Target Tracking and Circumnavigation Control for Multi-Unmanned Aerial Vehicle Systems Using Bearing Measurements

Abstract

This paper addresses the problem of target tracking and circumnavigation control for a bearing-only multi-Unmanned Aerial Vehicle (UAV) system. First, using the bearing measurements, an adaptive algorithm in the form of a Proportional Integral (PI) controller is developed to estimate the target state. Subsequently, a distributed circumnavigation control protocol is established to evenly encircle the target. Then, we use the local information from each UAV in the network to calculate the relative position of the target, and further enhance the accuracy of estimation and circumnavigation algorithms by employing a Kalman filter. Finally, numerical simulation experiments are conducted to validate the effectiveness of the proposed tracking control algorithm.

1. Introduction

In recent years, the problem of target tracking and circumnavigation control has garnered significant research attention due to its wide range of applications. These applications include satellite formation flying, dynamic encirclement by multiple Unmanned Aerial Vehicles (UAVs), and cooperative surrounding missions involving multiple vessels [1,2,3,4]. In general, the position of the target is initially unavailable for the agents, thus requiring an estimator to estimate the target state [5,6]. Particularly, in some military scenarios, to maintain radio silence, the passive target locating and tracking techniques, which uses only the bearings measured by camera, has become a hot research field [7,8,9].

Researchers commonly utilized target bearing data to estimate unknown states and update control algorithms, merging localization and circumnavigation control. This strategy enhances system performance via environmental observation and recognition, thus addressing a dual problem [10]. In early studies [11,12,13], the method employing an orthogonal projection matrix was introduced to handle unit bearing vectors, enabling an agent to estimate the target position and accomplish the circumnavigation mission of static or slowly moving targets. In [14], this approach has been extended to multi-agent systems for circumnavigation control of multiple static targets. In [15], the authors established a condition to ensure a safe distance between agents and targets, preventing agents from entering the proximity of targets during circumnavigation. In [16,17], corresponding control frameworks are proposed to enable the estimator’s application to a single or multiple UAVs for target encirclement. Later, the authors of [18,19,20] employed the linear filter and finite-time stability theory to enhance the convergence rate of the estimator. This enhancement allows the multi-agent system to rapidly enclose the targets. For multi-drone motion planning, especially for large drone teams, studying collision avoidance strategies between drones is fundamental and crucial [21,22]. Researchers in [23,24,25] further investigated the discrete-time and communication-free circumnavigation control problem, incorporating collision avoidance among agents to enhance the system’s applicability in real-world scenarios.

However, the aforesaid consequences [11,12,13,14,15,16,17,18,19,20,23,24,25] heavily rely on GPS devices for agents to acquire real-time global position data, which is crucial for target estimation. Yet, in hostile environments, GPS signals are susceptible to disruption or attacks, potentially compromising target tracking effectiveness [26]. Addressing this challenge, researchers proposed the concept of distributed bearing-only formation control [27,28,29]. Within this framework, agents only need to measure the relative bearings between each other, without reliance on GPS information. Target state estimation is often integrated with multi-agent formation control, utilizing bearing measurements for more sophisticated target tracking and cooperative motion. For instance, in [30], the bearing rigidity theory and a rotation matrix were used to establish a time-varying formation for encircling a moving target. Additionally, a dynamic compensation approach in [31] modified traditional target estimators to calculate relative target positions without global information. Building on this, refined target estimators in [32,33] relaxed the initial condition requirements, enabling adept navigation around one or multiple targets in circular or convex trajectories. Furthermore, the researchers in [34,35] analyzed target tracking and circumnavigation within GPS-free systems involving multiple vessels and second-order agents, proposing tailored algorithms for specific dynamic models. Despite their effectiveness, these estimation and control strategies heavily rely on precise sensor data, and may not adequately address high-speed target scenarios. Addressing these limitations is essential for enhancing performance and expanding applicability.

Motivated by the above discussion, this paper is devoted to solving the target tracking and circumnavigation control problem for a moving target within bearing-based networked multi-UAV systems. The main contributions of this paper are presented as follows:

(1)

A distributed target estimation algorithm and a circumnavigation control protocol are designed by using bearing measurements of the target. In contrast to the proportional (P) estimator used in [11,12,13,14,15,16,17,18,19,20,23,24,25,29,30,31,32,33], we introduce a PI control method to develop the estimator, which significantly enhances its robustness. It can be proven that the algorithm converges to a bounded range near the true value. Moreover, in comparison with [11,12,13,14,15,16,17,18,19,20,23,24,25], we take into account practical communication range restrictions and employ dynamic compensation to enable the system to track and encircle the target in a GPS-denied environment.

(2)

A Kalman filter approach is applied to improve the accuracy of target estimation and circumnavigation control. Each UAV in the system utilizes local information, including relative bearing and velocity data from neighbors, to calculate measurements of the target’s relative position. These measurements are then incorporated into a Kalman filter for target state estimation. This approach enhances circumnavigation tracking precision, enabling successful tracking of the dynamic target within the system. Compared to previous methods [11,12,13,14,15,16,17,18,19,20,23,24,25,28,29,30,31,32,33,34,35], our approach leverages effective communication and collaboration among agents to achieve superior target localization and circumnavigation performance, especially in high-speed target scenarios.

The rest of this paper is structured below. In Section 2, the problem to be investigated and necessary preliminaries are given. In Section 3, the distributed target-state estimator and circumnavigation control algorithm are designed. In Section 4, the precise estimator and modified circumnavigation control strategy by Kalman filtering method are developed. In Section 5, simulation results are presented. Finally, conclusions are summarized in Section 6.

2. Preliminaries and Problem Statement

2.1. Notations

All vectors in ${\mathbb{R}}^n$ are n dimensional column vectors. For a given vector or matrix X, $X^{\mathrm{T}}$ denotes its transposition. $I_n$ is an identity matrix with n dimensions, and when n is 2, it is simply written as I. For a unit vector $\phi \in {\mathbb{R}}^n$, let $P_{\phi }=I_n-\phi {\phi }^{\mathrm{T}}$ be the corresponding projection matrix and $\overline{\phi }\in {\mathbb{R}}^n$ be the unit vector perpendicular to $\phi$, obtained by ${\displaystyle \frac{\pi }{2}}$ clockwise rotation of $\phi$. Let $∥·∥$ be the Euclidean norm or the spectral norm of a matrix. The symbol ⊗ represents the Kronecker product. For a random variable Y, $E\left[Y\right]$ represents its expectation. For the convenience of expression, if there is no special description, the symbol $(t)$ is omitted in this paper.

