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

Authors: Masood Ghasemi; Sergey G. Nersesov
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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. Letℝ denote the set of real numbers, let ℝn 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⊂ℝn, respectively. Finally, we write V′(x) for the Fréchet derivative of V at x, ∥·∥∞ for ℓ∞ vector/matrix norm, and e∈ℝn for the ones vector of order n, that is, e≜[1,…,1]T.Next, consider the nonlinear dynamical system given by(1)x˙(t)=f(x(t)),x(0)=x0,t≥0,
where x(t)∈𝒟⊆ℝn,  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 sx(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, sx(t) is defined on [0,T(x)), sx(t)∈𝒩∖𝒟0 for all t∈[0,T(x)), and limt→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=ℝn.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
(2)V′(x)f(x)≤-c(V(x))α,x∈𝒟.
Then, 𝒟0 is finite-time stable. Moreover, if 𝒩 is as in Definition 1 and T:𝒩→[0,∞) is the settling-time function, then
(3)T(x0)≤1c(1-α)(V(x0))1-α,x0∈𝒩,
and T(·) is continuous on 𝒩. If, in addition, 𝒟=ℝn, V(·) is radially unbounded, and (2) holds on ℝn, 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].Adirected graphG 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.Theadjacency matrix of G is defined as A={aij},i,j=1,…,n, where aij,i≠j is equal to one if (uj,ui)∈ℰ(G) for ui,uj∈𝒱(G) and is equal to zero otherwise. Since, G has no self-loops, aii=0,i=1,…,n. The in-degree, respectively out-degree, of a vertex u, denoted by din(u), respectively dout(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 Din, respectively Dout, is defined as a diagonal matrix with din(u), respectively dout(u), for all u∈𝒱(G) as its diagonal elements.Apath on graph G of length m from u0∈𝒱(G) to um∈𝒱(G) is an ordered set of distinct vertices {u0,…,um} such that, for any pair of consecutive vertices, (ui-1,ui)∈ℰ(G) for all i=1,…,m. A directed path on a directed graph G of length m from u0∈𝒱(G) to um∈𝒱(G) is an ordered set of distinct vertices {u0,…,um} such that, for any directed pair of consecutive vertices, (ui-1,ui)∈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 ui∈𝒱(G), the reachable setℛi is a set containing ui and all vertices uj∈𝒱(G) such that there is a directed path from ui to uj. 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.TheKirchhoff matrix or directed Laplacian of a directed graph is defined by L=Din-A∈ℝn×n, where Din is the in-degree matrix and A is the adjacency matrix of the graph. The normalized directed Laplacian denoted by ℒ∈ℝn×n is obtained by dividing each row of L by the corresponding diagonal element of Din, if it is not zero. If an element of Din is zero, then the corresponding row of L consists of only zero elements.Proposition 3.
LetG 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 ofL sums up to zero, it follows that λ=0 and η=e is a valid solution to (λIn-L)η=0 which proves the result.Proposition 4 (see [25]).
LetG 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.
LetG 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.
SinceG 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
(4)Mi(qi(t))q¨i(t)=fi(qi(t),q˙i(t))+ui(t)+gi(qi(t),q˙i(t)),qi(0)=q0i,q˙i(0)=q˙0i,t≥0,i=1,…,m,
where qi∈ℝn is the vector of generalized coordinates of the ith agent, Mi(qi)∈ℝn×n is the positive definite inertia matrix of the ith agent, ui∈ℝn,t≥0, is the control input for the ith agent, fi(qi,q˙i)∈ℝn is the vector of Coriolis, centrifugal, conservative, and nonconservative forces acting on the ith agent, and gi(qi,q˙i)∈ℝn represents the vector of bounded uncertainties and disturbances affecting the dynamics of the ith agent. Alternatively, (4) can be written as
(5)q¨i(t)=f~i(qi(t),q˙i(t))+u~i(t)+g~i(qi(t),q˙i(t)),qi(0)=q0i,q˙i(0)=q˙0i,t≥0,i=1,…,m,
where
(6)f~i(qi,q˙i)≜Mi-1(qi)fi(qi,q˙i),u~i≜Mi-1(qi)ui,g~i(qi,q˙i)≜Mi-1(qi)gi(qi,q˙i).
