-varying disturbances within a finite time *.

*Therefore, the designed control law (0.33) can drive the velocities of the USV to track the desired velocities (0.13) within finite time *

**Remark 2***The control law (0.33) contains the variable *.

**Remark 3***From the proof of theorem (3), it can be seen that the convergence time *.

## 5. SIMULATION RESULTS

In order to verify the disturbance observation method and USV coverage control method proposed in this paper, this section presents the simulation results for a scenario involving eight USVs (

The kinematic and dynamic models of the USV are given by (0.1) and (0.3), respectively, and nominal physical parameters are as follows:

and the initial positions and angles of the USVs are randomly assigned. It is assumed that each USV carries the actuator with the same performance, and the performance function is defined as:

Firstly, the simulation results for observations of the time-varying disturbances are displayed in Figure 1, where

Figure 1. When the number of objects

Then, the simulation results of the angle errors and position errors at the desired velocities (0.13) are presented in Figure 2(a) and (b). These results indicate that each USV can drive the angle and position track the desired angle and position, respectively, at the expected velocities of the design.

Figure 2. The curves of angular velocity error, surge velocity error, angle error, and distance error with time

The errors between the angular and surge velocities of each USV and the desired angular and surge velocities are shown in Figure 2(c) and (d). it can be clearly observed that the designed control law (0.33) can drive the velocities of each USV to track the desired velocities designed in (0.13), with external disturbances within the finite time.

Finally, the curves of the metric function (0.7) describing the coverage effect are shown in Figure 3(a), and the comparison between the algorithm designed in this paper and the classic Lloyd algorithm for optimizing regional coverage is shown in Figure 3(b). It is worth noting that the coverage optimization algorithm designed in this paper can achieve a superior coverage effect. Moreover, It is important to highlight that the coverage algorithm proposed in this paper is based on an underactuated USV model with disturbances, while the classic Lloyd algorithm is based on the first-order integral model robot. The coverage process of the USVs is illustrated in Figure 4.

Figure 3. The curve of the metric function with time t (a), and comparison curve between our algorithm and traditional Lloyd algorithm (b).

## 6. CONCLUSIONS

This paper proposes a method for observing unknown disturbances and an optimal coverage controller to address the challenge of region coverage control for a USV network. The proposed disturbance observation method is capable of estimating unknown time-varying disturbances within a finite time. Furthermore, a robust coverage controller is designed to enable the USV network to track the desired velocities within a finite time, achieving an optimal coverage effect of the task region. Simulation results demonstrate the effectiveness of the proposed approach. However, it is important to acknowledge that this paper has certain limitations. Specifically, it only considers simple convex task regions. If the task region is non-convex or contains obstacles, the coverage optimization control proposed in this paper may not be applicable. Therefore, future work will focus on addressing the challenges of collision avoidance and extending the coverage control problem to non-convex regions with obstacles.

## DECLARATIONS

### Authors' contributions

Made significant contributions to the formal analysis and derivation of the content and has conducted the writing of the thesis and the completion of the first draft: Sun Q, Liu ZW

Contributed to the conceptualization of content, the review and editing of the article, and provided administrative, technical, and material support: Chi M, He D

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work was supported by the National Natural Science Foundation of China under Grant 61973133.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

## How to Cite

Sun, Q.; Liu Z. W.; Chi M.; Ge M. F.; He D. Robust coverage control of multiple USVs with time-varying disturbances. *Intell. Robot.* **2023**, *3*, 242-56. http://dx.doi.org/10.20517/ir.2023.15

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