Two-layer distributed formation-containment control strategy for linear swarm systems: Algorithm and experiments

Summary This article addresses the problem of designing a two-layer distributed formation-containment control scheme for linear time-invariant swarm systems with directed communication topology, where the states of the leaders attain a prespecified time-varying/stationary formation and the states of the followers converge into the convex hull spanned by the states of the leaders. To achieve formation-containment, a set of distributed control protocols is developed utilizing the neighboring state information, which enables the proposed scheme operate without using global information about the entire interaction topology. The conditions to achieve formation-containment are suggested and a theoretical proof of the proposed scheme is also derived exploiting the Lyapunov stability approach. An algorithm is written to provide systematic guidelines on how to implement the control protocols in practice. It is argued that the consensus problem, formation tracking problem, and containment control problem can all be viewed as special cases of formation-containment. A simulation case study has been presented to demonstrate the usefulness and effectiveness of the proposed scheme and moreover, lab-based hardware experiments involving networked mobile robots were performed to test the feasibility of the scheme in real-time implementation.

principles for systems with second-order dynamics and applied it in solving various formation control problems. The article 13 also argued that the existing leader-follower, behavioral and virtual leader formation control approaches can be unified in the general framework of consensus building.
Recently, with the development of algebraic graph theory, many consensus-based control strategies have been applied to deal with formation control problems. 14 Formation tracking control of first-order multiagent systems was analyzed in Reference 15. In Reference 13, a distributed formation control strategy was applied to a group of second-order swarm systems based on local neighbor-to-neighbor information exchange. A finite-time formation control protocol for second-order multiagent systems was proposed in Reference 16, where a time-invariant formation tracking is achieved within finite time. It is worth emphasizing that in some practical applications the dynamics of the agents can only be described by high-order models. Reference 17 discusses the formation stability problems for general high-order swarm systems and the result is extended to deal with formation tracking for multiple high-order autonomous agents by using a two-level consensus approach given in Reference 18. In Reference 19, adaptive group formation tracking problems was investigated but the theoretical results were not validated with real experiments. Experimental results on static formation of quadrotor swarm systems based on consensus approach were reported in Reference 20, while time-varying formation control of swarm systems is still an active area of research which requires further attention.
Another significant research topic pertaining to cooperative control of swarm systems is the containment control, which deals with multiple leaders and followers and the states of the followers are required to converge into a convex hull spanned by the states of the leaders. In Reference 21, containment control problems for both continuous-time and discrete-time multiagent systems with general linear dynamics under directed communication topologies were investigated. Containment control scheme was applied to a group of mobile autonomous agents with single-integrator kinematics under both fixed and switching communication networks in Reference 22. Besides, theoretical results for containment control of autonomous vehicles with double-integrator dynamics were validated through experiments in Reference 23. The article 24 studied the distributed containment control problem for networked Lagrangian systems with multiple dynamic leaders in presence of parametric uncertainties. However, it was assumed in Reference 24 that the leaders do not communicate among themselves, which is not usual in practical applications. In addition, 24 did not provide any experimental validation results.
Based on the formation control and containment control, the notion of formation-containment is proposed in the recent years, which requires the states of leaders to develop certain formation while the states of followers converge into the convex hull spanned by the states of the leaders. One of the potential applications of formation-containment control (FCC) is "search and rescue problem" of mobile robots in hazardous and extreme environments, where the leader robots with detection devices are able to achieve a predefined formation depending on the surrounding environment and the follower robots then move into a safe region formed by the leaders. The articles [25][26][27] have laid major contribution in the area of FCC of Euler-Lagrange systems: in Reference 25, a cooperative and adaptive FCC framework has been introduced for networked Euler-Lagrange systems without using relative velocity information; in Reference 26, the development of Reference 25 has been extended to consider the effect of input saturation; while in Reference 27, the FCC problem of Euler-Lagrange systems is solved in an event-triggered framework. FCC of first-order and second-order multiagent systems are investigated in References 28 and 29, respectively. LQR-based formation-containment of high-order multiagent systems is developed in Reference 30. The output formation-containment problem of coupled heterogeneous linear systems with undirected graph and directed graph are addressed in References 31 and 32, respectively. In Reference 33, the authors have proposed distributed consensus protocols to achieve leader-follower consensus in directed graphs under external disturbances. In practice, bidirectional communication is not robust and reliable when unexpected communication failure occurs. To counteract sudden communication channel disruptions during real-time implementation, it is required to study FCC with directed interaction topology. In some of the applications, a swarm system should not only accomplish the given formation-containment task, but also track the desired trajectories provided by a virtual leader, which may have nonzero control input. A virtual leader may be considered as a pseudo agent which generates the formation reference signal for the leader agents. To the best of authors' knowledge, distributed FCC problem for linear swarm systems with a virtual leader having its own control input independent of the agents and the network topology are not yet explored much.
