# Distributed Adaptive Coordinated Control of Multiple Euler–Lagrange Systems considering Output Constraints and Time Delays

Authors: Hongde Qin; Xiaojia Li; Yanchao Sun
Journal: Complexity
 (2021)

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## Abstract

In this paper, we mainly investigate the coordinated tracking control issues of multiple Euler–Lagrange systems considering constant communication delays and output constraints. Firstly, we devise a distributed observer to ensure that every agent can get the information of the virtual leader. In order to handle uncertain problems, the neural network technique is adopted to estimate the unknown dynamics. Then, we utilize an asymmetric barrier Lyapunov function in the control design to guarantee the output errors satisfy the time-varying output constraints. Two distributed adaptive coordinated control schemes are proposed to guarantee that the followers can track the leader accurately. The first scheme makes the tracking errors between followers and leader be uniformly ultimately bounded, and the second scheme further improves the tracking accuracy. Finally, we utilize a group of manipulator networks simulation experiments to verify the validity of the proposed distributed control laws.


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## Body

## 1. Introduction

With the rapid development of industrial technology, the industrial tasks are gradually becoming complicated and large-scale. When solving some complex industrial tasks, multiagent systems (MASs) gradually become the first choice due to its high reliability and economy, such as multiple robotic manipulator systems, spacecraft formation flying, and unmanned underwater vehicles [1–3]. More and more experts and scholars focus on the distributed coordinated control of MASs [4].So far, there mainly have been two consensus methods for the MASs coordinated control. The first control strategy is about the leaderless control method, which requires all state variables to gradually converge to a constant. For example, the authors used Lyapunov finite-time theory to propose a consensus control method to ensure the states converge to a constant based on undirected graphs in [5]. The authors in [6] proposed a leaderless consensus control strategy and analysed the stability problem for MAS. However, for the leaderless case, the convergence of the state variables of all agents is related to the initial state variables of each agent, which leads to the great restriction of the system movement. The second strategy is the leader-following control for MASs. In this case, followers utilize their own or neighbors’ information to track leader so that one only needs to design the movement of leader accurately. This method can not only simplify the design difficulty of the control algorithm but also reduce energy consumption and cost. Therefore, this approach is more suitable for MASs. In [7], a trajectory tracking algorithm was proposed by using the small gain feedback technique, so that the followers can effectively track the dynamic leader. In [8], the authors used the linear event-trigger feedback technique to make the state variables of followers and leaders tend to be consistent in finite time so as to achieve the tracking control. For the coordinated tracking control case, the leader’s motion is the most crucial part in MASs, which means that the whole system will stop working if leader fails. Actually, virtual leaders play the same role as the real leaders in MASs. At the same time, virtual leaders can change the number of leaders flexibly [9]. In [10], the authors discussed the current situation and illustrated the extensive application prospect of the leadership relationship within virtual working environment. Furthermore, the input signals were regarded as the virtual leaders and a distributed control algorithm was proposed by using Lyapunov theory; this control algorithm realized the coordinated movement of the followers and achieved the goal of tracking the target trajectory [11, 12].It is worth noting that all the above research studies about MASs are based on linear systems. However, linear systems have great limitations in MASs due to the existence of nonlinear uncertainties. Therefore, it is necessary to study the coordinated control problem of nonlinear systems [13, 14]. In fact, Euler–Lagrange (EL) equation is widely used in the field of nonlinear systems, such as autonomous underwater robots and manipulating robots systems [15–17]. In [18], the authors focused on the distributed coordinated tracking issues and designed two kinds of novel control algorithms for multiple EL systems. In multiple EL systems, distributed observers are often used to ensure the normal operation of the systems when only partial agents could get the state information of the leader. In [19], the authors utilized a new dynamic velocity observer to ensure all agents could get the leader’s velocity variables. In [20], the leader’s information could be acquired by the followers through the velocity observer and guaranteed the goal of coordinated tracking control. At present, there usually exist communication delays among different agents due to the limitation of communication bandwidth and packet losses. Therefore, the communication delays cannot be ignored in the study of multiple EL systems’ coordinated control [21]. Considering the communication delays, a new aperiodic sampled-data cluster formation control algorithm was proposed for the cluster formation control problem for MASs in [22]. In [23], the authors investigated the leader-following consensus problem of multiple EL systems which considered communication delays under the switching network. In [24], in order to solve the problem of communication delays, the authors proposed a novel coordinated tracking control algorithm based on Lyapunov stability theory. In actual engineering projects, state constraint problems or output constraint problems are often considered to meet certain requirements. At present, the state constraint or output constraint methods for nonlinear systems include the prescribed performance (PP) methods and the barrier Lyapunov function (BLF) technique [25]. PP methods mainly utilize error transformation to achieve the performance constraint [26, 27], which may cause control law singularity problem. In contrast, the BLF technique mainly applies the performance constraint variables to construct constrained boundary functions, which greatly reduces the difficulty of designing the control algorithms. Thus BLF technique is widely used in the state constraint control problem for nonlinear systems. In [28], the authors utilized Integral BLFs to make all state errors converge to the neighbor of origin. The authors used an asymmetric BLF to solve the time-varying output error limitation problem in [29]. The authors in [30] utilized the asymmetric time-varying BLF to construct an adaptive controller and applied the Lyapunov stability theory to guarantee the output error within the excepted range. It is feasible to use the time-varying BLF method to deal with the output constraints of multiple EL systems.Generally speaking, MASs are often affected by working environment, unknown dynamics, and unknown disturbances. The uncertainties will affect the work efficiency of MASs. To handle the uncertainties problem, the neural network (NN) technique is widely used for the MASs due to its good approximation ability [31, 32]. In [33], the authors designed a distributed adaptive NN controller to ensure the mobile robots can obtain the expected control effect. In the case that only partial agents could get the leader’s information, a control algorithm which used the NN technique to compensate the uncertainties and external disturbances was designed in [34]. It made the tracking errors among leader and followers tend to origin.In this study, the coordinated control problems for nonlinear multiple EL systems considering communication delays and output constraints are investigated. Compared with the existing papers, it uses a distributed observer to solve the communication delay problem of multiple EL systems. Then, the BLF technique is used to guarantee the time-varying output errors of the systems within the prescribed constraint boundary. We also utilize adaptive NNs to deal with the uncertain dynamics and the unknown disturbances of the multiple EL systems. The main contributions of the paper are summarized as follows.(1)
Considering communication delays among different followers, the input signal source is regarded as a virtual leader. In addition, a distributed observer is used to ensure that the virtual leader’s state information can be obtained by all followers.(2)
Two distributed control schemes are designed to guarantee the tracking errors are UUB and asymptotically converge to origin.(3)
The adaptive NN technique is utilized to compensate the uncertain dynamics of the multiple EL systems.(4)
Based on the BLFs, an asymmetric BLF is designed to make the output errors satisfy the time-varying output constraint requirements.In the following research, Section2 introduces the dynamic model and some basic lemmas. In Section 3, two distributed adaptive control algorithms and stability analysis are formulated. Section 4 presents several simulation examples to prove the validity of the control algorithms. Section 5 summarizes the whole paper.


