Distance-Based Formation Control for Quadrotors with Collision Avoidance via Lyapunov Barrier Functions

In this paper, group formation control with collision avoidance is investigated for heterogeneous multiquadrotor vehicles. Specifically, the distance-based formation and tracking control problem are addressed in the framework of leader-follower architecture. In this scheme, the leader is assigned the task of intercepting a target whose velocity is unknown, while the follower quadrotors are arranged to set up a predefined rigid formation pattern, ensuring simultaneously interagent collision avoidance and relative localization. The adopted strategy for the control design consists in decoupling the quadrotor dynamics in a cascaded structure to handle its underactuated property. Furthermore, by imposing constraints on the orientation angles, the follower will never be overturned. Rigorous stability analysis is presented to prove the stability of the entire closed-loop system. Numerical simulation results are presented to validate the proposed control strategy.


Introduction
1.1. Background and Related Works. Quadrotors are agile mechanical systems which have drawn the attention of researchers for the last decades [1][2][3], mainly because the use of these types of aerial vehicles is appealing for several real-world applications. Due to their excellent performances, quadrotors are likely to be adopted in various civil potential applications such as inspection of power lines, oil platforms, search and rescue operations, and surveillance [4,5]. On the other hand, it is worth pointing out that a team of quadrotors has many benefits over using a single quadrotor in several applications where it results in faster and more efficient operations. The coordination of multiple quadrotors to perform cooperative tasks has received considerable attention due to the challenges arising in their control design. Coordinated movement of quadrotors comes with several methodologies and approaches proposed over the last few years [6][7][8][9][10]. From the control structure perspective, the approaches used for coordinated control can be categorized into centralized and decentralized methods. In the centralized method, the control signals for all the agents are generated in the leader agent or in a central station [7,11]. Whereas, in the decentralized method also known as the distributed method, each member of the team uses its own local controller enhanced with local information from its close neighbors [6,8,10].
From the control mechanism perspective, there are roughly five approaches to cooperative control of multiple aerial vehicles: the behavioral-based approach [12,13] where each agent locally reacts to actions of its neighbors. This method is categorized under the decentralized approach and is suitable for a large number of agents in the group. The disadvantage of this approach, however, is that the complexity of the dynamics of the group of robots cannot be amenable to mathematical stability analysis. Another important approach is the virtual structure approach [14][15][16][17]. This approach considers all the agents in the group as a single entity, thus limiting the formation scenarios that can be anticipated to merely static formation patterns. Moreover, the priority of agents in the group team is to respect one of the two rules: following their individual trajectories [18] or maintaining the group formation. The artificial potential function-based approach [6,19,20] is a powerful tool as it allows the agents to evolve in a cohesive motion while avoiding obstacles. The disadvantage of this approach is that local minima may appear where the vector field vanishes, therefore forcing the agents to get trapped into undesired equilibrium points. The leader-following approach [11,21,22], on the other hand, seems to be particularly attractive and preferred over the aforementioned approaches due to its simplicity and scalability. In this approach, some agents are designated as leaders aimed at tracking predefined trajectories, while others are treated as followers whose actions are merely to follow the leader(s) in order to keep the formation. The followers can in turn be designated as leaders for other vehicles in the group resulting in a scalability of the formation. The advantage of this approach is that from the definition of the motion of one entity, the overall group behavior can subsequently be defined. Also, it should be pointed out that the internal formation stability is induced by the control laws of individual vehicles. A major drawback of this approach, however, is that there exists no feedback from the followers to the leaders. Consequently, if the followers are perturbed, the coordination shape of the group team cannot be maintained.
A leader-follower formation control algorithm for multiquadrotors was proposed in [22,23] that involves a linear inner-loop and nonlinear outer-loop control structure, where a combination of sliding mode control and LQR/PD technique was deployed for the trajectory tracking control of a follower to track a leader quadrotor. The formation shape is formed then maintained by keeping a constant distance and a desired angle between each follower and the leader. The authors in [24] proposed a control strategy for a string-like formation of multiple autonomous quadrotor vehicles. In this approach, the objective is to design a maneuvering control for the quadrotor group along a geometric path while respecting an assigned timing law that determines the formation rate of advancement. A formation control strategy has been proposed [11] using the leader-follower approach. The suggested approach differs from common strategies employed for leader-follower schemes and consists in stabilizing, at a predefined distance vector, the position of the leader in the body frame of a virtual follower which in turn generates a trajectory which is used as a reference trajectory to be tracked by the real follower vehicle. The proposed strategy allows for generating curvature-rich formation trajectories. The authors in [25] proposed a 3D leader-follower formation control strategy for a quadrotor with completely unknown dynamics. Their presented strategy relies principally on the use of the spherical coordinate method where the desired position of the follower quadrotor is specified using a desired separation distance along with a desired angle of incidence and bearing. The model unknown nonlinearities are compensated using the neural network techniques. However, a common drawback of many previous leader-follower formation controllers is that they ignore the physical quadrotor flight limitation. In fact, for a safe flight, it is required that the quadrotor avoid singularities in the rotation matrix represented by the Euler angles. Essentially, they often assume that the quadrotor system is always away from the singularity points and that the quadrotor does not rotate too much. From a practical viewpoint, the trajectory generated by the leader vehicle might be aggressive sometimes. Consequently, the follower quadrotor is most likely exposed to a fall in singularity points. This, in turn, may result in poor tracking performance of the proposed controller. An immediate solution to alleviate this problem is to constrain the variation of the quadrotor states within a specified range, then design a controller such that the quadrotor's state never transgresses these constraints.

