Static group-bipartite consensus in networked robot systems with integral action

With the increasing complexity of modern industry, the traditional single-target control of swarm robots can no longer meet application requirements. Hence, this article addresses compound task control for swarm robot systems, where the control aim and the dynamics of the robot are modeled by static group-bipartite consensus and Euler–Lagrange systems, respectively. After introducing the concept of static group-bipartite consensus in networked Euler–Lagrange systems, a distributed group-bipartite consensus control protocol with integral action is designed, and the criterion that ensures that static group-bipartite consensus is reached is presented. The most remarkable feature of the proposed algorithm is that it can accurately calculate the final state of system convergence. Finally, simulation examples are presented to verify the theoretical results.


Introduction
Various coordinate behaviors of networked multi-agent systems can emerge, such as complete consensus or synchronization, 1 group consensus (GC), 2 cluster consensus, 3 and bipartite consensus (BC), 4 among others. According to these coordinate behaviors, much research has been carried out from many perspectives, for example, with various network structures-cooperative 1 and competitive 5 forms; control methods-adaptive control, 6 finite-time control mode, 7,8 event-triggered methods, 9,10 impulse coupling, 11 and the Udwadia-Kalaba approach 12 ; dynamic modelsfirst-order integral models,[ 1,13 second-order integral models, 14 generalized linear systems, 3 nonlinear systems that satisfy the Lipschitz condition, 15 and ELSs. 12 Among these, group-bipartite consensus (GBC) and Euler-Lagrange (EL) dynamics are a typical coordinate behavior and dynamic model, respectively. GBC is defined in our previous work, 13 and such a consensus form can describe multiple complex control aims well. The GBC demonstrates multiple origin-symmetric states, which distinguishes from BC and GC, and can be seen as an effective combination of BC and GC. Based on its unique convergence state, GBC can effectively depict multiple symmetric control objectives in modern industrial production. For additional details on GBC, one can refer to Liu et al. 13 In addition, a class of industrial robot dynamics can be well modeled by the EL equations, including the dynamics of industrial manipulators, space robots, and self-driving cars. 16 Many modern industrial applications can involve the coordinate control of ELSs, such as target tracking in sensor networks 17 and environmental monitoring. 18 For this reason, much work has been done on coordinated control for networked ELSs. For example, the robust synchronization control problem of networked ELSs is studied by considering disturbances, network delays, and uniformly connected switching networks. 19 By using the estimator method, hierarchical control protocols are introduced to study the coordination of ELSs. 20 An optimal position algorithm is designed to minimize a global cost function of networked ELSs, where the presented control law is fully distributed. 21 The distributed boundary control problem of swarm flexible manipulators is investigated using semigroup theory. 22 The leader-following consensus issue for multiple robots is studied by using a distributed position feedback control strategy over switching networks. 23 An adaptive containment control law for networked ELSs is presented in Corral et al. 24 Among a variety of swarm robot models, 24-27 the networked ELSs model has a wide range of engineering applications, especially in the complex and integrated production process, where the flexibility, reliability, manipulability, and scalability are highly required. Moreover, the networked ELSs can represent a class of networked robotic systems due to their ability in dealing with complex systems involving multiple dynamics, such as multiple robotic manipulators, formation flying spacecrafts, and autonomous vehicles. However, the traditional single-target coordination control of the networked ELSs is no longer applicable to modeling requirements of swarm robots in modern industrial production, where the complex behaviors and characteristics such as multilevel, multi-structure, and multi-scale are emerged. Based on the above background, a natural problem arises: How to establish a reasonable compound task model for swarm robot systems described by networked ELSs? An effective way to solve this problem is how to reasonably improve the compound control target model proposed in networked multi-agent systems to apply to swarm robot systems. Note that the BC introduced by Altafini 4 is an excellent bilateral symmetrical compound task model, and many studies investigated the BC of networked robot systems using various control methods, such as the auxiliary system approach, 28 the adaptive strategy, 29 and the finite-time coordination strategy. 30 Furthermore, as a further extension of BC, GBC fully combines the characteristics of GC and BC and can well describe the multiple symmetric tasks for modern industrial production. However, for the GBC case, very little research has been conducted on this topic. By considering this background, this article studies the GBC problem in networked robot systems that are modeled by the EL equation. We propose a distributed GBC control protocol that ensures the achievement of static GBC. The control strategy that is proposed in this article has an integral action by which the final GBC states can be expressed explicitly. Compared with previous articles, the current article has the following three innovation points. (1) The concept of static GBC in networked ELSs is introduced, which can well describe the multiple static symmetry control aims in practice. (2) The final convergence state can be explicitly expressed by the initial value of the position and the vectors in the null subspace of the Laplacian matrix. (3) By decomposing the whole controlled system into subsystems, this article develops a GBC convergence stability analysis method.

