Robust coverage control of multiple USVs with time-varying disturbances

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 $$ {\cal V} $$, which consists of $$ n $$ USVs ($$ \mathcal{V}=\{1, 2, \ldots, n\} $$), is considered in a task region $$ Q \in R^2 $$. The kinematic model of the USV can be described as

where $$ \eta = {\left[ {x, y, \psi } \right]^{\rm T}} $$ are the position and the heading angle in the earth-fixed frame. $$ V = {\left[ {u, v, r } \right]^{\rm T}} $$ denote the surge, sway velocities, and the angular velocity in the body-fixed frame. Considering time-varying disturbances caused by the Marine environment, the dynamics of the USV are:

where $$ M = {M^{\rm T}} \in {R^{3 \times 3}} $$ is the inertial matrix, $$ C(V) \in {R^{3 \times 3}} $$ denotes the Coriolis and centripetal matrix, and $$ D(V) \in {R^{3 \times 3}} $$ denotes the damping matrix. $$ \tau {\rm{ = }}{\left[ {{\tau _u}\; \; 0\; \; {\tau _r}} \right]^{\rm T}} $$ is the control input, and $$ {\tau _d}{\rm{ = }}{\left[ {{\tau _{ud}}\; \; {\tau _{vd}}\; \; {\tau _{rd}}} \right]^{\rm T}} $$ is the unknown time-varying disturbances caused by wind, waves, currents, and other factors. In order to simplify the control design of USVs, assuming that the USV is symmetric, $$ M{\rm{ = }}diag\{ {m_{11}}, {m_{22}}, {m_{33}}\} $$,

and $$ D(V){\rm{ = }}diag\{ {d_{11}}, {d_{22}}, {d_{33}}\} $$. The simplified surface 3-DOF dynamic is:

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 $$ f(\parallel q - {p_i}\parallel ) $$ is used to describe the performance variation for the actuator of the $$ i $$-th USV ^[27,28],

where $$ \parallel q - {p_i}\parallel $$ is the 2-norm of the vector $$ q - {p_i} $$, $$ q \in Q $$ is any point within task region $$ Q $$, $$ {p_i} = {\left[ {{x_i}\; \; \; {y_i}} \right]^{\rm T}} $$ is the position of $$ i $$-th USV, and $$ k_f , \beta >0 $$ are constant coefficients.

The risk density function $$ \Phi (q) $$ is used to quantify the importance of each point $$ q $$ in the task region $$ Q $$

where $$ \phi (q) $$ is the constant risk density value, and $$ {s_j} = {[{x_j}, {y_j}]^{\rm T}} (j=1, ... , m) $$ represents the location of the $$ j-th $$ important object. $$ \phi (q, {s_j}){\rm{ = }}{\alpha _j} \cdot \exp ( - \frac{{\parallel q - {s_j}{\parallel ^2}}}{{2{\delta ^2}}}) (j = 1, 2, \ldots , m) $$ is used to describe the contribution of the $$ j $$-th object on risk density function $$ \Phi (q) $$^[29,30]. $$ {\alpha _j} >0 $$ is the constant coefficient. As the location point $$ q $$ approaches the important object $$ s_j $$, the value of function $$ {\phi (q, {s_j})} $$ increases. Moreover, the risk density function $$ \Phi (q) $$ evaluates the significance of a point based on its value. Therefore, higher values of $$ \Phi (q) $$ indicate points that require greater allocation of resources for monitoring purposes.

The generalized Voronoi partition method is introduced to assign areas for each USV, as described in previous studies^[8,9]. The region $$ Q $$ is divided according to the performance of the actuators carried by different USVs, and the $$ i $$-th USV is assigned a Voronoi partition $$ V_i $$

Next, the effect of USVs executing tasks within the task region $$ Q $$ is described. The metric function $$ H(P) $$ is used to quantify the coverage effect of USVs on the task region $$ Q $$^[31,32]

where $$ P = \left\{ {{p_1}, {p_2}, \ldots , {p_n}} \right\} $$, and each USV is responsible for its Voronoi partition. The larger the value of the measurement function $$ H(P) $$, the better the coverage effect of the USV network on the task region $$ Q $$.

