Sliding Mode Cooperative Control for Multirobot Systems: A Finite-Time Approach

Abstract

Finite-time stability in dynamical systems theory involves systems whose trajectories converge to an equilibrium state in finite time. In this paper, we use the notion of finite-time stability to apply it to the problem of coordinated motion in multiagent systems. We consider a group of agents described by Euler-Lagrange dynamics along with a leader agent with an objective to reach and maintain a desired formation characterized by steady-state distances between the neighboring agents in finite time. We use graph theoretic notions to characterize communication topology in the network determined by the information flow directions and captured by the graph Laplacian matrix. Furthermore, using sliding mode control approach, we design decentralized control inputs for individual agents that use only data from the neighboring agents which directly communicate their state information to the current agent in order to drive the current agent to the desired steady state. We further extend these results to multiagent systems involving underactuated dynamical agents such as mobile wheeled robots. For this case, we show that while the position variables can be coordinated in finite time, the orientation variables converge to the steady states asymptotically. Finally, we validate our results experimentally using a wheeled mobile robot platform.

1. Introduction

Finite-time stability of a dynamical system involves trajectories that converge to the desired equilibrium state in finite time and remain there for all further times. As a result, such systems do not possess the property of uniqueness of solutions in backward time, and thus their dynamics are non-Lipschitz. Sufficient conditions for finite-time stability were rigorously studied in [1, 2] using Hölder continuous Lyapunov functions. Finite-time stabilization of second-order systems was considered in [3, 4]. More recently, researchers have considered finite-time stabilization of higher-order systems [5] as well as finite-time stabilization using output feedback [6].

Sufficient conditions for finite-time stability developed in [1] were further extended to finite-time stability and stabilization of sets for nonlinear dynamical systems in [7]. In this paper, we use these results in the context of coordination in multiagent systems. We consider specifically a group of agents characterized by Euler-Lagrange dynamics along with a leader agent with an objective to reach and maintain the desired formation in finite time. Using graph-theoretic notions [8, 9], we describe the communication topology in the network of agents characterized by the information share among the agents. Based on the assumption of a weakly connected graph, we develop a directed graph Laplacian matrix whose properties provide the basis for defining the generalized error states. Furthermore, we design decentralized sliding mode controllers for individual vehicles to drive the error states of the entire multiagent system to the origin in finite time. We show that these sliding mode controllers use only information from the neighboring agents that directly communicate their generalized position and velocity information to the current agent.

Recently, finite-time consensus was studied in [10] for single integrator systems with an objective to reach the state of equipartition [11] (or consensus), that is, equal steady states for all agents. In contrast with [10], our approach is not limited by a particular steady state configuration. In fact, the developed framework places no limitation on the motion of the leader and the structure of the desired formation and includes, as a special case, all of the above types of coordinated motion such as flocking, cyclic pursuit, and rendezvous. Furthermore, we have extended our results to multiagent systems composed of underactuated dynamical agents.

We use sliding mode control technique [12] to design decentralized controllers for individual agents to reach the desired formation in finite time. Traditionally, sliding mode control is based on defining the sliding surface as a function of the system states and using finite-time stability theory for sets to ensure that all closed-loop systems trajectories reach this surface in finite time [13–17]. The smooth surface is usually designed in such a way that the closed-loop dynamics restricted to the surface is exponentially stable which ensures that trajectories, after reaching the surface, slide along it to the origin exponentially. The main disadvantage of sliding mode control is chattering around the surface during the sliding phase due to the use of discontinuous sign function which, in practice, is normally approximated by a continuous function. In this paper, we briefly review the results of [18] where finite-time coordination in multiagent systems is achieved by designing nonsmooth sliding surfaces to obtain non-Lipschitzian closed-loop dynamics restricted to the surface which leads to the convergence of the system trajectories to the origin in finite time. It appears that the sliding mode control designed for such nonsmooth surfaces can be unbounded in a certain region of the state space. In order to overcome this difficulty, we design auxiliary (smooth) sliding surfaces for the initial conditions in this region such that the sliding mode control predicated on these smooth surfaces is bounded. We show that the trajectories starting in this region will inevitably leave it, and hence switching from the sliding mode control designed for smooth surfaces to the one designed for nonsmooth surfaces guarantees convergence of the closed-loop system trajectories to the desired steady states in finite time.

The main focus of this paper is on extending the above framework to multiagent systems composed of underactuated dynamical agents. We use wheeled mobile robot model of an agent to show that while the position variables can be coordinated in finite time, the orientation variables can only be shown to converge to the steady state asymptotically. We demonstrate the efficacy of our approach using an example involving eight planar mobile robots and the leader with given communication topology and the formation structure. In addition, we provide an experimental validation of our results using wheeled mobile robot platform. Overall, the framework reveals fast converging times without excessive control effort.

2. Mathematical Preliminaries

In this section, we introduce notation and definitions and present some key results needed for developing the main results of this paper. Let ℝ denote the set of real numbers, let ℝ^𝑛 denote the set of 𝑛×1 column vectors, let (⋅)^𝑇 denote transpose, and let (⋅)⁻¹ denote matrix inverse. Furthermore, let 𝒟^∘, 𝒟, and 𝜕𝒟 denote the interior, the closure, and the boundary of the set 𝒟⊂ℝ^𝑛, respectively. Finally, we write 𝑉^′(𝑥) for the Fréchet derivative of 𝑉 at 𝑥, ‖⋅‖_∞ for ℓ_∞ vector/matrix norm, and 𝐞∈ℝ^𝑛 for the ones vector of order 𝑛, that is, 𝐞≜[1,…,1]^T.

Next, consider the nonlinear dynamical system given by 𝑥̇(𝑡)=𝑓(𝑥(𝑡)),𝑥(0)=𝑥₀,𝑡≥0, (1) where 𝑥(𝑡)∈𝒟⊆ℝ^𝑛,  𝑡≥0, is the system state vector, 𝒟 is an open set, 0∈𝒟, 𝑓(0)=0, and 𝑓(⋅) is continuous on 𝒟. We denote by 𝑠^𝑥(𝑡) the solution to (1) with the initial condition at 𝑥∈𝒟. We assume that (1) possesses unique solutions in forward time for all initial conditions except possibly the origin. Sufficient conditions for forward uniqueness in the absence of Lipschitz continuity can be found in [19], [20, Section  10], [21], and [22, Section  1].

Definition 1 (see [7]). Consider the nonlinear dynamical system (1). Let 𝒟₀⊂𝒟 be a positively invariant set with respect to (1). 𝒟₀ is finite-time stable if there exist an open neighborhood 𝒩⊆𝒟 of 𝒟₀ and a function 𝑇∶𝒩⧵𝒟₀→(0,∞), called the settling-time function, such that the following statements hold.

(i) Finite-Time Convergence. For every 𝑥∈𝒩⧵𝒟₀, 𝑠^𝑥(𝑡) is defined on [0,𝑇(𝑥)), 𝑠^𝑥(𝑡)∈𝒩⧵𝒟₀ for all 𝑡∈[0,𝑇(𝑥)), and lim_{𝑡→𝑇(𝑥)}dist(𝑠(𝑥,𝑡),𝒟₀)=0.

(ii) Lyapunov Stability. For every open neighborhood 𝒪₁⊆𝒩 of 𝒟₀, there exists an open neighborhood 𝒪₂⊆𝒪₁ of 𝒟₀ such that for every 𝑥∈𝒪₂⧵𝒟₀, 𝑠(𝑡,𝑥)∈𝒪₁ for all 𝑡∈[0,𝑇(𝑥)).

Finally, 𝒟₀ is globally finite-time stable if it is finite-time stable with 𝒩=𝒟=ℝ^𝑛.

The next result provides sufficient conditions for finite-time stability of an invariant set.

