# Practical Bipartite Tracking for Networked Robotic Systems via Fixed-Time Estimator-Based Control

Authors: Peng Su; Jinqiang Gan; Teng-Fei Ding; Chang-Duo Liang; Ming-Feng Ge
Journal: Complexity
 (2021)

---

## Abstract

In this paper, the fixed-time practical bipartite tracking problem for the networked robotic systems (NRSs) with parametric uncertainties, input disturbances, and directed signed graphs is investigated. A new fixed-time estimator-based control algorithm for the NRSs is presented to address the abovementioned problem. By applying a sliding surface and the time base generator (TBG) approach, a new stability analysis method is proposed to achieve the fixed-time practical bipartite tracking for the NRSs. We also derive the upper bound of the convergence time for employing the presented control algorithm to solve the practical bipartite tracking problem and further demonstrate that the convergence time is independent of the initial value. Finally, the simulation examples are given to verify the effectiveness of the presented algorithms.


---

## Body

## 1. Introduction

Recently, the cooperative control of the networked robotic systems (NRSs) [1–4] has received increasing attention. The concept of the NRS is to denote a team of the controllable autonomous robots aiming to accomplish single or multiple global tasks over local communication. Compared with a single robot, the NRS is capable of achieving more complex missions in a more effective and flexible way. The distributed control algorithms of multiagent systems have been widely used to obtain various collective behaviors, including target tracking [5, 6], formation control [7–11], containment control [12, 13], distributed optimization [14, 15], distribution networks [16, 17], tracking performance limitations [18, 19], and resilient control [20] to name a few. Besides, many excellent control algorithms are proposed including observer-based adaptive control [21], fault-tolerant control [22], output feedback fuzzy control [23, 24], and fuzzy-approximation-based asymptotic control [25] to solve tracking problem for nonlinear systems. Note that all the abovementioned results are focused on networked systems with only cooperation. However, in some cases of real-world applications, the NRS is required to be divided into two subgroups and executes tasks from two opposite directions. This implies that the subgroups have to compete with each other. Therefore, it is of great significance to study the NRSs with both cooperation and competition.Recently, bipartite consensus [26–28] and bipartite tracking [29–32] of multiagent systems involving both cooperation and competition have received increasing attention. Compared with conventional coordinated tracking behaviors, the bipartite tracking implies that all the robots converge to two opposite states of the leader with the identical state value and different signs. In [29], a class of bipartite tracking and containment control problems with signed digraphs has been addressed. Considering high-order multiagent system, bipartite tracking problems with uncertainties have been solved in [30]. The bipartite tracking problems subject to the time lag over matrix-weighted signed graphs for the NRSs have been studied in [31]. In [32], the bipartite tracking problems with a dynamic leader have been studied in linear multiagent system. It is worth mentioning that the aforementioned research mainly focused on first-, second-, and higher-order dynamics, as well as Lipschitz-type nonlinear systems. There are only a few results on bipartite tracking problems for the NRSs, and the control approaches for solving such problems are still lacked.On the contrary, the convergence time is an important performance index in the field of distributed control. It thus motivates the development of the finite-time and the fixed-time control to improve such convergence performance and, meanwhile, reject input disturbance [33, 34]. Different from the finite-time control, whose convergence time depends on the initial value [21, 35], the fixed-time control can force the states to reach the origin in fixed-time regardless of the initial value [36–42]. Due to such advantage, the fixed-time consensus for nonlinear heterogeneous multiagent systems has been studied in [37]. In [38], the fixed-time tracking for high-order multiagent systems has been taken into account. In [39], the fixed-time control has been applied to second-order nonlinear multiagent systems by integral sliding-mode approach. Especially, a time base generator (TBG) approach has been delivered to drive the states to approach a desired bounded range in fixed time through adjusting the TBG gain [40]. However, to the best knowledge of the authors, the fixed-time practical bipartite tracking problem for the NRSs remains open.By the abovementioned discussions, this paper aims to provide a general solution to the fixed-time practical bipartite tracking problem for the NRSs with parametric uncertainties, input disturbances, and directed signed graphs. A fixed-time estimator-based control algorithm is designed to solve this problem. The main contributions of this work are threefold:(i)
Different from the control approaches for achieving the bipartite tracking of multiagent system [29–32], in which the system model is described by first-, second-, and higher-order dynamics, as well as Lipschitz-type nonlinear dynamics. The proposed control algorithms address the bipartite tracking problem of the networked robotic system, in which the dynamics of system is modeled by Euler–Lagrange equation, which is more meaningful to describe the actual physical agents.(ii)
Compared with the existing results on collective behavior of the NRS [3, 4], in which the converge time is asymptotical, finite time, which is all related to the initial values of the system, the proposed fixed-time estimator-based control algorithm guarantees that the convergence time is fixed time, which is irrespective of the initial states of the system.(iii)
The presented control algorithm can provide a theoretical framework for stabilizing other complex uncertain networked systems in fixed time.The remaining parts are organized as follows. Section2 provides the preliminaries and the problem formulations. In Section 3, the fixed-time estimator-based control algorithm and its stability analysis are proposed. The simulation results are presented in Section 4 to test the algorithm. Finally, the conclusions are summed up in Section 5.Notation: letℛN be the N×1 real matrix, ℛN×N be the N×N real matrix, diag⋅ be the diagonal matrix, and sgn⋅ be the sign function. siglx is equal to xlsgnx with l>1. 1N=1,1,…,1T is the column vector of the N dimension. IN is the N dimension identity matrix. ⋅max and ⋅min are the maximum and minimum values of the given vector. λmin⋅ is the minimum eigenvalues of the given matrix. Besides, ⋅ is the Euclidean norm.


## 2. Preliminaries

### 2.1. Graph Theory

The communication of the NRS can be modeled as a directed signed graphG=V,E,A, where V=1,2,…,N is the set of vertexes, E⊆V×V is the set of edges, and A=aij∈ℛN×N is the adjacency matrix. An edge eij∈E (i.e., aij≠0) implies that the communication information flows directly from the vertex j to the vertex i, otherwise aij=0. aij>0 implies that the vertex i cooperates with the vertex j; aij<0 implies that the vertex i competes with the vertex j. Furthermore, assume that G has no self-loops, i.e., aii=0. A directed path is denoted as a series of edges from 1,2,3,4,…,η−1,η with distinct vertexes and the length η−1. A cycle of G is denoted as the starting and ending vertexes of the path being the same, namely, η=1. Furthermore, if a cycle has even number negative weights, it is termed as positive cycle; otherwise, a cycle is referred to as a negative cycle. A directed signed graph includes a directed spanning tree, and it implies that there is a rooted vertex which has a directed path to any other vertexes. The Laplacian matrix of G is defined as Ls=lij∈ℛN×N [26], where lii=∑j=1,j≠iNaij and lij=−aij, if i≠j. A directed signed graph G is detail-balanced if there exist positive constants h1,h2,…,hN such that hjaij=hiaij. A diagonal weighted matrix B=diaga10,…,aN0 represents the connection weight between the rooted robot and other robots. In detail, ai0>0 implies that the ith robot can directly receive the information from the rooted robot, otherwise ai0=0.Definition 1.
(see [26]). The directed signed graph is said to be structurally balanced if V can be grouped into two sets V+ and V− satisfying V=V+∪V−, V+∩V−=ϕ and aij≥0,∀νi,νj∈V∂∂∈+,−, aij≤0,∀νi∈V∂,νj∈V℘,∂≠℘∂,℘∈+,−.Assumption 1.
The directed signed graphG is structurally balanced and detail-balanced. The augmented graph G¯ (including the graph G and the virtual leader) contains a directed spanning tree with the leader as the rooted vertex.Lemma 1.
(see [26]). A directed signed graph is structurally balanced if and only if there is a diagonal matrix D=diagd1,d2,…,dN such that DAD is positive semidefinite, where di=1 if νi∈V+ and di=−1 if νi∈V−.Lemma 2.
(see [44]). Let Ms=Ls+B and MD=DMsD. If Assumption 1 holds, there exists a positive-definite diagonal matrix H=diagh1,h2,…,hN such that the matrix MDH is positive definite.


