—This paper investigates fully distributed leaderless formation maneuver control over a directed sensing topology. For formation maneuvering, generally, a leader-follower approach is adopted. Leaderless formation maneuvers have no designated leaders, unlike traditional approaches. Some of the leaderless approaches proposed recently in literature considered undirected sensing graphs, which means the agents need to sense in both directions. Also, in these approaches, although the control law implementation is distributed in nature, the design of the control laws requires a stabilizing diagonal matrix design, which makes the design of the control law centralized and knowledge of exact interaction among the agents is required for this. When an agent’s sensing capabilities are limited, directed sensing topology becomes crucial. First, we propose control laws for leaderless formation maneuvers in 2-D by manipulating the weights of complex-laplacian for single integrators with directed sensing graphs, and GAS (Global Asymptotic Stability) is established for it. In this approach, only translational, scale, and rotational formation maneuvers are possible. Then, we propose control laws for leaderless affine formation maneuvering for single integrators with directed sensing graphs for 2-D and 3-D, where translational, rotational, scale, and sheer formation maneuvering is possible. Also, an extension for the higher-order integrators is provided using a back-stepping-based design. Constant dis- turbances are also considered in the system dynamics. Finally, simulations are provided to validate the results. Index terms: Formation control, affine formation, leader- less maneuvers, cooperative control I. I NTRODUCTION Controlling robotic swarms has emerged as a prominent area of research within the control and robotics community [1]. Multi-agent formations have a wide range of applications. These include satellite formation flying, search and rescue operations, source seeking, underwater ocean data retrieval, and estimating the gradient of a field, etc. [2], [3], [4], [5]. Achieving and maintaining a specific geometric pattern using distributed control laws, and performing maneuvers maintaining the desired formation shape are two important aspects of formation control. Traditional formation control techniques rely on position measurements of agents, but dis- tributed formation control aims to achieve formation control using onboard sensors of the agents and relative measurements of their neighbors. This approach allows for the establishment of formations even in environments where GPS signals are unavailable [6], [7], [8]. Different types of distributed forma- tion control approaches are classified as displacement-based, distance-based, and bearing-based depending on whether the desired formation shape is specified in terms of displace- ment, distance, or bearing constraints [7], [9], [10]. With C. Garanayak ( chinmayg22@gmail.com ) is a Ph. D. student, and D. Mukherjee ( dm@ee.iitb.ac.in ) is an Asst. Professor with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai- 400076, India. The work of DM was supported in part by an ISRO funded project bearing code RD/0120-ISROC00-007.displacement-based control, only translational maneuvers were possible because the displacement constraints exhibit transla- tion invariance [11], [12]. Distance-based formation control enables the execution of both translational and rotational maneuvers [13]–[15], while a bearing-based approach allows for translation and scaling formation maneuvers [9]. [16]–[18] proposed complex laplacian-based formation control where the constraints specifying the formation are translational, rotational, and scale-invariant, which enabled translational, rotational, and scale formation maneuvers. However, this ap- proach is limited to 2-D. Affine formation control is another noteworthy approach to formation control [19]–[23], where the primary goal is to acquire a geometric shape that is an affine transformation of the intended reference shape. In this methodology, the interactions between agents are encapsulated through the utilization of a stress matrix, commonly referred to as a weighted Laplacian or signed Laplacian. Specifi- cally, every agent within the system endeavors to ensure that the weighted aggregate of their neighboring agents’ relative positions remains equal to zero. In [19] the necessary and sufficient conditions for affine formation stabilization in Rd were established. It is possible to achieve translational, scal- ing, shear, and rotational maneuvers through the control of leaders within the framework of the affine formation control paradigm, as all of these alterations can be regarded as affine transformations [20]–[23]. [20] focused on investigating affine maneuvering strategies for the double integrator and unicycle agents operating within an undirected interaction topology. [21] considered higher-order integrator agents with a directed interaction topology. In [22] the authors put forth a layered affine formation control strategy, wherein various layers de- lineate the transmission of information among the leaders and the followers. [23], also proposed a layered affine formation strategy for higher order integrator systems, which did not require any acceleration information (control input for double integrator systems) which were necessitated in [20]. However, to accomplish formation maneuvers, past method- ologies have entailed the independent control of a small number of leaders, with the followers employing distributed control laws to fulfill the objectives of formation control. This particular approach necessitated an additional control layer and required the leaders to access their position measurements via GPS for autonomous navigation or to be remotely controlled by human operators. Consequently, it is imperative to explore strategies that facilitate formation maneuvering without the presence of a leader, thereby eliminating the requirement for GPS control or position measurements. [24] proposed a method for performing leaderless maneuvers with rigid forma- tions, based on distance measurements. The design parameter used was a mismatch in distance measurements and ensured a constant translational velocity of the formation and rotation about its centroid. In [25], the positions of the agents were represented using complex numbers from the set C, and by perturbing the weights of the complex Laplacian maneuvering forces were generated to achieve translational, scaling, and rotational leaderless maneuvers for planar formations. In [26], the weights of the stress matrix were perturbed to generate affine formation maneuvers in Rd,d= 2,3. However, the control algorithms proposed in [24], [26], and [25] were for single integrators, and an undirected sensing graph was consid- ered. A directed sensing graph among agents is of paramount importance in order to accommodate the constraints imposed by an agent’s sensing capabilities. Also, although the control