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 𝒥i given by
(7)𝒥i⊂{L,1,…,m}∖{i},i=1,…,m,
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∈𝒥i. 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
(8)ℒij={1,i=j,i≠L,-1|𝒥i|,j∈𝒥i,i≠L,0,otherwise,
where |𝒥i| is the cardinality of 𝒥i,i=1,…,m, which is the number of elements in 𝒥i. 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
(9)ℒ=[001×mL1L2],
where
(10)L1=[ℒ1L,…,ℒmL]T,L2=[ℒ11⋯ℒ1m⋮⋱⋮ℒm1⋯ℒmm],
and ℒij,i=1,…,m,j=L,1,…,m, are given by (8). The next lemma presents the key feature of the matrix L2∈ℝm×m.Lemma 6 (see [18]).
Let the connectivity graphG have the normalized directed Laplacian matrix ℒ∈ℝ(m+1)×(m+1) given by (9). Then, L2∈ℝm×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 qL(t), t≥0, and q˙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 lij(t)∈ℝn,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
(11)eij(t)≜qi(t)-qj(t)-lij(t),i,j=L,1,…,m,t≥0,
where eij(t)∈ℝn,t≥0 is the position error of the ith agent with respect to the jth agent. Note that, at this point, eij,i,j=L,1,…,m, are defined regardless of whether j∈𝒥i or not. Furthermore, we introduce a generalized error state for the ith agent as
(12)zi(t)≜qi(t)-1|𝒥i|∑j∈𝒥i‍(qj(t)+lij(t)),i=1,…,m.
Note that in contrast with (11), in (12), the generalized position qi,i=1,…,m, is compared to only the generalized positions qj, where j∈𝒥i.Lemma 7 (see [18]).
Consider the connectivity graphG with the normalized directed Laplacian matrix ℒ∈ℝ(m+1)×(m+1) given by (9). Define z≜[z1T,…,zmT]T, where zi∈ℝn is given by (12) and ej≜[e1jT,…,emjT]T,j=L,1,…,m. Then, the following statements hold.(i)
Ifz=0, then eij=0,i=1,…,m,j=L,1,…,m.(ii)
Ifeij=0,i=1,…,m,j∈𝒥i, then z=0.The generalized error dynamics are obtained by taking the second time derivative of (12) and are given by
(13)z¨i(t)=vi(t)-l¨i(t)+f~i(z,z˙,qL(t),q˙L(t))-1|𝒥i|∑j∈𝒥i∖{L}‍f~j(z,z˙,qL(t),q˙L(t))+ℒiLq¨L(t)+hi(z,z˙,qL(t),q˙L(t)),zi(0)=z0i,z˙i(0)=z˙0i,i=1,…,m,t≥0,
where
(14)vi(t)≜u~i(t)-1|𝒥i|∑j∈𝒥i∖{L}‍u~j(t),i=1,…,m,(15)li(t)≜1|𝒥i|∑j∈𝒥i‍lij(t),i=1,…,m,hi(z,z˙,qL(t),q˙L(t))≜g~i(qi(t),q˙i(t))-1|𝒥i|∑j∈𝒥i∖{L}‍g~j(qj(t),q˙j(t)),22222222222222222222222222222222222i=1,…,m.
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:
(16)z(t)v=(L2⊗In)q(t)+L1⊗qL(t)-l(t),(17)z¨(t)=v(t)-l¨(t)+(L2⊗In)F(z,z˙,qL(t),q˙L(t))+L1⊗q¨L(t)+h(z,z˙,qL(t),q˙L(t)),z(0)=z0,z˙(0)=z˙0,t≥0,(18)v(t)=(L2⊗In)u~(t),
where
(19)q(t)≜[q1T(t),…,qmT(t)]T,l(t)≜[l1T(t),…,lmT(t)]T,v(t)≜[v1T(t),…,vmT(t)]T,u~(t)≜[u~1T(t),…,u~mT(t)]T,F(z,z˙,qL(t),q˙L(t))≜[f~1T(z,z˙,qL(t),q˙L(t)),…,f~mT(z,z˙,qL(t),q˙L(t))]T,h(z,z˙,qL(t),q˙L(t))≜[h1T(z,z˙,qL(t),q˙L(t)),…,hmT(z,z˙,qL(t),q˙L(t))]T.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 σi:ℝn×ℝn→ℝn given by
(20)σi(zi,z˙i)=z˙i+CiSi(zi)|zi|1/2,(zi,z˙i)∈ℝn×ℝn,
where Ci=diag[ci1,…,cin], cij>0, i=1,…,m, j=1,…,n, Si(zi)≜diag[sign(zi1),…,sign(zin)], zij∈ℝ, i=1,…,m, j=1,…,n, is the jth component of zi∈ℝn, |zi|1/2≜[|zi1|1/2,…,|zin|1/2]T. We define the ith sliding surface as the null space of σi(·,·), that is,
(21)𝒮i≜{(zi,z˙i)∈ℝn×ℝn:σi(zi,z˙i)=0}.
