# Multiple Time-Varying Formation of Networked Heterogeneous Robotic Systems via Estimator-Based Hierarchical Cooperative Algorithms

Authors: Ming-Feng Ge; Chang-Duo Liang; Xi-Sheng Zhan; Chao-Yang Chen; Guanghui Xu; Jie Chen
Journal: Complexity
 (2020)

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## Abstract

This paper investigates both the time-varying formation and multiple time-varying formation tracking problems of networked heterogeneous robotic systems (NHRSs) with parameter uncertainties and external disturbances in the task space. Each robot inside can be either redundant or nonredundant. Several novel estimator-based hierarchical cooperative (EBHC) algorithms are designed to achieve both the tracking task and the possible preset subtasks for redundant robots. Besides, the designed estimator algorithms guarantee that each robot can obtain the accurate information of their corresponding leaders. By employing Lyapunov stability and input-to-state stability, sufficient conditions on the asymptotic stability of the error closed-loop system are derived. Finally, two simulation examples are presented to verify the effectiveness of the proposed algorithms.


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## Body

## 1. Introduction

In the past decades, the researches on distributed cooperative control of multiagent systems have attracted increasing attentions for its superiority over the traditional centralized control, such as stronger robustness, less energy consumption, and more flexibility [1], which thus has brought out a series of related topics, including consensus [2–4], distributed optimization in smart grids [5], consensus tracking [6–8], optimal tracking [9–13], bipartite consensus/tracking [14, 15], formation [16–19], formation-containment [20–22], synchronization [23–26], and flocking [27]. As a special case of the multiagent system, Euler–Lagrange dynamics are more applicative in many practical rigid structure modeling, especially the networked robotic system (NRS), which has gained increasing popularity due to its extensive potential applications, including industrial manufacture, search missions, and rescue missions [28].Therefore, more and more researchers embark on the cooperative control of NRSs [3, 6, 29]. However, the aforementioned literature studies mainly focused on the joint-space control of NRSs. In real-world applications, the task-space algorithms are indeed more practical due to the consideration of the kinematics analysis, which thus leads to a few researches studying the task-space control of NRSs [30, 31]. It noteworthy that the heterogeneity of NRSs has not been taken into account in the above researches. Besides, when performing tasks, redundant robots are generally more functional than the nonredundant robots for their extra degrees of freedom (DOFs) and usually utilized to perform some corresponding subtasks [32]. Thus, in a system and an application view, it is of great significance to investigate the task-space cooperative control of networked heterogeneous robotic systems (NHRSs).With the development of consensus theory, there is a tendency for researchers to embark on the studies of formation problems. To be specific, Dong et al. have studied the time-varying formation tracking problem of multiagent systems under switching topologies in [18]. The time-invariant formation problems of multiagent systems with time delays have been solved in [33]. A distributed position estimation method is employed in [34] to realize the formation of multiagent systems. Ge et al. [35] have proposed a finite-time algorithm to achieve a switching type of time-varying formation for NRSs in the joint space. It should be pointed out that most of the aforementioned works focused on the time-invariant or time-varying formation in the case of a single leader, namely, all agents inside form a single formation. However, in many practical circumstances, the main results of the above literature studies cannot be directly applied if the networked systems are required to execute multiple tasks concurrently. A few works centralized on multiple tracking [7, 36, 37], multiple formation [16, 17], and group time-varying formation [19] are thus activated. To the best of our knowledge, the time-varying formation tracking problem of NHRSs in the task space remains unsolved, let alone the multiple time-varying formation tracking problem.On the contrary, in the practical robotic application, inexact physical parameters and external disturbances are ineluctable and usually have negative impacts on the performance of the NHRS. It thus becomes extremely challenging to study the time-varying formation tracking problem of NHRSs in the task space with parameter uncertainties and external disturbances.Based on the above discussions, this paper investigates both the time-varying formation and multiple time-varying formation problems of NHRSs in the task space with parameter uncertainties and external disturbances. The main contributions of this paper can be summed up threefold: (1) The time-varying formation and multiple time-varying formation tracking problems of NHRSs in the task space are firstly studied and successfully addressed by designing several novel estimator-based hierarchical cooperative (EBHC) algorithms. (2) The proposed finite-time distributed estimator algorithm can guarantee the accurate estimation of the states of the virtual leaders, namely, the availability of leaders’ information for each robot is assured. (3) Without interfering with the time-varying formation tracking task, the possible preset subtasks for redundant robots can also be accomplished.Notations: letℝn equal the n-dimensional Euclidean space; especially, ℝ represents the real number field. The Kronecker product is denoted by ⊗. Im is the m-order identity matrix, and 1m0m symbolizes an m-dimensional column vector with all its elements being 1 (0). ℒ2 and ℒ∞ denote the ℒ2 space and ℒ∞ space. ⋅ and sup⋅ denote the Euclidean norm and its supremum, respectively. For given ℓ>0 and ℐ=ℐ1,ℐ2,…,ℐnT, one has sigℐℓ=sgnℐ1ℐℓ,sgnℑ2ℑℓ,…,sgnℑnℑℓT.


## 2. Preliminaries and Problem Formulation

### 2.1. Graph Theory

The considered NHRS consists ofN robots. An undirected graph g=ν,ℰ,W is employed to model the interactions among the robots, where ν=1,2,…,N denotes the robot set, ℰ⊆ν×ν symbolizes the edge set, and W=wijN×N is the nonnegatively weighted adjacency matrix. An undirected edge vj,vi∈ℰ implies that robots vi and vj can directly receive the information from each other. The neighbour set of the robot i is denoted as Ni=j∈νj,i∈ℰ. Generally, when i≠j, one has wij>0 if j∈Ni; otherwise, wij=0. Besides, it is assumed that g is a simple graph, namely, wii=0 for all i∈ν. The Laplacian matrix ℒ=lijN×N of g is defined as lij=−wij,∀i≠j, and lii=∑j∈Niwij,fori,j∈ν. Taking the virtual leader into consideration, a pinning matrix ℬ=diagb1,b2,…,bN is introduced to indicate the interactions between the robots and the leader. Specifically, bi>0 if the information of the leader is accessible for the robot i; otherwise, bi=0. It is noteworthy that the leader will not receive any information from the robots. Then, an usual assumption on the interaction topology is given as follows.Assumption 1.
The communication topology between the robots is undirected. Besides, for each robot, there exists at least one path such that the considered robot can receive the information of the leader, namely, the information of the leader is globally reachable for all robots inside.


### 2.2. System Formulation

Consider the NHRS with the virtual leader. The dynamics and kinematics of the roboti are given as follows:(1)Hiqiq¨i+Ciqi,q˙iq˙i+giqi=μi+dit,xi=ψqi,x˙i=Jiqiq˙i,where qi∈ℝpi denotes the generalized coordinates in the joint space and q˙i and q¨i are its velocity and acceleration, respectively; xi∈ℝd represents the generalized coordinates in the task space with x˙i being its velocity; Hiqi∈ℝpi×pi, Ciqi,q˙i∈ℝpi×pi, and giqi∈Ciqi,q˙i∈ℝpi symbolize the positive-definite inertia matrix, the Coriolis-centrifugal matrix, and the gravitational torque, respectively; μi denotes the input torque; dit is the external disturbance; and Jiqi=∂ψqi/∂qi∈giqi∈ℝd×pi represents the Jacobian matrix, which is supposed to be bounded and nonsingular throughout the full text.Besides, some properties associated with system (1) are presented as follows for the later analysis.Property 1.
Hiqi is positive definite, and H˙iqi−2Ciqi,q˙i is skew symmetric, namely, for any given ξ∈ℝpi, one obtains that ξTHiqiξ>0 and ξTH˙iqi−2Ciqi,q˙iξ=0.Property 2.
Hiqi, Ciqi, and giqi are bounded for all possible qi, i.e., σ¯HIpi≤Hiqi≤σ¯HIpi, Ciqi,q˙i≤σ¯Cq˙i, and giqi≤σ¯g for all i∈ν, where σ¯H, σ¯H, σ¯C, and σ¯g are positive constants.Property 3.
The involved dynamic terms can be linearly parameterized for any given vectorsυ and ρ with proper dimensions, i.e., Hiqiυ+Ciqi,q˙iρ+giqi=Yiqi,q˙i,ρ,υϑi, where Yiqi,q˙i,ρ,υ is the dynamical regressor and ϑi is the unknown constant parameter vector.Remark 1.
Property 3 is obtained because the inertia matrixHiqi can be linearly represented by a set of properly chosen combinations of dynamic parameters. The specific evolutionary process is omitted here for the space limit. Refer to [38] for more details.
On the contrary, the virtual leader is described as(2)x˙r=vr,v˙r=ar,where xr, vr, and ar, respectively, denote the reference position, velocity, and acceleration of the leader. In practical robotic applications, NHRSs are always expected to track the certain desired trajectory with respect to the practical task. Then, a reasonable assumption on the acceleration of the leader is given as follows.Assumption 2.
It is assumed that the jerk of the virtual leader is bounded, namely,a˙r≤σ¯a˙, where σ¯a˙ is a positive constant.