Motivated by the issues stated above, the distributed formation-containment problem for LTI swarm systems on directed graph is investigated in this article. First, the entire swarm system is decomposed into the leaders' layer and the followers' layer and then the distributed protocols are designed based on neighboring state information which renders the controller distributed and independent of global information about the communication topology. Subsequently, the necessary conditions (known as the "formation feasibility condition") to achieve formation-containment by a swarm system are presented which also underpin the set of feasible formations for a particular swarm system. After that, a set of distributed adaptive control protocols is designed which constitutes the FCC scheme. It is also discussed that the consensus problem, formation tracking problem and containment control problem can all be viewed as special cases of formation-containment problem solved in this article. Finally, a Matlab simulation case study is presented followed by the experimental validation results involving real robots to show the usefulness and feasibility of the proposed FCC scheme.
By contrast to the many existing FCC techniques developed so far on swarm systems, the proposed scheme of this article offers several advantages as mentioned below: • In the present approach, we have considered a separate control input for the virtual leader (ie, the formation reference) independent of the other agents and their interactions, in contrast to the existing results on this topic which consider the virtual agent to be an autonomous system only. Due to considering the virtual leader's own control input, the time-varying formation of the leaders can also track any given bounded reference signal; • Another advantage of the proposed scheme is that it does not need to check the eigenvalues of the graph Laplacian matrix unlike many existing cooperative control strategies on swarm systems. The proposed methodology uses only the relative state information of the agents and thereby avoids explicit computation of the Laplacian matrix which renders the scheme scalable for large-scale networked systems; • The dynamics of the leaders and followers are not restricted to first/second-order models, which is often considered in the literature to avoid complexities. Hence this framework allows the controller design process to be carried out on the actual system dynamics of the agents. Furthermore, the communication topology among the agents can be both directed as well as undirected which enables the proposed scheme to be flexible, reconfigurable and robust to network topology changes.
Rest of the article is organized as follows. Section 2 provides the commonly used terminologies used in Graph Theory and also discusses the problem formulation. Section 3 contains the main contribution of the article, which derives a two-layer distributed FCC protocol for networked swarm systems with time-varying formation reference. An in-depth simulation case study on achieving formation-containment by a team of networked satellites is conducted in Section 4 to show the effectiveness of the proposed methodology. Section 5 provides experimental validation results involving multiple mobile robots to test the feasibility of the proposed scheme. Finally, Section 6 concludes the article mentioning the future research directions.

Notation
Notations are standard throughout. R >0 and R ≥0 denote, respectively, the sets of all positive and all nonnegative real numbers. Let I n denote the identity matrix of dimension n × n and 1 N be the vector with N number of entries all being 1. diag{a i } represents a diagonal matrix with the diagonal entries a i ∀i. For a real-valued function ‖g(t)‖ < ∞. min (X) denotes the minimum eigenvalue of a matrix X ∈ R n×n that has only real eigenvalues. For a real, symmetric matrix A ≥ 0, A 2 denotes the standard matrix multiplication The Kronecker product of two matrices A and B is denoted by A ⊗ B. ||.|| expresses the 2-norm of a vector or a matrix. A square matrix A ∈ R n×n is called a nonsingular M-matrix if all its off-diagonal elements are nonpositive and all eigenvalues of A have positive real parts. 34

Terminologies of graph theory
Consider a weighted and directed graph  = (, , ) with a nonempty set of N nodes  ∈ {1, 2, … , N}, a set of edges  ⊂  × , and the associated adjacency matrix  = [ a ij ] ∈ R N×N . An edge rooted at node i and ended at node j is denoted by (i, j), which means information can flow from node i to node j. a ij is the weight of edge (i, j) and a ij ≠ 0 if (i, j) ∈ .
