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

Abstract

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 models—first-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.

Preliminary

Graph theory

The associated graph notations and definitions are standard. $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$ is a graph that contains $\mathcal{V}=\{1,2,\dots ,n\}$ and $\mathcal{E}\in \mathcal{V}\times \mathcal{V}$ as its agent set and edge set, respectively, and has weighted adjacency matrix $\mathcal{A}=[a_{ij}]\in {ℝ}^{n\times n}$ . The entries of the matrix $\mathcal{A}=[a_{ij}]$ are defined as follows: $a_{ij}\ne 0$ if $(j,i)\in \mathcal{E}$ ; $a_{ij}=0$ otherwise. If we can select nodes $i_1,\dots i_k$ such that all the binaries $(i_1,i_2),(i_2,i_3),\dots ,(i_{k-1},i_k)$ are the edges of $\mathcal{G}$ , then $(i_1,i_2),(i_2,i_3),\dots ,(i_{k-1},i_k)$ is called a path from i ₁ to i_k in $\mathcal{G}$ . 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 $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_k\}$ is called a partition of the node set $\mathcal{V}$ if ${\mathcal{V}}_i\ne \varnothing$ and ${\cup }_{j=1}^k{\mathcal{V}}_j=\mathcal{V}$ , where $i=1,\dots ,k$ .

Suppose $\mathcal{V}$ has a partition $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_k\}$ . For convenience, we assume ${\mathcal{V}}_i=\{h_{i-1}+1,h_{i-1}+2,\cdots ,h_i\}$ , where $h_0=n_0=0$ , $n_1=h_1<h_2<\cdots h_k=n$ , $h_{i+1}-h_i+1=n_i$ , $i=1,2,\dots ,k$ . $\overline{i}$ 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 $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\cdots ,{\mathcal{V}}_k\}$ should have an acyclic structure ³ and, therefore, the following assumption is needed.

Assumption 1

The partition $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_k\}$ is acyclic.

Under Assumption 1, the network graph $\mathcal{G}$ is divided into k groups, namely, $\{{\mathcal{G}}_1,{\mathcal{G}}_2,\dots ,{\mathcal{G}}_k\}$ , where ${\mathcal{G}}_i$ is the topology that is associated with set ${\mathcal{V}}_i$ , $i=1,2,\dots ,k$ . Then, the corresponding adjacency matrix $\mathcal{A}$ has the form

$\left(\begin{array}{ccc}{\mathcal{A}}_{11}& \cdots & {\mathcal{A}}_{1k}\\ ⋮& \ddots & ⋮\\ {\mathcal{A}}_{k1}& \cdots & {\mathcal{A}}_{kk}\end{array}\right)$

where ${\mathcal{A}}_{ii}$ is associated with the communication weights between agents in the node set ${\mathcal{V}}_i$ and ${\mathcal{A}}_{ij}$ is associated with the communication weights between the nodes from ${\mathcal{V}}_j$ to ${\mathcal{V}}_i$ , $i,j\in \{1,2,\dots ,k\}$ , $i\ne j$ .

Meantime, to describe the bipartite characteristic of the GBC, the following assumption should also be introduced.

Assumption 2

Each ${\mathcal{G}}_i$ , $i=1,2,\dots ,k$ is structurally balanced.

For the i-th group ${\mathcal{G}}_i$ , if Assumption 2 holds, then node subset ${\mathcal{V}}_i$ can be divided into two parts, namely, ${\mathcal{V}}_i^{(1)}$ and ${\mathcal{V}}_i^{(2)}$ , that satisfy ${\mathcal{V}}_i^{(k)}\ne \varnothing$ , $k=1,2$ , and ${\mathcal{V}}_i^{(1)}\cup {\mathcal{V}}_i^{(2)}={\mathcal{V}}_i$ . In addition, for $j\in {\mathcal{V}}_i$ , by defining ${\varphi }_j=1$ if $j\in {\mathcal{V}}_i^{(1)}$ and ${\varphi }_j=-1$ if $j\in {\mathcal{V}}_i^{(2)}$ , all entries in ${\Phi }_i{\mathcal{A}}_{11}{\Phi }_i$ are nonnegative, $i=1,2,\dots ,k$ . The Laplacian matrix that is associated with $\mathcal{G}$ is formulated as $\mathcal{L}=[l_{ij}]\in {ℝ}^{n\times n}$ , where $l_{ij}=-a_{ij}$ , $i\ne j$ and $l_{ii}={\displaystyle {\sum }_{j\notin {\mathcal{V}}_{\overline{i}}}}{\varphi }_j{a}_{ij}+{\displaystyle {\sum }_{j\in {\mathcal{V}}_{\overline{i}}}}|a_{ij}|$ , $i,j=1,2,\dots ,n$ . Moreover, under Assumption 1, $\mathcal{L}$ is of block lower-triangular form ³