## 2. Materials and Methods

### 2.1. The Basic Knowledge

The basic mathematical symbols and definitions in this paper are shown in Table1.Table 1
Mathematical symbols in this study.
Mathematical symbolDefinitionℝThe set of real numbersℝnThe set ofn-dimensional column vectors with all elements being real numbersℝn×nThe set ofn×n real matricesIpn×n identity matrixdiagx1,…,xnDiagonal matrices whose diagonal elements arex1,…,xn⊗Kronecker productλmin⋅The minimum eigenvalue of a real symmetric matrixλmax⋅The maximum eigenvalue of a real symmetric matrixLpft:ℝ↦ℝn|∫0∞fτp1/p<∞tr⋅Matrix traceThe communication interactions among a virtual leader andn followers can be presented by the directed graph ζ=υ,ε,A. In the directed graph ζ=υ,ε,A,u1,2,…,n+1 is the set of nodes, ε⊆υ×υ denotes the set of edges, and A=aij∈ℝn+1×n+1 represents the non-negative adjacency matrix. In node set υ, υi denotes the ith follower. The edge υ,υi∈ε represents that the jth agent can get the information from the ith agent. We call υi as the parent node and its neighbor υj is called as the child node. The path of a directed graph is the sequence of nodes υil,…,υin, which satisfies υik,υik+1∈ε. The directed tree denotes a directed graph in which each node has only one parent node, but there is one root node which is different. When a directed tree contains all nodes of the graph, it can be described as a directed spanning tree. If the directed spanning tree is a part of the directed graph, it is said that the directed graph contains a directed spanning tree.Assumption 1.
A directed spanning tree is contained withinζ.
We define the elementaij=1, if and only if υj,υi∈ε and otherwise aij=0. The element an+1j=0 when j=1,…,n+1 A matrix is defined as A¯=aij¯∈ℝn×n, where a¯ij=aij/∑k=1n+1aik, i,j∈1,…,n.


### 2.2. Dynamics of Euler–Lagrange Systems

The multiple EL systems containn followers and a virtual leader. Then, we utilize the EL equation to describe the dynamic model of the ith follower as(1)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi+ωi,where qi∈ℝp,i=1,…,n denotes the generalized coordinate, τi∈ℝp denotes the control input torque, Miqi∈ℝp×p represents the inertia matrix which is symmetric and positive definite, Ciqiq˙i∈ℝp×p represent the centripetal and Coriolis torques, giqi∈ℝp denotes the gravitational force, and ωi∈ℝp is the external disturbance. We assume Miqi, Ciqi,q˙i, and giqi are all unknown.Assumption 2.
The disturbanceωi is bounded, which one satisfies ωi≤γ, where γ is a bounded positive constant.
The following properties of EL systems (1) are useful.Property 1.
The matrixMiqi−2Ciqiq˙i is skew-symmetric, i.e., ∀xi∈ℝp, xiTM˙iqi−2Ciqi,q˙ixi=0.Property 2.
Miqi is bounded, and it satisfies B¯Ip≤Miqi≤B¯Ip, where B¯ and B¯ are positive constants.
We assume thatqn+1 can be represented as follows [35]:(2)v˙=Sv,qn+1=Fv,where v∈ℝp denotes the auxiliary state variable, S∈ℝp×p and F∈Rn×p are constant real matrices, and qn+1 represents the state of the leader.Assumption 3.
When all followers can obtainS and F, v and v˙ are bounded.Remark 1.
If the coefficient matricesS, F and the auxiliary state variable v are chosen appropriately, any desired trajectory can be formed by the leader.


## 2.1. The Basic Knowledge

The basic mathematical symbols and definitions in this paper are shown in Table1.Table 1
Mathematical symbols in this study.
Mathematical symbolDefinitionℝThe set of real numbersℝnThe set ofn-dimensional column vectors with all elements being real numbersℝn×nThe set ofn×n real matricesIpn×n identity matrixdiagx1,…,xnDiagonal matrices whose diagonal elements arex1,…,xn⊗Kronecker productλmin⋅The minimum eigenvalue of a real symmetric matrixλmax⋅The maximum eigenvalue of a real symmetric matrixLpft:ℝ↦ℝn|∫0∞fτp1/p<∞tr⋅Matrix traceThe communication interactions among a virtual leader andn followers can be presented by the directed graph ζ=υ,ε,A. In the directed graph ζ=υ,ε,A,u1,2,…,n+1 is the set of nodes, ε⊆υ×υ denotes the set of edges, and A=aij∈ℝn+1×n+1 represents the non-negative adjacency matrix. In node set υ, υi denotes the ith follower. The edge υ,υi∈ε represents that the jth agent can get the information from the ith agent. We call υi as the parent node and its neighbor υj is called as the child node. The path of a directed graph is the sequence of nodes υil,…,υin, which satisfies υik,υik+1∈ε. The directed tree denotes a directed graph in which each node has only one parent node, but there is one root node which is different. When a directed tree contains all nodes of the graph, it can be described as a directed spanning tree. If the directed spanning tree is a part of the directed graph, it is said that the directed graph contains a directed spanning tree.Assumption 1.
A directed spanning tree is contained withinζ.
We define the elementaij=1, if and only if υj,υi∈ε and otherwise aij=0. The element an+1j=0 when j=1,…,n+1 A matrix is defined as A¯=aij¯∈ℝn×n, where a¯ij=aij/∑k=1n+1aik, i,j∈1,…,n.


## 2.2. Dynamics of Euler–Lagrange Systems

The multiple EL systems containn followers and a virtual leader. Then, we utilize the EL equation to describe the dynamic model of the ith follower as(1)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi+ωi,where qi∈ℝp,i=1,…,n denotes the generalized coordinate, τi∈ℝp denotes the control input torque, Miqi∈ℝp×p represents the inertia matrix which is symmetric and positive definite, Ciqiq˙i∈ℝp×p represent the centripetal and Coriolis torques, giqi∈ℝp denotes the gravitational force, and ωi∈ℝp is the external disturbance. We assume Miqi, Ciqi,q˙i, and giqi are all unknown.Assumption 2.
The disturbanceωi is bounded, which one satisfies ωi≤γ, where γ is a bounded positive constant.
The following properties of EL systems (1) are useful.Property 1.
The matrixMiqi−2Ciqiq˙i is skew-symmetric, i.e., ∀xi∈ℝp, xiTM˙iqi−2Ciqi,q˙ixi=0.Property 2.
Miqi is bounded, and it satisfies B¯Ip≤Miqi≤B¯Ip, where B¯ and B¯ are positive constants.
We assume thatqn+1 can be represented as follows [35]:(2)v˙=Sv,qn+1=Fv,where v∈ℝp denotes the auxiliary state variable, S∈ℝp×p and F∈Rn×p are constant real matrices, and qn+1 represents the state of the leader.Assumption 3.
When all followers can obtainS and F, v and v˙ are bounded.Remark 1.
If the coefficient matricesS, F and the auxiliary state variable v are chosen appropriately, any desired trajectory can be formed by the leader.


