Target Enclosing and Coverage Control for Quadrotors with Constraints and Time-Varying Delays: A Neural Adaptive Fault-Tolerant Formation Control Approach

This paper investigates the problem of formation fault-tolerant control of multiple quadrotors (QRs) for a mobile sensing oriented application. The QRs subject to faults, input saturation and time-varying delays can be controlled to perform a target-enclosing and covering task while guaranteeing the state constraints will not be exceeded. A distributed formation control scheme is proposed, using a radial basis function neural network (RBFNN)-based time-delay position controller and an adaptive fault-tolerant attitude controller. The Lyapunov–Krasovskii approach is used to analyze the time-varying delay. Barrier Lyapunov function is deployed to handle the prescribed constraints, and an auxiliary system combined with a command filter is designed to resolve the saturation problem. An RBFNN and adaptive estimators are deployed to provide estimates of disturbances, fault signals and uncertainties. It is proven that all the closed-loop signals are bounded under the proposed protocol, while the prescribed constraints will not be violated, which enhances the flight safety and QR formation’s applicability. Comparative simulations based on application scenarios further verify the effectiveness of the proposed method.


Introduction
Formation control technology, which is based on the theory of multi-agent systems (MAS), enables multiple unmanned aerial vehicles (UAVs) to efficiently complete a shared task and is widely used in aerial mapping, atmospheric environment monitoring and even coordinated military missions [1][2][3].
As a typical small-scale UAV, quadrotor (QR) is qualified to be a formation platform for a variety of applications due to its simple structure, strong maneuverability and hovering capability [4], particularly for mobile sensing tasks, such as target-enclosing and covering, which have been studied by several works so far. The main purpose of the former was to control several mobile sensors to rotate around or above a detected target to obtain detailed information from all angles [5][6][7]. The objective of the latter was to optimize the deployment location of multiple sensors to achieve effective coverage of the interest area, where the methods are mainly Voronoi partitioning-based [8,9], coverage cost functionbased [10], K-means-based [11] and reinforcement learning-based [12]. However, these methods cannot be directly applied to small-scale aerial platforms due to the contradiction between the complex location optimization algorithms and limited computing resources. In this paper, a consensus-based formation controller was designed. The UAV's movement and placement can be directly and flexibly set by time-varying formation functions and virtual leader trajectory, which ensures that the mobile sensing task, including the above two, can be performed when the formation tracking is realized by QR members.
In light of the aforementioned obstacles, we propose a novel QR formation FTC framework for a mobile sensing oriented application. The main contribution of this work is threefold. Firstly, based on a distributed adaptive FTC mechanism, the effect of timevarying multiplicative and additive faults can be effectively compensated for each QR, and the desired formation flight can still be achieved. Secondly, by applying the barrier Lyapunov function (BLF) technique and designing an auxiliary system, the attitude states of QRs can be constrained in the presence of input saturation, and our BLF analysis can also be applied to unconstrained scenarios without modifying the control structure. Compared to the methods in [27][28][29], the scope of application is expanded. Thirdly, the time-varying delay of each QR is different. Only delayed neighbor information is needed to realize formation flight; that is, the proposed protocol is distributed, and the time-varying formation configuration can be flexibly designed to adapt to target enclosing, area covering and other scenarios. Meanwhile, the disturbances and uncertainties are handled properly by radial basis function neural network (RBFNN) and an adaptive estimator; the application restrictions in real-word environments are relaxed compared to [32][33][34].
Notations : Let a ⊗ b denote the Kronecker product of matrices a and b, and σ min (•) and σ max (•) indicate the minimum and maximum singular value of a matrix. We denote |•| as the absolute value of a real number, • the Euclidean norm of a vector and • F the Frobenius norm of a matrix.

