Long Duration Coverage Control of Multiple Robotic Surface Vehicles Under Battery Energy Constraints

Dear Editor,

This letter addresses long duration coverage problem of multiple robotic surface vehicles (RSVs) subject to battery energy constraints, in addition to uncertainties and disturbances. An anti-disturbance energy-aware control method is proposed for performing coverage task of RSVs. Firstly, a centroidal Voronoi tessellation (CVT) is used to optimize the partition of the given coverage area. The optimal position for each vehicle corresponds to the centroid of the Voronoi cell. Secondly, by consisting two battery energy control barrier functions, an energy-aware kinematic guidance law is designed to drive each RSV to the optimal position. Finally, an anti-disturbance fixed-time kinetic control law is designed for each RSV to track the desired speed based on the fixed-time extended state observer and nonlinear tracking differentiator. By the control method, RSVs are capable of achieving the long duration coverage within the given coverage area. Simulation results verify the effectiveness of the proposed anti-disturbance energy-aware control method for multi-RSV.

There is a surge of interest in motion control of RSVs because they are vital tools in exploring the oceans [1]–[3]. Cooperative control enables RSVs to perform more challenging task over a single RSV in terms of enhanced efficiency and effectiveness [4], [5]. In particular, coverage control is a typical cooperative control of RSVs. Such motion control scenario can find numerous missions in practice such as maritime patrol and environment monitoring [1], [6]–[9]. Coverage control is proposed for unmanned ground vehicles [6], unmanned aerial vehicles [7], and robotic surface vehicles [8]. However, the aforementioned works [6]–[9] do not explicitly guarantee the persistent coverage performance. In practice, the vehicles may not execute task persistently because of the battery energy constraints.

Motivated by the above observations, this letter aims to design an anti-disturbance energy-aware control method for RSVs to execute long duration coverage task. The main contributions of this letter can be summarized as: 1) In contrast to the works in [4], [5] where the objectives are to maintain formation, the proposed control method for multi-RSV to achieve coverage task within a given coverage area. 2) In contrast to the existing works [6]–[9] where the persistent coverage performance may not be guaranteed definitely, an energy-aware control method based on two battery energy control barrier functions (BE-CBFs) are designed to achieve long duration coverage mission. 3) In contrast to the existing coverage control methods in [8], [9] where the convergence time of estimation errors can not be calculated in a prescribed time, a fixed time extended state observer (FTESO) is proposed such that the estimation errors are convergent to zero within a fixed time$.$

Problem formulation: Consider the dynamics of RSVs as

where $x_i$, $y_i$, and ${\psi}_{bi}$ are the position and orientation in an earth-fixed inertial frame; $u_i$, $v_i$, and $r_i$ are the velocities in the surge, sway, and yaw directions respectively in a body-fixed frame; $m_{ui}$, $m_{\upsilon i}$, and $m_{ri}$ are the mass of RSVs; $f_{ui}(\cdot)$, $f_{\upsilon i}(\cdot)$, and $f_{ri}(\cdot)$ represent unknown nonlinear functions; $\tau_{ui}$ and $\tau_{ri}$ denote the surge force and yaw moment; $\tau_{wui}$, $\tau_{w\upsilon i}$, and $\tau_{wri}$ denote the time-varying environmental disturbances. Denote $U_i = \sqrt{u^2_i+\upsilon^2_i}$ as the total speed of the ith RSV, and the dynamics (1) of the RSV can be rewritten as

where ${\psi_{i}} = {\psi_{bi}+\beta_{i}}$ is the course angle with $\beta_{i} = \text{atan2}(\upsilon_i, u_i)$.

The charging and discharging model of the battery is stated as $\dot{E}_i = -\delta K_{ch}+(1-\delta)K_{en}$, where $E_{i}$ is the battery energy level of the ith RSV; $K_{ch}>0$/$K_{en}>0$ represents the case of the worst/fastest battery discharging/charging; $\delta\in [0,1]$ is the switching coefficient of battery charging and discharging.

