𝐻_∞ Formation Control and Obstacle Avoidance for Hybrid Multi-Agent Systems

Abstract

A new concept of 𝐻_∞ formation is proposed to handle a group of agents navigating in a free and an obstacle-laden environment while maintaining a desired formation and changing formations when required. With respect to the requirements of changing formation subject to internal or external events, a hybrid multiagent system (HMAS) is formulated in this paper. Based on the fact that obstacles impose the negative effect on the formation of HMAS, the 𝐻_∞ formation is introduced to reflect the above disturbed situation and quantify the attenuation level of obstacle avoidance via the 𝐻_∞-norm of formation stability. An improved Newtonian potential function and a set of repulsive functions are employed to guarantee the HMAS formation-keeping and collision-avoiding from obstacles in a path planning problem, respectively. Simulation results in this paper show that the proposed formation algorithms can effectively allow the multiagent system to avoid penetration into obstacles while accomplishing prespecified global objective successfully.

1. Introduction

In recent years, there has been a spurt of interest in the area of cooperative control for multiple agents due to its challenging features and many applications, for example, formation control [1, 2], obstacles avoidance [3, 4], rendezvous [5], flocking [6], foraging [7], troop hunting, and payload transport. Referring to the existing literature, it is obvious that multiple agents can perform tasks faster and more efficiently than a single one. The existing approaches for cooperative control of MASs fall into several categories, including behavior-based, artificial potential, virtual structure, leader-follower, graph theory, and decentralized control methods. Other methods and research aspects of the cooperative control for MASs can be found in [8–10].

As one key branch of cooperative control, formation and obstacle avoidance problems of multiagent systems have been received significant attentions [1–4, 8, 11]. In this case, the MAS is usually required to follow a trajectory while maintaining a desired formation and avoiding obstacles. In some practical situations, the group of agents may be necessary to perform certain maneuvers, such as split, reunion and reconfiguration, in order to negotiate the obstacles [2, 12]. Although the formation problem for MASs is attracting increasingly research attention, there are still several open fields deserving further investigation, such as robustness, fragility, and effectiveness of formation. With respect to the most robustness analysis of MASs, such as [13–15], the agents are investigated under uncertain environments with external disturbances. However, in some sense, obstacles in the navigational path can also be regarded as a disturbance from environment, which would impair the performance of formation stability. Furthermore, the influence of obstacles is usually negligible when the distances between agents and obstacles exceed certain range threshold. Inspired by obstacle avoidance issue and 𝐻_∞ control theory, we introduce a new concept of 𝐻_∞ formation, which treats the effects of obstacles as certain exterior disturbances, to handle the formation problem of MASs in clustered environment. Then a Lyapunov approach is employed to deal with 𝐻_∞ analysis.

Artificial potential field (APF) method is widely used in coordination control of MASs due to its simplicity and efficiency [7, 16], which was first introduced by [17] for formation and obstacle avoidance of MASs. Since then, several literatures have attempted to improve the performance of APF method. In [18], bifurcation theory is used to reconfigure the formation through a simple free parameter change to reduce the computational expense. By introducing a new concept of artificial potential trenches in [11], the scalability and flexibility of robot formations are improved. The basic idea of potential field theory is to create a workshop, where the agents are counterbalanced with each other by the interactive potential force between them and suffered a repulsive force from obstacles to steer around them [19]. Despite all the advantages of APFs, the lack of accurate representations of obstacles with arbitrary shapes is regarded as one major limitation to generally extend to practical applications. A potential function based on generalized sigmoid functions, which can be generated from the combinations of implicit primitives or from sampled surface data is proposed in [20]. Using the optical flow, [21] have achieved the automatic detection of obstacles in virtual environment. The formation control with obstacle avoidance is highly related to the flocking problems in [6], where only the obstacles with simple shapes are taken into account. In this paper, we assume that the boundary functions of arbitrary obstacles can be known from the implicit functions which can be constructed from sensor readings or image data. By combining the artificial potential model and the negotiating results with obstacles, a resultant artificial repulsive force is developed to guarantee the obstacles avoidance.

In addition, it may happen that the MASs are desired to perform various formation shapes to achieve specified navigational objective. As a result, it is necessary for the MAS to possess the ability of changing formation shape during the navigation, such as split, rejoin, and reconfiguration. In this case, the MASs consist of both continuous variables and discrete events. In [2], a triple (group element 𝔤, shape variable 𝔯, control graph ℌ) is employed to model the mobile robots and meet the requirement of changing formations. Furthermore, a Petri-potential-fuzzy hybrid controller is presented for the motion planning of multiple mobile robots with multiple targets in a clustered environment in [22]. In this paper, a hybrid formation controller is proposed where the formation changes as events (tasks) occur. In practice, the correspondence between tasks and formation can be prespecified at the initialization step, and created intelligently by the embedded processors in each agent during the implementation. It is remarkable that the hybrid multiagent systems exhibit continuous-state dynamics and discrete behavior jumping between formations. Then, we formulate the HMAS by a hybrid machine owing to its advantages of illustrating inputs and outputs explicitly [23, 24].

The paper proceeds as follows. The formation control and obstacle avoidance problem are addressed in Section 2. In Section 3, a new concept of 𝐻_∞ formation and technical proofs are provided. In Section 4, we discuss the obstacle-avoidance functions. Simulation results to illustrate the results are presented in Section 5. Conclusions and future work are provided in Section 6.

