Distance-based formation tracking control of multi-agent systems with double-integrator dynamics*

Project supported by the National Natural Science Foundation of China (Grant No. 61603188).

Wu Zixing, Sun Jinsheng, Wang Ximing
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

 

† Corresponding author. E-mail: jssun67@163.com

Project supported by the National Natural Science Foundation of China (Grant No. 61603188).

Abstract

This paper addresses the distance-based formation tracking problem for a double-integrator modeled multi-agent system (MAS) in the presence of a moving leader in d-dimensional space. Under the assumption that the state of leader can be obtained over fixed graphs, a distributed distance-based control protocol is designed for each double-integrator follower agent. The protocol consists of three terms: a gradient function term, a velocity consensus term, and a leader tracking term. Different shape stabilizing functions proposed in the literature can be applied to the gradient function term. The proposed controller allows all agents to both achieve the desired shape and reach the same velocity with moving leader by controlling the distances and velocity. Finally, we analyze the local asymptotic stability of the equilibrium set with center manifold theory. We validate the effectiveness of our approach through two examples.

1. Introduction

Over the past decade, formation control of MAS is an active research area in cooperative control and robotics due to its widespread applications in a variety of areas, such as satellite formation flying, search missions using flying agents, cooperative transportation, sensor networks, and so forth.[1,2] The aim of multi-agent formation control is to drive a group of agents to achieve and to maintain a desired graphic pattern with (or without) references.

As reviewed in Ref. [3], the existing control strategies for multi-agent formation control can be classified into position-based, displacement-based, and distance-based strategies. Among these formation strategies, distance-based control strategy assumes that agents sense and control the relative distance of their neighboring agents, which allows the reduced requirements for the sensing capabilities for each agent. In other words, compared with the other two strategies, the main advantages of the distance-based control lie in that agents need less global information and do not require expensive sensors.

We focus only on works based on distance-based formations in this paper. A gradient control law based on the potential function was proposed in Refs. [4] and [5], and the target undirected formation was defined by a set of inter-agent distances. After that, the authors in Ref. [6] extended the result in Ref. [5] to n-dimensional space and proposed gradient-based formation controllers for single-integrator and double-integrator modeled agents. A unified approach formation controller with general forms of gradient functions was developed in Ref. [7] to guarantee the exponential stability of the distance-based rigid formation control MASs. In Ref. [8], the authors considered the formation tracking problem under a rigidity framework for single-integrator modeled MAS. Among the aforementioned literature, one common feature is that the controllers employ objective functions to share shape control tasks between agents. Designing a potential function is a good way to show the constraint relation between each agent in the formation and easy to achieve real-time control.

Note that the results above primarily concentrate on the ideas of distance-based formations for single-integrator modeled agents. The research on distance-based control of double-integrator modeled agents has attracted attention, but most of them require the velocities of agents to decay to zero (e.g., Refs. [6] and [9]). Motivated by the pioneering work,[10] recent papers[11,12] combined the flocking strategy and the distance-based shape control strategy so as to show that the desired formation of double-integrator modeled agents can be achieved while all agents share a common velocity. As listed in Table 1, the distance-based formation for single-integrator modeled MAS are optimally solved via a gradient function that depends on formation distance error systems, and then the results can be extended to double-integrator modeled MAS by adding velocity attenuation or consensus term. It should be pointed out that the extension to distance-based formation tracking control has not been addressed in the literature. Motivated by the fact that formations in movement have more practical applications in various areas such as security patrols,[23,24] the formation tracking control problem will be considered in this paper.

Table 1.

Review of distance-based undirected formation with gradient-flow method.

.

Therefore, the motivation is to develop a new formation tracking control strategy with a gradient function for followers. The gradient function will be considered as a general form which can encompass the most distance-based shape stabilizing controllers with different forms proposed in the literature. The main contributions of this paper can be summarized as follows. Firstly, a procedure to design distance-based control laws for the formation tracking problem will be constructed. We will give a characterization of shape stabilizing gradient functions for undirected distance-based formation and list the required conditions for distributed gradient-based controllers over a given undirected graph . Hence if there exists a potential function which satisfies the qualifications, the corresponding controller can achieve formation tracking. The recent findings in Table 1 will be extended to agents modeled by double integrator dynamics with a moving leader. Also, a novel control function will be illustrated and validated. Secondly, to address the problem of motion and formation control as a whole, we will propose a strategy based on the gradient method to design a motion not only in a plane but also in three-dimensional (3D) space. The asymptotic stability of the double-integrator MAS with a leader under the proposed distributed controller will be investigated. The LaSalle invariance principle will be employed to analyze the equilibrium set. Since the equilibrium set is not uniquely defined, the center manifold theory will be used to determine the stability when linearizing an equilibrium point, such that the distance-based formation control with a leader is solvable.

