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.

1. Introduction

Recently, the cooperative control of the networked robotic systems (NRSs) [14] 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 [711], 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 [2628] and bipartite tracking [2932] 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 [3642]. 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 [2932], 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. Section 2 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: let be the real matrix, be the real matrix, be the diagonal matrix, and be the sign function. is equal to with . is the column vector of the dimension. is the dimension identity matrix. and are the maximum and minimum values of the given vector. 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 graph , where is the set of vertexes, is the set of edges, and is the adjacency matrix. An edge (i.e., ) implies that the communication information flows directly from the vertex to the vertex , otherwise . implies that the vertex cooperates with the vertex ; implies that the vertex competes with the vertex . Furthermore, assume that has no self-loops, i.e., . A directed path is denoted as a series of edges from with distinct vertexes and the length . A cycle of is denoted as the starting and ending vertexes of the path being the same, namely, . 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 is defined as [26], where and , if . A directed signed graph is detail-balanced if there exist positive constants such that . A diagonal weighted matrix represents the connection weight between the rooted robot and other robots. In detail, implies that the th robot can directly receive the information from the rooted robot, otherwise .

Definition 1. (see [26]). The directed signed graph is said to be structurally balanced if can be grouped into two sets and satisfying , and , .

Assumption 1. The directed signed graph is structurally balanced and detail-balanced. The augmented graph (including the graph 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 such that is positive semidefinite, where if and if .

Lemma 2. (see [44]). Let and . If Assumption 1 holds, there exists a positive-definite diagonal matrix such that the matrix 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:where is the TBG and . For any given , we can design a proper TBG such that(1) is at least on (2) is continuous and nondecreasing from to , where is a scheduled time constant(3), where the derivative of at is actually its right derivative(4) and if

Lemma 3. (see [40]). Considering the following differential system,where TBG gain is defined in the form of (1) and denotes the state. Then, there exists a positive constant with respect to in (1) such that and on , where is given in (1).

Remark 1. An example of the TBG is presented as follows [40]:where 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 . Then, the main techniques to design the TBG function is to satisfy four properties given in (1). A typical TBG function with is shown in Figures 1 and 2 to enhance its visualization.

2.3. System Formulation

The dynamics of the th robot in the NRS is presented below [36]:where , , are, respectively, the generalized position, velocity, and acceleration, stands for the positive-definite inertia matrix, is the centrifugal-Coriolis matrix, represents the gravitational term, is the input disturbances satisfying , is a positive constant, and denotes the control input.

The parameters of the dynamical model can be described aswhere , , and are the desired parts of the dynamic model and , , and are the uncertain term of the dynamic model. Therefore, system (4) can be rewritten aswhere . Following [36], is upper bounded, namely,where and are positive constants and . On the contrary, the leader’s states , , and obey that and .

Assumption 2. The acceleration of the leader is upper bounded, i.e., .

2.4. Problem Formulation

The control objective is to design a proper control input for achieving the practical bipartite tracking of the NRS in a fixed time. Denote and 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 constant regardless of the initial value of the states such that , , , and on , , and , , , and on , , where can be sufficiently small by choosing appropriate control parameters.

Lemma 4. (see [33]). For the nonlinear system . Suppose that there exist a positive-definite and continuous function , real numbers , and such that . Then, the origin is the fixed-time stable equilibrium point of the considered system. The setting time is given as

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:where , , and are the estimated states with respect to and , are positive-definite diagonal matrixes, is a positive constant, and obeyswhere and . Besides, for all , taking the derivative of with respect to provides that

Let and . The fixed-time estimator-based control algorithm is designed aswhere (12) represents the local control layer and (13) stands for the estimator layer, is the positive-definite diagonal matrix, , , and are positive constants, , , and the TBG gains and are defined as, denoting TBG and , and is defined in Lemma 2.