2.2. Problem Statement

In this paper, we consider a multi-agent system consisting of n fixed-wing UAVs i, $i\in \Omega =\{1,\cdots ,n\}$, flying at different altitudes. Using the feedback linearization method, each UAV can be described as a single integrator model [16,36]:

$$\begin{array}{c}\hfill {\dot{p}}_i=u_i,i\in \Omega ,\end{array}$$

(1)

where $p_i\in {\mathbb{R}}^2$ and $u_i\in {\mathbb{R}}^2$ represent the position and the control input of UAV i. The UAVs are required to locate and circumnavigate a moving target with an unknown position $p_0\in {\mathbb{R}}^2$. We assume that the trajectory of the target $p_0$ is differential, and there exists a positive finite constant $\epsilon$ such that for $t\ge 0$, $\Vert {\dot{p}}_0\Vert \le \epsilon$.

We suppose that the UAVs operate in a GPS-denied environment. As a result, each UAV cannot acquire its own global position, and instead only maintains a local frame ${\sum }^i$ with the origin at $p_0$, while relying on the compass to ensure that its orientation aligns with the global frame ${\sum }^g$. In ${\sum }^i$, we define $p_{i0}$ as the relative position between the target and UAV i, which can be obtained as $p_{i0}=p_0-p_i$. Each UAV is equipped with a camera to measure its bearing relative to the target, and the bearing can be represented as the unit vector ${\phi }_{i0}$ in the direction $p_0-p_i$:

$$\begin{array}{c}\hfill {\phi }_{i0}={\displaystyle \frac{p_{i0}}{\Vert p_{i0}\Vert }},i\in \Omega .\end{array}$$

(2)

When measuring bearings of the target, we use the correlated double sampling method to eliminate sensor measurement noise. Moreover, the UAVs has a communication radius $\rho >0$, within which they can the exchange bearing information and their own velocity data. Let $N_i$ represent the set of neighboring UAVs within the communication range of UAV i, defined as $N_i=\{j\in \{1,\cdots ,n\}/\left\{i\right\}:\Vert p_j-p_i\Vert \le \rho \}$. Let ${\Delta }_{ij}\ge 0$ denote the counterclockwise angle from ${\phi }_{i0}$ to ${\phi }_{j0}$. The UAV $j\in N_i$ that achieves the minimum value of ${\Delta }_{ij}$ will be designated as $v_i$. Furthermore, if $N_i$ is nonempty, we let ${\Delta }_i={\Delta }_{i,v_i}$, $p_{i,v_i}=p_{v_i}-p_i$, ${\phi }_{i,{\nu }_i}={\displaystyle \frac{p_{i,{\nu }_i}}{\Vert p_{i,{\nu }_i}\Vert }}$, and ${\theta }_{{\nu }_i,0}$ represent the angle between the edge from UAV $v_i$ to the target and the edge from UAV $v_i$ to UAV i, as depicted in Figure 1. Note that when $N_i$ is empty, we have ${\Delta }_i=0$; and when ${\phi }_{i0}={\phi }_{v_i,0}$, we let ${\Delta }_i(t)=0$ if ${\Delta }_i(t^{-})=0^{+}$, ${\Delta }_i(t)=2\pi$ if ${\Delta }_i(t^{-})=2{\pi }^{-}$.

Since the UAVs cannot directly obtain the accurate relative position of the target, UAV i needs to maintain a real-time estimate ${\widehat{p}}_{i0}$ in its local frame ${\sum }^i$. Thus, the estimation error of the relative position of the target by UAV i is

$$\begin{array}{c}\hfill {\tilde{p}}_{i0}={\widehat{p}}_{i0}-p_{i0},i\in \Omega .\end{array}$$

(3)

Our research focuses on developing a localization algorithm and a control strategy for the multi-UAV system. The goal is to accurately and efficiently locate and circumnavigate a moving target using bearing measurements. We aim to achieve circumnavigation tracking while ensuring a uniform distribution of UAVs along the surrounding orbit. In terms of the control objective, we will formally present the problem statement for the target tracking and circumnavigation control problem.

Problem 1

(Target tracking and circumnavigation control problem). With bearing-only measurements to the target, for each UAV i, design an estimator and a controller such that the following objectives can be achieved:

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\Vert {\tilde{p}}_{i0}\Vert \le {\mathrm{U}}_0,i\in \Omega ,\end{array}$$

(4)

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|D_i-D*\right|\le {\mathrm{U}}_1,i\in \Omega ,\end{array}$$

(5)

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}|{\Delta }_i-{\displaystyle \frac{2\pi }{n}}|\le {\mathrm{U}}_2,i\in \Omega ,\end{array}$$

(6)

where ${\mathrm{U}}_{\mathtt{0}}$, ${\mathrm{U}}_{\mathtt{1}}$ and ${\mathrm{U}}_{\mathtt{2}}$ are nonnegative constants associated with ε, $D_i=\Vert p_{i0}\Vert$ denotes the relative distance between UAV i and the target. The parameter $D*$ represents the desired circumnavigation radius, which is a prescribed positive value.

2.3. Technical Lemmas

To design and analyze our algorithms, we present the following necessary lemmas.

Lemma 1

(see [34,37]). Let $V(·):{\mathbb{R}}_{+}\to {\mathbb{R}}^{\mathrm{n}\times \mathrm{r}}$ be regulated. Then, $\dot{x}=\left[\begin{array}{cc}-{\mathrm{c}}_{1}V{V}^{\mathrm{T}}& I_n\\ -{\mathrm{c}}_{2}V{V}^{\mathrm{T}}& 0\end{array}\right]x$ is exponentially asymptotically stable iff $V(·)$ is persistently exciting (PE), where $c_1$ and $c_2$ are positive constants.