Notation 1. Throughout the paper, let ℤ be the set of positive integers, and let 𝕁=[𝑡₀,+∞)(𝑡₀≥0). ℝ^𝑛 represents the real Euclidean 𝑛-dimensional vector space. For 𝑥=(𝑥₁,…,𝑥_𝑛)^⊺∈ℝ^𝑛, the norm of 𝑥 is ‖𝑥‖≜(𝑥^⊺𝑥)^1/2, where the symbol (⋅)^⊺ denotes the transpose of a matrix or a vector. 𝐼_𝑛 denotes the identity matrix of order 𝑛 (for simplicity 𝐼 if no confusion arises). 𝔏₂[0,∞) is the Lebesgue space of 𝑅^𝑛-valued vector-functions 𝑔(⋅), defined on the time interval [0,∞), with the norm ‖𝑔‖₂≜(∫^∞₀‖𝑔(𝑡)‖²)^1/2𝑑𝑡.

2. Problem Formulation and Preliminaries

Consider a multiagent system with 𝑁 nodes and an undirected graph topology 𝔊=(𝜈,𝜀); 𝜈 and 𝜀 are the set of vertices and the set of edges (i.e, 𝜀∈𝜈×𝜈), respectively. The notation (𝑖,𝑗) or (𝑗,𝑖) equivalently denotes the edge of the graph between node 𝑖 and node 𝑗. Furthermore, a graph is connected if there exists a path between every pair of distinct nodes; otherwise it is disconnected.

Before proceeding further, the following assumptions are made in this paper. (1)Each agent is equipped with sensors and computational hardware that allow it to detect the distances to the obstacles within the sensing range. Furthermore, the agent can access its position in the world coordinate system and broadcast to its neighboring agents. The wireless communication has a limited range and is assumed to be imperfect; that is, links may be broken. (2)The multiagent system has a task set Σ and a formation set 𝔽 which meet actual project needs before initiation. And suppose that all agents know the information of Σ and 𝔽, as well as the desired formation shape and trajectories in every step.

Referred to hybrid machine presented in [23, 24] and associated with the practical application, we consider a special class of hybrid multiagent system (HMAS) which is modeled by an elementary hybrid machine (EHM) [25] as follows: HMAS=𝑄,Σ,𝐷𝑦,𝐸,Δ⁰,𝑥(0). (1) The elements of HMAS are denoted as follows. 𝑄={𝑞₀,𝑞₁,…,𝑞_𝑚−1} is a set of vertices (discrete states); in formation control, each discrete state 𝑞_𝑖∈𝑄 corresponds to a desired formation shape Δ^𝑖, and we denote a set of formation shape by 𝔽∶={Δ⁰,Δ¹,…,Δ^𝑚−1}. Σ={ℋ_{𝑞𝑖𝑞𝑗},𝑞_𝑖,𝑞_𝑗∈𝑄} (𝑖,𝑗=0,…,𝑚−1) is a finite (task) set of event labels; (𝑞₀,𝑥(0)) is the initial desired formation and state of HMAS, respectively. 𝐸={(𝑞_𝑖,ℋ_{𝑞𝑖𝑞𝑗},𝑞_𝑗,𝑥⁰_𝑞0)∶𝑞_𝑖,𝑞_𝑗∈𝑄} is a set of edges (transition-paths), where 𝑞_𝑖 is the exited vertex and 𝑞_𝑗 is the entered one. If the event ℋ_{𝑞𝑖𝑞𝑗} is triggered, consequently the formation of HMAS transits from Δ^𝑖 to Δ^𝑗. For example, Figure 1 shows a sequence of admissible collective behaviors of a HMAS triggered by event set Σ={form,dismiss,split,unite,transform1,transform2}.

Figure 1 

Example of a hybrid multiagent system performing under different events.

Remark 1. In practical application, the multiagent system is always assigned multiple tasks in the navigation, and each task may correspond to multiple formation shapes. Similarly to the deterministic automaton described in [26], we suppose that the HMAS is deterministic; namely, there cannot be two transitions with the same event label. It is worth mentioning that the following theoretical analysis is available for nondeterministic HMAS; that is, there can be multiple transitions triggered by the same event.

𝐷𝑦 is the dynamics of HMAS and for each agent 𝑖 and 𝑞∈𝑄, which is described by 𝑥̇_𝑖(𝑡)=𝑓(𝑡,𝑥)+^𝑁_𝑗=1𝐽_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)+𝑢^𝑞_𝑖(𝑡)+𝐶_𝑖𝑤_𝑖(𝑡), (2) where 𝑖∈{1,2,…,𝑁}, 𝑡∈𝕁, 𝑥_𝑖(𝑡)∈ℝ^𝑛×1, 𝑥=[𝑥^⊺₁,𝑥^⊺₂,…,𝑥^⊺_𝑁]^⊺, and 𝐶_𝑖 are real constant matrices with appropriate dimensions. 𝑓(𝑡,𝑥)∶𝕁×ℝ^𝑁×𝑛→ℝ^𝑛×1 is continuously differentiable, representing the group motion (i.e., path of the HMAS). 𝐽(𝑡)=(𝐽_𝑖𝑗(𝑡))_𝑁×𝑁 is the time-varying coupling configuration matrix representing the communication strength and communicational topology of the HMAS. If there is an interconnection between agent 𝑖 and agent 𝑗(𝑗≠𝑖), then 𝐽_𝑖𝑗(𝑡)=𝐽_𝑗𝑖(𝑡)>0; otherwise, 𝐽_𝑖𝑗(𝑡)=𝐽_𝑗𝑖(𝑡)=0, and the diagonal elements of matrix 𝐽(𝑡) are defined by 𝐽_𝑖𝑖(𝑡)=−^𝑁_{𝑗=1,𝑗≠𝑖}𝐽_𝑖𝑗(𝑡)=−^𝑁_{𝑗=1,𝑗≠𝑖}𝐽_𝑗𝑖(𝑡). (3)𝑤_𝑖(𝑡) here denotes the obstacle-avoiding function, which will be derived from potential function in Section 6.