laws proposed in [25], [26] can be implemented in a distributed manner, the design of the laws requires a stabilizing diagonal matrix whose computation requires information on the overall interaction graph topology, hence making the design of the control laws centralized. Motivated by the aforementioned factors, we put forth a control strategy aimed at achieving leaderless formation maneuvers completely distributed way without the need for any stabilizing diagonal matrix for its implementation. Also, this strategy takes into account a directed sensing topology between agents, a topic that has yet to be thoroughly in- vestigated in the existing literature on leaderless formation maneuvers. First, we propose control laws for translational, scale, and rotational formation maneuvering by manipulating the weights of a complex Laplacian for planar, i.e., 2-D, agents taking into account a directed interaction topology among the agents. Then control laws for affine formation maneuvering are proposed for Rd, d= 2,3, agents with directed sensing graphs. Unlike [25], [26], we do not need any stabilizing diagonal matrix in our laws, making the design and implementation both decentralized. Also, our analysis takes into account the existence of persistent constant disturbances in the dynamics of agents, in contrast to the setups examined in previous works such as [24]–[26]. Such disturbances can arise from uncertainties in modeling or the presence of external disturbances within the environment. Furthermore, we expand upon the findings obtained for single integrator dynamics to encompass the inclusion of agents represented as higher-order integrators. Although [20] did consider double integrators and unicycles, and [22] addressed higher-order systems, the topology examined in both studies was undirected. In addition, [20] required d+ 1 leaders for formations in Rd, while [22] required at least one leader to possess absolute position information for maneuvering. In contrast, our approach is leaderless. [21] did consider a directed graph topology with higher-order integrator systems, but it also necessitated d+ 1 leaders in Rd. In contrast, our approach considers leaderless digraphs. In [20]–[22] a stabilizing diagonal matrix is needed which will require centralized information while our algorithm can be implemented without such need. Remark 1. Although a preliminary version of this work is published in [27] which considered directed graphs, the control laws in [27] also required a stabilizing diagonal matrix, and hence the design of the laws required centralized computation. In contrast, the laws proposed in this article (a) Reference Shape (b) Affine motions in 2-D Fig. 1. (a) Example of a reference shape in 2-D. p∗= (16⊗pc.m.) + [p∗ c1Tp∗ c2T. . . p∗ c6T]T. (b) Different affine motions are possible in 2-D. Brown arrows represent 2 translations, green arrows represent scale change, yellow arrows are for rotation and red arrows represent 2 types of shear motion. have both design and implementation distributed in nature. Also, we have included the complex laplacian-based leaderless formation maneuvering, which was not there in [27]. In addition, for the design of higher order systems to avoid computational and noise amplification issues, we have adopted a command filter-based back-stepping design here, whereas in [27], a direct derivative of the virtual control laws was considered. Notations: Iis used to represent the identity matrix. 1n represents a vector composed entirely of ones. ||.||is employed to indicate the induced 2-norm for a matrix and the 2-norm for a vector. The symbol |.|is used to denote the absolute value of a real or complex number. The spaces L2andL∞refer to the space of square-integrable functions and the space of bounded functions, respectively. AH, A∈Cn×n,represents the transpose of complex conjugate of matrix A. Ker{A} and Im {A}represents the null space and image of matrix A, respectively. The symbol ιrepresents the unit complex number. Nis the set of natural numbers. II. P RELIMINARIES AND PROBLEM FORMULATION A. Graph Theory The interaction between the agents is represented through a graph G(V,E), where the node set is denoted as V= {1,2, . . . , n }, and the edge set is denoted as E ⊆ V×V . In the case of a directed graph, the edge set consists of directed edges (j, i), where node jis the tail node and node iis the head node. Node jis referred to as the in-neighbor of node i, while node iis referred to as the out-neighbor of node j. The in-neighbor set for agent iis denoted as Ni:={j|(j, i)∈ E} . It should be noted that in a directed graph, the presence of (j, i)∈ E does not necessarily imply the presence of (i, j)∈ E. For a directed graph, the definitions of κ-reachable ,κ-rooted , and κ-spanning tree are taken from [19]. Definition 1.1: (κ-reachable ) A node vis called κ-reachable from a non-singleton set of nodes, U, if there exist κdisjoint paths from a node in Uto node v. Alternately stated, the above definition implies that we can find a path from a node in Uto node veven after removing anyκ−1nodes from the graph, except node v. Definition 1.2: (κ-rooted ) A digraph is called κ-rooted if there exists a subset of nodes, called root nodes, from which all other nodes are κreachable. Definition 1.3: (κ-spanning tree ) For a digraph G(V,E)with a setU={u1, u2, . . . , u k} ⊂ V , a spanning κ-tree is a subgraph T(V,E0), such that (1) Every node u∈ U has no in-neighbours, (2) Every node v̸∈ U hasκin-neighbours, (3) Every node v̸∈ U isκ-reachable from U. We now state an important existing result. Lemma 1. [19, Lemma 2.1] A digraph G(V,E)isκ-rooted if and only if it has a κ-spanning tree. B. Preliminaries for Complex Laplacian-based control We have nplanar agents for complex Laplacian-based formation control, i.e., agents are in 2-D. 1) Complex Laplacian matrix: A Complex Laplacian ma- trixL ∈Cn×nis a matrix whose entries are complex numbers. Letwij∈C̸= 0,be the weight associated with the edge (i, j)∈ E,then we have the complex laplacian Lwhose entries are given by L(i, j) =−wijfor(i, j)∈ E,otherwise L(i, j) = 0 for(i, j)̸∈ E,andL(i, i) =P k∈Niwik. Also, note that L1n= 0. 2) Desired shape: The2-D position of every agent i∈ V is a complex number pi∈C, where the real and imaginary parts describe the two coordinates of the Euclidean plane. All the positions piof the agents are stacked in a single vector p∈Cn, which is called a configuration . The pair (G, p), is called a framework F, where the position of each agent piis attached with the node i∈ V. The reference shape p∗is an arbitrary configuration of interest, which is given by p∗=pc.m.1n+p∗ c, (1) where pc.m.∈Cis the position of the center of mass of the configuration, and p∗ cgives the shape/ appearance to the formation starting from pc.m.. For simplicity and without loss of generality, we have pc.m.