### 2.3. Problem Formulation

The control objective is to design an appropriate control algorithm aiming at solving the time-varying formation tracking problem defined hereinafter. Firstly, the time-varying formation offset of thei-th robot is specified by hit∈ℝd, and the tracking errors are predefined as eit=xit−xrt−hit and e˙it=x˙it−vrt−h˙it.Definition 1.
The time-varying formation tracking problem of the NHRS is addressed if the tracking errors satisfy(3)limt⟶∞eit=0,limt⟶∞e˙it=0,where i∈ν.
Some useful lemmas that will be invoked in the later analysis are presented as follows.Lemma 1 (see [39]).
Given the second-order dynamical system(4)ξ˙=η,η˙=−Γαsigξρ1+βsigηρ2,where ξ,η∈ℝn and Γ is a positive-definite matrix with proper dimension. Then, the origin is a finite-time stable equilibrium of system (4) if α>0, β>0, 0<ρ1<1, and ρ2=2ρ1/ρ1+1.Lemma 2 (see [40]).
For givenft,x,μ being Lipschitz, the system x˙=ft,x,μ is input-to-state stable, if the corresponding unexcited system x˙=ft,x,0 is globally exponentially stable with the equilibrium being the origin, i.e., xt=0 as t⟶∞ if μt⟶0 as t⟶∞.


## 2.1. Graph Theory

The considered NHRS consists ofN robots. An undirected graph g=ν,ℰ,W is employed to model the interactions among the robots, where ν=1,2,…,N denotes the robot set, ℰ⊆ν×ν symbolizes the edge set, and W=wijN×N is the nonnegatively weighted adjacency matrix. An undirected edge vj,vi∈ℰ implies that robots vi and vj can directly receive the information from each other. The neighbour set of the robot i is denoted as Ni=j∈νj,i∈ℰ. Generally, when i≠j, one has wij>0 if j∈Ni; otherwise, wij=0. Besides, it is assumed that g is a simple graph, namely, wii=0 for all i∈ν. The Laplacian matrix ℒ=lijN×N of g is defined as lij=−wij,∀i≠j, and lii=∑j∈Niwij,fori,j∈ν. Taking the virtual leader into consideration, a pinning matrix ℬ=diagb1,b2,…,bN is introduced to indicate the interactions between the robots and the leader. Specifically, bi>0 if the information of the leader is accessible for the robot i; otherwise, bi=0. It is noteworthy that the leader will not receive any information from the robots. Then, an usual assumption on the interaction topology is given as follows.Assumption 1.
The communication topology between the robots is undirected. Besides, for each robot, there exists at least one path such that the considered robot can receive the information of the leader, namely, the information of the leader is globally reachable for all robots inside.


## 2.2. System Formulation

Consider the NHRS with the virtual leader. The dynamics and kinematics of the roboti are given as follows:(1)Hiqiq¨i+Ciqi,q˙iq˙i+giqi=μi+dit,xi=ψqi,x˙i=Jiqiq˙i,where qi∈ℝpi denotes the generalized coordinates in the joint space and q˙i and q¨i are its velocity and acceleration, respectively; xi∈ℝd represents the generalized coordinates in the task space with x˙i being its velocity; Hiqi∈ℝpi×pi, Ciqi,q˙i∈ℝpi×pi, and giqi∈Ciqi,q˙i∈ℝpi symbolize the positive-definite inertia matrix, the Coriolis-centrifugal matrix, and the gravitational torque, respectively; μi denotes the input torque; dit is the external disturbance; and Jiqi=∂ψqi/∂qi∈giqi∈ℝd×pi represents the Jacobian matrix, which is supposed to be bounded and nonsingular throughout the full text.Besides, some properties associated with system (1) are presented as follows for the later analysis.Property 1.
Hiqi is positive definite, and H˙iqi−2Ciqi,q˙i is skew symmetric, namely, for any given ξ∈ℝpi, one obtains that ξTHiqiξ>0 and ξTH˙iqi−2Ciqi,q˙iξ=0.Property 2.
Hiqi, Ciqi, and giqi are bounded for all possible qi, i.e., σ¯HIpi≤Hiqi≤σ¯HIpi, Ciqi,q˙i≤σ¯Cq˙i, and giqi≤σ¯g for all i∈ν, where σ¯H, σ¯H, σ¯C, and σ¯g are positive constants.Property 3.
The involved dynamic terms can be linearly parameterized for any given vectorsυ and ρ with proper dimensions, i.e., Hiqiυ+Ciqi,q˙iρ+giqi=Yiqi,q˙i,ρ,υϑi, where Yiqi,q˙i,ρ,υ is the dynamical regressor and ϑi is the unknown constant parameter vector.Remark 1.
Property 3 is obtained because the inertia matrixHiqi can be linearly represented by a set of properly chosen combinations of dynamic parameters. The specific evolutionary process is omitted here for the space limit. Refer to [38] for more details.
On the contrary, the virtual leader is described as(2)x˙r=vr,v˙r=ar,where xr, vr, and ar, respectively, denote the reference position, velocity, and acceleration of the leader. In practical robotic applications, NHRSs are always expected to track the certain desired trajectory with respect to the practical task. Then, a reasonable assumption on the acceleration of the leader is given as follows.Assumption 2.
It is assumed that the jerk of the virtual leader is bounded, namely,a˙r≤σ¯a˙, where σ¯a˙ is a positive constant.


## 2.3. Problem Formulation

The control objective is to design an appropriate control algorithm aiming at solving the time-varying formation tracking problem defined hereinafter. Firstly, the time-varying formation offset of thei-th robot is specified by hit∈ℝd, and the tracking errors are predefined as eit=xit−xrt−hit and e˙it=x˙it−vrt−h˙it.Definition 1.
The time-varying formation tracking problem of the NHRS is addressed if the tracking errors satisfy(3)limt⟶∞eit=0,limt⟶∞e˙it=0,where i∈ν.
Some useful lemmas that will be invoked in the later analysis are presented as follows.Lemma 1 (see [39]).
Given the second-order dynamical system(4)ξ˙=η,η˙=−Γαsigξρ1+βsigηρ2,where ξ,η∈ℝn and Γ is a positive-definite matrix with proper dimension. Then, the origin is a finite-time stable equilibrium of system (4) if α>0, β>0, 0<ρ1<1, and ρ2=2ρ1/ρ1+1.Lemma 2 (see [40]).
For givenft,x,μ being Lipschitz, the system x˙=ft,x,μ is input-to-state stable, if the corresponding unexcited system x˙=ft,x,0 is globally exponentially stable with the equilibrium being the origin, i.e., xt=0 as t⟶∞ if μt⟶0 as t⟶∞.