Graph theory
The associated graph notations and definitions are standard. G ¼ fV; E; Ag is a graph that contains V ¼ f1; 2; . . . ; ng and E 2 V Â V as its agent set and edge set, respectively, and has weighted adjacency matrix A ¼ ½a ij 2 R nÂn . The entries of the matrix A ¼ ½a ij are defined as follows: a ij 6 ¼ 0 if ðj; iÞ 2 E; a ij ¼ 0 otherwise. If we can select nodes i 1 ; . . . i k such that all the binaries ði 1 ; i 2 Þ; ði 2 ; i 3 Þ; . . . ; ði kÀ1 ; i k Þ are the edges of G, then ði 1 ; i 2 Þ; ði 2 ; i 3 Þ; . . . ; ði kÀ1 ; i k Þ is called a path from i 1 to i k in G. The graph has a spanning tree if there exists at least one node that is connected by a path to every other node. A set fV 1 ; V 2 ; . . . ; V k g is called a partition of the node set V if V i 6 ¼ : and Suppose V has a partition fV 1 ; V 2 ; . . . ; V k g. For convenience, we assume . . . ; k. i denotes the index of the group to which the i-th agent belongs. Moreover, according to the GBC that is discussed in Liu et al., 13 the partition fV 1 ; V 2 ; Á Á Á ; V k g should have an acyclic structure 3 and, therefore, the following assumption is needed.
Under Assumption 1, the network graph G is divided into k groups, namely, fG 1 ; G 2 ; . . . ; G k g, where G i is the topology that is associated with set V i , i ¼ 1; 2; . . . ; k. Then, the corresponding adjacency matrix A has the form where A ii is associated with the communication weights between agents in the node set V i and A ij is associated with the communication weights between the nodes from V j to V i , i; j 2 f1; 2; . . . ; kg, i 6 ¼ j.
Meantime, to describe the bipartite characteristic of the GBC, the following assumption should also be introduced.
For the i-th group G i , if Assumption 2 holds, then node subset V i can be divided into two parts, namely, V

Problem formulation
Suppose that the dynamics of robot i are modeled by the EL equation as where q i 2 R p , 0 < M i ðq i Þ 2 R pÂp , C i 2 R pÂp , and g i ðq i Þ 2 R p . t i is the control input for the i-th robot. Moreover, the EL dynamics has several usual properties. 31 Property 1. There exist positive constants m i ; m i , c i , and . . . ; V k g and under Assumption 2, we define the concept of static GBC in ELSs (1).