Lemma 2

(see [31]). For the dynamic system

$$\begin{array}{c}\hfill {\dot{\tilde{p}}}_{i0}(t)=A_i(t){\tilde{p}}_{i0}+f(t),i\in \Omega ,\end{array}$$

(7)

where the coefficient matrix $A_i(t)$ is continuous for all $\mathfrak{t}\in [0,\infty )$, if constants $r_i>0$ and $b_i>0$ exist such that for every solution of the homogeneous differential equation ${\dot{\tilde{p}}}_{i0}(t)=A_i(t){\tilde{p}}_{i0}(t)$, it holds that $\parallel {\tilde{p}}_{i0}(t)\parallel \le b_i\parallel {\tilde{p}}_{i0}(t_0)\parallel e^{-r_i(t-t_0)}$ for $0\le t_0<t<\infty$, then for all each $f(t)$ bounded and continuous on $[0,\infty )$, every solution of the nonhomogeneous Equation $(7)$ is also bounded for $t\in [0,\infty )$. If $∥f(t)∥\le \epsilon <\infty$, then the solution of the perturbed system satisfies

$$\begin{array}{c}\hfill ∥\begin{array}{c}{\tilde{p}}_{i0}(t)\end{array}∥\le b_i∥\begin{array}{c}{\tilde{p}}_{i0}(t_0)\end{array}∥e^{-r_i(t-t_0)}+{\displaystyle \frac{\epsilon b_i^2}{r_i}}(1-e^{-r_i(t-t_0)}),i\in \Omega .\end{array}$$

(8)

3. Distributed Estimation and Circumnavigation Algorithms

To successfully estimate the relative target position $p_{i0}$ and accomplish the control objectives described in Problem 1, an estimator ${\widehat{p}}_{i0}$ and the control input (control protocol) $u_i$ for each UAV are constructed as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{p}}}_{io}=-{\kappa }_1{P}_{{\phi }_{io}}{\widehat{p}}_{io}-{\kappa }_2{\int }_0^{\mathrm{t}}{P}_{{\phi }_{io}}(\tau ){\widehat{p}}_{io}(\tau )\mathrm{d}\tau -{\dot{p}}_i,i\in \Omega ,\end{array}$$

(9)

$$\begin{array}{c}\hfill u_i={\gamma }_1({\widehat{D}}_i-D*){\phi }_{i0}+{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0},i\in \Omega ,\end{array}$$

(10)

where ${\kappa }_1$, ${\kappa }_2$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ and $\alpha$ are positive design parameters, and ${\widehat{D}}_i=\Vert {\widehat{p}}_{i0}\Vert$ represents the estimated relative distance between the target and UAV i.

Remark 1.

In (9), the term ${\dot{p}}_i$ is equivalent to $u_i$ in Equation (10), and is introduced to compensate for the estimation inaccuracies resulting from the movement of UAV i. The term $-{\kappa }_1{P}_{{\phi }_{io}}{\widehat{p}}_{io}-{\kappa }_2{\int }_0^{\mathrm{t}}{P}_{{\phi }_{io}}(\tau ){\widehat{p}}_{io}(\tau )\mathrm{d}\tau$ denotes the estimation of target velocity, structured as a PI controller. In (10), the radial term ${\gamma }_1({\widehat{D}}_i-D*){\phi }_{i0}$ forces the agent to encircle the target with the desired distance $D*$, while the tangential term ${\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0}$ ensures that the UAV rotate around the target while achieving the desired formation with other agents. In motion control, if $D*$ is set too small, the angular velocity of the UAV may become very large. To limit the angular velocity, the tangential term is adjusted to be dependent on $D*$.

We will first demonstrate that the estimation error ${\tilde{p}}_{i0}$ globally converges to a neighborhood of zero, leading to the following lemma.

Lemma 3.

Assuming that the estimation error ${\tilde{p}}_{i0}(0)$ in (3) is bounded, if the control parameters satisfy ${\gamma }_{2}D*\alpha -\epsilon >\xi >0$, where ξ is a given positive scalar, then the estimator (9) and control protocol (10) will cause ${\tilde{p}}_{i0}$ to exponentially converge to a circular area of radius ${\displaystyle \frac{\epsilon {b_i}^2}{r_i}}$ centered at origin as $t\to \infty$, where $r_i$ and $b_i$ are defined in Lemma 2.

Proof. 

Observing that $P_{{\phi }_{i0}}{p}_{i0}=0$, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\tilde{p}}}_{i0}& ={\dot{\widehat{p}}}_{i0}-{\dot{p}}_{i0}\hfill \\ & =-{\kappa }_1{P}_{{\phi }_{i0}}{\tilde{p}}_{i0}-{\kappa }_2{\int }_0^t{P}_{{\phi }_{i0}}(\tau ){\tilde{p}}_{i0}(\tau )d\tau -{\dot{p}}_0,i\in \Omega .\hfill \end{array}\end{array}$$

(11)

Utilizing the property $P_{{\phi }_{i0}}={\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}$ and defining $\varpi =-{\kappa }_2{\int }_0^t{P}_{{\phi }_{i_0}}(\tau ){\tilde{p}}_{i0}(\tau )d\tau$ as an auxiliary variable, Equation (10) can be rewritten as

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\dot{\tilde{p}}}_{i0}\\ \dot{\varpi }\end{array}\right]=\left[\begin{array}{cc}-{\kappa }_1{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& I\\ -{\kappa }_2{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& 0\end{array}\right]\left[\begin{array}{c}{\tilde{p}}_{i0}\\ \varpi \end{array}\right]-\left[\begin{array}{c}{\dot{p}}_0\\ 0\end{array}\right],i\in \Omega .\end{array}$$

(12)

Obviously, System (12) can be divided into a nominal system

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\dot{\tilde{p}}}_{i0}\\ \dot{\varpi }\end{array}\right]=\left[\begin{array}{cc}-{\kappa }_1{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& I\\ -{\kappa }_2{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& 0\end{array}\right]\left[\begin{array}{c}{\tilde{p}}_{i0}\\ \varpi \end{array}\right]\end{array}$$

(13)

and a perturbation term $\left[\begin{array}{c}{\dot{p}}_0\\ 0\end{array}\right]$. According to Lemma 2, if we can prove the global exponential stability of System (13), then we will be able to find some constants $r_i$ and $b_i$ such that Equation (8) holds, thereby proving Lemma 3. And by following Lemma 1, this objective can be redirected to prove that the unit vector ${\overline{\phi }}_{i0}$ is persistently exciting.