## 3. Analysis for Time-Varying Formation Tracking

### 3.1. Estimator-Based Hierarchical Cooperative Algorithm I

In this section, EBHC algorithm I is proposed for addressing the time-varying formation tracking problem of NHRSs. For the sake of simplification, the redundant robot set and nonredundant robot set are severally defined asℸ=i∈νpi>d and ℶ=i∈νpi=d. Furthermore, Ji# is predefined as(5)Ji#=Ji−1,i∈ℶ,Ji†,i∈ℸ.Letx^i, v^i, a^i, and ϑ^i, respectively, be the estimators of xi, vi, ai, and ai, ∀i∈ν, where vi and ai are, respectively, the velocity and acceleration in the task space. An auxiliary joint-space velocity is properly defined as(6)ψ^⋅i=Ji#v^i−ωixi−x^i+Ipi−Ji#Jiφi,where ωi is a positive scalar and φi∈ℝpi denotes the negative gradients of some performance indices with respect to the redundant robots. It follows that the auxiliary acceleration can be presented as(7)ψ^¨i=J˙i#v^i−ωixi−x^i+Ji#a^i−ωix˙i−v^i+ddtIpi−Ji#Jiφi.Thereupon, the sliding vector is defined as(8)s^i=q˙i−ψ^⋅i.Then, EBHC algorithm I is presented as follows:(9)μi=Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ^i−JiTκoiJis^i−κsis^i−κdisgns^i,ϑ^⋅i=−ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,(10)x^⋅i=v^i,v^⋅i=−αsig∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr−hiρ1−βsig∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr−h˙iρ2+a^i+h¨i,a^⋅i=−γsgn∑j∈Niwija^i−a^j+bia^i−ar,where i∈ν, κoi∈ℝd×d, and κsi,κdi∈ℝpi×pi are the feedback gain matrix, Λi is the adaptive positive-definite matrix with appropriate dimensions, and α, β, γ, ρ1, and ρ2 are positive constants. To be specific, (9) is the adaptive smooth control layer and (10) is the distributed estimator layer.Letx˜i=x^i−xr−hi, v˜i=v^i−vr−h˙i, a˜i=a^i−ar, and ϑ˜i=ϑi−ϑ^i. Substituting (9) and (10) into (1) yields the following cascade closed-loop system:(11)Hiqis^⋅i+Ciqi,q˙is^i=−Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ˜i−JiTκoiJis^i−κsis^i−κdisgns^i+dit,ϑ˜⋅i=ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,x˜⋅i=v˜i,v˜⋅i=−αsig∑j∈Niwijx˜i−x˜j+bix˜iρ1−βsig∑j∈Niwijv˜i−v˜j+biv˜iρ2+a˜i,a˜⋅i=−γsgn∑j∈Niwija˜i−a˜j+bia˜i−a˙r.


### 3.2. Coordination Analysis and Boundedness Analysis for the Estimator Layer

Letx˜=colx˜1,x˜2,…,x˜N, v˜=colv˜1,v˜2,…,v˜N, a˜=cola˜1,a˜2,…,a˜N, aR=1N⊗ar, and J=ℒ+ℬ. Then, the closed-loop system of the estimator layer can be further rewritten in the following compact form:(12)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2+a˜,a˜⋅=−γsgnJ⊗Ida˜−a˙R.Theorem 1.
Suppose that Assumptions 1 and 2 hold. The distributed estimator algorithm (10) guarantees that there exists Tf>0 such that x˜i=0 and v˜i=0 when t≥Tf for all i∈ν if(13)α>0,β>0,γ>σ¯a˙,0<ρ1<1,ρ2=2ρ1ρ1+1.Besides, the statesqi and q˙i remain bounded when t0≤t<Tf for any given bounded initial values, where t0 denotes the initial time.Proof.
For the first presentation, we are going to prove thatx˜i and v˜i, ∀i∈ν, converge to the origin in finite time Tf≥t0.
Based on Assumption 1, it can be derived thatJ is positive definite. Then, the Lyapunov function is constructed for the third differential system in (12) as V1a˜=1/2a˜TJ⊗Ida˜. It follows that(14)V˙1a˜=a˜TJ⊗Ida˜⋅=−γJ⊗Ida˜1−a˜TJ⊗Ida˙R≤−γJ⊗Ida˜+a˙RJ⊗Ida˜≤−γ−σ¯a˙J⊗Ida˜.
Equation (13) implies that V˙1a˜≤0. Besides, note that V1a˜≤1/2J⊗Ida˜2. It then follows that V˙1a˜≤−γ−σ¯a˙2J⊗IdV11/2a˜. Thus, we can derive that a˜ converges to zero in finite time Jf1. Then, when t≥Jf1, (12) equals(15)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2.
Taking the linear transformation forX˜=J⊗Idx˜ and V˜=J⊗Idv˜ yields that(16)X˜⋅=V˜,V˜⋅=−J⊗IdαsigX˜ρ1+βsigV˜ρ2.
Then, according to Lemma 1 and (13), system (16) is globally finite-time stable with the origin being its stable equilibrium, i.e., there exists a settling time Tf2>0 such that X˜ and V˜ converge to the origin on t∈Jf1,Jf2. Note that Jf=Jf1+Jf2. It follows that x˜=0 and v˜=0 when t≥Tf, namely, x˜i=0 and v˜i=0 when t≥Tf for all i∈ν.
The second presentation focuses on the boundedness analysis. Note that(17)sgn∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr−hi∞≤1,sgn∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr−h˙i∞≤1.
Besides, Assumption 2 implies thatxr, vr, and ar are all globally bounded on t∈t0,Tf, namely, xr, vr, ar∈ℒ2∩ℒ∞ as t0≤t<Tf, and the formation offsets of robots are usually set to be bounded in practical robotic applications, i.e., hit,h˙it,h¨it∈ℒ2∩ℒ∞ for all i∈ν. For any given initial values x^it0,v^it0∈ℒ2∩ℒ∞, (10) implies that x^i,v^i∈ℒ2∩ℒ∞ when t0≤t<Tf. Besides, through (1), we know that xi,x˙i∈ℒ2∩ℒ∞ since qi,q˙i∈ℒ2∩ℒ∞ as t0≤t<Tf. Then, substituting the aforementioned variables into (6)–(8) yields that ψ^i,ψ^⋅i,s^i∈ℒ2∩ℒ∞ as t0≤t<Tf. It follows from Property 2 that Yiqi,q˙i,ψ^⋅i,ψ^¨i∈ℒ2∩ℒ∞ as t0≤t<Tf. Furthermore, (9) implies that ϑ^i∈ℒ2∩ℒ∞ as t0≤t<Tf for any given initial value ϑ^it0∈ℒ2∩ℒ∞. Hence, we can conclude that qi,q˙i,s^i,ϑ^i∈ℒ2∩ℒ∞ if the initial values qit0,q˙it0,x^it0,x^⋅t0,ϑ^it0∈ℒ2∩ℒ∞ as t0≤t<Tf,∀i∈ν. This completes the proof.


