Distributed Cooperative Control of Multiple Nonholonomic Mobile Robots

Journal of Intelligent & Robotic Systems, Jan 2016

In this paper, the distributed cooperative control problem is considered for multiple type (1,2) nonholonomic mobile robots. Firstly, a local change of coordinates and feedback is proposed to transform the original nonholonomic system to a new transformed system. Secondly, a distributed controller for the transformed system is designed by using information of the intrinsic system and its neighbors to make the state converge to the same value asymptotically. Furthermore, it shows that the same value can be confined to the origin, which means that the problem of cooperatively converging to a stationary point of a group of nonholonomic systems can be practically solved. Finally, due to the communication delays are inevitable in practice, new distributed controllers for the transformed system are also proposed making the state converge to the same value or zero asymptotically with considering communication delays. The proposed methods are then extended to the case where the nonholonomic mobile robot needs to form a prescribed formation other than agreeing on a same value. The stability of the proposed methods is proved rigorously. Simulation results confirm the effectiveness of the proposed methods.

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Distributed Cooperative Control of Multiple Nonholonomic Mobile Robots

J Intell Robot Syst Distributed Cooperative Control of Multiple Nonholonomic Mobile Robots Gang Wang 0 1 Chaoli Wang 0 1 Qinghui Du 0 1 Lin Li 0 1 Wenjie Dong 0 1 0 W. Dong Department of Electrical Engineering, The University of Texas Rio Grande Valley , Edinburg, TX 78539 , USA 1 Q. Du Department of Mathematics, Luoyang Normal University , Luoyang 471022 , China In this paper, the distributed cooperative control problem is considered for multiple type (1, 2) nonholonomic mobile robots. Firstly, a local change of coordinates and feedback is proposed to transform the original nonholonomic system to a new transformed system. Secondly, a distributed controller for the transformed system is designed by using information of the intrinsic system and its neighbors to make the state converge to the same value asymptotically. Furthermore, it shows that the same value can be confined to the origin, which means that the problem of cooperatively converging to a stationary point of a group of nonholonomic systems can be practically solved. Finally, due to the communication delays are inevitable in practice, new distributed controllers for the transformed system are also proposed making the state converge to the same value or zero asymptotically with considering communication delays. The proposed methods are then extended to the case where the nonholonomic mobile robot needs to form a prescribed formation other than agreeing on a same value. The stability of the proposed methods is proved rigorously. Simulation results confirm the effectiveness of the proposed methods. Distributed control; Nonholonomic mobile robots; Formation control; Cooperative control 1 Introduction In recent years, there has been an increasing research interest in the distributed synchronization control of multi-agent systems due to its potential applications in many areas, such as formation control [1, 2], design of distributed sensor networks [3], flocking control [4, 5], etc. Some seminal works are [6, 7], just to name a few. A large number of effective control approaches have focused on two control problems of networked systems, i.e., leaderless consensus problems and leader-following consensus problems. For leaderless consensus problems, controllers are designed to drive all the agents to a common value, which depends on initial conditions (see [8, 9]). As for leader-following consensus problems, controllers are designed to make all the follower nodes track the trajectory of the leader node (see [10, 11]). Besides, there are also many works investigated for different types of agent dynamics including first-order integrator systems [12, 13], second-order integrator systems [14, 15] and higherorder integrator systems [16, 17]. However, many practical cooperative control applications involve agents that are nonlinear and nonholonomic. The stabilization problem of nonholonomic system cannot be solved by many methods of classical linear system for the fact nonholonomic system fails to meet the three necessary conditions of the theorem of Brockett [18]. Thus the above mentioned methods cannot the solve the cooperative control of multiple nonholonomic agents. To solve the single nonholonomic system