As in standard sliding mode control theory, the control law vi is calculated by setting σ˙i(zi,z˙i)=0 for the nominal system and adding a signum function to address uncertainties. Using (13) and (20) specifically, we set
(22)vi(t)=l¨i(t)-f~i(z,z˙,qL(t),q˙L(t))+1|𝒥i|∑j∈𝒥i∖{L}‍f~j(z,z˙,qL(t),q˙L(t))-ℒiLq¨L(t)-12Cidi(zi,z˙i)-Kisign(σi(zi,z˙i)),(zi,z˙i)∈ℳi,i=1,…,m,(zj,z˙j)∈ℝn×ℝn,222222222222222222222222j=1,…,m,j≠i,
where Ki=[ki1,…,kin], kij∈ℝ, i=1,…,m, j=1,…,n,
(23)sign(σi(zi,z˙i))≜[sign(σi1(zi,z˙i))⋮sign(σin(zi,z˙i))],di(zi,z˙i)≜[z˙i1|zi1|-1/2⋮z˙in|zin|-1/2],σij(·,·), i=1,…,m, j=1,…,n, is the jth component of σi(·,·), and ℳi⊂ℝn×ℝn is the set where di(·,·) is bounded. Define ℳi, i=1,…,m, as
(24)ℳi≜{(zi,z˙i)∈ℝn×ℝn:∥di(zi,z˙i)∥∞≤λi},
where
(25)λi≜∥Ci∥∞+δi,δi>0.
Note that vector di(·,·) and, consequently, vi are bounded in ℳi⊂ℝn×ℝn while being unbounded in the complement of ℳi. Thus, for the complement of ℳi, we design an auxiliary sliding mode controller below. First, however, we establish sufficient conditions under which ℳi 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 kij, i=1,…,m, j=1,…,n, satisfy
(26)kij=ηij+sup(z,z˙,t)∈ℝnm×ℝnm×ℝ∥hi(z,z˙,qL(t),q˙L(t))∥∞,
with
(27)ηij>λi2-cijλi2>0,
then the set ℳi defined by (24) is invariant with respect to (13).For the case when(zi,z˙i)∈ℳi,i=1,…,m, decentralized control input (22) can be rewritten for the entire ensemble of m agents as
(28)v(t)=l¨(t)-12Cd(z,z˙)+(L2⊗In)F(z,z˙,qL(t),q˙L(t))-L1⊗q¨L(t)-Ksign(σ(z,z˙)),221(zi,z˙i)∈ℳi,i=1,…,m,
where
(29)K≜block-diag[K1,…,Km],C≜block-diag[C1,…,Cm],d(z,z˙)≜[d1T(z1,z˙1),…,dmT(zm,z˙m)]T,σ(z,z˙)≜[σ1T(z1,z˙1),…,σmT(zm,z˙m)]T,sign(σ(z,z˙))≜[(sign(σ1(z1,z˙1)))T,…,(sign(σm(zm,z˙m)))T]T.
Next, for the initial conditions (z0i,z˙0i)∈ℝn×ℝn∖ℳi, we design an auxiliary sliding surface 𝒮iaux⊂ℝn×ℝn along with the corresponding sliding mode controller. Consider specifically the vector function σiaux:ℝn×ℝn→ℝn given by
(30)σiaux(zi,z˙i)=z˙i,(zi,z˙i)∈ℝn×ℝn,
and define the sliding surface 𝒮iaux as
(31)𝒮iaux≜{(zi,z˙i)∈ℝn×ℝn:σiaux(zi,z˙i)=0}.