### 3.3. Convergence Analysis

In this section, we will show whether the time-varying formation tracking problem can be addressed under the proposed EBHC algorithm I or not. Firstly, the norm variables associated withψ^⋅i, ψ^¨i, and s^i are given as(18)ψ˙i=Ji#vr+h˙i−ωixi−xr−hi+Ipi−Ji#Jiφi,(19)ψ¨i=J˙i#vr+h˙i−ωixi−xr−hi+Ji#ar−ωix˙i−vr−h˙i+ddtIpi−Ji#Jiφi,(20)si=qi−ψ˙i.Based on Theorem 1, it can be figured out thatψ˙i, ψ¨i, and si remain bounded when t0≤t<Tf and ψ^˙i=ψ˙i, ψ^¨i=ψ¨i, and s^i=si when t≥Tf. Thus, when t≥Tf, the following closed-loop system arises based on (11):(21)s˙i=−Hi−1qiCiqi,q˙isi+Yiqi,q˙i,ψ˙i,ψ¨iϑ˜+JiTκoiJisi+κsisi+κdisgnsi−dit,ϑ˜˙i=ΛiYiTqi,q˙i,ψ˙i,ψ¨isi.Theorem 2.
Suppose that Assumptions 1 and 2 hold. The time-varying formation tracking problem of the NHRS can be addressed under the proposed EBHC algorithm I, if(22)λminκoi>0,λminκsi>0,λminκdi≥supt≥t0dit,namely, it can be obtained that the task-space coordinates xi and vi can eventually form the time-varying formations, i.e., xi⟶xr+hit and vi⟶vr+h˙it as t⟶∞.Proof.
Consider the following storage function as the Lyapunov function candidate for system (21):(23)Vsi=12siTHiqisi+12ϑ˜iTΛi−1ϑ˜i.
Differentiating (23) along (21) yields that(24)V˙si=siTHiqis˙i+12siTH˙iqisi+ϑ˜iTΛi−1ϑ˜⋅i=−siTCiqi,q˙isi+Yiqi,q˙i,ψ˙i,ψ¨iϑ˜+JiTκoiJisi+κsisi+κdisgnsi−dit+12siTH˙iqisi+ϑ˜iTYiTqi,q˙i,ψ˙i,ψ¨isi=12siTH˙iqi−2Ciqi,q˙isi−siTJiTκoiJi+κsisi−siTκdisgnsi+siTdit≤−siTJiTκoiJi+κsisi−siλminκdi−supt≥t0dit,where Property 1 and λminκdi≥supt≥t0dit in (22) have been invoked to make the above inequality hold. Thereupon, λminκoi>0 and λminκsi>0 imply that V˙si<0. Furthermore, (22) and (24) provide that si∈ℒ2∩ℒ∞. Then, one obtains that s˙i∈ℒ∞ through the closed-loop system (21). Thus, V¨si∈ℒ∞ is derived. According to Barbalat’s lemma [41] and the above stability analysis, it thus can be concluded that si⟶0 as t⟶∞.
According to (18) and (20), Jiqisi is presented as follows when t≥Tf:(25)Jiqisi=Jiqiqi−vr+h˙i−ωixi−xr−hi=x˙i−vr−h˙i+ωixi−xr−hi=e˙i+ωiei,which can be further rewritten in the following form:(26)e˙i+ωiei=Jiqisi.
Note thatJiqisi⟶0 as t⟶∞ for the boundedness of Jiqi. According to Lemma 2, it thus follows that the kinematics loop (26) is input-to-state stable with Jiqisi being the input and ei and e˙i being the states. Then, ∀i∈ν, we draw the conclusion that ei⟶0 and e˙i⟶0 as t⟶∞ through the fact that Jiqisi⟶0 as t⟶∞. This completes the proof.Remark 2.
Selecting appropriateφi for a redundant robot can execute some subtasks but does not conflict with the main task (i.e., the time-varying formation tracking task). For instants, denote qi1 as the first element of qi, i∈ℸ, and choose the performance index as Hiqi=0.412−qi12, then one has φi=−∂Hiqi/∂qi=0.812−qi1,0,…,0T. Such a setting mainly aims at forcing the first joint of the i-th robot toward 12 rad. A corresponding simulation experiment on redundant robots will be conducted to support the above presentation.Remark 3.
According to the study in [32], the subtask for the redundant robots is called completed if(27)limt⟶∞esi=limt⟶∞Ipi−Ji#qiJiqiq˙i−φi=0.
Then, it will be discussed whether the subtasks can be addressed or not under the proposed EBHC algorithm I. Note that whent≥Tf,(28)esi=Ipi−Ji#qiJiqiq˙i−φi=Ipi−Ji#qiJiqiq˙i−Ji#vr+h˙i−ωixi−xr−hi−Ipi−Ji#qiφi=Ipi−Ji#qiJiqiq˙i−ψ˙i=Ipi−Ji#qiJiqisi,where Ipi−Ji#qiJiqiIpi−Ji#qiJiqi=Ipi−Ji#qiJiqi, Ipi−Ji#qiJiqiJi#qi=0, and (18) and (20) have been invoked to obtain the above equation. It has been already obtained from Theorem 2 that si⟶0 as t⟶∞. Then, one can easily derive that Ipi−Ji#qiJiqisi⟶0 as t⟶∞, namely, limt⟶∞esi=0. This completes the proof.Remark 4.
Referencing [32], we can choose the performance index of redundant robots as Hiqi=−detJiJiT to maximize the manipulability of the redundant robots and tend to keep the robots away from singularities, namely, setting φi=−∂Hiqi/∂qi=∂detJiJiT/∂qi can help maintain the Jacobian matrices for the redundant robots to be nonsingular.


## 3.1. Estimator-Based Hierarchical Cooperative Algorithm I

In this section, EBHC algorithm I is proposed for addressing the time-varying formation tracking problem of NHRSs. For the sake of simplification, the redundant robot set and nonredundant robot set are severally defined asℸ=i∈νpi>d and ℶ=i∈νpi=d. Furthermore, Ji# is predefined as(5)Ji#=Ji−1,i∈ℶ,Ji†,i∈ℸ.Letx^i, v^i, a^i, and ϑ^i, respectively, be the estimators of xi, vi, ai, and ai, ∀i∈ν, where vi and ai are, respectively, the velocity and acceleration in the task space. An auxiliary joint-space velocity is properly defined as(6)ψ^⋅i=Ji#v^i−ωixi−x^i+Ipi−Ji#Jiφi,where ωi is a positive scalar and φi∈ℝpi denotes the negative gradients of some performance indices with respect to the redundant robots. It follows that the auxiliary acceleration can be presented as(7)ψ^¨i=J˙i#v^i−ωixi−x^i+Ji#a^i−ωix˙i−v^i+ddtIpi−Ji#Jiφi.Thereupon, the sliding vector is defined as(8)s^i=q˙i−ψ^⋅i.Then, EBHC algorithm I is presented as follows:(9)μi=Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ^i−JiTκoiJis^i−κsis^i−κdisgns^i,ϑ^⋅i=−ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,(10)x^⋅i=v^i,v^⋅i=−αsig∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr−hiρ1−βsig∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr−h˙iρ2+a^i+h¨i,a^⋅i=−γsgn∑j∈Niwija^i−a^j+bia^i−ar,where i∈ν, κoi∈ℝd×d, and κsi,κdi∈ℝpi×pi are the feedback gain matrix, Λi is the adaptive positive-definite matrix with appropriate dimensions, and α, β, γ, ρ1, and ρ2 are positive constants. To be specific, (9) is the adaptive smooth control layer and (10) is the distributed estimator layer.Letx˜i=x^i−xr−hi, v˜i=v^i−vr−h˙i, a˜i=a^i−ar, and ϑ˜i=ϑi−ϑ^i. Substituting (9) and (10) into (1) yields the following cascade closed-loop system:(11)Hiqis^⋅i+Ciqi,q˙is^i=−Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ˜i−JiTκoiJis^i−κsis^i−κdisgns^i+dit,ϑ˜⋅i=ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,x˜⋅i=v˜i,v˜⋅i=−αsig∑j∈Niwijx˜i−x˜j+bix˜iρ1−βsig∑j∈Niwijv˜i−v˜j+biv˜iρ2+a˜i,a˜⋅i=−γsgn∑j∈Niwija˜i−a˜j+bia˜i−a˙r.