control problem, many scholars have done a lot of relevant research in this area (see [19–23], etc.). But most of the methods focused on the single nonholonomic system cannot solve the cooperative control of multiple nonholonomic systems directly, because we consider multiple nonholonomic mobile robots and the associated controller is distributed in nature-for each robot has access to the state of its neighbors only. Motivated by those observations, the authors in [2, 24–28] have focused on the cooperative control of multiple nonholonomic agents. In [2], Lin, Francis, and Maggiore have studied the feasibility problem of achieving a specified formation among a group of nonholonomic unicycles by local distributed control. In [24], Dong and Farrell presented two controllers for cooperative control problems of nonholonomic systems. One distributed controller was proposed to make a group of nonholonomic mobile agents cooperatively converge to some stationary point; The other controller was proposed to make a group of mobile agents converge to and track a target point which moves along a desired trajectory under various communication scenarios. And they also extended the methods to solve the problem of cooperative control of multiple nonholonomic dynamic systems with uncertainty in [25]. In [26], Liu and Jiang proposed a new class of distributed nonlinear controller for leader-following formation control of unicycle robots by using nonlinear small-gain design methods. In [27], Dong studied the distributed tracking control of multiple nonholonomic chained systems. Different from their works in [24, 25], the assumption that all follower robots have access to the information of the leader robot is not needed. In other words, for each robot, the available information for feedback is its own information and its neighbours’ information. In [28], Cao, Jiang, and Yue have also investigated the consensus problems of multiple nonholonomic systems. Distributed controller was constructed by using the theory of cascaded systems. Different to previous assumptions on the group reference such as persistent excitation or converging to nonzero constant in [24], the condition on the group reference signal has been further relaxed. Campion, Basin, and D’Andre´a-Novel claimed that the interesting nonholonomic wheeled mobile robots are type (2, 0), (2, 1), (1, 1), (1, 2) robots in [29]. In this paper, we study distributed cooperative control problem of multiple type (1, 2) nonholonomic mobile robots. This kind of systems is more complicated, compared with type (2, 0), type (2, 1) and type (1, 1). The idea exploited in this paper can be used to investigate the same problem of the other three nonholonomic wheeled mobile robots. The main contributions of this paper are threefold. First, a local change of coordinates and feedback is proposed to transform the original nonholonomic system to a new transformed system. Second, distributed controllers for the new transformed system are designed by using its own information and its neighbours’ information to make the state converge to the same value or zero asymptotically with and without considering communication delays. Third, extension is provided to extend the proposed schemes to the case, where the nonholonomic mobile robot needs to form a stable formation other than agreeing on a same value. The remainder of this paper is organized as follows. In Section 2, some notions and preliminaries about the algebraic graph theory are briefly introduced, and the kinematic of type (1, 2) and the distributed cooperative control problem of type (1, 2) are presented. In Section 3, under two different communication scenarios, distributed controllers are designed to ensure that the state of each transformed system converges to the same value or zero asymptotically. Extensions are provided in Section 4. In Section 5, the simulation results are shown to illustrate the performance of the proposed methods. Some conclusions are given in the last Section. 