Contributions and Novelties.
Motivated by the earlier works, the main purpose of this work is to tackle the distance-based formation control problem for a group of homogeneous quadrotor vehicles modeled by an undirected graph. Contrary to the current state of the art, limited explicit communication among neighboring quadrotors is mainly motivated to avoid flooding the communication bandwidth in real-time applications, particularly when the number of cooperating quadrotors increases. To keep communication to a minimum, we propose a new distance-based formation control protocol for a group of multiple quadrotors in a leader-follower architecture, where sensor constraints are explicitly addressed in the design. In this architecture, we assume that the target's relative position with a leader in the group can be shared among other follower quadrotors, while its velocity is made unavailable. To learn this reference velocity, an estimation mechanism is implemented on each quadrotor so that a predefined formation is recovered so the quadrotors maintain a desired distance among their neighboring followers, while avoiding collision and ensuring relative localization. To deal with underactuation of the quadrotor vehicle in the formation, a decentralized intermediary translational control input is designed for relative position tracking, by which a bounded control thrust and bounded desired orientation are extracted. Then, a control torque for each individual quadrotor is designed to track the desired orientation while avoiding overturn of the quadrotors.
In comparison with existing works, the main contributions of this paper are listed as follows: (1) Compared to [6,26,27], our controller is completely designed independently from a global reference coordinate and does not require the local frame of all vehicles to be aligned. Only the distance requirements are used with local sensors to ensure formation maintenance (2) Our controller is decentralized and ensures simultaneously collision avoidance and self-localization. As opposed to [6,28], the control input for the translational dynamics is not designed based on a negative gradient of specific local/global potential functions; it is designed using a barrier Lyapunov function ensuring convergence to the appropriate distances with predefined performances 2 International Journal of Aerospace Engineering (3) The output constraints on the orientation angles are handled using the integral barrier Lyapunov (iBL) function [29] in the control design, thus relaxing the assumption on small orientation angles in [30][31][32] and therefore ensuring that the quadrotor will never overturn (4) Our design is modular in the sense that the stability of the cooperative system is easily shown by the fact that the closed-loop system is regarded as cascaded systems in which the position dynamic subsystem is the driven subsystem while the orientation subsystem is the driving subsystem 1.3. Organization. The paper is organized as follows. The upcoming section introduces basic concepts of rigid graph theory and results on the stability of cascade systems. Section 3 models the dynamic equations of motion for an individual quadrotor and presents the decentralized distance-based formation control objectives. The controllers for the translational and rotational dynamics of the quadrotors in the group formation are introduced in Section 4. The main result and the convergence analysis for the proposed controller are provided in Section 5. Numerical simulations are presented in Section 6. Finally, the conclusion is presented in Section 7.

Preliminaries
In this section, we present concepts of rigid graph theory for the control formulation taken from [33] and a lemma from [34] for the stability analysis of nonautonomous nonlinear systems, in cascade, which will be useful to establish the stability of the whole closed-loop system. An undirect graph G with pair ðv, εÞ is defined as a set of vertices V = 1, ⋯, n connected with a set of undirected edges E such that if the edge ði, jÞ in E,then ðj, iÞ ∈ ε. The number of edges that amounts for the connection between the vertices is l = 1, ⋯, nðn − 1Þ/2. The set of neighbors of vertex i is defined as N ðεÞ such that N ðεÞ = fj ∈ vjði, jÞj ∈ εg. A framework of G is the pair F = ðG, pÞ, where p ∈ ℝ 3n is the stack vector containing the position of all quadrotors. The rigidity function associated with F is denoted by Λð pÞ and is given by Λð pÞ = ½⋯, kp i − p j k 2 2 , ⋯ ⊤ , ði, jÞ ∈ ε, where k•k 2 2 denotes the Euclidian norm. The rigidity matrix M : ℝ 3n ⟶ ℝ l×3n of F = ðG, pÞ is defined as M = ð1/2Þð∂Λð pÞ/∂ð pÞÞ. It can be shown similarly to [35] that rank ðMÞ = 3n − 6. It follows that F is infinitesimally rigid [33] in ℝ 3 if the corresponding graph has at least 3n − 6 edges. An isometry of ℝ 3 is a bijective map I : ℝ 3 ⟶ ℝ 3 satisfying kν − ϖk = kI ðvÞ − I ðϖÞk, ν, ϖ ∈ ℝ 3 . Two frameworks are said to be isomorphic in ℝ 3 if they are related by isometry. The set of all frameworks that is isomorphic to F is denoted by IsoðFÞ. Frameworks ðG pÞ and ðG b pÞ are equivalent if Λð pÞ = λðpÞ and are congruent if kp i − p j k = kp i −p j k, ∀i, j ∈ V . If the infinitesimally rigid frameworks ðG pÞ and ðG b pÞ are equivalent but not congruent, they are called ambiguous. The collection of frameworks which are ambiguous to the infinitesimally rigid frameworks is denoted by AmbðFÞ. Lemma 1 (see [33]). Given a vector x ∈ ℝ 3 and let 1 n be the n × 1 vector of ones, then Mð pÞð1 n ⊗ xÞ = 0. Lemma 2. (see [33]). If the framework F = ðG, pÞ is minimally and infinitesimally rigid, the the matrix Mð pÞMð pÞ ⊤ is positive definite. Definition 3. A continuous function α : ½0, aÞ ⟶ ½0,∞Þ is said to belong to class K if it is strictly increasing and αð0Þ = 0. It is said to belong to class K ∞ if a = ∞ and lim r→∞ αðrÞ = ∞. A continuous function βð:,:Þ: ½0, aÞ × ½0, aÞ ⟶ ½0,∞Þ is said to belong to class KL if, for each fixed s, the mapping βðr, sÞ belongs to class K with respect to r and, for each fixed r, the mapping βðr, sÞ is decreasing with respect to s and βðr, sÞ ⟶ 0 as s ⟶ ∞.