## 3.2. Coordination Analysis and Boundedness Analysis for the Estimator Layer

Letx˜=colx˜1,x˜2,…,x˜N, v˜=colv˜1,v˜2,…,v˜N, a˜=cola˜1,a˜2,…,a˜N, aR=1N⊗ar, and J=ℒ+ℬ. Then, the closed-loop system of the estimator layer can be further rewritten in the following compact form:(12)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2+a˜,a˜⋅=−γsgnJ⊗Ida˜−a˙R.Theorem 1.
Suppose that Assumptions 1 and 2 hold. The distributed estimator algorithm (10) guarantees that there exists Tf>0 such that x˜i=0 and v˜i=0 when t≥Tf for all i∈ν if(13)α>0,β>0,γ>σ¯a˙,0<ρ1<1,ρ2=2ρ1ρ1+1.Besides, the statesqi and q˙i remain bounded when t0≤t<Tf for any given bounded initial values, where t0 denotes the initial time.Proof.
For the first presentation, we are going to prove thatx˜i and v˜i, ∀i∈ν, converge to the origin in finite time Tf≥t0.
Based on Assumption 1, it can be derived thatJ is positive definite. Then, the Lyapunov function is constructed for the third differential system in (12) as V1a˜=1/2a˜TJ⊗Ida˜. It follows that(14)V˙1a˜=a˜TJ⊗Ida˜⋅=−γJ⊗Ida˜1−a˜TJ⊗Ida˙R≤−γJ⊗Ida˜+a˙RJ⊗Ida˜≤−γ−σ¯a˙J⊗Ida˜.
Equation (13) implies that V˙1a˜≤0. Besides, note that V1a˜≤1/2J⊗Ida˜2. It then follows that V˙1a˜≤−γ−σ¯a˙2J⊗IdV11/2a˜. Thus, we can derive that a˜ converges to zero in finite time Jf1. Then, when t≥Jf1, (12) equals(15)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2.
Taking the linear transformation forX˜=J⊗Idx˜ and V˜=J⊗Idv˜ yields that(16)X˜⋅=V˜,V˜⋅=−J⊗IdαsigX˜ρ1+βsigV˜ρ2.
Then, according to Lemma 1 and (13), system (16) is globally finite-time stable with the origin being its stable equilibrium, i.e., there exists a settling time Tf2>0 such that X˜ and V˜ converge to the origin on t∈Jf1,Jf2. Note that Jf=Jf1+Jf2. It follows that x˜=0 and v˜=0 when t≥Tf, namely, x˜i=0 and v˜i=0 when t≥Tf for all i∈ν.
The second presentation focuses on the boundedness analysis. Note that(17)sgn∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr−hi∞≤1,sgn∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr−h˙i∞≤1.
Besides, Assumption 2 implies thatxr, vr, and ar are all globally bounded on t∈t0,Tf, namely, xr, vr, ar∈ℒ2∩ℒ∞ as t0≤t<Tf, and the formation offsets of robots are usually set to be bounded in practical robotic applications, i.e., hit,h˙it,h¨it∈ℒ2∩ℒ∞ for all i∈ν. For any given initial values x^it0,v^it0∈ℒ2∩ℒ∞, (10) implies that x^i,v^i∈ℒ2∩ℒ∞ when t0≤t<Tf. Besides, through (1), we know that xi,x˙i∈ℒ2∩ℒ∞ since qi,q˙i∈ℒ2∩ℒ∞ as t0≤t<Tf. Then, substituting the aforementioned variables into (6)–(8) yields that ψ^i,ψ^⋅i,s^i∈ℒ2∩ℒ∞ as t0≤t<Tf. It follows from Property 2 that Yiqi,q˙i,ψ^⋅i,ψ^¨i∈ℒ2∩ℒ∞ as t0≤t<Tf. Furthermore, (9) implies that ϑ^i∈ℒ2∩ℒ∞ as t0≤t<Tf for any given initial value ϑ^it0∈ℒ2∩ℒ∞. Hence, we can conclude that qi,q˙i,s^i,ϑ^i∈ℒ2∩ℒ∞ if the initial values qit0,q˙it0,x^it0,x^⋅t0,ϑ^it0∈ℒ2∩ℒ∞ as t0≤t<Tf,∀i∈ν. This completes the proof.


## 3.3. Convergence Analysis

In this section, we will show whether the time-varying formation tracking problem can be addressed under the proposed EBHC algorithm I or not. Firstly, the norm variables associated withψ^⋅i, ψ^¨i, and s^i are given as(18)ψ˙i=Ji#vr+h˙i−ωixi−xr−hi+Ipi−Ji#Jiφi,(19)ψ¨i=J˙i#vr+h˙i−ωixi−xr−hi+Ji#ar−ωix˙i−vr−h˙i+ddtIpi−Ji#Jiφi,(20)si=qi−ψ˙i.Based on Theorem 1, it can be figured out thatψ˙i, ψ¨i, and si remain bounded when t0≤t<Tf and ψ^˙i=ψ˙i, ψ^¨i=ψ¨i, and s^i=si when t≥Tf. Thus, when t≥Tf, the following closed-loop system arises based on (11):(21)s˙i=−Hi−1qiCiqi,q˙isi+Yiqi,q˙i,ψ˙i,ψ¨iϑ˜+JiTκoiJisi+κsisi+κdisgnsi−dit,ϑ˜˙i=ΛiYiTqi,q˙i,ψ˙i,ψ¨isi.Theorem 2.
Suppose that Assumptions 1 and 2 hold. The time-varying formation tracking problem of the NHRS can be addressed under the proposed EBHC algorithm I, if(22)λminκoi>0,λminκsi>0,λminκdi≥supt≥t0dit,namely, it can be obtained that the task-space coordinates xi and vi can eventually form the time-varying formations, i.e., xi⟶xr+hit and vi⟶vr+h˙it as t⟶∞.Proof.
Consider the following storage function as the Lyapunov function candidate for system (21):(23)Vsi=12siTHiqisi+12ϑ˜iTΛi−1ϑ˜i.
Differentiating (23) along (21) yields that(24)V˙si=siTHiqis˙i+12siTH˙iqisi+ϑ˜iTΛi−1ϑ˜⋅i=−siTCiqi,q˙isi+Yiqi,q˙i,ψ˙i,ψ¨iϑ˜+JiTκoiJisi+κsisi+κdisgnsi−dit+12siTH˙iqisi+ϑ˜iTYiTqi,q˙i,ψ˙i,ψ¨isi=12siTH˙iqi−2Ciqi,q˙isi−siTJiTκoiJi+κsisi−siTκdisgnsi+siTdit≤−siTJiTκoiJi+κsisi−siλminκdi−supt≥t0dit,where Property 1 and λminκdi≥supt≥t0dit in (22) have been invoked to make the above inequality hold. Thereupon, λminκoi>0 and λminκsi>0 imply that V˙si<0. Furthermore, (22) and (24) provide that si∈ℒ2∩ℒ∞. Then, one obtains that s˙i∈ℒ∞ through the closed-loop system (21). Thus, V¨si∈ℒ∞ is derived. According to Barbalat’s lemma [41] and the above stability analysis, it thus can be concluded that si⟶0 as t⟶∞.
According to (18) and (20), Jiqisi is presented as follows when t≥Tf:(25)Jiqisi=Jiqiqi−vr+h˙i−ωixi−xr−hi=x˙i−vr−h˙i+ωixi−xr−hi=e˙i+ωiei,which can be further rewritten in the following form:(26)e˙i+ωiei=Jiqisi.
Note thatJiqisi⟶0 as t⟶∞ for the boundedness of Jiqi. According to Lemma 2, it thus follows that the kinematics loop (26) is input-to-state stable with Jiqisi being the input and ei and e˙i being the states. Then, ∀i∈ν, we draw the conclusion that ei⟶0 and e˙i⟶0 as t⟶∞ through the fact that Jiqisi⟶0 as t⟶∞. This completes the proof.Remark 2.
Selecting appropriateφi for a redundant robot can execute some subtasks but does not conflict with the main task (i.e., the time-varying formation tracking task). For instants, denote qi1 as the first element of qi, i∈ℸ, and choose the performance index as Hiqi=0.412−qi12, then one has φi=−∂Hiqi/∂qi=0.812−qi1,0,…,0T. Such a setting mainly aims at forcing the first joint of the i-th robot toward 12 rad. A corresponding simulation experiment on redundant robots will be conducted to support the above presentation.Remark 3.
According to the study in [32], the subtask for the redundant robots is called completed if(27)limt⟶∞esi=limt⟶∞Ipi−Ji#qiJiqiq˙i−φi=0.
Then, it will be discussed whether the subtasks can be addressed or not under the proposed EBHC algorithm I. Note that whent≥Tf,(28)esi=Ipi−Ji#qiJiqiq˙i−φi=Ipi−Ji#qiJiqiq˙i−Ji#vr+h˙i−ωixi−xr−hi−Ipi−Ji#qiφi=Ipi−Ji#qiJiqiq˙i−ψ˙i=Ipi−Ji#qiJiqisi,where Ipi−Ji#qiJiqiIpi−Ji#qiJiqi=Ipi−Ji#qiJiqi, Ipi−Ji#qiJiqiJi#qi=0, and (18) and (20) have been invoked to obtain the above equation. It has been already obtained from Theorem 2 that si⟶0 as t⟶∞. Then, one can easily derive that Ipi−Ji#qiJiqisi⟶0 as t⟶∞, namely, limt⟶∞esi=0. This completes the proof.Remark 4.
Referencing [32], we can choose the performance index of redundant robots as Hiqi=−detJiJiT to maximize the manipulability of the redundant robots and tend to keep the robots away from singularities, namely, setting φi=−∂Hiqi/∂qi=∂detJiJiT/∂qi can help maintain the Jacobian matrices for the redundant robots to be nonsingular.