Static GBC in networked ELSs
This section presents a static GBC control protocol for ELSs (1). Toward this aim, the whole discussion will be conducted under Assumptions 1 and 2, and the following assumption is introduced. 13 . . . ; k, i 6 ¼ j and Assumption 4. 8i 2 f1; 2; . . . ; kg, G i has a spanning tree.
For the i-th robot agent, a slide vector s i is defined as Then, the corresponding auxiliary velocity _ q ri 2 R p with integral action can be expressed as follows The second sliding vector x i 2 R p has the form where a i > 0 is a constant. The adaptive law is expressed as where 0 < K i 2 R pÂp andq i is the estimate of q i . The adaptive law that is associated withq i is expressed as Remark 1. In contrast to Wang,32 in the current article, _ q r;i has three parts: the first part, namely, À P j2 i a ij ½sgnða ij Þ q i À q j , describes the communication between agents in the same group; the second part, namely, P j= 2 i a ij ½ j q i Àq j , indicates the influence from other groups, and the third part, namely, Àa i ð t 0 s i ðrÞ dr, is an integral action, which is an essential component in calculating the final convergence state of the controlled system. Obviously, if the second part, namely, P j= 2 i a ij ½ j q i À q j , of _ q r;i is removed, the adaptive control law that is designed in equation (5) is just the BC control law that is presented in Hu et al. 30 Moreover, if a ij ! 0, for i ¼ j, the control protocol t i that contains the auxiliary velocity _ q ri becomes the GC control strategy that is designed in Liu et al. 2 As stated in Liu et al., 13 GBC has the characteristics of BC and GC, and the structure of the adaptive control law that is presented in equation (5) reflects these characteristics appropriately.
Remark 2. The integral action item design is borrowed from Wang, 32 and the major differentiator is the integrand function, which contains GC and BC network topology structure information. For this reason, the control method that is constructed by equations (2) to (6) with integral action can be used to realize GBC in the swarm robot system (1).
Using equations (5) and (6), a controlled networked robot system (1) has the following form whereq ¼ q i Àq i . Now, the GBC stability criterion is presented in the following theorem.
Theorem 1. Suppose Assumptions 1 to 3 hold. Using equations (5) and (6) Proof. It is easy to see that under Assumption 3, F i L ii F i is a standard Laplacian matrix. Then, if Assumption 4 holds, the eigenvalues of the matrix F i L ii F i , which are denoted by l i;1 ; l i;2 ; . . . ; l i;n i , satisfy l i;1 ¼ 0, and their real parts Reðl i;j Þ > 0, j ¼ 2; 3; . . . ; n i . Moreover, all entries of the left eigenvector r i 2 R n i of F i L ii F i , which has a 0 eigenvalue, are nonnegative and Then, the matrix FLF has the following eigenvectors that are associated with eigenvalue 0 2 . .. ; k and j ¼ 1; 2; . .. ; k À 1.
The coordinate transform matrix D 2 R nÂn can be defined as . . . ; k. Moreover, D À1 has the form (2) can be reformulated as Hence, Combining equations (8) and (9) yields From the structure of the matrix D, one obtains where L 2 R ðnÀkÞÂðnÀkÞ and each of eigenvalues of L has a positive real part. By selecting the Lyapunov function . . . ; x T n Þ and U ¼ diagfa 1 ; a 2 ; . . . ; a n g. Then, system (4) has the vector form  (11) and (12), we obtain Then, the boundedness of S implies that ½ðp i FÞ . . . ; k. Then, by using the invertibility of the matrix D, one obtains that _ Q is bounded. Therefore, _ q r;i 2 L 2 and € q r;i 2 L 2 . This, in combination with Property 2, implies the boundedness of Y i , and we conclude that all _ x i are bounded, i ¼ 1; 2; . . . ; n. Therefore, € V i ðtÞ is bounded. Then, _ V i ðtÞ is uniformly continuous. By Barbalat's Lemma, _ V i ðtÞ ! 0 as t ! 1. Now, we conclude lim t!1 O ¼ 0 np . From equation (13), we have lim t!1 S ¼ 0 np . Then, lim t!1 C i S i ¼ 0 ðn i À1Þ . Therefore, lim t!1 C i Q i ¼ 0 ðn i À1Þ , i 2 f1; 2; . . . ; kg. This, in combination with equation (11), implies lim t!1 Q ¼ 0 np , namely, lim t!1 q i ¼ 0 p , i ¼ 1; 2; . . . ; n.
Additionally, from equations (10) and (11) and the selfsimilarity of each F i , we have . .

