Fully Distributed Robust Formation Flying Control of Drones Swarm Based on Minimal Virtual Leader Information

: This paper studies the robust formation ﬂying problem for a swarm of drones, which are modeled as uncertain second order systems. By making use of minimal virtual leader information, a fully distributed robust control scheme is proposed, which includes three parts. First, the output based adaptive distributed observer is adopted to recover the global ﬂying path vector as well as the coefﬁcients of the minimal polynomial of the system matrix of the virtual leader system for each drone based on neighboring information from the communication network. Second, based on the estimated minimal polynomial of the system matrix of the virtual leader system, an asymptotic internal model is conceived to deal with uncertain system parameters. Third, by combining the asymptotic internal model and a certainty equivalent dynamic state feedback control law, a local trajectory tracking controller is synthesized to solve the robust formation ﬂying problem. Numerical simulations are provided to validate the proposed control scheme.


Introduction
Animal swarming behaviors are universal in nature, such as the flocking of birds [1] and schooling of fish [2]. As revealed by scientists, movement in formation may help the animal swarm to save energy and increase efficiency for long distance migration [3]. Inspired by these intriguing swarming behaviors of living creatures, formation control for various types of unmanned swarm systems has been widely investigated by both the control and robotic communities, such as ground mobile robots [4], unmanned aerial vehicles [5], and autonomous underwater vehicles [6], just to name a few. There are two key control objectives for formation control. On the one hand, all the individuals in the swarm should track some common reference path, and on the other hand, the individuals should keep the desired relative position/attitude with respect to each other so that the swarm system as a whole exhibits certain spacial configuration. Autonomous formation control is the foundation of many practical tasks for unmanned swarm systems, such as cooperative load transportation [7], area exploration and surveillance [8,9], and cooperative hunting [10].
Roughly speaking, the existing control strategies for the formation control of swarm systems fall into three categories, namely, the behavior-based strategy, the leader-follower strategy, and the virtual structure strategy [11][12][13]. For the behavior-based strategy, the basic task modules are named "motor schemas", such as goal-searching, obstacle avoidance, and formation keeping. The control policy is determined by calculating the weighted sum of different motor schemas. A seminal work using behavior-based strategy was reported in [14]. Later, the behavior-based strategy was further developed in [15] by decomposing a complex formation maneuver into a sequence of sub-level maneuvers. Recently, a bioinspired behavior-based swarm strategy was studied in [16] for environment exploration, which did not rely on complicated sensing or computation. In the leader-follower strategy, asymptotic internal model should cover the generating modes of both the global flying path vector and the local formation vector. To this end, the minimal polynomials associated with the global and local reference vectors are multiplied together to obtain an integrated internal model, which covers the generating modes of both the global and local reference vectors. • The controllability of the matrix pair of the internal model is a prerequisite for synthesizing the dynamic state feedback control. For the time-invariant internal model, the matrix pair can take any form. Meanwhile, since the asymptotic internal model conceived in this paper is time-varying, we have adopted the canonical controllable form for the matrix pair of the asymptotic internal model so that the time-varying system matrices of the augmented closed-loop system associated with the time-varying asymptotic internal model is controllable for all the time being. • In the design of the dynamic state feedback control, the control gains should be properly assigned so that all the eigenvalues of the nominal closed-loop system will be placed at pre-specified locations in the complex plane.  In comparison with the existing results, the main contributions of this paper are summarized as follows.

•
The internal model approaches adopted in [27,31,32], and also in our previous work ( [26], Chapter 10) required that full/partial information of the exosystem should be known to each individual in advance. Meanwhile, in this paper, the virtual leader which generates the global flying path vector is initially completely unknown to all the drones. The information of the virtual leader will be transmitted to each drone through the output feedback adaptive distributed observer proposed in [37]. Based on the estimated minimal polynomial of the system matrix of the virtual leader system, an asymptotic internal model is conceived to deal with system uncertainties. • It is noteworthy that in [33][34][35][36], and also in our previous works ( [26], Chapters 8 and 11) and [38], all the elements of the system matrices need to be recovered by various adaptive distributed observers. While, in this paper, by invoking the output-based adaptive distributed observer, much less system parameters need to be transmitted over the communication network, which drastically reduces the communication burden in comparison with the existing results. • In [28,29,[33][34][35][36]39,40], the individual models are free of parameter uncertainty. In contrast, in this paper, both the uncertain velocity damping matrix and the uncertain control gain matrix are taken into consideration for the second order drone models. To deal with system parameter uncertainties, we have resorted to the p-copy internal model approach [41], which is robust against moderate variations of system parameters around their nominal values.
The rest of this paper is organized as follows. Section 2 summarizes the notation and preliminary that will be used subsequently in this paper. The robust formation flying problem is described in Section 3. The main results of this paper are presented in Section 4. Numerical simulations are provided in Section 5 to validate the proposed control approach. Finally, Section 6 concludes this paper.