Design and analysis: Step 1: The first step is optimizing the partition of the given coverage area. CVT is a powerful tool for the area partition. The centroid of each Voronoi cell can be defined as the optimal position for each RSV to track. Consider that N RSVs are randomly distributed in a closed and connected area $G\in{{\mathbb{R}}^2}$ that has to be covered. Associate a density function $\phi(g):g\in {{\mathbb{R}}^+}$ with each point $g \in G$. Define locational cost function of the multi-RSV as $H(P) = \sum_{i = 1}^N {{{\int_{{V_i}} {\left\|{{p_i} - g}\right\|}^2}}\phi(g)dg}$, where $p_i = [x_{i},y_{i}]^T$ is the position of ith RSV; $P = [{p_1},\cdots,{p_N}]$ is position set of RSVs; $\left\|{{p_i} - g}\right\|$ denotes the Euclidean distance metric. Next, the Voronoi cell of the ith RSV can be defined as ${V_i} = \{g\in G\left|\|{g - p_i}\|\leq \|{g - p_j}\|\right.,\forall i \ne j\}$. Then, it follows from [8] that the optimal position ${p}_{ci}$ corresponds to the centroid of the Voronoi cell in a CVT is:

where $M_{V_i}$ is the mass of the ${V_i}$ with $M_{V_i} = \int_{V_i}{\phi(g)dg}$.

Step 2: Define the target tracking error as $e_{pi} = p_i-{p}_{ci}$. Take the derivative of $e_{pi}$ along (2), and one has $\dot{e}_{pi} = q_i-\dot{p}_{ci}$, where $q_i = [q_{xi},q_{yi}]^T = U_i[\cos ({\psi_{bi}}),\sin ({\psi_{bi}})]^T$. To stabilize $e_{pi}$, an optimal total speed guidance law is designed as ${q}_{ci} = -{K_{qi}e_{pi}}/{\sqrt{\|e_{pi}\|^2+\Delta_{qi}^2}}+ \dot{p}_{ci}$, where $K_{qi}\in {\mathbb{R}}^2$ is the guidance law parameter vector and $\Delta_{qi}$ is a positive constant. Define the velocity tracking error as ${e_{qi}} = {q_i} - q_{ci}$ and the time derivative of the $e_{pi}$ can be rewritten

To implement the long duration coverage control task, the position of charging station $p_{bi}$ should satisfy $E_{\text{charge}} = \{{\varepsilon_i}(p_i)\in{{\mathbb{R}}^2}|\|p_i- p_{bi}\|\leq{d_{{ch}}}\}$, where ${\varepsilon _i}({p_i})$ is the minimum energy required for the ith RSV to return to the charging station and $d_{\text{ch}}$ is the radius of the charging station. In order to maintain a desired energy reserve, the following inequalities $E_i - E_{\min }>{\varepsilon _i}({p_i})$ and $E_{\max } - E_i>0$ hold, where $E_{\min}$ denotes the minimum capacity of battery storage and $E_{\max}$ is the maximum capacity of battery storage. It means that each RSV can return to the charging station before limited battery energy depletion and leave the charging station before overcharging. According to the definition of the CBF, the functions $h_{en}$ and $h_{cn}$ are defined as $h_{en}(z_i) = {E_i}-E_{\min}-{\varepsilon_i}(p_i)$ and $h_{ch}(z_i) = E_{\max}\delta(s_{i0}) - E_i$, where $z_i = [p_i^T,E_i]^T$, $s_{i0} = \|p_i - p_{i0}\|$, and $\delta(s_{i0})$ is a step function. If $s_{i0} > d_{{ch}}$, $\delta(s_{i0}) = 1$; otherwise, $\delta(s_{i0}) = 0$. The safe set ${\cal{C}}_{zi}$ in this letter is defined as ${{\cal{C}}_{zi}} = \{{z_i}\left|{h_{en}(z_i)\geq 0},{h_{ch}(z_i)\geq 0}\right.\}$.

It follows from the safe set that the functions $h_{en}$ and $h_{cn}$ can be described by an equivalent $h(z_i)$ as $h(z_i) = \text{min}\{ h_{en}(z_i),h_{ch}(z_i)\}$. Thus, the speed guidance law $q_{ci}^* = [q_{xi}^*,q_{yi}^*]^T$ can be obtained by the following quadratic programming problem:

where $\dot h(z_i) = (\frac{\partial h}{\partial p_i})^Tq_{ci}^*-\delta K_{ch}+(1-\delta)K_{en}$. The optimal total speed of the ith RSV is derived as $U_{ci} = q_{xi}^*\cos(\psi_{i})+q_{yi}^*\sin(\psi_{i})$.