$\mathcal{L}=\left(\begin{array}{ccc}{\mathcal{L}}_{11}& \cdots & 0\\ ⋮& \ddots & ⋮\\ {\mathcal{L}}_{k1}& \cdots & {\mathcal{L}}_{kk}\end{array}\right)$

Problem formulation

Suppose that the dynamics of robot i are modeled by the EL equation as

$M_i(q_i){\ddot{q}}_i+C_i(q_i,{\dot{q}}_i){\dot{q}}_i+g_i(q_i)={\tau }_i,i=1,2,\dots ,n$

1

where $q_i\in {ℝ}^p$ , $0<M_i(q_i)\in {ℝ}^{p\times p}$ , $C_i\in {ℝ}^{p\times p}$ , and $g_i(q_i)\in {ℝ}^p$ . ${\tau }_i$ is the control input for the i-th robot. Moreover, the EL dynamics has several usual properties. ³¹

Property 1

There exist positive constants ${\underset{\_}{m}}_i,$ ${\overline{m}}_i$ , c_i , and ${\overline{g}}_i$ such that ${\underset{\_}{m}}_i{I}_p\le M_i(q_i)\le {\overline{m}}_i$ , $\parallel C_i(x,y)z\parallel \le c_i\parallel y\parallel \parallel z\parallel$ , and $\parallel g_i\parallel \le {\overline{g}}_i$ for $\forall x,y,z\in {ℝ}^p$ .

Property 2

$M_i(q_i)-2C_i(q_i,{\dot{q}}_i)$ is a skew-symmetric matrix.

Property 3

$\forall x,y\in {ℝ}^p$ , there exists a constant vector ${\theta }_i$ such that $M_i(q_i)x+C_i(q_i,{\dot{q}}_i)y+g_i(q_i)=Y_i(q_i,{\dot{q}}_i,x,y){\theta }_i$ .

With the partition $\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_k\}$ and under Assumption 2, we define the concept of static GBC in ELSs (1).

Definition 1

ELSs (1) reach static GBC under the control law ${\tau }_u$ , $u=1,2,\dots ,n$ , if for $\overline{i}=\overline{j}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }|q_i-q_j|=0$ , $i,j\in {\mathcal{V}}_{\overline{i}}^{(1)}$ , and ${{\displaystyle \text{lim}}}_{t\to \infty }|q_i+q_j|=0$ for $i\in {\mathcal{V}}_{\overline{i}}^{(b)}$ and $j\in {\mathcal{V}}_{\overline{i}}^{(3-b)}$ and ${\text{lim}}_{t\to \infty }{\dot{q}}_e=0_p$ , where $i,j,e\in \{1,2,\dots ,n\}$ , $b\in \{1,2\}$ , and $0_p=(0,0,\dots {\mathrm{,0})}^{\text{T}}\in {ℝ}^p$ .

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

Assumption 3

${\Phi }_i{\mathcal{L}}_{ij}{\Phi }_j{1}_{n_j}=0$ , where ${\Phi }_i=\text{diag}\{{\varphi }_{h_{i-1}+1},{\varphi }_{h_{i-1}+2},\dots ,{\varphi }_{h_i}\}$ for $i,j=1,2,\dots ,k$ , $i\ne j$ and $1_{n_j}=(1,1,\dots {\mathrm{,1})}^{\text{T}}\in {ℝ}^{n_j}$ .

Assumption 4

$\forall i\in \{1,2,\dots ,k\}$ , ${\mathcal{G}}_i$ has a spanning tree.

For the i-th robot agent, a slide vector s_i is defined as

$\begin{array}{c}{s}_i={\dot{q}}_i+{\displaystyle \sum _{j\in \overline{i}}}{a}_{ij}\left[\text{sgn}(a_{ij})q_i-q_j\right]\\ +{\displaystyle \sum _{j\notin \overline{i}}}{a}_{ij}\left[{\varphi }_i{q}_i-q_j\right]\end{array}$

2

Then, the corresponding auxiliary velocity ${\dot{q}}_{ri}\in {ℝ}^p$ with integral action can be expressed as follows

$\begin{array}{c}{\dot{q}}_{r,i}=-{\displaystyle \sum _{j\in \overline{i}}}{a}_{ij}\left[\text{sgn}(a_{ij})q_i-q_j\right]+{\displaystyle \sum _{j\notin \overline{i}}}{a}_{ij}\left[{\varphi }_j{q}_i-q_j\right]\\ -{\alpha }_i{\displaystyle {\int }_0^t}{s}_i(r)dr\end{array}$

3

The second sliding vector ${\xi }_i\in {ℝ}^p$ has the form

${\xi }_i={\dot{q}}_i-{\dot{q}}_{r,i}=s_i+{\alpha }_i{\displaystyle {\int }_0^t}{s}_i(r)dr$