Notation and Preliminary
R and C denote the real and complex number field, respectively. ⊗ denotes the Kronecker product of matrices.
Given a square matrix A, let σ(A) denote the set of all the eigenvalues of A, and (σ(A)) denote the set of the real parts of all the elements in σ(A). Then, letδ A = max{ (σ(A))} and δ A = min{ (σ(A))}. 0 m×n denote a zero matrix of dimension m × n. Given a square matrix, A, let Φ c A (λ) and Φ m A (λ) denote the characteristic polynomial and minimal polynomial of A, respectively.
A digraph G = (V, E) consists of a finite set of nodes V = {1, . . . , N} and an edge set E = {(i, j), i, j ∈ V, i = j}. An edge from node i to node j is denoted by (i, j), and node i is called a neighbor of the node j. If the digraph G contains a sequence of edges of the from (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k , i k+1 ), then the set {(i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k , i k+1 )} is called a path of G from node i 1 to node i k+1 , and node i k+1 is said to be reachable from node i 1 . A digraph is said to contain a spanning tree if there exists a node i, which can reach any other node, and the node i is called the root of the spanning tree. The weighted adjacency matrix of Definition 1 (Definition 1.22 of [41]). Given any square matrixÂ 1 , a pair of matrices (G 1 , G 2 ) is said to incorporate a p-copy internal model of the matrixÂ 1 if the pair (G 1 , G 2 ) admits the following form: where (S 1 , S 2 , S 3 ) are arbitrary constant matrices of any dimensions so long as their dimensions are compatible, T is any nonsingular matrix with the same dimension as G 1 , and (G 1 , G 2 ) is described as follows: where for i = 1, . . . , p, β i is a constant square matrix of dimension d i for some integer d i , and σ i is a constant column vector of dimension d i such that The minimal polynomial ofÂ 1 divides the characteristic polynomial of β i .

Problem Description
In this paper, we consider the robust formation flying control of a swarm of N drones subject to uncertain system parameters. The mathematical model of the ith drone is given as followsṗ where p i (t), v i (t), u i (t) ∈ R 3 donate the position, velocity, and control input of the ith drone, respectively. E i ∈ R 3×3 denotes the uncertain velocity damping matrix, and Γ i ∈ R 3×3 denotes the uncertain control gain matrix.
and ∆E i ∈ R 3×3 denote the nominal part and uncertain part of E i , respectively, and and ∆Γ i ∈ R 3×3 denote the nominal part and uncertain part of Γ i , respectively. 18 denote the uncertainty vector of system parameter. To make system (3) controllable, it is assumed that rank(Γ o i ) = 3. The formation flying of the drones swarm is defined in the following way. For each drone, we design the reference path p ri (t) ∈ R 3 as where the global flying path vector p 0 (t) ∈ R 3 defines the reference flying path for the entire drones' swarm and can be considered as a virtual leader, and the local formation vector p f i ∈ R 3 defines the relative position between the ith drone and the virtual leader. The global flying path vector and the local formation vectors of all the drones together determine the specific way the drones swarm will fly in formation, and the formation flying tracking error e i (t) ∈ R 3 for the ith drone is defined by Suppose the global flying path vector is generated by the following linear of autonomous virtual leader systemṙ where r 0 ∈ R n r denotes the internal state of the virtual leader, and S 0 , W 0 are constant matrices with compatible dimensions. Since exponential decaying signals are meaningless as reference signals, without loss of generality, it is assumed that δ S 0 ≥ 0.

Remark 1.
The linear autonomous virtual leader system (6) can generate a large class of reference signals, such as polynomial signals with arbitrary coefficients in the following form: where a i ∈ R 3 , i = 0, 1, . . . , n are coefficients, or multi-tone sinusoidal signals with arbitrary amplitudes and initial phases in the following form: . . , n, are amplitudes, angular frequencies, and initial phases, respectively.
. Then, the mathematical model of (3) can be rewritten into the following compact forṁ Moreover , we define the nominal parts of the system matrices A i and B i as follows  (6) and the drones swarm (7) is described by Here, the node 0 is associated with the virtual leader, and the node i, i = 1, . . . , N, is associated with the ith drone. Let the weighted adjacency matrix of the digraph G be A = [a ij ] ∈ R (N+1)×(N+1) . We define a subgraph G s of G as G s = (V s , E s ) with V s = {1, . . . , N} and E s = E ∩ {V s × V s }. Let L s denote the Laplacian of G s and H = L s + D(a 10 , . . . , a N0 ). The following standard assumption is imposed on the communication graph G. Assumption 1. G contains a spanning tree with the node 0 as its root.
Note that Assumption 1 means that the information of the virtual leader can be transmitted to each drone through at least one directed communication path. Now, we are ready to formulate the robust formation flying problem as follows.
Problem 1 (Robust Formation Flying Problem). Given systems (6), (7) and the communication graph G, we need to design a distributed control law u i such that there exits an open neighborhood W of the origin of R 18 so that, for any w i ∈ W and any system initial condition, lim t→∞ e i (t) = 0.