## 3. The Distributed Adaptive Trajectory Tracking Control Law Design

### 3.1. Time-Varying BLF Design

The useful lemmas used in this study are shown as follows.Lemma 1.
(see [36]). If there is a continuous Lyapunov function Vw,t which satisfies w¯1w≤Vw,t≤w¯2w and the derivative V˙w,t=dVw,t/dt satisfies V˙w,t≤−kVw,t+c, where k and c are positive constants, then the variable wt is bounded.Lemma 2.
(see [37]). For ∀B∈ℝp×p, if matrix B is symmetric and positive definite, the following inequality can be obtained: ∀y∈ℝp, λminy2≤yTBy≤λmaxy2.Lemma 3.
(see [38]). ψt is a continuously differentiable function. For ∀t∈0,∞, if ψtψt≤Φ hold, is bounded where Φ is a small positive constant, it means that ψt is bounded ∀t∈0,∞.Lemma 4.
(see [39]). Consider x∈ℝ and x<ka, where ka is a positive constant, such that the inequality is shown as lnka2/ka2−x2≤x2/ka2−x2.In this study, if the trajectory tracking errors are too large, it may cause undesired losses. Therefore, the output states of the followers should be limited. If the time-varying bounds arek¯ct=k¯c1t,…,k¯cntT and k¯ct=k¯c1t,…,k¯cntT, then the output qit should remain in the region:(3)Ωqi≔qit∈R|k¯cit<qit<k¯cit,∀t≥0.When considering the influence of tracking errors on multiple EL systems, a time-varying BLF is used in this study to ensure that the outputqi satisfies the time-varying output constraints.First, an auxiliary variable is defined as(4)qri=Fηi.Then, we define the following error variables:(5)Z1i=qi−qri,(6)Z2i=q˙i−ri,where ri denotes a virtual control.Inspired by He et al. [40], we set the time-varying bounds of Z1i as(7)kait=qrit−k¯cit,(8)kbit=k¯cit−qrit.Consider that asymmetric BLF is an improvement of symmetric BLF, which can better adapt to the requirements of time-varying output constraints and ensure the high trajectory tracking accuracy for multiple EL systems. An asymmetric BLF is chosen as(9)V1it=hi2Inkbi2tkbi2t−Z1it2+1−hi2Inkai2tkai2t−Z1it2,where we define hi as(10)hi=0,Z1i≤0,1,Z1i>0,i=1,2,…,n.The output tracking error variables are transformed as follows:(11)εai=Z1ikai,εbi=Z1ikbi,εi=hiεbi+1−hiεai.Substituting (11) into (9), we obtain(12)V1it=12In11−εi2.From (12), we can know that V1it is positive definite and continuously differentiable when εi<1. Differentiating V1it yields(13)V˙1it=εbihi1−εbi2kbiZ2i+ri−q˙ri−Z1ik˙bikbi+εai1−hi1−εai2kaiZ2i+ri−q˙ri−Z1ik˙aikai.The virtual controlri is selected as(14)ri=q˙ri−k1iZ1i−K¯1iZ1i,where K¯1i is defined as(15)K¯1i=β+k˙aikai2+k˙bikbi2,β is a positive constant, and it can ensure r˙i is bounded when k˙ai and k˙bi are zero, where the gain matrix K1=diagk11,k12,…,k1i,…,k1n is symmetric positive definite. Substituting (14) and (15) into (13) yields(16)V˙1it=εbihi1−εbi2kbiZ2i+ri−q˙ri−Z1ik˙bikbi+εai1−hi1−εai2kaiZ2i+ri−q˙ri−Z1ik˙aikai≤−k1iεi21−εi2+XiZ1iZ2i,where Xi is defined as follows:(17)Xi=hikbi2−Z1i2+1−hikai2−Z1i2,i=1,2,3,…,n,and X=diagX1,X2,…,Xi,…,Xn.


### 3.2. Distributed Adaptive NN Tracking Controller 1 Design

When only part of the followers can get the state information, we use the following distributed observer [35]:(18)η˙i=Sηi−di∑j=1n+1aijηjt−T+di∑j=1n+1aijη˙jt−T+ηjt−T−ηi,where di=1/∑j=1n+1aij, ηir=ℝp represents the ith follower’s estimate of v and T represents the constant time delays among different followers. We assume that ηn+1=v.Lemma 5.
(see [41]). In this research, we utilize the distributed observer (18) to ensure the leader’s information can be acquired by all agents when Assumptions 1–3 hold. If Dη¯>λmaxS+Re holds, we can draw a conclusion that ηi−v is bounded, which can be expressed as(19)limt⟶∞ηi−v≤U0,where Re=−In⊗SA1n⊗vt−T−1n⊗v+A1n⊗v˙t−T+A1n⊗vt−T, η¯=η¯1T,…,η¯nTT, η¯i=ηi−v, U0 denotes a positive constant, and Dη¯=η¯−Aη¯t−T is the defined neutral operator.Remark 2.
SinceS and F are real matrices and the elements are independent of time and the state variables of the leader, we utilize matrices S and F to design the observer (18).
According to [42], we design following distributed adaptive control schemes of (1) as(20)τi=−K2iZ2i−XiZ1i−W^iTϕi,(21)W^⋅i=αϕiZ2iT−μW^i,where α and β are two positive constants and K2i is a gain matrix. W^i and ϕi are two related variables to NN.Theorem 1.
For the EL system (1) which considers communication delays and output constrains, if Assumptions 1–3 hold, the tracking error Z1i is UUB with the distributed observer (18) and the distributed adaptive control laws (20) and (21). At the same time, qit satisfies the time-varying output constraints, i.e., ∀t>0,k¯cit<qit<k¯cit.Proof.
Differentiating (6) yields(22)Z˙2i=q¨i−r˙i.
Substituting (22) into (1), we obtain(23)MiZ˙2i+CiZ2i=τi+ωi+diqi,q˙i,ri,r˙i,where(24)diqi,q˙i,ri,r˙i=−gi−Ciri−Mir˙i.
In this study,Miqi,Ciqi,q˙i,giqi are assumed to be unknown, and there are nonlinear uncertainties in the EL system (1). Considering that NN has good approximation ability for unknown nonlinear function, it is often used to deal with the uncertainty problem in nonlinear system. Therefore, the NN technique is used to solve the nonlinear uncertainties diqi,q˙i,ri,r˙i. The methods are shown as follows:(25)WiTϕiqi,q˙i,ri,r˙i=diqi,q˙i,ri,r˙i−Δi,where Wi represents the ideal weighted matrix, ϕi is the Gauss function, and Δi denotes the approximation error. In this research, Δi is assumed to satisfy Δi≤ΔMi, where ΔMi is a positive constant. The estimate of the nonlinear uncertainties di for the ith follower can be written as(26)d^iqi,q˙i,ri,r˙i=W^iTϕiqi,q˙i,ri,r˙i,where W^i is the estimate of Wi.
The distributed adaptive control schemes are designed as (20) and (21). Consider a Lyapunov function as follows:(27)V2i=V1i+12Z2iTMiZ2i+12αtrW˜iTW˜i,where W˜i=Wi−W˜i. Differentiating (27) and according to (20)–(26), we can obtain(28)V˙2i=V˙1i+Z2iTωi+Δi+Z2iTW˜iϕi−trW˜TϕiZ2i+μtrW˜iTW^i.
SinceΔi and ωi are bounded, there is a positive constant χ¯max satisfying Δi+ωi≤χ¯max. Furthermore, the inequality can be obtained as follows:(29)Z2iTωi+Δi≤12σZ2iTZ2i+σ2χ¯max,where σ is a positive constant.
BecauseZ2iTW˜iϕi is a scalar in (28), we have Z2iTW˜iϕi=trZ2iTW˜iϕi=trW˜iTϕiZ2i.
The matrices have the following properties:(30)trA+B=trA+trB,(31)trBA=trAB,(32)2trAB≤trBTB+trATA.
Further we have(33)trW˜iTW^i=trW˜iTWi−trW˜iTW˜i≤12trW˜iTW˜i+12trWiTWi−trW˜iTW˜i=12trWiTWi−12trW˜iTW˜i.
Substituting (16), (29), and (33) into (28), we have(34)V˙2i≤σ2χ¯max2−122λminK2i−σ−1Z2iTZ2i+2k1iεi21−εi2+μtrW˜iTW˜i−μtrWiTWi.
Thus, (34) can be written as(35)V˙2it≤−κ1V2i+υ1,where(36)κ1=min2λminK2i−σ−1B¯,α,μ,min2k1i,υ1=μ2trWiTWi+σ2χ¯max2.
We can knowV2i is UUB according to Lemma 1. Integrating (35) yields(37)V2it≤V2i0e−κ1t+υ1κ11−e−κ1t.
We can ensureκ1>0 and υ1>0 by properly selecting parameters, and we have(38)limt⟶∞V2it=υ1κ1=ρ.
It can be seen from (27) that(39)V1i≤V2i.
Moreover,(40)limt⟶∞12ln11−εi2≤ρ.
By substituting (10) and (11) into (40), we can obtain(41)−kai2t1−e−2p≤limt⟶∞Z1i≤kbi2t1−e−2p.
From (4), one has(42)qi−qn+1=qi−Fηi+Fηi−qn+1=qi−Fηi+Fηi−v.
According to (19), (41), and (42), we can obtain(43)limt⟶∞qi−qn+1≤limt⟶∞qi−Fηi+Flimt⟶∞ηi−v≤kbi2t1−e−2ρ+U0F.
From (46), we can conclude that the tracking errors among different followers and the virtual leader are bounded.
From (5), (7), (8), and (41), we can obtain(44)k¯cit<qit<k¯cit.
Equation (44) shows that the output qit satisfies the time-varying output constraints.