Taking the time derivative of and multiplying both sides of the equation by , we have

Then, substituting from (6) in (15) yields

Denote and . Substituting the proposed control input equation into (16) yields the following cascade closed-loop system:

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 constants and can be arbitrarily determined by selecting proper TBGs. Besides, and are predefined in (14).

The position error is represented in the form of . The velocity error is in the form of . Let , . Then, we can construct the following Lyapunov function candidate:

Denote . Let .

Theorem 1. Suppose that Assumptions 1-2 hold. If and , then estimator (13) yields thatwhere and are presented in (14). It is equivalent to that and converge to an arbitrarily small neighborhood of the leader’s state bilaterally within a bounded convergence time , where and are the user-designed parameters in the TBG gains and presented in (14), following the definition of and 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 . Since , one can obtain thatNext, for the given Lyapunov function candidate , where the matrix is defined in Lemma 2. Differentiating along (21) yieldsFor , by using Lemma 3, converges to a residual set at a prescribed time independent of initial states of robots, so , which is equal to . It indicates that the velocity error can converge to a desired level so long as is chosen properly in protocol. Moreover, and . That is, .
In the second step, according to (17), the compact form of the position error is expressed as . Then, one may further obtains thatNext, for the given following Lyapunov function candidate , differentiating along (23) yields thatIt thus follows from (22) that , , and the practical convergence error is disposed by sign function. For , using Lemma 3, one draws a conclusion that decreases to a set of residuals at that can be attained independent with initial states. Furthermore, one obtains that , which is equal to . This means that the position state error can be predesigned as a required level so long as an appropriate is applied in . Hence, , and . That is, . This completes the proof.

Remark 4. From Lemma 3, it obtains that the solution of (2) is . According to property of , it implies that ; it thus obtains that . When is small enough, the influence of the initial state is negligible. It thus implies that when , namely, the fixed-time convergence can be achieved by using the TBG-based approach.

Remark 5. It is worth noting that the closed-loop system , where is the tiny error in the convergence procedure. Based on the above analysis, can be approximately replaced by as . It is unnecessary that we employ in the stability analysis. Therefore, we employ the simplified closed-loop system 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 ensures that and globally converge to an user-defined small set within a fixed time , where and is derived as follows:

This implies that the fixed-time practical bipartite tracking problem can be solved in the fixed time , where and are user-defined parameters, as presented in Theorem 1.

Proof. The Lyapunov function candidate was chosen as follows:Taking the derivative of V with respect to time along (16), we obtainAfter substituting from the first equation of (17) into (28),Denote as the th element of the ; substituting (7) into (29), it follows thatSuppose that . Choosing [45], we haveApplying (31) to (30), it follows thatBy Lemma 4, it obtains that converge to zero when , which is predefined in (25). After the sliding surface is achieved within , the system dynamics are converted to . Then, the convergence of can be settled by the following two cases.Case 1 : the dynamics can be explicitly simplified as . Choose the Lyapunov function as follows: ; the time derivation of isInvoking Lemma 4 again, the tracking error converges to the arbitrary small within the time given by (26).Case 2 : applying (10), the dynamics can be definitely simplified as . Selecting the appropriate Lyapunov function candidate defined by Case 1, we obtainIt is obvious seen that the tracking errors exponentially converge to zero.

Remark 6. It is of great significance that we introduce a linear term in the function for the case of . Otherwise, the control design with may generate singularity problem due to the first derivative of including a negative power on , which may go to infinity while x goes to zero. This may lead to the desired control cost infinity.

Remark 7. The term of 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 Figure 3.

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 Assumption 1 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 Table 1, where , stands for the masses of links, represents the lengths of links, is the center of the link’s mass, and represents the moment of inertia of links.

The initial conditions of robots are chosen as , , , and , where

The position, velocity, and acceleration states of the leader are, respectively, selected as , , and . The control parameters are given as , , , , and . The input disturbance . From Figure 1, the Laplacian matrix iswhere