## 4. Further Results on Multiple Time-Varying Formation Tracking

In this section, a similar controller will be designed to address the multiple time-varying formation tracking problem of NHRSs.Firstly, some different definitions on graph theory should be further presented here. The adjacency matrixW is no longer nonnegatively weighted here. If the robot i cooperates with the robot j, then wij=wji>0; if the robot i competes with the robot j, then wij=wji<0. Besides, the considered NHRS can be divided into M subnetworks. The undirected graph gk=νk,ℰk,Wk, k∈1,2,…,M, which is nonnegatively weighted, is employed to describe the interaction of the k-th subnetwork. Besides, each robot can only belong to one of these subnetworks, namely, νk∩νp=∅ for k≠p∈1,2,…,M and ∪i=1Mνi=ν. nk is utilized to denote the number of robots in the k-th subnetwork. Each subnetwork corresponds to a single virtual leader. Taking the M virtual leaders into consideration, the pinning matrix ℬk is employed to illustrate the interactions between the robots in the subnetwork k and their corresponding leader. Then, ℬ=diagℬ1,ℬ2,…,ℬM. Thereupon, two different assumptions on the interaction topology are given as follows.Assumption 3.
For each subnetwork with the corresponding leader, there exists at least one path, through which the information of the leader can be obtained by the robots inside.Assumption 4.
The total information that each robot receives from any other different subnetworks is zero, i.e.,∑j=rk¯rk¯lij=0, ∀i∈νp, k∈1,2,…,M−p, where rk¯=1+∑i=1kni−1 and rk¯=∑i=1kni. Note that n0=0.
The virtual leader associated with thek-th subnetwork is described as(29)x˙r,k=vr,k,v˙r,k=ar,k,where xr,k, vr,k, and ar,k, respectively, denote the position, velocity, and acceleration of the k-th leader, k∈1,2,…,M.Assumption 5.
The jerks of all the leaders are bounded, namely, there exists a uniform upper boundσ¯a˙r such that ar,k≤σ¯a˙r, ∀k∈1,2,…,M.
The control objective in this section mainly focuses on the multiple time-varying formation tracking problem of the NHRS defined hereinafter.Definition 2.
The multiple time-varying formation tracking problem of the NHRS is addressed if the tracking errors satisfy(30)limt⟶∞ei,kt=0,limt⟶∞e˙i,kt=0,for all i∈νk, k∈1,2,…,M, where ei,kt=xi−xr,k−hi and e˙i,kt=x˙i−vr,k−h˙i denote the tracking errors.


### 4.1. Estimator-Based Hierarchical Cooperative Algorithm II

In this section, a similar controller is designed to address the multiple time-varying tracking problem of NHRSs. The definitions ofℸ, ℶ, Ji#, ψ^⋅i, ψ^¨i, s^i, ψ˙i, ψ¨i, and si are the same as in Section 3. Then, EBHC algorithm II is presented as follows:(31)μi=Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ^i−JiTκoiJis^i−κsis^i−κdisgns^i,ϑ^⋅i=−ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,(32)x^⋅i=v^i,v^⋅i=−αsig∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr,k−hiρ1−βsig∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr,k−h˙iρ2+a^i+h¨i,a^˙i=−γsgn∑j∈Niwija^i−a^j+bia^i−ar,k,where i∈νk, k∈1,2,…,M, and the remaining control parameters are the same as in (9) and (10).


### 4.2. Convergence Analysis for Multiple Time-Varying Tracking

In this section, it will be tested whether the multiple time-varying tracking problem of NHRSs can be solved or not. Redefinex˜i,k=x^i−xr,k−hi, v˜i,k=v^i−vr,k−h˙i, and a˜i,k=a^i−ar,k, i∈νk. Based on Assumption 4, substituting (31) and (32) into (1) yields the following cascade closed-loop system:(33)Hiqis^⋅i+Ciqi,q˙is^i=−Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ˜i−JiTκoiJis^i−κsis^i−κdisgns^i+μi,d,ϑ˜⋅i=ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,x˜⋅i,k=v˜i,k,v˜⋅i,k=−αsig∑j∈Niwijx˜i,k−x˜j,p+bix˜i,kρ1,−βsig∑j∈Niwijv˜i,k−v˜j,p+biv˜i,kρ2+a˜i,k,a˜⋅i,k=−γsgn∑j∈Niwija˜i,k−a˜j,p+bia˜i,k−a˙r,where i∈νk and j∈νp, k,p∈1,2,…,M.Remark 5.
Assumption 4 plays an important role in deriving the closed-loop system (33). To be specific, it can be obtained by mathematics deformation that(34)∑j∈Niwijx^i−x^j+hj−hi=∑j∈Niwijx^i−xr,k−hi−x^j+xr,p+hj+xr,k−xr,p=∑j∈Niwijx˜i,k−x˜j,k+ℒi,:⊗IdxR,where i∈νk and j∈νp, k,p∈1,2,…,M, ℒi,: denotes the i-th row vector of ℒ, and(35)xR=colxr,1,xr,1,…,xr,1︸n1,xr,2,xr,2,…xr,2︸n2,…,xr,M,xr,M,…,xr,M︸nM.
It is noteworthy that Assumption 4 guarantees thatℒi,:⊗IdxR=0. Besides, similar unitization has been employed in the deformation of ∑j∈Niwijv^i−v^j+h˙j−h˙i and ∑j∈Niwija^i−a^j.Theorem 3.
Suppose Assumptions 3–5 hold, then the multiple time-varying formation tracking problem can be solved under the designed EBHC algorithm II (31) and (32), if (13) and (22) hold and(36)mink=1Nλkℒ+ℬ>0,namely, xi⟶xr,k+hit and vi⟶vr,k+h˙it as t⟶∞ can be eventually obtained, ∀i∈νk,k∈1,2,…,M.Proof.
Note the adaptive smooth control layer is the same as in the previous part. Thus, we just need to redo the cooperative analysis and boundedness analysis of the distributed estimator layer. Similarly, redefine the following compact vectors:(37)x˜=colx˜1,1,…,x˜n1,1,…,x˜1+∑i=1M−1ni,M,…,x˜N,M,v˜=colv˜1,1,…,v˜n1,1,…,v˜1+∑i=1M−1ni,M,…,v˜N,M,a˜=cola˜1,1,…,a˜n1,1,…,a˜1+∑i=1M−1ni,M,…,a˜N,M.
It thus follows that(38)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2+a˜,a˜⋅=−γsgnJ⊗Ida˜−a˙R,where aR=x¨R (xR is defined in Remark 5). The fact that g is undirected guarantees the symmetry of ℒ. It thus follows from Assumption 3 and (36) that J=ℒ+ℬ is positive definite. Then, by similar analysis in Theorem 1, it can be derived that x^i and v^i converge to xr,k and vr,k in finite time, namely, there exists Tf>t0 such that x˜i,k=0 and v˜i,k=0, ∀i∈νk, k∈1,2,…,M. Besides, qi,q˙i,s^i,ϑ^i∈ℒ2∩ℒ∞ as t0≤t≤Tf.
The remaining convergence analysis of the adaptive smooth control layer is the same as that in Theorem 2 and is omitted here. Besides, the subtasks for the redundant robots can be fulfilled by the analysis in Remark 3. This completes the proof.