### 3.3. Improved Distributed Adaptive NN Tracking Controller 2 Design

According to [42], the authors proposed a new algorithm to decrease the tracking errors in (47) by using a discontinuous sign function, and it can be proved by similar steps to (49)–(58). However, this approach will bring about the additional chattering. Considering the chattering problem, we propose an improved continuous control scheme to improve the tracking accuracy. At the same time, auxiliary variable Zit can converge to origin asymptotically.For the EL system (1), we propose following modified distributed adaptive control strategy based on controller 1 as(45)τi=−K2iZ2i−XiZ1i−W^iTϕi−ιisatZ2i,(46)W^⋅i=αϕiZ2iT,where ιi is a positive constant, and it satisfies that(47)ιi≥ωi+Δi,(48)satZ2i=sgnZ2iδ,Z2iδ≥1,Z2iδ,otherwise,where δ is a small positive constant.Remark 3.
By introducing the continuous functionsatZ2i, it not only makes the tracking errors among different followers and the leader asymptotically converge to zero but also avoids the chattering phenomenon caused by discontinuous sign function sgnZ2i/δ.Theorem 2.
For the EL system (1) which considers communication delays and output constrains, when Assumptions 1–3 hold, each follower can gradually track the leader and the tracking error Z1i asymptotically converging to origin with the distributed continuous control laws (45) and (46). The time-varying output constraints are not breached, that is, ∀t>0,k¯cit<qit<k¯cit.Proof.
WhenZ2i/δ>1, we know satZ2i=sgnZ2i/δ. Substituting (45) and (46) into (1), we can obtain(49)MiZ˙2i+CiZ2i=−XiZ1i−K2iZ2i+W¯iTϕi+ωi+Δi−ιisgnZ2iδ.
Select the sameV1i and V2i as those in Theorem 1. Taking the derivative of V2i and from (16), (47), and (48), it can be obtained that(50)V˙2i≤−Z2iTK2iZ2i−k1iεi21−εi2+Z2iTωi+Δi+Z2iTW˜iϕ−trW˜iTϕZ2i−ιiZ2iδ.
Similar to (28)–(34), it holds that(51)V˙2i≤−Z2iTK2iZ2i−k1iεi1−εi2+Z2iωi+Δi−ιiZ2iδ.
For a small positive constantδ, we can find that Z2i≤Z2i/δ. Substituting (47) into (51), we have(52)V˙2i≤−Z2iTK2iZ2i−k1iεi21−εi2.
Sincek1i is a positive constant and matrix K2i is symmetric positive definite, it can be found that(53)V˙2i≤0.
Moreover, we have(54)limt⟶∞V2it=V∞∈0,V0.
Due to the fact thatV2i is bounded, we find Zμ,Z2i∈L∞. According to (6), we have Z˙1i∈L∞. When εi<1, εi2/1−εi2≥εi2. Integrating both sides of (54), it can be obtained that(55)k1i∫0∞εi21−εi2dt+λminK2i∫0∞Z2iTZ2idt≤V2i0−V2i∞.
Therefore, we haveZ1i∈L2 and Z1i∈L2∩L∞. According to Barbalet lemma, we have(56)limt⟶∞Z1it=0.
From (56), it can be found that(57)limt⟶∞qi−Fηi=0.
Following the similar steps as (42) and (43), we have(58)limt⟶∞qi−qn+1≤limt⟶∞qi−Fηi+Flimt⟶∞η−v≤U0F.
Theith follower has a good tracking performance on the virtual leader, and the upper boundary of the tracking error is described by (58).
WhenZ2i/δ<1, it can be obtained that satZ2i=Z2i/δ. According to (49) and (50), we have(59)V˙2it≤−Z2iTK2iZ2i−k1iεi21−εi2−Z2iTιiZ2iδ−ωi+Δi≤−Z2iTK2iZ2i−k1iεi21−εi2−ZιiZ2iδ−ωi+Δi.
Ifδ≤ιiZ2i/ωi+Δi, we have V˙2it≤−Z2iTK2iZ2i−k1iεi2/1−εi2. From the procedures in (53)–(58), we can find that the tracking error between the ith follower and the virtual leader is bounded.
According to (5), (7), (8), and (56), we can obtain the same result as (44). It shows that the output qit satisfies the time-varying output constraints.Remark 4.
The results of this paper can provide some reference for formation control of MASs. In the future, the formation control problems for MASs considering time-varying communication delays and full-state constraints will be studied.