By setting σ˙iaux(zi,z˙i)=0, we obtain
(32)viaux=l¨i(t)-f~i(z,z˙,qL(t),q˙L(t))+1|𝒥i|∑j∈𝒥i∖{L}‍f~j(z,z˙,qL(t),q˙L(t))-ℒiLq¨L(t)-Kiauxsign(σiaux(zi,z˙i)),(zi,z˙i)∉ℳi,i=1,…,m,(zj,z˙j)∈ℝn×ℝn,j=1,…,m,j≠i,
where Kiaux≜diag[ki1aux,…,kinaux],i=1,…,m,
(33)kijaux=ηijaux+sup(zi,z˙i,t)∈ℝn×ℝn×ℝ∥h(z,z˙,qL(t),q˙L(t))∥∞,22222222222222222222222222222222221j=1,…,n,
with ηijaux>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
(34)vi(t)={l¨i(t)-f~i(z,z˙,qL(t),q˙L(t))+1|𝒥i|∑j∈𝒥i∖{L}‍f~j(z,z˙,qL(t),q˙L(t))-ℒiLq¨L(t)-12Cidi(zi,z˙i)-Kisign(σi(zi,z˙i)),(zi,z˙i)∈ℳi,(zj,z˙j)∈ℝn×ℝn,j=1,…,m,j≠i,l¨i(t)-f~i(z,z˙,qL(t),q˙L(t))+1|𝒥i|∑j∈𝒥i∖{L}‍f~j(z,z˙,qL(t),q˙L(t))-ℒiLq¨L(t)-Kiauxsign(σiaux(zi,z˙i)),(zi,z˙i)∉ℳi,(zj,z˙j)∈ℝn×ℝn,j=1,…,m,j≠i,
where i=1,…,m, σi(·,·) and σiaux(·,·) are given by (20) and (30), respectively, the entries of control gain matrices Ki∈ℝn×n satisfy (26) and (27), and the entries of control gain matrices Kiaux∈ℝn×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 controllervi,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 Section3 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]
(35)x˙i(t)=vxi(t)cosθi(t)-dωi(t)sinθi(t),xi(0)=x0i,i=1,…,p,t≥0,(36)y˙i(t)=vxi(t)sinθi(t)+dωi(t)cosθi(t),yi(0)=y0i,(37)θ˙i(t)=ωi(t),θi(0)=θ0i,(38)v˙xi(t)=mdm~ωi2(t)+1m~r(τ1i(t)+τ2i(t))+g1i(η1i,η2i),vxi(0)=v0xi,(39)ω˙i(t)=-mdI~ωi(t)vxi(t)+a2I~r(τ2i(t)-τ1i(t))+g2i(η1i,η2i),ω(0)=ω0,
where (xi,yi) is the position of the ith robot's mass center C, θi is the ith robot's orientation (see Figure 1), vxi is the tangential component of the velocity of the robot's mass center, τ1i and τ2i are differential torques applied to each wheel, g1i(·,·) and g2i(·,·) are uncertainty functions affecting the robot dynamics, η1i≜[xi,yi,θi]T, η2i≜[vxi,ωi]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 I~≜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
(40)x¨i(t)=f1i(η1i(t),η2i(t))+u1i(t)cosθi(t)-du2i(t)sinθi(t)+g~1i(η1i(t),η2i(t)),xi(0)=x0i,x˙i(0)=x˙0i,i=1,…,p,t≥0,y¨i(t)=f2i(η1i(t),η2i(t))+u1i(t)sinθi(t)+du2i(t)cosθi(t)+g~2i(η1i(t),η2i(t)),yi(0)=y0i,y˙i(0)=y˙0i,
where
(41)u1i≜1m~r(τ1i+τ2i),i=1,…,p,u2i≜a2I~r(τ2i-τ1i),i=1,…,p,f1i(η1i,η2i)≜(mm~-1)dωi2cosθi+(md2I~-1)ωivxisinθi,i=1,…,p,f2i(η1i,η2i)≜(mm~-1)dωi2sinθi-(md2I~-1)ωivxicosθi,i=1,…,p,g~1i(η1i,η2i)≜g1i(η1i,η2i)cosθi-dg2i(η1i,η2i)sinθi,i=1,…,p,g~2i(η1i,η2i)≜g1i(η1i,η2i)sinθi+dg2i(η1i,η2i)cosθi,i=1,…,p.