The goal of achieving task region coverage with the USVs is to drive the location configuration of USVs to maximize the metric function $$ H(P) $$. If two USVs have common edges within their respective Voronoi partitions, they can establish communication with each other.

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 1Each USV can accurately measure its position and angle information $$ \eta = {\left[ {x, y, \psi } \right]^{\rm T}} $$ in the earth-fixed frame. The surge, sway velocities, and the angular velocity information $$ V = {\left[ {u, v, r } \right]^{\rm T}} $$ can be obtained in the body-fixed frame through relevant sensors and other equipment. Each USV can communicate with its neighbors.

From the Voronoi partition, there is

where $$ {H_i}{\rm{ = }}\int\limits_{{V_i}} {f(\parallel q - {p_i}\parallel )} \Phi (q)dq, (i \in {{\mathcal V}}). $$

Take the derivative of the $$ H_i $$ with respect to $$ p_i $$

where $$ {G_{vi}} = \int\limits_{{V_i}} {{\alpha _i}{\beta _i}f(\parallel q - {p_i}\parallel )} \Phi (q)dq $$, $$ {C_{vi}} = \frac{1}{{{G_{vi}}}}\int\limits_{{V_i}} {{\alpha _i}{\beta _i}qf(\parallel q - {p_i}\parallel )} \Phi (q)dq $$ are the generalized mass and generalized centroid of the Voronoi partition $$ V_i $$.

Let

Next, the expected velocities of the $$ i $$-th USV will be designed to drive the position of the $$ i $$-th USV $$ (i \in \mathcal V) $$ so as to optimize the coverage effect on the Voronoi partition $$ V_i $$. The expected surge velocity $$ {u_{d}} $$ and angular velocity $$ {r_{d}} $$ of the USV are designed as follows:

where constant coefficient $$ {k_{r1}}, {k_{u1}}>0, $$$$ b_1, b_2 $$ are positive constant.

Lemma 1^[33] Consider the system

where $$ {f}:[0, \infty ) \times {R^n} \times {R^m} \to {R^n} $$ is piecewise continuous in $$ t $$ and locally Lipschitz in $$ x $$ and $$ u $$. The input $$ u(t) $$ is a piecewise continuous, bounded function of $$ t $$ for all $$ t \ge 0 $$. If the unforced system $$ \dot x = f(t, x, 0) $$ has a globally exponentially stable equilibrium point at the origin $$ x=0 $$, then the system $$ \dot x = f(t, x, u) $$ is input-to-state stable.

Lemma 2^[33] For the cascade system

where $$ {f_1}:[0, \infty ) \times {R^{{m_1}}} \times {R^{{m_2}}} \to {R^{{m_1}}} $$ and $$ {f_2}:[0, \infty ) \times {R^{{m_2}}} \to {R^{{m_2}}} $$ are piecewise continuous in $$ t $$ and locally Lipschitz in $$ {\left[ {{x_1}, {x_2}} \right]^{\rm T}}. $$ If the system $$ {{\dot x}_1} = {f_1}(t, {x_1}, {x_2}) $$, with $$ x_2 $$ as input, is input-to-stable and the origin of $$ {{\dot x}_2} = {f_1}(t, {x_2}) $$ is globally uniformly asymptotically stable, then the origin of the cascade system is globally uniformly asymptotically stable.

Theorem 1Consider 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 $$ r_{id} $$ in (0.13), it gives

It can be inferred that the error $$ {\psi _{ie}} $$ has a stable equilibrium point at the origin, which is globally exponentially stable.