## 3.1. Time-Varying BLF Design

The useful lemmas used in this study are shown as follows.Lemma 1.
(see [36]). If there is a continuous Lyapunov function Vw,t which satisfies w¯1w≤Vw,t≤w¯2w and the derivative V˙w,t=dVw,t/dt satisfies V˙w,t≤−kVw,t+c, where k and c are positive constants, then the variable wt is bounded.Lemma 2.
(see [37]). For ∀B∈ℝp×p, if matrix B is symmetric and positive definite, the following inequality can be obtained: ∀y∈ℝp, λminy2≤yTBy≤λmaxy2.Lemma 3.
(see [38]). ψt is a continuously differentiable function. For ∀t∈0,∞, if ψtψt≤Φ hold, is bounded where Φ is a small positive constant, it means that ψt is bounded ∀t∈0,∞.Lemma 4.
(see [39]). Consider x∈ℝ and x<ka, where ka is a positive constant, such that the inequality is shown as lnka2/ka2−x2≤x2/ka2−x2.In this study, if the trajectory tracking errors are too large, it may cause undesired losses. Therefore, the output states of the followers should be limited. If the time-varying bounds arek¯ct=k¯c1t,…,k¯cntT and k¯ct=k¯c1t,…,k¯cntT, then the output qit should remain in the region:(3)Ωqi≔qit∈R|k¯cit<qit<k¯cit,∀t≥0.When considering the influence of tracking errors on multiple EL systems, a time-varying BLF is used in this study to ensure that the outputqi satisfies the time-varying output constraints.First, an auxiliary variable is defined as(4)qri=Fηi.Then, we define the following error variables:(5)Z1i=qi−qri,(6)Z2i=q˙i−ri,where ri denotes a virtual control.Inspired by He et al. [40], we set the time-varying bounds of Z1i as(7)kait=qrit−k¯cit,(8)kbit=k¯cit−qrit.Consider that asymmetric BLF is an improvement of symmetric BLF, which can better adapt to the requirements of time-varying output constraints and ensure the high trajectory tracking accuracy for multiple EL systems. An asymmetric BLF is chosen as(9)V1it=hi2Inkbi2tkbi2t−Z1it2+1−hi2Inkai2tkai2t−Z1it2,where we define hi as(10)hi=0,Z1i≤0,1,Z1i>0,i=1,2,…,n.The output tracking error variables are transformed as follows:(11)εai=Z1ikai,εbi=Z1ikbi,εi=hiεbi+1−hiεai.Substituting (11) into (9), we obtain(12)V1it=12In11−εi2.From (12), we can know that V1it is positive definite and continuously differentiable when εi<1. Differentiating V1it yields(13)V˙1it=εbihi1−εbi2kbiZ2i+ri−q˙ri−Z1ik˙bikbi+εai1−hi1−εai2kaiZ2i+ri−q˙ri−Z1ik˙aikai.The virtual controlri is selected as(14)ri=q˙ri−k1iZ1i−K¯1iZ1i,where K¯1i is defined as(15)K¯1i=β+k˙aikai2+k˙bikbi2,β is a positive constant, and it can ensure r˙i is bounded when k˙ai and k˙bi are zero, where the gain matrix K1=diagk11,k12,…,k1i,…,k1n is symmetric positive definite. Substituting (14) and (15) into (13) yields(16)V˙1it=εbihi1−εbi2kbiZ2i+ri−q˙ri−Z1ik˙bikbi+εai1−hi1−εai2kaiZ2i+ri−q˙ri−Z1ik˙aikai≤−k1iεi21−εi2+XiZ1iZ2i,where Xi is defined as follows:(17)Xi=hikbi2−Z1i2+1−hikai2−Z1i2,i=1,2,3,…,n,and X=diagX1,X2,…,Xi,…,Xn.


## 3.2. Distributed Adaptive NN Tracking Controller 1 Design

When only part of the followers can get the state information, we use the following distributed observer [35]:(18)η˙i=Sηi−di∑j=1n+1aijηjt−T+di∑j=1n+1aijη˙jt−T+ηjt−T−ηi,where di=1/∑j=1n+1aij, ηir=ℝp represents the ith follower’s estimate of v and T represents the constant time delays among different followers. We assume that ηn+1=v.Lemma 5.
(see [41]). In this research, we utilize the distributed observer (18) to ensure the leader’s information can be acquired by all agents when Assumptions 1–3 hold. If Dη¯>λmaxS+Re holds, we can draw a conclusion that ηi−v is bounded, which can be expressed as(19)limt⟶∞ηi−v≤U0,where Re=−In⊗SA1n⊗vt−T−1n⊗v+A1n⊗v˙t−T+A1n⊗vt−T, η¯=η¯1T,…,η¯nTT, η¯i=ηi−v, U0 denotes a positive constant, and Dη¯=η¯−Aη¯t−T is the defined neutral operator.Remark 2.
SinceS and F are real matrices and the elements are independent of time and the state variables of the leader, we utilize matrices S and F to design the observer (18).
According to [42], we design following distributed adaptive control schemes of (1) as(20)τi=−K2iZ2i−XiZ1i−W^iTϕi,(21)W^⋅i=αϕiZ2iT−μW^i,where α and β are two positive constants and K2i is a gain matrix. W^i and ϕi are two related variables to NN.Theorem 1.
For the EL system (1) which considers communication delays and output constrains, if Assumptions 1–3 hold, the tracking error Z1i is UUB with the distributed observer (18) and the distributed adaptive control laws (20) and (21). At the same time, qit satisfies the time-varying output constraints, i.e., ∀t>0,k¯cit<qit<k¯cit.Proof.
Differentiating (6) yields(22)Z˙2i=q¨i−r˙i.
Substituting (22) into (1), we obtain(23)MiZ˙2i+CiZ2i=τi+ωi+diqi,q˙i,ri,r˙i,where(24)diqi,q˙i,ri,r˙i=−gi−Ciri−Mir˙i.
In this study,Miqi,Ciqi,q˙i,giqi are assumed to be unknown, and there are nonlinear uncertainties in the EL system (1). Considering that NN has good approximation ability for unknown nonlinear function, it is often used to deal with the uncertainty problem in nonlinear system. Therefore, the NN technique is used to solve the nonlinear uncertainties diqi,q˙i,ri,r˙i. The methods are shown as follows:(25)WiTϕiqi,q˙i,ri,r˙i=diqi,q˙i,ri,r˙i−Δi,where Wi represents the ideal weighted matrix, ϕi is the Gauss function, and Δi denotes the approximation error. In this research, Δi is assumed to satisfy Δi≤ΔMi, where ΔMi is a positive constant. The estimate of the nonlinear uncertainties di for the ith follower can be written as(26)d^iqi,q˙i,ri,r˙i=W^iTϕiqi,q˙i,ri,r˙i,where W^i is the estimate of Wi.
The distributed adaptive control schemes are designed as (20) and (21). Consider a Lyapunov function as follows:(27)V2i=V1i+12Z2iTMiZ2i+12αtrW˜iTW˜i,where W˜i=Wi−W˜i. Differentiating (27) and according to (20)–(26), we can obtain(28)V˙2i=V˙1i+Z2iTωi+Δi+Z2iTW˜iϕi−trW˜TϕiZ2i+μtrW˜iTW^i.
SinceΔi and ωi are bounded, there is a positive constant χ¯max satisfying Δi+ωi≤χ¯max. Furthermore, the inequality can be obtained as follows:(29)Z2iTωi+Δi≤12σZ2iTZ2i+σ2χ¯max,where σ is a positive constant.
BecauseZ2iTW˜iϕi is a scalar in (28), we have Z2iTW˜iϕi=trZ2iTW˜iϕi=trW˜iTϕiZ2i.
The matrices have the following properties:(30)trA+B=trA+trB,(31)trBA=trAB,(32)2trAB≤trBTB+trATA.
Further we have(33)trW˜iTW^i=trW˜iTWi−trW˜iTW˜i≤12trW˜iTW˜i+12trWiTWi−trW˜iTW˜i=12trWiTWi−12trW˜iTW˜i.
Substituting (16), (29), and (33) into (28), we have(34)V˙2i≤σ2χ¯max2−122λminK2i−σ−1Z2iTZ2i+2k1iεi21−εi2+μtrW˜iTW˜i−μtrWiTWi.
Thus, (34) can be written as(35)V˙2it≤−κ1V2i+υ1,where(36)κ1=min2λminK2i−σ−1B¯,α,μ,min2k1i,υ1=μ2trWiTWi+σ2χ¯max2.
We can knowV2i is UUB according to Lemma 1. Integrating (35) yields(37)V2it≤V2i0e−κ1t+υ1κ11−e−κ1t.
We can ensureκ1>0 and υ1>0 by properly selecting parameters, and we have(38)limt⟶∞V2it=υ1κ1=ρ.
It can be seen from (27) that(39)V1i≤V2i.
Moreover,(40)limt⟶∞12ln11−εi2≤ρ.
By substituting (10) and (11) into (40), we can obtain(41)−kai2t1−e−2p≤limt⟶∞Z1i≤kbi2t1−e−2p.
From (4), one has(42)qi−qn+1=qi−Fηi+Fηi−qn+1=qi−Fηi+Fηi−v.
According to (19), (41), and (42), we can obtain(43)limt⟶∞qi−qn+1≤limt⟶∞qi−Fηi+Flimt⟶∞ηi−v≤kbi2t1−e−2ρ+U0F.
From (46), we can conclude that the tracking errors among different followers and the virtual leader are bounded.
From (5), (7), (8), and (41), we can obtain(44)k¯cit<qit<k¯cit.
Equation (44) shows that the output qit satisfies the time-varying output constraints.