Moreover, the formation controller in this paper is derived by extending the one in [1] into multiformation case (𝑞∈𝑄): 𝑢^𝑞_𝑖(𝑡)=^𝑁_{𝑗=1𝑗≠𝑖}2𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗×𝑆_𝑎𝐿²_𝑎𝑒^{−‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2/𝐿2𝑎}−𝑆_𝑟𝐿²_𝑟𝑒^{−‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2/𝐿2𝑟}+𝑆_𝑟1𝐿²_𝑟+1𝐿²_𝑎×𝑒^{−(1/𝐿2𝑟+1/𝐿2𝑎)‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2}, (4) where Δ^𝑞_𝑖𝑗∈ℝ^𝑛×1 and Δ^𝑞=(Δ^𝑞_𝑖𝑗)_𝑁×𝑁∈𝔽 is the formation-shape matrix of the multiagent system with Δ^𝑞_𝑖𝑗=−Δ^𝑞_𝑗𝑖 and Δ^𝑞_𝑖𝑖=0. Parameters 𝑆_𝑎, 𝑆_𝑟, 𝐿_𝑎, and 𝐿_𝑟 are positive constants representing the strengths and effect ranges of the attractive and repulsive forces, respectively; and with the following constraint we have 𝑆_𝑎𝑆_𝑟>𝐿²_𝑎𝐿²_𝑟𝑒^{(−1/𝐿2𝑟−1/𝐿2𝑎)‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2}−𝐿1+^2𝑎𝐿²_𝑟𝑒^{−(‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2/𝐿2𝑟)}, (5) where 𝐿_𝑎>𝐿_𝑟.

Remark 2. Compared to the formation controller given in [1], the formation controller in this paper is designed to achieve more complicated control objects, such as formation switching in clustered environment. Furthermore, it is worth mentioning that the obstacle-avoiding function 𝑤_𝑖(𝑡) (𝑖=1,…,𝑁) as a part of the controller is an important contribution for this paper.

In order to simplify (4), define 𝜑^𝑞_𝑖𝑗(𝑡)𝑆=2^𝑎𝐿²_𝑎𝑒^{−(‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2/𝐿2𝑎)}−𝑆_𝑟𝐿²_𝑟𝑒^{−(‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2/𝐿2𝑟)}+𝑆_𝑟𝐿²_𝑟+𝑆_𝑟𝐿²_𝑎×𝑒^{(−1/𝐿2𝑟+1/𝐿2𝑎)(‖𝑥𝑗(𝑡)−𝑥𝑖(𝑡)−Δ𝑞𝑖𝑗‖2)}. (6) According to (5), one can verify 𝜑^𝑞_𝑖𝑗(𝑡)>0, and the necessity of this constraint can be addressed by referring to that the force vector and position vector are unidirectional.

Then rewrite the formation controller (4) as follows: 𝑢^𝑞_𝑖(𝑡)=^𝑁_{𝑗=1𝑗≠1}𝜑^𝑞_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗,𝑞∈𝑄. (7)

Remark 3. Let 𝜑^𝑞_𝑖𝑗(𝑡) be a continuous function with respect to ‖𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗‖, and it is easy to prove that if the bounds of ‖𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗‖ exist, then 𝜑^𝑞_𝑖𝑗(𝑡) is bounded on all sets of ‖𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗‖ (𝑖,𝑗=1,…,𝑁 and 𝑞∈𝑄). Furthermore, regard the fact that most multiagent systems are implemented in finite horizon which means the limitation of interagent distances exists. Throughout the paper, we assume that the lower bound of 𝜑^𝑞_𝑖𝑗(𝑡) exists, and we denote it by 0<𝜑≤min_{𝑖,𝑗=1,…,𝑁𝑞=0,…,𝑚−1}𝜑^𝑞_𝑖𝑗(𝑡), (8) where 𝜑>0 can be guaranteed by choosing appropriate values of 𝐿_𝑎, 𝐿_𝑟, 𝑆_𝑎, and 𝑆_𝑟 in the constraint (5).

3. Analysis of 𝐻_∞ Formation Stability

Now, this section will analyze 𝐻_∞ formation stability of the above-developed framework of HMAS in a free and an obstacle-laden environment, respectively.