2 Problem Statement 2.1 Basic Graph Theory and Notations In this subsection, some notions and preliminaries about the algebraic graph theory are briefly introduced. Let G = {V, E } denote a directed graph, where V = {1, . . . , N } is the set of nodes corresponding to each robot, and E ⊆ V × V is the set of edges. (i, j ) ∈ E means that robot j can obtain information from robot i, but not necessarily vice versa for a directed graph. In this paper, self-loop is not allowed in the graph, that is, (i, i) ∈/ E . Ni = {j ∈ V|(j, i) ∈ E } denotes the neighbors of robot i. A matrix A = [aij ] ∈ RN×N denotes the adjacency matrix of G, where aij > 0 iff (j, i) ∈ E , else aij = 0. It is assumed that the topology is fixed which means A is time-invariant. A matrix L = D − A is called the Laplacian matrix of G, where D = diag(d1, . . . , dN ) is the in-degree matrix with di = jN=1 aij . A direct path from robot i to robot j is a sequence of successive edges in the form {(i, l), (l, m), . . . , (k, j )}. Graph G is strongly connected if any two robots (i, j ) with i = j , there is a direct path from robot i to robot j . A directed graph G has a spanning tree, if there exists a robot i such that there is a direct path from robot i to every other robot in the graph, where the robot i is called the root of graph G. A directed graph G is balanced if 1T L = 0, where 1 is a vector with element one. Bidirectional graph is a special case of a directed graph, if (i, j ) ∈ E , then (j, i) ∈ E . Meanwhile, it is stipulated that aij = aji in bidirectional graph. 2.2 Kinematic of the Mobile Robots Consider a group of N (N ≥ 2) type (1, 2) nonholonomic mobile robots as shown in Fig. 1 Each robot has two steering wheels (conventional centered orientable wheels) and one castor wheel (conventional off-centered orientable wheel). (xi , yi ) denotes the position Pi of the center of the ith (i = 1, 2, . . . , N ) robot’s mass, θi denotes the angle between xi1−axis and X−axis, and βi1 and βi2 denote angles between Fig. 1 Type (1,2) nonholonomic mobile robot the orientation of the plane of steering wheels and xi1−axis, lr (> 0) is half of the width of the ith robot. The nonholonomic constraint of the ith robot is defined by [29] (cos βi1, sin βi1, lr sin βi1)H (θi )ξ˙i = 0, (− cos βi2, − sin βi2, lr sin βi2)H (θi )ξ˙i = 0, ⎡ cos θi sin θi 0 ⎤ H (θi ) = ⎣ − sin θi cos θi 0 ⎦ . 0 0 1 In addition, Eq. 1 can be specifically written as x˙i = −lr νi1[sin βi1 sin(θi +βi2)+sin βi2 sin(θi + βi1)], y˙i = lr νi1[sin βi1 cos(θi + βi2) + sin βi2 cos(θi + βi1)], θ˙i = νi1 sin(βi2 − βi1), β˙i1 = νi2, β˙i2 = νi3, where qi = [xi , yi , θi , βi1, βi2]T is the state of the ith robot, and νi1, νi2, νi3 are the velocity of castor wheel and two angular velocities of steering wheels of the ith robot, respectively. 2.3 Cooperative Control Problem The chained form systems were first introduced in [30] as a class of systems to which one could convert a number of interesting examples, and for which it was easy to derive steering control laws. However, only the systems that have two input and one chain were focused on. In our manuscript, the type (1, 2) nonholonomic mobile robot has three inputs and two chains. Thus, the state feedback and coordinate transformation proposed in [30] cannot be utilized directly. The sufficient conditions for converting a multiple-input and multiple-chain system with nonholonomic constraints into a chained form via state feedback and a coordinate transformation were presented in [31, 32]. Here, we invoke the coordinate and state transformation which is similar to that in [32]. Then, to simplify the distributed cooperative controller design, a novel change of states by adding 0t ω(s)ds based on chained form is proposed as follows. zi4 = −xi sin θi + yi cos θi − 2lr ssiinn(ββii12s−inββi1i2) zi5 = xi cos θi + yi sin θi − lr ssiinn((ββii21+−ββii21)) −2lr νi2 sin2 (βi2−βi1) + lr νi1 sin(βi1 + βi2), where ω = ρ sin t , and ρ , γ1, γ2 are positive constants. Taking derivative of Eq. 3, we have z˙i1 = ui1 − ω, z˙i2 = −γ1zi2ω2 + ωzi4 + (ui1 − ω)(zi4 − γ1ωzi2), z˙i3 = −γ2zi3ω2 + ωzi5 + (ui1 − ω)(zi5 − γ2ωzi3), z˙i4 = ui2 + γ1ω˙ zi2 + γ1ωui1zi4 − γ12ω2ui1zi2, z˙i5 = ui3 + γ2ω˙ zi3 + γ2ωui1zi5 − γ22ω2ui1zi3. Remark 1 It should be noted that because of the local nature of the state and feedback transformations (3), the laws designed for the transformed system (4) do not guarantee global stability properties for the original model (2) of the ith type (1,2) nonholonomic mobile robot. Indeed, since the coordinate transformation and state feedback are well defined over the subset i = {(xi , yi , θi , βi1, βi2) ∈ R5|βi1 = βi2 mod π }. We have that only within such a domain can we obtain “global” stability. Definition 1 The distributed cooperative control problem of multiple type (1,2) nonholonomic mobile robots (2) discussed in this paper is to design the distributed control input ui = [ui1, ui2, ui3]T for the ith system (4) using zi = [zi1, zi2, zi3, zi4, zi5]T and the relative state zl of its neighbors for l ∈ Ni such that zi is bounded and limt→∞(zi (t ) − zj (t )) = 0 for 1 ≤ i = j ≤ N . Remark 2 The control laws are required to make the state zi of each transformed system converge to the same value c(t ) with c(t ) = [c1, c2(t ), c3(t ), c4, c5]T , where c1, c4, and c5 are constants