= 0, and hence p∗=p∗ c. Now, we define the desired formation shape in the case of Complex Laplacian-based formation control. Definition 1. (Desired formation shape) The formation or the framework is at the desired shape when we have p∈ M :={p=k11n+k2p∗|k1, k2∈C}. (2) Here, k1accounts for the translation, and k2accounts for the scaling and rotation of the reference shape p∗. However, in our case, we are not interested in a static formation shape as in [16]. Rather, we are interested in formation maneuvering, which refers to a trajectory in M. Remark 2. For the static complex Laplacian-based formation control reported in [16], the weights wijof the complex laplacian must satisfy Lp∗= 0,making p∗to be in the null space of L.Note that this condition is satisfied if and only if the graph is 2-rooted [16]. As 1nandp∗are in the null space of L, we have Lhas two zero eigenvalues and the corresponding eigenvectors are given by 1nandp∗. (a) 3-rooted graph (b) 3-rooted directed regu- lar graph Fig. 2. (a) Example of a 3-rooted graph. All nodes are 3-reachable from root set:R={r1, r2, r3}. (b) Example of a directed graph which is 3-rooted. Any three nodes can be considered as root nodes, and every node has 3 in- neighbours C. Preliminaries for Affine formation control 1) Signed-laplacian: A signed-Laplacian matrix (or stress matrix) associated with graph Gis denoted by L ∈Rn×n whose entries can be positive and negative real numbers. If wij̸= 0is the weight associated with an edge (j, i), the(i, j)- th entry of the signed-Laplacian Lis given by L(i, j) =−wij ifj∈ N i, otherwise it is 0.The diagonal entries are given byL(i, i) =P k∈Niwik.For a digraph, Lis usually not a symmetric matrix, but it still satisfies the condition L1n= 0, like its undirected counterpart. 2) Desired affine trajectory: Let us consider a set of n agents in the Euclidean space Rd, where dcan be either 2 or 3. The position of each agent i∈ V is denoted as pi∈Rd. We can collect all the positions piin a vector p∈Rnd, which is referred to as the configuration. A framework Fis defined as the pair (G, p), where Grepresents the underlying control/sensing topology among the agents. A shape of interest or a reference shape p∗∈Rnd(as depicted in Figure 1(a)) can be arbitrarily given: p∗= (1n⊗pc.m.) +p∗ c where the center of mass is denoted as pc.m.∈Rd, and the shape of the formation is denoted as p∗ c∈Rnd, with respect topc.m.. The desired shape is constructed based on the reference shape p∗, as described by the following definition: Definition 2. (Desired affine trajectory) The desired shape is achieved when p∈ S :={p= (In⊗A)p∗+ (1n⊗ b)|A∈Rd×d, b∈Rd},i.e., the desired shape is an affine transformation of the reference shape p∗,hence it is called affine formation control. Affine maneuver indicates the desired shape is not a static one as in [19]; rather, it can be a trajectory in S. Definition 3. (Generic Configuration) A configuration, p, is generic if there does not exist a non-zero polyno- mial g(y1, . . . , x nd), having integer coefficients, such that g(p1 1, . . . , pd 1, . . . , p1 n, . . . , pd n) = 0 where pl kis the l-th element ofpk[19]. Physically interpreted, generic configurations do not have any degeneracy, i.e., points on a line in R2and planar formations in R3are examples of non-generic configurations. Remark 3. For static affine formation control as reported in [19], the weights wijof the signed Laplacian, L, are designed in a way so that (L⊗Id)p∗= 0, thereby making the reference shape p∗an eigenvector of L ⊗Idcorresponding to the zero eigenvalue. When p∗is an eigenvector, the desired affine shape p∈ S becomes an eigenvector of L⊗Idwith zero eigenvalue, implying that Sis the null space/kernel of L ⊗Id[19]. D. Problem formulation 1) Complex Laplacian-based formation maneuvering: Sin- gle integrator kinematics of the agents is given by: ˙pi=ui+δi, i∈ V, (3) where δi∈Cis the constant disturbance input to agent i, and ui∈Cis its control input. Assumption 1: (a) The desired configuration ( p) is generic. (b) The directed sensing graph (G(V,E))among the agents is 2-rooted with each root node having at least 2in-neighbors. Problem 1: With agents whose dynamics are represented by (3), sensing over a directed graph, and subject to Assumption 1, the objective is to design a control input uifor agent i∈ V such that a desired trajectory can be achieved asymptotically. In other words, we aim for p(t)and˙p(t)to converge to M astapproaches infinity. 2) Affine formation maneuvering: Single integrator kine- matics of the agents is given by: ˙pi=ui+δi, i∈ V, (4) where δi∈Rdis the constant disturbance input to agent i, andui∈Rdis its control input, d= 2 or3is the dimension of ambient space. Dynamics of ρ-th order integrator is given by: ˙yi,l=yi,l+1+δi,l, l≤ρ−1 ˙yi,ρ=ui+δi,ρ, i∈ V (5) where ui∈Rdis the control input for i-th agent, and δi,l∈Rd are the disturbances terms. We are now well placed to state the main problem addressed in this paper, along with the key assumptions. Assumption 2: (a) The desired configuration ( p) is generic. (b) The directed sensing graph (G(V,E))among the agents isd+ 1-rooted with each root node having at least d+ 1in- neighbours where dis the dimension of the ambient space. (c) The number of agents, n, satisfies n≥d+ 2. Remark 4. For static affine formation control over a digraph, the necessary and sufficient condition was that the digraph should be d+ 1-rooted [19]. It is important to note that for ad+ 1-rooted graph, the root nodes need not have any in- neighbours. The extra condition of d+ 1 in-neighbors of the root nodes is imposed to solve the problem of affine maneu- vering over a directed topology. Part (c) of the assumption is used so that part (b) can be satisfied. Remark 5. Note that the additional condition does not change thed+1- rooted structure of the graph because from Definition1.3 and Lemma 1, it is clear that if graph G(V,E)isd+ 1- rooted, it contains a d+ 1-spanning tree T(V,E0)and after addition of extra edges to the root nodes, the new graph G(V,¯E)will also contain the same d+ 1-spanning tree T(V,E0), thereby ensuring that it is d+ 1-rooted. Remark 6. The assumption used here is weaker than that used in [26] for solving the undirected case, i.e., the assumption of a universally rigid interaction topology. Universally rigid configurations ensure d+ 1 connectedness and hence are always d+ 1-rooted [19, Remark 2.1], while the converse is not always true. Problem 2: With agents whose dynamics are represented by (4) or (5), sensing over a directed graph and subject to Assumption 2, the objective is to design control input uifor agent ito achieve an asymptotic convergence of the desired affine trajectory, i.e., p(t)→ S and˙p(t)→ S ast→ ∞ . III. M AIN RESULTS In this section, we first provide the key idea used that makes the leaderless formation maneuvers for directed graphs tractable. Then, we propose the control laws to solve Problem 1 and Problem 2 and analyze the stability. A. Key idea used for the directed leaderless maneuvers Here, we first provide an intuitive idea of why the control laws presented in [25], [26] for agents sensing over undirected graphs will not work for the directed