Basic Concepts on Graph Theory
An undirected graph G = {V, E , W } represents the communication topology of the N QRs, which contains a set of nodes V = {q 1 , q 2 , ..., q N }, a set of edges E ⊆ (q i , q j ) : q i , q j ∈ V and a weighted adjacency matrix W = [a ij ] ∈ R N×N . If agent i is connected by an edge with agent j, that is (q i , q j ) ∈ E , then a ij = a ji > 0. Otherwise a ij = a ji = 0, and a ii = 0 for all i ∈ Σ = {1, 2, ..., N}. The set of neighbors of node q i is defined by N i = q j ∈ V : (q i , q j ) ∈ E . The out-degree of node q i is defined by Deg out (q i ) = ∑ j∈N i a ij . The degree matrix of graph G is represented by D = diag{Deg out (q i ), i ∈ Σ}, and the Laplacian matrix of graph G is represented by L = D − W. The undirected graph G is said to be connected if a path exists between any two nodes q i , q j ∈ V, where the path represents a series of diverse adjacent points from q i to q j . If q i can access information from the leader, then the connection weight between them b i > 0. Otherwise b i = 0, and the matrix form is B = diag{b 1 , b 2 , ..., b N }. Throughout this brief, the following assumption is made for the communication topology. Assumption 1. The undirected graph for N QRs is connected, and there exists at least one path between the leader and follower.

Problem Formulation and Modeling
Consider a group of N QRs following a virtual leader labeled as 0, of which the interaction topology is described by an undirected graph G; it is assumed that graph G is connected. Taking practical factors into account, the dynamic model of QR can be formulated by using Newton's laws [42]: T are position, velocity and attitude of the i-th QR in inertia frame, respectively. Ω i = [p i , q i , r i ] T represents the angular velocity in a body-fixed frame. In addition, m i and J i = diag J xi , J yi , J zi represents the total mass and inertia matrix, respectively. T i represents the total thrust, e 3 = [0, 0, 1] T and g represents the gravity constant, and F iP ( represents the lumped uncertainty term including disturbance and inaccurate modeling in (1).
Unknown time-varying function ∆F i ∈ R 3×3 and D iA represent parameter perturbation and external disturbance in (2), respectively. R it , R ir and S(Ω i ) are shown below: where s ( * ) sin( * ), c ( * ) cos( * ). The model of input saturation is expressed as follows: represents the control input free from limits but subject to actuator faults, which are expressed as follows are time-varying additive and multiplicative actuator faults, respectively. U iAC (t) ∈ R 3 is generated by the attitude controller to be designed. Considering that most sensors and PTZ systems have constraint requirements for rotational motion, the attitude states of QRs will be constrained and are defined as follows where C im (t) ∈ R, m = 1, 2 represents the time-varying constraints.
The formation center is regarded as the virtual leader, which is specified by P 0 = 1 N ∑ N i P i , and its trajectory is are 2nd-order differentiable functions defining the motion mode of i-th QR with respect to the geometric center, k = 1, 2, 3. Based on consensus theory, we give the following definition: Definition 1. The formation tracking flight is said to be achieved when Except the communication delay τ ij (t) between i-th QR and j-th QR, this paper also considers the self delay τ ii (t) of i-th QR caused by calculation or measurement. τ ij (t) and τ ii (t) are generally regarded as uniform delay τ i (t) in the MAS consensus control problem [43]. Assumption 2. The time-varying delay has upper bound, that is, τ i ≤ τ M , i ∈ Σ.

Control Objective
As depicted in Figure 1, the objective of this work is to design a formation control scheme for the QR mobile sensing platforms to perform the following tasks. The first one is a covering task, in which the QRs can follow the virtual leader to track a moving target and fully cover the target's adjacent area to carry out sensing or surveillance. The second task is target enclosing, in which the QRs can be controlled to gather and rotate above the moving target to monitor or observe it. The detailed control objectives of proposed formation control protocol are as follows: • Consensus-based time-varying formation control protocol (10) is designed based on the demands of the target-enclosing and covering tasks; • Distributed adaptive FTC mechanism is deployed to compensate the fault signals (4); • BLF and auxiliary system are designed to ensure that the constraint requirements (5) of sensor payload will not be violated in the presence of input saturation (3); • The influence of time-varying communication delay τ i can be eliminated by the L-K technique; • The problem of uncertainties and disturbances in (1) and (2) can be neutralized RBFNN (12) and adaptive estimators (49).