Next, we define two quantities,

Then, one has

Taking the derivative of the quantity $$ x_2 $$ yields

From equation (0.12), using $$ {\psi _{ir}} - {\psi _{ie}} $$ to replace $$ {\psi _i} $$, one has

According to equation (0.11), $$ \cos {\psi_{ir}} = \frac{{{x_{ei}}}}{{{E_i}}}, \sin {\psi_{ir}} = \frac{{{y_{ei}}}}{{{E_i}}} $$, the expression yields

Note that

For the system

one has

Since $$ {e_i^{\rm T}}{\dot C_{Vi}} \le {\rm{||}}{e_i^{\rm T}}{\rm{||}} \cdot {\rm{||}}{\dot C_{Vi}}{\rm{||}} $$, it follows that

The globally exponentially stable equilibrium point lies at the origin for the system $$ {\dot x_2} = f(t, {x_2}, 0) $$. Based on Lemma (1), it can be inferred that the system $$ {\dot x_2} = f(t, {x_2}, {x_1}) $$ is input-to-state stable.

As the system $$ {\dot x_1} = f(t, {x_1}) $$ exhibits global exponential convergence, it can be concluded that the cascade system $$ (x_1, x_2) $$ is globally uniformly asymptotically stable according to Lemma (2). This suggests that $$ \mathop {\lim }\limits_{t \to \infty } {e_i} = 0, (i \in \mathcal V). $$

Because $$ \frac{{d{H_i}}}{{d{p_i}}}{\rm{ = }}2{G_{Vi}}{({C_{Vi}} - {p_i})^{\rm T}} $$, it can be noted that if the position $$ p_i $$ of the $$ i $$-th USV is located at the generalized centroid $$ C_{Vi} $$ of its Voronoi partition $$ V_i $$($$ e_i=0 $$), there are $$ \frac{{d{H_i}}}{{d{p_i}}}{\rm{ = }} 0 $$ and the $$ i $$-th USV achieves the optimal coverage of partition $$ V_i(i \in \mathcal V) $$. When each USV achieves the optimal coverage of its respective Voronoi partition, the USV network attains the optimal coverage effect of the task region $$ Q $$.

Remark 1It can be shown that the quantity $$ ||{{\dot C}_{Vi}}|| $$ exists and is bounded, and the quantity $$ {{\dot \psi }_{ir}} $$ exists and is bounded when $$ E \ne 0 $$. The quantity $$ {{\dot \psi }_{ir}} $$ has a singularity at $$ E=0 $$. At the singularity, $$ E=0 $$, the USV has achieved optimal coverage.

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 2The first time derivative of the unknown time-varying disturbances $$ {\tau _d}{\rm{ = }}{\left[ {{\tau _{ud}}\; \; {\tau _{vd}}\; \; {\tau _{rd}}} \right]^{\rm T}} $$ are bounded $$ (\parallel {\dot \tau _d}\parallel < B) $$.

Lemma 3(^[34]) Consider the nonlinear system $$ (y_1, y_2) $$. If the system satisfied the following equations:

where $$ b_1>0 $$, $$ b_2>0 $$, and $$ \delta $$ is a bounded quantity. Then the system will converge to the origin and stabilize within a finite time.

Considering the dynamic model (0.3), the control input $$ \tau {\rm{ = }}{\left[ {{\tau _u}\; \; 0\; \; {\tau _r}} \right]^{\rm T}} $$ is designed to drive the velocities of the USV tracking the desired velocities (0.13) under the unknown time-varying disturbances $$ {\tau _d}{\rm{ = }}{\left[ {{\tau _{ud}}\; \; {\tau _{vd}}\; \; {\tau _{rd}}} \right]^{\rm T}} $$. It is worth noting that the designed control input $$ \tau $$ can achieve the convergence of the velocities tracking error within a finite time.