Since we have property (3), the HMAS (1) is equivalent to 𝑥̇_𝑖(𝑡)=𝑓(𝑡,𝑥)+^𝑁_𝑗=1𝐽_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+𝑢^𝑞_𝑖(𝑡)+𝐶_𝑖𝑤_𝑖(𝑡). (9)

Before moving on, we need to note that the formation switching in the controller will introduce discontinuities to the right hand side of (9). With respect to the dwell-time theory in [27], if the switching of a family of individually stable systems is sufficiently slow, then overall systems remain stable. As a result, we assume that the intervals between consecutive switching signals, that is, dwell time, are large enough. Due to the introduction of average dwell-time, this assumption does not represent a restriction because this concept allows the formation switching mechanism to be more flexible provided that the average interval between consecutive switching is no less than certain fixed positive constant.

To investigate the formation control of MAS, we introduce a measurement error 𝑋_𝑖𝑗(𝑡) given as 𝑋_𝑖𝑗(𝑡)=𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡). (10) It follows from (5), (9), and (10) that the time derivative of 𝑋_𝑖𝑗(𝑡) is as follows: 𝑋̇_𝑖𝑗(𝑡)=^𝑁_𝑘=1𝐽_𝑗𝑘(𝑡)𝑋_𝑗𝑘(𝑡)−𝐽_𝑖𝑘(𝑡)𝑋_𝑖𝑘(𝑡)+𝐶_𝑗𝑤_𝑗(𝑡)−𝐶_𝑖𝑤_𝑖(𝑡)+^𝑁_𝑘=1𝜑^𝑞_𝑗𝑘(𝑡)𝑋_𝑗𝑘(𝑡)−Δ^𝑞_𝑗𝑘−𝜑^𝑞_𝑖𝑘(𝑡)𝑋_𝑖𝑘(𝑡)−Δ^𝑞_𝑖𝑘. (11)

For a formation of multiple agents moving in a clustered environment, it is inevitable to encounter various obstacles which affect the performance of formation or even break the whole system down. Naturally, the multiagent system is desirable to be able to adapt to the environment. In general, the agents are only affected by the obstacles when they enter a certain region. At other times, the influence being exerted from obstacles can be negligible. With the above analysis, the obstacles can be treated as exogenous disturbances deriving from the environment, and 𝐻_∞ analysis can be employed to investigate the formation stability of HMAS.

Associating the system (2) with the formation controller (7), we define a disagreement function similar to [28] as follows: Φ𝑋_𝑖𝑗(𝑡)=1𝑁^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑋_𝑖𝑗(𝑡)−Δ^𝑞_𝑖𝑗², (12) which demonstrates the formation performance of HMAS for 𝑞∈𝑄.

For the HMAS given in (2), 𝐻_∞ formation stability means to find a formation controller (7) such that the following conditions in (DF1) and (DF2) hold.

(DF1). The formation of HMAS (1) is asymptotically stable when 𝑤(𝑡)=0, where 𝑤(𝑡)=[𝑤^⊺₁,𝑤^⊺₂,…,𝑤^⊺_𝑁]^⊺, which is equivalent to the asymptotical formation stability of the HMAS (1) in the absence of obstacles. That is to say that, all agents asymptotically converge to the desired formation positions; that is, ‖𝑋_𝑖𝑗(𝑡)−Δ^𝑞_𝑖𝑗‖→0 as 𝑡→∞, where 𝑞∈𝑄.

(DF2). The formation controller ensures a certain level of 𝐻_∞ formation performance as follows: sup_{𝑋𝑖𝑗(0)𝜔𝑖≠0}∫^∞₀Φ𝑋_𝑖𝑗(𝑡)𝑑𝑡(1/4𝑁)∑^𝑁_𝑖=1∑^𝑁_𝑗=1𝑋_𝑖𝑗(0)−Δ^𝑞_𝑖𝑗²+(1/𝑁)∑^𝑁_𝑖=1𝑤_𝑖(𝑡)²₂<𝛾 (13) for initial states 𝑥_𝑖(0)∈ℝ^𝑛×1 and 𝑞∈𝑄, where 𝛾≥1 is a constant, 𝑋_𝑖𝑗(0)=𝑥_𝑗(0)−𝑥_𝑖(0), and Φ(𝑋_𝑖𝑗(𝑡)) is given by (12). If the conditions (DF1) and (DF2) hold, then the formation of HMAS (1) is said to achieve the 𝐻_∞ formation stability.

Remark 4. It is worthwhile noting that (DF2) is a standard condition arising from the 𝐻_∞ control theory, which implies that 𝑋_𝑖𝑗, (𝑖,𝑗=1,…,𝑁), converges to Δ^𝑞_𝑖𝑗 in the sense of 𝔏₂.

Remark 5. According to (DF2), the attenuation level 𝛾≥1 shows the sensitivity of obstacle avoidance in HMAS (1). Giving a smaller 𝛾≥1 means that the intensity and range of reaction of HMAS towards obstacles are smaller.

Theorem 6. Given a positive scalar 𝛾≥1, the 𝐻_∞ formation problem of the HMAS (1) under the conditions (DF1) and (DF2) is solved at initial positions 𝑥_𝑖(0)∈ℝ^𝑛, if −𝜋_𝑖𝑗+𝐼𝑁𝐶00−^𝑖𝑁0−𝜋_𝑖𝑗𝜋_𝑖𝑗+𝜒_𝑖𝑗200𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗2−𝜒_𝑖𝑗0−𝐶^⊺_𝑖𝑁00−2𝛾𝑁²<0 (14) holds for all 𝑖,𝑗=1,2,…,𝑁, where 𝜋_𝑖𝑗=𝜑+𝐽_𝑖𝑗𝐼2,𝜒_𝑖𝑗=𝜑−𝐽_𝑖𝑗𝐼2, (15) and 𝜑 is defined in (8).