Main Results
The desired formation control scheme is proposed in Figure 2, which can be divided into a RBFNN-based time-delay position controller (NTDPC) (outer-loop) and an adaptive fault-tolerant attitude controller (AFTAC) (inner-loop). The inputs of outer-loop, including time-delayed neighbor information (P jτ , V jτ ) j∈N i , time-delayed self information P iτ , V iτ and time-delayed leader information P dτ , Λ Fτ are entered to NTDPC. In the mean time, the lumped uncertainties F iP (P i , V i , t) are compensated by the RBFNN approximation law. Then, the command attitude signals φ iC , θ iC and total thrust T iC are calculated from the outputs of NTDPC. The inputs of inner-loop, including command attitude signals φ iC , θ iC , ψ iC , are transferred to AFTAC. Meanwhile, the external disturbances D iA , actuator faults Γ i , δ i and model uncertainties ∆F i are compensated by adaptive estimation laws. Finally, the control inputs U iA and T id are applied to i-th QR for formation flight. It should be pointed out that the derivatives of φ iC , θ iC are obtained from Command Filter_1 for the sake of reducing computational burden.

RBFNN Approximation
Suppose an unknown smooth nonlinear function f (x) : R m → R can be approximated over a prescribed compact set Σ R ∈ R m as follows where Ψ(x) = [ψ 1 (x), · · · , ψ l (x)] T : Σ R → R l denotes the radial basis function vector, of which the element is expressed as follows where ς k ∈ R m and µ k ∈ R are the center and spread. ∈ R is the bounded RBFNN approximation error on Σ R , that is, | | ≤¯ with¯ is an unknown constant. W * ∈ R l is the ideal RBFNN weight vector expressed as follows whereŴ represents the estimation of W * .

Design of NTDPC
For i-th QR, the local tracking errors are defined as follows Then the error dynamics of system (2) can be expressed in a compact form as follows where e P = [e T 1P , e T 2P , ..., e T NP ] T , e V = [e T 1V , e T 2V , ..., e T NV ] T , To obtain the approximation of the lumped uncertainty F iP , we adopt an adaptive RBFNN with time-delayed states P i (t − τ i ) and V i (t − τ i ) as inputs and approximation value as output, which is expressed asF Then F P can be expressed as In addition, the RBFNN weights estimation error is denoted asW = W * −Ŵ. Now we design the control inputs U iP of i-th QR position subsystem (2) and update laws of RBFNN weightsŴ i as: Combining (14) and (15), we havė [44], in which ∆ P and ∆ V are formation tracking errors, Lemma 2. According to [46], we can conclude that the following inequality is always valid: Under Assumptions 1-3, with the control law (14) and update law (15), the timevarying formation tracking for N QRs position systems (2) subject to time-varying delays and uncertainties can be achieved if the positive design constants M k P k V and k P , k V , K W , M 1 , M 2 are chosen appropriately to make the symmetric matrix M be positive definite, which is Proof. Consider the Lyapunov-Krasovskii candidate function as Taking the time derivative of V 1 and V 4 we havė By Lemma 2, we obtain the time derivatives of V 2 and V 3 as followṡ where where Thus, e is uniformly ultimately bounded (UUB) according to [47]. Moreover, e P , e V are bounded stable referring to the definition of e, and following Lemma 1, the formation tracking errors ∆ P and ∆ V are also UUB. So, the desired position control for formation flight can be realized by control law (14) and RBFNN update law (15).

Remark 2.
In matrix M, k p , k v , K P , K V and K W are control and adaptive parameters, and K 0 , M 1 and M 2 are constants to be selected. τ M is the upper bound of time delays. Except τ M , all of the above parameters are adjustable to ensure the solvability of M. Besides, one can see that all of the diagonal elements of M are positive when τ M being a certain value. Therefore, M is solvable in Theorem 1.

Design of AFTAC
where ψ iC is a free variable and can be set to ψ iC = 0 for simplicity.

Remark 3.
It is feasible to ensure U i3P + g is constantly positive to avoid singularity because U i3P is bounded by selecting suitable gain constant k P , k V andŴ T i Ψ i ,∆ i0τ is in a cetain range when calculating θ iC in (28).
To deploy the attitude control scheme, the following assumptions and lemma need to be made: The initial state of the attitude subsystem needs to be within the constraints C im , m = 1, 2, which is the (3 − m)-th order differentiable.
Assumption 5. The actuators will not completely fail during operation, and the fault signals Γ ik (t) and δ i (t) change continuously within certain ranges, that is, where Γ ik,min , Γ ik,max are known constants with k = 1, 2, 3, and Assumption 6. The model uncertainty factor ∆F i (t) and its derivatives and unknown disturbances D iA are bounded, which are expressed as tr Lemma 3. By [48], we know that the following inequality holds: where z i2 = Ω i − α i is angular velocity tracking error, α i is the command filtered signal of designed virtual control law α iC , in which the command filter limits the magnitude, rate and bandwidth of α iC and is shown in Figure 3. In order to deal with the constraints on the attitude states, we adopt the tan-type BLF as follows