Step 3: Define $\psi_{ci} = \text{atan2}(q_{yi}^*, q_{xi}^*)$ and $e_{\psi i} = \psi_{i}-\psi_{ci}$, and the dynamics of $e_{\psi i}$ is $\dot{e}_{\psi i} = r_i+\dot{\beta}_{i}-\dot{\psi}_{ci}$. To stabilize $e_{\psi i}$, a yaw guidance law is designed as

where $k_{ri}\in {\mathbb{R}}$ is the guidance law parameter and $\Delta_{ri}$ is a positive constant. Define ${e_{ri}} = {r_i} - r_{ci}$, and the dynamics of the yaw target tracking error is rewritten as

Step 4: Due to the fact that the uncertainties and disturbances can affect the performance of the control system seriously [10]–[12], it is necessary to recover the uncertainties and disturbances. First, to obtain a smoother motion profile for RSV, let ${U}_{ci}$ and ${ r}_{ci}$ pass through two nonlinear tracking differentiators as

where $\hat{U}_{ci}$ is the estimate of the ${U}_{ci}$; $\hat{r}_{i}$ is the estimate of the ${r}_{ci}$; ${U}_{di}$ is the estimate of the $\dot{U}_{ci}$, and ${r}_{di}$ is the estimate of the $\dot{r}_{ci}$. $k_{i1}^U\in{\mathbb{R}}^+$, $k_{i2}^U\in{\mathbb{R}}^+$, $k_{i3}^U>2$, $k_{i1}^r\in{\mathbb{R}}^+$, $k_{i2}^r\in{\mathbb{R}}^+$, and $k_{i3}^r>2$ are the design parameters. Define $e_{Ui}^c = \hat{U}_{i}-{U}_{ci}$, $e_{ri}^c = \hat{r}_{i}-{r}_{ci}$, $e_{Ui}^d = {U}_{di}-\dot{U}_{ci}$, and $e_{ri}^d =$ ${r}_{di}- \dot{r}_{ci}$. It follows from (8) that the dynamics of the $e_{Ui}^c$, $e_{ri}^c$, $e_{Ui}^d$, and $e_{ri}^d$ are obtained as:

From (2), the kinetics of the ith RSV can be rewritten as $\dot{U}_i = g_{ui}+m_{ui}^{-1}\tau_{ui}$ and $\dot{r}_i = g_{ri}+m_{ri}^{-1}\tau_{ri}$, where $g_{ui} = m_{ui}^{-1}(\cos ({\beta_{i}})(f_{ui}+ \tau_{wui})- 2\sin^2(\frac{\beta_i}{2})\tau_{ui}+\sin ({\beta_{i}})(f_{\upsilon i}+\tau_{w\upsilon i})m_{ui}/m_{\upsilon i})$ and $g_{ri} = m_{ri}^{-1}(f_{ri}+ \tau_{wri})$. Two fixed-time ESOs are designed to estimate the unknown functions $g_{ui}$ and $g_{ri}$

where $k_{ia}^u\in {{\mathbb{R}}^+}$ and $k_{ia}^r\in {{\mathbb{R}}^+}(a = 1,2,3,4)$ are the observer gains; $\hat {U}_i$, $\hat {r}_i$, $\hat g_{ui}$, and $\hat g_{ri}$ are the estimates of $U_i$, $r_i$, $g_{ui}$, and $g_{ri}$ respectively; ${\tilde U_i} = {\hat U_i} - {U_i}$, ${\tilde r_i} = {\hat r_i} - {r_i}$, $\alpha_{ui} \in (1-\varepsilon_{ui},1)$, $\beta_{ui} = 1/\alpha_{ui}$, $\alpha_{ri} \in (1- \varepsilon_{ri},1)$, and $\beta_{ri} = 1/\alpha_{ri}$ with $\varepsilon_{ui}$ and $\varepsilon_{ri}$ being the small constants; $k_{i0}^u>0$ and $k_{i0}^r >0$.

Assumption 1: For unknown functions $g_{ui}$ and $g_{ri}$, there are $g_{ui}^*\in {\mathbb{R}}^+$ and $g_{ri}^*\in {\mathbb{R}}^+$ such that $|\dot g_{ui}|\le g_{ui}^*$ and $|\dot g_{ri}|\le g_{ri}^*$.