Assume that there are no repeated edges and no self-loops. Node j is called a neighbor of node i if (i, j) ∈ . Define the in-degree matrix as D = diag{d i } ∈ R N×N with d i = ∑ N j=1 a ij . The Laplacian matrix L ∈ R N×N of  is defined as L = D − . A directed graph has or contains a directed spanning tree if there exists a node, called the root, such that the root has a directed path to every other nodes. Lemma 1 (35). Consider a nonsingular M-matrix L. Then there exists a diagonal matrix G = diag{g 1 , … , g N } with g i > 0 ∀i ∈ {1, … , N} such that GL + L T G > 0

Problem formulation
Consider a swarm system of N agents, which consists of M followers and rest N − M leaders. Let F = {1, 2, … , M} and E = {M + 1, M + 2, … , N} be the sets of the followers and leaders, respectively. The dynamics of each agent is described bẏx where x i = x i (t) ∈ R n is the state of the ith agent and u i = u i (t) ∈ R m is the associated control input for all t ∈ R ≥0 . A ∈ R n×n and B ∈ R n×m are constant matrices with rank(B) = m and (A, B) being stabilizable. The desired trajectory of a formation (ie, the formation reference) is generated by a separate agent, called the virtual leader. It may be considered as the (N + 1)th leader with the dynamicṡx where x N+1 = x N+1 (t) ∈ R n and u N+1 = u N+1 (t) ∈ R m are its state and control input, respectively, for all t ∈ R ≥0 and K 1 ∈ R m×n is a constant matrix to be chosen by the designer to meet the formation feasibility condition (discussed later in Theorem 1).

Assumption 1.
The bounded control input u N+1 (t) is unknown to all leaders and followers and there exists a constant ≥ 0 such that ‖u N+1 (t)‖ ≤ ∀t ∈ R ≥0 .
Being inspired from the developments of References 30 and 31, a two-layer framework is adopted to handle the distributed formation-containment problem, which consists of the leaders' formation layer and the followers' containment layer. In the first layer, the leaders are supposed to achieve a predetermined time-varying formation and move together following a desired reference. In the second layer, all the followers are expected to converge into a convex hull spanned by the leaders. The two-layer formation-containment is said to be met if and only if the control objectives in both the layers are achieved. The communication topology among each agent is denoted by  which satisfies the following assumption.

Assumption 2.
Suppose the neighbors of a leader are only leaders or the virtual leader, and the neighbors of a follower are either leaders or followers.
Under Assumption 2, the Laplacian matrix L associated with the graph  can be partitioned as where , and L 4 ∈ R (N−M)×1 . Desired formation of the leaders is specified by h i can be time-varying as well and in that case, h i (t) ∈ R n for all t ∈ R ≥0 . Desired formation h i is prespecified to the ith leader. For simplicity of presentation, explicit dependence of h i on time t ∈ R ≥0 is omitted.
The leaders are said to achieve time-varying formation tracking with the virtual leader if for any given bounded initial states and any j ∈ E, Definition 2. The followers are said to achieve containment if for any given bounded initial states and any k ∈ F, there exist nonnegative constants kj satisfying ∑ N j=M+1 kj = 1 such that Definition 3. The swarm system (1) is said to achieve formation-containment if for any given bounded initial states, any k ∈ F and j ∈ E, there exist nonnegative constants kj satisfying ∑ N j=M+1 kj = 1 such that (4) and (5) hold simultaneously. Remark 1. According to Definitions 1 to 3, it can be argued that if M = 0, the two-layer formation-containment problem reduces to "formation tracking" problem. If M = 0 and h E ≡ 0, the formation-containment problem specializes to "consensus seeking" problem. Moreover, if h E ≡ 0 and leaders have no neighbors, then we can conclude that the formation-containment problem specializes to "containment" problem. Therefore, the consensus seeking, formation tracking and containment control can all be viewed as special cases of FCC problem.