### 2.2. Time Base Generator

The time base generator (TBG) [43] is defined as a function based on time satisfying predetermined restrictions on its initial and final values. Let the TBG gain be presented as follows:(1)gt=ξ˙t1−ξt+δ,where ξt is the TBG and 0<δ≤1. For any given tf>0, we can design a proper TBG ξt such that(1)
ξt is at least C2 on 0,+∞(2)
ξt is continuous and nondecreasing from ξ0=0 to ξtf=1, where tf<+∞ is a scheduled time constant(3)
ξ˙0=ξ˙tf=0, where the derivative of ξt at t=0 is actually its right derivative(4)
ξt=1 and ξ˙t=0 if t>tfLemma 3.
(see [40]). Considering the following differential system,(2)x˙t=−gtxt,x0=x0,where TBG gain gt is defined in the form of (1) and xt denotes the state. Then, there exists a positive constant tf>0 with respect to ξt in (1) such that limt⟶tfxt≤δ/1+δx0 and xt≤δ/1+δx0 on t∈tf,∞, where δ is given in (1).Remark 1.
An example of the TBG is presented as follows [40]:(3)ξt=10tf6t6−24tf5t5+15tf4t4,0≤t≤tf,1,t>tf,where tf>0 is a positive constant and denotes the user-designed convergence time, which is an important parameter to ensure the convergence of the states before the time tf. Then, the main techniques to design the TBG function is to satisfy four properties given in (1). A typical TBG function with tf=4 is shown in Figures 1 and 2 to enhance its visualization.Figure 1
The TBG function.Figure 2
The derivative of the TBG function.


### 2.3. System Formulation

The dynamics of theith robot in the NRS is presented below [36]:(4)Miqiq¨i+Ciqi,q˙iq˙i+giqi+dit=τi,where t≥0, i∈V, qi,q˙i,q¨i∈ℛn are, respectively, the generalized position, velocity, and acceleration, Miqi∈ℛn×n stands for the positive-definite inertia matrix, C1qi,q˙i∈ℛn×n is the centrifugal-Coriolis matrix, giqi∈ℛn represents the gravitational term, dit∈ℛn is the input disturbances satisfying supt≥0dit≤dM, dM is a positive constant, and τi∈ℛn denotes the control input.The parameters of the dynamical model can be described as(5)Miq=Mi0qi+ΔMiqi,Ciqi,q˙i=Ci0qi,q˙i+ΔCiqi,q˙i,giqi=gi0qi+Δgiqi,where Mi0qi, Ci0qi,q˙i, and gi0qi are the desired parts of the dynamic model and ΔMiqi, ΔCiqi,q˙i, and Δgiqi are the uncertain term of the dynamic model. Therefore, system (4) can be rewritten as(6)Mi0qiq¨i+Ci0qi,q˙iqi+gi0qi=τi+ρt,where ρt=−ΔMiqiq¨i−ΔCiqi,q˙iq˙i−Δgiqi−dit. Following [36], ρt is upper bounded, namely,(7)ρt≤b0+b1q˙i2+b2τi,where b0 and b1 are positive constants and 0<b2<1. On the contrary, the leader’s states q0, v0, and u0∈ℛn obey that q0.=v0 and v0.=u0.Assumption 2.
The acceleration of the leader is upper bounded, i.e.,supt≥0u0t≤umax.


### 2.4. Problem Formulation

The control objective is to design a proper control inputτi for achieving the practical bipartite tracking of the NRS in a fixed time. Denote qi and vi as the position and the velocity of the followers. Then, a useful definition and Lemma 4 are listed as follows.Definition 2.
The practical fixed-time bipartite tracking problem is addressed if there exists a positive constantT regardless of the initial value of the states such that limt⟶Tqit−q0t≤ρ1, limt⟶Tvit−v0t≤ρ2, qit−q0t≤ρ1, and vit−v0t≤ρ2 on t∈T,∞, ∀i∈V+, and limt⟶Tqit+q0t≤ρ1, limt⟶Tvit+v0t≤ρ2, qit+q0t≤ρ1, and vit+v0t≤ρ2 on t∈T,∞, ∀i∈V−, where ρ1,ρ2>0 can be sufficiently small by choosing appropriate control parameters.Lemma 4.
(see [33]). For the nonlinear system x˙=fx,x0=x0,x∈ℛN. Suppose that there exist a positive-definite and continuous function Vx, real numbers α,β>0,0<γ1<1, and γ2>1 such that V˙x+αVγ1x+βVγ2x≤0. Then, the origin is the fixed-time stable equilibrium point of the considered system. The setting time is given as(8)T≤Tmax=1α1−γ1+1βγ2−1.Remark 2.
In missile guidance, hitting the target within a specified time and its predefined neighborhood is a demanding objective. Compare with the traditional control approaches, which can only achieve asymptotic, exponentially, and finite-time stability, because the value of their convergence time depends on the initial condition. Besides, for the complex network, it is very difficult to deal with the switched information between the cyber layer and the physical layer. In the paper, we employ the estimator-based control algorithm to deal with information of each layer. It is a feasible plan for complex cyber-physical system to simplify the proof and make the system stable within fixed time.Remark 3.
The estimator-based control approach is early employed to address the target tracking problem of the networked robotic system in [6]. Based on this idea, the predefined-time formation tracking problem of networked surface vehicles [7], the multiformation tracking problem of networked heterogeneous robotic systems [11], and lag-bipartite tracking problem of networked robotic systems [31] have been successfully addressed. It thus believes that the proposed control algorithm can be generalized to other systems and other tracking control problems.