Theorem 2 (see [7]). Consider the nonlinear dynamical system (1) and let 𝒟₀⊂𝒟 be an invariant set with respect to (1). Assume there exists a continuously differentiable function 𝑉∶𝒟→ℝ and real numbers 𝑐>0 and 𝛼∈(0,1) such that 𝑉(𝑥)=0, 𝑥∈𝒟₀, 𝑉(𝑥)>0, 𝑥∈𝒟⧵𝒟₀, and 𝑉^′(𝑥)𝑓(𝑥)≤−𝑐(𝑉(𝑥))^𝛼,𝑥∈𝒟. (2) Then, 𝒟₀ is finite-time stable. Moreover, if 𝒩 is as in Definition 1 and 𝑇∶𝒩→[0,∞) is the settling-time function, then 𝑇𝑥₀≤1𝑐()𝑉𝑥1−𝛼^01−𝛼,𝑥₀∈𝒩, (3) and 𝑇(⋅) is continuous on 𝒩. If, in addition, 𝒟=ℝ^𝑛, 𝑉(⋅) is radially unbounded, and (2) holds on ℝ^𝑛, then 𝒟₀ is globally finite-time stable.

Next, we present some basic definitions and results from the graph theory needed in this paper. For more details, see [8, 23, 24].

A directed graph 𝐺 of order 𝑛 is a pair (𝒱,ℰ), where 𝒱 (vertex set) is a set of 𝑛 elements called vertices (or nodes) and ℰ (edge set) is a set of ordered pairs of vertices called edges. For 𝑢,𝑣∈𝒱(𝐺), the ordered pair (𝑢,𝑣)∈ℰ(𝐺) denotes an edge from 𝑢 to 𝑣. We assume that 𝑢≠𝑣 for any (𝑢,𝑣)∈ℰ(𝐺), that is, the graph has no self-loops. If (𝑢,𝑣)∈ℰ(𝐺) implies (𝑣,𝑢)∈ℰ(𝐺) for any (𝑢,𝑣)∈ℰ(𝐺), then the graph is called undirected graph or simply graph.

The adjacency matrix of 𝐺 is defined as 𝐴={𝑎_𝑖𝑗},𝑖,𝑗=1,…,𝑛, where 𝑎_𝑖𝑗,𝑖≠𝑗 is equal to one if (𝑢_𝑗,𝑢_𝑖)∈ℰ(𝐺) for 𝑢_𝑖,𝑢_𝑗∈𝒱(𝐺) and is equal to zero otherwise. Since, 𝐺 has no self-loops, 𝑎_𝑖𝑖=0,𝑖=1,…,𝑛. The in-degree, respectively out-degree, of a vertex 𝑢, denoted by 𝑑_in(𝑢), respectively 𝑑_out(𝑢), is the number of edges (𝑣,𝑢), respectively (𝑢,𝑣), where (𝑣,𝑢)∈ℰ(𝐺), respectively (𝑢,𝑣)∈ℰ(𝐺). Moreover, the in-degree matrix of the directed graph 𝐺, respectively out-degree matrix of the directed graph 𝐺, denoted by 𝐷_in, respectively 𝐷_out, is defined as a diagonal matrix with 𝑑_in(𝑢), respectively 𝑑_out(𝑢), for all 𝑢∈𝒱(𝐺) as its diagonal elements.

A path on graph 𝐺 of length 𝑚 from 𝑢₀∈𝒱(𝐺) to 𝑢_𝑚∈𝒱(𝐺) is an ordered set of distinct vertices {𝑢₀,…,𝑢_𝑚} such that, for any pair of consecutive vertices, (𝑢_𝑖−1,𝑢_𝑖)∈ℰ(𝐺) for all 𝑖=1,…,𝑚. A directed path on a directed graph 𝐺 of length 𝑚 from 𝑢₀∈𝒱(𝐺) to 𝑢_𝑚∈𝒱(𝐺) is an ordered set of distinct vertices {𝑢₀,…,𝑢_𝑚} such that, for any directed pair of consecutive vertices, (𝑢_𝑖−1,𝑢_𝑖)∈ℰ(𝐺) for all 𝑖=1,…,𝑚. A graph 𝐺 is connected if there exists a path between any two of its vertices. A directed graph 𝐺 is strongly connected if there exists a directed path between any two of its vertices. Also, a directed graph 𝐺 is weakly connected if it is not strongly connected and its associated undirected graph obtained by removing directions on edges is connected. For any vertex 𝑢_𝑖∈𝒱(𝐺), the reachable set ℛ_𝑖 is a set containing 𝑢_𝑖 and all vertices 𝑢_𝑗∈𝒱(𝐺) such that there is a directed path from 𝑢_𝑖 to 𝑢_𝑗. The reachable set ℛ⊂𝒱(𝐺) is called maximal if there exists a vertex 𝑢∈ℛ, such that ℛ is the biggest reachable set in 𝒱(𝐺) containing 𝑢. A maximal reachable set is called a reach.

The Kirchhoff matrix or directed Laplacian of a directed graph is defined by 𝐿=𝐷_in−𝐴∈ℝ^𝑛×𝑛, where 𝐷_in is the in-degree matrix and 𝐴 is the adjacency matrix of the graph. The normalized directed Laplacian denoted by ℒ∈ℝ^𝑛×𝑛 is obtained by dividing each row of 𝐿 by the corresponding diagonal element of 𝐷_in, if it is not zero. If an element of 𝐷_in is zero, then the corresponding row of 𝐿 consists of only zero elements.

Proposition 3. Let 𝐺 be a graph of order 𝑛 and let 𝐿 denote its Laplacian matrix. Then, zero is an eigenvalue of 𝐿 and the associated eigenvector is 𝐞=[1,…,1]^T.

Proof. Since every row of 𝐿 sums up to zero, it follows that 𝜆=0 and 𝜂=𝐞 is a valid solution to (𝜆𝐼_𝑛−𝐿)𝜂=0 which proves the result.

Proposition 4 (see [25]). Let 𝐺 be a directed graph, let 𝐿 denote its directed Laplacian matrix, and suppose that 𝐺 has 𝑘 reaches. Then, the algebraic and geometric multiplicity of the zero eigenvalue of 𝐿 is equal to 𝑘.

Corollary 5. Let 𝐺 be a strongly connected directed graph and let 𝐿 denote its directed Laplacian matrix. Then, the algebraic and geometric multiplicity of the zero eigenvalue of 𝐿 is equal to one.

Proof. Since 𝐺 is strongly connected, it follows that 𝐺 has only one reach which includes all vertices of 𝐺. Now, the result is a direct consequence of Proposition 4.

