Research Article
Adaptive coordinated tracking control for multi-robot system with directed communication topology
International Journal of Advanced Robotic Systems November-December 2017: 1–10 ª The Author(s) 2017 DOI: 10.1177/1729881417743227 journals.sagepub.com/home/arx
Qin Wang, Zuwen Chen and Yang Yi
Abstract In this article, an adaptive coordinated tracking control problem for a group of nonholonomic chained systems has been discussed under the assumption that the desired trajectory is available only to a part of neighboring agents. Firstly, coordinated tracking controllers under the directed communication topology graph containing a spanning tree are designed for two subsystems, adaptive control gains are employed in the linear subsystem based on the state information of neighboring agents, then the global information of the topology graph is not required to be known in the control laws. Furthermore, the backstepping strategy is applied in the rest of chained-form subsystem such that the state of all robots converge to the desired reference trajectory. And then, the results are further evolved with the case of switching topology. Finally, an application is introduced and the simulation results are given to show the validity of the proposed theoretical results. Keywords Multi-robot system, distributed control, adaptive control, coordinated tracking Date received: 26 July 2017; accepted: 7 October 2017 Topic: Special Issue – Intelligent Control Methods in Advanced Robotics and Automation Topic Editor: Lino Marques Associate Editor: Junzhi Yu
Introduction With the rapid development of distributed computing technology and modern control theory, distributed cooperative control of multi-agent systems has caused researchers’ tremendous attentions in the last few years. This research direction includes consensus, formation, rendezvous, containment, and tracking control.1 These problems have been discussed in pioneering works via different control strategies, such as leader–follower,2 virtual structure,3 behaviorbased,4 and graph theory methods.5 Obviously, the topic of coordinated tracking has a wide range of applications in engineering and physics, such as tracking control of mobile robots, satellites clustering, unmanned helicopter, and autonomous underwater vehicles.6–8 The main objective of coordinated tracking control is to make a group of autonomous vehicles to track a target in a cooperative manner via distributed control protocols such
that some challenging tasks can be completed and many inevitable physical constraints including sensor ranging are removed. Hence, this distributed control behavior not only reduces the operational costs but also improves the robustness and adaptivity of the systems. In most of the previous technical notes concerned with decentralized control, the coordinated control problem of multi-agent systems which are usually considered as linear dynamic systems has been studied by many researchers.9 In the study by Qu et al.,10 by
College of Information Engineering, Yangzhou University, Yangzhou, People’s Republic of China Corresponding author: Qin Wang, College of Information Engineering, Yangzhou University, Yangzhou 225009, People’s Republic of China. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 the use of matrix theory, a sufficient and necessary condition is given to guarantee the convergence of the closedloop system. Two different distributed tracking controllers for linear dynamic systems in the study by Li et al.11 are proposed to guarantee that the states reach an agreement. Ni and Cheng12 considered leader-following consensus problem of high-order linear systems, the Lyapunov inequality and Riccati inequality are introduced to solve the consensus problem. In practical applications, the practical physical model cannot be described by the linear model, and the most practical model may be nonlinear model. Moreover, most of the mobile models have to satisfy nonholonomic constraints. The design of control laws for nonholonomic systems is involved, mainly due to Brockett’s condition which indicates that continuous pure state feedback law is not applied. Therefore, when agents with nonholonomic constraint are considered, the coordinated tracking control problem becomes more challenging. Some proposed effective control strategies, such as time varying,13 discontinuous feedback,14 and hybrid control method,15 establish the foundation for our further study. The distributed tracking controller is designed in the study by Yang et al.16 with the aid of the dynamic oscillator strategy. In the study by Dong,17 the state of each follower converges to the leader by utilizing the adaptive control technique. Some other methods including small gain and dynamic feedback linearization are employed in the study by Liu and Jiang18 to deal with the formation problem of nonholonomic mobile robots. Cao et al.19 also considered the consensus problem of high-order chained-form systems, the cascading theory is applied to construct hybrid distributed controllers. In the studies by Peng et al.20 and Qiu et al.,21 distributed adaptive controller is designed to steer a class of mobile robots to achieve a prespecified geometric pattern and track a reference trajectory. Xu et al.22 consider the output consensus of nonholonomic system, a time-varying control law is given to demonstrate the stability of the overall closed-loop system. However, for the tracking control problem in the study by Zhang et al.9, and Ni and Cheng,12 the input information of a reference trajectory is either available to all the other neighboring agents or equal to zero, which is impractical and restrictive due to the constraints of large scale of network. All the above studies13–22consider the nonholonomic chained-form system, then the global information of the communication topology graph is required to be known. Motivated by the previous studies and investigations on coordinated tracking control of multiple mobile robots, the distributed coordinated tracking problem for a group of nonholonomic chained systems is discussed in this article. Compared to the result in the study by Xu et al.,22 the contributions of this work are stated as follows. First, the time-varying control strategy is applied in the cooperative control law instead of strict time-invariant continuous feedback linear system. Second, the controllers are designed on the basis of the neighboring state information of the agent
International Journal of Advanced Robotic Systems with the coupling adaptive gains. The global information of communication topology is not required to be known in the controllers. Furthermore, it is worth mentioning that the tracking problem is solved under a practical assumption that the information of the reference trajectory is available to a part of neighboring systems rather than all neighboring systems.