## 2.1. Graph Theory

The communication of the NRS can be modeled as a directed signed graphG=V,E,A, where V=1,2,…,N is the set of vertexes, E⊆V×V is the set of edges, and A=aij∈ℛN×N is the adjacency matrix. An edge eij∈E (i.e., aij≠0) implies that the communication information flows directly from the vertex j to the vertex i, otherwise aij=0. aij>0 implies that the vertex i cooperates with the vertex j; aij<0 implies that the vertex i competes with the vertex j. Furthermore, assume that G has no self-loops, i.e., aii=0. A directed path is denoted as a series of edges from 1,2,3,4,…,η−1,η with distinct vertexes and the length η−1. A cycle of G is denoted as the starting and ending vertexes of the path being the same, namely, η=1. Furthermore, if a cycle has even number negative weights, it is termed as positive cycle; otherwise, a cycle is referred to as a negative cycle. A directed signed graph includes a directed spanning tree, and it implies that there is a rooted vertex which has a directed path to any other vertexes. The Laplacian matrix of G is defined as Ls=lij∈ℛN×N [26], where lii=∑j=1,j≠iNaij and lij=−aij, if i≠j. A directed signed graph G is detail-balanced if there exist positive constants h1,h2,…,hN such that hjaij=hiaij. A diagonal weighted matrix B=diaga10,…,aN0 represents the connection weight between the rooted robot and other robots. In detail, ai0>0 implies that the ith robot can directly receive the information from the rooted robot, otherwise ai0=0.Definition 1.
(see [26]). The directed signed graph is said to be structurally balanced if V can be grouped into two sets V+ and V− satisfying V=V+∪V−, V+∩V−=ϕ and aij≥0,∀νi,νj∈V∂∂∈+,−, aij≤0,∀νi∈V∂,νj∈V℘,∂≠℘∂,℘∈+,−.Assumption 1.
The directed signed graphG is structurally balanced and detail-balanced. The augmented graph G¯ (including the graph G and the virtual leader) contains a directed spanning tree with the leader as the rooted vertex.Lemma 1.
(see [26]). A directed signed graph is structurally balanced if and only if there is a diagonal matrix D=diagd1,d2,…,dN such that DAD is positive semidefinite, where di=1 if νi∈V+ and di=−1 if νi∈V−.Lemma 2.
(see [44]). Let Ms=Ls+B and MD=DMsD. If Assumption 1 holds, there exists a positive-definite diagonal matrix H=diagh1,h2,…,hN such that the matrix MDH is positive definite.


## 2.2. Time Base Generator

The time base generator (TBG) [43] is defined as a function based on time satisfying predetermined restrictions on its initial and final values. Let the TBG gain be presented as follows:(1)gt=ξ˙t1−ξt+δ,where ξt is the TBG and 0<δ≤1. For any given tf>0, we can design a proper TBG ξt such that(1)
ξt is at least C2 on 0,+∞(2)
ξt is continuous and nondecreasing from ξ0=0 to ξtf=1, where tf<+∞ is a scheduled time constant(3)
ξ˙0=ξ˙tf=0, where the derivative of ξt at t=0 is actually its right derivative(4)
ξt=1 and ξ˙t=0 if t>tfLemma 3.
(see [40]). Considering the following differential system,(2)x˙t=−gtxt,x0=x0,where TBG gain gt is defined in the form of (1) and xt denotes the state. Then, there exists a positive constant tf>0 with respect to ξt in (1) such that limt⟶tfxt≤δ/1+δx0 and xt≤δ/1+δx0 on t∈tf,∞, where δ is given in (1).Remark 1.
An example of the TBG is presented as follows [40]:(3)ξt=10tf6t6−24tf5t5+15tf4t4,0≤t≤tf,1,t>tf,where tf>0 is a positive constant and denotes the user-designed convergence time, which is an important parameter to ensure the convergence of the states before the time tf. Then, the main techniques to design the TBG function is to satisfy four properties given in (1). A typical TBG function with tf=4 is shown in Figures 1 and 2 to enhance its visualization.Figure 1
The TBG function.Figure 2
The derivative of the TBG function.


## 2.3. System Formulation

The dynamics of theith robot in the NRS is presented below [36]:(4)Miqiq¨i+Ciqi,q˙iq˙i+giqi+dit=τi,where t≥0, i∈V, qi,q˙i,q¨i∈ℛn are, respectively, the generalized position, velocity, and acceleration, Miqi∈ℛn×n stands for the positive-definite inertia matrix, C1qi,q˙i∈ℛn×n is the centrifugal-Coriolis matrix, giqi∈ℛn represents the gravitational term, dit∈ℛn is the input disturbances satisfying supt≥0dit≤dM, dM is a positive constant, and τi∈ℛn denotes the control input.The parameters of the dynamical model can be described as(5)Miq=Mi0qi+ΔMiqi,Ciqi,q˙i=Ci0qi,q˙i+ΔCiqi,q˙i,giqi=gi0qi+Δgiqi,where Mi0qi, Ci0qi,q˙i, and gi0qi are the desired parts of the dynamic model and ΔMiqi, ΔCiqi,q˙i, and Δgiqi are the uncertain term of the dynamic model. Therefore, system (4) can be rewritten as(6)Mi0qiq¨i+Ci0qi,q˙iqi+gi0qi=τi+ρt,where ρt=−ΔMiqiq¨i−ΔCiqi,q˙iq˙i−Δgiqi−dit. Following [36], ρt is upper bounded, namely,(7)ρt≤b0+b1q˙i2+b2τi,where b0 and b1 are positive constants and 0<b2<1. On the contrary, the leader’s states q0, v0, and u0∈ℛn obey that q0.=v0 and v0.=u0.Assumption 2.
The acceleration of the leader is upper bounded, i.e.,supt≥0u0t≤umax.


## 2.4. Problem Formulation

The control objective is to design a proper control inputτi for achieving the practical bipartite tracking of the NRS in a fixed time. Denote qi and vi as the position and the velocity of the followers. Then, a useful definition and Lemma 4 are listed as follows.Definition 2.
The practical fixed-time bipartite tracking problem is addressed if there exists a positive constantT regardless of the initial value of the states such that limt⟶Tqit−q0t≤ρ1, limt⟶Tvit−v0t≤ρ2, qit−q0t≤ρ1, and vit−v0t≤ρ2 on t∈T,∞, ∀i∈V+, and limt⟶Tqit+q0t≤ρ1, limt⟶Tvit+v0t≤ρ2, qit+q0t≤ρ1, and vit+v0t≤ρ2 on t∈T,∞, ∀i∈V−, where ρ1,ρ2>0 can be sufficiently small by choosing appropriate control parameters.Lemma 4.
(see [33]). For the nonlinear system x˙=fx,x0=x0,x∈ℛN. Suppose that there exist a positive-definite and continuous function Vx, real numbers α,β>0,0<γ1<1, and γ2>1 such that V˙x+αVγ1x+βVγ2x≤0. Then, the origin is the fixed-time stable equilibrium point of the considered system. The setting time is given as(8)T≤Tmax=1α1−γ1+1βγ2−1.Remark 2.
In missile guidance, hitting the target within a specified time and its predefined neighborhood is a demanding objective. Compare with the traditional control approaches, which can only achieve asymptotic, exponentially, and finite-time stability, because the value of their convergence time depends on the initial condition. Besides, for the complex network, it is very difficult to deal with the switched information between the cyber layer and the physical layer. In the paper, we employ the estimator-based control algorithm to deal with information of each layer. It is a feasible plan for complex cyber-physical system to simplify the proof and make the system stable within fixed time.Remark 3.
The estimator-based control approach is early employed to address the target tracking problem of the networked robotic system in [6]. Based on this idea, the predefined-time formation tracking problem of networked surface vehicles [7], the multiformation tracking problem of networked heterogeneous robotic systems [11], and lag-bipartite tracking problem of networked robotic systems [31] have been successfully addressed. It thus believes that the proposed control algorithm can be generalized to other systems and other tracking control problems.