First, the observer is designed to approximate the unknown external disturbance $$ {\tau _{d}} $$. Let

and the observer of the USV is designed as

where $$ {\Delta } = - {b_{1}}|{M}({{\bar V}} - {V}){|^{\frac{1}{2}}}sign({M}({{\bar V}} - {V}) + {{\bar {\tau} }_{d}}, $$$$ b_{1}, b_{2}>0 $$ are constant coefficients, $$ {{\bar V}} = {\left[ {{{\bar u}}, {{\bar v}}, {{\bar r}}} \right]^{\rm{T}}} $$ is the estimate of the velocity vector $$ V = {\left[ {u, v, r} \right]^{\rm T}} $$, and $$ {\bar \tau} _{d} = {\left[ {{\bar\tau _{ud}}\;\;{\bar\tau _{vd}}\;\;{\bar\tau _{rd}}} \right]^{\rm{T}}} $$ is the estimate of the unknown time-varying disturbance vector $$ {\tau _{d}}{\rm{ = }}{\left[ {{\tau _{ud}}\; \; {\tau _{vd}}\; \; {\tau _{rd}}} \right]^{\rm T}} $$.

Theorem 2.For the unknown time-varying disturbances $$ {\tau _{d}} $$ in the USV dynamic model (0.3), they can be estimated by the designed observer (0.30) within a finite time.

Proof.Let

and differentiate with respect to time, considering variables $$ \Omega $$ and $$ {\tilde \tau _d} $$,

According to assumption (2), the derivative of unknown time-varying disturbance $$ {{\dot \tau }_d} $$ exists and is bounded. It can be concluded from (0.32) by using Lemma (3) that the states $$ \Omega $$ and $$ {\tilde \tau _d} $$ are finite-time stable. Thus, the estimator $$ {\bar \tau} _{d} $$ converges to the actual disturbance $$ {\tau} _{d} $$ within a finite time.

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 $$ Q $$. Subsequently, a robust controller is designed for the USV to track the desired velocities (0.13) within a finite time while compensating for external disturbances.

Let $$ {u_{e}} = {u} - {u_{d}} $$ be the error and $$ {r_{e}} = {r} - {r_{d}} $$ be the error. The controller of the USV is designed as

where $$ \left\{ \begin{array}{l} \nabla {u} = - {k_{{\tau _u}1}}si{g^{{\lambda _1}({\xi _1})}}({u_{e}}) - {k_{{\tau _u}2}}si{g^{{\eta _1}({\varsigma _1})}}({u_{e}}), \\ \nabla {r} = - {k_{{\tau _r}1}}si{g^{{\lambda _2}({\xi _2})}}({r_{e}}) - {k_{{\tau _r}2}}si{g^{{\eta _2}({\varsigma _2})}}({r_{e}}), \end{array} \right. $$$$ {k_{{\tau _u}1}}, {k_{{\tau _u}2}}, {k_{{\tau _r}1}}, {k_{{\tau _r}2}} > 0 $$, $$ {\mathop{\rm s}\nolimits} i{g^r}(s) = {\mathop{\rm sgn}} (s)|s{|^r} $$ and

$$ \xi_1 >0, {{\xi _2}}>0 $$ and $$ 0 < {\varsigma _1} < \frac{1}{2}, 0 < {\varsigma _2} < \frac{1}{2} $$.

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 $$ Q $$ can finally be achieved.

Proof.Taking the derivative of time with respect to $$ u_{e} $$, we can obtain

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 $$ T_1 $$. When $$ t>T_1 $$, one has

When $$ |{u_{e}}| > = 1 $$, Substituting (0.34) into (0.37) yields $$ {\dot u_{e}} = - {k_{{\tau _u}1}}si{g^{1 + {\xi _1}}}({u_{e}}) - {k_{{\tau _u}2}}sig({u_{e}}). $$ Let $$ y=|{u_{e}}| $$, it can be obtained that $$ \dot y = - {k_{{\tau _u}1}}{y^{1 + {\xi _1}}} - {k_{{\tau _u}2}}y. $$ When $$ |{u_{e}}| < 1 $$, Substituting (0.34) into (0.37) yields $$ {{\dot u}_{e}} = - {k_{{\tau _u}1}}si{g}({u_{e}}) - {k_{{\tau _u}2}}si{g^{{1-\varsigma _1}}}({u_{e}}). $$ Let $$ y=|{u_{e}}| ^{\varsigma_1} $$, we can get $$ \dot y = - {k_{{\tau _u}1}}{\varsigma _1}y - {k_{{\tau _u}2}}{\varsigma _1} $$. Suppose error $$ u_{e} $$ starts at infinity $$ ({{y(0)} \to \infty }) $$. The time $$ T_2 $$ it takes for error $$ u_{e} $$ to converge from infinity to zero is

So, it follows that the error $$ u_{e} $$ converges to 0 in time $$ T_2 $$. The same analysis yields that the error $$ r_{e} $$ converges to 0 in time $$ T_2 $$.