3. Coordination Control for Euler-Lagrange Systems

In this section, we briefly review a general framework for coordination control of a set of fully actuated Euler-Lagrange systems in pursuit of a leader [18]. We design specifically decentralized sliding mode controllers for individual agents that guarantee finite-time coordination. Consider a set of 𝑚 agents whose individual dynamics are given by 𝑀_𝑖𝑞_𝑖(𝑡)𝑞̈_𝑖(𝑡)=𝑓_𝑖𝑞_𝑖(𝑡),𝑞̇_𝑖(𝑡)+𝑢_𝑖(𝑡)+𝑔_𝑖𝑞_𝑖(𝑡),𝑞̇_𝑖(𝑡),𝑞_𝑖(0)=𝑞_0𝑖,𝑞̇_𝑖(0)=𝑞̇_0𝑖,𝑡≥0,𝑖=1,…,𝑚, (4) where 𝑞_𝑖∈ℝ^𝑛 is the vector of generalized coordinates of the 𝑖th agent, 𝑀_𝑖(𝑞_𝑖)∈ℝ^𝑛×𝑛 is the positive definite inertia matrix of the 𝑖th agent, 𝑢_𝑖∈ℝ^𝑛,𝑡≥0, is the control input for the 𝑖th agent, 𝑓_𝑖(𝑞_𝑖,𝑞̇_𝑖)∈ℝ^𝑛 is the vector of Coriolis, centrifugal, conservative, and nonconservative forces acting on the 𝑖th agent, and 𝑔_𝑖(𝑞_𝑖,𝑞̇_𝑖)∈ℝ^𝑛 represents the vector of bounded uncertainties and disturbances affecting the dynamics of the 𝑖th agent. Alternatively, (4) can be written as 𝑞̈_𝑖(𝑡)=𝑓_𝑖𝑞_𝑖(𝑡),𝑞̇_𝑖(𝑡)+𝑢_𝑖(𝑡)+𝑔_𝑖𝑞_𝑖(𝑡),𝑞̇_𝑖(𝑡),𝑞_𝑖(0)=𝑞_0𝑖,𝑞̇_𝑖(0)=𝑞̇_0𝑖,𝑡≥0,𝑖=1,…,𝑚, (5) where 𝑓_𝑖𝑞_𝑖,𝑞̇_𝑖≜𝑀_𝑖⁻¹𝑞_𝑖𝑓_𝑖𝑞_𝑖,𝑞̇_𝑖,𝑢_𝑖≜𝑀_𝑖⁻¹𝑞_𝑖𝑢_𝑖,𝑔_𝑖𝑞_𝑖,𝑞̇_𝑖≜𝑀_𝑖⁻¹𝑞_𝑖𝑔_𝑖𝑞_𝑖,𝑞̇_𝑖. (6) We associate with the network of agents, including the leader agent, the directed graph 𝐺 and use subscripts 1,…,𝑚 to refer to the agents and “L” to refer to the leader agent. Next, define the set 𝒥_𝑖 given by 𝒥_𝑖⊂{L,1,…,𝑚}⧵{𝑖},𝑖=1,…,𝑚, (7) to denote the set of all agents whose position and velocity vectors are available for the 𝑖th agent, that is, (𝑗,𝑖)∈ℰ(𝐺) for all 𝑗∈𝒥_𝑖. We assume that the directed graph 𝐺 of the ensemble of agents is such that there exists at least one directed path from the leader to any agent, which implies that 𝐺 is weakly connected. Note that the leader, where the information flow starts, does not sense any agent, that is, there is no directed path from any agent to the leader. Moreover, we assume that the communication links between all agents remain unchanged. Thus, the graph topology of the agent network is static. The normalized directed Laplacian matrix ℒ∈ℝ^{(𝑚+1)×(𝑚+1)} of the graph is given by ℒ_𝑖𝑗=−11,𝑖=𝑗,𝑖≠L,𝒥_𝑖,𝑗∈𝒥_𝑖,𝑖≠L,0,otherwise, (8) where |𝒥_𝑖| is the cardinality of 𝒥_𝑖,𝑖=1,…,𝑚, which is the number of elements in 𝒥_𝑖. Note that, since the leader does not sense any agent, the first row of the Laplacian matrix is zero. Alternatively, the Laplacian matrix (8) can be rewritten as ℒ=00_1×𝑚𝐿₁𝐿₂, (9) where 𝐿₁=ℒ_1L,…,ℒ_𝑚L^T,𝐿₂=ℒ₁₁⋯ℒ_1𝑚ℒ⋮⋱⋮_𝑚1⋯ℒ_𝑚𝑚, (10) and ℒ_𝑖𝑗,𝑖=1,…,𝑚,𝑗=L,1,…,𝑚, are given by (8). The next lemma presents the key feature of the matrix 𝐿₂∈ℝ^𝑚×𝑚.

Lemma 6 (see [18]). Let the connectivity graph 𝐺 have the normalized directed Laplacian matrix ℒ∈ℝ^{(𝑚+1)×(𝑚+1)} given by (9). Then, 𝐿₂∈ℝ^𝑚×𝑚 is invertible.

We consider an objective where the agents whose dynamics are given by (4) are required to achieve and maintain coordinated motion with respect to the leader in finite time. We consider specifically a formation leader whose position and velocity profiles are known functions of time and are given by 𝑞_L(𝑡), 𝑡≥0, and 𝑞̇_L(𝑡), 𝑡≥0. In this case, the desired steady state value of the 𝑖th agent relative position with respect to the 𝑗th agent is uniquely defined by a known vector 𝑙_𝑖𝑗(𝑡)∈ℝ^𝑛,𝑡≥0,𝑖,𝑗=L,1,…,𝑚, characterizing the time varying difference between the 𝑖th and the 𝑗th agents' generalized position vectors. Thus, the error variable that needs to be driven to zero is given by 𝑒_𝑖𝑗(𝑡)≜𝑞_𝑖(𝑡)−𝑞_𝑗(𝑡)−𝑙_𝑖𝑗(𝑡),𝑖,𝑗=L,1,…,𝑚,𝑡≥0, (11) where 𝑒_𝑖𝑗(𝑡)∈ℝ^𝑛,𝑡≥0 is the position error of the 𝑖th agent with respect to the 𝑗th agent. Note that, at this point, 𝑒_𝑖𝑗,𝑖,𝑗=L,1,…,𝑚, are defined regardless of whether 𝑗∈𝒥_𝑖 or not. Furthermore, we introduce a generalized error state for the 𝑖th agent as 𝑧_𝑖(𝑡)≜𝑞_𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖}𝑞_𝑗(𝑡)+𝑙_𝑖𝑗(𝑡),𝑖=1,…,𝑚. (12) Note that in contrast with (11), in (12), the generalized position 𝑞_𝑖,𝑖=1,…,𝑚, is compared to only the generalized positions 𝑞_𝑗, where 𝑗∈𝒥_𝑖.

Lemma 7 (see [18]). Consider the connectivity graph 𝐺 with the normalized directed Laplacian matrix ℒ∈ℝ^{(𝑚+1)×(𝑚+1)} given by (9). Define 𝑧≜[𝑧^T₁,…,𝑧^T_𝑚]^T, where 𝑧_𝑖∈ℝ^𝑛 is given by (12) and 𝑒_𝑗≜[𝑒^T_1𝑗,…,𝑒^T_𝑚𝑗]^T,𝑗=L,1,…,𝑚. Then, the following statements hold.(i) If 𝑧=0, then 𝑒_𝑖𝑗=0,𝑖=1,…,𝑚,𝑗=L,1,…,𝑚. (ii) If 𝑒_𝑖𝑗=0,𝑖=1,…,𝑚,𝑗∈𝒥_𝑖, then 𝑧=0.

The generalized error dynamics are obtained by taking the second time derivative of (12) and are given by 𝑧̈_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝑙_𝑖̈(𝑡)+𝑓_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑓_𝑗𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)+ℒ_𝑖L𝑞̈_L(𝑡)+ℎ_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡),𝑧_𝑖(0)=𝑧_0𝑖,𝑧̇_𝑖(0)=𝑧̇_0𝑖,𝑖=1,…,𝑚,𝑡≥0, (13) where 𝑣_𝑖(𝑡)≜𝑢_𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑢_𝑗(𝑡)𝑙,𝑖=1,…,𝑚,_𝑖(𝑡)≜1𝒥_{𝑖𝑗∈𝒥𝑖}𝑙_𝑖𝑗(𝑡)ℎ,𝑖=1,…,𝑚,_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)≜𝑔_𝑖𝑞_𝑖(𝑡),𝑞̇_𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑔_𝑗𝑞_𝑗(𝑡),𝑞̇_𝑗(𝑡),𝑖=1,…,𝑚. (14)(15) Equations (12), (13), and (14) for individual agents can be rewritten using Kronecker algebra to represent the multiagent formation consisting of 𝑚 agents as follows: 𝑧(𝑡)𝑣=𝐿₂⊗𝐼_𝑛𝑞(𝑡)+𝐿₁⊗𝑞_L(𝑡)(𝑡),𝑧̈(𝑡)(𝑡)−𝑙̈(𝑡)+𝐿−𝑙=𝑣₂⊗𝐼_𝑛𝐹𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)+𝐿₁⊗𝑞̈_L(𝑡)+ℎ𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡),𝑧(0)=𝑧₀,𝑧̇(0)=𝑧̇₀𝑣(𝑡)=𝐿,𝑡≥0,₂⊗𝐼_𝑛𝑢(𝑡), (16)(17)(18) where 𝑞(𝑡)≜𝑞^T₁(𝑡),…,𝑞^T_𝑚(𝑡)^T,𝑙(𝑡)≜𝑙^T₁(𝑡),…,𝑙^T_𝑚(𝑡)^T,𝑣(𝑡)≜𝑣^T₁(𝑡),…,𝑣^T_𝑚(𝑡)^T,𝑢(𝑡)≜𝑢^T₁(𝑡),…,𝑢^T_𝑚(𝑡)^T,𝐹𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)≜𝑓^T₁𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡),…,𝑓^T_𝑚𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)^T,ℎ𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)≜ℎ^T₁𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡),…,ℎ^T_𝑚𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)^T. (19)