## 3. Main Results

### 3.1. Fixed-Time Estimator-Based Control Algorithm

In this section, the fixed-time estimator-based control algorithm for achieving practical bipartite tracking of the NRS is proposed. Before constructing the control input, we design the following sliding-mode surface:(9)si=q˜˙i+K1iϕq˜i+K2isiglq˜i,where q˜i=qi−q^i, q˜˙i=q˙i−q^˙i, q^i and q^˙i are the estimated states with respect to qi and q˙i∈ℛn, K1i,K2i∈ℛn×n are positive-definite diagonal matrixes, l>1 is a positive constant, and ϕq˜i=ϕq˜i1,…,ϕq˜in obeys(10)ϕq˜ik=sigpq˜ik,q˜ik≥Δ,pΔp−1q˜ik,q˜ik<Δ,∀k∈1,…,n,where 0<Δ≤1 and 0<p<1. Besides, for all k∈1,…,n, taking the derivative of ϕq˜ik with respect to q˜ik provides that(11)∇ϕq˜ik=∂ϕq˜ik∂q˜ik=pq˜ikp−1,q˜ik≥Δ,pΔp−1,q˜ik<Δ.Let∇ϕq˜i=col∇ϕq˜i1,…,∇ϕq˜in and ζq˜i=collq˜i1l−1,…,lq˜inl−1. The fixed-time estimator-based control algorithm is designed as(12)τi=τi0+τi1+τi2,τi0=Mi0qiq^¨i+Ci0qi,q˙iq˙i+gi0qi,τi1=−K0isigrsi−Mi0qiK1idiag∇ϕqi+K2idiagζq˜iq˜˙i,τi2=−sisiτaux,τaux=11−b2k+b0+b1q˙i2+b2τi0+τi1,(13)q^˙it=−η1t+1∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0+v^it−c1sgn∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0,v^˙it=−η2t+1∑j=1Naijv^it−sgnaijv^jt+ai0v^i−div0−c2sgn∑j=0Naijv^it−sgnaijv^jt+ai0v^i−div0,where (12) represents the local control layer and (13) stands for the estimator layer, K0i is the positive-definite diagonal matrix, r>1, k, c1 and c2 are positive constants, hmax=maxh1,…,hN, hmin=minh1,…,hN, and the TBG gains η1t and η2t are defined as(14)η1t=hmaxξ˙1t2λminMDH⊗In1−ξ1t+δ1,η2t=hmaxξ˙2t2λminMDH⊗In1−ξ2t+δ2,0<δ1,δ2≪1, ξ1tandξ2t denoting TBG and MD, and H is defined in Lemma 2.Taking the time derivative ofsi and multiplying both sides of the equation by Mi0q, we have(15)Mi0qis˙i=Mi0qiq˜¨i+Mi0qiK1idiag∇ϕq˜i+K2idiagζq˜iq˜˙i.Then, substitutingMi0qiq¨i from (6) in (15) yields(16)Mi0qis˙i=τi+ρt−Mi0qiq^¨i−C0qi,q˙iq˙i−g0qi+Mi0qiK1idiag∇ϕq˜i+K2idiagζq˜iq˜˙i.Denotee^it=q^it−diq0 and ε^it=v^it−div0. Substituting the proposed control input equation τi into (16) yields the following cascade closed-loop system:(17)s˙i=Mi0qi−1−k+b0+b1q˙i2+b2τi0+τi11−b2sisi+ρt−K0isigrsi,e^˙it=−η1t+1∑j=1Naije^it−sgnaije^jt+ai0e^i+ε^it−c1sgn∑j=1Naije^it−sgnaije^jt+ai0e^i,ε^˙it=−η2t+1∑j=1Naijε^it−sgnaijε^jt+ai0ε^i−c2sgn∑j=0Naijε^it−sgnaijε^jt+ai0ε^i−diu0t.Obviously, the fixed-time practical bipartite tracking problem for the NRSs is solved if the closed-loop system (17) is fixed-time stable.