In this article, we introduce a distributed time-varying FCC protocol given below: Above, c i (t) andĉ i (t) denote the time-varying coupling weights associated with the ith agent with c i (0) ≥ 0 andĉ i (0) ≥ 0; K 1 ∈ R m×n , K 2 ∈ R m×n and Γ ∈ R n×n are constant matrices to be chosen; and are positive constants to be selected; i and i are smooth functions of i and i , respectively; and f(.) represents a nonlinear function to be determined later.
If two-layer formation-containment is achieved, the leaders will first attain the desired formation (time-varying/stationary) and keep tracking the reference trajectory generated by the virtual leader, and after that, the followers will converge into the convex hull spanned by the leaders. This article primarily focuses on the following issues: (i) under which conditions the formation-containment can be achieved, and (ii) the steps to design a distributed (ie, distributed and adaptive) control protocol to achieve two-layer formation-containment.

FCC PROTOCOL DESIGN
In this section, the problem of designing a distributed FCC protocol for LTI swarm systems, represented by (1), is investigated.

Assumption 3.
The interaction topology among the leaders has a spanning tree with the virtual leader being the root node. Let Applying the distributed FCC protocols (7) and (6), the closed-loop dynamics of the swarm system (1) can be written in a compact form as Let the global formation tracking error vector of the leaders be T . Now can be written in the Kronecker product form as If formation tracking is achieved asymptotically by the leaders, then we have lim t→∞ (t) = 0 which implies from (11) lim From Assumption 3, it follows that L −1 3 L 4 = −1 N−M and plugging this in (12) yields Subsequently, the global containment error vector of the followers, denoted by Note that x E , x F and are all time-dependent vectors. If the containment phase is reached asymptotically by the followers, then we can write lim t→∞ (t) = 0. This implies from (14) lim Therefore, the swarm system (1) achieves the formation-containment by applying the distributed FCC protocol (7) and (6), if for any given bounded initial states, The following theorem is the main contribution of this article which establishes the conditions to be satisfied by the swarm systems (1) to achieve formation-containment applying the distributed FCC protocol proposed in (7) and (6), provided the agents and the network hold certain properties. Moreover, the virtual leader has its own control input u N+1 independent of the dynamics of the agents and the network topology. Theorem 1. Suppose, Assumptions 1 to 4 hold and for a given ≥ 0, the constants and satisfy the relation If the following formation feasibility condition is satisfied for a given choice of K 1 ∈ R m×n and h i (t) ∈ R n ∀t ∈ R ≥0 , then the two-layer formation-containment is achieved by the distributed FCC protocol given in (6) and (7) with K 2 = −R −1 B T P, Γ = PBR −1 B T P,̂i = T i P i and i = T i P i where P > 0 is the solution of the algebraic Riccati equation for given Q > 0 and R > 0. The nonlinear function f(.) used in (7) and (6) is designed as: where ∈ R n represents either i or i for all i ∈ {1, 2, … , N}.
Proof. The proof has been divided into two parts: Part I establishes the formation tracking by the leaders; while Part II deals with containment of the followers. (Part I: Formation tracking) We first derive the closed-loop dynamics of the leaders approaching formation tracking with the help of the proposed distributed and adaptive control law (6). Substituting (6) into (1) and then, substituting further the resulting expression into the derivative of (11), we obtain Achieving formation tracking by the leaders is posed as an asymptotic stability problem of the closed-loop dynamics (21) involvinġanḋc i . In order to prove asymptotic stability of (21) using Lyapunov stability approach, we consider the following Lyapunov function candidate where g i > 0 for all i, c i (t) > 0 for all t ≥ 0 and for all i since Γ > 0, i > 0 for all i since P > 0, and > 0. Therefore, V 1 > 0.