Step 5: Let ${\tilde g_{ui}} = {\hat g_{ui}}-{g_{ui}}$ and ${\tilde g_{ri}} = {\hat g_{ri}}-{g_{ri}}$. It follows from equation (10) that the error dynamics can be expressed as:

Define the tracking errors as $e_{Ui} = U_i-\hat{U}_{ci}$ and $e_{ri} = r_i-\hat{r}_{ci}$. Taking the derivatives of $e_{Ui}$ and $e_{ri}$ yields

Two anti-disturbance fixed-time kinetic control laws are designed

where $k_{ia}^{\tau}\in {{\mathbb{R}}^+} (a = 1,2,3,4)$ are the control gains. Then, the tracking error systems (12) can be put into $\dot{e}_{Ui} = -k_{i1}^{\tau}{\lceil{e}_{Ui}\rfloor}^{\alpha_{ui}^{\tau}}-k_{i2}^{\tau}{\lceil{e}_{Ui}\rfloor}^{\beta_{ui}^{\tau}}- {{\tilde g}_{ui}}$ and $\dot{e}_{ri} = -k_{i3}^{\tau}{\lceil{e}_{ri}\rfloor}^{\alpha_{ri}^{\tau}}-k_{i4}^{\tau}{\lceil{e}_{ri}\rfloor}^{\beta_{ri}^{\tau}}- {{\tilde g}_{ri}}$.

Lemma 1: The tracking error subsystem consisting of (4) and (7): $[{e}_{qi}, {e}_{ri}]$ $\mapsto$ $[{e}_{pi}, {e}_{\psi i}]$ is input-to-state stable (ISS).

Proof: Choose the Lyapunov function as $V_{i1}^e = \sum_{i = 1}^{N}(1/2)({e}_{pi}^T{e}_{pi}+ {e}_{\psi i}^2)$. The time derivative of $V_{i1}^e$ is $\dot{V}_{i1}^e\leq \sum_{i = 1}^{N} (-\lambda_{\max}(K_{i1})||H_{i1}||^2/ \sqrt{||H_{i1}||^2+\delta_{i \max}}+||h_{i1}||||H_{i1}||)$, where $H_{i1} = [{e}_{pi}^T, {e}_{\psi i}]$, $K_{i1} = \text{diag}\{K_{qi}, K_{ri}\}$, $\delta_{i \max} = \max\{\delta_{i q}, \delta_{i r}\}$, $h_{i1} = [{e}_{qi}, {e}_{r i}]$, and $\lambda_{\min}(K_{i1})$ is the minimum eigenvalue of a square matrix $K_{i1}$. Note that as $||H_{i1}||^2/$ $\sqrt{||H_{i1}||^2+\delta_{i \max}}\geq ||h_{i1}||/(a_{i1}\lambda_{\min}(K_{i1}))$, $\dot{V}_{i1}^e\;\leq\; -\;\sum_{i = 1}^N\{\lambda_{\min}(K_{i1})(1- a_{i1})\}$.■

Lemma 2: Under Assumption 1, with the FTESO (10), the states $\hat{U}_i$, $\hat{g}_{ui}$, $\hat{r}_{i}$, and $\hat{g}_{ri}$ can achieve the estimations in a fixed time.

Proof: Take the following part from error dynamics (11):