which are unknown and depend on robots’ initial conditions and communication between robots, and c2(t ), c3(t ) are bounded functions. Furthermore, if limt→∞(ui1(t )−ω(t )) = 0, c1 = 0, c4 = 0, and c5 = 0, then c2 = 0, c3 = 0 (see Lemma 2). Since the system (2) discussed in this paper is nonholonomic, by the theorem of Brockett [18], the state qi of each original system (2) cannot be stabilized at a stationary point by a smooth pure state feedback controller which is a smooth function of its own state qi and the states ql of its neighbors for l ∈ Ni . To overcome this difficulty, we design cooperative control laws such that the state zi of each transformed system (4) converges to a moving vector c(t ). Then, we will state that c(t ) can also be confined to the origin, which means that cooperatively converging to a stationary point of a group of nonholonomic systems (2) can be practically solved. For details, please refer to the remarks after Theorem 2. An additional assumption on the communication topology is given below. Assumption 1 The communication digraph G has a spanning tree and G with weight matrix A is balanced. Remark 3 Note that this assumption is very common which has appeared in relevant literature such as Dong [33]. And it is much more relaxed than undirected connected graph as has been made in Hou, Cheng, and Tan [8], Ou, Du, and Li [34], Feng and Wen [35]. The following lemmas are useful in our design and analysis of distributed controllers. Lemma 1 (Dong and Farrell [24]) If the digraph G has a spanning tree and the Laplacian matrix L of the digraph G with weight matrix A = [aij ](aij ≥ 0), then for any μ ∈ [0, Re(λ2(L))), where λ2 is the nonzero eigenvalue of L with the smallest real part, w satisfies wT L = 0 and wT 1 = 1. Lemma 2 (Dong [33]) If the digraph G has a spanning tree and the Laplacian matrix L of the digraph G with weight matrix A is balanced, the matrix LT +L is semidefinite. Furthermore, if limt→∞ xT (LT + L)x = 0 for a vector x = [x1, x2, . . . , xN ]T , then lim (xi (t ) − xj (t )) = 0, 1 ≤ i = j ≤ N . t→∞ Before proceeding further, the following additional lemma is required. Lemma 3 For the ith transformed system (4), if ui1 − ω, zi4, zi5 are bounded and converge to zero asymptotically, then zi2, zi3 are bounded and converge to zero asymptotically. Proof Consider the Lyapunov function candidate Differentiating V1 along with solutions of system (4), we get V˙1 = −ω2 γ1zi22 + γ2zi23 + ωzi2zi4 + ωzi3zi5 +zi2(ui1 − ω)(zi4 − γ1ωzi2) +zi3(ui1 − ω)(zi5 − γ2ωzi3) ≤ −2γ ω2V1 + 2ϕ1V1 + 2ϕ2√V1, ϕ1 = γ¯ |ω||ui1 − ω|, ϕ2 = √12 (|zi4| + |zi5|)|ui1|, Due to boundedness of ω, and limt→∞(ui1 − ω) = 0, limt→∞ zi4(t ), zi5(t ) = 0, we have limt→∞ ϕ1(t ), ϕ2(t )=0. In order to facilitate the following analysis, we take σ = √V1, then D+σ ≤ −γ ω2σ + ϕ1σ + ϕ2, where D+ is the upper Dini derivative. Thus, we get σ (t ) ≤ e 0t (−γ ω2(s)+ϕ1(s))ds σ (0) + 0t e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ. With this observation in mind, since limt→∞ ϕ1(t )=0, γ ρ2 there always exists T1 > 0 such that ϕ1(t ) ≤ 4 for all t ≥ T1. Define function ϕ¯1(t ) = sup0≤τ ≤t ϕ1(τ ), the following equation can be achieved ≤ ϕ¯1(T1)T1 + 4 (t − T1). ≤ −γ ρ2 2t − sin42t + ϕ¯1(T1)T1 + γ ρ2 γ ρ2 γ ρ2 ≤ − 4 t + 4 + ϕ¯1(T1)T1 − 4 T1. = elilmimtt→→∞∞ e0t (−0t(γ−ωγ2ω(s2)(+s)ϕ+1ϕ(s1)()sd)s)dσs(σ0()0=) 0. Next, we will show that t e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ = 0. Due to limt→∞ ϕ2(t )=0, ∀η > 0, we can always find that T2 ≥ T1 such that ϕ2(t ) ≤ η for all t ≥ T2. Define function ϕ¯2(t ) = sup0≤τ ≤t ϕ2(τ ), we get t e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ = 00T2 e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ + Tt2 e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ +η Tt2 e τ (−γ ω2(s)+ϕ1(s))ds dτ ≤ ϕ¯2(T2) 0T2te 0τ(γ ω2(s)−ϕ1(s))dsdτ e 0t (−γ ω2(s)+ϕ1(s))ds ≤ ζ + η Tt2 e−γ ρ2 t−2τ − 41 sin 2t+ 14 sin 2τ + γ ρ42 (t−τ )dτ ψ e 0t (−γ ω2(s)+ϕ1(s))ds with ψ t e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ ≤ ε, ∀t ≥ max{T2, T3} which implies limt→∞ 0t e τt (−γ ω2(s)+ϕ1(s))ds ϕ2(τ )dτ = 0. Therefore, it can be concluded that the right-side of inequality (9) will converge to zero as t → ∞. Consequently, σ (t ) is bounded and tends to zero asymptotically, which also suggests that V1, zi2, zi3 are bounded and converge to zero asymptotically. This completes the proof. Remark 4 It should be noted that the proof of Lemma 3 is different from that in Lemma 6 of [24] and Lemma 2 of [25]. The requirements for the convergence of ϕ1, ϕ2 must be exponential in [24] and [25], which are relaxed to be asymptotical here, and the proof here is much more rigorous. Lemma 