Proof. Without the loss of generality, construct a common Lyapunov function in the form of 1𝑉=4𝑁^𝑁𝑁_𝑖=1𝑗=1𝑋_𝑖𝑗(𝑡)−Δ^𝑞_𝑖𝑗², (16) where 𝑞∈𝑄. If the derivative of 𝑉 with respective to (9) is constantly negative for all subsystems, then the formation of HMAS (1) is stable. For the sake of convenience, 𝑋_𝑖𝑗(𝑡) is implicitly rewritten as 𝑋_𝑖𝑗, as well as 𝐽_𝑖𝑗(𝑡), 𝜑^𝑞_𝑖𝑗(𝑡), and 𝑤_𝑖(𝑡) in the proof.
From the above discussion, one has 𝑉̇=12𝑁^𝑁𝑁_𝑖=1𝑗=1𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝑋̇_𝑖𝑗=12𝑁^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝐽_𝑗𝑘𝑋_𝑗𝑘−𝐽_𝑖𝑘𝑋_{𝑖𝑘𝑣1}+1𝑁^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1𝜑^𝑞_𝑗𝑘𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝑋_𝑗𝑘−Δ^𝑞_{𝑗𝑘𝑣2}+12𝑁^𝑁𝑁_𝑖=1𝑗=1𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝐶_𝑗𝑤_𝑗−𝐶_𝑖𝑤_𝑖𝑣3. (17)
In respect that the coupling configuration matrix 𝐽(𝑡) is symmetric, and 𝑋_𝑖𝑗=−𝑋_𝑗𝑖, Δ^𝑞_𝑖𝑗=−Δ^𝑞_𝑗𝑖, 𝑋_𝑖𝑖=0, and Δ^𝑞_𝑗𝑗=0, one has 𝑣₁1=−𝑁^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1𝑋_𝑗𝑖−Δ^𝑞_𝑗𝑖^⊺𝐽_𝑗𝑘𝑋_𝑗𝑘1=−𝑁^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑘<𝑗}𝑋_𝑗𝑖−Δ^𝑞_𝑗𝑖^⊺𝐽_𝑗𝑘𝑋_𝑗𝑘−1𝑁^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑘>𝑗}𝑋_𝑗𝑖−Δ^𝑞_𝑗𝑖^⊺𝐽_𝑗𝑘𝑋_{𝑗𝑘𝑣11}. (18)
Renaming 𝑗 in the 𝑣₁₁ as 𝑘, thus one has 𝑣₁1=−𝑁^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑘<𝑗}𝑋_𝑗𝑖−Δ^𝑞_𝑗𝑖^⊺−𝑋_𝑘𝑖−Δ^𝑞_𝑘𝑖^⊺𝐽_𝑗𝑘𝑋_𝑗𝑘. (19)
One can find the fact that 𝑋^⊺_𝑗𝑖−𝑋^⊺_𝑘𝑖=𝑋^⊺_𝑗𝑘 and Δ^𝑞_𝑗𝑖−Δ^𝑞_𝑘𝑖=Δ^𝑞_𝑗𝑘; then 𝑣₁1=−𝑁^𝑁𝑁_𝑖=1^𝑁_{𝑘=1𝑘<𝑗}𝑋_𝑗𝑘−Δ^𝑞_𝑗𝑘^⊺𝐽_𝑗𝑘𝑋_𝑗𝑘=−^𝑁𝑁_{𝑖=1𝑗>𝑖}𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝐽_𝑖𝑗𝑋_𝑖𝑗. (20)
Using the standard completing the square argument, it follows from (20) that 𝑣₁1=−2^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝐽_𝑖𝑗𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗²−12^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝐽_𝑖𝑗𝑋_𝑖𝑗²+12^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝐽_𝑖𝑗Δ^𝑞_𝑖𝑗². (21)
Furthermore, the 𝑣₂ is similarly analyzed as follows: 𝑣₂=−^𝑁𝑁_{𝑖=1𝑗>𝑖}𝜑^𝑞_𝑖𝑗𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗. (22)
With respect to (8), one can easily obtain 𝑣₂≤−𝜑^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗=𝜑2^𝑁−1𝑁_{𝑖=1𝑗>𝑖}2𝑋^⊺_𝑖𝑗Δ^𝑞_𝑖𝑗−𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗²−𝑋_𝑖𝑗²−Δ^𝑞_𝑖𝑗². (23)