Next, we present a general framework for the sliding mode control design that guarantees finite-time coordination for the multiagent system (17). Consider a vector function 𝜎_𝑖∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 given by 𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖=𝑧̇_𝑖+𝐶_𝑖𝑆_𝑖𝑧_𝑖𝑧_𝑖^1/2,𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛, (20) where 𝐶_𝑖=diag[𝑐_𝑖1,…,𝑐_𝑖𝑛], 𝑐_𝑖𝑗>0, 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, 𝑆_𝑖(𝑧_𝑖)≜diag[sign(𝑧_𝑖1),…,sign(𝑧_𝑖𝑛)], 𝑧_𝑖𝑗∈ℝ, 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, is the 𝑗th component of 𝑧_𝑖∈ℝ^𝑛, |𝑧_𝑖|^1/2≜[|𝑧_𝑖1|^1/2,…,|𝑧_𝑖𝑛|^1/2]^T. We define the 𝑖th sliding surface as the null space of 𝜎_𝑖(⋅,⋅), that is, 𝒮_𝑖≜𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛∶𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖=0. (21) As in standard sliding mode control theory, the control law 𝑣_𝑖 is calculated by setting 𝜎̇_𝑖(𝑧_𝑖,𝑧̇_𝑖)=0 for the nominal system and adding a signum function to address uncertainties. Using (13) and (20) specifically, we set 𝑣_𝑖(𝑡)=𝑙_𝑖̈(𝑡)−𝑓_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)+1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑓_𝑗𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−ℒ_𝑖L𝑞̈_L(𝑡)−12𝐶_𝑖𝑑_𝑖𝑧_𝑖,𝑧̇_𝑖−𝐾_𝑖sign𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖,𝑧_𝑖,𝑧̇_𝑖∈ℳ_𝑖,𝑖=1,…,𝑚,𝑧_𝑗,𝑧̇_𝑗∈ℝ^𝑛×ℝ^𝑛,𝑗=1,…,𝑚,𝑗≠𝑖, (22) where 𝐾_𝑖=[𝑘_𝑖1,…,𝑘_𝑖𝑛], 𝑘_𝑖𝑗∈ℝ, 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, sign𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖≜sign𝜎_𝑖1𝑧_𝑖,𝑧̇_𝑖⋮sign𝜎_𝑖𝑛𝑧_𝑖,𝑧̇_𝑖,𝑑_𝑖𝑧_𝑖,𝑧̇_𝑖≜𝑧̇_𝑖1𝑧_𝑖1^−1/2⋮𝑧̇_𝑖𝑛𝑧_𝑖𝑛^−1/2, (23)𝜎_𝑖𝑗(⋅,⋅), 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, is the 𝑗th component of 𝜎_𝑖(⋅,⋅), and ℳ_𝑖⊂ℝ^𝑛×ℝ^𝑛 is the set where 𝑑_𝑖(⋅,⋅) is bounded. Define ℳ_𝑖, 𝑖=1,…,𝑚, as ℳ_𝑖≜𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛∶𝑑_𝑖𝑧_𝑖,𝑧̇_𝑖∞≤𝜆_𝑖, (24) where 𝜆_𝑖≜𝐶_𝑖∞+𝛿_𝑖,𝛿_𝑖>0. (25) Note that vector 𝑑_𝑖(⋅,⋅) and, consequently, 𝑣_𝑖 are bounded in ℳ_𝑖⊂ℝ^𝑛×ℝ^𝑛 while being unbounded in the complement of ℳ_𝑖. Thus, for the complement of ℳ_𝑖, we design an auxiliary sliding mode controller below. First, however, we establish sufficient conditions under which ℳ_𝑖 is positively invariant with respect to (13).

Proposition 8 (see [18]). Consider the error dynamics (13) with the feedback controller (22). If the sliding mode controller gains 𝑘_𝑖𝑗, 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, satisfy 𝑘_𝑖𝑗=𝜂_𝑖𝑗+sup_{̇(𝑧,𝑧,𝑡)∈ℝ𝑛𝑚×ℝ𝑛𝑚×ℝ}ℎ_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)_∞, (26) with 𝜂_𝑖𝑗>𝜆²_𝑖−𝑐_𝑖𝑗𝜆_𝑖2>0, (27) then the set ℳ_𝑖 defined by (24) is invariant with respect to (13).

For the case when (𝑧_𝑖,𝑧̇_𝑖)∈ℳ_𝑖,𝑖=1,…,𝑚, decentralized control input (22) can be rewritten for the entire ensemble of 𝑚 agents as 𝑣(𝑡)=𝑙̈(𝑡)−12𝐶𝑑(𝑧,𝑧)̇+𝐿₂⊗𝐼_𝑛𝐹𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−𝐿₁⊗𝑞̈_L(𝑡)(𝜎(𝑧̇,𝑧−𝐾sign𝑧,))_𝑖,𝑧̇_𝑖∈ℳ_𝑖,𝑖=1,…,𝑚, (28) where 𝐾≜block−diag𝐾₁,…,𝐾_𝑚,𝐶≜block−diag𝐶₁,…,𝐶_𝑚,𝑑(𝑧,𝑧)̇≜𝑑^T₁𝑧₁,𝑧̇₁,…,𝑑^T_𝑚𝑧_𝑚,𝑧̇_𝑚^T,𝜎(𝑧,𝑧)̇≜𝜎^T₁𝑧₁,𝑧̇₁,…,𝜎^T_𝑚𝑧_𝑚,𝑧̇_𝑚^T,(𝜎(𝑧̇≜𝜎sign𝑧,))sign₁𝑧₁,𝑧̇₁^T,…,sign𝜎_𝑚𝑧_𝑚,𝑧̇_𝑚^TT. (29) Next, for the initial conditions (𝑧_0𝑖,𝑧̇_0𝑖)∈ℝ^𝑛×ℝ^𝑛⧵ℳ_𝑖, we design an auxiliary sliding surface 𝒮_𝑖aux⊂ℝ^𝑛×ℝ^𝑛 along with the corresponding sliding mode controller. Consider specifically the vector function 𝜎_𝑖aux∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 given by 𝜎_𝑖aux𝑧_𝑖,𝑧̇_𝑖=𝑧̇_𝑖,𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛, (30) and define the sliding surface 𝒮_𝑖aux as 𝒮_𝑖aux≜𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛∶𝜎_𝑖aux𝑧_𝑖,𝑧̇_𝑖=0. (31) By setting 𝜎̇_𝑖aux(𝑧_𝑖,𝑧̇_𝑖)=0, we obtain 𝑣_𝑖aux=𝑙_𝑖̈(𝑡)−𝑓_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)+1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑓_𝑗𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−ℒ_𝑖L𝑞̈_L(𝑡)−𝐾_𝑖auxsign𝜎_𝑖aux𝑧_𝑖,𝑧̇_𝑖,𝑧_𝑖,𝑧̇_𝑖∉ℳ_𝑖𝑧,𝑖=1,…,𝑚,_𝑗,𝑧̇_𝑗∈ℝ^𝑛×ℝ^𝑛,𝑗=1,…,𝑚,𝑗≠𝑖, (32) where 𝐾_𝑖aux≜diag[𝑘_𝑖1aux,…,𝑘_𝑖𝑛aux],𝑖=1,…,𝑚, 𝑘_𝑖𝑗aux=𝜂_𝑖𝑗aux+sup_{𝑧𝑖,̇𝑧𝑖,𝑡∈ℝ𝑛×ℝ𝑛×ℝ}ℎ𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)_∞,𝑗=1,…,𝑛, (33) with 𝜂_𝑖𝑗aux>0. The next result establishes that the sliding mode controllers (22) and (32) guarantee that the error states (𝑧,𝑧)̇ of (17) converge to the origin in finite time, thus ensuring finite-time coordination among the agents.