Preliminaries and problem statement Graph theory and notations In this article, the interconnection topology of multi-agent systems is modeled as a directed graph. The communication topology of the N agents can be described as the weighted graph G ¼ ðV; E; AÞ, where V ¼ fvi ; v 2 ; . . . ; vN g is a finite vertex set which represents each individual agent, and E V V is a set of edge with an ordered pair ðvi ; vj Þ which describes the information exchange between nodes i and j. ðj; iÞ 2 E is a directed edge which means the state information of agent j is available to agent i. If the node j can receive the information from the node i, then the set of all neighboring nodes j is defined by Ni ¼ fj 2 V : ðj; iÞ 2 Eg. The N N adjacent matrix AðGÞ ¼ aij associated with edges of the directed graph is denoted such that aij > 0 if ðvj ; vi Þ 2 EðGÞ; aii ¼ 0 for i ¼ 1; 2; . . . ; N , otherwise. The Laplacian matrix L ¼ ½lij 2 RNN is defined as L ¼P D A, where D ¼ diagfd1 ; d2 ; . . . ; dN g, lii ¼ di ¼ Nj¼1;j6¼i aij and P lij ¼ aij , i 6¼ j which satisfies that Nj¼1 lij ¼ 0. A directed path from vi 1 to vil is a sequence of ordered edges in the form ðik ; ikþ1 Þ, k ¼ 1; . . . ; l 1 with distinct vertices ik 2 V, k ¼ 1; . . . ; l. A directed graph has a spanning tree if there exists at least one node, which has directed paths to all other nodes, and such node is called a root node. Throughout this article, the following notations are used. 1N represents a column vector with all elements equal to one. For a vector x ¼ ðx 1 ; x 2 ; . P . . ; xN ÞT 2 RN , let jjxjj 1 denote its 1-norm, where jjxjj1 ¼ Ni¼1 jxi j. The sign function is denoted by signðxÞ ¼ 1 if the variable x is negative; signðxÞ ¼ 1 if the variable x is positive; and otherwise, signðxÞ ¼ 0. For the matrix L 2 RN N , λ max ðLÞ and λ min ðLÞ denote the maximum and minimum eigenvalue of the matrix L, respectively. Assumption 1. Graph G contains a directed spanning tree, and C ¼ diagðc 1 ; . . . ; cN Þ is the connection matrix, where the vector c ¼ ½c 1 ; . . . ; cN T 2 RN ; ci > 0; i ¼ 1; . . . ; N , and at least one of the elements of c is nonzero (equivalently, at least one of the N systems can connect with system 0).
Problem statement Many kinematic systems in our life such as nonholonomic wheeled mobile robots can be transformed into chained form by coordinate transformation.22 In this article, we
Wang et al.
3
consider a group of chained-form nonholonomic systems as follows 8 > < x_ 1i ¼ u 1i (1) x_pi ¼ xpþ1;i u1i ; p ¼ 2; 3; :::; n 1 > : x_ni ¼ u2i where x ¼ ðx 1i ; x 2i ; . . . ; xni ÞT and u ¼ ðu1i ; u2i ÞT are the state and control input of agent i, i ¼ 1; 2; . . . ; N , respectively. In chained-form systems, the formation tracking control problem is taken into consideration in this article. Assume that the desired reference trajectory x 0 ¼ ðx 10 ; x 20 ; . . . ; xn0 ÞT is described by the following chained form 8 > < x_ 10 ¼ u10 (2) x_p0 ¼ xpþ1;0 u 10 ; p ¼ 2; 3; :::; n 1 > : x_n0 ¼ u20 T
where u0 ¼ ðu 10 ; u20 Þ is the known time-varying reference input. Assumption 2. The reference inputs u0 ¼ ðu10 ; u20 ÞT are bounded, and the states x 10 ; xn0 ; xp0 are also bounded. Let us define the tracking error as x^k ¼ xki xk0 ; k ¼ 1; 2; :::; n. It is easy to derive the following differential equations by utilizing equations (1) and (2) x^_1 ¼ u1i u10 x^_2 ¼ x_ 2i x_ 20 ¼ x 3i u1i x 30 u10 ¼ x^3 u 1i þ x 30 ðu1i u10 Þ .. . x^_n1 ¼ x_ni x_n0 ¼ x^n u1i þ xn0 ðu1i u10 Þ x^_n ¼ u2i u20 (3) The Coordinated Tracking Control Problem is to design the control laws u1i and u2i for each agent i, i ¼ 1; 2; :::; N based on xki and its neighboring state xkj , k ¼ 1; 2; :::; n; j 2 Ni , such that, for an initial coordinated tracking state error x^k ð0Þ ¼ xki ð0Þ xk0 ð0Þ, the state of each agent can converge to the desired reference trajectory, that is lim ðxki ðtÞ xkj ðtÞÞ ¼ 0
(4)
lim x^k ðtÞ ¼ lim ðxki ðtÞ xk0 ðtÞÞ ¼ 0
(5)
t!1
t!1
t!1
Lemma 1. If graph G has a directed spanning tree, zero is an eigenvalue of L with associated right eigenvector 1N , the other nonzero eigenvalues have positive real parts, and the matrix ½L þ C is positive definite.11
Controller design and convergence analysis In this section, we will design a coordinated tracking controller under a directed communication topology containing directed spanning tree by two parts. In the first part, we add a new auxiliary variable in the traditional cooperative tracking control protocol instead of linear control strategy. An adaptive control law is also proposed for updating the linear subsystem gain. In the second part, the controller for the rest of system in chained form is constructed by the backstepping design method.