Since the sub-Laplacian matrix L 3 corresponding to the interaction topology of the leaders is a nonsingular M-matrix (exploiting the property given in Lemma 2), there exists a diagonal positive definite matrix G ∈ R (N−M)×(N−M) such that holds via Lemma 1. As the scalar parameters g i can take any positive values as decided by the designer, we may choose (23) is satisfied. The time derivative of V 1 along any trajectories of (21) is obtained as: We will first convert the summation terms present in (24) into the Kronecker product form aiming to finally expresṡ V 1 in a quadratic form with respect to . Note that and where min 0 represents the smallest positive eigenvalue of . We will now simplify each of the terms appearing in expressions (25) and (26). From (20), it is straightforward to show that Applying Cauchy-Schwarz inequality 37 on (27), we obtain We now simplify the following term containing F( ) since (27) and (28), and rest of the parameters c i , i , g i , are all positive. Subsequently, we simplify another component, as shown below, involving the virtual leader's input u N+1 Applying the simplified conditions (29) and (30) into (26) and selecting ≥ , the expression forV 1 reduces tȯ Now, the formation feasibility condition (18) can be expressed in a compact form and upon premultiplying both sides of (32) by (C + )G ⊗ P, we get Plugging (33) into (31), the expression ofV 1 further reduces tȯ Now using a common matrix property 38 we simplify the term assuming X = , ≥ and substituting (35) into (34), the expression ofV 1 becomesV Let us introduce a change of variable = ( √ (C + )G ⊗ I ) in (36) and it yieldṡ This impliesV 1 ≤ 0 using the ARE given in (19), andV 1 = 0 when = 0. Therefore, by applying the LaSalle's invariance principle, 37 the closed-loop dynamics (21) can be guaranteed to be asymptotically stable.  (6), it is now the turn of the followers to undergo the containment phase. We will start with deriving the closed-loop containment error dynamics (̇) of the followers. Inserting the containment control law (7) into (10) and invoking (11)-(14), we obtain the dynamics oḟanḋĉ i as mentioned below: Similar to Part I, achieving containment by the followers has been cast as an asymptotic stability problem of the containment error dynamics (40). Following the Lyapunov approach adopted in Part I to establish asymptotic stability, in Part II also, we consider a similar Lyapunov function candidate which consists of i > 0 for all i ∈ {1, … , M},ĉ i (t) > 0 for all t ≥ 0 and for all i since Γ > 0,̂i > 0 for all i since P > 0, and > 0. Thus, V 2 > 0. Since the sub-Laplacian matrix L 1 corresponding to the leaders' interaction graph is a nonsingular M-matrix according to Lemma 2, there exists a diagonal positive definite matrix Ξ ∈ R M×M such that holds via Lemma 1. As the scalar parameters i can take any positive values, we may propose Ξ = diag{ 1 , 2 , … , M } > 0 such that (42) is satisfied. The next step is to findV 2 along the trajectory of (40) as shown below: Similar to Part I,V 2 needs to be expressed in a quadratic form with respect to . In order to do so, we expand the summation-terms present in (43) and rearrange them in the Kronecker product form: and where min 1 represents the smallest positive eigenvalue of [ . We will now simplify the components of (45) which are not in the quadratic form with respect to . The terms involving the nonlinear function F( ) can be linked with an upper bound using (28) Since the leaders have already achieved the formation (in Part I) specified by h E , the following relation holds which is equivalent to Premultiplying both sides of (49) by (C + ) ΞL 2 L 3 ⊗ PBK 2 , we have Substituting (46), (47) and (50) into (45), we derive a simplified expression ofV 2 as noted beloẇ Now applying the property X 2 + Y 2 ≥ 2XY of positive definite matrices, 38 we obtain the following implication Inequality (51) can be simplified aṡ by plugging (52) and selecting ≥ plugging it into (53), we haveV which impliesV 2 ≤ 0 using the ARE given in (19) andV 2 = 0 only when Υ = 0. Now, invoking LaSalle's invariance principle, 37 it can ensured that the containment error dynamics (40) is asymptotically stable which ultimately implies lim t→∞ (t) = 0.
The above analysis proves that the containment error asymptotically goes to zero which in turn ensures that the followers will asymptotically converge into the convex hull spanned by the leaders. Combining Parts I and II, it can be concluded that the swarm system (1) satisfying Assumptions 1 to 4 accomplishes two-layer time-varying formation-containment tracking by applying the proposed distributed FCC protocol (6) and (7). This completes the proof. ▪ While analyzing the two-layer formation-containment problem and deriving the FCC protocol in Theorem 1, we observe a number of significant aspects which we will describe in the following remarks.