Next, using the feedback linearizing controller given by
(42)[u1iu2i]=[cosθi-dsinθisinθidcosθi]-1[-f1i(η1i,η2i)+ν1i-f2i(η1i,η2i)+ν2i],i=1,…,p,
where ν1i∈ℝ  i=1,…,p, and ν2i∈ℝ  i=1,…,p, are the new feedback control inputs, (40) can be written as
(43)x¨i(t)=ν1i(t)+g~1i(η1i(t),η2i(t)),xi(0)=x0i,x˙i(0)=x˙0i,i=1,…,p,y¨i(t)=ν2i(t)+g~2i(η1i(t),η2i(t)),yi(0)=y0i,y˙i(0)=y˙0i,
while (37) can be rewritten as
(44)θ˙i(t)=-1d(x˙i(t)sinθi(t)-y˙i(t)cosθi(t)),θi(0)=θ0i,t≥0,i=1,…,p.Figure 1
Diagram of a mobile robot.The objective is to coordinate the motion of a set ofp robots and a leader. Define the leader's position in the plane as (xL(t),yL(t)), t≥0, and its orientation as θL(t), t≥0, and define the generalized position vector for the leader as qL(t)=[xL(t),yL(t),θL(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 vxL(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 vxL>0. As in Section 3, we define the set 𝒥i given by
(45)𝒥i⊂{L,1,…,p}∖{i},i=1,…,p,
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(46)zi(t)≜[zxi(t)zyi(t)]=[xi(t)-1|𝒥i|∑j∈𝒥i‍(xj(t)+lxij)yi(t)-1|𝒥i|∑j∈𝒥i‍(yj(t)+lyij)],i=1,…,p,t≥0,
where lxij∈ℝ and lyij∈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
(47)z¨xi(t)=ν~1i(t)+ℒiLx¨L(t)+g~1i(z~(t),η2(t),qL(t),q˙L(t)),zxi(0)=z0xi,z˙xi(0)=z˙0xi,i=1,…,p,t≥0,z¨yi(t)=ν~2i(t)+ℒiLy¨L(t)+g~2i(z~(t),η2(t),qL(t),q˙L(t)),zyi(0)=z0yi,z˙yi(0)=z˙0yi,i=1,…,p,
where
(48)z~(t)≜[zx1(t),zy1(t),θ1(t),…,zxp(t),zyp(t),θp(t)]T,η2(t)≜[η21(t),…,η2p(t)]T,ν~1i(t)≜ν1i(t)-1|𝒥i|∑j∈𝒥i∖{L}‍(ν1j(t)),ν~2i(t)≜ν2i(t)-1|𝒥i|∑j∈𝒥i∖{L}‍(ν2j(t)).
Note that, for t≥0, leader’s position and orientation satisfy
(49)x˙L(t)=vxL(t)cosθL(t)-dωL(t)sinθL(t),y˙L(t)=vxL(t)sinθL(t)+dωL(t)cosθL(t),ωL(t)=-1d(x˙L(t)sinθL(t)-y˙L(t)cosθL(t)),
where ωL(t)≜θ˙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 Ts, it follows from Lemma 7 that xi(t)-xj(t)-lxij=0,yi(t)-yj(t)-lyij=0,t≥Ts,i,j=L,1,…,p, which implies that x˙i(t)=x˙j(t),y˙i(t)=y˙j(t),t≥Ts,i,j=L,1,…,p. Thus, the orientation dynamics (37) for the ith robot can be rewritten as
(50)θ˙i(t)=-1d(x˙j(t)sinθi(t)-y˙j(t)cosθi(t)),θi(Ts)=θsi,t≥Ts,i=1,…,p,j=L,1,…,p.
Let eθiL(t)≜θi(t)-θL(t),t≥Ts,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
(51)e˙θiL(t)=-1d[x˙L(t)sin(eθiL(t)+θL(t))1111-y˙L(t)cos(eθiL(t)+θL(t))]-ωL(t),eθiL(Ts)=esθiL,t≥Ts,i=1,…,p.