This paper is organized as follows. Section 2 recalls some basic concepts on graphs and shape control. Section 3 formulates the main problem to be addressed. In Sections 4 and 5, detailed analysis on the proposed controller and stability of the formation tracking algorithm are provided, respectively. Section 6 presents two sets of simulation examples to demonstrate the controller performance. A conclusion is then given in Section 7.

2. Preliminaries
2.1. Basic Concepts in Graph Theory and Rigidity Theory

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 aij is a positive weight if (i, j) ∈ ε, and aij = 0 otherwise. For an undirected graph, aij = aji, ∀ij. The Laplacian matrix of is defined as Ln = [lij] ∈ ℝn × n, where , lij = −aijij. Note that if (j, i) ∉ ε then lij = −aij = 0.

Consider n single-integrator modeled agents in d-dimensional space where qi ∈ ℝd and ui ∈ ℝd 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 Jgq(q). We define the rigidity function associated with the framework as the function : given by The k-th component of , , coincides with the relative distance for the k-th edge zkij = ||qiqj||, 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* ∈ ℝdn 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 Eq* is asymptotically stable under the control law.

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

3. Problem statement
3.1. Agent dynamics

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 where qi ∈ ℝd, pi ∈ ℝd, and ui ∈ ℝd 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, The pair (qr, pr) ∈ ℝd × ℝd is the state of the leader. The selection of ur(qr, pr) is part of the tracking control design for the collective objective of a group of followers. The choice of ur leads to a corresponding motivation. For example, ur ≡ 0 means the leader moves along a straight line with the desired velocity pr(0).

3.2. The distance-based formation tracking control problem

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: where L12 ϵ Rn×1, , and .

Let p* ∈ ℝdn be given and define the desired formation Eq*,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 pi(t) − pr(t) → 0 as t → ∞. For a given realization we define the desired formation

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.

4. Distance-based distributed formation tracking control law
4.1. Description of the desired formation tracking control law

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 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 where kq is a positive constant, and ρij 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 where kp is a positive constant.

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

4.2. Collective dynamics

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

Let the following vector Φ be the gradients of potential functions of edges 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

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

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

Let qij = qiqj and xij = xixj. The relative positions and velocities remain the same in the moving frame, i.e., xij = qij, vjvi = pjpi. Thus, eij(qij) = eij(xij) and .

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

The system is converted to a matrix form: where x ∈ ℝdn and v ∈ ℝdn are respectively the stacked vectors of all xi and vi. Similarly, the collective potential functions satisfy and , respectively.

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

5. Stability analysis
5.1. Equilibrium set

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) = Σi||vi||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.

5.2. Stability analysis of formation

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 where θ ∈ ℝk, and ρ ∈ ℝvk. 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, Jf1(0, 0) = 0, Jf2(0,0) = 0, where Jf(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 where is a sufficiently small neighborhood of the origin, h(0) = 0 and Jh(0) = 0.

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

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

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 Then by linearizing Eq. (46) at a point in , one has where . Thus, rank , which implies that According to Lemma 1, the framework is infinitesimally rigid, we have that Thus, Ker (Jgx(x)) = Ker(Jgx(x)TJgx(x)) = d(d − 1)/2, which means that Jgx(x)TJgx(x) has d(d − 1)/2 eigenvalues.

Next, it can be shown that JF(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.

6. Illustrative examples

In this section, two illustrative examples are provided to verify the effectiveness of the proposed distance-based formation tracking control method for four double-integrator agents in two-dimensional (2D) space and eight double-integrator agents in three-dimensional (3D) space, respectively. The local convergence of the formation system with the tracking controllers is demonstrated.

Fig. 1. (color online) Flipped situation for the fixed topology with four agents and of five edges.

Figures 2 and 3 show the interagent distance errors and velocity errors of agents. The trajectories of all agents under heterogeneous controller are depicted in Fig. 4.

Fig. 2. (color online) Asymptotic convergence of the distance errors for 2D rectangular formation tracking control.
Fig. 3. (color online) Asymptotic convergence of the velocity errors for 2D rectangular formation tracking control.
Fig. 4. (color online) The simulation on tracking control of a 4-agent rectangular formation shape.
Table 2.

Selection for each edge of the corresponding potential function ρij.

.
Fig. 5. (color online) Asymptotic convergence of the distance errors for 3D cube formation tracking control.
Fig. 6. (color online) Asymptotic convergence of the velocity errors for 3D cube formation tracking control.
Fig. 7. (color online) The simulation on tracking control of an 8-agent cube formation shape in 3D space.
7. Conclusion

We have investigated the distance-based formation tracking control for a group of double-integrator modeled agents. It is assumed that the state of the leader is time-varying and all agents with the leader communicate with each other according to an undirected fixed topology. With the target formation defined by inter-agent distances, the formation tracking controller is developed. The proposed controller can be decomposed to three terms, where the gradient-based term with potential function exhibited the formation target. Next, the asymptotic stability of the system with the proposed control law has been demonstrated. Two examples of the 2D plane and 3D space are provided to illustrate the validity of the theoretical results. The research on time-varying graphs and global stability analysis is ongoing.

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