Control law design for the linear subsystem Consider the linear subsystem in equation (1) x_ 1i ¼ u 1i
(6)
The control law u1i is designed by adopting the adaptive scheme for updating coupling gain and adding a new auxiliary variable in the traditional cooperative tracking protocol as follows X u1i ¼ aij ðx 1i x 1j Þ ci ðx 1i x 10 Þ þ z 1i (7) j2Ni
" z_1i ¼ 1i
# X aij ðz 1i z 1j Þ þ ci ðz 1i z 10 Þ j2Ni
"
# X 1i sign aij ðz 1i z 1j Þ þ ci ðz 1i z 10 Þ
(8)
j2Ni
" _ 1i ¼
X
#2 aij ðz 1i z 1j Þ þ ci ðz 1i z 10 Þ
j2Ni
X þ aij ðz 1i z 1j Þ þ ci ðz 1i z 10 Þ j2N
(9)
i
where u10 ¼ z 10 , and aij is the element of the adjacency matrix of G. The parameter ci > 0 if the system 0 is available to system i, 1 i N and ci ¼ 0, otherwise. 1i ðtÞ denotes the time-varying adaptive gain. Theorem 1. For the system (6) with controller (7) to (9), if assumptions 1 and 2 hold, then we have limt!1 ðx 1i ðtÞ x 10 ðtÞÞ ¼ 0 and limt!1 ðz 1i ðtÞ z 10 ðtÞÞ ¼ 0, for all i, 1 i N . Proof. The closed-loop system can be written as in the following form z_1 ¼ 1 Lz 1 1 sign½Lz 1 þ Cðz 1 1N z 10 Þ 1 Cðz 1 1N z 10 Þ
(10)
x_ 1 ¼ z 1 Lx 1 Cðx 1 1N x 10 Þ
(11)
where 1 ¼ diagð 11 ; 12 ; . . . ; 1N Þ; x 1 ¼ ðx 11 ; x 12 ; . . . ; x 1N ÞT , C ¼ diagðc 1 ; c 2 ; . . . ; cN Þ;. z 1 ¼ ðz 11 ; z 12 ; . . . ;
4
International Journal of Advanced Robotic Systems
z 1N ÞT . Let z^1 ¼ z 1 1N z 10 . It follows that z 1 1 sign½ðL þ CÞ^ z 1 1N z_ 10 z^_1 ¼ 1 ðL þ CÞ^ (12) x 1 þ z^1 x^_1 ¼ ðL þ CÞ^
(13)
where L1N ¼ 0 is applied. Choosing the following Lyapunov candidate function
V_ 1 ¼ z^T1 ðL þ CÞz^_1 þ
N 1 1X V1 ¼ z^T1 ðL þ CÞ^ z 1 þ ð Þ 2 2 2 i¼1 1i
(14)
where is a positive constant. It is directly checked that V1 0. Taking the time derivative of V1 along the solution of equation (14) as follows
N X ð 1i Þ_ 1i i¼1
¼
1 ^ zT1 ðL
2
þ CÞ z^1 z^T1 ðL þ CÞ 1 sign½ðL þ CÞ^ z 1
z^T1 ðL þ CÞ1N z_10 þ
N X ð 1i Þ_ 1i i¼1
^ zT1 ðL
2
þ CÞ ^ z 1 jj^ zT1 ðL þ CÞjj 1 þ supjz_10 jjj^ zT1 ðL þ CÞjj 1
¼ ^ zT1 ðL þ CÞ 2 z^1 ð supju10 jÞjj^ zT1 ðL þ CÞjj 1
In light of Lemma 1 in the study by Li et al.,11 it yields that the matrix ðL þ CÞ 2 is positive definite. Choosing large enough such that supju10 j, it can obtain that V_ 1 ^ zT1 ðL þ CÞ 2 z^1 ¼ W ð^ xÞ
(15)
By noting that W ð^ xÞ is positive definite, then it follows that V_ 1 0, implying that V1 ðtÞ is not increasing. Therefore, in view of equation (14), we know that 1i ; ^z 1 are bounded. Since u10 is bounded, it implies that z^_1 is bounded from equation (12). As V1 ðtÞ is nonincreasing, it indicates that V1 ðþ1Þ exists as t ! þ1. Integrating ð þ1 W ð^ zðtÞÞ dt V1 ð^ zð0ÞÞ equation (15), it has 0 ð þ1 W ð^ zðtÞÞ dt exists and is finite. V1 ðþ1Þ. Hence, 0
Because z^1 and ^ z_1 are bounded, it is easy to be found that W_ ð^ zÞ is also bounded, which accordingly guarantees the uniform continuity of W ð^ zÞ. Therefore, by Barbalat’s Lemma, it can obtain that W ð^ zÞ ! 0 as t ! þ1, it implies that z^1 ! 0 as t ! þ1. Equation (13) can be treated as a stable system, hence x^1 exponentially converges to zero as z^1 converges to zero. Consequently, it shows that limt!1 x^1 ðtÞ ¼ limt!1 ðx 1i ðtÞ x 0 ðtÞÞ ¼ 0 and limt!1 ðz 1i ðtÞ z 10 ðtÞÞ ¼ 0.