## 4.1. Estimator-Based Hierarchical Cooperative Algorithm II

In this section, a similar controller is designed to address the multiple time-varying tracking problem of NHRSs. The definitions ofℸ, ℶ, Ji#, ψ^⋅i, ψ^¨i, s^i, ψ˙i, ψ¨i, and si are the same as in Section 3. Then, EBHC algorithm II is presented as follows:(31)μi=Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ^i−JiTκoiJis^i−κsis^i−κdisgns^i,ϑ^⋅i=−ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,(32)x^⋅i=v^i,v^⋅i=−αsig∑j∈Niwijx^i−x^j+hj−hi+bix^i−xr,k−hiρ1−βsig∑j∈Niwijv^i−v^j+h˙j−h˙i+biv^i−vr,k−h˙iρ2+a^i+h¨i,a^˙i=−γsgn∑j∈Niwija^i−a^j+bia^i−ar,k,where i∈νk, k∈1,2,…,M, and the remaining control parameters are the same as in (9) and (10).


## 4.2. Convergence Analysis for Multiple Time-Varying Tracking

In this section, it will be tested whether the multiple time-varying tracking problem of NHRSs can be solved or not. Redefinex˜i,k=x^i−xr,k−hi, v˜i,k=v^i−vr,k−h˙i, and a˜i,k=a^i−ar,k, i∈νk. Based on Assumption 4, substituting (31) and (32) into (1) yields the following cascade closed-loop system:(33)Hiqis^⋅i+Ciqi,q˙is^i=−Yiqi,q˙i,ψ^⋅i,ψ^¨iϑ˜i−JiTκoiJis^i−κsis^i−κdisgns^i+μi,d,ϑ˜⋅i=ΛiYiTqi,q˙i,ψ^⋅i,ψ^¨is^i,x˜⋅i,k=v˜i,k,v˜⋅i,k=−αsig∑j∈Niwijx˜i,k−x˜j,p+bix˜i,kρ1,−βsig∑j∈Niwijv˜i,k−v˜j,p+biv˜i,kρ2+a˜i,k,a˜⋅i,k=−γsgn∑j∈Niwija˜i,k−a˜j,p+bia˜i,k−a˙r,where i∈νk and j∈νp, k,p∈1,2,…,M.Remark 5.
Assumption 4 plays an important role in deriving the closed-loop system (33). To be specific, it can be obtained by mathematics deformation that(34)∑j∈Niwijx^i−x^j+hj−hi=∑j∈Niwijx^i−xr,k−hi−x^j+xr,p+hj+xr,k−xr,p=∑j∈Niwijx˜i,k−x˜j,k+ℒi,:⊗IdxR,where i∈νk and j∈νp, k,p∈1,2,…,M, ℒi,: denotes the i-th row vector of ℒ, and(35)xR=colxr,1,xr,1,…,xr,1︸n1,xr,2,xr,2,…xr,2︸n2,…,xr,M,xr,M,…,xr,M︸nM.
It is noteworthy that Assumption 4 guarantees thatℒi,:⊗IdxR=0. Besides, similar unitization has been employed in the deformation of ∑j∈Niwijv^i−v^j+h˙j−h˙i and ∑j∈Niwija^i−a^j.Theorem 3.
Suppose Assumptions 3–5 hold, then the multiple time-varying formation tracking problem can be solved under the designed EBHC algorithm II (31) and (32), if (13) and (22) hold and(36)mink=1Nλkℒ+ℬ>0,namely, xi⟶xr,k+hit and vi⟶vr,k+h˙it as t⟶∞ can be eventually obtained, ∀i∈νk,k∈1,2,…,M.Proof.
Note the adaptive smooth control layer is the same as in the previous part. Thus, we just need to redo the cooperative analysis and boundedness analysis of the distributed estimator layer. Similarly, redefine the following compact vectors:(37)x˜=colx˜1,1,…,x˜n1,1,…,x˜1+∑i=1M−1ni,M,…,x˜N,M,v˜=colv˜1,1,…,v˜n1,1,…,v˜1+∑i=1M−1ni,M,…,v˜N,M,a˜=cola˜1,1,…,a˜n1,1,…,a˜1+∑i=1M−1ni,M,…,a˜N,M.
It thus follows that(38)x˜⋅=v˜,v˜⋅=−αsigJ⊗Idx˜ρ1−βsigJ⊗Idv˜ρ2+a˜,a˜⋅=−γsgnJ⊗Ida˜−a˙R,where aR=x¨R (xR is defined in Remark 5). The fact that g is undirected guarantees the symmetry of ℒ. It thus follows from Assumption 3 and (36) that J=ℒ+ℬ is positive definite. Then, by similar analysis in Theorem 1, it can be derived that x^i and v^i converge to xr,k and vr,k in finite time, namely, there exists Tf>t0 such that x˜i,k=0 and v˜i,k=0, ∀i∈νk, k∈1,2,…,M. Besides, qi,q˙i,s^i,ϑ^i∈ℒ2∩ℒ∞ as t0≤t≤Tf.
The remaining convergence analysis of the adaptive smooth control layer is the same as that in Theorem 2 and is omitted here. Besides, the subtasks for the redundant robots can be fulfilled by the analysis in Remark 3. This completes the proof.