## 3.3. Improved Distributed Adaptive NN Tracking Controller 2 Design

According to [42], the authors proposed a new algorithm to decrease the tracking errors in (47) by using a discontinuous sign function, and it can be proved by similar steps to (49)–(58). However, this approach will bring about the additional chattering. Considering the chattering problem, we propose an improved continuous control scheme to improve the tracking accuracy. At the same time, auxiliary variable Zit can converge to origin asymptotically.For the EL system (1), we propose following modified distributed adaptive control strategy based on controller 1 as(45)τi=−K2iZ2i−XiZ1i−W^iTϕi−ιisatZ2i,(46)W^⋅i=αϕiZ2iT,where ιi is a positive constant, and it satisfies that(47)ιi≥ωi+Δi,(48)satZ2i=sgnZ2iδ,Z2iδ≥1,Z2iδ,otherwise,where δ is a small positive constant.Remark 3.
By introducing the continuous functionsatZ2i, it not only makes the tracking errors among different followers and the leader asymptotically converge to zero but also avoids the chattering phenomenon caused by discontinuous sign function sgnZ2i/δ.Theorem 2.
For the EL system (1) which considers communication delays and output constrains, when Assumptions 1–3 hold, each follower can gradually track the leader and the tracking error Z1i asymptotically converging to origin with the distributed continuous control laws (45) and (46). The time-varying output constraints are not breached, that is, ∀t>0,k¯cit<qit<k¯cit.Proof.
WhenZ2i/δ>1, we know satZ2i=sgnZ2i/δ. Substituting (45) and (46) into (1), we can obtain(49)MiZ˙2i+CiZ2i=−XiZ1i−K2iZ2i+W¯iTϕi+ωi+Δi−ιisgnZ2iδ.
Select the sameV1i and V2i as those in Theorem 1. Taking the derivative of V2i and from (16), (47), and (48), it can be obtained that(50)V˙2i≤−Z2iTK2iZ2i−k1iεi21−εi2+Z2iTωi+Δi+Z2iTW˜iϕ−trW˜iTϕZ2i−ιiZ2iδ.
Similar to (28)–(34), it holds that(51)V˙2i≤−Z2iTK2iZ2i−k1iεi1−εi2+Z2iωi+Δi−ιiZ2iδ.
For a small positive constantδ, we can find that Z2i≤Z2i/δ. Substituting (47) into (51), we have(52)V˙2i≤−Z2iTK2iZ2i−k1iεi21−εi2.
Sincek1i is a positive constant and matrix K2i is symmetric positive definite, it can be found that(53)V˙2i≤0.
Moreover, we have(54)limt⟶∞V2it=V∞∈0,V0.
Due to the fact thatV2i is bounded, we find Zμ,Z2i∈L∞. According to (6), we have Z˙1i∈L∞. When εi<1, εi2/1−εi2≥εi2. Integrating both sides of (54), it can be obtained that(55)k1i∫0∞εi21−εi2dt+λminK2i∫0∞Z2iTZ2idt≤V2i0−V2i∞.
Therefore, we haveZ1i∈L2 and Z1i∈L2∩L∞. According to Barbalet lemma, we have(56)limt⟶∞Z1it=0.
From (56), it can be found that(57)limt⟶∞qi−Fηi=0.
Following the similar steps as (42) and (43), we have(58)limt⟶∞qi−qn+1≤limt⟶∞qi−Fηi+Flimt⟶∞η−v≤U0F.
Theith follower has a good tracking performance on the virtual leader, and the upper boundary of the tracking error is described by (58).
WhenZ2i/δ<1, it can be obtained that satZ2i=Z2i/δ. According to (49) and (50), we have(59)V˙2it≤−Z2iTK2iZ2i−k1iεi21−εi2−Z2iTιiZ2iδ−ωi+Δi≤−Z2iTK2iZ2i−k1iεi21−εi2−ZιiZ2iδ−ωi+Δi.
Ifδ≤ιiZ2i/ωi+Δi, we have V˙2it≤−Z2iTK2iZ2i−k1iεi2/1−εi2. From the procedures in (53)–(58), we can find that the tracking error between the ith follower and the virtual leader is bounded.
According to (5), (7), (8), and (56), we can obtain the same result as (44). It shows that the output qit satisfies the time-varying output constraints.Remark 4.
The results of this paper can provide some reference for formation control of MASs. In the future, the formation control problems for MASs considering time-varying communication delays and full-state constraints will be studied.