Alternatively, (51) can be rewritten as
(52)e˙θiL(t)=-1dx˙L(t)(sineθiL(t)cosθL(t)+coseθiL(t)sinθL(t))+1dy˙L(t)×(coseθiL(t)cosθL(t)-sineθiL(t)sinθL(t))-ωL(t)=-1dsineθiL(t)(x˙L(t)cosθL(t)+y˙L(t)sinθL(t))+1dcoseθiL(t)×(-x˙L(t)sinθL(t)+y˙L(t)cosθL(t))-ωL(t)=-1dvxL(t)sineθiL(t)+ωL(t)(coseθiL(t)-1)=-2dvxL(t)sin(eθiL(t)2)cos(eθiL(t)2)-2ωL(t)sin2(eθiL(t)2)=-2dsin(eθiL(t)2)[vxL(t)cos(eθiL(t)2)+dωL(t)sin(eθiL(t)2)]=-1dvxL(t)sineθiL(t)[1+dωL(t)vxL(t)tan(eθiL(t)2)],2222222222222eθiL(Ts)=esθiL,t≥Ts,i=1,…,p.
In the next proposition, we show that the zero equilibrium point of (52) is uniformly asymptotically stable with respect to eθiL.Proposition 11.
Consider the error dynamics given by (52) and define
(53)𝒟i≜{eθiL∈ℝ:|eθiL|<2atan((1-δi)vxL*dωL*)}⊂(-π,π),i=1,…,p,
where vxL*≜inft≥0(vxL(t))>0, ωL*≜supt≥0(|ωL(t)|), and δi>0, i=1,…,p, is an arbitrarily small constant. Then, the zero solution of (52) is uniformly asymptotically stable with respect to eθiL and with an estimate of the domain of attraction given by 𝒟i.Proof.
Note that{0} is the only equilibrium point of (52) in 𝒟i. Consider a Lyapunov function candidate given by
(54)Vi(eθiL)=1-coseθiL,eθiL∈𝒟i,i=1,…,p,
and note that Vi(0)=0 and Vi(eθiL)>0,eθiL∈𝒟i, i=1,…,p. Furthermore, since 𝒟i⊂(-π,π),i=1,…,p, there exists γi>0,i=1,…,p, such that |sineθiL|>γi|eθiL|,eθiL∈𝒟i,i=1,…,p. Thus, the Lyapunov derivative along the trajectories of (52) satisfies
(55)V˙i(eθiL(t))=sineθiL(t)e˙θiL(t)=-1dvxL(t)sin2eθiL(t)[1+dωL(t)vxL(t)tan(eθiL(t)2)]≤-1dvxL*δisin2eθiL(t)≤-γi2dvxL*δieθiL2(t),eθiL∈𝒟i,i=1,…,p.
It follows from Theorem  4.1 of [27] that the zero solution to (52) is uniformly asymptotically stable with respect to eθiL and with an estimate of the domain of attraction given by 𝒟i,i=1,…,p. This implies that θi(t)→θL(t) as t→∞ for all i=1,…,p.Proposition11 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 Figure2. This communication topology remains unchanged over time. For the given configuration, L1 and L2 matrices are given as follows:
(56)L1=[-12-12000000],L2=[1-12000000-121000000-1201-120000-14-14-141-140000-120-12100000-13-1301-130000-13-13-131000000-12-121].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 thatL2 is invertible. Next, the leader is set to be moving counter-clockwise on a circular path of radius 0.5 according to xL(t)=0.5cos(πt/30), yL(t)=0.5sin(πt/30), t≥0. The values of control gains are set to be Ki=0.5×I2 and Kiaux=0.5×I2, i=1,…,8. Finally, the matrices Ci∈ℝ2×2 characterizing the design of the sliding surface for each agent are chosen to be Ci=0.5×I2, i=1,…,8. In this case, λi=0.75, i=1,…,8, ensure that condition (27) is satisfied.All robots are identical with the values of the parameters set to bem=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 byη11(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 η2i=[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 zxi and zyi,i=1,…,8, converge to zero faster than eθiL,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 τ1i and τ2i 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 functionσ~i:ℝn×ℝn→ℝn given by
(57)σ~i(zi,z˙i)=z˙i+C~izi,(zi,z˙i)∈ℝn×ℝn,
where C~i=diag[c~i1,…,c~in], c~ij>0, i=1,…,p, j=1,…,n. We define the ith sliding surface as the null space of σ~i(·,·); that is,
(58)𝒮~i≜{(zi,z˙i)∈ℝn×ℝn:σ~i(zi,z˙i)=0}.
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 τ1i and τ2i 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 Figure9. 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 Figure10. 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 Section5 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 Figures11, 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.



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