Remark 2. The formation feasibility condition (18) imposed in Theorem 1 is a common restriction in the consensus-based time-varying formation control literature. 2,30,39 This criteria needs to be checked a priori to confirm whether a desired formation shape, specified by h E , is possible to be achieved by the leaders. In this regard, the gain matrix K 1 plays a major role in selecting the formation reference h E as K 1 is the only parameter in the formation feasibility criteria (18) that can be freely adjusted. If it fails then the formation shape needs to be changed. In case of static formation, it is easier to find K 1 for given (A, B) of the leaders, but it may not be straightforward in case of a time-varying formation to find an appropriate K 1 matrix to satisfy (18). In particular, it depends on the experience of a designer. However, in many practical applications, for example, in robotic applications employing nonholonomic mobile robots, the nonlinear dynamics of a robot can be feedback linearized into single or double integrator dynamics and then, it becomes much easier to find an appropriate K 1 matrix that can satisfy the formation feasibility criteria (18) for a particular choice of h E . In this article, we have considered two case studies to address the issue of finding K 1 for a given leader's dynamics and a desired formation shape.
Remark 3. The nonlinear function f(⋅), defined in (20), is included in the proposed FCC protocols (6) and (7) to handle the time-varying dynamics of the virtual leader. By contrast to most of the existing results in the formation-containment literature, the present work does not presume the virtual leader to be an autonomous system; it may have a separate control (either reference or disturbance) input (u N+1 ). But, the nonlinear function f(⋅) makes the FCC action discontinuous, which may produce "chattering effect" due to frequent "zero-crossing" and causes severe performance degradation. In order to avoid the chattering phenomenon in practical applications, the boundary layer technique proposed in Reference 9 can be exploited which redefines the nonlinear function f(⋅) to make the control action continuous by removing the zero-crossing criteria. Although in this method, the formation tracking or containment errors cannot be made exactly zero but can be made infinitesimally small, which is sufficient to achieve satisfactory performance and most importantly, this technique minimizes the chattering effect. fix the desired formation reference h E for leaders; 4: choose an appropriate feedback gain K 1 to widen the set of feasible formation; 5: if formation feasibility condition (18) is satisfied then 6: compute controller parameters K 2 = −R −1 B T P and Γ = PBR −1 B T P by solving ARE (19) for given Q < 0 and R > 0; 7: design the smooth functionŝi and i as described in Theorem 1; 8: choose positive constants and according to (17) for a given > 0; 9: construct the nonlinear function f (.) according to (20); 10: construct the distributed formation-containment control (FCC) protocol given in (6) and (7); 11: else 12: back to Step 4; 13: end if 14: end for In the proposed FCC scheme as derived in Theorem 1, the states of the followers converge into the convex hull spanned by the states of the leaders. However, it doesn't explicitly mention that whether the positions of the followers in a convex hull can also be controlled. In this context, the following theorem expands Theorem 1 to show an explicit relationship among the states of followers x i (t), i ∈ F, the time-varying formation of the leaders h i (t), i ∈ E, and the trajectory of the virtual leader x N+1 (t). Theorem 2. Suppose the assumptions of Theorem 1 hold and the swarm system (1) achieves formation-containment by applying the distributed FCC protocol (6) and (7). The states of followers satisfy the following relationship where l ij denotes the entries of −L −1 1 L 2 .
Proof. If the assumptions of Theorem 1 are satisfied, the formation-containment of swarm system given in (1) is accomplished by applying the adaptive control laws (7) and (6). From the proof of Theorem 1, it can be readily observed that if (39) and (55) hold simultaneously, then we have from (13) and (15). Subsequently, it follows from Lemma 2 that Substituting (58) into (57), we obtain which is equivalent to (56) ∀i ∈ F. This completes the proof. ▪ Theorem 2 reveals that the states of the followers are jointly determined by the communication graph, the time-varying formation of the leaders, and the formation reference. According to (56), it can be shown that the states of the followers will also give rise to a time-varying formation determined by the convex combination of the formation h E of the leaders with respect to the formation reference x N+1 . Therefore, the states of the followers not only converge into the convex hull spanned by the leaders, but also maintain a time-varying formation inside the convex hull.