Define ${F}_{i1} = [\tilde{U}_{i},\tilde{{g}}_{ui}]^T$ and ${\zeta}_{i1} = [\tilde{U}_{i}, ((\tilde{{g}}_{ui})^{1/\alpha_{ui}})]^T$. Consider a Lyapunov function $V_{i1}^u({\zeta}_{i1}) = {\zeta}_{i1} ^T P_{i1}{\zeta}_{i1}$, where $P_{i1}$ is a symmetric positive definite matrix satisfying $P_{i1}A_{i1}+A_{i1}^TP_{i1} = -I$ and $A_{i1}$ is a Hurwitz matrix with $A_{i1} = [-k_{i1}^u,1;-k_{i3}^u,0]$. Setting $\alpha_{ui} = 1$, the equation (14) becomes $\dot{F}_{i1} = A_{i1}{F}_{i1}$. Taking the time derivatives of $V_{i1}^u({F}_{i1})$ gives $\dot{V}_{i1}^u({F}_{i1}) = F_{i1}^T(P_{i1}A_{i1}+A_{i1}^TP_{i1})F_{i1}\; =\; -F_{i1}^TF_{i1}\;\leq\; 0$. Define $\sigma_{ui}$ as a small constant, and (14) is locally asymptotically stable by choosing $\alpha_{ui}$ in the interval $(1-\sigma_{ui},1)$. Then, from Lemma 2 in [5], ${V}_{i1}^u({\zeta}_{i1})$ is homogeneous of degree $\varrho_{i1}$ with $(0,\alpha_{ui})$ and its derivative $\dot{V}_{i1}^u({\zeta}_{i1})$ is homogeneous of degree $\varrho_{i1}+\alpha_{ui}-1$. One has $V_{i1}^u({F}_{i1})\leq \lambda_{\max}(P_{i})||F_{i1}||^2$ and $\dot{V}_{i1}^u({F}_{i1})\leq$ $-||F_{i1}||^2$. It renders that $\dot{V}_{i1}^u({F}_{i1})\leq -{V}_{i1}^u({F}_{i1})/\lambda_{\max}(P_{i1})$. The second part of (11) is expressed as ${\dot{\tilde{U}}}_i = -k_{i2}^u {\lceil{\tilde U}_i\rfloor}^{\beta_{u i}} + {{\tilde g}_{ui}}$ and ${\dot{\tilde{g}}}_{ui} = -k_{i4}^u {\lceil{\tilde U}_i\rfloor}^{2\beta_{u i}-1}$. Define ${\zeta}_{i2} = [\tilde{U}_{i},$ $((\tilde{{g}}_{ui})^{1/\beta_{ui}})]^T$. Consider a Lyapunov function $V_{i2}({\zeta}_{i2}) = {\zeta}_{i2} ^T P_{i2}{\zeta}_{i2}$. Similar to the analysis in (14), one has $\dot{V}_{i2}({\zeta}_{i2})\leq$ $-{V}_{i2}({\zeta}_{i2})^{(\varrho_{i2}+\beta_{ui}-1)/\varrho_{i2}}/ \lambda_{\max}(P_{i2})$. For

there exists $\iota_{ui}$ such that ${V}_{i2}^u({\zeta}_{i2}(t_0))>\iota_{ui}$ with $\iota_{ui}$ being $0<\iota_{ui}< \lambda_{\min}(P_{i2})$. It renders that ${V}_{i2}^u$ reaches $\iota_{ui}$ within a fixed-time $T_{i1}^u = \varrho_{i2}\lambda_{\max}(P_{i2})/(\beta_{ui}-1)\iota_{ui}^{(\beta_{ui}-1)/\varrho_{i2}}$. Besides, the inequalities $||{\zeta}_{i2}||^2\leq {V}_{i2}^u({\zeta}_{i2})/\lambda_{\min}(P_{i2})\leq \iota_{ui}/\lambda_{\min}(P_{i2})\leq 1$ and ${V}_{i1}^u({\zeta}_{i1})\leq \lambda_{\max}(P_{i1})||{\zeta}_{i1}||^2$ hold. If ${V}_{i1}^u({\zeta}_{i1})<\iota_{ui}$ and $||{\zeta}_{i1}|| = 1$, $||{\zeta}_{i1}||^2\leq ||{\zeta}_{i2}||^2\leq 1$ and ${V}_{i1}({\zeta}_{i1})\leq$ $\lambda_{\max}(P_{i1})$. The settling time of the system (14) satisfies $T_{i2}^u\leq$ $\varrho_{i1} \lambda_{\max}^{(\varrho_{i1}+\alpha_{ui}-1)/\varrho_{i1}}(P_{i1})/(1-\alpha_{ui})$. As a result, the states of the system (15) are convergent to zero within a fixed time $T_{i1}^u+T_{i2}^u$. If $t\geq$ $T_{i1}^u+T_{i2}^u$, the system (11) becomes ${\dot{\tilde{g}}}_{ui} = k_{i0}^u\text{sgn}({{\tilde U}_i})-{{\dot g}_{ui}}$. Consider a Lyapunov function ${V}^u_{i3} =( 1/2)({\tilde{g}}_{ui})^2$. The derivative of ${V}_{i3}^u$ can be obtained as $\dot{V}_{i3}^u\leq -(k_{i0}^u-{g}^*_{ui})\sqrt{(2{V}_{i3}^u)}$. In summary, ${\tilde{g}}_{ui}$ is convergent to zero within the time $T_{i3}^u\leq \sqrt{2{V}_{i3}^u(T_{i1}^u+T_{i2}^u)}/(k_{i0}^u-{g}^*_{ui})$. Thus, the states ${\tilde{U}}_{i}$ and ${\tilde{g}}_{ui}$ are convergent to zero within a fixed time $T_{ui}\leq T_{i1}^u+T_{i2}^u+T_{i3}^u$ regardless of the initial conditions. Similarly, the states ${\tilde{r}}_{i}$ and ${\tilde{g}}_{ri}$ achieve the estimations within the fixed time $T_{ri}\leq T_{i1}^r+T_{i2}^r+T_{i3}^r$ with $T_{i1}^r = \varrho_{i4}\lambda_{\max}(P_{i4})/(\beta_{ri}-1)\iota_{ri}^{(\beta_{ri}-1)/\varrho_{i4}}$, $T_{i2}^r\leq \varrho_{i3} \lambda_{\max}^{(\varrho_{i3}+\alpha_{ri}-1)/\varrho_{i3}}(P_{i3})/(1-\alpha_{ri})$, and $T_{i3}^r\leq \sqrt{2{V}_{i3}^r(T_{i1}^r+T_{i2}^r)}/(k_{i0}^r-{g}^*_{ri})$ regardless of the initial conditions. The definitions of $P_{i2}$, $P_{i3}$, $P_{i4}$, $A_{i2}$, $A_{i3}$, $A_{i4}$, $\varrho_{i2}$, $\varrho_{i3}$, $\varrho_{i4}$, and $\iota_{ri}$ analogize the definitions of $P_{i1}$, $A_{i1}$, $\varrho_{i1}$, and $\iota_{ui}$.■