It is known that if there exist ${\epsilon }_{i1},{\epsilon }_{i2},T_i>0$ such that

$$\begin{array}{c}\hfill {\epsilon }_{i1}\le {\int }_{t_0}^{t_0+T_i}{\left(U^{\mathrm{T}}{\overline{\phi }}_{i0}\left(\tau \right)\right)}^{2}d\tau \le {\epsilon }_{i2},i\in \Omega ,\end{array}$$

(14)

is satisfied for all unit constant vectors $U\in {\mathbb{R}}^2$ and all $t_0>0$, then ${\overline{\phi }}_{i0}$ is PE (see [9]). Since ${(U^{\mathrm{T}}{\overline{\phi }}_{i0}(\tau ))}^2\le 1$, we can always find an upper bound ${\epsilon }_{i2}=T_i$. Next, we shall prove that a lower bound ${\epsilon }_{i1}$ also exists. Let ${\beta }_{Ui}$ be the angle measured counterclockwise from U to ${\overline{\phi }}_{i0}$; its derivative is

$$\begin{array}{c}\hfill {\dot{\beta }}_{Ui}={\displaystyle \frac{{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i)-{{\overline{\phi }}_{i0}}^{\mathrm{T}}{\dot{p}}_0}{D_i}},i\in \Omega .\end{array}$$

(15)

Since $\Vert {{\overline{\phi }}_{i0}}^{\mathrm{T}}{\dot{p}}_0\Vert \le \epsilon$ and ${\gamma }_3{\Delta }_i\ge 0$, we have ${\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i)-{{\overline{\phi }}_{i0}}^{\mathrm{T}}{\dot{p}}_0>0$. Therefore, if there exists an upper bound $D_{imax}$ such that $D_i\le D_{imax}$, then we can obtain ${\dot{\beta }}_{Ui}\ge {\displaystyle \frac{\zeta }{D_{imcx}}}>0$; thus, one can always find a ${\epsilon }_{i1}>0$ and a $T_i>0$ such that (14) is satisfied.

The final step is to show that $D_i$ is indeed bounded. Defining $\widetilde{D_i}=D_i-{\widehat{D}}_i$, we derive the derivative of $D_i$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{D}}_i& ={{\phi }_{i0}}^{\mathrm{T}}({\dot{p}}_0-{\dot{p}}_i)\hfill \\ & =-{\gamma }_1(D_i-D*)+{\gamma }_1{\tilde{D}}_i+{{\phi }_{i0}}^{\mathrm{T}}{\dot{p}}_0,i\in \Omega ,\hfill \end{array}\end{array}$$

(16)

and introducing the error ${\delta }_i=D_i-D*$ between the distance of UAV i from the target and the desired distance results in

$$\begin{array}{c}\hfill {\delta }_i={\delta }_i(0)e^{-{\gamma }_{1}t}+{\int }_0^t{e}^{-{\gamma }_1(t-\tau )}({\gamma }_1{\tilde{D}}_i(\tau )+{\phi }_{i0}^{\mathrm{T}}(\tau ){\dot{p}}_0(\tau ))d\tau ,i\in \Omega .\end{array}$$

(17)

According to the triangular relationship, we have the following inequality:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\tilde{D}}_i(t)|& =|\Vert {\widehat{p}}_{i0}(t)\Vert -\Vert p_{i0}(t)\Vert |\hfill \\ & \le \left|\right|{\widehat{p}}_{i0}(t)-p_{i0}(t)\left|\right|=\left|\right|{\tilde{p}}_{i0}(t)\left|\right|,i\in \Omega .\hfill \end{array}\end{array}$$

(18)

Choose the Lyapunov candidate function $V={\displaystyle \frac{1}{2}}({\tilde{p}}_{i0}^{\mathrm{T}}{\tilde{p}}_{i0}+{\varpi }^{\mathrm{T}}\varpi )$. Given that the matrix $\left[\begin{array}{cc}-{\kappa }_1{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& I\\ -{\kappa }_2{\overline{\phi }}_{i0}{\overline{\phi }}_{i0}^{\mathrm{T}}& 0\end{array}\right]$ is negative semi-definite, we have $\dot{V}\le 0$. This implies that the estimation error always remains ${\tilde{p}}_{i0}(t)\le {\tilde{p}}_{i0}(0)$. Since ${\tilde{p}}_{i0}(0)$ is bounded, we can assume an upper bound ${\rho }_i$ that satisfies ${\tilde{p}}_{i0}(t)\le {\rho }_i$. Therefore, adding $D*$ on both sides of Equation (17), we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_i(t)& \ge D*+{\delta }_i(0)e^{-{\gamma }_{1}t}-(1-e^{-{\gamma }_{1}t})({\rho }_i+{\displaystyle \frac{\epsilon }{{\gamma }_1}}),\hfill \\ \hfill D_i(t)& \le D*+{\delta }_i(0)e^{-{\gamma }_{1}t}+(1-e^{-{\gamma }_{1}t})({\rho }_i+{\displaystyle \frac{\epsilon }{{\gamma }_1}}),i\in \Omega .\hfill \end{array}\end{array}$$

(19)

From (19), we can see $D_i$ is bounded, and we have $D_{imax}=D*+max\{{\delta }_i(0),{\rho }_i+{\displaystyle \frac{\epsilon }{{\gamma }_1}}\}$. Hence, we have completed the proof of Lemma 3. □

Remark 2.

For a proportional-integral (PI) controller, the integral term prevents the error from growing indefinitely by capping it. Given that the target speed ${\dot{p}}_0$ has been regarded as a bounded disturbance, the estimation error $\Vert {\tilde{p}}_{i0}(t)\Vert$ will consequently remain bounded throughout the target estimation process. By integrating the error over time, the PI controller drives the error towards an acceptable range.

Remark 3.

Based on (19), we can also determine a lower bound $D_{imin}\ge D*-max\{{\delta }_i(0),{\rho }_i+{\displaystyle \frac{\epsilon }{{\gamma }_1}}\}$ for $D_i$. This suggests that by reasonably setting the value of the desired radius $D*$, we can ensure that $D_i(t)>0$ for all $t\ge 0$, thereby guaranteeing UAV i can consistently obtain the bearing vector ${\phi }_{i0}$ while avoiding collisions between the UAVs and the target at all times.

Next, on the proof of Lemma 3, we will show that the distance error ${\delta }_i$ globally converges to a ball centered at zero for all the UAVs.

Theorem 1.

Under the estimation strategy (9), the controller (10) makes the relative distance $D_i$ between the target and UAV i converge exponentially fast to a bounded circular area centered at $D*$ with radius $\epsilon ({\displaystyle \frac{1}{{\gamma }_1}}+{\displaystyle \frac{b_i^2}{r_i}})$.

Proof. 