SIMULATION CASE STUDY WITH NETWORKED ARTIFICIAL SATELLITES
In this section, a simulation case study is presented to illustrate the effectiveness of the FCC scheme developed in the previous section. Consider a network of twelve satellites and the associated communication topology is shown in Figure 1 where it is assumed that the weights of the links vary in the range [0, 1]. Figure 1 portrays that the first layer consists of six satellites considered as leaders (7)(8)(9)(10)(11)(12) while the second layer contains rest six satellites which act as followers (1 to 6). The virtual target (labeled with 13) generates the tracking reference to be tracked by the leaders' formation. The dynamics of each satellite can be described by (1)  In this case study, the virtual leader is governed by (2) with the control input u N+1 = 0. Note that u N+1 = 0 has been made zero only for simplicity of the pictorial representation of the simulation results. The control objective of this theoretical case study is to let the assembly of six leader satellites maintain a circular time-varying formation having hexagonal shape and keep rotating around a specific target or reference generated by the virtual leader. Then the positions of the six followers are required to converge into the convex hull spanned by the leaders. The time-varying circular formation The Second Layer (followers) 8 9 10 11 12 The First Layer (leaders) Virtual leader (target) F I G U R E 1 A two-layer directed communication topology among twelve satellites is shown of which six are considered as leaders (labeled with 7-12) while the rest six are the followers (labeled with 1-6) and the virtual leader (or the target) is labeled with 13 [Colour figure can be viewed at wileyonlinelibrary.com] of the leaders is specified by is chosen such that the formation tracking feasibility condition (19) in Theorem 1 is satisfied with this desired h i (t) ∀i ∈ {7, 8, … , 12}. Thus, if the predefined time-varying formation h i (t) is achieved, the six leaders will be placed at the six vertices of a parallel hexagon and keep rotating around the reference signal with an angular velocity of 1 rad/s. The followers will also converge into the parallel hexagon formed by the leaders.  Figure 3 shows the desired formation reference trajectory. The figures suggest that the six leader satellites have attained a hexagonal formation (Green colored) in the 3D plane which keeps rotating around the desired target (marked by the Red star in Figure 2A-D) and the follower satellites eventually converge into this hexagonal region formed by the leaders while maintaining the same angular velocity as that of the leaders. Time variation of the coupling weightŝ c i (t) of the followers and c i (t) of the leaders are plotted in Figure 4 which shows that the coupling weights converge to finite steady values within finite time. 2-norms of the formation tracking error of the leaders ( ) and the containment error of the followers ( ) are depicted in Figure 5A,B. It is now concluded that the desired time-varying formation-containment is achieved under the influence of the distributed FCC protocol as given in (6) and (7).

EXPERIMENTAL VALIDATION RESULTS WITH NONHOLONOMIC MOBILE ROBOTS
In the hardware experiment, we used small-scale, autonomous mobile robots, Mona, 40 to test the feasibility of the proposed FCC scheme. Mona is a low-cost, open-source, miniature robot which has been developed for swarm robotic applications. As shown in Figure 6, Mona has two 6V low power DC geared-motors connected directly to its wheels having 32mm diameter and the robot has a circular chassis having 80mm diameter. The main processor embedded in this robot is an AVR microcontroller equipped with an external clock having 16 MHz frequency. Each robot is equipped with NRF24101 wireless transceiver module for interrobot communication. The robots are operated and controlled by The experimental platform includes a 1.2m × 1.2m blue arena and a digital camera connected to a host PC which operates the entire tracking system. The position tracking system used in this experiment is an open-source multirobot localization platform. 41 The tracking system is not only capable of following a robot's position but also capable of measuring its velocity and detecting its orientations by identifying the unique circular tags attached on top of the robots. The state information is transmitted to the computer via the ROS communication framework and then the relative state information is sent to the corresponding robot using RF transceiver module. In the experiment, six nonholonomic mobile robots (Mona) were considered to perform a formation-containment activity. Each mobile robot can be described by the following set of dynamic equations in terms of the global coordinates where x xi and x yi together represent the position of the ith robot in the X − Y plane in terms of the Cartesian coordinate, i is the angle between the heading direction and the X axis, v i denotes the linear velocity and i represents the angular velocity. For each mobile robot, we introduce a new control input q i such thaṫ v i = q i ∀i ∈ {1, … , 6}. Define v xi = v i cos i and v yi = v i sin i which represent, respectively, the X-axis and Y -axis components of the linear velocity v i . Therefore,̇x xi = v xi anḋx yi = v yi . Subsequently, by taking the time-derivative of v xi and v yi are computed as

Now a feedback linearizing control law is designed below by assuming