4

where ${\alpha }_i>0$ is a constant. The adaptive law is expressed as

${\tau }_i=Y_i(q_i,{\dot{q}}_i,{\ddot{q}}_{ri},{\dot{q}}_{r,i}){\widehat{\theta }}_i-K_i{\xi }_i$

5

where $0<K_i\in {ℝ}^{p\times p}$ and ${\widehat{\theta }}_i$ is the estimate of ${\theta }_i$ . The adaptive law that is associated with ${\widehat{\theta }}_i$ is expressed as

${\dot{\widehat{\theta }}}_i=-{\Lambda }_i{Y}_i^{\text{T}}(q_i,{\dot{q}}_i,{\ddot{q}}_{r,i},{\dot{q}}_{ri}){\xi }_i$

6

where $0<{\Lambda }_i\in {ℝ}^{p\times p}$ .

Remark 1

In contrast to Wang, ³² in the current article, ${\dot{q}}_{r,i}$ has three parts: the first part, namely, $-{\displaystyle {\sum }_{j\in \overline{i}}}{a}_{ij}\left[\text{sgn}(a_{ij})q_i-q_j\right]$ , describes the communication between agents in the same group; the second part, namely, ${\displaystyle {\sum }_{j\notin \overline{i}}}{a}_{ij}\left[{\varphi }_j{q}_i-q_j\right]$ , indicates the influence from other groups, and the third part, namely, $-{\alpha }_i{\displaystyle {\int }_0^t}{s}_i(r)dr$ , 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, ${\displaystyle {\sum }_{j\notin \overline{i}}}{a}_{ij}\left[{\varphi }_j{q}_i-q_j\right]$ , of ${\dot{q}}_{r,i}$ 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 $a_{ij}\ge 0$ , for $\overline{i}=\overline{j}$ , the control protocol ${\tau }_i$ that contains the auxiliary velocity ${\dot{q}}_{ri}$ 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

$M_i(q_i){\xi }_i=C_i(q_i,{\dot{q}}_i){\xi }_i-Y_i(q_i,{\dot{q}}_i,{\ddot{q}}_{r,i},{\dot{q}}_{r,i}){\stackrel{\sim }{\theta }}_i-K_i{\xi }_i$

7

where $\stackrel{\sim }{\theta }={\theta }_i-{\widehat{\theta }}_i$ . Now, the GBC stability criterion is presented in the following theorem.

Theorem 1

Suppose Assumptions 1 to 3 hold. Using equations (5) and (6) for the ELSs (1), the systems (1) reach static GBC if Assumption 4 holds. Furthermore, the final convergence states of the ELSs (1) are ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_i={\varphi }_i\left[({\pi }_{\overline{i}}^{\text{T}}\Phi )\otimes I_p\right]Q(0)$ and ${{\displaystyle \text{lim}}}_{t\to \infty }{\dot{q}}_i=0_p$ , where $Q={\left(q_1^{\text{T}},q_2^{\text{T}},\dots ,q_n^{\text{T}}\right)}^{\text{T}}$ .

Proof

It is easy to see that under Assumption 3, ${\Phi }_i{\mathcal{L}}_{ii}{\Phi }_i$ is a standard Laplacian matrix. Then, if Assumption 4 holds, the eigenvalues of the matrix ${\Phi }_i{\mathcal{L}}_{ii}{\Phi }_i$ , which are denoted by ${\lambda }_{i\mathrm{,1}},{\lambda }_{i\mathrm{,2}},\dots ,{\lambda }_{i,n_i}$ , satisfy ${\lambda }_{i\mathrm{,1}}=0$ , and their real parts $Re({\lambda }_{i,j})>0$ , $j=2,3,\dots ,n_i$ .

Moreover, all entries of the left eigenvector ${\rho }_i\in {ℝ}^{n_i}$ of ${\Phi }_i{\mathcal{L}}_{ii}{\Phi }_i$ , which has a 0 eigenvalue, are nonnegative and ${\rho }_i^{\text{T}}{1}_{n_i}=1$ . Let

$\Phi =\text{diag}\{{\Phi }_1,{\Phi }_2,\dots ,{\Phi }_k\}$

Then, the matrix $\Phi \mathcal{L}\Phi$ has the following eigenvectors that are associated with eigenvalue 0²

$\begin{array}{l}{\pi }_1={\left({\rho }_1^{\text{T}}{\mathrm{,0}}_{n-h_{1_2}}^{\text{T}}\right)}^{\text{T}}\\ {\pi }_2={\left({({\gamma }_1^{(2)})}^{\text{T}},{\rho }_2^{\text{T}}{\mathrm{,0}}_{n-h_2}^{\text{T}}\right)}^{\text{T}}\\ ⋮\\ {\pi }_k={\left({({\gamma }_1^{(k)})}^{\text{T}},({\gamma }_2^{(k)}{)}^{\text{T}},\dots ,{\rho }_k^{\text{T}}\right)}^{\text{T}}\end{array}$

which satisfy ${\gamma }_j^{(i)}\in {ℝ}^{n_i}$ and ${({\gamma }_j^{(i)})}^{\text{T}}{1}_{n_i}=0$ for $i=1,2,\dots ,k$ and $j=1,2,\dots ,k-1$ .