From (17), and using the triangular inequality (18), we have

$$\begin{array}{c}\hfill \mid {\delta }_i(t)\mid \le \mid {\delta }_i(0)\mid e^{-{\gamma }_{1}t}+{\int }_0^t{e}^{-{\gamma }_1(t-\tau )}({\gamma }_1\Vert {\tilde{p}}_{i0}(\tau )\Vert +\mid {\phi }_{i0}^{\mathrm{T}}(\tau ){\dot{p}}_0(\tau )\mid )d\tau ,i\in \Omega .\end{array}$$

(20)

Using Inequality (8) in Lemma 2, (20) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mid {\delta }_i(t)& \mid \le \mid {\delta }_i(0)\mid e^{-{\gamma }_{1}t}+{\displaystyle \frac{1}{{\gamma }_1}}(1-e^{-{\gamma }_{1}t})\epsilon \hfill \\ & +{\gamma }_1{e}^{-{\gamma }_{1}t}{\int }_0^t{e}^{{\gamma }_1\tau }[b_i\Vert {\tilde{p}}_{i0}(0)\Vert e^{-r_i\tau }+{\displaystyle \frac{\epsilon b_i^2}{r_i}}(1-e^{-r_i\tau })]d\tau \hfill \\ & \le \mid {\delta }_i(0)\mid e^{-{\gamma }_{1}t}+{\displaystyle \frac{1}{{\gamma }_1}}(1-e^{-{\gamma }_{1}t})\epsilon +{\displaystyle \frac{{\gamma }_1{b}_i(r_i\Vert {\tilde{p}}_{i0}(0)\Vert -\epsilon b_i)}{r_i({\gamma }_1-r_i)}}(e^{-r_{i}t}-e^{-{\gamma }_{1}t})\hfill \\ & +{\displaystyle \frac{\epsilon b_i^2}{r_i}}(1-e^{-{\gamma }_{1}t}),i\in \Omega ,\hfill \end{array}\end{array}$$

(21)

and letting $t\to \infty$, we finally obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mid {\delta }_i(t)|\le \epsilon ({\displaystyle \frac{1}{{\gamma }_1}}+{\displaystyle \frac{b_i^2}{r_i}}),i\in \Omega .\end{array}$$

(22)

Therefore, as indicated by (22), the distance between the target and each UAV i converges to a neighborhood around the desired radius $D*$ as $t\to \infty$. □

Finally, we aim to demonstrate that the UAVs will arrange themselves in a regular polygon formation along the circumference of a circle, thereby achieving an optimal enclosing performance.

Theorem 2.

For all $r_i$ and $b_i$, assume that the communication radius satisfies

$$\begin{array}{c}\hfill \rho \ge 2[D*+\epsilon ({\displaystyle \frac{1}{{\gamma }_1}}+{\displaystyle \frac{b_i^2}{r_i}})].\end{array}$$

(23)

Then, under estimator (9) and control law (10), each ${\Delta }_i$ converges exponentially to a bounded circular area centered at ${\displaystyle \frac{2\pi }{\mathrm{n}}}$.

Proof. 

According to Theorem 1, ${\delta }_i$ converges exponentially to a ball centered at zero with radius $\epsilon ({\displaystyle \frac{1}{{\gamma }_1}}+{\displaystyle \frac{b_i^2}{r_i}})$. Therefore, under (23), there exists a time $0<T_0<\infty$ such that for all $t>T_0$, we have $N_i=\{1,\cdots ,n\}/\left\{i\right\}$. This implies that each UAV i will always maintain a neighbor ${\nu }_i$, while simultaneously ensuring that ${\sum }_{i=1}^n{\Delta }_i=2\pi$. Then, we calculate the derivative of ${\Delta }_i$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\Delta }}_i& ={\dot{\beta }}_{U{\nu }_i}-{\dot{\beta }}_{Ui}\hfill \\ & ={\displaystyle \frac{{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_{{\nu }_i})-{\overline{\phi }}_{{\nu }_i,0}^{\mathrm{T}}{\dot{p}}_0}{D_{{\nu }_i}}}-{\displaystyle \frac{{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i)-{\overline{\phi }}_{i0}^{\mathrm{T}}{\dot{p}}_0}{D_i}}\hfill \\ & ={\gamma }_2{\gamma }_3({\Delta }_{{\nu }_i}-{\Delta }_i)+{\gamma }_2{\gamma }_3({\displaystyle \frac{{\Delta }_i{\delta }_i}{D_i}}-{\displaystyle \frac{{\Delta }_{{\nu }_i}{\delta }_{{\nu }_i}}{D_{{\nu }_i}}})+{\gamma }_{2}D*\alpha ({\displaystyle \frac{1}{D_{{\nu }_i}}}-{\displaystyle \frac{1}{D_i}})\hfill \\ & +{({\displaystyle \frac{{\overline{\phi }}_{i0}}{D_i}}-{\displaystyle \frac{{\overline{\phi }}_{{\nu }_i,0}}{D_{{\nu }_i}}})}^{\mathrm{T}}{\dot{p}}_0,i\in \Omega .\hfill \end{array}\end{array}$$

(24)

Define $\Delta ={[{\Delta }_1,\cdots ,{\Delta }_n]}^{\mathrm{T}}$, and let $L(t)$ be the Laplacian matrix of the time-varying networked graph $G=(\Omega ,E)$, where $E(t)=\{({\nu }_i(t),i):i\in \Omega \}$. We can rewrite Equation (24) as

$$\begin{array}{c}\hfill \dot{\Delta }(t)=-{\gamma }_2{\gamma }_{3}L(t)\Delta (t)+f_0(t)+f_1(t)+f_2(t),\end{array}$$

(25)

where ${\left[f_0(t)\right]}_i={\gamma }_2{\gamma }_3({\displaystyle \frac{{\Delta }_i{\delta }_i}{D_i}}-{\displaystyle \frac{{\Delta }_{{\nu }_i}{\delta }_{{\nu }_i}}{D_{{\nu }_i}}})$, ${\left[f_1(t)\right]}_i={\gamma }_{2}D*\alpha ({\displaystyle \frac{1}{D_{{\nu }_i}}}-{\displaystyle \frac{1}{D_i}})$, ${\left[f_2(t)\right]}_i={({\displaystyle \frac{{\overline{\phi }}_{i0}}{D_i}}-{\displaystyle \frac{{\overline{\phi }}_{{\nu }_i,0}}{D_{{\nu }_i}}})}^{\mathrm{T}}{\dot{p}}_0$ and ${\left[L(t)\right]}_{ij}=\left\{\begin{array}{ccc}1& & j=i\\ -1& & j={\nu }_i\\ 0& & otherwise\end{array}\right.$. Since G is a strongly connected graph, the $\Delta (t)$ in equation $\dot{\Delta }(t)=-{\gamma }_2{\gamma }_{3}L(t)\Delta (t)$ will achieve consensus: ${\Delta }_i\to {\displaystyle \frac{2\pi }{n}}$. Thus, let $\tilde{\Delta }={[{\Delta }_1-{\displaystyle \frac{2\pi }{\mathrm{n}}},\cdots ,{\Delta }_n-{\displaystyle \frac{2\pi }{\mathrm{n}}}]}^{\mathrm{T}}$, and we can obtain

$$\begin{array}{c}\hfill \dot{\tilde{\Delta }}(t)=-{\gamma }_2{\gamma }_{3}L(t)\tilde{\Delta }(t)+f_0(t)+f_1(t)+f_2(t).\end{array}$$