Lemma 4.
(see [34]). Consider the following cascade system: If the equilibrium x = 0 is GAS/LES and the equilibrium z = 0 is LES and assume also that there exists a constant ℓ 1 ≥ 0 and a C 1 -K function γð: If in addition there exists a positive semidefinite radially unlimited function WðxÞ and some positive constants ℓ 2 and ℓ 3 such that kxk ≥ ℓ 2 then the equilibrium point ðx, yÞ = ð0, 0Þ is LES.

Problem Formulation
The proposed formation scheme is to coordinate the motion of n quadrotors under the leader-follower architecture. The leader quadrotor in the group is indexed by n, while the rest of the follower vehicles update their state with locally available information. The formation control scheme to be addressed in this paper will be formulated as formation maneuvering and target interception problems as in [33].
Hereafter, the interaction among the vehicles is modeled by undirected graphs G ≔ ðV , EÞ where V = f1, ⋯, i, ⋯, ng is the set of vertices (quadrotors) and E ⊂ V × V is the set of edges representing the connectivity among the vehicles. Define the set of neighbors of the i-th quadrotor as N i = fj ∈ V jði, jÞ ∈ Eg.
3 International Journal of Aerospace Engineering 3.1. Quadrotor Model Dynamics. To develop the quadrotor's equations of motion, we first defined two reference frames. The first one is the earth-fixed inertial frame defined as I = fO I , x I , y I , z I g, and the second is a body-fixed frame denoted by B = fO, x b , y b , z b g whose origin O is located at the quadrotor's center of gravity (CG) as shown in Figure 1.
Let p i ∈ ℝ 3 be the three-dimensional position of the i-th quadrotor and the rotation matrix R i ∈ SOð3Þ parameterized with respect to the three Euler angles Θ i = ½ϕ i , θ i , ψ i ⊤ with ϕ i , θ i , and ψ i being the roll, pitch, and yaw angles, respectively, and where the special Euclidean group SOð3Þ is the group of 3 × 3 orthogonal matrices with a determinant of one, i.e., SOð3Þ = fR i ∈ ℝ 3×3 jR ⊤ i R i = I, det ðR i Þ = 1g. Following the standard results, the associated orientation dynamics are governed by where : ð5Þ The kinematic and dynamic equations of motion of the i-th quadrotor are given by where v i = ½v xi , v yi , v zi ⊤ ∈ ℝ 3 is the linear velocity vector in inertial coordinates, Ω i = ½ω xi , ω yi , ω zi ⊤ ∈ ℝ 3 , m ∈ ℝ denotes the total mass of the quadrotor i, J i ∈ ℝ 3 is the inertia matrix with respect to the frame B, e z = ð0, 0, 1Þ ⊤ is the unit vector, and g is the gravitational acceleration. The thrust magnitude is denoted as f i ∈ ℝ, and τ i ∈ ℝ 3 is the moment vector; both are considered as the control inputs of the quadrotor. The matrix SðΩ i Þ is a skew-symmetric matrix containing the elements of the body-fixed angular velocity Ω i . Denote by Δ xi and Δ Ωi the unstructured uncertainties in the translational dynamics and the rotational dynamics of a quadrotor, respectively. We further assume that the uncertainties Δ xi and Δ Ωi are small time varying and bounded, that is, where δ xi and δ ωi are known constants. From the dynamic equations (7)-(9), it is worth noting that the quadrotor is an underactuated system with only one degree of freedom for the thrust actuation in the body frame, while the attitude dynamics are fully actuated and can be used to drive the thrust force to any desired orientation.