Theorem 9 (see [18]). Consider the error dynamics given by (13). The sliding mode control law given by 𝑣_𝑖(𝑡)=𝑙_𝑖̈(𝑡)−𝑓_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡+1)𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑓_𝑗𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−ℒ_𝑖L𝑞̈_L(𝑡)−12𝐶_𝑖𝑑_𝑖𝑧_𝑖,𝑧̇_𝑖−𝐾_𝑖sign𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖,𝑧_𝑖,𝑧̇_𝑖∈ℳ_𝑖,𝑧_𝑗,𝑧̇_𝑗∈ℝ^𝑛×ℝ^𝑛,𝑙̈𝑗=1,…,𝑚,𝑗≠𝑖,_𝑖(𝑡)−𝑓_𝑖𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)+1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝑓_𝑗𝑧,𝑧,̇𝑞_L(𝑡),𝑞̇_L(𝑡)−ℒ_𝑖L𝑞̈_L(𝑡)−𝐾_𝑖auxsign𝜎_𝑖aux𝑧_𝑖,𝑧̇_𝑖,𝑧_𝑖,𝑧̇_𝑖∉ℳ_𝑖,𝑧_𝑗,𝑧̇_𝑗∈ℝ^𝑛×ℝ^𝑛,𝑗=1,…,𝑚,𝑗≠𝑖, (34) where 𝑖=1,…,𝑚, 𝜎_𝑖(⋅,⋅) and 𝜎_𝑖aux(⋅,⋅) are given by (20) and (30), respectively, the entries of control gain matrices 𝐾_𝑖∈ℝ^𝑛×𝑛 satisfy (26) and (27), and the entries of control gain matrices 𝐾_𝑖aux∈ℝ^𝑛×𝑛 satisfy (33), guarantees finite-time convergence of the error state (𝑧,𝑧)̇ of (17) to the origin, and hence, the ensemble of 𝑚 agents and a leader reaches the desired coordination in finite time.

Note that each decentralized controller 𝑣_𝑖,𝑖=1,…,𝑚, given by (34) uses only local information from the 𝑖th agent and the neighboring agents that directly communicate their state information to the 𝑖th agent. Furthermore, the earlier statement that the leader does not receive information from the agents is not restrictive. In fact, if the agents do communicate their state values to the leader, then as long as the leader's position and velocity are known by at least one agent, the analysis will be the same as presented above.

Remark 10. Sliding mode control always results in chattering around the surfaces due to the signum function in (34). In practice, the discontinuous sign function is often approximated by a continuous function such as the hyperbolic tangent or by a linear saturation function to avoid chattering.

4. Finite-Time Coordination for Underactuated Systems

In this section, we extend the results of Section 3 to the case of underactuated mechanical systems. As an example system, we consider a model of a mobile robot subject to a nonholonomic constraint such as the no-slip condition in the normal direction to the motion path. The equations of motion for a mobile robot shown in Figure 1 are given by [26] 𝑥̇_𝑖(𝑡)=𝑣_𝑥𝑖(𝑡)cos𝜃_𝑖(𝑡)−𝑑𝜔_𝑖(𝑡)sin𝜃_𝑖(𝑡),𝑥_𝑖(0)=𝑥_0𝑖𝑦̇,𝑖=1,…,𝑝,𝑡≥0,_𝑖(𝑡)=𝑣_𝑥𝑖(𝑡)sin𝜃_𝑖(𝑡)+𝑑𝜔_𝑖(𝑡)cos𝜃_𝑖(𝑡),𝑦_𝑖(0)=𝑦_0𝑖,𝜃̇_𝑖(𝑡)=𝜔_𝑖(𝑡),𝜃_𝑖(0)=𝜃_0𝑖,𝑣̇_𝑥𝑖(𝑡)=𝑚𝑑𝑚𝜔²_𝑖(𝑡)+1𝜏𝑚𝑟^1𝑖(𝑡)+𝜏_2𝑖(𝑡)+𝑔_1𝑖𝜂_1𝑖,𝜂_2𝑖,𝑣_𝑥𝑖(0)=𝑣_0𝑥𝑖,𝜔̇_𝑖(𝑡)=−𝑚𝑑𝐼̃𝜔_𝑖(𝑡)𝑣_𝑥𝑖(𝑡)+𝑎2̃𝜏𝐼𝑟^2𝑖(𝑡)−𝜏_1𝑖(𝑡)+𝑔_2𝑖𝜂_1𝑖,𝜂_2𝑖,𝜔(0)=𝜔₀, (35)(36)(37)(38)(39) where (𝑥_𝑖,𝑦_𝑖) is the position of the 𝑖th robot's mass center 𝐶, 𝜃_𝑖 is the 𝑖th robot's orientation (see Figure 1), 𝑣_𝑥𝑖 is the tangential component of the velocity of the robot's mass center, 𝜏_1𝑖 and 𝜏_2𝑖 are differential torques applied to each wheel, 𝑔_1𝑖(⋅,⋅) and 𝑔_2𝑖(⋅,⋅) are uncertainty functions affecting the robot dynamics, 𝜂_1𝑖≜[𝑥_𝑖,𝑦_𝑖,𝜃_𝑖]^T, 𝜂_2𝑖≜[𝑣_𝑥𝑖,𝜔_𝑖]^T, 𝑑>0 is the distance between the mass center 𝐶 and mid-axle point 𝐴, 𝑚 is the total mass of the robot, 𝐼 is the moment of inertia of the robot about the axis orthogonal to the plane passing through 𝐶, 𝑎 is the length of robot's axle, 𝑚≜𝑚+2𝐽/𝑟² and 𝐼̃≜𝐼+𝑚𝑑²+𝑎²/𝑟²𝐽,  𝐽 is the rotational inertia of the wheels about their axis of rotation, 𝑟 is the radius of the wheels, and 𝑝 is the number of mobile robots. Differentiating (35) and (36) yields 𝑥̈_𝑖(𝑡)=𝑓_1𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡)+𝑢_1𝑖(𝑡)cos𝜃_𝑖(𝑡)−𝑑𝑢_2𝑖(𝑡)sin𝜃_𝑖(𝑡)+𝑔_1𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡),𝑥_𝑖(0)=𝑥_0𝑖,𝑥̇_𝑖(0)=𝑥̇_0𝑖,𝑦̈𝑖=1,…,𝑝,𝑡≥0,_𝑖(𝑡)=𝑓_2𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡)+𝑢_1𝑖(𝑡)sin𝜃_𝑖(𝑡)+𝑑𝑢_2𝑖(𝑡)cos𝜃_𝑖(𝑡)+𝑔_2𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡),𝑦_𝑖(0)=𝑦_0𝑖,𝑦̇_𝑖(0)=𝑦̇_0𝑖, (40) where 𝑢_1𝑖≜1𝜏𝑚𝑟^1𝑖+𝜏_2𝑖𝑢,𝑖=1,…,𝑝,_2𝑖≜𝑎2̃𝜏𝐼𝑟^2𝑖−𝜏_1𝑖𝑓,𝑖=1,…,𝑝,_1𝑖𝜂_1𝑖,𝜂_2𝑖≜𝑚𝑚−1𝑑𝜔²_𝑖cos𝜃_𝑖+𝑚𝑑²𝐼̃−1𝜔_𝑖𝑣_𝑥𝑖sin𝜃_𝑖𝑓,𝑖=1,…,𝑝,_2𝑖𝜂_1𝑖,𝜂_2𝑖≜𝑚𝑚−1𝑑𝜔²_𝑖sin𝜃_𝑖−𝑚𝑑²𝐼̃−1𝜔_𝑖𝑣_𝑥𝑖cos𝜃_𝑖𝑔,𝑖=1,…,𝑝,_1𝑖𝜂_1𝑖,𝜂_2𝑖≜𝑔_1𝑖𝜂_1𝑖,𝜂_2𝑖cos𝜃_𝑖−𝑑𝑔_2𝑖𝜂_1𝑖,𝜂_2𝑖sin𝜃_𝑖𝑔,𝑖=1,…,𝑝,_2𝑖𝜂_1𝑖,𝜂_2𝑖≜𝑔_1𝑖𝜂_1𝑖,𝜂_2𝑖sin𝜃_𝑖+𝑑𝑔_2𝑖𝜂_1𝑖,𝜂_2𝑖cos𝜃_𝑖,𝑖=1,…,𝑝. (41) Next, using the feedback linearizing controller given by 𝑢_1𝑖𝑢_2𝑖=cos𝜃_𝑖−𝑑sin𝜃_𝑖sin𝜃_𝑖𝑑cos𝜃_𝑖⁻¹−𝑓_1𝑖𝜂_1𝑖,𝜂_2𝑖+𝜈_1𝑖−𝑓_2𝑖𝜂_1𝑖,𝜂_2𝑖+𝜈_2𝑖,𝑖=1,…,𝑝, (42) where 𝜈_1𝑖∈ℝ  𝑖=1,…,𝑝, and 𝜈_2𝑖∈ℝ  𝑖=1,…,𝑝, are the new feedback control inputs, (40) can be written as 𝑥̈_𝑖(𝑡)=𝜈_1𝑖(𝑡)+𝑔_1𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡),𝑥_𝑖(0)=𝑥_0𝑖,𝑥̇_𝑖(0)=𝑥̇_0𝑖𝑦̈,𝑖=1,…,𝑝,_𝑖(𝑡)=𝜈_2𝑖(𝑡)+𝑔_2𝑖𝜂_1𝑖(𝑡),𝜂_2𝑖(𝑡,𝑦_𝑖)(0)=𝑦_0𝑖,𝑦̇_𝑖(0)=𝑦̇_0𝑖, (43) while (37) can be rewritten as 𝜃̇_𝑖(𝑡)=−1𝑑𝑥̇_𝑖(𝑡)sin𝜃_𝑖(𝑡)−𝑦̇_𝑖(𝑡)cos𝜃_𝑖(𝑡,𝜃_𝑖)(0)=𝜃_0𝑖,𝑡≥0,𝑖=1,…,𝑝. (44)