## 5. Simulation

In this section, two simulation examples are provided to illustrate the effectiveness of the proposed EBHC algorithms I and II. The simulation examples hereinafter are performed in theXY plane, and the considered NHRS consists of 2-DOF robots and 3-DOF robots, namely, d=2 and pi=2or3. Besides, the mechanical structures of both the nonredundant robot and redundant robot are shown in Figure 1.Figure 1
Mechanical structures of 2-DOF (a) and 3-DOF (b) robots.
(a)(b)Example 1.
EBHC algorithm I (9) and (10) are used for (1) to address the time-varying formation tracking problem of the NHRS consisting of four robots in the case of one virtual leader. It is set that robot 1 is redundant and the remain ones are nonredundant, whose physical parameters are selected as m1=col3.0,2.0,1.0kg, l1=col5.0,4.0,3.0m, and r1=0.5l1, and mi=col3.8+0.1i,3.3+0.1ikg, li=col6.6+0.2i,5.6+0.2im, and ri=0.5li for i∈2,3,4. The interactions among the NHRS and the virtual leader are shown in Figure 2(a). To be specific, the node L denotes the leader, node 1 represents the redundant robot, nodes 2–4 are the nonredundant robots, and robots 1 and 2 can directly receive the information from the leader. It thus follows that ℬ=diag2,0,0,3 and(39)ℒ=1−100−1100002−200−22.
Let the external disturbances bedit=−0.1cost,sintT for 2-DOF robots and dit=−0.1sint,cost,costT for 3-DOF robots. The control parameters are selected and shown in Table 1, which guaranteed the establishment of (13) and (22).
Besides, the virtual leader is chosen as(40)xr=7+2.5sin0.4t,4+2.5cos0.4tT,vr=2.5×0.4cos0.4t,−2.5×0.4sin0.4tT,ar=−2.5×0.42sin0.4t,−2.5×0.42cos0.4tT,which makes Assumption 2 hold. The NHRS is required to form a time-varying square and move along the trajectory of the leader with the leader being its centre, where the time-varying square is specified by(41)hi=0.4cos2i−1π4+t,0.4sin2i−1π4+tT,h˙i=−0.4sin2i−1π4+t,0.4cos2i−1π4+tT,h¨i=−0.4cos2i−1π4+t,−0.4sin2i−1π4+tT,for i∈1,2,3,4. On the contrary, the elements of the initial values x^i0, x^˙i0 , qi0, q˙i0, and ϑ^i0 are randomly chosen in 0,15, −1,1, −20,20, −5,5, and −2,2, respectively.
The simulation results of Example 1 are shown in Figures3–8. For details, from Figures 3(c), 3(d), 4(c), and 4(d), it can be observed that the estimators x^i, i∈1,2,3,4, converge to xr+hi almost when t≥25.5s. Also, Figures 3(a), 3(b), 4(a), and 4(b) imply that the task-space coordinates of each robot can asymptotically track the trajectory of xr+hi. Similarly, Figures 5 and 6 indicate the tracking performance of vi and v^i. Figure 7 provides the trajectories of all the robots inside and the leader, from which we can observe that the four robots form a square, and the square is rotating itself with a certain angular velocity while moving along the trajectory of the leader. It is noted that the tracking errors ei and e˙i are bounded by −0.08,0.08, which are tiny enough to support our main results.
On the contrary, we choose (redundant) robot 1 to verify that the subtask can also be finished but does not conflict with the tracking task. The negative gradient of the performance index corresponding to robot 1 is designed asφ1=col0.812−q11,0,0, where q11 denotes the first joint of robot 1. The designed φ1 aims at forcing the first joint of robot 1 toward 12 rad. It is clearly shown in Figure 8 that q11 is forced to 12 rad compared with the case that no subtask control is involved. Next, we choose φ1=∂/∂q1detJ1J1T to verify the singularity avoidance method presented in Remark 4. It can be observed in Figure 9 that the manipulability of robot 1 is enhanced with subtask control. It thus can be obtained that both the tracking task and the subtask can be addressed under the proposed EBHC algorithm I.Figure 2
Interaction topologies of Example 1 (a) and Example 2 (b).
(a)(b)Table 1
Control parameters of EBHC algorithms I and II.
Control parametersαβρ1ρ2ωiκoiκsiκdiΛi2-DOF8120.52/30.810I212I215I25I53-DOF8120.52/30.810I215I312I38I9Figure 3
Evolution ofei1 (a)ei2 (b)x˜i1 (c), and x˜i2 (d) in Example 1.
(a)(b)(c)(d)Figure 4
Evolution ofxi1−hi1 (a), xi2−hi2 (b), x^i1−hi1 (c), and x^i2−hi2 (d) in Example 1.
(a)(b)(c)(d)Figure 5
Evolution ofe˙i1 (a), e˙i2 (b), v˜i1 (c), and v˜i2 (d) in Example 1.
(a)(b)(c)(d)Figure 6
Evolution ofvi1−h˙i1 (a), vi2−h˙i2 (b), v^i1−h˙i1 (c), and v^i2−h˙i2 (d) in Example 1.
(a)(b)(c)(d)Figure 7
Task-space time-varying formation tracking performance of the NHRS shown in theXY plane in Example 1.Figure 8
Joint-space coordinates of robot 1 with (a) and without (b) subtask control.
(a)(b)Figure 9
Manipulability of robot 1 with (a) and without (b) singularity avoidance control.Example 2.
EBHC algorithm II (31) and (32) are employed to address the multiple time-varying formation tracking problem of the NHRS consisting of seven robots in the case of two virtual leaders, among which robots 1 and 7 are redundant and the remain five robots are nonredundant. The NHRS considered here can be divided into two subnetworks, where robots 1–4 and robots 5–7 belong to the first and the second subnetwork, respectively. The physical parameters of robots 1 are the same as in Example 1, and choose mi=col3.8+0.1i,3.3+0.1ikg, li=col6.6+0.2i,5.6+0.2im, and ri=0.5li for i∈2,3,4,5,6, and m7=col3.2,2.2,1.2kg, l7=col5.4,4.4,3.4m, and r7=0.5l1. The interactions among the robots and the leaders are depicted in Figure 2(b), which implies that the pinning matrix ℬ and the Laplacian matrix ℒ are given as ℬ=diag2,0,0,3,9,0,0 and(42)ℒ=1−100000−1100000002−21−1000−22−110001−16−6000−11−69−300000−33.
The external disturbances, the control parameters, leader 1, and the formation offsets of robots1−4 are the same as in Example 1. Besides, leader 2 is described as(43)xr,2=3.5cos0.5t,8+2.5sin0.5tT,vr,2=−3.5×0.5sin0.5t,2.5×0.5cos0.5tT,ar,2=−3.5×0.52cos0.5t,−2.5×0.52sin0.5tT.
The second subnetwork is expected to form a time-varying regular triangle, which is specified by(44)h5=0.10,6+2sintT,h˙5=0.10,2costT,h¨5=0.10,−2sintT,h6=0.1−33−3sint,−3−sintT,h˙6=0.1−3cost,−costT,h¨6=0.13sint,sintT,h7=0.133+3sint,−3−sintT,h˙7=0.13cost,−costT,h¨7=0.1−3sint,sintT.
Finally, the initial values are randomly chosen in the same corresponding ranges given in Example 1.
The simulation results are shown in Figures10–14. Similarly, Figures 10 and 11 indicate that xi can eventually track the desired position, and the convergence of x^i−hi is completed in finite time. Besides, Figure 12 depicts the tracking performance of vi and v^i. Figure 13 indicates that the NHRS can achieve the multiple time-varying formation in the XY plane under the proposed EBHC algorithm II, which is also depicted in Figure 14. In detail, the robots belonging to the first subnetwork form the time-varying square as same as in Example 1, and the robots belonging to the second subnetwork form a time-varying regular triangle with its size changing periodically. Note that the tracking errors in Figures 10 and 12 do not converge to zero. This is mainly because although it has been theoretically proved that the states converge to zero asymptotically, the final simulation results converge to the neighbourhood of the origin due to the presence of external disturbances, parametric uncertainties, and kinematic redundancy and the employment of the sign function. This phenomenon is very general in the applications of the sliding mode control technique on regulating complex systems with various uncertainties. Thus, it is theoretically correct to obtain the simulation results shown in Figures 10 and 12. It thus can be concluded that the time-varying formation tracking problems of NHRSs with both a single leader and multiple leaders can be solved under the proposed EBHC algorithms.Figure 10
Evolution ofei1 (a), ei2 (b), x˜i1 (c), and x˜i2 (d) in Example 2.
(a)(b)(c)(d)Figure 11
Evolution ofxi1−hi1t (a), xi2−hi2t (b), x^i1−hi1 (c), and x^i2−hi2 (d) in Example 2.
(a)(b)(c)(d)Figure 12
Evolution ofe˙i1 (a), e˙i2 (b), vi1−h˙i1t (c), and vi2−h˙i2t (d) in Example 2.
(a)(b)(c)(d)Figure 13
Task-space time-varying formation tracking performance of the NHRS shown in theXY plane in Example 2.Figure 14
Tracking trajectories of the first subnetwork (a) and the second subnetwork (b) in theXY plane.
(a)(b)


## 6. Conclusion

This paper mainly focused on the time-varying formation tracking problem of NHRSs in the task space considering the parameter uncertainties and external disturbances. Two novel EBHC algorithms, where both redundant robots and nonredundant robots have been taken into account, have been proposed to realize the time-varying formation in the case of a single virtual leader and multiple leaders, respectively. Besides, the designed distributed estimator algorithms are able to guarantee the availability of the corresponding leaders’ information for each robot in finite time. The sufficient conditions have been derived by invoking Lyapunov stability and input-to-state stability. The presented simulation results have given satisfactory performance of proposed EBHC algorithms. Future works will focus on the time-varying formation tracking problem of NHRSs with time delays.



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