An undirected graph with n agents and m edges is denoted by with nonempty node set and edges set . j is called a neighbour of i if (i, j) ∈ ε, and the neighbor set of node i is denoted by . The weighted adjacency matrix associated with is defined such that a_ij is a positive weight if (i, j) ∈ ε, and a_ij = 0 otherwise. For an undirected graph, a_ij = a_ji, ∀i ≠ j. The Laplacian matrix of is defined as L_n = [l_ij] ∈ ℝ_{n × n}, where , l_ij = −a_iji ≠ j. Note that if (j, i) ∉ ε then l_ij = −a_ij = 0.

Consider n single-integrator modeled agents in d-dimensional space 1 where q_i ∈ ℝ^d and u_i ∈ ℝ^d denote the position and control input, respectively, of agent i with respect to a global coordinate system.

A graph that has a unique realization cannot be susceptible to continuous flexing and must be rigid.^[25] A detailed discussion of graph rigidity can be found in Ref. [25]. One useful tool to characterize some other concepts related to graph rigidity is the rigidity matrix J_gq(q). We define the rigidity function associated with the framework as the function : given by 2 The k-th component of , , coincides with the relative distance for the k-th edge z_kij = ||q_i − q_j||, which connects agents i and j. The Jacobian of function is the gradient of and we denoted it by . The matrix is referred to as the rigidity matrix of .

Let q* ∈ ℝ^dn be given. Let and denote, respectively, the desired distance and distance error of edge k which links the agents i and j in graph . The desired formation frame can be defined as a set of inter-agent distance realizations that are related to q*: Then, the distance-based formation shape control problem of the single-integrator (1) MAS is equivalent to designing a decentralized control law such that E_q* is asymptotically stable under the control law.

The gradient-based method is widely used to achieve the desired formation E_q*. One can construct a local potential function V_i for agent i: 4 where the function ψ_ij is defined as follows.

Consider that the MAS consists of n double-integrator modeled agents and an additional agent labeled n + 1, which acts as the collective objective or the unique leader of the team. For ease of development, let the first n agents indexed by 1,2, . . ., n be the followers and the agent indexed by n + 1 be the team leader. The motion of follower i is given by 8 where q_i ∈ ℝ^d, p_i ∈ ℝ^d, and u_i ∈ ℝ^d denote the position, velocity, and control input, respectively, of follower i with respect to a global coordinate system. The motion of the leader is given a priori as a reference, and it is assumed that the velocity of the leader is piecewise continuously differentiable, 9 The pair (q_r, p_r) ∈ ℝ^d × ℝ^d is the state of the leader. The selection of u_r(q_r, p_r) is part of the tracking control design for the collective objective of a group of followers. The choice of u_r leads to a corresponding motivation. For example, u_r ≡ 0 means the leader moves along a straight line with the desired velocity p_r(0).

Denote by the interaction topology among n + 1 agents. Similarly, and are, respectively, the adjacency matrix and the Laplacian matrix associated with . Furthermore, the Laplacian matrix of graph can be written as follows: 10 where L₁₂ ϵ R^n×1, , and .

Let p* ∈ ℝ^dn be given and define the desired formation E_q*,p* for followers (8) as Denote the desired distance and the distance error between the leader and follower i as: and , respectively. In order to achieve the goal of tracking the leader while maintaining the relative inter-follower position, it is assumed that each follower can detect the relative position of the leader, thus agents in target formation should satisfy the following constraints.

1) Inter-follower positions: , ∀(i,j) ∈ ε;

2) Follower-to-leader positions: , .

Let if the real formation converges to the target formation , the desired formation pattern can be simultaneously achieved. The control objective is to design the suitable control laws for followers to drive and velocity error p_i(t) − p_r(t) → 0 as t → ∞. For a given realization we define the desired formation 11

In this paper, the control objective for a group of double-integrator modeled agents is to achieve a formation that satisfies a given set of desired distance constraints and moving with the leader. Before starting the main problem, we list the assumption as follows.

We employ the gradient control laws and construct distance potential functions for stabilizing rigid formation shapes. To achieve the tracking control target, the control input for follower can be divided into three terms 12 where is a gradient-based term, is a velocity consensus term, and is a tracking term that acts as a navigational feedback due to the leader, respectively. More specifically, the details are as follows.

i) The gradient-based control term can be constructed as 13 where k_q is a positive constant, and ρ_ij is defined in Eq. (5).

ii) For each agent, the velocity consensus equation, defining how the neighbor of each agent influences and changes its velocity, is 14 where k_p is a positive constant.

iii) The tracking term is considered as a navigational feedback, which is given by 15 where c₁, c₂ are positive constant control gains and ρ_ir = ∂ψ_ir/∂e_ir is a gradient function regarding the distance error between leader and followers with . The accelerated velocity of leader u_r is globally known.