Figure 1 

Diagram of a mobile robot.

The objective is to coordinate the motion of a set of 𝑝 robots and a leader. Define the leader's position in the plane as (𝑥_L(𝑡),𝑦_L(𝑡)), 𝑡≥0, and its orientation as 𝜃_L(𝑡), 𝑡≥0, and define the generalized position vector for the leader as 𝑞_L(𝑡)=[𝑥_L(𝑡),𝑦_L(𝑡),𝜃_L(𝑡)]^T, 𝑡≥0. Note that the leader obeys the same nonholonomic constraint as all mobile robots such as no slip in the normal direction to the path. Without loss of generality, we assume that 𝑣_𝑥L(𝑡)>0,𝑡≥0, which physically implies that, during the motion of the leader, its mass center 𝐶 is always ahead of the mid-axle point 𝐴. In other words, in order for the leader robot to turn around, it does not back up the wheels, but uses the differential torques applied to the wheels while keeping 𝑣_𝑥L>0. As in Section 3, we define the set 𝒥_𝑖 given by 𝒥_𝑖⊂L,1,…,𝑝⧵{𝑖},𝑖=1,…,𝑝, (45) where “L” stands for the leader, to denote the set of all vehicles whose position and velocity vectors are available for the 𝑖th agent. We assume that the directed graph of the ensemble of vehicles is such that there exists at least one directed path from the leader to any agent. Note that the leader does not sense any vehicle. Moreover, we assume that the graph topology of the vehicle network is time invariant. The normalized directed Laplacian matrix ℒ∈ℝ^{(𝑝+1)×(𝑝+1)} of the graph is defined by (8).

Next, define the position error states as 𝑧_𝑖(𝑡)≜𝑧_𝑥𝑖(𝑡)𝑧_𝑦𝑖(𝑡)=𝑥_𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖}𝑥_𝑗(𝑡)+𝑙_𝑥𝑖𝑗𝑦_𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖}𝑦_𝑗(𝑡)+𝑙_𝑦𝑖𝑗,𝑖=1,…,𝑝,𝑡≥0, (46) where 𝑙_𝑥𝑖𝑗∈ℝ and 𝑙_𝑦𝑖𝑗∈ℝ are the steady state differences between the horizontal and vertical positions, respectively, of the 𝑖th and the 𝑗th robots. The position error dynamics are obtained by taking the second time derivative of (46) given by 𝑧̈_𝑥𝑖(𝑡)=𝜈_1𝑖(𝑡)+ℒ_𝑖L𝑥̈_L(𝑡)+𝑔_1𝑖𝑧(𝑡),𝜂₂(𝑡),𝑞_L(𝑡),𝑞̇_L(𝑡),𝑧_𝑥𝑖(0)=𝑧_0𝑥𝑖,𝑧̇_𝑥𝑖(0)=𝑧̇_0𝑥𝑖𝑧̈,𝑖=1,…,𝑝,𝑡≥0,_𝑦𝑖(𝑡)=𝜈_2𝑖(𝑡)+ℒ_𝑖L𝑦̈_L(𝑡)+𝑔_2𝑖𝑧(𝑡),𝜂₂(𝑡),𝑞_L(𝑡),𝑞̇_L(𝑡,𝑧_𝑦𝑖)(0)=𝑧_0𝑦𝑖,𝑧̇_𝑦𝑖(0)=𝑧̇_0𝑦𝑖,𝑖=1,…,𝑝, (47) where 𝑧(𝑡)≜𝑧_𝑥1(𝑡),𝑧_𝑦1(𝑡),𝜃₁(𝑡),…,𝑧_𝑥𝑝(𝑡),𝑧_𝑦𝑝(𝑡),𝜃_𝑝(𝑡)^T,𝜂₂(𝑡)≜𝜂₂₁(𝑡),…,𝜂_2𝑝(𝑡)^T,𝜈_1𝑖(𝑡)≜𝜈_1𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝜈_1𝑗(𝑡),𝜈_2𝑖(𝑡)≜𝜈_2𝑖(𝑡)−1𝒥_{𝑖𝑗∈𝒥𝑖⧵{L}}𝜈_2𝑗(𝑡). (48) Note that, for 𝑡≥0, leader’s position and orientation satisfy 𝑥̇_L(𝑡)=𝑣_𝑥L(𝑡)cos𝜃_L(𝑡)−𝑑𝜔_L(𝑡)sin𝜃_L(𝑡),𝑦̇_L(𝑡)=𝑣_𝑥L(𝑡)sin𝜃_L(𝑡)+𝑑𝜔_L(𝑡)cos𝜃_L(𝑡),𝜔_L(𝑡)1=−𝑑𝑥̇_L(𝑡)sin𝜃_L(𝑡)−𝑦̇_L(𝑡)cos𝜃_L(𝑡), (49) where 𝜔_L(𝑡)≜𝜃̇_L(𝑡), 𝑡≥0. Since, the dynamics of (47) conform with the structure of the error dynamics (13), it follows from Theorem 9 that the sliding mode control (34) guarantees that the solutions to (47) converge to the origin in finite time, and hence all robots reach the desired formation on the plane in finite time. After all robots reached the desired formation at the settling time 𝑇_𝑠, it follows from Lemma 7 that 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−𝑙_𝑥𝑖𝑗=0,𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)−𝑙_𝑦𝑖𝑗=0,𝑡≥𝑇_𝑠,𝑖,𝑗=L,1,…,𝑝, which implies that 𝑥̇_𝑖(𝑡)=𝑥̇_𝑗(𝑡),𝑦̇_𝑖(𝑡)=𝑦̇_𝑗(𝑡),𝑡≥𝑇_𝑠,𝑖,𝑗=L,1,…,𝑝. Thus, the orientation dynamics (37) for the 𝑖th robot can be rewritten as 𝜃̇_𝑖(𝑡)=−1𝑑𝑥̇_𝑗(𝑡)sin𝜃_𝑖(𝑡)−𝑦̇_𝑗(𝑡)cos𝜃_𝑖(𝑡,𝜃_𝑖)𝑇_𝑠=𝜃_𝑠𝑖,𝑡≥𝑇_𝑠,𝑖=1,…,𝑝,𝑗=L,1,…,𝑝. (50) Let 𝑒_𝜃𝑖L(𝑡)≜𝜃_𝑖(𝑡)−𝜃_L(𝑡),𝑡≥𝑇_𝑠,𝑖=1,…,𝑝, measure the orientation error between the 𝑖th robot and the leader. Using (50) with 𝑗=L, the orientation error dynamics are given by 𝑒̇_𝜃𝑖L(𝑡)1=−𝑑𝑥̇_L(𝑡)sin𝑒_𝜃𝑖L(𝑡)+𝜃_L(𝑡−𝑦̇_L)(𝑡)cos𝑒_𝜃𝑖L(𝑡)+𝜃_L(𝑡)−𝜔_L(𝑡),𝑒_𝜃𝑖L𝑇_𝑠=𝑒_{𝑠𝜃𝑖L},𝑡≥𝑇_𝑠,𝑖=1,…,𝑝. (51) Alternatively, (51) can be rewritten as 𝑒̇_𝜃𝑖L(𝑡)1=−𝑑𝑥̇_L(𝑡)sin𝑒_𝜃𝑖L(𝑡)cos𝜃_L(𝑡)+cos𝑒_𝜃𝑖L(𝑡)sin𝜃_L(𝑡)+1𝑑𝑦̇_L(𝑡)×cos𝑒_𝜃𝑖L(𝑡)cos𝜃_L(𝑡)−sin𝑒_𝜃𝑖L(𝑡)sin𝜃_L(𝑡)−𝜔_L(𝑡)1=−𝑑sin𝑒_𝜃𝑖L(𝑡)𝑥̇_L(𝑡)cos𝜃_L(𝑡)+𝑦̇_L(𝑡)sin𝜃_L(𝑡)+1𝑑cos𝑒_𝜃𝑖L(𝑡)×−𝑥̇_L(𝑡)sin𝜃_L(𝑡)+𝑦̇_L(𝑡)cos𝜃_L(𝑡)−𝜔_L(𝑡)1=−𝑑𝑣_𝑥L(𝑡)sin𝑒_𝜃𝑖L(𝑡)+𝜔_L(𝑡)cos𝑒_𝜃𝑖L(𝑡)2−1=−𝑑𝑣_𝑥L(𝑡)𝑒sin^𝜃𝑖L(𝑡)2𝑒cos^𝜃𝑖L(𝑡)2−2𝜔_L(𝑡)sin²𝑒_𝜃𝑖L(𝑡)22=−𝑑𝑒sin^𝜃𝑖L(𝑡)2𝑣_𝑥L(𝑡)𝑒cos^𝜃𝑖L(𝑡)2+𝑑𝜔_L(𝑡)𝑒sin^𝜃𝑖L(𝑡)21=−𝑑𝑣_𝑥L(𝑡)sin𝑒_𝜃𝑖L(𝑡)1+𝑑𝜔_L(𝑡)𝑣_𝑥L(𝑡)𝑒tan^𝜃𝑖L(𝑡)2,𝑒_𝜃𝑖L𝑇_𝑠=𝑒_{𝑠𝜃𝑖L},𝑡≥𝑇_𝑠,𝑖=1,…,𝑝. (52) In the next proposition, we show that the zero equilibrium point of (52) is uniformly asymptotically stable with respect to 𝑒_𝜃𝑖L.