Remark 2.
In the statement of Problem 1, it is required that the robust formation flying problem should be solved by the distributed control scheme for any w i ∈ W, which physically means that for any uncertain system parameter that belongs to a moderate range of the nominal value of this parameter, the trajectory tracking error for the drone should always be driven to zero asymptotically.

Main Results
In this paper, we solve Problem 1 by a fully distributed robust control scheme, which consists of three steps. First, given the virtual leader system (6), we design an output based adaptive distributed observer to estimate necessary leader information for each drone. Second, based on the estimated leader information, we conceive an asymptotic internal model to deal with uncertain system parameters. Finally, we synthesize a local trajectory tracking controller by combining the asymptotic internal model and a certainty equivalent dynamic state feedback control law. The detailed design process and the stability analysis of the closed-loop system are given as follows.

Design of the Output Based Adaptive Distributed Observer
There have been, so far, various distributed observer designs for the virtual leader system in the form of (6). While, to make use of the minimal leader information of (6), we adopt the output-based adaptive distributed observer as in [37], which is repeated as follows.

Design of the Asymptotic Internal Model
Since p f i is constant, it follows that p ri (t) can be generated by the following augmented virtual leader system˙r and G 01 = I 3 ⊗ g 01 , G 02 = I 3 ⊗ g 02 .
Then (g 01 , g 02 ) is controllable and the characteristic polynomial of g 01 is given by Since the minimal polynomial ofS 0 is either λΦ . Then, by Definition 1 and Remark 1.23 of [41], (G 01 , G 02 ) incorporates a 3-copy internal model ofS 0 .

Remark 3.
In the local trajectory tracking controller (18), (18c) defines the certainty equivalent trajectory tracking error. The dynamic compensator (18b) is the asymptotic internal model, which is time-varying since the matrix G i1 (t) is time-varying. (18a) gives the certainty equivalent dynamic state feedback control law. The external and internal signal transmission for the fully distributed robust control scheme is illustrated by Figure 2.
x j virtual leader neighboring information flow minimal VL information Figure 2. The external and internal signal transmission for the fully distributed robust control scheme.
Here, VL, ADO, AIM, and DFC are short for virtual leader, adaptive distributed observer, asymptotic internal model, and dynamic feedback control, respectively.

Stability Analysis
The main result of this paper is given as follows.

Numerical Simulations
In this section, a numerical simulation is given to validate the effectiveness of the proposed control scheme. Consider a swarm of N = 5 drones. The communication graph G is shown in Figure 3. It can be directly checked that Assumption 1 is satisfied. The nominal parts and uncertain parts of the matrices in (3) are given by The desired formation for drones swarm is in a wedge-shape, which is shown in Figure 4.  The local formation vectors in (4) are designed to be Note that p f 1 = col(0, 0, 0) means that the trajectory of drone 1 coincides with the trajectory of the global flying path. The global flying path vector is designed to be p 0 (t) = col(0, t, sin ωt) with ω = 0.3 rad/s. p 0 (t) can be generated by the linear autonomous virtual leader system (6), where the constant matrices in (6) are given by The initial value of the linear autonomous virtual leader system is r 0 (0) = col(0, 1, 0, ω). Besides, the minimal polynomial of S 0 is Φ m S 0 (λ) = λ 4 + ω 2 λ 2 . In the simulation example, the initial positions for the drones are given by and the initial velocities of all the drones are set to be zero.
By selecting the gains of the adaptive distributed observers and the dynamic state feedback control to be • Gain Set 1: µ α = 0.1, µ ζ = 5, κ i = 0.5, • Gain Set 2: µ α = 1, µ ζ = 10, κ i = 0.5, • Gain Set 3: µ α = 1, µ ζ = 10, κ i = 2 the performance of the fully distributed robust control scheme is shown by Figures 5 and 6, where Figure 5 shows the trajectory tracking errors of the drones under different control gains, and Figure 6 shows the swarm path profiles under different control gains. For all these three cases, it can be seen that the control objective has been successfully achieved. By comparing the results obtained under gain sets 1 and 2, it is found that the gains of the adaptive distributed observers µ α , µ ζ do not affect much the control performance. Meanwhile, by comparing the results obtained under gain sets 2 and 3, it can be observed that the average oscillation amplitudes of the tracking errors are smaller for larger dynamic state feedback control gains κ i .

Conclusions
This paper solves the robust formation flying control problem for a swarm of drones by a fully distributed robust control scheme. By utilizing the output-based adaptive distributed observer to recover the necessary information of the virtual leader, an asymptotic internal model is conceived, which together with a certainty equivalent dynamic state feedback controller solves the robust formation flying problem for the drones' swarm in a distributed way. In this paper, the communication network is assumed to be static and reliable. In the future, it would be interesting to further consider the case of unreliable communication networks.