State-constrained bipartite tracking of interconnected robotic systems via hierarchical prescribed-performance control

This paper investigates the collaborative design problem aiming to achieve state-constrained bipartite tracking of the interconnected robotic systems (IRSs) with prescribed performance. In practical applications, the physical limitations of the robots are inevitable. Besides, it is difficult to ensure that the target trajectory is known for each robot of the IRSs in advance. Thus, it is important to follow the target trajectory and meanwhile obey the state constraint being generated from the physical limitations and the external environment of the IRSs. To this end, we propose a new hierarchical state-constrained estimator-based control framework with the characteristics of low computation complexity and high task adaptability. With limited accessibility of the target trajectory, we newly present the estimator to observe it at each time interval through the interconnections among the robots. The state constraint is never violated throughout the convergence process by using the presented control algorithm. The theoretical proof and simulation results are presented to validate the feasibility of the control framework.


Introduction
In the last decades, the multi-agent systems were widely used in designing unmanned mobile vehicles, unmanned marine surface vehicles and unmanned air vehicles due to their high performance [1,2]. Recently, the cooperation problems for networked robotic systems have become an academic hot spot due to their high autonomy, intelligence and flexibility [3,4]. In practical applications, the joint angles of the robots are limited by complex working conditions and physical limitations such that their states cannot exceed a certain range, namely, the states of the robots are constrained. In recent works, distributed consensus and formation control algorithms for multi-agent systems with state constraints have been studied [5][6][7]. Besides, some application-oriented strategies for dealing with state constraint have been presented. Discretetime robotic control for human-robot interactions with state constraint has been obtained [8]. Discrete-time control for double-integrator systems with state con-straint has been presented [9]. Resilient feedback control of high-speed trail with protection constraints has been designed [10]. Optimal power allocation with constraint of high-speed rail has been achieved [11]. However, there still lacks an effective method for controlling the interconnected robotic systems (IRSs) with state constraint and non-cooperative individuals.
Nevertheless, there are possibly cooperative and non-cooperative pairs of individuals in real-world multi-agent systems [12,13]. For example, the biological individuals cooperate with their companions and simultaneously compete with other individuals in order to pursue abundant resources and benefits. In modern manufacturing, the robots synchronously work on two opposite surfaces of the same workpiece to acquire the identical consequence. In cooperative and competitive systems, the final aim still lies on achieving consensus. Therefore, the consensus protocols in different cases are highly demanded, including resilient consensus, scaled consensus and other distributed tracking strategies. In [14], a projection-based resilient constrained consensus protocol has been presented for solving the problem of resilient constrained consensus. In [15], a gradient projection approach and distributed consensus protocols have been used to solve the scaled consensus problem for the multi-agent network. The problems of bipartite consensus, bipartite tracking and bipartite formation have been solved by describing the cooperation and competition individuals as a signed graph [16][17][18][19]. However, to the best knowledge of the authors, the control methods for achieving state-constrained bipartite tracking of the IRSs is still a blank.
Recently, the control methods with prescribed performance have attracted great attention, and the prescribed performance control has been used to guarantee that the errors of the states stay within a prescribed region [20,21]. The tracking performance is an important index in the field of tracking control. In various tracking missions including missile guidance and others based on accurate locations, how to guarantee the prescribed performance of the tracking process is an important issue. An adaptive sliding mode control with performance function has been proposed for regulating vehicular platoons [22]. A simple nonlogarithmic error mapping function has been presented to guarantee that the errors of the underwater vehicles are limited to a certain range [23]. In [24], the control algorithm for nonlinear strict-feedback multiagent systems has been proposed to ensure the pre-defined bounds of the overshoot, the convergence rates and the steady-state errors. A novel distributed adaptive control algorithm for uncertain higher-order nonlinear multi-agent systems has been proposed to ensure the uniform boundedness of all closed-loop signals without violation of the output constraints [25]. In [26], the distributed control problem with asymmetric timevarying output constraints has been solved for uncertain nonlinear multi-agent systems. However, all the abovementioned researches were mainly focused on the single physical systems without interconnected individuals or the multi-agent systems without considering physical constraints. Therefore, there still lacks a costeffective method for achieving prescribed performance of the interconnected robotic systems (IRSs) with state constraints.
In this work, a hierarchical prescribed-performance control method is proposed to achieve state-constrained bipartite tracking of the IRSs. The contributions of this paper are listed as follows.
• Different from the researches considering state constraints of the single physical systems [27,28], the states of the robots keep constrained within a certain range for the IRSs for the convergence process in this paper. That is, the states of each robot do not exceed the predefined upper bound and lower bound. • In contrast with the bipartite consensus problems considered in [29,30], the state-constrained bipartite tracking problem is successfully solved for the IRSs. • Compared with the existing work for obtaining prescribed performance of multi-agent systems [31,32], a hierarchical control frame is newly presented to guarantee that the tracking errors of the IRSs are limited to a prescribed range as well as the complexity and coupling of the IRSs are properly handled in the whole control process.
Besides, the differences between the presented result and other related results are summarized as follows. Different from [33], we consider the bipartite consensus behavior and guarantee the state constraints of the robotic systems. Unlike [34], we achieve the state constraints of the mechanical systems via prescribedperformance control. Furthermore, we ensure that the state constraints are never violated during convergence process to reduce the overshoot of the closed-loop system. Different from the other prescribed-performance algorithms for bipartite consensus presented in [35][36][37], we focus on the bipartite consensus problems of the IRSs with Lagrangian-type dynamics. In [35][36][37], for achieving prescribed performance bipartite consensus, the authors select a proper performance boundary function to regulate the tracking errors. Unlike the abovementioned works, we guarantee the boundedness of the tracking errors by combining the local tracking errors and the performance boundary function. In this way, the state constraints are never violated during the whole process. Meanwhile, the target trajectory can be estimated by the estimator-based algorithm.
The remaining parts are organized as follows. Section 2 provides the preliminaries including prerequisite knowledge and the problem formulations. In Sect. 3, the state-constrained estimator-based control algorithm and its stability analysis are proposed. The simulation results are presented in Sect. 4 to testify the effectiveness of the algorithm. Finally, the conclusion is summarized in Sect. 5.