sensing graph. In [25], the formation maneuvers are generated by perturbing the edge weights for complex Laplacian. Similarly, in [26], the signed Laplacian/stress matrix weights were perturbed in order to induce affine motions. The assumption of only a d-rooted graph, which is the necessary and sufficient condition for static formation control based on complex-laplacian [16], [18], and the assumption of only a d+ 1-rooted graph, which is the necessary and sufficient condition for static affine formation control [19] is not sufficient for generating formation ma- neuvers adopting a similar approach of perturbing the edge weights as in [25], [26]. This is because the alteration of weights leads to changes in the forces acting on an agent, which are in equilibrium, thereby causing it to move in a particular direction [25], [26]. However, in a directed graph, sensing is unidirectional, resulting in some nodes not having any in-neighbors. Unlike the case for a d-rooted undirected graph considered in [25], for a directed d-rooted graph, the root nodes do not have any in-neighbors, so they will not experience any force by modification of the edge weights in case of complex-laplacian. Hence, we need to modify the assumption, and we have added another condition to thed-rooted graph, i.e., each root node will also need to have at least din neighbors. Similarly, for affine formation maneuvering, a d+ 1-rooted graph where the root node must have at least d+ 1 in-neighbors assumption is used. This condition entails the addition of extra edges to the sensing graph, thus enabling a solution to the problem of directed sensing topology. In Fig.2(a), the root nodes do not possess any in-neighbors. Consequently, perturbing edge weights will not impact the force exerted on these root nodes, leading to a lack of movement. Directed 3-regular graphs, exemplified in Fig.2(b), inherently satisfy these requirements as each agent can access measurements from three neighbors. The topology in Fig.2(a) does not facilitate affine maneuvers, whereas the one in Fig.2(b) does. Importantly, the stipulated condition does not restrict the graph to a regular digraph like the one depicted in Fig.2(b); any arbitrary d+ 1-rooted digraph meeting the additional conditions specified by Assumption 1 will suffice. B. Leaderless formation maneuvers by manipulating complex laplacian In this section, we solve Problem 1, i.e., we discuss how to generate only translational, rotational, and scale leaderless formation maneuvers by manipulating weights of complex Laplacian for directed graphs. Unlike the case in [25], where such motions can be generated by manipulating weights of complex Laplacian for undirected graphs, directed graphs are considered here. Also, we do not need to know global interaction topology among the agents as our proposed law does not use the stabilizing diagonal matrix Kas used in [25]. 1) Control strategy for single-integrators: The following control law is proposed for complex laplacian-based formation maneuvering ( ∀i∈ V, j∈ Ni) ui=−X j∈Ni¯wijζi+X i∈Nj¯wjiζj−1 hX j∈Niˆwij(pj−pi)−ˆδi, ˙ζi=−aζi+X j∈Niwij(pj−pi) (6a) ˙ˆδi=X j∈Ni¯wijζi−X i∈Nj¯wjiζj+1 hX j∈Niˆwij(pj−pi) +vi, (6b) where ¯wijis the complex conjugate of wij.It is important to observe that here we have ζi, pi∈C,∀i∈ V,andζi∈C is an auxiliary state variable. Also, the information wjiζjhas to be transmitted from the out-neighbors of agent i, which are sensing the agent i. The velocity information, i.e., vi,can be obtained by integrating acceleration measurements obtained by accelerometer sensors. The complex weights wij,(i, j)∈ E are calculated from the following equation [16],which was required for static formation control, X j∈Niwij(p∗ i−p∗ j) = 0 . (7) For calculating ˆwijthe following equation is used X j∈Niˆwij(p∗ i−p∗ j) =v∗ i, (8) where v∗ iis the desired velocity of the i-th agent. The perturbation weight ˆwijis responsible for generating maneu- vering force. Note that as we have at least 2neighbors for each agent i,the solution of (8) always exists. The desired velocity ( v∗) can be split into translation, rotation, and scale components, i.e., v∗=v∗ t1n+γp∗+iwp∗, where v∗ t∈Cis the common translational velocity, γ∈R>0keeps track of whether the formation grows or contracts, i.e., it controls the scale, and w∈Rcontrols the angular speed and hence controls the rotational motion of the formation. All agents’ desired reference velocities are stacked together to obtain v∗:= [v∗ 1. . . v∗ n]T. Now, we define a new state variable x:= [pTζT]T∈C2n. In the compact form (6) can be written as: ˙x=˙p ˙ζ = 0 −LH L − aIn |{z} Ap ζ −1 hˆL 0 0 0 |{z} ˆAp ζ +˜δ 0 |{z} ˜∆, =⇒˙x=Ax−1 hˆAx+˜∆, (9a) ˙ˆδ=LHζ+1 hˆLp+v, (9b) where ˆδ= [ˆδT 1ˆδT 2. . .ˆδT n]T, δ= [δT 1δT 2. . . δT n]T,˜δ= δ−ˆδ, u= [uT 1uT 2. . . uT n]T, v= [vT 1vT 2. . . vT n]T, and A:= 0−LH L − aIn and ˆA:=ˆL0 0 0 ,and˜∆ :=˜δ 0 .Here, LHdenotes the transpose of complex conjugate of the matrix L. We may rewrite ˆL=ˆLt+ˆLr+ˆLs, where ˆLt,ˆLr, and ˆLs denotes the translational, rotational, and scale component of ˆL. Lemma 2. The eigenvalues of Aare given by λ= −a±√ a2−4σ 2, where σis an eigenvalue of LHL,andA has two zero eigenvalues whose corresponding eigenvectors are[ΩT0T]T∈C2nwhere Ω∈Cnis the eigenvector corresponding to the zero eigenvalue of LHL. Proof. Let[ΩT¯ΩT]T∈C2nbe an eigenvector of A,and λbe the corresponding eigenvalue. Hence, A[ΩT¯ΩT]T=  0−LH L − aInΩ ¯Ω =λΩ ¯Ω . Hence, −LH¯Ω = λΩ,and LΩ−a¯Ω = λ¯Ω,implies −LHLΩ =−λ(λ+a)Ω,implies Ω∈Cnis an eigenvector of LHLand the corresponding eigenvalue is σ:=−λ(λ+a).Hence, λ=−a±√ a2−4σ 2.Note that eigenvalues of LHLare real and non negative. As, the nullity and null space of LHLis same as that of L,we have 2 zero eigenvalues of A.Also, when λ= 0,LΩ = 0 ,implying ¯Ω = 0 . 2) Stability analysis: Here, we present our main theorem regarding the leaderless translation, scale, and rotation forma- tion maneuvering. From Lemma 2, we have [ΩT0T]T∈C2n is an eigenvector corresponding to the zero eigenvalues of A, and we have Ω∈ M =span{1n, p∗}.The following Lemma will be useful in our future analysis. Lemma 3. For a vector v= [vT 1vT 2]T∈C2n, when v1∈ M , we have ˆAv= [vT 30T]Twhere v3∈ M. Proof. As we have v1∈ M,we can write v1=c11n+c2p∗, where c1, c2∈C.Now we have ˆAv=ˆL0 0 0v1 v2 =ˆLv1 0 =v3 0 . (10) Now we have ˆLv1=c1ˆL1n+c2ˆLp∗= 0+ c2(ˆLtp∗+ˆLrp∗+ ˆLsp∗) =c2(v∗ t1n+γp∗+ιωp∗) =c2v∗ t1n+c2(γ+ιω)p∗:= v3∈ M. Theorem 1. For a group of single integrators described by (3)and the control law (6), subject to Assumption 1, when the control gain parameter, h, is sufficiently high, then p(t)→ M and˙p(t)→ −1 hˆLp(t)→ M ast→ ∞ . Proof. Let us consider the subspace M⊥to be the orthogonal complement of M.The dimensions of MandM⊥are2 andn−2, respectively, over the complex field. Here, M= Ker{L}. Also, we have Ker {L} =Ker{LHL},and hence, M=Ker{LHL}.Now, we add zeros to these subspaces to extend them and get ˆM:=ImBM 0n×2 ,ˆM⊥:=ImBM⊥ BW . where {BM},{BM⊥}, and{BW}are a set of basis for M, M⊥, andW, respectively. Here, W ⊂Chaving a dimension ofn−2.Hence, ˆM,ˆM⊥⊂C2n,having dimension of 2 and2n−2, respectively. We may observe that these two sub- spaces are mutually orthogonal, i.e., for any vector s∈ˆMand ¯s∈ˆM⊥we have ⟨s,¯s⟩= 0.Also, ˆM ⊕ ˆM⊥=C2n.Now the projection matrix for a vector v∈C2nto the subspace ˆM is denoted by PˆM:=PM0 0 0 ,where PMis the projection for a vector v′∈Cnto the subspace M. Also, from Lemma 2, it is clear that ˆMis the kernel of A. Now, we can rewrite x= [pTζT]T∈C2nand˜∆as x=PˆMx+PˆM⊥x=x∥+x⊥, ˜∆ =PˆM˜∆ +PˆM⊥˜∆ = ˜∆∥+˜∆⊥, where x∥andx⊥are the components of x∈C2nalong ˆM and ˆM⊥, respectively. Hence, pandˆδare written as p=PMp+PM⊥p=p∥+p⊥ (11a) ˆδ=PSˆδ+PS⊥ˆδ=ˆδ∥+ˆδ⊥, (11b) where p∥,ˆδ∥, p⊥,ˆδ⊥are the components of p,ˆδ∈Cnalong MandM⊥, respectively. Observe that x∥= [pT ∥0T]T∈ C2n, and x⊥= [pT ⊥ζT]T∈C2n.From (9), the closed loop dynamics is ˙x=Ax−1 hˆAx+˜∆, (12a) ˙ˆδ=−LHζ+1 hˆLp+v. (12b) We consider (12a) as a nominal system with input with ˜∆ being the input. First, we show that the error term, ˜∆, goes to zero exponentially. From (12b) and (12a) , ˜δhas dynamics: ˙˜δ=˙δ−˙ˆδ=−˙ˆδ=−LHζ−1 hˆLp−v =−˜δ=⇒˜δ(t) = exp( −In×nt)˜δ(t0). (13)From (13), it is clear that ˜δgoes to zero exponentially, and hence ˜∆goes to zero exponentially. Now, the dynamics of x⊥ is given by: ˙x⊥=PˆM⊥Ax−1 hPˆM⊥ˆAx+˜∆⊥ =PˆM⊥A(x∥+x⊥)−1 hPˆM⊥ˆA(x∥+x⊥) +˜∆⊥.(14) We will now consider only the nominal system (without input ˜∆⊥) and show that x⊥(t)→0and as t→ ∞ .Asx∥is in the Null space of A,we have Ax∥= 0.Also, PˆM⊥ˆAx∥=PM⊥0 0 0ˆL0 0 0p∥ ζ∥ = PM⊥ˆLp∥ 0 . Also, from Lemma 3 we have ˆLp∥∈ M . Hence, PM⊥ˆLp∥= 0,implying PˆM⊥ˆAx∥= 0.Hence, (14) becomes ˙x⊥=PˆM⊥Ax⊥−1 hPˆM⊥ˆAx⊥. (15) Now, let a basis for ˆMbe given by the columns of BˆM:= [v1v2]∈C2n×2.We can obtain a set of vectors orthog- onal to the columns of BˆM, which form a basis for the space ˆM⊥and let these vectors be given by the columns ofBˆM⊥= [u1u2. . . u 2n−2].Now we define T−1= [BˆMBˆM⊥]∈C2n×2n, also these basis vectors are chosen such that TP ˆM⊥T−1=02×2 0 0 I2n−2×2n−2 .Now, we apply change of coordinates, T, tox, i.e., [zT 1zT 2]T=Tx=T(x∥+x⊥). Here, z1∈C2andz2∈R2n−2. Observe that Tx∥= [zT 10T]T andTz⊥= [0TzT 2]T. Also, TAT−1=A11 0 0A22 ,where A11∈C2×2sharing the zero eigenvalues of A, andA22∈ C2n−2×2n−2which shares all eigenvalues of Awith negative real parts. Applying the same co-ordinate change to (15) and obtain d dt0 z2 =TP ˆM⊥AT−10 z2 −1 hTP ˆM⊥ˆAT−10 z2 .(16) Now, form (27) and using the fact TP ˆM⊥AT−1= (TP ˆM⊥T−1)(TAT−1) =0 0 0A22 ,we have ˙z2=A22z2−1 h(TP ˆM⊥ˆAT−1)†z2, (17) where the †symbol indicates the last 2n−2×2n−2 block is taken because it is compatible with the size of z2. Now, consider the Lyapunov candidate V=zT 2Qz2, where Qis positive definite and satisfies the Lyapunov equation QA22+AT 22Q=−2I2n−2.Since all the eigenvalues of A22 are negative, such a matrix Qalways exists. Now, taking the derivative of Valong the trajectories of the dynamics of z2 using (28), we get: ˙V≤ −2||z2||2+2 h||Q(TPS⊥ˆAT−1)†||||z2||2 =⇒˙V≤ −2||z2||2+2 h||Q||||(TP ˆM⊥ˆAT−1)†||||z2||2. We have ||(TP ˆM⊥ˆAT−1)||=||(TP ˆM⊥ˆAT−1)†||and||Q|| ≤ 1 ||A22||≤1 |λ(A22)|max=1 |λ(A)|max≤1 a, because from Lemma 2, we have a≤ |λ(A)|max.Hence, ˙V≤ −2||z2||2+2||ˆA|| ha||z2||2(∵||PS⊥|| ≤1) =⇒˙V≤ −2||z2||2 h h−||ˆL|| a! (∵||ˆA||=||ˆL||).(18) Also, from (18) we obtain ˙V≤ − kV where k:= h−α hλmax(Q), α:=||ˆL|| a.Hence, with sufficiently large h, the nominal system is exponentially stable. Now, the system (14) is globally Lipschitz with respect to x⊥and input ˜∆and hence is input to state stable [28, Lemma 4.6]. Using (13), as the input ˜∆goes to zero exponentially, x⊥→0exponentially. Hence, from (12a), we obtain x(t)→x∥(t) =PˆMx=PM0 0 0p ζ =⇒p(t)→p∥∈ M t→ ∞ . As,˙ζ=Lp−aζ,˙ζ→ − aζ, ast→ ∞ ,hence ζ→0, as t→ ∞ .Hence, ˙p(t)→ −1 hˆLp(t)→ −1 hˆLp(t)∥∈ M , t→ ∞. Remark 7. Note that we have to choose h > α :=||ˆL|| a,and by choosing a sufficiently large h,global information on the interaction topology can be avoided. In [25], [26] also a suffi- ciently large or sufficiently small perturbation parameter was chosen, however, in addition, [25], [26] required a stabilizing diagonal matrix whose design requires the knowledge of exact interaction among the agents, which is not required here. Remark 8. From Theorem 1, we know that using the control law (6), we have ˙p(t)→ −1 hˆLp(t)→ M . We provide an analytical solution for p(t)in the next Theorem. Theorem 2. The following two mutually exclusive cases hold when his sufficiently high, (a)(i) When we have at least one of the speeds ωorγ is nonzero, then the matrix ˜A:=A −1 hˆAhas an eigen- value 0, and the corresponding eigenvector is [1T n0T n]T, one eigenvalue is −1 h(γ+ιω)whose corresponding eigenvector is [(v∗ t γ+ιω1n+p∗)T0T n]T, where v∗ tis the designed translational velocity. The other eigenvalues have a negative real part. (ii) Also, p(t)→l11n+l2 v∗ t γ+ιω ∈ M ast→ ∞ , where l{1,2}∈Care determined by the initial condition p(0). (b)(i) When ω=γ= 0 andv∗ t̸= 0 , then the matrix has a zero eigenvalue with algebraic multiplicity two, but the geometric multiplicity equals 1, and the chain of gen- eralized (complex) associated with the zero eigenvalues are {[−1 hv∗ t1T n0T n]T,[(p∗)T0T n]T}and the other eigenvalues have negative real part. (ii) Also, ˙pi(t)→ −1 hl2v∗ t,∀i∈ V,as t→ ∞ ,where l2∈C,is determined by the initial condition, i.e.,p(0). Proof. From Lemma 2 and Remark 2, we have Ahas two zero eigenvalues having algebraic and geometric multiplicity equals to two. The eigenvectors are [1T n0T]Tand[(p∗)T0T]T, respectively. Also, from Theorem 1, we already know that˜Ahas2n−2eigenvalues with negative real parts. Now, we analyze the first case, i.e., case (i). ForA −1 hˆA, we have (A −1 hˆA)[1T n0T n]T= 0−1 hˆL0 0 01n 0n = −1 hˆL1n 0n = 0. Also, (A −1 hˆA)" v∗ t γ+ιω1n+p∗ 0n# = −1 hˆLp∗ 0n = −1 hˆLtp∗−1 hˆLrp∗−1 hˆLsp∗ 0n = −1 hv∗ t1n−1 h(γ+ιω) 0n =−1 h(γ+ιω)" v∗ t (γ+ιω)1n+p∗ 0n# . Hence, (i)of(a)is established. Hence, we can write the solution of ˙x=˜Axin the following form x(t) =l11n 0n + l2" v∗ t (γ+ιω)1n+p∗ 0n# e−1 h(γ+ιω)t+P2n m=3fmeλmt,where the eigenvalues λmare such that Re (λm)<0, m≥3,and the functions fm(t, lm, gm, gm−1, . . . , g m−ϱ), ϱ∈N,depending on the algebraic and geometric multiplicity of λmand takes the form of a linear combination, i.e., lm(gm+gm−1t+gm−2t2 2!