Control
Objective. The leader-follower approach considered in this paper assumes a group of n quadrotor vehicles with the n-th vehicle set to be the leader which can merely measure the target's relative position p d ðtÞ ∈ ℝ 3 . Additionally, it is also assumed that the only information that the quadrotors can sense is exclusively acquired from their local sensors. The control objective is to design controllers for the group of vehicles to maintain interdistance specification d ⋆ ij ∈ ε among the vehicles to attain a rigid desired formation, while guaranteeing formation maintenance and intervehicle collision, that is, if the desired formation is modeled by the framework F * = ðG * , p * Þ, which is assumed to be infinitesimally and minimally rigid, where F * = ðv * , E * Þ and ⊤ are the collection of the targets' positions. Also, considering that the actual formation of quadrotors is modeled by FðtÞ = ðG * , pÞ, then the formation control problem boils down to ensure the following: Mathematically speaking, the control objective reduces to design a desired thrust magnitude for the system dynamics of each quadrotor vehicle (7) such that where v d ðtÞ ∈ ℝ 3 is a bounded, continuous function designating the translational velocity of the formation and is a positive limit upper-bound. An important issue in the  International Journal of Aerospace Engineering formation of multiple quadrotor vehicles is to ensure safe navigation by avoiding collision among interacting vehicles. To this end, the distance that separates the quadrotors should be kept greater than a safety zone. Similarly, the quadrotors should also be retained within the connectivity network to achieve the group tasks. The decentralized controller should also satisfy the following constraint: where e ij ðtÞ = kp i − p j k − d ⋆ ij and ρ ij ðtÞ > 0, ρ ij ðtÞ > 0 are prescribed functions.In addition to the translational motion of the quadrotor, an essential part of the control is the attitude innermost layer of the controller. While the position controller calculates the desired direction of the thrust vector, the attitude controller will ensure the alignment of the real thrust vector with the desired one. A second control objective is to ensure a fast control loop for the attitude innermost layer. Thus, our goal is to satisfy the following limit: where R di ðtÞ represents the desired attitude that is calculated based on the desired trust vector f di which will be derived later in the next section.

Decentralized Control Design for the Position Outer-Loop.
In this section, we will use the backstepping technique to derive a decentralized cooperative controller for the position control loop in the sense that each quadrotor requires the relative position of its neighbors and its own velocity, while ensuring collision avoidance among the vehicles. For that purpose, the vector term f i R i e z will be considered as the control input signal for the cooperative position-tracking task. It is common to assign to that input signal a desired value f di R di e z that we wish to reach in the final stabilization process. R di denotes the desired orientation of the follower quadrotor while R di e z represents the direction of the force amplitude f di Assumption 5. The actuator dynamics of the follower quadrotor are negligible with respect to the rigid body dynamics of the quadrotor.
Due to the dynamics consideration, Assumption 5 reveals that the desired thrust magnitude f di can be instantaneously reached by the force control input f i ; therefore, it is straightforward to conclude that f di R di e z = f i R di e z . Hence, the first two equations in (7) can be rewritten in terms of R di and R i as follows: where the coupling termh di is the rotation matrix error. Thus, system (15) can be viewed as a nominal system, perturbed by a bounded term of the orientation error, which is assumed to converge to zero faster than the closed-loop translational dynamics by ensuring thatR i = I or, in other words, by designing an attitude control law that ensures that R i converges to R di as time goes to infinity. For this reason and to design the position control law, we will make the following assumption.
Assumption 6. Since the closed-loop attitude dynamics is faster than the translational dynamics, then it is logical to Remark 7. It is important to mention that Assumption 6 is only useful to help for the design of the control law position without explicitly accounting for the orientation tracking errorR i in the translational dynamics. The stability analysis of the resulting translational error dynamics with the coupling term will be investigated in detail later on in Section 5.
To help for the development of the decentralized cooperative controller, we first define the following state variable: where kp ij k = kp i − p j k from the definition of the relative distance, it can be easily seen that ℓ ij = e ij ðe ij + 2d ⋆ ij Þ which can be shown to be constrained as where ℓ ⋆ ðtÞ = 2ℓðtÞ is a performance function chosen similar to [36], such that it enforces transient and steady state performance specifications on the state variable ℓ ij . As such, Owing to the decreasing property of the performance function, one can see that −ℓ ij ℓ 0 and ℓ ij ℓ 0 serve as the maximum allowed overshoot and the minimum allowed undershot of ℓ ij , respectively. The control design proceeds as follows: Step 1. It is important to transform the control problem with the constraint (18) into an unconstrained one. To this end, the following barrier function is defined; it is a subtle treatment for ensuring collision avoidance and connectivity maintenance [37]:  (19), if the variable z ij is bounded, then ℓ ij /ℓ remains within the interval ð−ℓ ij , ℓ ij Þ for all t ≥ 0; therefore, the constraints in (18) The time derivative of the state variable of z ij considering (17) and (19) is where For the dynamic system (20), design the following velocity profile: where k > 0 is a design parameter, λ min ð:Þ denotes the minimum eigenvalue, andv d is the estimate of the target velocity in which position is only available to the leader vehicle. The estimate of the target velocity is the output of the following filter designed in a similar way as in [38]: where ℏ 1i ðφ i Þ = ½c 1 tanh ðφ 1i /c 1 Þ, c 2 tanh ðφ 2i /c 2 Þ, c 3 tanh ðφ 3i /c 3 Þ ⊤ and ℏ 2i ðφ i Þ = ½k 1 tanh ðφ 1i /k 1 Þ, k 2 tanh ðφ 2i /c 2 Þ, k 3 tanh ðφ 3i /c 3 Þ ⊤ , and c m , k m > 0, m = 1, 2, 3 are design parameters, K = diag ðk 1 , k 2 , k 3 Þ, and ρ e = p n ðtÞ −p d ðtÞ is the interception error between the leader quadrotor and the target, withp d ðtÞ =v d ðtÞ.
Step 2. In this step, define the velocity tracking error as Taking the time derivative of (24) along the solutions of the second equation of (15) results in Note that the termh i ð f i , R i , R di Þ is temporarily ignored at this stage of the design. It will, however, be investigated in a later stage of the stability analysis. It becomes easy from (25) to select the thrust force f i that stabilizes v ei to the origin. It is also important to point out that the total thrust force to be designed must be limited such that actuators' physical limitations are not violated. By defining the saturated input U i = f i R di e z and the nominal input as U 0i = f 0i R di e z , then this unsaturated input is designed as where k 1i and k 2i are design parameters and ξ i is an auxiliary state system generated as More discussion about the auxiliary system to overcome the saturation effect of the control input can be found in [39]. b Δ xi is the estimate of the disturbance Δ xi , generated using the following disturbance observer [40]: where K xi is s a symmetric and positive definite matrix.
The following proposition summarizes the distributed cooperative control for the translational dynamics of the set of quadrotors in the formation.