It can be observed from theorem (2) that the designed observer (0.30) can estimate the external time-varying disturbances within a finite time $$ T_1 $$. Based on the above analysis, it can be concluded that the designed controller (0.33) can force the velocity error of the USV to converge to zero in $$ T_2 $$ after the external disturbance is estimated.

Therefore, the designed control law (0.33) can drive the velocities of the USV to track the desired velocities (0.13) within finite time $$ T_1+T_2 $$. Therefore, when $$ t>T $$, the velocities of the USV reach the expected velocities (0.13). Theorem (1) implies that maximizing the metric function (0.7) at the desired velocities (0.13) results in achieving optimal coverage of the task area $$ Q $$.

Remark 2The control law (0.33) contains the variable $$ \frac{{d{{\dot \psi }_{ir}}}}{{dt}} $$, which exists and is bounded when $$ E \ne 0 $$. However, it has a singularity at $$ E=0 $$. At the singularity, $$ E=0 $$, the USV has achieved optimal coverage.

Remark 3From the proof of theorem (3), it can be seen that the convergence time $$ T_2 $$ of the velocity errors is only related to the designed controller parameters $$ {{{k_{{\tau _u}1}}, {k_{{\tau _u}2}}}}, {\varsigma _1}, {\varsigma _2} $$ and is independent of the initial state of the system. Thus, it is fixed time stable. However, considering the observation errors of the time-varying disturbances converge to zero within a finite time $$ T_1 $$, the final conclusion is that the controller (0.33) forces velocity errors of the USV to converge to zero within a finite time $$ T_1+T_2 $$.

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 ($$ n=8 $$) and three important objects ($$ m=3 $$). Consider a $$ 100m \times 100m $$ task area $$ Q $$; the risk density function is given by (0.5). The basic constant risk density $$ \phi(q) =0.1 $$ and the contribution function $$ {\phi _j}(q, s_j) (j=1, 2, 3) $$ of objects to the risk density are defined as follows:

The kinematic and dynamic models of the USV are given by (0.1) and (0.3), respectively, and nominal physical parameters are as follows: $$ m_{11} = m_{22}=m_{33}=2000, d_{11} = 20, d_{22}=30, d_{33} = 35 $$. For the $$ i $$-th USV ($$ i \in \mathcal V $$), the time-varying disturbances

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 $$ {\tau _{ue}} = [{{\bar \tau }_{1ud}}, {{\bar \tau }_{2ud}}, \ldots, {{\bar \tau }_{8ud}}] - [{\tau _{1ud}}, {\tau _{2ud}}, \ldots, {\tau _{8ud}}] $$ represents the observation error of the disturbance in surge velocity for each of the eight USVs, and $$ {\tau _{ve}} $$ and $$ {\tau _{re}} $$ are similar. It can be noted that the observation errors $$ {\tau _{ue}} $$, $$ {\tau _{ve}} $$, and $$ {\tau _{re}} $$ converge to zero in finite time. This indicates that, under the designed disturbance vector observer, each USV effectively estimates the unknown time-varying disturbances, even though the input disturbances of each USV are time-varying and distinct.

Figure 1. When the number of objects $$ m=3 $$ and the number of USV $$ n=8 $$, the curves of observation errors with time $$ t $$ for time-varying disturbances. USVs: unmanned surface vehicles.

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 $$ t $$.

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).

Figure 4. The coverage evolution process of the USVs to the task area, where the diamond represents the position of the USV, and different colors represent different risk density function values of the task area environment.

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