4 If ui1 − ω, uj1 − ω, zi4, zj4, zi5, zj5 are bounded and ui1 − ω, uj1 − ω, zi4 − zj4, zi5 − zj5 asymptotically converge to zero for 1 ≤ i = j ≤ N , then zi2, zi3, zj2, zj3 are bounded and zi2 − zj2 and zi3 − zj3 converge to zero asymptotically. gi11 = γ1ω2 + (ui1 − ω)γ1ω, gi12 = ui1zi4, gi21 = γ2ω2 + (ui1 − ω)γ2ω, gi22 = ui1zi5. Since ui1 − ω, ω, zi4, zi5 are all bounded, thus gi11, gi12, gi21, gi22 are bounded. Furthermore, it can be proved that zi2 and zi3 are bounded by Eq. 14. Let eij2 = zi2 − zj2 for 1 ≤ i = j ≤ N , we have e˙ij2 = −γ1zi2ω2 + ωzi4 + (ui1 − ω)(zi4 − γ1ωzi2) +γ1zj2ω2 −ωzj4 − (uj1 − ω)(zj4 − γ1ωzj2) = −γ1ω2eij2 + ϕij1(t ), where ϕij1(t ) = ω(zi4 − zj4) + (ui1 − ω)(zi4 − γ1ωzi2) − (uj1 − ω)(zj4 − γ1ωzj2). Since zi4 − zj4 and ui1 − ω asymptotically converge to zero, and ω, zi2, zi4, zj2, zj4 are bounded, thus ϕij1(t ) converge to zero asymptotically. Choose the following Lyapunov function Proof First, we will prove that zi2 and zi3 are bounded for i = 1, . . . , N . By Eq. (4) and using means of variation of constants and initial integral methods, we have Using the mimicking argument as the proof of Lemma 3, it can be easily proved that limt→∞ eij2(t ) = 0, namely, zi2 − zj2 converges to zero asymptotically for 1 ≤ i = j ≤ N . Also, with the similar technique, the conclusion that zi3 − zj3 asymptotically converges to zero can be given. 3 Controller Design and Stability Analysis 3.1 Closed-loop System Stability In this subsection, we will design the distributed control input ui for the ith system (4) using zi and the relative state zl of its neighbors for l ∈ Ni such that zi is bounded and limt→∞(zi (t ) − zj (t )) = 0 for 1 ≤ i = j ≤ N . The structure of system (4) suggests zi1, zi4, zi5 can be directly controlled via ui1, ui2, ui3. Now we are ready to choose the distributed controller ui as Remark 5 The first term of Eq. 17 is a weighted sum of the relative state information between system i and its neighbors. And the terms ω, −γ1ω˙ zi2 − γ1ωui1zi4 + γ12ω2ui1zi2, −γ2ω˙ zi3 − γ2ωui1zi5 + γ22ω2ui1zi3 are the canceling terms, which are designed to cancel the extra parts. Substituting control input (17) into system (4), we can get the following closed-loop error system for zi1, zi4, zi5 z˙i1 = − z˙i4 = − z˙i5 = − Theorem 1 Consider the closed-loop system consisting of N transformed systems (4) satisfying Assumption 1, the proposed distributed controller (17). Then the state zi of the ith transformed system (4) in the closed-loop system is bounded and limt→∞(zi (t ) − zj (t )) = 0 for 1 ≤ i = j ≤ N . Proof By Eq. 18, we have Z˙ 1 = −LZ1, Z˙ 4 = −LZ4, Z˙ 5 = −LZ5, where Zq = [z1q , z2q , . . . , zNq ] for q = 1, 4, 5, and L is the Laplacian matrix of G. Therefore By Lemma 1, we have limt→∞ Z1(t ) = 1wT Z1(0) =: c11, limt→∞ Z4(t ) = 1wT Z4(0) =: c41, limt→∞ Z5(t ) = 1wT Z5(0) =: c51. It is apparent that limt→∞(z1i (t ) − z1j (t )) = 0, limt→∞(z4i (t ) − z4j (t )) = 0, limt→∞(z5i (t ) − z5j (t )) = 0 for 1 ≤ i = j ≤ N . By utilizing Eqs. 17 and 20, we can prove that ul1 − ω is bounded and converges to zero asymptotically for l = 1, . . . , N . Then according to Lemma 4, we have that zl2 and zl3 are bounded. In addition, the conclusion that zi2 − zj2 and zi3 − zj3 converge to zero asymptotically for 1 ≤ i = j ≤ N can also be given by Lemma 4, namely, limt→∞(zl2(t ) − c2(t )) = 0, limt→∞(zl3(t ) − c3(t )) = 0 for l = 1, . . . , N , where c2(t ) and c3(t ) are unknown but bounded functions. Remark 6 A distributed control law for system (4) is given by Eq. 17. Control law (17) can make zl for l = 1, . . . , N converge to c(t ) asymptotically with c(t ) = [c1, c2, c3, c4, c5]T . By Eq. 3, it is easy to prove that where q¯l = [xl , yl , θl ]T for l = 1, . . . , N . The following theorem shows that we can make zi converge to zero. We redesign the distributed controller ui as ui1 = − jN=1 aij (zi1 − zj1) − pi zi1 + ω, ui2 = − jN=1 aij (zi4 − zj4) − qi zi4 − γ1ω˙ zi2 −γ1ωui1zi4 + γ12ω2ui1zi2, ui3 = − jN=1 aij (zi5 − zj5) − ki zi5 − γ2ω˙ zi3 −γ2ωui1zi5 + γ22ω2ui1zi3, where pi ≥ 0, qi ≥ 0, ki ≥ 0, and iN=1 qi > 0, iN=1 ki > 0. Remark 7 These terms pi zi1, qi zi4, ki zi5 in Eq. 22 can also be considered as relative information between robot i and a virtual robot with its state being zero. Theorem 2 Consider the closed-loop system consisting of N transformed systems (4) satisfying Assumption 1, the proposed distributed controller (22) with the parameters satisfying pi ≥ 0, qi ≥ 0, ki ≥ 0, and iN=1 pi > 0, iN=1 qi > 0, iN=1 ki > 0. Then the state zi of the ith transformed system (4) in the closed-loop system is bounded and converges to zero asymptotically, i.e., limt→∞zi (t ) = 0 for i = 