Control law design for the remainder of chained-form subsystem Noting that the remainder of chained-form subsystem in equation (1), the control law u 2i is designed by utilizing the backstepping design method.
Theorem 2. Consider the rest of chained-form subsystem in equation (1), if assumptions 1 and 2 hold, then the coordinated tracking problem of chained-form subsystems is solved with the following controllers " # X _ aij ð~ xnj x~ni Þ ci x~ni u2i ¼ u20 þ xn x~n1 u1i þ k 2 j2Ni
(16) x3i ¼ k 1
x^2i x 30 ðu1i u10 Þ ; u1i u1i
x~3i ¼ x^3i x3i
x_3i x^2i u1i k 1 x~3i x 40 ðu1i u10 Þ þ ; u1i u1i u1i u1i x~4i ¼ x^4i x4i
(17)
x4i ¼
(18)
k 1 x~p1;i xp0 ðu 1i u10 Þ x_p1;i x~p2;i u1i þ ; u1i u1i u1i u1i x~pi ¼ x^pi xpi
xpi ¼
(19) where p ¼ 5; 6; :::; n, i ¼ 1; 2; :::; N , k 1 > ðn 2Þλ max ðL þ CÞ, k 2 > 0. Proof. Firstly, consider the error dynamic chained-form subsystem in equation (3) x^_2 ¼ x^3 u1i þ x 30 ðu1i u10 Þ
(20)
In order to simplify the variables, we will omit partial second variable i of right subscript. Take x3 as a virtue control input to make this subsystem (20) stable, we have x^_2 ¼ x3 u1i þ x 30 ðu1i u10 Þ
(21)
Wang et al.
5
There exists a function to satisfy V2 ð^ xÞ 0, then we choose V2 ¼ x^2 =2 as a Lyapunov candidate function to verify the stability of equation (20). Since u1i 6¼ 0, setting the virtue control in equation (17), we get V_ 2 ðxÞ ¼ x^2 x3 u 1i þ x^2 x30 ðu1i u10 Þ ¼ k 1 x^22 ¼ 2k1 V2 < 0 (22)
which ensures the exponential stability of the subsystem (20). Obviously, as V 2 decays with 2k 1 , x^2 decays with k 1 . The attenuation index of u1i is λ max ðL þ CÞ, where λ max ðL þ CÞ is the maximum eigenvalue of matrix ðL þ CÞ. Based on equation (17), we require that x^2 converges faster than u 1i , hence choosing the parameter k 1 > λ max ðL þ CÞ, which guarantees the boundedness of x3 . Secondly, we introduce the error variable x~3 ¼ x^3 x3 and extend equation (21), applying this error variable x 3 þ x3 Þu1i þ x 30 ðu 1i u10 Þ x^_2 ¼ ð~
(23)
x~_3 ¼ x^4 u1i þ x 40 ðu1i u 10 Þ x_3
(24)
x^_n1 ¼ ðu2i u20 Þu1i þ xn;0 ðu 1i u10 Þ
(28)
x~_n ¼ u2i u20 x_n
(29)
Choosing the Lyapunov candidate function Vn ¼ Vn1 þ x~n2 =2 and using equation (16), we have V_ n ¼ 2k 1 Vn1 þ x~n x~n1 u 1i þ x~n ½u2i u20 x_n " # X ¼ 2k 1 Vn1 þ k 2 x~n aij ð~ xnj x~ni Þ ci x~ni
P Let Vn ¼ Ni¼1 Vn;i , and define x^2 ¼ ½^ x 21 ; . . . ; x^2N T , x^p ¼ ½^ xp1 ; x^pN T ; p ¼ 3; 4; :::; n, then we have N X V_ n;i i¼1
(25)
¼ 2k 1
Using Lyapunov candidate function V3 ¼ V2 þ and setting the virtue control in equation (18), then take the time derivative, we have V_ 3 ¼ V_ 2 þ x~3 x~_3 ¼ x^2 ð~ x 3 þ x3 Þu1i þ x^2 x 30 ðu1i u10 Þ þ x~3 ½ x 4 u1i þ x 40 ðu1i u10 Þ x_3
(26) (27)
Regard xn as a virtue control input x~_n1 ¼ xn u1i þ xn0 ðu1i u10 Þ x_n1 Using Lyapunov candidate function Vn1 ¼ Vn2 þ 2 x~n1 =2, then the time derivative of Vn1 is given as follows
2 ¼ 2k 1 Vn2 k 1 x~n1 ¼ 2k 1 Vn1 < 0
Vn1;i
N X
x~n;i