Theorem 1: Under Assumption 1, by using the optimal total speed guidance law (5), the yaw guidance law (6), the nonlinear tracking differentiator (8), the fixed-time ESOs (10), and the anti-disturbance kinetic fixed-time control laws (12), the RSVs described by (2) are able to cover the centroid of the Voronoi cell (3) at any initial sates with the safe set ${{\cal{C}}_{zi}}$. Besides, all the errors in the closed-loop system are uniformly ultimately bounded.

Proof: According to Lemma 1, the tracking errors ${e}_{qi}$, ${e}_{pi}$, and ${e}_{\psi i}$ are uniformly ultimately bounded (UUB). By [5], it can obtain that the errors $e_{Ui}^c$, $e_{ri}^c$, $e_{Ui}^d$, and $e_{ri}^d$ can converge to a small neighborhood of the origin. Construct the Lyapunov function as $V_{i4} =( 1/2)({e}_{Ui}^2+ {e}_{ri}^2)$. Taking the time derivative of the $V_{i4}$ along (12), it can be obtained that $\dot{V}_{i4}\;\leq \;-2^{(1+\alpha_{ui}^{\tau})/2}k_{i1}^{\tau}(1/2 |{e}_{Ui}|^2)^{1+\alpha_{ui}^{\tau}}\;-\; 2^{(1+\beta_{ui}^{\tau})/2}k_{i2}^{\tau} \times$ $(1/2 |{e}_{Ui}|^2)^{1\;+\;\beta_{ui}^{\tau}}\;-$ $2^{(1+\alpha_{ri}^{\tau})/2}k_{i3}^{\tau}((1/2) |{e}_{ri}|^2)^{1\;+\;\alpha_{ri}^{\tau}}\;-\;2^{(1+\beta_{ri}^{\tau})/2}\;k_{i4}^{\tau}\;((1/$ $2 )|{e}_{ri}|^2)^{1+\beta_{ri}^{\tau}}+ g_{ui}^*|{e}_{Ui}|+g_{ri}^*|{e}_{ri}|$. Then, the following inequality holds $\dot{V}_{i4}\leq -k_{i1}(V_{i4})^{\alpha_i}-2^{1-\beta_i}k_{i2}(V_{i4})^{\beta_i}+g_{ui}^*|{e}_{Ui}|+g_{ri}^*|{e}_{ri}|$, where $k_{i1}>0$, $k_{i2}>0$, $1>\alpha_i>0$, and $\beta_i>1$. It follows from Lemma 2 that the errors $|{{\tilde g}_{ui}}|$ and $|{{\tilde g}_{ri}}|$ converge to zero within a fixed-time and $\dot{V}_{i4}\leq -k_{i1}(V_{i4})^{\alpha_i}-2^{1-\beta_i}k_{i3}(V_{i4})^{\beta_i}$, where $k_{i3} = 2^{1-\beta_i}k_{i2}$. Thus, all the tracking errors in the entire closed-loop system are UUB.■