## 4. Results and Discussion

### 4.1. Parameter Setting

In this section, we utilize some examples to verify the validity of the proposed schemes in practical aspect. In this study, we consider 4 2-degree-of-freedom robotic manipulators as an example for the simulation experiment. The communication topology is shown in Figure1, where 5 represents the virtual leader and 1–4 represent the four followers. The structure of robotic manipulator is shown in Figure 2.Figure 1
Communication topology.Figure 2
The structure of robotic manipulator.The dynamic equation of theith follower can be expressed as(60)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi+wi,i=1,…,4,where(61)qi=qi1,qi2T,Miqi=Ξi1+Ξi2+2Ξi3cosqi2Ξi2+Ξi3cosqi2Ξi2+Ξi3cosqi2Ξi2,Ciqi,q˙i=−Ξi3sinqi2q˙i2−Ξi3sinqi2q˙i1+q˙i2Ξi3sinqi2q˙i10,giqi=Ξi4cosq˙i1+Ξi5gcosqi1+qi2Ξi5gcosqi1+qi2,ωi=5%⋅Miqiq¨i+Ciqi,q˙iq˙i+giqi.In this section,q1i,q2i represent the rotation angle of two joints of the manipulator. Ξi1=Ji1+mi2li12, Ξi1=0.25mi2li22+Ji2, Ξi3=0.5mi2li1li2, Ξi4=0.5mi1+mi2li1, Ξi5=0.5mi2li22, and g=9.8m/s2 denotes gravitational acceleration, where mi1 and mi2 denote masses of links, li1 is the length of links, li2 represents the distance from center of mass to motors, and Ji1 and Ji2 are moments of inertia.Table2 shows the indicators of robotic manipulators.Table 2
The parameters of the robotic manipulators.
ParameterManipulator 1Manipulator 2Manipulator 3Manipulator 4mi1kg1.010.9611.04mi2kg1.121.141.031.09li1m10.950.981li2m0.9610.951Ji1kg⋅m20.230.210.190.21Ji2kg⋅m20.410.400.420.41We set the time-varying output constraints as follows:(62)k¯ct=2.2+15e−0.8t,2.2+15e−0.5t,2.2+15e−0.35t,2.2+15e−0.25tT,k¯ct=1.8+15e−0.8t,1.8+15e−0.5t,1.8+15e−0.35t,1.8+15e−0.25tT.The boundary value of tracking errorZ1i is expressed as(63)kait=qrit−k¯cit,kbit=k¯cit−qrit.Thus, we have(64)−kait<Z1it<kbit.The initial angle of robotic manipulators is set as follows.q110=π/5, q120=−π/3, q210=2π/5, q220=−π/6, q310=3π/5, q320=π/6, q410=4π/5, q420=π/3, and q˙i10=q˙i20=0.For theith follower i=1,…,4, the activation equation of the neural network system can be written as(65)ϕiZ=ϕi1z,…,ϕi6zT.We select Gauss function as the activation function, and its form is(66)ϕijz=expz−cij2σij2,j=1,…,6,where z=qi,q˙i,ri,r˙iT∈ℝ8. Suppose all followers utilize the same activation equation. Cij denotes the NN center and it is distributed over −5,54×−0.5,0.54. σij is Gaussian function’s width, and we define σij=2, W^i0=06×2.The desired trajectory of the virtual leader is designed as follows:(67)q51t=q51_ampsinωt+ϕ51_pha+q51_bias,q52t=q52_ampsinωt+ϕ52_pha+q52_bias,where q51_amp=π/6, q51_pha=π/2, q51_bias=π/2, q52_amp=2π/3, ϕ52_pha=π/2, q52_bias=0, and ω=0.1π.The state variableq5 of the virtual leader can be expressed as(68)v˙=Sv,q5=Fv,where(69)v=sinωtsinωt+π21,ω=0.1π,S=00.1π0−0.1π00000,F=0π6π202π30.In practical physical systems, we often need to limit the amplitude of the control inputsτi. The following saturation equation is used to limit the amplitude:(70)satτi=−τimaxτi<−τimax,τiτi≤τimax,τimaxτi>τimax,where τimax denotes a positive constant, and we let τimax=50Nm.In this section, we choose the communication delay asT=0.5s.


### 4.2. Simulation Performance for Controller 1

For controller 1, we choose the parameters ask1i=10, K2i=20I2, γ=1, and v=10. Figures 3–12 introduce the simulation results.Figure 3
The generalized coordinatesqi1,i=1,…,4 of manipulators.Figure 4
The generalized coordinatesqi2,i=1,…,4 of manipulators.Figure 5
The auxiliary variablesZi1,i=1,…,4.Figure 6
The auxiliary variablesZi2,i=1,…,4.Figure 7
The control input torqueτi1,i=1,…,4 of manipulators.Figure 8
The control input torqueτi2,i=1,…,4 of manipulators.Figure 9
Constraint effect ofZ11 under Algorithm 1.Figure 10
Constraint effect ofZ21 under Algorithm 1.Figure 11
Constraint effect ofZ31 under Algorithm 1.Figure 12
Constraint effect ofZ41 under Algorithm 1.Figures3 and 4 illustrate the state variables qi1 and qi2 among the followers and the virtual leader, from which we can see that each follower can effectively track the leader after about 5 s. Figures 5 and 6 show that the auxiliary variables Zi1 and Zi2 are bounded. At the same time, the auxiliary variables Zi1 do not exceed 0.1rad, and Z2i do not exceed 0.05 after 3 s. From Figures 7 and 8, it can be obtained that the control inputs are continuous and change within 10 Nm. From Figures 9–12, we can find that the time-varying output constraints of the robotic manipulators are always satisfied.


### 4.3. Simulation Performance for Algorithm 2

For the improved algorithm by using discontinuous sign function on Algorithm 1, letk1i=20, K2i=20I2, γ=1, ιi=80, and δ=0.5. Figures 13 and 14 show that the control input torque of all robotic manipulators has chattering problem.Figure 13
The control input torqueτi1,i=1,…,4 of manipulators.Figure 14
The control input torqueτi2,i=1,…,4 of manipulators.For Algorithm 2, we setk1i=20, K2i=20I2, γ=1, ιi=80, and δ=0.5. Figures 15–24 show the performance of simulation results.Figure 15
The generalized coordinatesqi1,i=1,…,4 of manipulators.Figure 16
The generalized coordinatesqi2,i=1,…,4 of manipulators.Figure 17
The auxiliary variablesZ1i,i=1,…,4.Figure 18
The auxiliary variablesZ2i,i=1,…,4.Figure 19
The control input torqueτi1,i=1,…,4 of manipulators.Figure 20
The control input torqueτi2,i=1,…,4 of manipulators.Figure 21
Constraint effect ofZ11 under Algorithm 2.Figure 22
Constraint effect ofZ21 under Algorithm 2.Figure 23
Constraint effect ofZ31 under Algorithm 2.Figure 24
Constraint effect ofZ41 under Algorithm 2.From Figures15 and 16, it is obvious that the tracking performance is better, and the tracking errors are smaller compared with the cases in Figures 5 and 6. Comparing Figures 17 and 18 with Figures 7 and 8, we can find that the fluctuation range of Z1i and Z2i is smaller after 1 s. From Figures 19 and 20, we can obtain that the control input torques of the robotic manipulators are stable and continuous. From Figures 21–24, we can see that under the control Algorithm 2, the error variables of the followers always satisfy the time-varying output constraints.Based on the above simulation results, we can discover that all followers have very good tracking effect on the leader, and the tracking error always satisfies the time-varying constraints. In addition, the tracking errorsZ1i based on Algorithm 2 are smaller than those based on Algorithm 1, and Algorithm 2 avoids the unexpected chattering problem arising from sign function.