Proposition 8.
Consider the formation of a group of quadrotors, given the translational dynamics for each quadrotor in group (15), under Assumptions 5 and 6, if the nominal thrust magnitude f 0i and its direction R di e z are selected according to (30) and (31) wherev d = ½v∧ xd , v∧ yd , v∧ zd ⊤ ∈ ℝ 3 is the estimate of the target velocity using filter (23), then the equilibrium ðz ij , v ⊤ ei , ξ ⊤ i Þ ⊤ = 0 of the closed-loop system is globally asymptotically stable (GAS). Furthermore, the velocity vector of the leader quadrotor uniformly semiglobally practical asymptotically converges to v d ðtÞ.
Proof. The proof proceeds in two folds: In a first stage, we show that the desired formation is attained with prescribed transient performance, while avoiding interagent collision and connectivity intermittent by ensuring that the leader vehicle tracks an estimate of the target's state quite accurately. In a second stage, we show that the estimate of the target's velocity 6 International Journal of Aerospace Engineering converges to a vicinity of its true value. To proceed, we first define the following candidate Lyapunov function: where z = ½⋯, z ij , ⋯ ⊤ ∈ ℝ p , ði, jÞ ∈ ε. Differentiating V 1 along the solutions of (20) results in where then adding and subtracting the term z ⊤ ϒ M ϑ d and substituting the expression ϑ di , i = 1, ⋯, n, from the velocity protocol (22), using Lemmas 1 and 2, we obtain where b 0 = λ min ðϒ MM ⊤ ϒ Þ and j is a parameter chosen such that k > 1/2b 0 j. Note that the second term on the right-hand side of inequality (34) will be dealt with in a further step of the backstepping procedure. In the sequel, let us consider the following positive definite Lyapunov function: whereΔ xi = Δ xi − b Δ xi represents the estimation error of the disturbances affecting the transitional dynamics of the follower quadrotor. Take the time derivative of (35) along the solutions of (25), (27), (28), and (34) and substituting the control (30) and (31) we obtain where a z = 2 min fðkb 0 − 1/2jÞ, ðk 2i − ðj/2Þb 0 Þ, K xi , K ξ g. From (36), it is clear that _ V 2 ≤ 0; therefore, V 2 ðtÞ is decreasing for all t ≥ 0 from which it can be concluded that z, v ei , Δ xi , and ξ i are all bounded. Thus, there exists a positive constant z such that z ∈ Ψ z ≜ , fz ∈ ℝ p : kzk ≤ zg, ∀t ≥ 0, which from (19), by noting that Thus, it can be concluded that lim t→∞ ℓ ij = 0, ∀ði, jÞ ∈ E. Also, from (22), it can be deduced that lim t→∞ ϑ di ðtÞ =v d .
The second stage of the proof is to show that the estimate of the leader's velocity converges to a neighborhood of the target's true velocity. In this direction, following [38], a Lyapunov function candidate is proposed as The time derivative of V 3 along the solutions of filter (23) yields Since we showed that v ei , ∀i = 1, ⋯, n, is bounded and so is ϑ di , also by construction,v d is bounded. It follows that there exists a constant V n such thatkv n k ≤ V n ; therefore, where r e = ½r e1 , ⋯, r en ⊤ . Letting b 0 be a given positive constant, we obtain for kr ei k ≥ b 0 the following bound for _ V 3 since h 1i ðφ i Þ ⊤ h 2i ðφ i Þ ≥ 0; therefore, _ V 3 ≤ 0; this shows that the solution of filter (23) is uniform semipractically asymptotically stable [23]. This completes the proof.