+ ···+gm−ϱtϱ ϱ!).Now, as fmwill be a polynomial function in t, hence fmeλm→0ast→ ∞ ,andp(t)→l11n+l2 v∗ t γ+ιω . Now, we analyse the case (b),which corresponds to γ= 0 andω= 0,hence we have, (A −1 hˆA)p∗=−1 hˆAp∗= −1 hˆLp∗ 0n = −1 hv∗ t1n 0n and (A −1 hˆA) −1 hv∗ t1n 0n =−1 hv∗ t˜A 1n 0n = 0. Hence, we have (˜A)2[p∗T0T]T= 0,but(˜A)[p∗T0T]T̸= 0, which proofs (i)if part (b).Now, we can write the solution x(t)in the following form x(t) =−l1v∗ t h1n 0n +l2 (p∗−l1v∗ t h1nt) 0n +2nX m=3fmeλmt, where the eigenvalues λmare such that Re (λm)<0, m≥ 3,and the functions fm(t, lm, gm, gm−1, . . . , g m−ϱ), ϱ∈N, depending on the algebraic and geometric multiplicity of λm. Hence, p(t)→ −1 hv∗ t(l1+l2t)1n+l2p∗and˙p(t)→ −l2 hv∗ t1n ast→ ∞ . Remark 9. Different from the affine formation maneuver case, which will be discussed in Section III-C, for the com- plex laplacian-based maneuvering discussed in Section III-B although affine formation maneuvers are not possible, i.e., the sheer maneuvering is not possible, the assumption of d+ 1 rooted graph is not required. Only a d-rooted graph assumption is required here. Hence, each agent only needs to sense a minimum of din-neighbors instead of d+ 1 in- neighbors, reducing the sensing requirements for each agent compared to the affine formation maneuvering case. C. Leaderless Affine formation maneuvers In this subsection, we present the proposed control laws for solving the Problem 2. 1) Control strategy for single integrators: We propose the following control law for affine maneuvering of single integrators ( ∀i∈ V, j∈ Ni,) ui=−X j∈Niwijζi+X i∈Njwjiζj−1 hX j∈Niˆwij(pj−pi)−ˆδi, ˙ζi=−aζi+X j∈Niwij(pj−pi), (19a) ˙ˆδi=X j∈Niwijζi−X i∈Njwjiζj+1 hX j∈Niˆwij(pj−pi) +vi, (19b) where ζi∈Rdis an auxiliary state variable, a > 0is a constant, and viis the measured velocity of the i-th agent. In the control law (19), the information wjiζjhas to be transmitted from the out-neighbors of i; hence inter-agent communication is required. The term, ˆδi, in (19) is an integral term that is used for tackling constant disturbance affecting the dynamics. The weights wijandˆwijare calculated from the following equations: X j∈Niwij(p∗ i−p∗ j) = 0 ∀i∈ V, (20a) X j∈Niˆwij(p∗ i−p∗ j) =v∗ i, (20b) where the desired velocity of the i-th agent is denoted as v∗ i. Note that as we have d+1unknowns with dequations in (20b) for each agent i, the solution of (20b) will be the summation of a particular solution and any vector in the null space, i.e., the solution of eqn. (20a) which is one-dimensional vector space in Rd+1. Based on Assumption 1, it is evident that each agent has d+ 1in-neighbors. Additionally, we have a generic desired configuration. Therefore, for each agent with index i, it is always possible to find dvectors in Rdof the form p∗ ij, j∈ Ni, which are linearly independent. (According to Lemma 1 and Definition 1.3, in a d+ 1rooted digraph, all nodes except the root nodes have d+ 1 in-neighbors, and our additional assumption on root nodes ensures that the root nodes also have d+ 1in-neighbors.) Consequently, solutions (which are not unique) to (20a) and (20b) always exist. It is essential to observe that the solution to (20b) is the perturbation weights that are responsible for the generation of maneuvering force, and as the modified interaction topology ensures each root- node also has d+ 1 in-neighbors, the solution for (20b) for root nodes also exists. Now, define new state variable x:= [pTζT]T∈R2nd. Incompact form (19) can be written as: ˙x=˙p ˙ζ = 0 −¯LT ¯L − aInd |{z } ¯A:=A⊗Id p ζ −1 h¯ˆL 0 0 0 |{z} ¯ˆA:=ˆA⊗Id p ζ +˜δ 0 |{z} ˜∆, =⇒˙x=¯Ax−1 h¯ˆAx+˜∆, (21a) ˙ˆδ=¯LTζ+1 h¯ˆLp+v, (21b) where ˆδ= [ˆδT 1ˆδT 2. . .ˆδT n]T, δ= [δT 1δT 2. . . δT n]T,˜δ= δ−ˆδ, u= [uT 1uT 2. . . uT n]T, v= [vT 1vT 2. . . vT n]T, and ¯L:=L ⊗Id,¯ˆL:=ˆL ⊗Id,¯A:=A ⊗ Id,¯ˆA:=ˆA ⊗ Id withA:= 0−LT L − aIn and ˆA:=ˆL0 0 0 ,and˜∆ :=˜δ 0 . For designing ˆL,the desired velocity ( v∗) can be split into translation, rotation, scale and shear components, as shown in Fig.1(b), i.e., v∗=v∗ t1+v∗ t2+v∗ r+v∗ s+v∗ s1+v∗ s2, and weights can be designed for each component by solving (20b) independently i.e., for R2ˆL=ˆLt1+ˆLt2+ˆLr+ˆLs+ˆLs1+ˆLs2, where the subscripts {t1, t2}stand stands for two translational directions, {r}for rotational, {s}for scale and {s1, s2}for shear components which form a basis for all affine transfor- mations in R2. The following lemma is about the eigenvalues and eigen- vectors of the matrix ¯A. Lemma 4. The eigenvalues of Aare given by λ= −a±√ a2−4σ 2, where σis an eigenvalue of LTL.¯Ahasd(d+1) zero eigenvalues and the corresponding eigenvectors are given by[ΩT0T]T∈R2nd, where then Ω =ω⊗Id,andω∈Rn lies in the null space of LTL. Proof. The proof is similar to the proof of Lemma 2. 2) Stability analysis for single integrator: Next, stability analysis with formation control law (19) for single integrators is carried out. The following lemma will be useful in our analysis. Lemma 5. For an affine transformation ˆp= (In⊗A)p∗+ (1n⊗b), with arbitrary A∈Rd×dandb∈Rd, we have¯ˆLˆp= (In⊗A)v∗∈ S,where¯ˆL=ˆL ⊗ Idand v∗= [v∗ 1Tv∗ 2T. . . v∗ nT]Tis the vector containing desired velocities of the agents. Proof. The following identities are true for Kronecker prod- ucts of matrices: (M1⊗M2)(M3⊗M4) = (M1M3⊗M2M4), implies (M1⊗I)(I⊗M2) = (M1⊗M2) = (I⊗M2)(M1⊗I), with appropriate sizes of identity matrices. It then follows that ¯ˆLˆp= (ˆL ⊗Id)((In⊗A)p∗+ (1n⊗b)) = (ˆL ⊗Id)(In⊗A)p∗+ (ˆL ⊗Id)(1n⊗b) = (ˆL ⊗A)p∗+ (ˆL1n⊗b) = (In⊗A)(ˆL ⊗Id)p∗(∵ˆL1n= 0) = (In⊗A)v∗(from (20b) ,we have v∗= (ˆL ⊗Id)p∗) As, we have chosen v∗∈ S so(In⊗A)v∗∈ S,and¯ˆLˆp∈ S. Next, we state the main theorem for single integrators. Theorem 3. For a group of single integrators described by (4)and the control law (19), subject to Assumption 2, if the control gain parameter, h, is sufficiently large then p(t)→ S and˙p(t)→ −1 h¯ˆLp(t)→ S ast→ ∞ . Proof. Let us consider the subspace S⊥to be the orthogonal complement of S.The dimensions of SandS⊥ared(d+ 1) andd(n−d−1)[19, Theorem 4.1,Theorem 4.2], respectively. Here,S=Ker{¯L}. Also, we have Ker {¯L}=Ker{¯LT¯L},and hence, S=Ker{¯LT¯L}.Now, we add zeros to this subspaces to extend it to get ˆS:=ImBS 0nd×d(d+1) ,ˆS⊥:=ImBS⊥ BW , where {BS},{BS⊥},and{BWare a set of basis for S,S⊥, andW, respectively. Hence, W ⊂Rndhaving dimension of nd−d(d+ 1).We may observe that the sub-spaces ˆS,ˆS⊥, are mutually orthogonal, i.e., for any vector s∈ˆSand¯s∈ ˆS⊥we have ⟨s,¯s⟩= 0.Also, ˆS ⊕ ˆS⊥=R2nd.Now the projection matrix for a vector v∈R2ndto the subspace ˆS is denoted by PˆS:=PS0 0 0 ,where PSis the projection for a vector v′∈Rndto the subspaces S. Also, from Lemma 4, it is clear that ˆSis the kernel of ¯A. Now, we can rewrite x= [pTζT]T∈R2ndand˜∆as x=PˆSx+PˆS⊥x=x∥+x⊥, ˜∆ =PˆS˜∆ +PˆS⊥˜∆ = ˜∆∥+˜∆⊥, where x∥andx⊥are the components of x∈R2ndalong ˆS andˆS⊥, respectively. Therefore, pandˆδare written as p=PSp+PS⊥p=p∥+p⊥, (22a) ˆδ=PSˆδ+PS⊥ˆδ=ˆδ∥+ˆδ⊥, (22b) where p∥,ˆδ∥, p⊥,ˆδ⊥are the components of p,ˆδ∈Rndalong SandS⊥, respectively. Observe that x∥= [pT ∥0T]T∈R2nd, andx⊥= [pT ⊥ζT]T∈R2nd.From (21), the closed loop dynamics is ˙x= (A ⊗Id)x−1 h(ˆA ⊗Id)x+˜∆ (23a) ˙ˆδ=−(LT⊗Id)ζ+1 h(ˆL ⊗Id)p+v. (23b) We consider (23a) as a nominal system with input with ˜∆ being the input. First, we show that the error term, ˜∆, goes to zero exponentially. From (23b) and (23a) , ˜δhas dynamics: ˙˜δ=˙δ−˙ˆδ=−˙ˆδ= (LT⊗Id)ζ−1 h(ˆL ⊗Id)p−v =−˜δ=⇒˜δ(t) = exp( −Idn×dnt)˜δ(t0). (24) From (24), it is clear that ˜δgoes to zero exponentially, and hence ˜∆goes to zero exponentially. Now, the dynamics of x⊥is given by: ˙x⊥=PˆS⊥(A ⊗Id)x−1 hPˆS⊥(ˆA ⊗Id)x+˜∆⊥ =PˆS⊥¯A(x∥+x⊥)−1 hPˆS⊥¯ˆA(x∥+x⊥) +˜∆⊥.