## 4.1. Parameter Setting

In this section, we utilize some examples to verify the validity of the proposed schemes in practical aspect. In this study, we consider 4 2-degree-of-freedom robotic manipulators as an example for the simulation experiment. The communication topology is shown in Figure1, where 5 represents the virtual leader and 1–4 represent the four followers. The structure of robotic manipulator is shown in Figure 2.Figure 1
Communication topology.Figure 2
The structure of robotic manipulator.The dynamic equation of theith follower can be expressed as(60)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi+wi,i=1,…,4,where(61)qi=qi1,qi2T,Miqi=Ξi1+Ξi2+2Ξi3cosqi2Ξi2+Ξi3cosqi2Ξi2+Ξi3cosqi2Ξi2,Ciqi,q˙i=−Ξi3sinqi2q˙i2−Ξi3sinqi2q˙i1+q˙i2Ξi3sinqi2q˙i10,giqi=Ξi4cosq˙i1+Ξi5gcosqi1+qi2Ξi5gcosqi1+qi2,ωi=5%⋅Miqiq¨i+Ciqi,q˙iq˙i+giqi.In this section,q1i,q2i represent the rotation angle of two joints of the manipulator. Ξi1=Ji1+mi2li12, Ξi1=0.25mi2li22+Ji2, Ξi3=0.5mi2li1li2, Ξi4=0.5mi1+mi2li1, Ξi5=0.5mi2li22, and g=9.8m/s2 denotes gravitational acceleration, where mi1 and mi2 denote masses of links, li1 is the length of links, li2 represents the distance from center of mass to motors, and Ji1 and Ji2 are moments of inertia.Table2 shows the indicators of robotic manipulators.Table 2
The parameters of the robotic manipulators.
ParameterManipulator 1Manipulator 2Manipulator 3Manipulator 4mi1kg1.010.9611.04mi2kg1.121.141.031.09li1m10.950.981li2m0.9610.951Ji1kg⋅m20.230.210.190.21Ji2kg⋅m20.410.400.420.41We set the time-varying output constraints as follows:(62)k¯ct=2.2+15e−0.8t,2.2+15e−0.5t,2.2+15e−0.35t,2.2+15e−0.25tT,k¯ct=1.8+15e−0.8t,1.8+15e−0.5t,1.8+15e−0.35t,1.8+15e−0.25tT.The boundary value of tracking errorZ1i is expressed as(63)kait=qrit−k¯cit,kbit=k¯cit−qrit.Thus, we have(64)−kait<Z1it<kbit.The initial angle of robotic manipulators is set as follows.q110=π/5, q120=−π/3, q210=2π/5, q220=−π/6, q310=3π/5, q320=π/6, q410=4π/5, q420=π/3, and q˙i10=q˙i20=0.For theith follower i=1,…,4, the activation equation of the neural network system can be written as(65)ϕiZ=ϕi1z,…,ϕi6zT.We select Gauss function as the activation function, and its form is(66)ϕijz=expz−cij2σij2,j=1,…,6,where z=qi,q˙i,ri,r˙iT∈ℝ8. Suppose all followers utilize the same activation equation. Cij denotes the NN center and it is distributed over −5,54×−0.5,0.54. σij is Gaussian function’s width, and we define σij=2, W^i0=06×2.The desired trajectory of the virtual leader is designed as follows:(67)q51t=q51_ampsinωt+ϕ51_pha+q51_bias,q52t=q52_ampsinωt+ϕ52_pha+q52_bias,where q51_amp=π/6, q51_pha=π/2, q51_bias=π/2, q52_amp=2π/3, ϕ52_pha=π/2, q52_bias=0, and ω=0.1π.The state variableq5 of the virtual leader can be expressed as(68)v˙=Sv,q5=Fv,where(69)v=sinωtsinωt+π21,ω=0.1π,S=00.1π0−0.1π00000,F=0π6π202π30.In practical physical systems, we often need to limit the amplitude of the control inputsτi. The following saturation equation is used to limit the amplitude:(70)satτi=−τimaxτi<−τimax,τiτi≤τimax,τimaxτi>τimax,where τimax denotes a positive constant, and we let τimax=50Nm.In this section, we choose the communication delay asT=0.5s.


## 4.2. Simulation Performance for Controller 1

For controller 1, we choose the parameters ask1i=10, K2i=20I2, γ=1, and v=10. Figures 3–12 introduce the simulation results.Figure 3
The generalized coordinatesqi1,i=1,…,4 of manipulators.Figure 4
The generalized coordinatesqi2,i=1,…,4 of manipulators.Figure 5
The auxiliary variablesZi1,i=1,…,4.Figure 6
The auxiliary variablesZi2,i=1,…,4.Figure 7
The control input torqueτi1,i=1,…,4 of manipulators.Figure 8
The control input torqueτi2,i=1,…,4 of manipulators.Figure 9
Constraint effect ofZ11 under Algorithm 1.Figure 10
Constraint effect ofZ21 under Algorithm 1.Figure 11
Constraint effect ofZ31 under Algorithm 1.Figure 12
Constraint effect ofZ41 under Algorithm 1.Figures3 and 4 illustrate the state variables qi1 and qi2 among the followers and the virtual leader, from which we can see that each follower can effectively track the leader after about 5 s. Figures 5 and 6 show that the auxiliary variables Zi1 and Zi2 are bounded. At the same time, the auxiliary variables Zi1 do not exceed 0.1rad, and Z2i do not exceed 0.05 after 3 s. From Figures 7 and 8, it can be obtained that the control inputs are continuous and change within 10 Nm. From Figures 9–12, we can find that the time-varying output constraints of the robotic manipulators are always satisfied.


## 4.3. Simulation Performance for Algorithm 2

For the improved algorithm by using discontinuous sign function on Algorithm 1, letk1i=20, K2i=20I2, γ=1, ιi=80, and δ=0.5. Figures 13 and 14 show that the control input torque of all robotic manipulators has chattering problem.Figure 13
The control input torqueτi1,i=1,…,4 of manipulators.Figure 14
The control input torqueτi2,i=1,…,4 of manipulators.For Algorithm 2, we setk1i=20, K2i=20I2, γ=1, ιi=80, and δ=0.5. Figures 15–24 show the performance of simulation results.Figure 15
The generalized coordinatesqi1,i=1,…,4 of manipulators.Figure 16
The generalized coordinatesqi2,i=1,…,4 of manipulators.Figure 17
The auxiliary variablesZ1i,i=1,…,4.Figure 18
The auxiliary variablesZ2i,i=1,…,4.Figure 19
The control input torqueτi1,i=1,…,4 of manipulators.Figure 20
The control input torqueτi2,i=1,…,4 of manipulators.Figure 21
Constraint effect ofZ11 under Algorithm 2.Figure 22
Constraint effect ofZ21 under Algorithm 2.Figure 23
Constraint effect ofZ31 under Algorithm 2.Figure 24
Constraint effect ofZ41 under Algorithm 2.From Figures15 and 16, it is obvious that the tracking performance is better, and the tracking errors are smaller compared with the cases in Figures 5 and 6. Comparing Figures 17 and 18 with Figures 7 and 8, we can find that the fluctuation range of Z1i and Z2i is smaller after 1 s. From Figures 19 and 20, we can obtain that the control input torques of the robotic manipulators are stable and continuous. From Figures 21–24, we can see that under the control Algorithm 2, the error variables of the followers always satisfy the time-varying output constraints.Based on the above simulation results, we can discover that all followers have very good tracking effect on the leader, and the tracking error always satisfies the time-varying constraints. In addition, the tracking errorsZ1i based on Algorithm 2 are smaller than those based on Algorithm 1, and Algorithm 2 avoids the unexpected chattering problem arising from sign function.


## 5. Conclusions

In this study, we propose two practical control strategies to address distributed coordinated tracking problem for the multiple EL systems subjected to communication delays and time-varying constraints. The distributed observer is used to cope with the communication delays for multiple EL systems. We utilize the NN technique to compensate nonlinear uncertainties. At the same time, an asymmetric BLF is used to guarantee that the output errors are always within the output constraints. The adaptive control Algorithm 1 is designed to ensure that the tracking errors is designed to ensure that the tracking errors among the followers and the virtual leader can be bounded. Based on Algorithm 1, the improved Algorithm 2 can make the tracking errors smaller. The simulation results indicate that the proposed methods can effectively solve the problem of communication delays and make the tracking errors meet the prescribed output constraints.



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*Source: 5594053-2021-05-19.xml*