Control Design for the Attitude Inner-Loop.
In this stage, the last two equations of (9) are considered. The moment vector τ i will be designed to globally asymptotically and 7 International Journal of Aerospace Engineering locally exponentially stabilize the tracking error ψ i − ψ di and the errors between the virtual controls of the roll and pitch angles and their actual values at the origin.
Step 1. Controlling the yaw angle.
The desired orientation R di can be deduced from the expression of the rotation matrix R i ∈ SO (2) using its Euler angle parameterization subject to the constraint given by the specification of the desired yaw angle extracted from the translational system as shown in Figure 2. Indeed, in Figure 2, the angleψ d represents the desired angle associated with the vectorv d ðtÞ projected on the x-y plan. Clearly, when v i ðtÞ matches withv d ðtÞ, the yaw rate angleψ i converges to zero. Following [41], the desired yaw angle is calculated by extracting yaw rate information from the cross-and innerproduct of the desired velocityv d ðtÞ and the actual velocity v i ðtÞ as follows: which implies thatψ i ðtÞ = −arctan 2ðsin ðψ i ðtÞÞ, cos ðψ i ðtÞÞÞ. Therefore, the desired yaw angle can be obtained by the following expression: Note also that the time derivative ofψ can be calculated as indicated in (43). Therefore, we can write where Ω ei = Ω i − α i is the angular velocity error, with α i = ½α e ϕi , α e θi , αψ i ⊤ yet to be determined later. γ ωi ðϕ i , θ i Þ = ½00 ðcos ϕ i /cos θ i Þ and αψ i is the stabilizing function, which will be chosen such that it cancels out unneeded terms in (42). As such, the stabilizing function can be chosen as follows: where κ 1 is a positive gain. Substituting (44) into (42) results in the following closed-loop dynamics: Clearly, if the last term of (45) is exponentially vanishing, then the tracking errorψ i is also exponentially stable; this can be seen by associating the quadratic Lyapunov function V ψ i = ð1/2Þψ 2 i , whose time derivative along the solutions of (45) gives where κ = κ 1 − 1/2 > 0. Hence, the virtual control law (44) renders the closed-loop (45) system input-to-state stability (ISS) from γ ωi ðϕ i , θ i ÞΩ ei toψ i . Therefore, if Ω ei is globally exponentially stable GES, then so isψ i .

Remark 9.
It has been shown in [42] that overturn of the quadrotor can only be avoided if the last component of R i e z = ½ρ m i ⊤ , m = 1 ⋯ 3 ði:e:, ρ m i = cos θ i cos ϕ i Þ is ensured to be strictly positive. This constraint further requires that e ζi = ζ i − ζ di (with ζ i = ½ρ i , m ⊤ , m = 1, 2, and ζ d = ½ρ d , m ⊤ , m = 1, 2, is a vector that contains the first two component of R d e z ) exponentially decays to zero in order to guarantee that θ i ⟶ θ d and ϕ i ⟶ ϕ d exponentially. Since ρ i, 3 , it is sufficient to show through an adequate control law that kζ i k ≤ 1: Remark 10. The virtual control (44) is well defined as long as θ i , ϕ i ≠ π/2; this singularity is avoided as long as the term cos θ cos ϕ is strictly positive or alternatively kζ i k ≤ 1: Step 2. Controlling the pitch and roll angles.
The pitch and roll angles are contained in the column vector R i e z and are denoted by the vector ζ i . The time derivative of ζ i along the solutions of the first equation of (9) and (5) gives the following dynamics: ðΘ i Þ can be obtained by a long but simple calculation as The matrix A i ðΘ i Þ is invertible, due to the fact that det ðA i ðΘ i ÞÞ = 1/cos θ i cos ϕ i , which implies that as long as the singularity (i.e., θ i = ϕ i = π i /2) is avoided, the matrix is nonsingular. The control objective is therefore to ensure on top of forcing the pitch and roll angles to track their immediate desired values and to also prevent them from violating the constraint that leads to singularity. In other words, the state variable ζ i is required to remain in the set kζ i k < 1 for all t ≥ 0: To address this problem, the integral barrier Lyapunov function iBLF [29] will be employed to guarantee nonviolation of the aforementioned constraint.
Motivated by the work in [29], in order to design the virtual control ω i,xy , the following integral barrier Lyapunov candidate functional is proposed: where ζ ei = ζ i − ζ d . It can be seen that V ζi ðζ ei ðtÞ, ζ d ðtÞÞ is positive definite and continuously differentiable and satisfies the decrescent condition in the set kζ i k < 1 such that for σ i = η i ζ ei , we have International Journal of Aerospace Engineering Based on the definition in (49), the time derivative of the iBLF V ζi ðζ ei ðtÞ, ζ d ðtÞÞ along the solutions of (47) yields where with the vector 1 being all components 1. We can obtain the stabilizing function α i,xy as where κ 2 is a positive gain to be designed. Under (52), the time derivative of V ζi is given by at this step according to (53); the time derivative of V ζi becomes negative definite only when Ω ei exponentially converges to zero. The design of the force moment is not finished yet. One more step is required to ensure that Ω ei exponentially converges to zero.
Step 3. Force moment control design.
The angular velocity error dynamics Ω ei based on the second equation of (9) has the following form: At this stage, the control input for the force moment τ i can be readily designed to compensate for the coupling terms created in the previous steps and the external perturbation to which the quadrotor is subject. To this end, let the control input be designed as follows: where K i is a diagonal positive gain matrix and K Ω is a symmetric and positive definite matrix.
Remark 11. The disturbance term is compensated for by an estimate b Δ Ω i which is the output of a nonlinear-observer inspired by [40]. The particularity of this type of observer is to guarantee an exponential convergence of the unknown disturbance to its true value. Furthermore, as we want the inner control loop (i.e., attitude dynamics) to be fast convergent as compared to the dynamic behavior of the outer control loop (i.e., translational dynamics), our choice is hence justified by ensuring complete cancellation of the uncertain terms.
For the convenience of stability analysis, we apply the control input (55) along with the virtual controls (52) and (44) and then the error attitude dynamics becomes The control design for the attitude inner-loop dynamics has been completed. We summarize the results in the following proposition.