(26)

It is obvious that the disturbances $f_0(t)$, $f_1(t)$, $f_2(t)$ are bounded, and according to Lemma 2, the $\tilde{\Delta }$ will converge exponentially fast to a ball centered at zero. □

Based on Lemma 3, and Theorems 1 and 2, we have successfully achieved the estimation and control objectives (4)–(6). However, when the target’s speed is high, the estimation error given by (9) and the control error for circumnavigation radius and interval angle given by (10) are unsatisfactory. Therefore, in the subsequent section of this paper, we will enhance our estimation and circumnavigation strategy by incorporating the Kalman filtering method to improve the encircling tracking performance.

4. Kalman Filtering Strategy

In this section, we will first utilize the local information exchange between neighboring UAV ${\nu }_i$ and UAV i to obtain an observation of the target’s relative position. Then, by employing the Kalman filtering method, along with estimator (9), we can achieve a more accurate estimation of the target state, thereby improving the overall performance of collaborative target encircling tracking in the multi-UAV system.

As shown in Figure 1, when $N_i\ne \varnothing$, using trigonometric relationships, we have

$$\begin{array}{c}\hfill D_i\mid sin{\Delta }_i\mid =D_{i,{\nu }_i}\mid sin{\theta }_{{\nu }_i,0}\mid ,i\in \Omega ,\end{array}$$

(27)

where $D_{i,{\nu }_i}=\Vert p_{i,{\nu }_i}\Vert$. By Lemma 5 in [38], for the angle $\theta \in [0,\pi ]$ between any two unit vectors $x,y\in {\mathbb{R}}^2$, we have $sin\theta =\Vert P_x-P_y\Vert$. Thus, we can rewrite (27) as

$$\begin{array}{c}\hfill D_i\Vert P_{{\phi }_{i0}}-P_{{\phi }_{{\nu }_i,0}}\Vert =D_{i,{\nu }_i}\Vert P_{{\phi }_{i,{\nu }_i}}-P_{{\phi }_{{\nu }_i,0}}\Vert ,i\in \Omega .\end{array}$$

(28)

By calculating the derivative of ${\phi }_i,{\nu }_i$, we can obtain

$$\begin{array}{c}\hfill {\dot{\phi }}_{i,{\nu }_i}={\displaystyle \frac{P_{{\phi }_{i,{\nu }_i}}({\dot{p}}_{{\nu }_i}-{\dot{p}}_i)}{D_{i,{\nu }_i}}},i\in \Omega .\end{array}$$

(29)

In a practical environment, the values of ${\phi }_i,{\nu }_i$ and ${\dot{\phi }}_i,{\nu }_i$ can be measured directly by UAV i. Then, by substituting Equation (28), we can finally obtain ${\widehat{p}}_{i0}^{observe}$ as the observation of the target’s relative position for each UAV i:

$$\begin{array}{c}\hfill {\widehat{p}}_{i0}^{observe}=D_i{\phi }_{i0}={\displaystyle \frac{\Vert P_{{\phi }_i,{\nu }_i}\left({\dot{p}}_{{\nu }_i}-{\dot{p}}_i\right)}{\Vert {\dot{\phi }}_i,{\nu }_i\Vert }}{\displaystyle \frac{\Vert P_{{\phi }_i,{\nu }_i}-P_{{\phi }_{{\nu }_i,0}}\Vert }{\Vert P_{{\phi }_{i,0}}-P_{{\phi }_{{\nu }_i,0}}\Vert }}{\phi }_{i0},i\in \Omega .\end{array}$$

(30)

Remark 4.

From Equations (27)–(30), we can easily see that the relative position observation ${{\widehat{p}}_{i0}}^{observe}$ is calculated using the measurements of ${\phi }_{i0}$, ${\phi }_{i,{\nu }_i}$, and their rate of change. In the theoretical analysis section, we do not consider the measurement noises for ${\phi }_{i0}$, ${\phi }_{i,{\nu }_i}$, and their rates of change. However, the error of ${{\widehat{p}}_{i0}}^{observe}$ can be affected by communication packet loss and bit errors between drones. We can model the observation noise as white Gaussian noise $\vartheta (s)$ with zero mean and covariance $\mathbb{E}\left[\vartheta \left(s\right)\vartheta \left(t\right)\right]={\sigma }_{\vartheta }^2\delta \left(s-t\right)$, where $\delta \left(·\right)$ is a Dirac function and ${\sigma }_{\vartheta }$ is the standard deviation of the noise.

By utilizing ${\widehat{p}}_{i0}^{observe}$ with a Kalman filter, we can apply the following equations [39]:

$$\begin{array}{c}\hfill \begin{array}{c}\dot{X}=AX+Bu+w,w\sim (0,Q)\\ Z=HX+\upsilon ,\upsilon \sim (0,R)\\ \widehat{X}(0)=E\left[X(0)\right]\\ P(0)=E[(X(0)-\widehat{X}(0)){(X(0)-\widehat{X}(0))}^{\mathrm{T}}]\\ K=P{H}^{\mathrm{T}}{R}^{-1}\\ \widehat{X}=A\widehat{X}+Bu+K(Z-H\widehat{X})\\ \dot{P}=-P{H}^{\mathrm{T}}{R}^{-1}HP+AP+P{A}^{\mathrm{T}}+Q\end{array},\end{array}$$

(31)

Here, X and Z represent the estimated and observed states of the target, determined by ${[{\widehat{p}}_{i0},{\dot{\widehat{p}}}_{i0}]}^{\mathrm{T}}$ and ${\widehat{p}}_{i0}^{observe}$, respectively. B and u represent the control input matrix and the control input. A and H are the state transition and observation models. The terms w and $\upsilon$ denote process and measurement noises, following distributions $(0,Q)$ and $(0,R)$. Q and R are the covariance matrices of process and measurement noises. P represents the state covariance matrix, K is the Kalman gain factor, and $\widehat{X}$ denotes the estimated target state obtained through Kalman filtering.