Proposition 11. Consider the error dynamics given by (52) and define 𝒟_𝑖≜𝑒_𝜃𝑖L∈ℝ∶𝑒_𝜃𝑖L<2atan1−𝛿_𝑖𝑣^∗_𝑥L𝑑𝜔^∗_L⊂(−𝜋,𝜋),𝑖=1,…,𝑝, (53) where 𝑣^∗_𝑥L≜inf_𝑡≥0(𝑣_𝑥L(𝑡))>0, 𝜔^∗_L≜sup_𝑡≥0(|𝜔_L(𝑡)|), and 𝛿_𝑖>0, 𝑖=1,…,𝑝, is an arbitrarily small constant. Then, the zero solution of (52) is uniformly asymptotically stable with respect to 𝑒_𝜃𝑖L and with an estimate of the domain of attraction given by 𝒟_𝑖.

Proof. Note that {0} is the only equilibrium point of (52) in 𝒟_𝑖. Consider a Lyapunov function candidate given by 𝑉_𝑖𝑒_𝜃𝑖L=1−cos𝑒_𝜃𝑖L,𝑒_𝜃𝑖L∈𝒟_𝑖,𝑖=1,…,𝑝, (54) and note that 𝑉_𝑖(0)=0 and 𝑉_𝑖(𝑒_𝜃𝑖L)>0,𝑒_𝜃𝑖L∈𝒟_𝑖, 𝑖=1,…,𝑝. Furthermore, since 𝒟_𝑖⊂(−𝜋,𝜋),𝑖=1,…,𝑝, there exists 𝛾_𝑖>0,𝑖=1,…,𝑝, such that |sin𝑒_𝜃𝑖L|>𝛾_𝑖|𝑒_𝜃𝑖L|,𝑒_𝜃𝑖L∈𝒟_𝑖,𝑖=1,…,𝑝. Thus, the Lyapunov derivative along the trajectories of (52) satisfies 𝑉̇_𝑖𝑒_𝜃𝑖L(𝑡)=sin𝑒_𝜃𝑖L(𝑡)𝑒̇_𝜃𝑖L(𝑡)1=−𝑑𝑣_𝑥L(𝑡)sin²𝑒_𝜃𝑖L(𝑡)1+𝑑𝜔_L(𝑡)𝑣_𝑥L(𝑡)𝑒tan^𝜃𝑖L(𝑡)21≤−𝑑𝑣^∗_𝑥L𝛿_𝑖sin²𝑒_𝜃𝑖L(𝑡)𝛾≤−^2𝑖𝑑𝑣^∗_𝑥L𝛿_𝑖𝑒²_𝜃𝑖L(𝑡),𝑒_𝜃𝑖L∈𝒟_𝑖,𝑖=1,…,𝑝. (55) It follows from Theorem  4.1 of [27] that the zero solution to (52) is uniformly asymptotically stable with respect to 𝑒_𝜃𝑖L and with an estimate of the domain of attraction given by 𝒟_𝑖,𝑖=1,…,𝑝. This implies that 𝜃_𝑖(𝑡)→𝜃_L(𝑡) as 𝑡→∞ for all 𝑖=1,…,𝑝.

Proposition 11 along with Theorem 9 ensures finite-time coordination of the agents’ positions and asymptotic coordination of the agents’ orientations.

5. Numerical Example

In this section, we provide a numerical simulation for the case of planar coordinated motion of eight mobile robots and a leader. The desired formation and the information flow between agents are shown in Figure 2. This communication topology remains unchanged over time. For the given configuration, 𝐿₁ and 𝐿₂ matrices are given as follows: 𝐿₁=−12−12000000,𝐿₂=11−2−10000002−110000002101−2−100004−14−1411−410000−210−21100000−3−13101−301000−3−13−1311000000−2−121. (56)

Figure 2 

The desired formation of the set of eight robots and a leader with information flow topology.

Note that the directed graph of the formation satisfies the required property that there exits at least one directed path from the leader to any agent. Also, note that 𝐿₂ is invertible. Next, the leader is set to be moving counter-clockwise on a circular path of radius 0.5 according to 𝑥_L(𝑡)=0.5cos(𝜋𝑡/30), 𝑦_L(𝑡)=0.5sin(𝜋𝑡/30), 𝑡≥0. The values of control gains are set to be 𝐾_𝑖=0.5×𝐼₂ and 𝐾_𝑖aux=0.5×𝐼₂, 𝑖=1,…,8. Finally, the matrices 𝐶_𝑖∈ℝ^2×2 characterizing the design of the sliding surface for each agent are chosen to be 𝐶_𝑖=0.5×𝐼₂, 𝑖=1,…,8. In this case, 𝜆_𝑖=0.75, 𝑖=1,…,8, ensure that condition (27) is satisfied.