Remark 4.
When there is no attitude constraint on i-th QR, then C im → ∞; thus, C im → ∞, m = 1, 2, and we have that is, our BLF analysis method is also available for the unconstrained circumstance.
For simplicity of notation, define ν im = z im

2C 2 im
, m = 1, 2 and take the derivative of V i1B with respect to time, and we havė The designed virtual control law α iC is shown as below where µ i1 is a positive small constant, K i1α > 2K i1C > 0, K i1α is a design parameter, and

Remark 5.
To makeV i1B be negative definite, the terms − (33) will be canceled by terms (34), respectively. Noticing that , this will generate the negative definite BLF-form term in (33).
The auxiliary system E i1 is designed aṡ

Remark 6.
When saturation occurs, the auxiliary system will respond to it. Otherwise, ∆α i = 0, thenĖ i1 = −K i1E E i1 ; thus, E i1 will converge into E i1 E i1 , after which, if saturation occurs again, E i1 can be reset so that E i1 > E i1 . Then, the auxiliary system can be made responsive again.
As can be seen from (33), The Lyapunov function for this step is constructed as Taking the time derivative of (37) and notice that where ν T i1 R ir z i2 will be compensated later. Similarly, the BLF for m = 2 is The dynamics of the angular velocity tracking error z i2 is derived aṡ where ∆U iA = U iA − U iAF . According to (4), the following inequality holds: where U iA is the given input, U iAC is produced by our desired control law and Γ i and δ i are known from Assumption 5; thus, U iA can be determined. Define U iAC as whereΓ i is the estimation of the multiplicative fault Γ i , Φ i will be designed later. Notice that whereΓ i =Γ i − Γ i is the estimation error of multiplicative fault Γ i . Then, the desired control law Φ i is designed as where i2C > 0, µ i2 > 0 are small constants, K i2Φ > 0 is a design parameter and The update law for ∆F i ,δ i ,Γ i andD iA are constructed as Similarly, auxiliary system E i2 is designed aṡ , similarly as (36), and we can obtaiṅ are estimation errors of model uncertainty and additive fault, respectively. The Lyapunov candidate function for this step is derived as Taking the time derivative of V * i2 and observing that where (5), possesses the following properties:

Theorem 2. Under the Assumptions 4-6, with the adaptive estimation laws (46)-(49) and control laws (43), (45), the attitude subsystem (3) of i-th QR subject to input saturation (3), actuator faults (4) and state constraints
i. The attitude state constraints (5) of i-th QR will not be exceeded during formation flight. ii. The attitude and angular velocity tracking error will exponentially converge into the set iii. The estimation errorsΓ i ,δ i , ∆F i ,D iA and the closed-loop signals E im will be bounded, m = 1, 2.

Proof. By (58), we have
; thus, V * i2 has upper bound, which means the BLF is bounded. Besides, ; hence, during formation flight, no violation of attitude state constraints will occur. Additionally, where m = 1, 2, which indicates that z im will exponentially converge into z im 2S i s i , and the estimation errors and closed-signals mentioned above will also be bounded.

Simulations
To demonstrate the effectiveness of the proposed scheme, some comparative simulations were carried out, which were programmed via Matlab 2016a and performed on a PC with a 4-core Intel i7-4980HQ@2.8 GHz CPU and 16 GB of RAM. The application scenario of using 5 QRs to enclose and cover a moving ground target is considered. Suppose a target is detected at t = 0s and moving along T g = [15sin(0.026t), 15cos(0.026t), 0] T . Meanwhile, the QRs will follow the virtual leader to fly right above the target and cover its adjacent area to monitor or sense. Then, the QR formation will converge towards its center at T 1 , start spinning at T 2 and lower the altitude at T 3 to enclose the target closely. The target's adjacent area is defined as a circular area with the radius being 3.5 m and centered on the target. The coverage area of i-th QR is centered on [P i1 , P i2 , 0] , with the radius being P i3 tan θ S 2 , and θ S = 50 • represents the angle of view of the sensor payload. The trajectory of the virtual leader is set as and ω 0 = 0.8(rad/s). The topology graph is shown in Figure 4, which is undirected and connected, with the weights being a 12 = a 21 = 1, a 23 = a 32 = 1, a 34 = a 43 = 1, a 45 = a 54 = 1 and b 1 = b 5 = 1.