To facilitate the subsequent analysis, we use the following artificial potential function to show the sum of energy of all agents and the leader: 19 Then, the controller (12) can be expressed as 20

Let the following vector Φ be the gradients of potential functions of edges 21 where m and l are the numbers of edges in graph and . Noting that and combining the expression of Q(x) with the rigidity matrix , one can derive the following form of the overall shape controller 22

Then, the MAS model combining the controller (12) can be written as 23

The system (23) is aligned to a global system. However, followers do not necessarily know the origin of the global coordinate system. One can further transform the origin into a moving frame according to the leader. The position and velocity of follower i in the moving frame are given by 24

Let q_ij = q_i − q_j and x_ij = x_i − x_j. The relative positions and velocities remain the same in the moving frame, i.e., x_ij = q_ij, v_j − v_i = p_j − p_i. Thus, e_ij(q_ij) = e_ij(x_ij) and .

Using the control protocol (12) in the moving frame, the closed-loop MAS can be expressed as 25

The system is converted to a matrix form: 26 where x ∈ ℝ^dn and v ∈ ℝ^dn are respectively the stacked vectors of all x_i and v_i. Similarly, the collective potential functions satisfy and , respectively.

Let 27 then the collective dynamics of agents can be decomposed as the following structural dynamics: 28 where .

Before presenting the result on the stability analysis of distance-based formation behavior under control algorithm (12), we define the Hamiltonian function of system (25) as where K(v) = Σ_i||v_i||² is a velocity mismatch function. Note that H(x, v) is a nonnegative valued function. Then, we present the following theorem which summarizes the stability analysis result.

(I) All agents asymptotically move with the same velocity.

(II) Almost all solutions of system (25) asymptotically converge to an equilibrium point , where is minima of .

It should also be mentioned that if the agents are initialized on a collinear configuration in two-dimensional (2D) space, or on a coplanar configuration in 3D space, then they remain collinear or coplanar. These realizations and the flip vision are described as degenerate equilibrium in Ref. [11]. These extra equilibria are not expected. In this paper, since we focused on the local convergence of a gradient-based system, it is feasible to avoid the degenerate equilibrium by assuming that the initial formation is closer to the target formation than the degenerate equilibrium.

In the following, we concentrate on the analysis of asymptotic stability of the set Ω₁. In this section, we first briefly introduce some basic concepts in center manifold theory that could be used to determine the stability of a nonlinear system when linearizing an equilibrium point. More details are available in Ref. [28]. We then consider the corresponding stability analysis using the center manifold theory.

Consider a system in normal form 41 where θ ∈ ℝ^k, and ρ ∈ ℝ^{v − k}. A is a matrix having eigenvalues only on the imaginary axis, B is a matrix having eigenvalues with negative real parts, and the functions f₁(θ, ρ) and f₂(θ, ρ) satisfy f₁(0, 0) = 0, f₂(0, 0) = 0, J_f1(0, 0) = 0, J_f2(0,0) = 0, where J_f(x) denotes the Jacobian of f(x) evaluated at a point x. An invariant manifold is a center manifold for Eq. (41) if it can be locally represented as 42 where is a sufficiently small neighborhood of the origin, h(0) = 0 and J_h(0) = 0.

It can be shown that a center manifold always exists and the dynamic (41) restricted to the center manifold are 43

The stability of the system (41) can be analyzed from the dynamics on the center manifold using the following lemma.

The set of correct equilibrium points can be written as 45

The advantage of using rather than Ω₁ in the ensuing stability analysis is that is a compact set.

Consider Eq. (28) and define F(x, v) as 46 Then by linearizing Eq. (46) at a point in , one has 47 where . Thus, rank , which implies that According to Lemma 1, the framework is infinitesimally rigid, we have that 48 Thus, Ker (J_gx(x)) = Ker(J_gx(x)^TJ_gx(x)) = d(d − 1)/2, which means that J_gx(x)^TJ_gx(x) has d(d − 1)/2 eigenvalues.

Next, it can be shown that J_F(x, v) in Eq. (47) has d(d − 1)/2 zero eigenvalues and the rest have negative real parts. The proof is omitted here, as it follows a similar idea to the proof of Lemma 5 in Ref. [10].

We are now in a position to give the main result.