All robots are identical with the values of the parameters set to be 𝑚=2.83 kg, 𝐼=0.0226 kg/m², 𝐽=5.1×10⁻⁵ kg/m², 𝑎=0.315 m, 𝑑=0.078 m, and 𝑟=0.045 m. The above values correspond to the experimental wheeled mobile robot parameters which will be discussed in the next section.

For the initial positions and orientations of the robots given by 𝜂₁₁(0)=[1,−0.5,𝜋/2]^T, 𝜂₁₂(0)=[1.5,−1.5,−𝑝𝑖]^T, 𝜂₁₃(0)=[−1,1,−𝜋]^T, 𝜂₁₄(0)=[−1.5,−3.5,𝜋/2]^T, 𝜂₁₅(0)=[0.5,−3,2𝜋/3]^T, 𝜂₁₆(0)=[−5,−1,𝜋/2]^T, 𝜂₁₇(0)=[−3,−3.5,5𝜋/6]^T, and 𝜂₁₈(0)=[−5.5,−3,𝜋/3]^T and the initial velocities given by 𝜂_2𝑖=[0,0]^T,𝑖=1,…,8, Figure 3 shows the position phase portrait with robot orientations and Figure 4 shows position errors defined in (46) and orientation errors with respect to the leader for eight robots. It can be seen from Figures 3 and 4 that, with the settling time of approximately 5 seconds, the agents reach the desired formation shown in Figure 2. Furthermore, Figure 4 shows that the errors 𝑧_𝑥𝑖 and 𝑧_𝑦𝑖,𝑖=1,…,8, converge to zero faster than 𝑒_𝜃𝑖L,𝑖=1,…,8, since the former are directly controlled by the feedback control law (34) ensuring finite-time convergence, while the latter is not directly controlled but converges to zero asymptotically. Finally, Figure 5 shows the time history of the control torques acting on both wheels of each robot with 𝜏_1𝑖 and 𝜏_2𝑖 being the control torques acting on the left and the right wheels, respectively, of the 𝑖th robot.

Figure 3 

Position phase portrait with orientations of eight robots and a leader.

Figure 4 

Time history of the errors versus time.

Figure 5 

Control torques versus time.

Next, we show a comparison between the developed finite-time sliding mode control algorithm and the traditional sliding mode controller predicated on a smooth linear sliding surface. Consider specifically a vector function 𝜎_𝑖∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 given by 𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖=𝑧̇_𝑖+𝐶_𝑖𝑧_𝑖,𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛, (57) where 𝐶_𝑖=diag[𝑐̃_𝑖1,…,𝑐̃_𝑖𝑛], 𝑐̃_𝑖𝑗>0, 𝑖=1,…,𝑝, 𝑗=1,…,𝑛. We define the 𝑖th sliding surface as the null space of 𝜎_𝑖(⋅,⋅); that is, 𝒮_𝑖≜𝑧_𝑖,𝑧̇_𝑖∈ℝ^𝑛×ℝ^𝑛∶𝜎_𝑖𝑧_𝑖,𝑧̇_𝑖=0. (58) Accordingly, the sliding mode controller can be obtained using the above sliding surface. Next, for this controller, we consider the planar coordinated motion of eight mobile robots and a leader with the same parameter values and the initial conditions as given in the above numerical example. Figure 6 shows the position phase portrait with robot orientations and Figure 7 shows position errors defined in (46) and orientation errors with respect to the leader for eight robots. Finally, Figure 8 shows the time history of the control torques acting on both wheels of each robot with 𝜏_1𝑖 and 𝜏_2𝑖 being the control torques acting on the left and the right wheels, respectively, of the 𝑖th robot. The parameters of the sliding surface and the controller are selected such that the settling time for convergence to the desired formation is approximately 5 seconds, that is, the same settling time as for the finite-time sliding mode controller. However, the peak of the control effort for the sliding mode controller predicated on the linear surface (57) is almost three times higher than that of the finite-time sliding mode controller (see Figures 5 and 8) which shows the efficacy of the proposed finite-time control algorithm.

Figure 6 

Position phase portrait with orientations of eight robots and a leader.

Figure 7 

Time history of the errors versus time.

Figure 8 

Control torques versus time.

6. Experimental Validation

We validate the above results using experimental wheeled mobile robots. The experimental setup consists of an overhead camera, two wheeled mobile robots, and a PC as shown in Figure 9. We use a black and white 640×480-resolution COHU-2672 camera equipped with a Data Transitions DT3120 frame grabber to capture overhead image frames in order to identify robot position that will be used in our feedback measurements. The camera is installed 2.86 m above the floor. To ensure that there is no delay throughout the experiment, the camera is set to capture the frames at a sampling rate of 2 fps. Since the captured image is distorted, a calibration process is used to allow the true real-world position of every point in the image to be calculated.

Figure 9 

Experimental setup.

Two wheeled mobile robots used for the experiment are shown in Figure 10. The parameters of these robots are the same as those used in Section 5. In order to detect each robot, three square white labels are attached to its upper surface. Using MATLAB image processing toolbox, the center points of these labels are calculated. The robot orientation is determined by a triangle that these labels form. Moreover, by knowing the robot's center of mass location with respect to any of these labels, the position of the center of mass with respect to the inertial reference frame is obtained by interpolation among the positions of the labels. Finally, the velocity of the robot is determined by taking the time derivative of the filtered position coordinates. The filtering is simply an averaging process since the experiment is implemented at low speed values for the robot.

Figure 10 

Experimental differential drive wheeled mobile robots.

The robot wheels are driven by two DC motors which are controlled by a Pandaboard single-board computer running a Linux operating system and an Arduino microcontroller board. The control program is built in The MathWorks Simulink environment. Through the Simulink Real-Time Workshop, the program is compiled and built as a real-time executable for the Pandaboard target [28]. The communication between the PC computer and the Pandaboard is done via wireless connection. The communication signals contain computed control velocity commands, specifically, the angular velocities for the wheels. These control velocity commands are generated by integrating the differential torques obtained using the coordination control algorithm.

For the following experiment, we consider the coordination problem discussed in Section 5 with the first and the second agents in Figure 2 being our experimental wheeled mobile robots. Figure 11 shows the position phase portrait with the robots orientations and Figure 12 shows position and orientation errors of the robots. Figure 13 shows the time history of the control torques acting on both wheels of the robot with 𝜏₁ and 𝜏₂ being the control torques acting on the left and the right wheels, respectively.

Figure 11 

Position phase portrait with orientation of the robots.

Figure 12 

The time history of the errors.

Figure 13 

Control torques versus time.

In order to suppress chattering induced mostly by the sampling rate of the camera, we used a less sharp (than in the simulations in the previous section) arctangent estimate of the signum function in the controller. Therefore, it is expected that the robots in the experiment converge to the desired formation slower than those in the simulation. As can be seen from Figures 11, 12, and 13, the experimental results match the trajectory profiles from the simulation (Figures 3, 4, and 5), although with a longer convergence time due to the change in the signum function approximation and saturation. This validates the proposed control algorithm even in the presence of implementation issues.

7. Conclusion

In this paper, we developed a decentralized sliding mode control technique that guarantees finite-time coordination in multiagent systems. We considered specifically a group of dynamical agents along with a leader with an objective to reach and maintain the desired formation in finite time. Using graph-theoretic notions, we characterized such formation via a directed graph Laplacian matrix and developed individual controllers that use only local information from the current agent and the neighboring agents that directly communicate their information to the current agent. The key feature of our sliding mode control is in the design of nonsmooth sliding surfaces that allow for finite-time convergence of the error states to the origin. The above results were developed for multiagent systems composed of fully actuated Euler-Lagrange systems and were further extended to address underactuated agent dynamics. A numerical example along with experimental validation was presented to show the efficacy of the approach.

Acknowledgment

This research was supported in part by the College of Engineering at Villanova University.