Remark 7.
In practical applications, the motion information of some non-cooperative targets may not be directly obtained. In this case, the estimated motion information can be obtained by other means and used for formation control, but it is not within the scope of this study. More details can be seen in [49,50]. (Λ F , P d ) can be designed carefully according to different sensing tasks, sensor performances and quality-of-service policies. The (Λ F , P d ) chosen in this paper is a basic example to demonstrate the effectiveness of the proposed method. , Ω i = 0 (rad/s). The constraints on attitude state A i is C i1 = A iC + 0.065 (rad) and the angular velocity state Ω i is constrained by C i2 = 13π 36 (rad/s). The control input saturation of attitude controller is set as

(62)
The time-varying multiplicative and additive actuator fault signals are considered as follows withΓ ik = 1,δ ik = 0, i, k = 1, 2, 3 as the initial estimation values. The simulation results of trajectory, position, attitude, attitude constraints, angular velocity constraints, control inputs, RBFNN, disturbance estimation, multiplicative fault estimation and additive fault estimation are demonstrated in Figures 5-14, respectively. The trajectory and position snapshots of QRs and a moving target are illustrated in Figure 5. It can be seen that the QRs can successfully form the desired formation pattern Λ F and track the desired trajectory P d , thereby achieving the full coverage and close-range enclosing. Figure 6 shows the position tracking errors with and without RBFNN. In the case of with RBFNN, the tracking errors converge to the neighborhood of zero rapidly under the influence of lumped uncertainties. The effectiveness of RBFNN is demonstrated by the fact that tracking error cannot be reduced to near zero and continues to oscillate in the absence of RBFNN. Figure 7 demonstrates the robust learning ability of RBFNNs, convergence of approximation errors takes only a few seconds and oscillation at the beginning is caused by randomly selected initial weights. Figure 8 depicts the tracking performance of AFTAC, which, despite initial misalignments, tracks the command signal exceptionally well. Furthermore, Figures 9 and 10 show the norm of attitude A i and norm of angular velocity Ω i always satisfy the predefined constraints C i1 and C i2 during the whole process. In Figure 9, the unconstrained AFTAC in [51] is compared under identical conditions, and the parameters of the comparison AFTAC are adjusted to achieve relatively good tracking performance. One can observe that the comparison AFTAC tracks the command signal closely throughout the whole process, but it cannot guarantee the state constraints will always be met; the constraints are occasionally exceeded, particularly when the command signal changes rapidly. The comparison results demonstrate that the specific system states can be constrained within a certain range to meet safety or sensor payload requirements, which is an advantage of our method. Figure 11 depicts the input signals of QRs, which contain large spikes at the beginning, T 1 and T 2 . These spikes are effectively filtered out by input saturation, where the actuator's limitations are fully reflected. As demonstrated by the proof of Theorem 2, the upper bound of external disturbance and actuator fault signals are effectively estimated in Figures 12-14.     Figure 10. Ω i , α i and constraints C i2 , i = 1, 2, 3, 4, 5.

Conclusions
This article presents a distributed formation control scheme for a group of QRs subject to constraints and time-varying delays. The proposed scheme consists of NTDPC for position control and state-constrained AFTAC for attitude regulating. In NTDPC, an adaptive RBFNN is utilized to compensate the lumped uncertainties, and a Lyapunov-Krasovskii analysis is applied to handle the time-varying delay. Based on the backstepping technique, AFTAC employs a tan-type BLF to handle the state constraints, an auxiliary system combined with a command filter to deal with input saturation and adaptive estimators to compensate fault signals and disturbances. To determine the efficacy of the proposed method, comparative simulations were conducted. We demonstrate that the proposed method can be applied for a mobile sensing task; the formation tracking errors are UUB; the estimation errors of actuator faults, uncertainties, and disturbances are also bounded; and the predefined constraints will never be violated during formation flight. However, the current method has some limitations, such as symmetric state constraints and a fixed network topology. Additional research will yield asymmetric state constraints and a mechanism for switching topologies.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.