1, . . . , N . Proof With the distributed controller ui defined in Eq. 22, we have Choose the Lyapunov function V = 2 1 N i=1 Differentiating V along the solutions of Eq. 23 yields 1 Z1T (LT + L)Z1 − 21 Z4T (LT + L)Z4 V˙ = − 2 − 12 Z5T (LT + L)Z5 N − i=1 pi zi21 + qi zi24 + ki zi25 , where Zq = [z1q , z2q , . . . , zNq ] for q = 1, 4, 5, and L is the Laplacian matrix of G. Since LT + L is positive semidefinite, V˙ ≤ 0, hence that V (t ) is bounded and zi1, zi4, zi5 are bounded. According to the definition (24), Barbalat’s Lemma [36] can be employed to prove that that limt→∞ V˙ (t ) = 0. Thus we obtain, limt→∞ pl zl21, ql zl24, kl zl25 = 0, l = 1, . . . , N limt→∞ ZT (LT + L)Z1 = 0, 1 limt→∞ Z4T (LT + L)Z4 = 0, limt→∞ Z5T (LT + L)Z5 = 0. Since there is at least one integer m such that pm > 0, limt→∞ zm1(t ) = 0. By applying Lemma 2, limt→∞(zi1(t ) − zj1(t )) = 0 for 1 ≤ i = j ≤ N . Hence, limt→∞ zl1(t ) = 0 for l = 1, . . . , N . And limt→∞ zl4(t ) = 0, limt→∞ zl5(t ) = 0 can also be proved in the similar argument. By utilizing Eqs. 22 and 26, we can prove that ul1 − ω is bounded and converges to zero asymptotically for l = 1, . . . , N . Then according to Lemma 3, we have zl2, zl3 are bounded and converge to zero asymptotically. In summary, the state zl of the lth transformed system (4) in the closed-loop system is bounded and converges to zero asymptotically, i.e., limt→∞zl (t ) = 0 for l = 1, . . . , N . Remark 8 By Eq. 3 and limt→∞zl (t ) = 0, we have limt→∞[θl (t ) − ρ(1 − cos t )] = 0, which means that θl converges to a neighborhood Bd of the origin with radius ρ. And from the second equation and third equation of Eq. 3, we can also get xl , yl are bounded and asymptotically converge to zero provided zl2, zl3 are bounded and converge to zero asymptotically. From the fourth equation and fifth equation of Eq. 3, it can also be proved that if limt→∞zl (t ) = 0, then limt→∞βl1(t ) = kl1π and limt→∞βl2(t ) = kl2π , kl1, kl2 ∈ Z. Thus, the problem of cooperatively converging to a stationary point of a group of nonholonomic systems (2) is practically solved. In addition, if ρ decreases, then the θl becomes small. However, the performance of xl , yl becomes bad, i.e., the convergence rate of zl2, zl3 to zero decreases. Therefore, there is a tradeoff between small θl and a large convergence rate of xl , yl when one chooses ρ. 3.2 Closed-loop System Stability with Communication Delays In practice, there are always time delays due to communication and other factors. In this subsection, we will consider communication delays in the control design and analysis. For simplicity, in this paper we assume that all communication delays are constant. Assumption 2 The communication digraph G is bidirectional and strongly connected. Under Assumption 2, the distributed controller is designed as ui1(t) = − jN=1 aij (zi1(t) − zj1(t − τi )) + ω(t), ui2(t) = − jN=1 aij (zi4(t) − zj4(t − τi )) − γ1ω˙ (t)zi2(t) −γ1ω(t)ui1(t)zi4(t) + γ12ω2(t)ui1(t)zi2(t), ui3(t) = − jN=1 aij (zi5(t) − zj5(t − τi )) − γ2ω˙ (t)zi3(t) −γ2ω(t)ui1(t)zi5(t) + γ22ω2(t)ui1(t)zi3(t), where communication delay τi (≥ 0) is a positive constant. Fig. 3 Profiles of the states zi1 with controller (17) and controller (22) Theorem 3 Consider the closed-loop system consisting of N transformed systems (4) satisfying Assumption 2, and the proposed distributed controller Fig. 4 Profiles of the states zi2 with controller (17) and controller (22) Substituting the distributed controller (27) into system (4), we can get the following closed-loop error system for zi1, zi4, zi5 (27). Then the state zi of the ith transformed system (4) in the closed-loop system is bounded and limt→∞(zi (t ) − zj (t )) = 0 for 1 ≤ i = j ≤ N . Proof Let 1 V˙ (t ) = − 2 1 − 2 1 − 2 where the fact that the communication graph G is bidirectional has been used. By the invariance principle Differentiate V along the solutions of Eq. 28 yields Fig. 6 Profiles of the states zi4 with controller (17) and controller (22) Fig. 5 Profiles of the states zi3 with controller (17) and controller (22) [37], zl1, zl4, and zl5 will converge to constants for l = 1, . . . , N . The following proof is the same as the proof in Theorem 1, but omitted here. Remark 9 In practice, there are always time delays due to communication and other factors. In our manuscript, we take time delays into account in our design of distributed protocol and we allow the delays to be arbitrarily large. In the theorem, communication delays only appear in the neighbors states. This Fig. 7 Profiles of the states zi5 with controller (17) and controller (22) assumption is reasonable because the communication delay is the dominated delay