(25)We will now consider only the nominal system (without input ˜∆⊥) and show that x⊥(t)→0and as t→ ∞ .Asx∥is in the Null space of ¯A,we have ¯Ax∥= 0.Also, PˆS⊥¯ˆAx∥=PS⊥0 0 0ˆL ⊗Id0 0 0p∥ ζ∥ = PS⊥¯ˆLp∥ 0 , and from Lemma 5 we have¯ˆLp∥∈ S. Hence, PS⊥¯ˆLp∥= 0, implying PˆS⊥¯ˆAx∥= 0.Hence, (25) becomes ˙x⊥=PˆS⊥¯Ax⊥−1 hPˆS⊥¯ˆAx⊥. (26) Now, let a basis for ˆSbe given by the columns of BˆS:= [v1v2. . . v d(d+1)]∈R2nd×d(d+1).We can obtain a set of vectors orthogonal to the columns of BˆS, which form a basis for the space ˆS⊥and let these vectors be given by the columns ofBˆS⊥= [u1u2. . . u 2dn−d(d+1)].Now we define T−1= [BˆSBˆS⊥]∈R2nd×2nd, also these basis vectors are chosen such that TPˆS⊥T−1=0d(d+1) 0 0 I2dn−d(d+1) .Now, we apply change of coordinates, T, tox, i.e., [zT 1zT 2]T=Tx=T(x∥+x⊥). Here, z1∈Rd(d+1)andz2∈R2dn−d(d+1). Observe that Tx∥= [zT 10T]TandTz⊥= [0TzT 2]T. Also, T¯AT−1=A11 0 0A22 ,where A11∈R(d2+d)×(d2+d)sharing the zero eigenvalues of ¯A, andA22∈R2dn−d(d+1)×2dn−d(d+1)which shares all eigenvalues of ¯Awith negative real parts. Applying the same co-ordinate change to (26) and obtain d dt0 z2 =TPˆS⊥¯AT−10 z2 −1 hTPˆS⊥¯ˆAT−10 z2 .(27) Now, form (27) and using the fact TPˆS⊥¯AT−1= (TPˆS⊥T−1)(T¯AT−1) =0 0 0A22 , ˙z2=A22z2−1 h(TPˆS⊥¯ˆAT−1)†z2, (28) where the †symbol indicates the last 2dn−d(d+ 1)×2dn− d(d+1) block is taken because it is compatible with the size of z2.Now, consider the Lyapunov candidate V=zT 2Qz2, where Qis positive definite and satisfies the Lyapunov equation QA22+AT 22Q=−2I2dn−d(d+1).Since all the eigenvalues of A22are negative, such a matrix Qalways exists. Now, taking the derivative of Valong the trajectories of the dynamics of z2using (28), we get: ˙V≤ −2||z2||2+2 h||Q(TPS⊥¯ˆAT−1)†||||z2||2 =⇒˙V≤ −2||z2||2+2 h||Q||||(TPˆS⊥¯ˆAT−1)†||||z2||2. We have ||(TPˆS⊥¯ˆAT−1)||=||(TPˆS⊥¯ˆAT−1)†||and||Q|| ≤ 1 ||A22||≤1 |λ(A22)|max=1 |λ(¯A)|max≤1 a, because from Lemma 4, we have a≤ |λ(¯A)|max.Hence, ˙V≤ −2||z2||2+2||¯ˆA|| a||z2||2(∵||PS⊥|| ≤1) =⇒˙V≤ −2||z2||2 h h−||ˆL|| a! (∵||¯ˆA||=||¯ˆL||).(29) Hence, if h >||ˆL|| a:=α,˙Vis negative definite. Also, from (29) we obtain ˙V≤ −kV (30) where k:=h−α hλmax(Q)and hence the nominal system is ex- ponentially stable. Now, the system (25) is globally Lipschitz with respect to x⊥and input ˜∆and hence is input to state stable [28, Lemma 4.6]. Using (24), as the input ˜∆goes to zero exponentially, x⊥→0exponentially. Hence, from (23a), we obtain x(t)→x∥(t) =PˆSx=PS0 0 0p ζ =⇒p(t)→p∥andζ(t)→0t→ ∞ . Hence, from (21a) it is clear that ˙p(t)→ −1 h(ˆL ⊗Id)p(t) =−1 h¯ˆLp∥(t)∈ S, t→ ∞ . Remark 10. After subtracting the term1 h¯ˆAfrom ¯Ain(21), withh > α :=||ˆL|| a, we end up modifying the d(d+ 1) zero eigenvalues of ¯Athat span S. The eigenvectors corresponding to these modified eigenvalues now also span the space of affine transformations, i.e., S. Moreover, the non-zero eigenvalues of ¯Athat originally resided in the left half of the complex plane are also changed so that they do not migrate to the right half of the complex plane. Remark 11. In [25], [26] a diagonal matrix is required for the control design for undirected graphs, and a fully decentralized design without the need for any diagonal matrix was not addressed for undirected graphs in the literature. Although we have considered a directed interaction topology here, using our proposed laws in (6)and(19) we can solve this problem for undirected graphs as well. 3) Affine maneuvering for higher-order integrators: We next design a control law and establish the stability for affine maneuvering of higher order integrator systems in the presence of constant disturbances as described in (5). We adopt an adaptive back-stepping-based approach for the design. Let y∗ i,l+1,1≤l≤ρ−1be the desired input for stabilizing l-th order term for the i-th agent. Then, the error term is defined as ˜yi,l+1:=yi,l+1−y∗ i,l+1.Further, to avoid the use of the derivatives of virtual control input, i.e.,y∗ i,l+1,1≤l≤ρ−1, in the control law, for avoiding the amplification of noise and computational issues, motivated by design methods in [29], we use a second order command filter H(s) :=w2 n s2+2ζwns+w2n,with ζ= 1 . The filteredsignal is denoted by y∗ if,l+1,1≤l≤ρ−1, and here y∗ if,l+1=H(s)y∗ i,l+1(s).In frequency domain, the derivative ofy∗ if,l+1is obtained by ˙y∗ if,l+1(s) :=sH(s)y∗ i,l+1. Here, wn is the filter’s natural frequency, and the filter’s dc gain and damping ratio are 1, which ensures no overshoot for tracking the signal. The dynamics of the command filter in the state space form is: ˙xi,l+1=wn(F⊗Id)xi,l+1+wn(B⊗Id)y∗ i,l+1,(31a) [(y∗ if,l+1)T( ˙y∗ if,l+1)T]T= (C⊗Id)xi,l+1, (31b) where xi,l+1:= [ηT 1i,l+1ηT 2i,l+1]Tare the state variables asso- ciated with the command filter, y∗ if,l+1is the filtered output of y∗ i,l+1,andF:=0 1 −1−2 , B:= [0 1]T, C:=1 0 0wn . The output is considered to be [(y∗ if,l+1)T( ˙y∗ if,l+1)T]T. The equilibrium point for the filter is calculated as xeq i,l+1=−(F⊗ Id)−1(B⊗Id)y∗ i,l+1=−(F−1B⊗Id)y∗ i,l+1= ([1 ,0]T⊗ Id)y∗ i,l+1.The following coordinate transform for agent iis defined, ˜xi,l+1:= [˜ηT 1i,l+1˜ηT 2i,l+1]T=xi,l+1−xeq i,l+1= [(η1i,l+1−y∗ i,l+1)TηT 2i,l+1]T.Hence, the filter error dynamics of the filter states becomes d˜xi,l+1 dt=wn(F⊗Id)˜xi,l+1. As the matrix Fis Hurwitz, this filter error dynamics is exponentially stable. Now, the desired state and control laws that are used for i-th agent are given by: y∗ i,2=−X j∈Niwijζi+X i∈Njwjiζj −1 hX j∈Niˆwij(yj,1−yi,1)−ˆδi, (32a) ˙ζi=−aζi+X j∈Niwij(yj,1−yi,1), (32b) y∗ i,l+1=−kl˜yi,l+ ˙y∗ if,l−ˆδi,2,2≤l≤ρ−1, (32c) ui=−kρ˜yi,ρ+ ˙y∗ if,ρ−ˆδi,ρ, i∈ V, (32d) ˙ˆδi,1=X j∈Niwijζi−X i∈Njwjiζj+ ˙yi,1, (32e) ˙ˆδi,l= ˜yi,l,2≤l≤ρ. (32f) Let¯yi,1= [yT i,1ζT i]Tand¯y1= [¯yT 1,1. . .¯yT i,1]T. With the above control law (32) for the ρ-th order integrator system (5), we have the following closed loop dynamics in compact form for nagents: ˙¯y1= (A ⊗Id)¯y1−1 h(ˆA ⊗Id)¯y1+˜δ1, 0 +˜y2 0 ,(33a) ˙yl=−kl˜yl+wn˜η2,l+˜δl+ ˜yl+1,2≤l≤ρ−1,(33b) ˙yρ=−kρ˜yρ+wn˜η2,ρ+˜δρ, (33c) ˙ˆδ1= (LT⊗Id)ζ+1 h(ˆL ⊗Id)y1+ ˙y1, (33d) ˙ˆδl= ˜yl,2≤l≤ρ, (33e) where yl,ˆδl∈Rdn,2≤l≤ρ,are the stacked state vectors of all agents, and ˜η2,l:= [˜ηT 21,l. . .