Then, for 𝑣₃, one has 𝑣₃2=−𝑁^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝐶_𝑖𝑤_𝑖. (24)
Now, we consider the formation stability of HMAS (1) with 𝑤(𝑡)=0. From the inequalities (21) and (23), 𝑉̇ becomes 𝑉̇≤^𝑁−1𝑁_{𝑖=1𝑗>𝑖}Ω^⊺_𝑖𝑗−𝜋_𝑖𝑗000−𝜋_𝑖𝑗𝜋_𝑖𝑗+𝜒_𝑖𝑗20𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗2−𝜒_𝑖𝑗Ω_𝑖𝑗, (25) where Ω_𝑖𝑗=[(𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗)^⊺,𝑋^⊺_𝑖𝑗,(Δ^𝑞_𝑖𝑗)^⊺]^⊺, 𝜋_𝑖𝑗, and 𝜒_𝑖𝑗 are given in (15). By the Schur complement formula, the achievement of inequality (14) is equivalent to the following inequalities satisfied as follows: −𝜋_𝑖𝑗+𝐼𝑁000−𝜋_𝑖𝑗𝜋_𝑖𝑗+𝜒_𝑖𝑗20𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗2−𝜒_𝑖𝑗𝐶<0,_𝑖𝑁00^⊺𝜋_𝑖𝑗−𝐼𝑁000𝜋_𝑖𝑗−𝜋_𝑖𝑗+𝜒_𝑖𝑗2𝜋0−^⊺𝑖𝑗+𝜒^⊺_𝑖𝑗2𝜒_𝑖𝑗⁻¹𝐶_𝑖𝑁00<2𝛾𝑁². (26) By nonnegative matrix theory, one can easily find that −𝜋_𝑖𝑗000−𝜋_𝑖𝑗𝜋_𝑖𝑗+𝜒_𝑖𝑗20𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗2−𝜒_𝑖𝑗≤−𝜋_𝑖𝑗+𝐼𝑁000−𝜋_𝑖𝑗𝜋_𝑖𝑗+𝜒_𝑖𝑗20𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗2−𝜒_𝑖𝑗<0, (27) which implies 𝑉̇<0. This proves that condition (DF1) holds for the HMAS (1) with 𝑤(𝑡)=0.
Next, we prove the 𝐻_∞ performance constraint (DF2) for all nonzero 𝑤_𝑖(𝑡)∈𝔏₂[0,∞) and a prescribed 𝛾≥1. Define 𝑉=^∞₀Φ𝑋_𝑖𝑗(𝑡)1−𝛾𝑁^𝑁_𝑖=1𝑤^⊺_𝑖(𝑡)𝑤_𝑖(𝑡)−𝛾𝑑𝑡2𝑁^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑥_𝑗(0)−𝑥_𝑖(0)−Δ^𝑞_𝑖𝑗², (28) and one has 𝑉=^∞₀Φ𝑋_𝑖𝑗(𝑡)−𝛾𝑁^𝑁_𝑖=1𝑤^⊺_𝑖(𝑡)𝑤_𝑖(𝑡)+𝑉̇(𝑥(𝑡(𝑥(∞+))𝑑𝑡−𝑉))1−𝛾2𝑁^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑥_𝑗(0)−𝑥_𝑖(0)−Δ^𝑞_𝑖𝑗²≤^∞₀Φ𝑋_𝑖𝑗(𝑡−𝛾)𝑁^𝑁_𝑖=1𝑤_𝑖(𝑡)²+𝑉̇(𝑥(𝑡))_Ξ𝑑𝑡. (29)
Combining (21), (23), and (24), one obtains Ξ=^𝑁𝑁_𝑖=1𝑗=11𝑋2𝑁^𝑖𝑗−Δ^𝑞_𝑖𝑗²−𝛾𝑁²𝑤^⊺_𝑖𝑤_𝑖−1𝑁𝑋_𝑖𝑗−Δ^𝑞_𝑖𝑗^⊺𝐶_𝑖𝑤_𝑖+Ω^⊺_𝑖𝑗−𝜋_𝑖𝑗2𝜋000−^𝑖𝑗2𝜋_𝑖𝑗+𝜒_𝑖𝑗40𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗4−𝜒_𝑖𝑗2Ω_𝑖𝑗=^𝑁𝑁_𝑖=1𝑗=1𝜉^⊺_𝑖𝑗×−𝜋_𝑖𝑗2+𝐼𝐶2𝑁00−^𝑖𝜋2𝑁0−^𝑖𝑗2𝜋_𝑖𝑗+𝜒_𝑖𝑗400𝜋^⊺_𝑖𝑗+𝜒^⊺_𝑖𝑗4−𝜒_𝑖𝑗20−𝐶^⊺_𝑖𝛾2𝑁00−𝑁²𝜉_𝑖𝑗, (30) where 𝜉_𝑖𝑗=[Ω^⊺_𝑖𝑗,𝑤^⊺_𝑖(𝑡)]^⊺. From the LMI (14), it is easy to prove Ξ<0 which implies 𝑉<0 and immediately leads to the inequality (13).
Therefore, the formation of HMAS (1) has the property of 𝐻_∞ criteria (𝐃𝐅𝟏) and (𝐃𝐅𝟐). This completes the proof.