Thus, we can improve the accuracy of the target state estimation in Equation (9) and obtain ${\widehat{X}}_{i0}^{Kal}={[{\widehat{p}}_{i0}^{Kal},{\widehat{\dot{p}}}_{i0}^{Kal}]}^{\mathrm{T}}$ from $\widehat{X}$ in (31), where ${\widehat{p}}_{i0}^{Kal}$ and ${\widehat{\dot{p}}}_{i0}^{Kal}$ represent the relative position and relative velocity of the target after the Kalman filtering process. Then, by ${\widehat{\dot{p}}}_{i0}^{Kal}$, each UAV can estimate the global velocity of the target:

$$\begin{array}{c}\hfill {\widehat{\dot{p}}}_0^{Kal}={\dot{p}}_i+{\widehat{\dot{p}}}_{i0}^{Kal},i\in \Omega ,\end{array}$$

(32)

and by letting ${\widehat{D}}_i^{Kal}=\Vert {\widehat{p}}_{i0}^{Kal}\Vert$, we can modify the controller (10) by

$$\begin{array}{c}\hfill {\tilde{u}}_i={\gamma }_1({\widehat{D}}_i^{\mathit{Kal}}-D*){\phi }_{i0}+{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0}+{\widehat{\dot{p}}}_0^{Kal},i\in \Omega .\end{array}$$

(33)

Remark 5.

The updated control law (33), in comparison to the preliminary controller (10), replaces ${\widehat{D}}_i$ with ${\widehat{D}}_i^{Kal}$, allowing agent i to employ more accurate relative distance estimation for achieving encirclement. Additionally, the additional term ${\widehat{\dot{p}}}_0^{Kal}$ is introduced to compensate the circumnavigation error caused by the target’s movement. If ${\widehat{\dot{p}}}_0^{Kal}$ is similar to the actual target velocity, the problem can be treated as a static target circumnavigation problem.

Theorem 3.

Through the estimated target state ${\widehat{X}}_{i0}^{Kal}$ and the modified control law (33), when $t\to \infty$, we can finally obtain $E[{\tilde{p}}_{i0}]=0$, $E\left[{\delta }_i\right]=0$ and $E\left[{\Delta }_i\right]=0$.

Proof. 

Similar to the proof process of Corollary 1 and Corollary 2 in [16], based on Lemma 3, along with the unbiasedness of the Kalman filter, we can directly conclude that $E[{\widehat{X}}_{i0}^{Kal}-X_{i0}]=0$, for $i\in \Omega$. Here, $X_{i0}={[p_{i0},{\dot{p}}_{i0}]}^{\mathrm{T}}$ is the real state of the target. And from (32), we can also acquire

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E[{\widehat{\dot{p}}}_0^{Kal}-{\dot{p}}_0]& =E[{\dot{p}}_i+{\dot{p}}_{i0}^{Kal}-({\dot{p}}_{i0}+{\dot{p}}_i)]\hfill \\ & =E[{\dot{p}}_{i0}^{Kal}-{\dot{p}}_{i0}]=0,i\in \Omega .\hfill \end{array}\end{array}$$

(34)

Therefore, as $t\to \infty$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E[{\tilde{u}}_i-{\dot{p}}_0]& =E[{\gamma }_1({\widehat{D}}_i^{Kal}-D*){\phi }_{i0}+{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0}+{\widehat{\dot{p}}}_0^{Kal}-{\dot{p}}_0]\hfill \\ & =E[{\gamma }_1({\widehat{D}}_i^{Kal}-D*){\phi }_{i0}+{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0}]\hfill \\ & ={\gamma }_1(D_i^{Kal}-D*){\phi }_{i0}+{\gamma }_{2}D*(\alpha +{\gamma }_3{\Delta }_i){\overline{\phi }}_{i0},i\in \Omega .\hfill \end{array}\end{array}$$

(35)

Thus, the target tracking problem is transferred into the problem of encircling a static target, which means we can let $\epsilon =0$ in Inequality (22), and the disturbances $f_0(t)$, $f_1(t)$, and $f_2(t)$ in Equation (26) will finally vanish. That is, as $t\to \infty$, $E[{\delta }_i]=0$ and $E[{\Delta }_i]=0$. □

Remark 6.

In some cases, there may be situations where the observed value of ${\widehat{p}}_{i0}^{observe}$ is lost. This can happen when $N_i=\varnothing$, or when the target, neighbor UAV, and UAV i are lined up in a straight line. In such situations, if the observed value is lost during the filtering process, we will use the estimated value ${\widehat{p}}_{i0}$ instead of ${\widehat{p}}_{i0}^{observe}$. This allows us to continue estimating the state of the target.

At last, we demonstrate the implementation process of our algorithm.

Step 1:

Define two integers, k and N, where k is the number of simulation steps, and N denotes a pre-set threshold value. Additionally, initialize the estimation of the relative target position ${\widehat{p}}_{i0}$, the error covariance matrix $P_k$, prediction covariance matrix Q, and measurement covariance matrix R in the filter.

Step 2:

When $k\le N$, we estimate the target state using ${[{\widehat{p}}_{i0},{\dot{\widehat{p}}}_{i0}]}^{\mathrm{T}}$ in Equation (9). Then, we update ${\widehat{\dot{p}}}_{i0}^{Kal}$ as $-{\kappa }_1{P}_{{\phi }_{i0}}{\widehat{p}}_{i0}-{\kappa }_2{\sum }_{t_i=1}^k{P}_{{\phi }_{i0}}(t_i){\widehat{p}}_{i0}(t_i)$, ${\widehat{D}}_i^{Kal}$ as $\parallel {\widehat{p}}_{i0}\parallel$ in control law (33). Finally, apply the updated control law to drive each UAV i.

Step 3:

When $k>N$, we start the Kalman filtering process. Each UAV i begins to calculate ${\widehat{p}}_{i0}^{observe}$ in Equation (30); if the observed value is not available, we let ${\widehat{p}}_{i0}^{observe}={\widehat{p}}_{i0}$. Next, for agent i, by calculating ${\widehat{X}}_{i0}^{Kal}$, we using the modified controller (33) to eventually complete the target encircling tracking.

5. Numerical Simulations

In this section, we will demonstrate several numerical examples to illustrate the theoretical results obtained in the previous sections.