The coordinate transform matrix $\mathcal{D}\in {ℝ}^{n\times n}$ can be defined as

$\mathcal{D}=\left(\begin{array}{c}{\pi }_1\Phi \\ ⋮\\ {\pi }_k\Phi \\ {\mathcal{D}}_1\\ ⋮\\ {\mathcal{D}}_k\end{array}\right)$

8

where ${\mathcal{D}}_i=\left[0^{(n_i-1)\times h_{i-1}},{\Psi }_i{\mathrm{,0}}^{(n_i-1)\times n-h_i}\right]\in {ℝ}^{(n_i-1)\times n}$ and

$\begin{array}{c}{\Psi }_i=\left(\begin{array}{ccccc}{\varphi }_{h_{i-1}+1}& -{\varphi }_{h_{i-1}+2}& 0& \cdots & 0\\ ⋮& & \ddots & & \\ {\varphi }_{h_{i-1}+1}& 0& 0& \cdots & -{\varphi }_{h_i}\end{array}\right)\\ \in {ℝ}^{(n_i-1)\times n_i}\end{array}$

$i=1,2,\dots ,k$ . Moreover, ${\mathcal{D}}^{-1}$ has the form

${\mathcal{D}}^{-1}=\text{diag}\{{\Delta }_1,{\Delta }_2,\dots ,{\Delta }_k\}$

where ${\Delta }_i=\left(\begin{array}{cc}{\Phi }_i{1}_{n_i},& *\end{array}\right),$ , namely, the first column of the matrix ${\Delta }_i$ is ${\Phi }_i{1}_{n_i}$ , $i=1,2,\dots ,k$ .

Define $Q_i={\left(q_{h_{i-1}+1}^{\text{T}},q_{h_{i-1}+2}^{\text{T}},\dots ,q_{h_i}^{\text{T}}\right)}^{\text{T}}$ , $S_i={\left(s_{h_{i-1}+1}^{\text{T}},s_{h_{i-1}+2}^{\text{T}},\dots ,s_{h_i}^{\text{T}}\right)}^{\text{T}}$ , and $S={\left(S_1^{\text{T}},S_2^{\text{T}},\dots ,S_k^{\text{T}}\right)}^{\text{T}}$ . Then, $Q={\left(Q_1^{\text{T}},Q_2^{\text{T}},\dots ,Q_k^{\text{T}}\right)}^{\text{T}}$ . The system (2) can be reformulated as

$\dot{Q}=S-\left(\mathcal{L}\otimes I_p\right)Q$

9

Hence,

$\begin{array}{l}{\dot{Q}}_1=S_1-\left({\mathcal{L}}_{11}\otimes I_p\right)Q_1\\ {\dot{Q}}_2=S_2-\left({\mathcal{L}}_{21}\otimes I_p\right)Q_1-\left({\mathcal{L}}_{22}\otimes I_p\right)Q_2\\ ⋮\\ {\dot{Q}}_k=S_k-\left({\mathcal{L}}_{k1}\otimes I_p\right)Q_1-\left({\mathcal{L}}_{k2}\otimes I_p\right)Q_2\\ \hspace{1em}-\cdots -\left({\mathcal{L}}_{kk}\otimes I_p\right)Q_k\end{array}$

10

Combining equations (8) and (9) yields

$\left(\mathcal{D}\otimes I_p\right)\dot{Q}=\left(\mathcal{D}\otimes I_p\right)S-\left[(\mathcal{D}\mathcal{L}{\mathcal{D}}^{-1})\otimes I_p\right]\mathcal{D}Q$

11

From the structure of the matrix $\mathcal{D}$ , one obtains

$\mathcal{D}\mathcal{L}{\mathcal{D}}^{-1}=\text{diag}\{0_{k\times k},\overline{\mathcal{L}}\}$

12

where $\overline{\mathcal{L}}\in {ℝ}^{(n-k)\times (n-k)}$ and each of eigenvalues of $\overline{\mathcal{L}}$ has a positive real part. By selecting the Lyapunov function $V_i(t)=\frac{1}{2}{\xi }_i^{\text{T}}{M}_i(q_i){\xi }_i+\frac{1}{2}{\stackrel{\sim }{\theta }}_i^{\text{T}}{\Lambda }_i{\stackrel{\sim }{\theta }}_i$ for equation (7), we obtain ${\dot{V}}_i(t)=-{\xi }_i^{\text{T}}{K}_i{\xi }_i\le 0$ under Property 2. Then, ${\xi }_i\in L_2\cap L_{\infty }$ and ${\stackrel{\sim }{\theta }}_i\in L_{\infty }$ . Let $\Omega =\left({\xi }_1^{\text{T}},{\xi }_2^{\text{T}},\dots ,{\xi }_n^{\text{T}}\right)$ and $\Upsilon =\text{diag}\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$ . Then, system (4) has the vector form

$S=\Omega -\Upsilon \otimes I_p{\displaystyle {\int }_0^r}S(r)dr$

13

with the input $\Omega$ and the state ${\displaystyle {\int }_0^r}S(r)dr$ . Since $\Upsilon$ is a positive matrix, the states ${\displaystyle {\int }_0^r}S(r)dr$ and $S(t)$ are both bounded.