Proposition 12.
Consider the attitude dynamics (9), given a desired attitude R d e z extracted from the positioning controller (31), under the proposed attitude control (55), the error of the attitude dynamics described by equations (56) and (57) is exponentially stable for any initial condition In particular, for any initial condition ζ i ðt 0 Þ ∈ Z i , the constraint kζ i ðtÞk < 1 is satisfied ∀t > 0.
Proof. The stability analysis of the attitude dynamics is conducted by considering the following Lyapunov function of the associated attitude variables: where δ 0 is a positive constant to be determined later. Differentiating both side of (59) along the solutions of 10 International Journal of Aerospace Engineering the closed-loop system consisting of (56) and (57) gives In light of the last inequality in (60) and in order to ensure that _ V i AT is negative definite, one can select the matrix K i such that λ min ðK i Þ > 1/2. Also, the observer gain K Ωi is selected such that λ min ðK Ωi Þ > 2ð1/δ 0 Þ, whereδ 0 can always be picked as a sufficiently small constant. Note, however, that δ 0 is not needed in the implementation of the proposed controller. Now referring to Lemma 2 in [29] to approximate the functional V ζi , it is shown that for all kζ i ðtÞk ≤ 1, the following inequality holds: Based on (61), the inequality (60) rewrites where β i ≔ min f2κ 1 , 2κ 2 , 2ðk 2 /kJ i kÞ, 2ðk 2 /δ 0 Þg. It implies that where . This implies that the closedloop system (56) and (57) is locally exponentially stable at the  11 International Journal of Aerospace Engineering origin. Furthermore, it can be seen that kζ i k is bounded by an exponentially decreasing function. Then, from (61), it can be concluded that V ζi is also bounded by an exponentially decreasing function such that V ζi ≤ c 0i with c 0i ≔ 2V ATi ð0Þ. On the other hand, since by construction V ζi ⟶ ∞ as kζ i k ⟶ 1, it can be inferred from the fact that V ζi is bounded for all t ≥ 0 and the state ζ i ðtÞ always satisfies the constraint jζ i ðtÞk≤1, provided that kζ i ð0Þk ≤ 1. This completes the proof.
Remark 13. The controller τ i in (55) involves the derivative of the virtual control α i , which appears to be difficult to calculate. So the robust differentiator [43] can be introduced to handle finite convergence of _ α i to its true value.

Control Law Synthesis: Closed-Loop System Stability Analysis
Before moving on to the stability analysis, we summarize as follows the motion control hierarchy developed in this work. First, a decentralized formation controller is designed for the open-loop translational dynamics of each individual quadrotor, which is based only on exchanging the relative position among neighbor quadrotors to guarantee safe formation maneuver and target interception. Second, the flight controller of the individual quadrotor receives the appropriate signal commands and ensures the relevant aerodynamic thrust and desired attitude necessary to achieve track-following of the leader whose only available information is its position. The resulting guidance and inner-and outer-loop control structures are illustrated in Figure 3.

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International Journal of Aerospace Engineering The total motion control system can now be analyzed by considering the complete system composed of the decentralized controller for the translational dynamics (15) and the orientation dynamics (56)-(57). The main result is presented in the following theorem: Theorem 14. Consider a group of n quadrotors each of which is described by the dynamics (7)- (9). If the set of quadrotors are organized at an initial minimally and infinitesimally rigid configuration, Assumptions 5 and 6 hold, and if the initial conditions for the roll and pitch angles are away from the singularity angles (i.e., π/2), then the decentralized control law and update laws consisting of (30), (31), (27), and (28) along with the constrained attitude control (55) guarantee achievement of the control objectives (13) and (14). In particular, the combined closed-loop inner-and outer-loop dynamics are forward complete, and the origin of the whole system is GAS.
Proof. The closed-loop of the overall system consists of the translational and the rotational dynamics of the n quadrotor in the formation group. In previous sections, each subsystem has been treated separately to investigate its stability property, assuming the interconnection terms among the subsystems are neglected. In this proof, the closed-loop system is rewritten with their corresponding interconnected terms:   In order to proceed with the stability analysis of the cascaded subsystems, it is first necessary to prove that the solutions of the whole systems do not escape to infinity in a finite time. One way to show that is to prove forward completeness [44] of the closed-loop systems (64) and (65). Consider the following Lyapunov function: Taking the time derivative of V f along the closed-loop systems (56) and (57) yields where β = min fβ i g, i = 1, ⋯n; C 0 is a positive constant; and σð:Þ is a class-K ∞ function. Hence, according to [44], the closed-loop systems (56) and (57) are forward complete. This result implies that we can proceed and analyze the overall system as a cascaded system. Note from (64), it can easily be verified that Fð0,0,0Þ = 0. Furthermore, to show steadiness of systems (64) and (65), it is sufficient to show that both conditions in Lemma 4 are satisfied. Clearly, from previous sections, we showed that the nominal system Σ X (without the coupling term) is locally exponentially stable (LES), also for the subsystem Σψ. The fact that these two cascaded subsystems are LES is because the term kH e ðt, XÞ ζ e k verifies the linear growth condition (2). Indeed, it is easy to figure out that the coupling term satisfies Furthermore, taking V 2 = W, it can be easily shown that conditions (3) are satisfied. This completes the stability analysis of the cascaded subsystems.