N X aij ð~ xnj x~ni Þ ci x~n;i x~n;i
i¼1 N X
j¼1
Vn1;i k 2 x~Tn ðL þ CÞ~ xn
i¼1
In this step, x4 is a function of x_3 =u1i , that also is the function of x^2 =u21i , based on equation (18). Using the same description method as before, we choose the parameter k 1 > 2λ max ðL þ CÞ that guarantees the boundedness of x4 . For all further steps, similarly introduce an error variable x~n1 ¼ x^n1 xn1 , and we have
V_ n1 ¼ V_ n2 þ x~n1 x~_n1 ¼ 2k 1 Vn2 þ x~n2 x~n1 u1i þ x~n1 ½ xn u1i þ xn0 ðu1i u10 Þ x_n1
þ k2
¼ 2k 1
¼ k 1 x^22 þ x^2 x~3 u 1i þ x~3 ½ x 4 u1i þ x 40 ðu1i u10 Þ x_3 2 2 ¼ k 1 ð^ x 2 þ x~3 Þ ¼ 2k 1 V3 < 0
x~_n1 ¼ x^n u1i þ xn0 ðu1i u10 Þ x_n1
N X i¼1
x~23 =2
xn1 þ xn1 Þu1i þ xn1;0 ðu1i u 10 Þ x^_n1 ¼ ð~
(30)
j2Ni
V_ n ¼
Similarly, regard x4 as a virtue control input x~_3 ¼ x4 u1i þ x 40 ðu1i u 10 Þ x_3
Using a similar method as before, choosing the parameter k 1 > ðn 2Þλ max ðL þ CÞ that assure the convergence and boundedness of the virtue control xn . Finally, in the last step, introduce an error variable x~n ¼ x^n xn , and we have
Since L þ C is positive definite matrix. Hence, we get V_ n 0. As Vn is nonincreasing and bounded from below by zero, it indicates that limt!1 Vn exists and x~p is ð þ1 V_ n ð^ xðtÞÞ dt ¼ Vn ðþ1Þ Vn ð^ xð0ÞÞ bounded, hence 0
exists and is finite. And by the recursive formula from equations (17) to (19), it can be obtained that x^_2 ; x~_p ; p ¼ 3; 4; :::; n; are bounded. The derivative of V_ n is P V€n ¼ 2 k 1 Ni¼1 V_ n1;i 2k 2 x~Tn ðL þ CÞx~_n which is bounded, it implies that V_ n is uniformly continuous. Therefore, with the aid of Barbalat’s Lemma, it can be obtained that V_ n ! 0, p ¼ 3; 4; :::; n 1 as t ! 1. Moreover x^2 ! 0, x~p ! 0, p ¼ 3; 4; :::; n 1, and x~n ¼ 0, that is, x^p ! 0, p ¼ 2; 3; :::; n 1, and x^n ¼ 0. Therefore, we can get limt!1 ðxpi xp0 Þ ¼ 0, p ¼ 2; 3; :::; n; i ¼ 1; 2; :::; N . Remark 1. Compared to the static controller in the study by Li et al.,11 the proposed adaptive control law (7) to (9) does not need to calculate the minimal eigenvalue of L, then the global information of the topology graph is not required to be known. On the other hand, the coupling gains need to be dynamically updated in equations (7) to (9), implying that the adaptive control law (7) to (9) is more complex than the static controller.
6
International Journal of Advanced Robotic Systems
Coordinated tracking for switching communication graph
5
In the preceding sections, the fix communication graph is considered, If the communication graph GðtÞ is switching at time t, the previous theorem also holds and is extended in the following results.
1
4 0
Theorem 3. For the system (6) ð1 i N Þ with controller (7) to (9), if the switching communication topology GðtÞ contains a directed spanning tree at each instant t, assumption 1 holds, then we have limt!1 x^1 ðtÞ ¼ limt!1 ðx 1i ðtÞ x 10 ðtÞÞ ¼ 0 and limt!1 ðz 1i ðtÞ z 10 ðtÞÞ ¼ 0, for all i, 1 i N .
2
3
Figure 1. The communication topology graph G 1 .