among all other time delays. The first term of Eq. 22 can be treated as a weighted sum of the relative state information between the current states of system i and the delayed state information of its neighboring. By applying the invariance principle, it is proved that our proposed cooperative control laws are still effective even existing communication delay. Assumption 2 is stronger than Assumption 1, since the existence of delays is in the communication. Corresponding to Theorem 2, we have the following delayed version result. Theorem 4 Consider the system consisting of N transformed systems (4) satisfying Assumption 2, and use distributed controller given by with the parameters satisfying pi ≥ 0, qi ≥ 0, ki ≥ 0, and iN=1 pi > 0, iN=1 qi > 0, iN=1 ki > 0, where communication delay τi (≥ 0) is a positive constant. Then the state zi of the ith transformed system (4) in the closed-loop system is bounded and converges to zero asymptotically, i.e., limt→∞zi (t ) = 0 for i = 1, . . . , N . Proof The proof is analogous as that of Theorem 2 and Theorem 3 and is omitted here. 4 Extensions In practical applications, multiple type (1, 2) nonholonomic mobile robots may need to achieve a prescribed formation other than rendezvousing at a common value. It is shown that, if convergence to a common value is feasible, then other formations can also be obtained by the simple transformation. Definition 2 The formation control problem discussed in this paper is to design a distributed controller for the ith system (2), based on its state information Fig. 8 The communication graph G2 with time-delays Fig. 10 Profiles of the states zi2 with communication delays τ = 0.5s and τ = 2.5s qi and the relative state ql of its neighbors for l ∈ Ni such that iiNN==11 xβii1== k1iN=π1, lpi mixt,→li∞mt→ ∞iN=1 βiNi=1 1=yik=2π, iNk=11, kp2iy∈, Z xyii −− yxjj − −cossinχχ csoinsχχ ppiiyx −− ppjjyx limt→∞(θi (t ) − θj (t )) = 0, 1 ≤ i = j ≤ N = 0, where χ is a free variable, and pix , piy are the prescribed displacements between the state value xi , yi of robot i and the system consensus value, which is Fig. 11 Profiles of the states zi3 with communication delays τ = 0.5s and τ = 2.5s unknown and depends on robots’ initial conditions and communication between robots. Let z¯i2 = (xi − pix ) cos θi + (yi − piy ) sin θi , z¯i3 = (xi − pix ) sin θi − (yi − piy ) cos θi , sin βi1 sin βi2 z¯i4 = −z¯i3 − 2lr sin(βi2 − βi1) + γ1ωz¯i2, Fig. 12 Profiles of the states zi4 with communication delays τ = 0.5s and τ = 2.5s Fig. 13 Profiles of the states zi5 with communication delays τ = 0.5s and τ = 2.5s where ω = ρ sin t , and ρ , γ1, γ2 are positive constants. Taking derivative of Eq. 34, we have Section 2.3. Let N = 4 and the initial values of each system be z˙¯i1 = u¯ i1 − ω, z˙¯i2 = −γ1z¯i2ω2 + ωz¯i4 + (u¯ i1 − ω)(z¯i4 − γ1ωz¯i2), z˙¯i3 = −γ2z¯i3ω2 + ωz¯i5 + (u¯ i1 − ω)(z¯i5 − γ2ωz¯i3), z˙¯i4 = u¯ i2 + γ1ω˙ z¯i2 + γ1ωu¯ i1z¯i4 − γ12ω2u¯ i1z¯i2, z˙¯i5 = u¯ i3 + γ2ω˙ z¯i3 + γ2ωu¯ i1z¯i5 − γ22ω2u¯ i1z¯i3. Lemma 5 If limt→∞(Z¯ i (t ) − Z¯ j (t )) = 0 for 1 ≤ i = j ≤ N , then Eq. 32 holds, where Z¯ i (t ) = [z¯i1, z¯i2, z¯i3, z¯i4, z¯i5]T . Furthermore, if limt→∞Z¯ l (t ) = 0 for l = 1, . . . , N , then Eqs. 32 and 33 hold. By replacing zij in Eqs. 17, 22, 27, and 31 with z¯ij for j = 1, . . . , 5, similar control algorithms can be obtained. By Lemma 5, the formation control problem is also solved. 5 Simulations We consider some examples to illustrate the proposed design schemes and verify the established theoretical results. Consider the system (4) discussed in Case 1 The communication graph G1 without communication delays is described in Fig. 2. Note that this communication graph G1 satisfies Assumption 1. The corresponding adjacency matrix A1 is given by A1 = ⎢⎢ ⎣ Two simulations are respectively implemented for the distributed control law (17) and the distributed control law (22) with p1 = 0.5, q2 = 0.5, k3 = 0.5 and other control parameters are all zero. We choose the parameter ρ = 1 in local change of coordinates and feedback (3). The simulations are conducted by the Matlab “ode45” method. The trajectories of states versus time plotted using solid line and dash-dot line shown in Figs. 3, 4, 5, 6 and 7 are corresponding to the distributed controller (17) and the distributed controller (22), respectively. Note that the states do not converge to zero directly, but are the same as its neighbors’. It demonstrates that if zi1, zi4, zi5 converge to nonzero constants, then zi2 and zi3 are bounded. Furthermore, if zi1, zi4, and zi5 converge to zero asymptotically, then zi2 and zi3 also converge to zero asymptotically. Case 2 The communication graph