### 3.2. Stability Analysis for the Distributed Estimator Layer

We firstly focus on the convergence analysis of the distributed estimator layer. It is worthy to point out two positive constantsTf1 and Tf2 can be arbitrarily determined by selecting proper TBGs. Besides, ξ1t and ξ2t are predefined in (14).The position error is represented in the form ofe^t=e^1Tt,e^2Tt,…,e^NTtT. The velocity error is in the form of ε^t=ε^1Tt,ε^2Tt,…,ε^NTtT. Let e¯t=D⊗Ine^t, ε¯t=D⊗Inε^t. Then, we can construct the following Lyapunov function candidate:(18)V1t=12MD⊗Ine¯tTMDH⊗In−1MD⊗Ine¯t,V2t=12MD⊗Inε¯tTMDH⊗In−1MD⊗Inε¯t.DenotezijNn×Nn=MDTH−1⊗In. Let zmin=minzii+Σj≠izij:zii+Σj≠izij≠0,∀i,j∈Nn,minzij:zij≠0,i≠j,∀i,j∈Nn.Theorem 1.
Suppose that Assumptions1-2 hold. If c1≥hmax/hmin2δ2/zmin1+δ2V20 and c2≥Numaxhmax/hmin, then estimator (13) yields that(19)limt⟶Tf1+Tf2e^it≤1N2δ1zmin1+δ1V10,limt⟶Tf2ε^it≤1N2δ2zmin1+δ2V20,(20)e^it≤1N2δ1zmin1+δ1V10,∀t>Tf1+Tf2,ε^it≤1N2δ2zmin1+δ2V20,∀t>Tf2,where δ1 and δ2 are presented in (14). It is equivalent to that q^i and v^i converge to an arbitrarily small neighborhood of the leader’s state bilaterally within a bounded convergence time Tf1+Tf2, where Tf1 and Tf2 are the user-designed parameters in the TBG gains η1 and η2 presented in (14), following the definition of gt and tf provided in Section 2.2.Proof.
The proof is processed in two steps. In the first step, according to (17), the compact form of the velocity error is obtained as ε^˙t=−η2tMs⊗Inε^t−Ms⊗Inε^t−c2sgnMs⊗Inε^t−D⊗In1N⊗u0. Since DD=IN, one can obtain that(21)ε¯˙=−η2t+1MD⊗Inε¯−c2sgnMD⊗Inε¯−1N⊗u0.
Next, for the given Lyapunov function candidateV2t, where the matrix H is defined in Lemma 2. Differentiating V2t along (21) yields(22)V˙2t=MD⊗Inε¯tTMDH⊗In−1MD⊗Inε¯˙t=−η2t+1MD⊗Inε¯TH−1⊗InMD⊗Inε¯−MD⊗Inε¯TH−1⊗Inc2sgnMD⊗Inε¯−1N⊗u0≤−η2t+1hmax−1MD⊗Inε¯2−c2hmax−1MD⊗Inε¯+hmax−1MD⊗Inε¯1N⊗u0≤−2η2thmax−1λminMDH⊗InV2t+Numaxhmax−1−c2hmax−1MD⊗Inε¯≤−ξ˙2t1−ξ2t+δ2V2t.
Fort∈0,Tf2, by using Lemma 3, V2t converges to a residual set Ω2=V2t|V2t≤δ2/1+δ2V20 at a prescribed time Tf2 independent of initial states of robots, so limt⟶Tf2ε¯it≤1/N2δ2/zmin1+δ2V20, which is equal to limt⟶Tf2ε^it≤1/N2δ2/zmin1+δ2V20. It indicates that the velocity error can converge to a desired level so long as δ2 is chosen properly in protocol. Moreover, ε^it≤1/N2δ2/zmin1+δ2V20,∀t>Tf2 and limt⟶+∞ε^it=0. That is, v^i⟶v0,∀i∈V+;v^i⟶−v0,∀i∈V−.
In the second step, according to (17), the compact form of the position error is expressed as e^˙t=−η1tMs⊗Ine^t−Ms⊗Ine^t+v^t−D⊗In1N⊗v0−c1sgnMs⊗Ine^. Then, one may further obtains that(23)e¯˙t=−η1t+1MD⊗Ine¯+D⊗Inv^−D⊗In1N⊗v0−c1sgnMD⊗Ine¯.
Next, for the given following Lyapunov function candidateV1t, differentiating V1t along (23) yields that(24)V˙1t=MD⊗Ine¯tTMDH⊗In−1MD⊗Ine¯˙t=−η1t+1MD⊗Ine¯TH−1⊗InMD⊗Ine¯+MD⊗Ine¯TD⊗Inv^−D⊗In1N⊗v0−c1sgnMD⊗Ine¯≤−η1thmax−1MD⊗Ine¯2+hmax−1ε¯MD⊗Ine¯−c1hmax−1MD⊗Ine¯≤−2η1thmax−1λminMDH⊗InV1t+ε¯hmax−1−c1hmax−1MD⊗Ine¯≤−ξ˙1t1−ξ1t+δ1V1t.
It thus follows from (22) that ε^it≤1/N2δ2/zmin1+δ2V20, ∀t>Tf2, and the practical convergence error is disposed by sign function. For t∈Tf2,Tf1+Tf2, using Lemma 3, one draws a conclusion that V1t decreases to a set of residuals Ω1=V1t|V1t≤δ1/1+δ1V10 at Tf1+Tf2 that can be attained independent with initial states. Furthermore, one obtains that limt⟶Tf1+Tf2e¯it≤1/N2δ1/zmin1+δ1V10, which is equal to limt⟶Tf1+Tf2e^it≤1/N2δ1/zmin1+δ1V10. This means that the position state error can be predesigned as a required level so long as an appropriate δ1 is applied in η1t. Hence, e^it≤1/N2δ1/zmin1+δ1V10, ∀t>Tf1+Tf2 and limt⟶+∞e^it=0. That is, q^i⟶q0,∀i∈V+;q^i⟶−q0,∀i∈V−. This completes the proof.Remark 4.
From Lemma3, it obtains that the solution of (2) is xt=x01−ξt/1+δ. According to property of ξt, it implies that ξtf=1; it thus obtains that limt⟶tfxt≤δ/1+δx0. When δ is small enough, the influence of the initial state is negligible. It thus implies that xt=0 when t≥tf, namely, the fixed-time convergence can be achieved by using the TBG-based approach.Remark 5.
It is worth noting that the closed-loop systemsi=q˙i−v^i+δr+K1iϕq˜i+K2isiglq˜i, where δr is the tiny error in the convergence procedure. Based on the above analysis, q^˙i can be approximately replaced by v^i as t>Tf1+Tf2. It is unnecessary that we employ δr in the stability analysis. Therefore, we employ the simplified closed-loop system si=q˙i−v^i+K1iϕq˜i+K2isiglq˜i in the subsequent analysis for avoiding redundant proof.


### 3.3. Stability Analysis for the Fixed-Time Practical Bipartite Tracking for the NRSs

In this section, the stability analysis for the proposed fixed-time estimator-based control algorithm is studied.Theorem 2.
Considering the robot dynamic system with parametric uncertainties and input disturbances described by (4), the input torque τi ensures that q˜i and q˜˙i globally converge to an user-defined small set Δ within a fixed time Ts+Te, where Ts and Te is derived as follows:(25)Ts≤2m1+m21k+nr−1/2r−1λminK0im1+m2r,(26)Te≤21−p/21−pλminK1i+2n1−l/2l−1λminK2i.This implies that the fixed-time practical bipartite tracking problem can be solved in the fixed timeT≤Ts+Te+Tf1+Tf2, where Tf1 and Tf2 are user-defined parameters, as presented in Theorem 1.Proof.
The Lyapunov function candidate was chosen as follows:(27)V=12siTMi0qisi.
Taking the derivative of V with respect to time along (16), we obtain(28)V˙=siTMi0qisi..
After substitutings˙i from the first equation of (17) into (28),(29)V˙=siT−k+b0+b1q˙i2+b2τi0+τi11−b2sisi+ρ−K0isigrsi=−k+b0+b1q˙i2+b2τi0+τi11−b2sisiTsi+siTρ−siTK0isigrsi=−k+b0+b1q˙i2+b2τi0+τi11−b2si+siTρ−siTK0isigrsi=−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+siTρ−siTK0isigrsi≤−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+siρ−siTK0isigrsi.
Denotesij as the jth element of the si; substituting (7) into (29), it follows that(30)V˙≤−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+sib0+b1q˙i2+b2τi−siTK0isigrsi≤−siTK0isigrsi−ksi−b2siτaux+b2siτi2≤−siTK0isigrsi−ksi≤−λminK0i∑j=1nsijr+1−ksi≤−n1−r/2λminK0isir+1−ksi.
Suppose thatm1≤Mi0−1qi≤m2. Choosing Mi0=2/m1+m2In [45], we have(31)V=1m1+m2si2.
Applying (31) to (30), it follows that(32)V˙+km1+m2V+n1−r/2λminK0im1+m2r+1/2Vr+1/2≤0.
By Lemma4, it obtains that si converge to zero when t⟶Ts, which is predefined in (25). After the sliding surface si=0 is achieved within Ts, the system dynamics are converted to q˜˙i=−K1iϕq˜i−K2isiglq˜i. Then, the convergence of q˜i can be settled by the following two cases.Case 1q˜i>Δ: the dynamics can be explicitly simplified as q˜˙i=−K1isigpq˜i−K2isiglq˜i. Choose the Lyapunov function as follows: Vca1=1/2q˜iTq˜i; the time derivation of Vca1 is(33)V˙ca1=q˜iTq˜˙i=−q˜iTK1isigpq˜i−q˜iTK2isiglq˜i≤−λminK1iq˜ip+1−n1−l/2λminK2iq˜il+1≤−2p+1/2λminK1iVca1p+1/2−21+l/2n1−l/2λminK2iVca11+l/2.Invoking Lemma4 again, the tracking error q˜ converges to the arbitrary small Δ within the time Te given by (26).Case 2q˜i<Δ: applying (10), the dynamics can be definitely simplified as q˜˙i=−pΔp−1K1iq˜i−K2isiglq˜i. Selecting the appropriate Lyapunov function candidate defined by Case 1, we obtain(34)V˙ca2=−pΔp−1q˜iTK1iq˜i−q˜iTK2isiglq˜i≤−2pΔp−1λminK1iVca2.
It is obvious seen that the tracking errorsq˜i exponentially converge to zero.Remark 6.
It is of great significance that we introduce a linear term in the functionϕq˜i for the case of q˜i<Δ. Otherwise, the control design with ϕq˜i may generate singularity problem due to the first derivative of ϕq˜i including a negative power p−1 on q˜i, which may go to infinity while x goes to zero. This may lead to the desired control cost infinity.Remark 7.
The term of−η1t+1∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0 in protocol (13) is TBG-based, which guarantees that the position and velocity error finally converge to a required boundary within the prescribed time. The TBG-based protocol generates a benign margin for convergence time, which has practicability in projects.