Proof. In any time instant t, we have the following closedloop systems 30
z^_1 ¼ 1 ðL þ CÞt z^1 1 sign½ðL þ CÞt ^ z 1 1 N z_10 (31)
1 1 V1t ¼ z^T1 ðL þ CÞt z^1 þ 2 2
ð 1i Þ 2
15
(32)
Choosing Lyapunov candidate function N X
20
y (m)
x^_1 ¼ ðL þ CÞt x^1 þ z^1
Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 0
25
10 5 0
(33)
–5
i¼1
–10
And taking the time derivative of V1t along the solution of equation (31) with the aid of the equations (7) to (9). One gets
where > 0, and λ min;t ðL þ CÞ is the smallest eigenvalue of symmetric matrix ðL þ CÞt at each time instant t. Because the symmetric matrix ðL þ CÞt2 is positive definite, λ min;t ðL þ CÞ is greater than zero for the each time interval t. Therefore, by Barbalat’s Lemma, similar to the proof of theorem 1, it proves that V_ 1t 0 and z 1i asymptotically converge to z 10 for 1 i N . Furthermore, on the basis of equation (32), it is easy to get that x 1i asymptotically converges to x 10 for 1 i N . Theorem 4. Consider the remaining subsystem in equation (1) ð1 i N Þ, if the switching communication topology GðtÞ contains a directed spanning tree at each instant t and at least one of the N systems can connect with system 0, assumption 1 holds, and the system is driven by the controller (16) and virtue control law (17) to (19), where the control parameter aij > 0, k 2 > 0, and k 1 > ðn 2Þλ max ðL þ CÞ. Then, the coordinated tracking among chained-form subsystems is achieved, that is lim ðxki ðtÞ xkj ðtÞÞ ¼ 0;
t!1
lim x^k ðtÞ ¼ lim ðxki ðtÞ xk0 ðtÞÞ ¼ 0
t!1
t!1
where k ¼ 2; 3; :::; n, i ¼ 1; 2; :::; N ; j 2 Ni .
5
0
–5
–10
–15
10
15
x (m)
Figure 2. The movement trajectories of five robots with a reference path.
10 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
5
0 ω1i−ω10 (rad/s)
V_ 1t ^ zT1 ðL þ CÞt2 ^ z 1 λ 2min;t ðL þ CÞ^ zT1 ^ z 1 (34)
–15 –20
–5
–10
–15
–20 0
5
10
15
20
25
30
t (s)
Figure 3. The angular velocity tracking error of five robots.
Proof. Similar process of proof in Theorem 2, the variables xpi ; x~pi ; p ¼ 3; 4; :::; n; i ¼ 1; 2; :::; N are estimated recursively by backstepping design method, the condition that the parameter k 1 > ðn 2Þλ max;t ðL þ CÞ is satisfied to
Wang et al.
(a)
7
(b)
10
15
Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
8 6 4
Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
10 5 x2i–x20
x1i–x10
2 0 –2
0 –5
–4 –6
–10
–8 –15
–10 0
5
10
15
20
25
0
30
5
10
15
25
30
t (s)
t (s)
(c)
20
5
0
x3i–x30
–5
–10 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
–15
–20 0
5
10
15
20
25
30
t (s)
ensure the virtue control bounded, where λ max;t ðL þ CÞ is the maximum eigenvalue of Laplacian matrix L þ C at each time instant t. Hence, in the last step, since L þ C is positive definite at each time instant, we get V_ n 0. Therefore, by Barbalat’s Lemma, it is easy to obtain that x^p ! 0; p ¼ 2; 3; :::; n 1 and x^n ¼ 0. Finally, we conclude that limt!1 ðxpi xp0 Þ ¼ 0; p ¼ 2; 3; :::; n; i ¼ 1; 2; :::; N . Remark 2. In control laws (7) and (16), aij and ci are control parameters. The parameter of the control law affects the convergence velocity of the system. Therefore, the convergence rate of ðxi x0 Þ relies on the switching communication topology graph at different time intervals.
Applications Tracking control of wheeled mobile robots has extensive application in cooperative transportation and target tracking. In this section, an application example is showed in formation control with a desired trajectory by converting the wheeled mobile robots into nonholonomic chained form and integrating with the proposed control laws.