G2 with communication delays is described in Fig. 8. Note that this communication graph G2 satisfies Assumption 2. The corresponding adjacency matrix A2 is given by ⎡ 0 0.7 0.4 0.5 ⎤ 0.7 0 0.6 0 A2 = ⎢⎢ 0.4 0.6 0 0 ⎥⎥ . ⎣ ⎦ 0.5 0 0 0 To simplify the simulation, we assume all the communication delays are common to each system, namely τ1 = τ2 = τ3 = τ4 = τ . The simulation is implemented for the distributed control law (27). We choose the parameter ρ = 1 in local change of coordinates and feedback (3), p1 = 1.5, q2 = 1.5, k3 = 1.5 and other control parameters are all zero. In order to better analyze the influence of communication delays for the system, τ is set to be 0.5s, 2.5s in the two simulations, respectively. The simulations are performed by the Matlab “dde23” method. The trajectories of states versus time plotted using solid line and dash-dot line shown in Figs. 9, 10, 11, 12 and 13 are corresponding to the time delay τ = 0.5s and τ = 2.5s, respectively. Figs. 9–13 verify the fact that the states of every system (4) converge to zero asymptotically even with communication delays. It also indicates that the asymptotical convergence of the states can also be achieved for large constant delays. However, the cooperative performance is bad if communication delays are large. 6 Conclusion In this paper, the distributed cooperative control problem has been investigated for type (1, 2) nonholonomic mobile robots. Four distributed controllers are designed to ensure that the state of the transformed system converges to the common value or zero asymptotically with and without considering communication delays. Extension is also provided to extend the proposed schemes to the case, where the nonholonomic mobile robot needs to form a stable formation other than rendezvousing at a common value. The stability of the proposed methods is proved rigorously. Simulation results confirm the effectiveness of the proposed methods. It is our future work to solve the consensus problem for multiple nonholonomic mobile robots based on visual servoing. Acknowledgments This paper was partially supported by The Scientific Innovation program (13ZZ115), National Natural Science Foundation (61374040, 61203143), Hujiang Foundation of China (C14002), Graduate Innovation program of Shanghai (54-13-302-102). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Gang Wang received the B.Sc. degree in information and computing science from University of Shanghai for Science and Technology, Shanghai, China, in 2012, where he is currently pursuing the Ph.D. degree in systems analysis and integration. His current research interests include distributed control of nonlinear systems, adaptive control and robotics. Chaoli Wang received his Ph.D. degree in Control Theory and Engineering at Beijing Univ. of Aero. and Astro. in China in 1999, following his M.Sc. degree and B.Sc. degree in Applied Maths at Lanzhou University in Lan Zhou, China, respectively in 1992 and 1986. Currently, he is a Professor with the Department of Electrical Engineering at the University of Shanghai for Science and Technology. Wangs research interests include nonlinear control, robust control, robot dynamic and control, visual servoing feedback control and pattern identification. Qinghui Du received the Ph.D. degree from the Department of Control Science and Engineering at University of Shanghai for Science and Technology, China, in 2015. She is currently a Lecturer with the Department of Mathematics, Luoyang Normal University, Luoyang, China. Her main research interests include the control of stochastic nonholonomic systems. Lin Li received B.E. degree in automation from Qufu Normal University, Qufu, China, in 2004, and the Ph.D. degree in control theory and control engineering from Beihang University, Beijing, China, in 2010. She is currently with University of Shanghai for Science and Technology as an Associate Professor. Her current research interests include robust control and filtering, adaptive control and the cooperative control of multi-agent systems. Wenjie Dong received his MS degree in automatic control from Beijing University of Aeronautics and Astronautics in 1996 and PhD degree in electrical engineering from the University of California, Riverside, in 2009. He is an Associate Professor in the Department of Electrical Engineering, the University of Texas Rio Grande Valley, Edinburg, TX. 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Gang Wang, Chaoli Wang, Qinghui Du, Lin Li, Wenjie Dong. Distributed Cooperative Control of Multiple Nonholonomic Mobile Robots, Journal of Intelligent & Robotic Systems, 2016, 525-541, DOI: 10.1007/s10846-015-0316-x