˜ηT 2n,l]T, i.e., the stacked filter error vector of all agents. Theorem 4. Using the control law in (32) for a group of ρ-th order systems described by (5)under Assumption 1 when the gain parameter h > α andk1>1, kρ>1, kl>1.5∀l∈ {2, . . . , ρ −1},affine maneuvering is achieved, i.e., yi,1(t)→ Sandyi,2(t)→ −1 h¯ˆLyi,1∈ S ∀ i∈ V. Proof. We first show the error system ˜yl,2≤l≤ρgoes to zero asymptotically as t→ ∞ .Choosing the positive definite and radially unbounded Lyapunov candidate V= 1 2Pρ l=2˜yT l˜yl+1 2Pρ l=2˜δT l˜δl,and taking its derivative along the trajectories of we get: ˙V=ρX l=2˜yT l( ˙yl−˙y∗ l)−ρX l=2˜δT l(˙ˆδl). (34) Using (33b),(33c),(33e) in the expression for ˙Vwe get: ˙V≤ −ρX l=2kl||˜yl||2+ρ−1X l=2||˜yl||||˜yl+1||+ρX l=2||˜yl||||˙˜η1,l|| =⇒˙V≤ −ρX l=2kl||˜yl||2+ρ−1X l=2||˜yl||2 2+||˜yl+1||2 2 +ρX l=2||˜yl||2 2+||˙˜η1,l|| 2 =⇒˙V≤ −ρX l=2(kl−βl)||˜yl||2+ρX l=2||˙˜η1,l||2 2 =⇒˙V≤ −ρX l=2(kl−βl) ||˜yl||2−||˙˜η1,l||2 2(kl−βl) ,(35) where βl= 1.5,3≤l≤ρ−1andβ2= 1, βρ= 1.Setting kl> βl, when we have ||˜yl||>||˙˜η1,l||√ 2(kl−βl)we have ˙V≤0 from (35). Hence, from [28], we have the closed loop system is input-to-state stable, where the input is given by||˙˜η1,l||√ 2(kl−βl). However, as the filter dynamics is exponentially stable, and we haved2˜xi,l+1 dt=wn(F⊗Id)d˜xi,l+1 dt, hence as Fis Hurwitz, ˙˜xi,l+1goes to zero exponentially as t→ ∞ , hence ˙˜η1,lgoes to zero exponentially t→ ∞ . Now, from 35, we have the system 33 is Lyapunov stable. Hence, ˜yl∈ L∞and˜δ∈ L∞.Also, taking derivative of (32e) and using (32a) , we have ˙y∗ i,2∈ L∞and similarly from (32c) and (32f) we have ˙y∗ l∈ L∞. Then from (33b) we have ˙yl∈ L∞,implies ylis uniformly- continuous. Now, integrating both sides of (35), we get: V(∞)−V(0)−ρX l=2Z∞ 0||˙˜η1,l||2 2dt ≤ −ρX l=2(kl−βl)Z∞ 0||˜yl||2dt. (36) As˙˜η1,lgoes to zero exponentially fast, we have ||˙˜η1,l|| ≤ λ1e−λ2tfor some λ1, λ2>0,implying ||˙˜η1,l||2≤λ2 1e−2λ2t. Hence,R∞ 0||˙˜η1,l||2 2=λ2 1 4λ2<∞,∀l∈ {2, . . . , ρ }.Hence, the L.H.S. of (36) is bounded and so will be the R.H.S., implying ˜yl∈ L2.By Barbalat’s lemma [28, Lemma 8.2] we have ˜yl→ 0ast→ ∞ ,2≤l≤ρ.Now, system (33a) can be thought of as a nominal system with input [˜yT 20T]T.Here, the input -10 -5 0 5 10 15 20 X-axis (m)-10-505101520Y-axis (m)Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Ahent 6Fig. 3. Trajectory of single integrators performing translation and scale maneuvers using control law (6). ˜y2goes to zero as t→ ∞ ,and using similar arguments in the Theorem 3, exponential stability of the nominal system can be established, and hence using the arguments of input-to- state stability, we have yi,1(t)→ S andyi,2(t)→ −1 h¯ˆLyi,1∈ S ∀i∈ V. IV. S IMULATIONS In this section, simulations are provided for illustration. A. Complex Laplacian based maneuvering The reference shape (p∗)is considered to be a hexagon in C,which is defined by the the positions p∗ 1=−cos(π/3) + ιsin(π/3), p∗ 2=cos(π/3) + ιsin(π/3), p∗ 3= 1 , p∗ 4= cos(π/3)−ιsin(π/3), p∗ 5=−cos(π/3)−ιsin(π/3), p∗ 6=−1. The initial position of the six agents are p1(0) = 1 .6 + ι2.5, p2(0) = 3 .7 +ι0.4, p3(0) = 5 .3 +ι6.7, p4(0) = −2.1−ι3.5, p5(0) = −4.7−ι5.8, p6(0) = −6.2−ι7.5.The constant disturbances for single integrator dynamics are taken to be δ1= 1 + ι2, δ2=−1 +ι0.1, δ3= 0.2 +ι0.1, δ4= 0.9 +ι0.4, δ5= 2.3−ι0.5, δ6= 0.5 +ι2.1.The interaction topology among the agents is a 2-rooted graph with each agent having the previous two agents as the in-neighbors, i.e., N1={5,6},N2={1,6},N3={1,2},N4={2,3},N5= {3,4},N6={4,5}.Then, the perturbation weights for the control law are designed by solving (8) through the choice of appropriate velocity v∗ i, i∈ V for translation, rotation, scale maneuvers. The parameter his set as 10. In Fig. 3, the agents perform a translational maneuver for the first 30sec., followed by a scaling maneuver for 20sec. Here, the velocities for designing the perturbation weights are v∗ i= 1−ι1∀ifor first 30sec., followed by v∗ i=p∗ ifor the next20sec. In Fig. 4, the agents are performing rotational motion, keeping agent 3as their center. The desired velocities for different agents are v∗ 1=ι(p∗ 1−p∗ 3), v∗ 2=ι(p∗ 1−p∗ 3), v∗ 3= 0, v∗ 4= ι(p∗ 4−p∗ 3), v∗ 5=ι(p∗ 5−p∗ 3), v∗ 1=ι(p∗ 6−p∗ 3). B. Affine formation maneuvering The reference shape (p∗)is considered to be a hexagon inR2,which is defined by the the positions p∗ 1= -10 -5 0 5 10 X-axis (m)-10-5051015Y-axis (m) Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6Fig. 4. Trajectory of single integrators performing a rotation maneuver making agent 3as their center using control law (6). [−cos(π/3)sin(π/3)]T, p∗ 2= [cos(π/3)sin(π/3)]T, p∗ 3= [1 0]T, p∗ 4= [cos(π/3)−sin(π/3)]T, p∗ 5= [−cos(π/3)− sin(π/3)]T, p∗ 6= [−1 0]T.The initial position of the six agents are p1(0) = [ −1.6 2.5]T, p2(0) = [3 .7 4.0]T, p3(0) = [5.3 0.6]T, p4(0) = [2 .1−3.5]T, p5(0) = [ −1.7− 1.8]T, p6(0) = [ −0.1 2]T.The constant disturbances for single integrator dynamics are taken to be δ1= [3 4]T, δ2= [1− 1]T, δ3= [1 0]T, δ4= [−1 0]T, δ5= [2−1]T, δ6= [1 2]T. The interaction topology among the agents is a 3-rooted graph, which is the same as in Fig. 2(b), i.e., each agent has the previous three agents as the in-neighbors. Then, the pertur- bation weights for the control law are designed by solving (20b) through the choice of appropriate velocity v∗ i, i∈ V for translation, rotation, scale, and shear transformations. The parameter his set as 10.Fig. 5 and Fig. 6 illustrate the simulations related to the single integrators. In Fig. 7, a sim- ulation corresponding to higher-order (third-order) integrators is shown. In Fig. 5, the agents perform a translational maneuver for the first 70 sec., followed by a scaling maneuver for 10 sec., which is further followed by a translational maneuver for 20 sec., and finally, a rotational maneuver is performed for 6sec. Here, the velocities for designing the perturbation weights are v∗ i= [−1−1]T∀ifor first 70sec., followed by v∗ i=−p∗ i for the next 10sec. Subsequently, v∗ i= [−1−1]T∀ifor next 20sec., and finally v∗ i=0 1 −1 0 p∗ i. In Fig. 6, first, a translational maneuver is performed by the single integrators, followed by a shearing maneuver. Here, we have taken v∗ i= [−0.5−0.8]T, and v∗ i=0 1 1 0 p∗ i∀ifor the two phases, respectively. In Fig. 7 the agents perform scaling and rotational maneuvers simultaneously which gives rise to a spiral pattern. Here, v∗ i= −1 2p∗ i+0 1 −1 0 p∗ i∀i. V. C ONCLUSION In this work, we have proposed control laws for leaderless maneuverings, considering a directed interaction topology be- tween the agents. With a modification of the interaction graph, i.e., by adding in-neighbors to the root nodes, the maneuvering -10 0 10 20 30 40 50 60 X-axis (m)-10010203040Y-axis (m)Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6Fig. 5. Trajectory of single integrators performing translation, scale, and rotation maneuvers using control law (19). -5 0 5 10 15 20 25 X-axis (m)-50510152025303540Y-axis (m)Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Fig. 6. Trajectory of single integrators performing translation and shear maneuvers using control law (19). -20 -10 0 10 20 30 X-axis (m)-40-30-20-10010203040Y-axis (m) Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Fig. 7. Trajectory of 3-rd order integrators performing scaling and rotational maneuver yielding a spiral pattern using control law (32). force is generated to the root nodes. The control laws are fully distributed in nature. For maneuvering no stabilizing diagonal matrix is required. Hence, the full knowledge of the interaction topology among the agents is not required, which was not the case in previously reported literature. In complex laplacian- based maneuvering, translation, rotation, and scale formation maneuvering are reported for planar agents. In affine formation maneuvering translation, rotation, scale, and sheer formation maneuverings are reported for both 2-D and 3-D cases. For higher-order systems, control laws are also proposed, and stability is analyzed.