## 3.1. Fixed-Time Estimator-Based Control Algorithm

In this section, the fixed-time estimator-based control algorithm for achieving practical bipartite tracking of the NRS is proposed. Before constructing the control input, we design the following sliding-mode surface:(9)si=q˜˙i+K1iϕq˜i+K2isiglq˜i,where q˜i=qi−q^i, q˜˙i=q˙i−q^˙i, q^i and q^˙i are the estimated states with respect to qi and q˙i∈ℛn, K1i,K2i∈ℛn×n are positive-definite diagonal matrixes, l>1 is a positive constant, and ϕq˜i=ϕq˜i1,…,ϕq˜in obeys(10)ϕq˜ik=sigpq˜ik,q˜ik≥Δ,pΔp−1q˜ik,q˜ik<Δ,∀k∈1,…,n,where 0<Δ≤1 and 0<p<1. Besides, for all k∈1,…,n, taking the derivative of ϕq˜ik with respect to q˜ik provides that(11)∇ϕq˜ik=∂ϕq˜ik∂q˜ik=pq˜ikp−1,q˜ik≥Δ,pΔp−1,q˜ik<Δ.Let∇ϕq˜i=col∇ϕq˜i1,…,∇ϕq˜in and ζq˜i=collq˜i1l−1,…,lq˜inl−1. The fixed-time estimator-based control algorithm is designed as(12)τi=τi0+τi1+τi2,τi0=Mi0qiq^¨i+Ci0qi,q˙iq˙i+gi0qi,τi1=−K0isigrsi−Mi0qiK1idiag∇ϕqi+K2idiagζq˜iq˜˙i,τi2=−sisiτaux,τaux=11−b2k+b0+b1q˙i2+b2τi0+τi1,(13)q^˙it=−η1t+1∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0+v^it−c1sgn∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0,v^˙it=−η2t+1∑j=1Naijv^it−sgnaijv^jt+ai0v^i−div0−c2sgn∑j=0Naijv^it−sgnaijv^jt+ai0v^i−div0,where (12) represents the local control layer and (13) stands for the estimator layer, K0i is the positive-definite diagonal matrix, r>1, k, c1 and c2 are positive constants, hmax=maxh1,…,hN, hmin=minh1,…,hN, and the TBG gains η1t and η2t are defined as(14)η1t=hmaxξ˙1t2λminMDH⊗In1−ξ1t+δ1,η2t=hmaxξ˙2t2λminMDH⊗In1−ξ2t+δ2,0<δ1,δ2≪1, ξ1tandξ2t denoting TBG and MD, and H is defined in Lemma 2.Taking the time derivative ofsi and multiplying both sides of the equation by Mi0q, we have(15)Mi0qis˙i=Mi0qiq˜¨i+Mi0qiK1idiag∇ϕq˜i+K2idiagζq˜iq˜˙i.Then, substitutingMi0qiq¨i from (6) in (15) yields(16)Mi0qis˙i=τi+ρt−Mi0qiq^¨i−C0qi,q˙iq˙i−g0qi+Mi0qiK1idiag∇ϕq˜i+K2idiagζq˜iq˜˙i.Denotee^it=q^it−diq0 and ε^it=v^it−div0. Substituting the proposed control input equation τi into (16) yields the following cascade closed-loop system:(17)s˙i=Mi0qi−1−k+b0+b1q˙i2+b2τi0+τi11−b2sisi+ρt−K0isigrsi,e^˙it=−η1t+1∑j=1Naije^it−sgnaije^jt+ai0e^i+ε^it−c1sgn∑j=1Naije^it−sgnaije^jt+ai0e^i,ε^˙it=−η2t+1∑j=1Naijε^it−sgnaijε^jt+ai0ε^i−c2sgn∑j=0Naijε^it−sgnaijε^jt+ai0ε^i−diu0t.Obviously, the fixed-time practical bipartite tracking problem for the NRSs is solved if the closed-loop system (17) is fixed-time stable.