Consider a group of N wheeled mobile robots to track a presupposed target in a plane. The kinematic model of each mobile indexed by i 2 N is given as follows x_fi ¼ vi cosθi ;
y_fi ¼ vi sinθi ;
θ_ i ¼ oi
(35)
where ðxfi ; yfi Þ is the position of robot i in an inertial coordinate system. θi is the orientation of robot i and vi ; oi denote the linear velocity and angular velocity, respectively. The trajectory of tracked target ðxl0 ; yl0 ; θ0 Þ is described as x_l0 ¼ v 0 sinθ 0 ;
y_l0 ¼ v 0 sinθ 0 ;
θ_ 0 ¼ o 0
(36)
The position of desired formation is given by constant vectors ðpxi ; pyi Þ in local coordinate frame. Without loss of generality, if the origin of the local coordinate frame is the center of trajectory of desired formation, then we conclude P P that Ni¼1 pxi ¼ 0 and Ni¼1 pyi ¼ 0. Control goals: Design coordinated tracking controllers for each robot, on the basis of neighboring state information and tracked trajectory information, such that the multi-robot system can obtain desired formation
8
International Journal of Advanced Robotic Systems and track the desired trajectory. Therefore, coordinated tracking control goals can be rewritten as follows
5
lim ½ðxfj xfi Þ ðpxj pxi Þ ¼ 0 t!1
lim ½ðyfj yfi Þ ðpyj pyi Þ ¼ 0;
1
1 i 6¼ j N
4
t!1
0
(37) lim
t!1
N X xfi i¼1
N
! xl0
¼ 0;
lim
t!1
N X yfi i¼1
N
! yl0
¼0
2
(38)
30 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 0
25 20 15 10 5 0
(40)
The systems in equations (35) and (36) can transform into equations (1) and (2) with n ¼ 3 through the coordinate transformation in equations (39) and (40). The following corollary can be easily derived through simple derivative and calculation to the above results, which the process of the proof is omitted here. Corollary 1. By the changes of variables, the systems in equations (35) and (36) are converted to the nonholonomic chained system in equations (1) and (2) with n ¼ 3, under theorems 1 and 2, the coordinated tracking can be achieved, that is limt!1 ðxpi ðtÞ xp0 ðtÞÞ ¼ 0; p ¼ 1; 2; 3; i ¼ 1; 2; :::; N . Moreover, robots obtain desired formation and track the desired trajectory, that is, equations (37) and (38) are also reached.
Numerical illustrations To verify the effectiveness of proposed theoretical results, we give the corresponding value to carry on the simple simulation. Consider the example described in the above section. There exist five robots, and their desired trajectory coordinates are given by ðpx1 ; py1 Þ ¼ ð6:5; 7:5Þ, ðpx2 ; py2 Þ ¼ ð9:0; 4:0Þ, ðpx3 ; py3 Þ ¼ ð1 :0 ; 10Þ, ðpx4 ; py4 Þ ¼ ð9:5; 2Þ, and ðpx5 ; py5 Þ ¼ ð5:0; 8:5Þ. By
–5 –10 –15 –20
–15
–10
–5
0
5
10
15
x (m)
Figure 6. The movement trajectories of five robots with a reference path.
10 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
5
0 ω1i−ω10 (rad/s)
In the same way, letting 8 x 10 ¼ θ 0 > > > > > > < x 20 ¼ xl0 sinθ 0 þ yl0 cosθ0 x 30 ¼ xl0 cosθ 0 þ yl0 sinθ 0 > > > > u10 ¼ o0 > > : u20 ¼ vi þ x 20 o0
Figure 5. The communication topology graph G 2 .
y (m)
For completing the above control goal, in this subsection, it is necessary to construct a new coordinate transformation as follows 8 x 1i ¼ θi > > > > > > < x 2i ¼ ðxfi pxi Þ sinθi þ ðyfi pyi Þ cosθi (39) x 3i ¼ ðxfi pxi Þ cosθi þ ðyfi pyi Þ sinθi > > > u1i ¼ oi > > > : u2i ¼ vi þ x 2i oi
3
–5
–10
–15
–20 0
5
10
15
20
25
t (s)
Figure 7. The angular velocity tracking error of five robots.
30
Wang et al.
9
10
15 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
8 6 4
Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
10 5 x2i−x20
x1i−x10
2 0 –2
0 –5
–4 –6
–10
–8 –10
–15 0
5
10
15
20
25
30
0
5
10
15
t (s)
20
25
30
t (s) 5
0
x3i−x30
–5
–10 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5
–15
–20 0
5
10
15
20
25
30
t (s)
Figure 8. The state tracking errors of five robots.
choosing bounded reference input in equation (32) with v 0 ¼ 5 and o 0 ¼ 5, the tracked trajectory is generated by ðxl0 ; yl0 ; θ 0 Þ ¼ ð sinð5tÞ; cosð5tÞ; 5tÞ. Let the communication topology graph be shown in Figure 1. The coordinated tracking control can be achieved by corollary 1 combining with previous results in the section “ Controller design and convergence analysis.” The control parameters are selected as aij ¼ 1, ci ¼ 1, k1 ¼ 2, and k 2 ¼ 1. The formation trajectories of five robots with a tracked reference path are shown in Figure 2. Figures 3 and 4 show that the angular velocity and the original three different state tracking of five robots converge to the reference one, which confirm the result in theorems 1 and 2. If the communication topology graph is different in each time interval, the control laws are valid in theorem 3. Assume that the switching communication topology graph shown in Figures 1 and 5 satisfies the following rules G1 ift roundðtÞ 0 G¼ G2 ift roundðtÞ < 0 Figure 6 shows the formation trajectory of five robots with a tracked reference path for switching topology graph
case. In the case of topology graph switching, Figures 7 and 8 show that the tracking error of angular velocity and three original states still converge to zero, which verifies the effectiveness of the proposed coordinated tacking control algorithm.