Numerical Simulations
To show the effectiveness of the proposed constrained distance-based formation controller for a group of quadrotors, numerical simulations are carried out in this section using the MATLAB software platform. In order to make the simulation close to reality, a Gaussian white noise is added to the measured signals including distance measurements among neighboring quadrotors, local velocity, and attitude using rand ð:Þ function. In the simulation, we 14 International Journal of Aerospace Engineering consider three quadrotors, where a reference trajectory is only known by the leader in the group as shown in Figure 4. In this configuration, all quadrotors are equidistantly distributed around the leader, such that d ⋆ ij = 1 m. For the simulations, the quadrotor follower system parameters are selected as m i = 8:2 kg, the inertia moment is J i = diag ð0:18, 0:34,0:28Þ kg · m 2 , and we assumed g = 9:8 m/s 2 . The initial conditions for the quadrotor followers are taken as p 1f ð0Þ = ½−5,−2, 0 ⊤ , p 2f ð0Þ = ½10, 3, 4 ⊤ , p 3f ð0Þ = ½10, 3, 4 ⊤ , v if ð0Þ = ½0, 0, 0 ⊤ , and Ω if ð0Þ = ½0, 0, 0 ⊤ where the subscript if stands for follower vehicle i = 1, ⋯, 3. The initial values of Θ i ð0Þ are chosen such that ½ϕ i ð0Þ, θ i ð0Þ, ψ i ð0Þ ⊤ = ½4,−2, 1 ⊤ ðπ/3iÞ, i = 1, ⋯, 3. The initial values chosen for the roll and pitch angles, ϕ i ð0Þ and θ i ð0Þ, i = 1, ⋯, 3, respectively, are to illustrate the ability of the proposed formation control design to handle large roll and pitch angles without violating the constraints that lead to singularity. The target reference trajectory and the desired heading of the leader quadrotor are chosen as follows: To test the robustness of the controller, external disturbances acting on the quadrotor due to wind have been added, which can be modelled by the forces Δ xi = ½1 + 0:5 cos ð2tÞ, 1 + 0:5 cos ð2tÞ, 0:5 cos ð2tÞ ⊤ and Δ Ωi = ½0:5 + 0:5 sin ð2tÞ, 1:5 + 0:2 sin ð2tÞ, 0:1 sin ð2tÞ ⊤ . The control gain parameters for the translational dynamics are k = 10, k 1i = 5, k 2i = 15, K ξi = 7, K xi = 15. The safety distance before collision and connectivity range specifications are set according to the performance parameters ℓ 0 = 0:5, ℓ ∞ = 1 m, and a = 0:1. The control gains for the attitude dynamics are chosen such that κ 1 = 10, κ 2 = 15, K i = diag ð10,10,10Þ, and K Ωi = diag ð5, 5, 5Þ. Figure 5 shows the evolution of the three-dimensional position of three quadrotors following the reference trajectory. It can be seen that the proposed controller provides global asymptotic convergence towards the reference trajectory. This can also be verified in Figure 6, where it can be shown how the distance between quadrotors asymptotically converges to the prescribed fixed value. Figure 7 corroborates these results through the convergence of the attitude angles to their desired values. Clearly, the attitude angles of all quadrotors in the group aggregate on the angular value during the formation. It can also be noticed that the pitch and roll angles are within the authorized range of variation and thus do not allow the quadrotor to overturn during the maneuver. Figure 8 depicts the asymptotic convergence of the angular velocity tracking error to zero, which validates the results of Theorem 14. Finally, the thrust forces and the torques applied to the quadrotors can be seen in Figures 9 and 10. From these plots, it can be concluded that the numerical results illustrate that the leader-follower strategy has a successful tracking performance.

Conclusion
In this paper, a strategy for formation maintenance with target interception control of a quadrotor-type aerial vehicle was investigated. The proposed decentralized controller guaranteed formation establishment with prescribed transient and steady state performances while avoiding interagent collision and connectivity losses. A distinctive feature of the proposed approach is that the group of quadrotors does not need explicit communication exchange; it only relies on the relative position of the neighboring vehicles and local sensor measurements. This makes the proposed decentralized control protocol extremely simple and easy to implement. Furthermore, the integral barrier Lyapunov function and the backstepping technique were instrumental to the implementation of the attitude controller which ensured that the quadrotor avoids overturning during the formation maneuver. The future work is to carry out experiments to test the proposed formation control design.

Data Availability
All the data are in the manuscript that validate the concept of the design.

Conflicts of Interest
The authors declare that they have no conflict of interest.

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International Journal of Aerospace Engineering