## 3.2. Stability Analysis for the Distributed Estimator Layer

We firstly focus on the convergence analysis of the distributed estimator layer. It is worthy to point out two positive constantsTf1 and Tf2 can be arbitrarily determined by selecting proper TBGs. Besides, ξ1t and ξ2t are predefined in (14).The position error is represented in the form ofe^t=e^1Tt,e^2Tt,…,e^NTtT. The velocity error is in the form of ε^t=ε^1Tt,ε^2Tt,…,ε^NTtT. Let e¯t=D⊗Ine^t, ε¯t=D⊗Inε^t. Then, we can construct the following Lyapunov function candidate:(18)V1t=12MD⊗Ine¯tTMDH⊗In−1MD⊗Ine¯t,V2t=12MD⊗Inε¯tTMDH⊗In−1MD⊗Inε¯t.DenotezijNn×Nn=MDTH−1⊗In. Let zmin=minzii+Σj≠izij:zii+Σj≠izij≠0,∀i,j∈Nn,minzij:zij≠0,i≠j,∀i,j∈Nn.Theorem 1.
Suppose that Assumptions1-2 hold. If c1≥hmax/hmin2δ2/zmin1+δ2V20 and c2≥Numaxhmax/hmin, then estimator (13) yields that(19)limt⟶Tf1+Tf2e^it≤1N2δ1zmin1+δ1V10,limt⟶Tf2ε^it≤1N2δ2zmin1+δ2V20,(20)e^it≤1N2δ1zmin1+δ1V10,∀t>Tf1+Tf2,ε^it≤1N2δ2zmin1+δ2V20,∀t>Tf2,where δ1 and δ2 are presented in (14). It is equivalent to that q^i and v^i converge to an arbitrarily small neighborhood of the leader’s state bilaterally within a bounded convergence time Tf1+Tf2, where Tf1 and Tf2 are the user-designed parameters in the TBG gains η1 and η2 presented in (14), following the definition of gt and tf provided in Section 2.2.Proof.
The proof is processed in two steps. In the first step, according to (17), the compact form of the velocity error is obtained as ε^˙t=−η2tMs⊗Inε^t−Ms⊗Inε^t−c2sgnMs⊗Inε^t−D⊗In1N⊗u0. Since DD=IN, one can obtain that(21)ε¯˙=−η2t+1MD⊗Inε¯−c2sgnMD⊗Inε¯−1N⊗u0.
Next, for the given Lyapunov function candidateV2t, where the matrix H is defined in Lemma 2. Differentiating V2t along (21) yields(22)V˙2t=MD⊗Inε¯tTMDH⊗In−1MD⊗Inε¯˙t=−η2t+1MD⊗Inε¯TH−1⊗InMD⊗Inε¯−MD⊗Inε¯TH−1⊗Inc2sgnMD⊗Inε¯−1N⊗u0≤−η2t+1hmax−1MD⊗Inε¯2−c2hmax−1MD⊗Inε¯+hmax−1MD⊗Inε¯1N⊗u0≤−2η2thmax−1λminMDH⊗InV2t+Numaxhmax−1−c2hmax−1MD⊗Inε¯≤−ξ˙2t1−ξ2t+δ2V2t.
Fort∈0,Tf2, by using Lemma 3, V2t converges to a residual set Ω2=V2t|V2t≤δ2/1+δ2V20 at a prescribed time Tf2 independent of initial states of robots, so limt⟶Tf2ε¯it≤1/N2δ2/zmin1+δ2V20, which is equal to limt⟶Tf2ε^it≤1/N2δ2/zmin1+δ2V20. It indicates that the velocity error can converge to a desired level so long as δ2 is chosen properly in protocol. Moreover, ε^it≤1/N2δ2/zmin1+δ2V20,∀t>Tf2 and limt⟶+∞ε^it=0. That is, v^i⟶v0,∀i∈V+;v^i⟶−v0,∀i∈V−.
In the second step, according to (17), the compact form of the position error is expressed as e^˙t=−η1tMs⊗Ine^t−Ms⊗Ine^t+v^t−D⊗In1N⊗v0−c1sgnMs⊗Ine^. Then, one may further obtains that(23)e¯˙t=−η1t+1MD⊗Ine¯+D⊗Inv^−D⊗In1N⊗v0−c1sgnMD⊗Ine¯.
Next, for the given following Lyapunov function candidateV1t, differentiating V1t along (23) yields that(24)V˙1t=MD⊗Ine¯tTMDH⊗In−1MD⊗Ine¯˙t=−η1t+1MD⊗Ine¯TH−1⊗InMD⊗Ine¯+MD⊗Ine¯TD⊗Inv^−D⊗In1N⊗v0−c1sgnMD⊗Ine¯≤−η1thmax−1MD⊗Ine¯2+hmax−1ε¯MD⊗Ine¯−c1hmax−1MD⊗Ine¯≤−2η1thmax−1λminMDH⊗InV1t+ε¯hmax−1−c1hmax−1MD⊗Ine¯≤−ξ˙1t1−ξ1t+δ1V1t.
It thus follows from (22) that ε^it≤1/N2δ2/zmin1+δ2V20, ∀t>Tf2, and the practical convergence error is disposed by sign function. For t∈Tf2,Tf1+Tf2, using Lemma 3, one draws a conclusion that V1t decreases to a set of residuals Ω1=V1t|V1t≤δ1/1+δ1V10 at Tf1+Tf2 that can be attained independent with initial states. Furthermore, one obtains that limt⟶Tf1+Tf2e¯it≤1/N2δ1/zmin1+δ1V10, which is equal to limt⟶Tf1+Tf2e^it≤1/N2δ1/zmin1+δ1V10. This means that the position state error can be predesigned as a required level so long as an appropriate δ1 is applied in η1t. Hence, e^it≤1/N2δ1/zmin1+δ1V10, ∀t>Tf1+Tf2 and limt⟶+∞e^it=0. That is, q^i⟶q0,∀i∈V+;q^i⟶−q0,∀i∈V−. This completes the proof.Remark 4.
From Lemma3, it obtains that the solution of (2) is xt=x01−ξt/1+δ. According to property of ξt, it implies that ξtf=1; it thus obtains that limt⟶tfxt≤δ/1+δx0. When δ is small enough, the influence of the initial state is negligible. It thus implies that xt=0 when t≥tf, namely, the fixed-time convergence can be achieved by using the TBG-based approach.Remark 5.
It is worth noting that the closed-loop systemsi=q˙i−v^i+δr+K1iϕq˜i+K2isiglq˜i, where δr is the tiny error in the convergence procedure. Based on the above analysis, q^˙i can be approximately replaced by v^i as t>Tf1+Tf2. It is unnecessary that we employ δr in the stability analysis. Therefore, we employ the simplified closed-loop system si=q˙i−v^i+K1iϕq˜i+K2isiglq˜i in the subsequent analysis for avoiding redundant proof.


## 3.3. Stability Analysis for the Fixed-Time Practical Bipartite Tracking for the NRSs

In this section, the stability analysis for the proposed fixed-time estimator-based control algorithm is studied.Theorem 2.
Considering the robot dynamic system with parametric uncertainties and input disturbances described by (4), the input torque τi ensures that q˜i and q˜˙i globally converge to an user-defined small set Δ within a fixed time Ts+Te, where Ts and Te is derived as follows:(25)Ts≤2m1+m21k+nr−1/2r−1λminK0im1+m2r,(26)Te≤21−p/21−pλminK1i+2n1−l/2l−1λminK2i.This implies that the fixed-time practical bipartite tracking problem can be solved in the fixed timeT≤Ts+Te+Tf1+Tf2, where Tf1 and Tf2 are user-defined parameters, as presented in Theorem 1.Proof.
The Lyapunov function candidate was chosen as follows:(27)V=12siTMi0qisi.
Taking the derivative of V with respect to time along (16), we obtain(28)V˙=siTMi0qisi..
After substitutings˙i from the first equation of (17) into (28),(29)V˙=siT−k+b0+b1q˙i2+b2τi0+τi11−b2sisi+ρ−K0isigrsi=−k+b0+b1q˙i2+b2τi0+τi11−b2sisiTsi+siTρ−siTK0isigrsi=−k+b0+b1q˙i2+b2τi0+τi11−b2si+siTρ−siTK0isigrsi=−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+siTρ−siTK0isigrsi≤−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+siρ−siTK0isigrsi.
Denotesij as the jth element of the si; substituting (7) into (29), it follows that(30)V˙≤−b2siτaux−sik+b0+b1q˙i2+b2τi0+τi1+sib0+b1q˙i2+b2τi−siTK0isigrsi≤−siTK0isigrsi−ksi−b2siτaux+b2siτi2≤−siTK0isigrsi−ksi≤−λminK0i∑j=1nsijr+1−ksi≤−n1−r/2λminK0isir+1−ksi.
Suppose thatm1≤Mi0−1qi≤m2. Choosing Mi0=2/m1+m2In [45], we have(31)V=1m1+m2si2.
Applying (31) to (30), it follows that(32)V˙+km1+m2V+n1−r/2λminK0im1+m2r+1/2Vr+1/2≤0.
By Lemma4, it obtains that si converge to zero when t⟶Ts, which is predefined in (25). After the sliding surface si=0 is achieved within Ts, the system dynamics are converted to q˜˙i=−K1iϕq˜i−K2isiglq˜i. Then, the convergence of q˜i can be settled by the following two cases.Case 1q˜i>Δ: the dynamics can be explicitly simplified as q˜˙i=−K1isigpq˜i−K2isiglq˜i. Choose the Lyapunov function as follows: Vca1=1/2q˜iTq˜i; the time derivation of Vca1 is(33)V˙ca1=q˜iTq˜˙i=−q˜iTK1isigpq˜i−q˜iTK2isiglq˜i≤−λminK1iq˜ip+1−n1−l/2λminK2iq˜il+1≤−2p+1/2λminK1iVca1p+1/2−21+l/2n1−l/2λminK2iVca11+l/2.Invoking Lemma4 again, the tracking error q˜ converges to the arbitrary small Δ within the time Te given by (26).Case 2q˜i<Δ: applying (10), the dynamics can be definitely simplified as q˜˙i=−pΔp−1K1iq˜i−K2isiglq˜i. Selecting the appropriate Lyapunov function candidate defined by Case 1, we obtain(34)V˙ca2=−pΔp−1q˜iTK1iq˜i−q˜iTK2isiglq˜i≤−2pΔp−1λminK1iVca2.
It is obvious seen that the tracking errorsq˜i exponentially converge to zero.Remark 6.
It is of great significance that we introduce a linear term in the functionϕq˜i for the case of q˜i<Δ. Otherwise, the control design with ϕq˜i may generate singularity problem due to the first derivative of ϕq˜i including a negative power p−1 on q˜i, which may go to infinity while x goes to zero. This may lead to the desired control cost infinity.Remark 7.
The term of−η1t+1∑j=1Naijq^it−sgnaijq^jt+ai0q^i−diq0 in protocol (13) is TBG-based, which guarantees that the position and velocity error finally converge to a required boundary within the prescribed time. The TBG-based protocol generates a benign margin for convergence time, which has practicability in projects.