Remark 3.
By adding an integral action into the adaptive control law (5) in Algorithm 3, the networked robot systems (1) reach the static GBC that is defined in Definition 1 under the conditions in Theorem 1. Moreover, the final static GBC state can be expressed explicitly by the eigenvectors of the corresponding Laplacian matrix and the initial values of the positions. It is easy to see that when F i ¼ I n i , then the i-th group realizes complete consensus for i 2 f1; 2; . . . ; kg. Moreover, when all F i ¼ I n i , the static GBC problem that is considered in the current article becomes the GC problem. When k ¼ 1, the static GBC becomes the BC. Therefore, the GC that is studied in Liu et al. 2 and the BC that is investigated in Liu et al. 33 are both involved in our study.

Remark 4.
The acyclic partition indicates that the front groups can affect the back groups but not the reverse. Therefore, the convergence analysis of the controlled systems can proceed according to the subgroup indices. The first group cannot receive any information from other groups; therefore, we can study the BC problem of the first group. Then, by using the BC result in the first group and the second system in equation (10), we can analyze the BC problem in the second group. Repeating this process, we can also obtain the stability analysis results for the controlled system instead of the proof of Theorem 1.

Simulations
Corresponding simulations will be conducted by Matlab in this section. The networked ELSs that are considered in this section are composed of seven revolute joint manipulators, and the equations and parameters are the same as in Example 1 in Liu et al. 2 Figure 1 presents the structure of the i-th robot node. In this case, the parameter constant vector estimateq i of the i-th robot has the formq i ¼ ðq i1 ;q i2 ; q i3 ;q i4 ;q i5 Þ T . The communication status is presented in Figure 2. Obviously, the node set V ¼ f1; 2; . . . ; 7g has an acyclic partition, namely, V 1 ¼ f1; 2g, V 2 ¼ f3; 4g, and V 3 ¼ f4; 5; 6g. The control design weights K i and L i , i ¼ 1; 2; . . . ; 7, are set as K i ¼ 22diagf2:3; 1:5g and L i ¼ 10I 2 for i ¼ 1; 2; . . . ; 7. It is easy to obtain that 1 ¼ 3 ¼ 5 ¼ 1 and 2 ¼ 4 ¼ 6 ¼ 7 ¼ À1. Then, Assumption 3 holds, and we have F ¼ diagf1; À1; 1; Algorithm 1. Group-bipartite network topology algorithm.
In contrast, some groups often have cooperative topologies, and other groups often exhibit cooperative and competitive coexistence. Figure 14 presents a graph topology         with the acyclic partition V 1 ¼ f1; 2g, V 2 ¼ f3; 4g, and V 3 ¼ f4; 5; 6g, in which the corresponding graphs G 1 and G 3 are cooperatively and competitively coexisting, while G 2 is of cooperative type. Then, It is easy to see that the final convergence state should be BC between the first and third groups and convergence of the second group to complete consensus. We obtain 1 ¼ 3 ¼ 4 ¼ 5 ¼ 1 and 2 ¼ 6 ¼ 7 ¼ À1. Notably, 3 ¼ 4 ¼ 1 implies the same convergence state of the agents in V 2 . Then, F ¼ diagf1; À1; 1; 1; 1; À1; À1g and We select the same initial value and control design weights as discussed above. Figures. 15 to 18 present the    position and velocity evolution processes over time, which indicates that the first and third groups reach BC and that the second group reaches complete consensus.

Conclusions
By introducing the concept of static GBC into swarm robots that are modeled by EL dynamics, a mathematical model of the compound task has been constructed in this article. Accordingly, this article has investigated the static GBC problem for ELSs. By utilizing the structure of the acyclic partition network topology, a static GBC control protocol has been presented, and geometric criteria for ensuring that multiple symmetric consensus is achieved in networked robot systems have been established. The range of topologies that have been discussed in this article is somewhat limited, and this issue will be considered in a more general network topology range in our future work.