Combining equations (11) and (12), we obtain

$\begin{array}{l}\left[({\pi }_1\Phi )\otimes I_p\right]\dot{\mathbf{Q}}=\left[({\pi }_1\Phi )\otimes I_p\right]S,\\ \left[({\pi }_2\Phi )\otimes I_p\right]\dot{Q}=\left[({\pi }_2\Phi )\otimes I_p\right]S\\ ⋮\\ \left[({\pi }_k\Phi )\otimes I_p\right]\dot{Q}=\left[({\pi }_k\Phi )\otimes I_p\right]S\end{array}$

and

$\left(\begin{array}{c}{\Psi }_1{\dot{Q}}_1\\ ⋮\\ {\Psi }_k{\dot{Q}}_k\end{array}\right)=\left(\begin{array}{c}{\Psi }_1{S}_1\\ ⋮\\ {\Psi }_k{S}_k\end{array}\right)-\left(\overline{\mathcal{L}}\otimes I_p\right)\left(\begin{array}{c}{\Psi }_1{Q}_1\\ ⋮\\ {\Psi }_k{Q}_k\end{array}\right)$

Then, the boundedness of $S$ implies that $\left[({\pi }_i\Phi )\otimes I_p\right]\dot{Q}\in L_2$ and ${\Psi }_i{S}_i\in L_2$ , $i=1,2,\dots ,k$ . Since $-\overline{\mathcal{L}}$ is Hurwitz, ${\Psi }_i{Q}_i$ and ${\Psi }_i{\dot{Q}}_i$ are all bounded, $i=1,2,\dots ,k$ . Then, by using the invertibility of the matrix $\mathcal{D}$ , one obtains that $\dot{Q}$ is bounded. Therefore, ${\dot{q}}_{r,i}\in L_2$ and ${\ddot{q}}_{r,i}\in L_2$ . This, in combination with Property 2, implies the boundedness of Y_i , and we conclude that all ${\dot{\xi }}_i$ are bounded, $i=1,2,\dots ,n$ . Therefore, ${\ddot{V}}_i(t)$ is bounded. Then, ${\dot{V}}_i(t)$ is uniformly continuous. By Barbalat’s Lemma, ${\dot{V}}_i(t)\to 0$ as $t\to \infty$ . Now, we conclude ${{\displaystyle \text{lim}}}_{t\to \infty }\Omega =0_{np}$ . From equation (13), we have ${{\displaystyle \text{lim}}}_{t\to \infty }S=0_{np}$ . Then, ${{\displaystyle \text{lim}}}_{t\to \infty }{\Psi }_i{S}_i=0_{(n_i-1)}$ . Therefore, ${{\displaystyle \text{lim}}}_{t\to \infty }{\Psi }_i{Q}_i=0_{(n_i-1)}$ , $i\in \{1,2,\dots ,k\}$ . This, in combination with equation (11), implies ${{\displaystyle \text{lim}}}_{t\to \infty }Q=0_{np}$ , namely, ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_i=0_p$ , $i=1,2,\dots ,n$ .

Additionally, from equations (10) and (11) and the self-similarity of each ${\Phi }_i$ , we have

$\begin{array}{l}\left({\Phi }_1\otimes I_p\right){\dot{Q}}_1=\left({\Phi }_1\otimes I_p\right)S_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\Phi }_1{\mathcal{L}}_{11}{\Phi }_1\otimes I_p\right)\left({\Phi }_1\otimes I_p\right)Q_1\\ \left({\Phi }_2\otimes I_p\right){\dot{Q}}_2=\left({\Phi }_2\otimes I_p\right)S_2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\Phi }_2{\mathcal{L}}_{21}{\Phi }_1\otimes I_p\right)Q_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\Phi }_2{\mathcal{L}}_{22}{\Phi }_2\otimes I_p\right)\left({\Phi }_2\otimes I_p\right)Q_2\\ ⋮\\ \left({\Phi }_k\otimes I_p\right){\dot{Q}}_k=\left({\Phi }_k\otimes I_p\right)S_k\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\Phi }_k{\mathcal{L}}_{k1{\Phi }_1}\otimes I_p\right)\left({\Phi }_1\otimes I_p\right)Q_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\Phi }_k{\mathcal{L}}_{k2}{\Phi }_2\otimes I_p\right)\left({\Phi }_2\otimes I_p\right)Q_2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\cdots -\left({\Phi }_k{\mathcal{L}}_{kk}{\Phi }_k\otimes I_p\right)\left({\Phi }_k\otimes I_p\right)Q_k\end{array}$