Remark 7. By studying the LMIs (14), the variables 𝜒_𝑖𝑗>0 in (15) indicate the connectivity strength among agents. In particular, due to the monotone decreasing property of 𝜑_𝑖𝑗 with respect to norm ‖𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ_𝑖𝑗‖ and the discussion in Remark 3, this connectivity strength is inversely proportional to the relatively active scope of MASs.

4. Design Obstacle-Avoiding Functions

In this section, collision avoidance in trajectory tracking is achieved using mutual repulsion between agents and obstacles, which is resulted from the Newtonian potential-based model. By regarding the agents and obstacles as conductors with uniform charges, the repulsive force inversely proportional to the distance of them can be derived in closed form.

Based on practical applications in robotics and haptic rendering, an ideal potential field should possess all of the following attributes.(i)With respect to the property of obstacle avoidance, the magnitude of potential and corresponding repulsion should be infinite at the boundary of obstacles and drop off with distance. And the range of potential is bounded, which is accordant with the limited detective scope of agent’s built-in explorer in this paper. (ii)The shapes of equipotential surface should be similar with the obstacle surface and spherical symmetrical at the boundary of potential field. (iii)The first and second derivatives of potential function should exist and be continuous, so that the resulting force field is smooth.

Before moving on, for making this paper self-contained, we revisit the general definition of external disturbance in the 𝐻_∞ problem. 𝐻_∞ techniques are usually used to evaluate the incremental gain of external input signal in any direction and at any frequency. In the context of 𝐻_∞ theory, the external signals with finite energy are often investigated; that is, ‖𝑤(𝑡)‖≤∞. More explicitly, the finite energy signal 𝑤(𝑡) is said to belong to 𝐿₂[0,∞), which implies ^∞₀‖𝑤(𝑡)‖²𝑑𝑡^1/2=^∞₀^𝑁_𝑖=1𝑤^⊺_𝑖(𝑡)𝑤_𝑖(𝑡)𝑑𝑡^1/2<∞. (31)

In order to utilize 𝐻_∞ theory to design obstacle-avoidance controller, we assume that multiagent systems ultimately get far from the obstacles as time evolves. Incorporated with attributes (ii) this assumption leads to lim_𝑡→∞𝑤(𝑡)=0. That is, the obstacle-avoidance functions 𝑤_𝑖(𝑡) are available in finite time intervals. In addition, for avoiding obstacles, the agent-obstacle distances are intuitively greater than zero, and consequently the supremum of function 𝑤_𝑖(𝑡) (𝑖=1,…,𝑁) exists. Based on above analysis on obstacle-avoidance function, one can easily derive the conclusion that 𝑤(𝑡) belongs to 𝐿₂[0,∞).

Now, we will discuss the obstacle-avoiding function beginning with the instance of mass points (when the bulks of agents and obstacles are close to each other) and then extending to bulky obstacles (over 10 times bigger than agent) with arbitrary shapes.

Consider that an agent 𝑖 navigates in an obstacle-laden environment with 𝑀∈ℤ₊ obstacles, and assume that 𝑠_𝑙∈ℝ^𝑛 is the position of obstacle 𝑙 (𝑙∈{1,2,…,𝑀}). The potential at agent 𝑖 due to obstacle 𝑙 is 𝑃^𝑜_𝑖𝑙(𝑡)=𝜌_𝑖𝑙𝑥_𝑖(𝑡)−𝑠_𝑙, (32) where 𝜌_𝑖𝑙 is the repulsion coefficient for obstacles avoidance and is defined as follows: 𝜌_𝑖𝑙=𝜌,𝑥_𝑖(𝑡)−𝑠_𝑙𝑥<𝛿0,_𝑖(𝑡)−𝑠_𝑙≥𝛿, (33) where 𝜌 is a positive scalar and 𝛿 is the maximal sensing range of agent. When the relative distance of agent 𝑖 and obstacle 𝑙 is shorter than that of the detective scope 𝛿, agent 𝑖 will receive a signal of possible collision and the obstacle-avoiding function will work. In other words, the repulsive potentials between agents and obstacles act only when they get close to certain range. Now, the obstacle-avoiding function for agent 𝑖 is introduced based on the negative gradient of the potential (32) in the following form: 𝑤_𝑖(𝑡)=−∇_𝑥𝑖𝑃^𝑜_𝑖𝑙𝑥_𝑖,𝑠_𝑙=^𝑀_𝑙=1𝜌_𝑖𝑙𝑥_𝑖(𝑡)−𝑠_𝑙𝑥_𝑖(𝑡)−𝑠_𝑙³. (34)

Then we extend the obstacle-avoiding function for the bulky objects with arbitrary shapes. In practical situations, particularly in many exploration applications, the implicit functions of obstacles to be modeled are not available. But by samples of boundary surfaces obtained from camera, laser range finder and sonar, and with the help of some techniques such as signal sampling and image processing, the implicit functions can be obtained, and then they construct the APFs [20]. For convenience, in this paper, assume that the boundary function of obstacle 𝑙 is known as 𝐁_𝑙, and 𝛽_𝑙 here is the position of arbitrary point on the boundary of obstacle. In fact, most of obstacles can be mathematically approximated with polyhedron. In this paper we focus on the convex polyhedra obstacles. Moreover, the results can be easily extended to obstacles with arbitrary shapes. The repulsion between agent 𝑖 and an arbitrary point of the obstacle’s boundary is 𝐹^𝑜_𝑖𝑙(𝑡)=𝜌_𝑖𝑙𝑥_𝑖(𝑡)−𝛽_𝑙𝑥_𝑖(𝑡)−𝛽_𝑙³, (35) where 𝜌_𝑖𝑙 are defined in (33). Then for the agent 𝑖 the obstacle-avoiding function is defined as 𝑤_𝑖(𝑡)=^𝑀_{𝑙=1𝐁𝑙}𝑑𝐹^𝑜_𝑖𝑙, (36) where ∮_𝐁𝑙 represents the surface integrals on boundary of obstacles.