## 4. Simulation Results

In this section, simulation experiments are carried out to verify the effectiveness of the proposed algorithm. Assume that the NRS contains eight two-DOF robotic manipulators with interactions displayed in Figure3.Figure 3
The directed signed graph.The communication mode includes one leader and two subnetworks. There only exist cooperative relationships among the robots in each subnetwork by the solid lines, while there are the competitive relationships between the two subnetworks by the dotted lines. Besides, there are an even number of negative weights in each loop. It thus obtained that Assumption1 holds. In addition, bidirection communication between two vertexes stands for different information delivered between two robots.Furthermore, we propose a simple application of our control strategy, in which two parts of robots cooperate and compete with each other to avoid obstacles in the task. In the phase 1, the robots cooperate with each other and track the target before detecting the obstacle. In the phase 2, when the robots detect obstacles, the eight robots which have both cooperative and competitive relationships can be split into two clusters to bypass the obstacle. In the phase 3, the robots cooperate with each other and track the target again after passing the obstacle.In the practical application, the communication data are determined by the hardware setting of the communication system. However, in the theoretical research, we need to employ the communication data of the directed signed graph to verify the effective of the control algorithm.The physical parameters of the eight robotic manipulators are listed in Table1, where mk,k∈1,2, stands for the masses of links, lk represents the lengths of links, rk is the center of the link’s mass, and Jk represents the moment of inertia of links.Table 1
The physical parameters of the eight robotic manipulators.
ith robotmkkglkmrkmJkkg⋅m211.32, 1.082.76, 2.481.38, 1.240.84, 0.5521.23, 1.022.64, 2.421.32, 1.210.71, 0.5031.26, 1.042.68, 2.441.34, 1.220.75, 0.5241.32, 1.082.76, 2.481.38, 1.240.84, 0.5551.23, 1.022.64, 2.421.32, 1.210.71, 0.5061.26, 1.042.68, 2.441.34, 1.220.75, 0.5271.32, 1.082.76, 2.481.38, 1.240.84, 0.5581.23, 1.022.64, 2.421.32, 1.210.71, 0.50The initial conditions of robots are chosen asq0=q10T,q20TT, v0=v10T,v20TT, q^0=q^10T,q^20TT, and v^0=v^10T,v^20TT, where(35)q10=3,−2,−2,3,−2,−2,3,−2T,q20=−2,3,−1,2,3,−1,−2,3T,v10=−1,−3,2,−1,−3,2,−1,−3T,v20=−3,6,−5,−7,3,1,−2,7T,q^10=1,2,−3,1,2,−3,1,2T,q^20=−1,8,5,−7,1,−3,2,4T,v^10=−1,−3,2,−1,−3,2,−1,−3T,v^20=2,2,−3,2,2,−3,2,2T.The position, velocity, and acceleration states of the leader are, respectively, selected asq0=sint;sintT, v0=cost;costT, and u0=−sint;−sintT. The control parameters are given as p=0.5,q=1.2,r=1.5,k=1,b0=12,b1=2.8,b2=0.5,δ1=δ2=0.01, c1=0.01,c2=1, K0i=diag4,3, K1i=diag2,2, and K2i=diag2,2. The input disturbance dit=0.1cost;sintT. From Figure 1, the Laplacian matrix Ls is(36)Ls=300−2100002.5−0.5−101000−1100000−0.5−0.50200100.50001.500−100.50002.5−200000.50−12−0.50000−0.50−11.5,where B=diag0,1,0,0,0,1,0,0 and H=diag2,1,2,0.5,1,0.5,0.25,0.5.Based on the aforementioned parameter setting, the simulation results of the proposed algorithms are shown in Figures4–9. From Figures 4 and 5, it is shown that the robots’ position and velocity states are divided into two groups to track the leader’s trajectory, where robots belonging to subnetwork A converge to the states of the leader within the predesigned time, while the robots belonging to subnetwork B approach to the opposite state of the leader within the predesigned time. In Figures 6–8, the tracking errors are, respectively, provided to illustrated the tracking performances of the position states. From Figure 7, the estimators q^i can track the neighbourhood of leader state q0 within Tf1+Tf2, which proves the effectiveness of the estimator layer. From Figure 8, the true value qi can converge to the neighbourhood of estimators q^i within Ts+Te, which demonstrates the effectiveness of the local control layer. In Figures 6 and 9, the tracking errors ei and e˙i converge to the neighbourhood of the origin within the convergence time T, which indicate the effectiveness of the proposed algorithms.Figure 4
Pictures (a) and (b) reveal the evolution ofqi for links 1 and 2, respectively. Pictures (c) and (d) reveal the evolution of q^i for links 1 and 2, respectively.
(a)(b)(c)(d)Figure 5
Pictures (a) and (b) reveal the evolution ofvi for links 1 and 2, respectively. Pictures (c) and (d) reveal the evolution of v^i for links 1 and 2, respectively.
(a)(b)(c)(d)Figure 6
Pictures (a) and (b) reveal the evolution of the position tracking errors for links 1 and 2.
(a)(b)Figure 7
Pictures (a) and (b) reveal the evolution of the errorse^i for links 1 and 2.
(a)(b)Figure 8
Pictures (a) and (b) reveal the evolution of the errorsq˜i for links 1 and 2.
(a)(b)Figure 9
Pictures (a) and (b) reveal the evolution of the velocity tracking errors for links 1 and 2.
(a)(b)


## 5. Conclusion

In this paper, the fixed-time practical bipartite tracking problem for the NRSs has been investigated under parametric uncertainties, input disturbances, and directed signed graph. A new fixed-time estimator-based control algorithm is proposed to solve this problem. Some necessary conditions are established. The validity of the proposed algorithm was demonstrated in the simulation graph. Future work will concentrate on bipartite tracking problem with uniform time delay and input saturation.



---

*Source: 6593015-2021-05-27.xml*