Conclusions In this article, the distributed adaptive coordinated tracking control problem for a group of nonholonomic chained systems has been discussed under the condition that the communication topology graph has a spanning tree. At least one of the agents can receive the state information of the desired reference trajectory whose control input is bounded. The design of control laws is divided by two parts. The control law of the linear subsystem is proposed with the aid of adaptive control method, and the control protocol of the remaining subsystem in chained form is addressed on the basis of the backstepping techniques. The state of each agent exponentially converges to the reference trajectory by the virtue of Lyapunov techniques, Barbalat’s Lemma and graph theory. Finally, the coordinated tracking
10 controllers design is also investigated under the switching topology. Simulation results show the validity of the proposed controllers.
International Journal of Advanced Robotic Systems
10.
Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Nature Science Foundation under grants 61503329, 61573307, 61473249, and 61773335; Jiangsu Planned Projects for Postdoctoral Research Funds 1601024B; and the Natural Science Foundation of Jiangsu Province BK20171289.
11.
12.
13.
14.
References 1. Ren W, Beard RW, and Atkins EM. Information consensus in multivehicle cooperative control. IEEE Control Syst 2007; 27(2): 71–82. 2. Tanner HG, Pappas GJ, and Kumar V. Leader-to-formation stability. IEEE Trans Robot Autom 2004; 20(3): 443–445. 3. Ren W and Beard RW. Formation feedback control for multiple spacecraft via virtue structures. Control Theory Appl 2004; 151(3): 357–368. 4. Balch TR and Arkin RC. Behavior-based formation control for multiagent robot teams. IEEE Trans Robot Autom 1998; 14(6): 926–936. 5. Lin Z, Francis B, and Maggiore M. State agreement for coupled nonlinear systems with time-varying interaction. Soc Ind Appl Mathem 2007; 46(1): 288–307. 6. Yoo SJ and Kim TH. Distributed formation tracking of networked mobile robots under unknown slippage effects. Automatica 2015; 54(C): 100–106. 7. Karimoddini A, Lin H, Chen BM, et al. Hybrid threedimensional formation control for unmanned helicopters. Automatica 2013; 49(2): 424–433. 8. Wang Y, Yan W, and Li J. Passivity-based formation control of autonomous underwater vehicles. IET Control Theory Appl 2012; 6(4): 518–525. 9. Zhang H, Lewis F, and Das A. Optimal design for synchronization of cooperative systems: state feedback, observer, and
15.
16.
17.
18.
19.
20.
21.
22.
output feed-back. IEEE Trans Autom Control 2011; 56(8): 1948–1952. Qu Z, Wang J, and Hull RA. Cooperative control of dynamical systems with application to autonomous vehicles. IEEE Trans Autom Control 2008; 53(4): 894–911. Li Z, Liu X, Ren W, et al. Distributed tracking control for linear multiagent systems with a leader of bounded unknown input. IEEE Trans Autom Control 2013; 58(2): 518–523. Ni W and Cheng D. Leader-following consensus of multiagent systems under fixed and switching topologies. Syst Control Lett 2010; 59(3): 209–217. Dong WJ and Farrell JA. Cooperative control of multiple nonholonomic mobile agents. IEEE Trans Autom Control 2008; 53(6): 1434–1448. Khoo S, Xie L, and Man Z. Leader–follower consensus control of a class of nonholonomic systems. In: Proceedings of the 11th international conference on control, automation, robotics and vision, Singapore, 7–10 December 2010, pp.1381–1386. IEEE. Kolmanovsky I and McClamroch NH. Hybrid feed-back laws for a class of cascade nonlinear control systems. IEEE Trans Autom Control 1996; 41(9): 1271–1282. Yang QK, Fang H, Cao M, et al. Distributed trajectory tracking control for multiple nonholonomic mobile robots. IFAC PapersOnLine 2016; 49(4): 31–36. Dong WJ. Tracking control of multiple-wheeled mobile robots with limited information of a desired trajectory. IEEE Trans Robot 2012; 28(1): 262–268. Liu T and Jiang Z. Distributed formation control of nonholonomic mobile robots without global position measurements. Automatica 2013; 49(2): 592–600. Cao KC, Jiang B, and Yue D. Consensus of multiple nonholonomic chained form systems. Syst Control Lett 2014; 72: 61–70. Peng Z, Yang S, Wen G, et al. Adaptive distributed formation control for multiple nonholonomic wheeled mobile robots. Neurocomputing 2016; 173(P3): 1485–1494. Qiu YL and Xiang LY. Distributed adaptive coordinated tracking for coupled nonholonomic mobile robots. IET Control Theory Appl 2014; 8(18): 2336–2445. Xu YJ, Tian YP, and Chen YQ. Output consensus for multiple non-holonomic systems under directed communication topology. Int J Syst Sci 2015; 46(3): 451–463.