Therefore,

$\left(\begin{array}{c}{1}_{n_1}{\pi }_1^{\text{T}}{\Phi }_1\otimes I_p\\ 1_{n_2}{\pi }_2^{\text{T}}{\Phi }_2\otimes I_p\\ ⋮\\ 1_{n_k}{\pi }_k^{\text{T}}{\Phi }_k\otimes I_p\end{array}\right)\dot{Q}=\left(\begin{array}{c}{1}_{n_1}{\pi }_1^{\text{T}}{\Phi }_1\otimes I_p\\ 1_{n_2}{\pi }_2^{\text{T}}{\Phi }_2\otimes I_p\\ ⋮\\ 1_{n_k}{\pi }_k^{\text{T}}{\Phi }_k\otimes I_p\end{array}\right)S$

14

Integrating equation (14) yields

$\begin{array}{l}\left(\begin{array}{c}{1}_{n_1}{\pi }_1^{\text{T}}{\Phi }_1\otimes I_p\\ 1_{n_2}{\pi }_2^{\text{T}}{\Phi }_2\otimes I_p\\ ⋮\\ 1_{n_k}{\pi }_k^{\text{T}}{\Phi }_k\otimes I_p\end{array}\right)\left(Q(t)-Q(0)\right)\\ =\left(\begin{array}{c}{1}_{n_1}{\pi }_1^{\text{T}}{\Phi }_1\otimes I_p\\ 1_{n_2}{\pi }_2^{\text{T}}{\Phi }_2\otimes I_p\\ ⋮\\ 1_{n_k}{\pi }_k^{\text{T}}{\Phi }_k\otimes I_p\end{array}\right){\displaystyle {\int }_0^t}S(r)dr\end{array}$

Equation (4) and ${{\displaystyle \text{lim}}}_{t\to \infty }\Omega =0_{np}$ imply ${{\displaystyle \text{lim}}}_{t\to \infty }S(r)\text{d}r=0_{np}$ . Then, from equation (14), we have

$\begin{array}{l}\underset{t\to \infty }{{\displaystyle \text{lim}}}\left[{(1}_{n_1}\otimes {\rho }_1^{\text{T}}){\Phi }_1\otimes I_p\right]Q_1(t)\\ =\left[{(1}_{n_1}\otimes {\rho }_1^{\text{T}}){\Phi }_1\otimes I_p\right]Q_1(0)\\ \underset{t\to \infty }{{\displaystyle \text{lim}}}\left[\left(1_{n_2}\otimes {\rho }_2^{\text{T}}\right){\Phi }_2\otimes I_p\right]Q_2(t)\\ =\left[{(1}_{n_2}\otimes {({\gamma }_1^{(2)})}^{\text{T}}){\Phi }_1\otimes I_p\right]Q_1(0)\\ \hspace{1em}+\left[{(1}_{n_2}\otimes {\rho }_2^{\text{T}}){\Phi }_2\otimes I_p\right]Q_2(0)\\ \underset{t\to \infty }{{\displaystyle \text{lim}}}\left[\left(1_{n_k}\otimes {\rho }_k^{\text{T}}\right){\Phi }_k\otimes I_p\right]Q_k(t)\\ =\left[{(1}_{n_k}\otimes {({\gamma }_1^{(k)})}^{\text{T}}){\Phi }_1\otimes I_p\right]Q_1(0)\\ \hspace{1em}+\left[{(1}_{n_k}\otimes {({\gamma }_2^{(k)})}^{\text{T}}){\Phi }_2\otimes I_p\right]Q_2(0)\\ \hspace{1em}+\cdots +\left[{(1}_{n_k}\otimes {\rho }_k^{\text{T}}){\Phi }_k\otimes I_p\right]Q_k(0)\end{array}$

Since $1_{n_i}^{\text{T}}{\rho }_i=1$ , we have ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_i={\varphi }_i\left[({\pi }_{\overline{i}}{\Phi }_i)\otimes I_p\right]Q(0)$ , $i=1,2,\dots ,n$ . Thus, static GBC is reached.

Algorithm 1.

Group-bipartite network topology algorithm.