Simulation: Consider a multi-RSV system including six RSVs. The environment has the $xy$ dimensions of $100\;{\rm{m}}\times100\;{\rm{m}}$ and $\phi(q) = 1$. The initial states of six charging station and six RSVs are set to $p_{bi} = [2,85-15(i-1)]^T \; (i = 1,\ldots,6)$, $p_{1} = [28,15]^T$, $p_{2} = [65, 43]^T$, $p_{3} = [37,20]^T$, $p_{4} = [12,15]^T$, $p_{5} = [55,50]^T$, $p_{6} = [51,12]^T$, ${\psi}_{b1} = {\psi}_{b6} = 0$, ${\psi}_{b2} = \pi/2$, ${\psi}_{b3} = -4\pi/5$, ${\psi}_{b4} = \pi/4$, ${\psi}_{b5} = \pi$, $q_{i} = [0,0]^T$, ${r}_{i} = 0$, $E_1 = 3600\;{\rm{mV}}$, $E_2 = 3900\;{\rm{mV}}$, $E_3 = 4100\;{\rm{mV}}$, $E_4 = 3800\;{\rm{mV}}$, $E_5 = 4000\;{\rm{mV}}$, and $E_6 = 3700\;{\rm{mV}}$. The parameters in this case are set to $K_{ch} = 10$, $K_{en}$, $k = 0.02$, $d_{ch}$, $c = 0.5$, $E_{\max} = 4200\;{\rm{mV}}$, and $E_{\min} = 3000\;{\rm{mV}}$, $k_{i1}^U = k_{i2}^U = k_{i1}^r = k_{i2}^U = 10$, $k_{i3}^U = k_{i3}^r = 2.5$, $k_{i1}^u = k_{i2}^u = k_{i1}^r = k_{i2}^r = 20$, $k_{i3}^u = k_{i4}^u = k_{i3}^r = k_{i4}^r = 100$, $k_{i0}^u = k_{i0}^r = 0.05$, $\alpha_{ui} = \alpha_{ri} = 0.9$, and $\beta_{ui} = \beta_{ri} = 1.1$, $K_{qi} = \text{diag}\{1,1\}$, $k_{ri} = 0.5$, $k_{i1}^\tau = k_{i2}^\tau = k_{i3}^\tau = k_{i4}^\tau = 1$, $\alpha_{ui}^\tau = \alpha_{ri}^\tau = 0.9$, $\beta_{ui}^\tau = \beta_{ri}^\tau = 1.1$, and $\Delta_{qi} = \Delta_{ri} = 1$. The long duration coverage behavior and the energy change curve of the RSV battery are depicted in Fig. 1. It shows that the optimal coverage of the given coverage area is achieved for the first time at $125\;{\rm{s }}$. At $220\;{\rm{s}}$, the red RSV returns to the charging station for charging, and other RSVs continue the long duration coverage task. Next, the red RSV leaves the charging station to perform the coverage task. Then, at $630$ s, the RSVs complete the optimal coverage for the second time.

Figure  1.  Simulation results. (a) The coverage behavior; (b) The energy change curve.

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Conclusion: In this letter, an anti-disturbance energy-aware control method for RSVs is proposed based on a CVT, two BE-CBFs, and FTESO to achieve long duration coverage task. By the control method, RSVs are capable of achieving the long duration coverage within the given coverage area.

Acknowledgments: This work was supported in part by the National Natural Science Foundation of China (51939001, 52301408), the National Science and Technology Major Project (2022ZD0119902), the Key Basic Research of Dalian (2023JJ11CG008), the Dalian Science and Technology Innovation Fund (2022JJ12GX034), and the Dalian Outstanding Young Scientific and Technological Talents Project (2022RY07).