Consider a formation of five circumnavigating UAVs and an unknown dynamic target in a GPS-denied indoor environment; the initial positions of the target and UAVs are assigned as $p_0={[0,0]}^{\mathrm{T}}$, $p_1={[-35,-40]}^{\mathrm{T}}$, $p_2={[36,-50]}^{\mathrm{T}}$, $p_3={[-2,-48]}^{\mathrm{T}}$, $p_4={[30,49]}^{\mathrm{T}}$ and $p_5={[-26,34]}^{\mathrm{T}}$. The initial estimates of the relative target position are ${\widehat{p}}_{10}={[-5,-15]}^{\mathrm{T}}$, ${\widehat{p}}_{20}={[31,-25]}^{\mathrm{T}}$, ${\widehat{p}}_{30}={[17,-36]}^{\mathrm{T}}$, ${\widehat{p}}_{40}={[5,21]}^{\mathrm{T}}$ and ${\widehat{p}}_{50}={[-11,27]}^{\mathrm{T}}$. The desired circumnavigating radius is set as $D*=5$.

The designed parameters in Estimator (9) and Controller (10) are given as ${\kappa }_1=4$, ${\kappa }_2=0.12$, ${\gamma }_1=1.2$, ${\gamma }_2=1$, ${\gamma }_3=0.4$ and $\alpha =1$. Let $\Delta t$ represent the simulation step size, and set $\Delta t$ to be 0.01 s. We define two different speed levels for the target: a velocity of $0.1v_t$ for low speed and a velocity of $v_t$ for high speed, where $v_t={[2-2cos(0.1t),2+sin(0.05t)]}^{\mathrm{T}}$.

We first assess the impact of the rough estimator (9) and control protocol (10) on tracking the low-speed target. The simulation results, as depicted in Figure 2, reveal that while the drones can maintain an accurate estimation of the target’s relative position, the effectiveness of the circumnavigation is not satisfactory. It is evident from the results that, with only the rough distributed estimation and circumnavigation algorithm, the target state estimation ultimately converges to a neighborhood of the accurate value. The distance error ${\delta }_i$ between agent i and the target also converges to a neighborhood of zero, and the separation angle ${\Delta }_i$ between two neighboring UAVs converges to a neighborhood of the desired value ${\displaystyle \frac{2\pi }{n}}$. However, it is apparent that the results are not ideal, as the circumnavigation errors exhibit obvious fluctuations.

Now, we adopt the filtering strategy. We set $N=2500$, and utilize the observed value ${\widehat{p}}_{i0}^{observe}$ from Equation (30) and the modified control law (33). Additionally, let the state transition matrix $\Phi =\left[\begin{array}{ccc}1& \Delta t& {({\displaystyle \frac{\Delta t}{2}})}^2\\ 0& 1& \Delta t\\ 0& 0& 1\end{array}\right]\otimes I$ and the observation matrix $\overline{H}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\otimes I$, we select the following target dynamic model [40,41]:

$$\begin{array}{lll}X(k)\hfill & =& {[x(k)u(k)a(k)]}^{\mathrm{T}},\hfill \\ X(k)& =& \Phi X(k-1)+w(k),\\ Z(k)& =& \overline{H}X(k)+\upsilon (k),\end{array}$$

where the state vector $X(k)$ has a dimension of $6\times 1$. The $2\times 1$ vectors $x(k)$, $u(k)$, $a(k)$ refer to the target state variables to be estimated, respectively, representing the position, velocity, and acceleration in the k-th step of the simulation. $Z(k)$ represents the observed values of the target state. $w(k)$ and $\upsilon (k)$ represent process noise and measurement noise, respectively, assumed to be zero-mean Gaussian white noise. We set $P_k(0)=I_6$, $Q=0.2I_6$ and $R=I$. The simulation results in low-speed target scenario are shown in Figure 3.

By comparing Figure 3 with Figure 2, we can observe that the application of the Kalman filter has further improved the accuracy of estimating the target’s relative position and the circumnavigation control, enabling the multi-UAV system to accurately encircle and track the low-speed moving target.

Then, we begin to discuss the high-speed scenario; the corresponding results are depicted in Figure 4, Figure 5 and Figure 6. From these results, it can be seen that the system also performs well in circumnavigation tracking of high-speed moving targets. It is able to accurately estimate the target’s state and execute the circumnavigation maneuvers precisely.

Considering the potential disturbance to the UAVs caused by air currents, which could lead to interference in bearing measurements, we conducted a simulation experiment. The simulation involved perturbing the bearing unit vectors ${\phi }_{i0}$ and ${\phi }_{ij}$ with Gaussian noise $n(·)$. The noise follows $E[n(·)]=0$ and $E[n(t)n(s)]=0.035\delta (t-s)$. The simulation results, as shown in Figure 7, indicate that the system still maintains satisfactory circumnavigation performance around the high-speed target, demonstrating a significant level of robustness against noise.

Finally, to substantiate the performance of our algorithm, we have conducted comparisons with the methods in reference [16,31]. In reference [16], the authors employed a single UAV equipped with GPS technology. They used the Virtual Intersecting Location Algorithm (VILA) to calculate a virtual observation value of the target state. Subsequently, by utilizing a Kalman filter and an initial estimation algorithm, the UAV estimated the target state and successfully accomplished the task of encircling the target. We selected the estimator (3) and the controller (17) from that reference, with the UAV’s velocity v set to 20 and the estimation parameter $k_E$ set to 200. The circumnavigation tracking results for both low-speed and high-speed targets are shown in Figure 8.

Additionally, in reference [31], the authors considered a scenario where five agents track a moving target in a GPS-denied environment. We adopted the estimation method (18) from that reference, and set the parameter $k_i$ to 12. The initial states of all agents were adjusted to match our situation. The simulation results of the comparative analysis are depicted in Figure 9. For the high-speed target, simulated control indicators show significant deviations from their desired values. Therefore, corresponding simulation illustrations are not be presented here. It is evident that neither of the two existing methods can ensure accurate target encircling tracking when the target has a high speed.

Through Figure 8 and Figure 9, it can be observed that existing typical algorithms can only successfully achieve encircling tracking in low-speed target scenario, while they perform poorly, or even fail, in encircling tracking of the high-speed target. By comparing with existing algorithms, the effectiveness of the algorithm designed in this paper is further validated.

6. Conclusions

In this paper, we have developed a distributed estimation algorithm that utilizes bearing measurements of the target. Based on the estimated target state, we have designed a distributed circumnavigation control law to guide the UAVs to complete the mission of encircling the target. Moreover, we have leveraged the collaboration among UAVs by utilizing relative bearing and velocity information in the network to calculate relative positions of the target. Additionally, we incorporated a Kalman filtering strategy to refine our estimator and controller, thereby achieving more accurate target encircling tracking. In a future work, we will consider the high-speed target tracking problem for the case with more practice UAV dynamics in 3D space.