such that the nonlinear dynamics of the robots can be linearized into the forṁx i = Ax i + Bu i , as given in (1) where u yi ] T represent the state vector and the control input vector of the feedback linearized system, respectively. In this experimental demonstration, the four leader robots will form a square-shaped region (ie, the convex hull) in the experimental arena and the two follower robots is expected to converge into this safe region. Figure 7 depicts the directed interaction topology among the six mobile robots. The state of the virtual leader (labeled with no. 7) is arbitrarily chosen as [0.7, 0, 0.7, 0] T . The control objective of this experiment is to let the four leader robots to achieve a planar square formation, tracking the virtual target and keep surrounding the followers. The time-varying circular formation for the leader robots is specified by where r = 0.3 m and = 0.1 rad/s denote, respectively, the radius and angular velocity of the desired time-varying formation. It can be verified that the formation tracking feasibility condition (18) of Theorem 1 is satisfied in the present scenario and hence, when the predefined time-varying formation h i (t) is achieved, the leaders will automatically be placed at the four vertices of a square. After attaining the desired formation, the assembly of the leader robots keeps on rotating around the tracking reference with a constant angular velocity of 0.1 rad/s and two follower robots will eventually converge into this area.  Figure 8A shows the initial orientation (at t = 0 s) of the robots on the arena. Figure 8B depicts the progress of the mission at t = 20 s from which it is apparent that the four leader robots are moving in the right track to attain the expected square formation and subsequently, Figure 8C reveals that the leader robots have truly achieved the desired square formation (at t = 40 s) and the two follower robots have also converged into the area spanned by the leaders. Finally, Figure 8D portrays that the assembly of the leaders has moved toward the center of the arena in order to track the position of the given target without disturbing the formation and the followers have also moved accordingly being surrounded by the leaders. This then concludes that the formation-containment mission has been achieved successfully under the application of the proposed distributed FCC scheme. Figure 9 complements the hardware experiment result presented in Figure 8A-D by plotting the position trajectories of all six robots in the XY plane during the course of achieving the formation tracking and containment in the hardware experiment. Figure 9 traces the movement of the robots staring from the initial positions until they converge at the desired locations to attain the leaders' formation around the given target (marked by the Red star) and the followers' containment within the area spanned by the positions of the leader robots. Finally, time evolvement of the 2-norms of the formation tracking error i (t) of four leaders and the containment error i (t) of two followers are shown in Figure 10A,B. The figures reveal that both the formation tracking and containment errors die down to zero within 80 seconds which is in agreement with the fact that the experiment also took 80 seconds to accomplish the mission as reflected in Figure 8D. It can hence be ascertained that both formation tracking and containment have been achieved by the team of six mobile robots under the influence of the proposed distributed FCC scheme. Even though we have used lab-based experimental set-up, small-scale robots and open source software modules, but the experimental results reflect usefulness of the proposed scheme in finding potential applications in robotics. High precision performance is also possible to be achieved by the proposed scheme with improvised camera tracking system, high-end embedded microcontrollers and advanced robots.

CONCLUSIONS
In this article, a formation-containment problem for linear swarm systems connected via directed graphs is investigated. A two-layer distributed FCC scheme is proposed for LTI swarm systems that can be decomposed into leaders' formation layer and the followers' containment layer. The proposed FCC protocols are adaptive and constructed based on relative state information of the agents which enables the scheme to work without using global information about the interaction topology. Theoretical basis of the proposed scheme has been established using the Lyapunov stability framework and the control parameters are chosen via solving an algebraic Riccati equation. It is also proved that the states of followers not only converge into the convex hull spanned by the leaders but also maintain certain formation determined by the leaders and the virtual agent. An algorithm is also written for the control practitioners to identify the design steps to implement the proposed scheme in practice. A simulation case study has been conducted on achieving the formation-containment tasks of networked satellites to highlight the usefulness of the proposed FCC scheme and finally, experimental validation results involving networked mobile robots are also included to show the feasibility of the proposed scheme in real-time implementation. In future scope, the present FCC scheme may be extended to consider the effects of time delays and nonlinear dynamics of the robotic systems. Furthermore, robust distributed controller design techniques can also be explored by utilizing the ideas of References 42-45.