Require: input the Laplacian matrix: $\mathcal{L}$ , input the partition matrix: $\Phi$ , input the mass vector: m, input the length vector: $\ell$ , input the centroid position: c, input the inertia tensor: J, input disturbance matrix: K, input the coupling coefficient: $\gamma$ , input the initial generalized coordinates: $q(0)$ , input the initial velocity: $\dot{q}(0)$ Ensure: Return the generalized coordinates information q and the velocity information $\dot{q}$  1: Input the Laplacian matrix $\mathcal{L}$ ;  2: Extract the number n of agents from the matrix $\mathcal{L}$ ;  3: Input the partition matrix $\Phi$ ;  4: Input the mass vector m;  5: Input the length vector $\ell$ ;  6: Input the centroid position c;  7: Input the inertia tensor J;  8: Calculate the inertia matrix M, the Coriolis and centrifugal matrix C and the gravitational torque g for the given m, $\ell$ , c, and J;  9: Input the partition matrix $\Phi$ ; 10: Input the disturbance matrix K; 11: Input the coupling coefficient $gamma$ ; 12: Input the initial states $q(0)$ and $\dot{q}(0)$ ; 13: Input the $t_{\text{final}}$ ; 14: Get present coordinates $x(0)=q(0)$ and present velocity $v(0)=\dot{q}(0)$ 15: for $t=0:0.001:t_{\text{final}}$ do 16:  Calculate the regressor matrix $Y(t)$ by combining the given m, $\ell$ , c, J with the present states $x(t)$ and $v(t)$ ; 17:  Calculate the sliding vector $s(t)$ by using the present states $x(t)$ and $v(t)$ from equation: $s(t)=v(t)+\Phi \mathcal{L}\Phi x(t)$ ; 18:  Obtain the vector $q(t)$ , $\dot{q}(t)$ , and $\widehat{\theta }(t)$ by submitting M, C, g, K, $Y(t)$ , and $s(t)$ into equation (7) by using Runge–Kutta’s method; 19:  Update the information of the present states $x(t+0.001)=q(t)$ and $v(t+0.001)=\dot{q}(t)$ . 20: end for 21: return the sate information of the generalized coordinates information q and the velocity information $\dot{q}$ ;

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 ${\Phi }_i=I_{n_i}$ , then the i-th group realizes complete consensus for $i\in \{1,2,\dots ,k\}$ . Moreover, when all ${\Phi }_i=I_{n_i}$ , the static GBC problem that is considered in the current article becomes the GC problem. When $k=1$ , 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 estimate ${\widehat{\theta }}_i$ of the i-th robot has the form ${\widehat{\theta }}_i=({\widehat{\theta }}_{i1},{\widehat{\theta }}_{i2},{\widehat{\theta }}_{i3},{\widehat{\theta }}_{i4},{\widehat{\theta }}_{i5}{)}^{\text{T}}$ . The communication status is presented in Figure 2. Obviously, the node set $\mathcal{V}=\{1,2,\dots \mathrm{,7}\}$ has an acyclic partition, namely, ${\mathcal{V}}_1=\{1,2\}$ , ${\mathcal{V}}_2=\{3,4\}$ , and ${\mathcal{V}}_3=\{4,5,6\}$ . The control design weights K_i and ${\Lambda }_i$ , $i=1,2,\dots \mathrm{,7}$ , are set as $K_i=22\text{diag}\{2.3,1.5\}$ and ${\Lambda }_i=10I_2$ for $i=1,2,\dots \mathrm{,7}$ . It is easy to obtain that ${\varphi }_1={\varphi }_3={\varphi }_5=1$ and ${\varphi }_2={\varphi }_4={\varphi }_6={\varphi }_7=-1$ . Then, Assumption 3 holds, and we have $\Phi =\text{diag}\{1,-1,1,-1,1,-1,-1\}$ . Accordingly, the Laplacian matrix can be expressed as

$\mathcal{L}=\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 1& 1& 0& 0& 0\\ -1& -1& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& -1\\ 0& 0& 1& 1& 1& 0& 1\end{array}\right)$

and therefore,

$\Phi \mathcal{L}\Phi =\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 1& -1& -1& 1& 0& 0& 0\\ -1& 1& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 0& 1& -1\\ 0& 0& -1& 1& -1& 0& 1\end{array}\right)$

Figure 1.

The structure of the i-th robot.

Figure 2.

Graph topology with all subgroups being structurally balanced.