In view of above analysis, the obstacles laden in the terrain are static. However, it is worth mentioning that all the obstacle-avoidance functions are available for moving obstacles except for those with high-speed. In this case, the position vector 𝛽_𝑙(𝑡) is a vector-valued function over time 𝑡∈𝕁.

5. Numerical Examples

In this section, some simulation results illustrate the performance of the proposed control laws to achieve formation and obstacles avoidance. To avoid triviality, the advantage of formation controllers (4) compared with others is omitted in this paper, and the reader is referred to [1].

Firstly, drive 4 agents with the initialization positions (−1.2,−1.1)^⊺, (−2.0,3.8)^⊺, (5.3,2.0)^⊺, and (4.2,2.0)^⊺ to form a shape of square and realize obstacle avoidance in the navigation. The formation-shape matrix Δ^𝑞 is set as Δ¹₁₂=√22,√22^⊺,Δ¹₂₃=√22√,−22^⊺,Δ¹₃₄=−√22√,−22^⊺,Δ¹₄₁=−√22,√22^⊺,Δ¹₁₃=√2,0^⊺,Δ¹₂₄=0,−√2^⊺, (37) and two obstacles locate at [2.50,5.00]^⊺ and [−0.82,15.90]^⊺. The target trajectory of 𝑓(𝑡,𝑥) in HMAS (1) is given by 𝑓(𝑡,𝑥)=𝑥−0.1000.1^𝑖+()(),cos0.4∗𝑡sin0.25∗𝑡 (38) and the time-varying configuration matrix switches from one mode to another. They are as follows: 𝐽¹=,𝐽−0.60.300.30.3−0.60.3000.3−0.60.30.300.3−0.6₂=.−0.90.30.30.30.3−0.90.30.30.30.3−0.90.30.30.30.3−0.9 (39) For the formation controller (5), let 𝑆_𝑎=8.1, 𝐿_𝑎=0.31, 𝑆_𝑟=0.69, 𝐿_𝑅=0.3, and 𝛿=3, 𝜌=0.5 in obstacle-avoiding function (34).

According to Figures 2 and 3, one can demonstrate the control laws presented in this paper realize the formation keeping while avoiding collision with obstacles (marked by black stars in the figures) in complex environment. For the sake of measuring and visualising the formation effectiveness, we introduce the following formation error: 𝑒(𝑡)=^𝑁−1𝑁_{𝑖=1𝑗>𝑖}𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)−Δ^𝑞_𝑖𝑗, (40) which is shown in Figure 4. Then, the sensitivity of obstacle-avoiding function is investigated in Figure 5, which depicts the curves of left term in inequality (13) under different repulsion coefficient 𝜌 and repulsion rang 𝛿. From Figure 5, one can verify the fact that the performance index 𝛾 in inequality (13) reflects the sensitivity of HMAS towards obstacles; that is, smaller 𝛾 requires HMAS to be less active (smaller 𝜌 and 𝛿) towards obstacle.

Figure 2 

Formation of MAS without obstacle-avoidance functions case.

Figure 3 

Formation of MAS with obstacle-avoidance functions case.

Figure 4 

Formation error of MAS controlled by obstacle-avoidance functions.

Figure 5 

Investigation of parameters 𝛾, 𝜌, and 𝛿.

Next, in an attempt to demonstrate the effectiveness of obstacle-avoidance function (36), a rectangular obstacle stands in the way of the HMAS path. Figures 6 and 7 show that the HMAS steers around the obstacle smoothly with the help of obstacle-avoidance functions and keeps a predefined formation in the whole process, as well as the corresponding formation error.

Figure 6 

Obstacle avoidance of MAS in 2D.

Figure 7 

Formation error for obstacle avoidance of MAS in 2D.

In many practical situations, the HMAS occasionally encounters the trench-shape obstacles which are impossible for the whole system keeping original shape to pass through. In such case, the HMAS has to transform formation to a feasible one. In this example, the task is to transition from a square formation to a straight line to pass the trench safely. When the HMAS detects the trench existed in the tracking path, specified task is triggered and corresponding formation is determined according to the predetermined task-formation scheme. As shown in Figure 8, the HMAS breaks formation, goes in a straight line when it meets the trench, and transforms back into the original formation after passing the trench.

Figure 8 

Formation change for HMAS in the presence of trench-shaped obstacle.

6. Conclusion

In this paper, new formation and obstacle-avoidance protocols of multiagent systems are presented. A notion of 𝐻_∞ formation has been first defined to characterize the performance of obstacle-avoiding, and the 𝐻_∞ performance index is concreted as a sensitivity of obstacle-avoidance in this paper. Then a hybrid formation controller with a task set and a formation set is introduced to handle distinct obstacles. By designing diverse task-formation mapping, the HMAS can accomplish various complex missions. Then obstacle-avoidance functions using potential field model are specified to realize multiagent systems avoiding arbitrarily-shaped obstacles on the path. According to the simulation results, not only the HMAS can steer around the obstacles with proposed approach, but also the reconfiguration of formation can be achieved in the complex environment.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61174158, 61034004, 51075306, and 61272271, Special Financial Grant from the China Postdoctoral Science Foundation (no. 201104286), China Postdoctoral Science Foundation funded project 2012M510117, Natural Science Foundation Program of Shanghai (no. 12ZR1434000), and the Fundamental Research Funds for the Central Universities.