It is easy to see that $\Phi \mathcal{L}\Phi$ has three 0 eigenvalues and that the left eigenvectors with 0 are ${\pi }_1{=(\mathrm{1,0,0,0,0,0,0})}^{\text{T}}$ , ${\pi }_2{=(\mathrm{0,0,1,0,0,0,0})}^{\text{T}}$ , and ${\pi }_3=(\frac{2}{3},-\frac{2}{3},\frac{1}{3},-\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}{)}^{\text{T}}$ . From Theorem 1, the final static GBC states are ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_1={\varphi }_1\left({\pi }_1\Phi \otimes I_2\right)Q(0)$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_2={\varphi }_2\left({\pi }_1^{\text{T}}\Phi \otimes I_2\right)Q(0)$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_3={\varphi }_3\left({\pi }_2^{\text{T}}\Phi \otimes I_2\right)Q(0)$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_4={\varphi }_4\left({\pi }_2^{\text{T}}\Phi \otimes I_2\right)Q(0)$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_5={\varphi }_5\left({\pi }_3^{\text{T}}\Phi \otimes I_2\right)Q(0)$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_6={\varphi }_5\left({\pi }_3^{\text{T}}\Phi \otimes I_2\right)Q(0)$ , and ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_7={\varphi }_1\left({\pi }_3^{\text{T}}\Phi \otimes I_2\right)Q(0)$ . Here, the initial value $Q(0)$ is taken as $Q(0)=(0.3377,0.9001,$ $0.9561,0.5752,$ $0.6477,0.4509,$ $0.0811,0.9294,$ $0.6443,0.3786,$ $0.3012,0.4709,$ ${0.4302,0.1848)}^{\text{T}}$ . Following Theorem 1, the final static GBC states are ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_1{=(0.3377,0.9001)}^{\text{T}}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_2=-{(0.3377,0.9001)}^{\text{T}}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_3{=(0.9561,0.5752)}^{\text{T}}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_4=-{(0.9561,0.5752)}^{\text{T}}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_5{=(1.0764,1.3513)}^{\text{T}}$ , ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_6=-{(1.0764,1.3513)}^{\text{T}}$ , and ${{\displaystyle \text{lim}}}_{t\to \infty }{q}_7=-{(1.0764,1.3513)}^{\text{T}}$ . Figures 3 and 4 show the evolution trajectories of the first coordinate and the second coordinate of the position state of seven two-joint manipulators, respectively. It can be clearly seen that the relevant position state converges to the result of the above calculation, which fully conforms to the conclusion of Theorem 1. Figures 5 and 6 show the evolution trajectories of the first coordinate and the second coordinate of the relevant velocity state, respectively. These trajectories converge to 0, which is also completely consistent with the conclusion of Theorem 1. Figures 7 to 13 describe the trajectories of the estimates ${\widehat{\theta }}_i$ , $i=1,2,\dots \mathrm{,7}$ . Although all ${\widehat{\theta }}_i$ , $i=1,2,\dots \mathrm{,7}$ , converge to constant vectors, the final states of ${\widehat{\theta }}_i$ are not equal to the true values, namely, ${\theta }_i$ , $i=1,2,\dots \mathrm{,7}$ . However, static GBC is still achieved in networked robot systems (1) under the proposed control protocol.

Figure 3.

Trajectories of q_i (the first coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 4.

Trajectories of q_i (the second coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 5.

Trajectories of ${\dot{q}}_i$ (the first coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 6.

Trajectories of ${\dot{q}}_i$ (the second coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 7.

Evolutionary process of ${\widehat{\theta }}_1$ .

Figure 8.

Evolutionary process of ${\widehat{\theta }}_2$ .

Figure 9.

Evolutionary process of ${\widehat{\theta }}_3$ .

Figure 10.

Evolutionary process of ${\widehat{\theta }}_4$ .

Figure 11.

Evolutionary process of ${\widehat{\theta }}_5$ .

Figure 12.

Evolutionary process of ${\widehat{\theta }}_6$ .

Figure 13.

Evolutionary process of ${\widehat{\theta }}_7$ .

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 ${\mathcal{V}}_1=\{1,2\}$ , ${\mathcal{V}}_2=\{3,4\}$ , and ${\mathcal{V}}_3=\{4,5,6\}$ , in which the corresponding graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_3$ are cooperatively and competitively coexisting, while ${\mathcal{G}}_2$ is of cooperative type. Then,

$\mathcal{L}=\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& -1& 0& 0& 0\\ -1& -1& -1& 1& 0& 0& 0\\ -1& -1& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& -1\\ 0& 0& 1& -1& 1& 0& 1\end{array}\right)$

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 ${\varphi }_1={\varphi }_3={\varphi }_4={\varphi }_5=1$ and ${\varphi }_2={\varphi }_6={\varphi }_7=-1$ . Notably, ${\varphi }_3={\varphi }_4=1$ implies the same convergence state of the agents in ${\mathcal{V}}_2$ . Then, $\Phi =\text{diag}\{1,-1,1,1,1,-1,-1\}$ and

$\Phi \mathcal{L}\Phi =\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& -1& 0& 0& 0\\ -1& 1& -1& 1& 0& 0& 0\\ -1& 1& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 0& 1& -1\\ 0& 0& -1& 1& -1& 0& 1\end{array}\right)$

15

Figure 14.

Graph topology with the second group being cooperative.

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.

Figure 15.

Evolutionary process of q_i (the first coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 16.

Evolutionary process of q_i (the second coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 17.

Evolutionary process of ${\dot{q}}_i$ (the first coordinate), $i=1,2,\dots \mathrm{,7}$ .

Figure 18.

Evolutionary process of ${\dot{q}}_i$ (the second coordinate), $i=1,2,\dots \mathrm{,7}$ .

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.