Notations Explanations
diagonal matrix I n n dimension identity matrix max{·} maximum values of the given vector · the Euclidean norm sgn(·) the sign function Z + Set of positive integers

Graph theory
The interconnections of the IRSs can be modeled as a directed signed graph . . , N } are respectively the sets of vertexes, E ⊆ V × V is the set of edges, and A = [a i j ] ∈ R N ×N is the adjacency matrix. An edge e i j ∈ E (namely, a i j = 0) implies that the information flows directly from the vertex j to the vertex i; a i j = 0 otherwise. a i j > 0 implies that the vertex i has the cooperative relationship with the vertex j. a i j < 0 implies that the vertex i has the competitive relationship with the vertex j. Furthermore, G is said to be a strongly connected graph if there exists a path between any two distinct vertexes. Assume that G has no self-loops, namely, a ii = 0. A cycle of G is denoted as a path whose beginning vertex and ending vertex are the same one. Furthermore, a directed signed graph includes the directed spanning tree, in which there exists a rooted vertex having a directed path to each of the other vertexes. The Laplacian matrix of the G is defined as represents the connection weight between the leader (namely, the vertex 0) and the other vertexes. In detail, b i > 0 implies that the i-th robot can directly receive the information of the leader; otherwise b i = 0, ∀i ∈ V. Then we firstly present some preliminaries following [17].
Definition 1 A directed signed graph is structurally balanced if V can be divided into two sets V 1 and V 2 , satisfying the following conditions: 2) a i j > 0 implies that the vertexes i and j are connected and belong to the same set; 3) a i j < 0 implies that the connected vertexes belong to the different sets; 4) a i j = 0 implies that there are no connections between them.

Remark 1
The signed graph is used to describe the patterns of amity and enmity in [38]. Different from the network containing multiple groups, the network with two factions is considered, in which the relationship between the two groups is competition and the relationship within the same groups is cooperation.

Assumption 1
The directed signed graph G contains a directed spanning tree with the leader being the rooted vertex and is structurally balanced.

System specification
The IRSs include N robots being connected through an interconnected network. The model of the i-th robot is presented as follows.
where t ≥ 0, i ∈ V, M i (q i ) ∈ R n×n represents the positive-definite inertia matrix, q i ,q i ,q i ∈ R n are respectively the generalized position, velocity and acceleration, C i (q i ,q i ) ∈ R n×n stands for the centrifugal-Coriolis matrix, g i (q i ) ∈ R n represents the gravitational term, d i (t) ∈ R n is the input disturbance satisfying sup t≥0 d i (t) ≤ d M , d M is a positive constant, and τ i ∈ R n denotes the control input to be designed later. Then (2.1) can be rewritten as follows. where

Property 1
The inertia matrix is a symmetric and positive definite matrix, and satisfies On the other hand, the position q 0 and the velocity v 0 of the leader satisfy thatq 0 = v 0 and v 0 is bounded.

State constraints and decaying functions
Due to the fact that most of the mechanical systems in applications are constrained to specific physical boundaries and external environment, the joint constraints of the mechanical systems are taken into consideration. We then propose the asymmetric time-varying state constraints as follows.
and q uk (t) represent the lower bound and upper bound of the state constraints, respectively. The constraints of the robots can be identical or different. Without loss of generality, we consider that the constraints of different joints are different while the same joint of each robot has the same constraint. Then the following initial condition holds.

Assumption 2
In real-world applications, the boundary functions q lk (t) and q uk (t) are artificially predetermined and are specified to be continuous-time. Then we assume that the derivatives of the boundary functions q lk (t) and q uk (t) are bounded. In this way, there exist two constants a l and a u satisfying that Then, we present the following decaying function where ζ 0 > 0, ω > 0 and 0 < ζ ∞ < 1 are the preset parameters. It can be easily concluded that ζ(t) monotonically decreases with the decreasing rate e −ωt from the initial value ζ 0 to the final value ζ ∞ . Therefore, we present the following two decaying functions in order to constrain the position error and the velocity error.

Remark 2
The decaying function ζ(·) is used to guarantee that the tracking errorẽ i can be limited to the prescribed performance. Besides, it is decided by ζ 0 , ζ ∞ and ω satisfying that ζ 0 > 0, 0 < ζ ∞ < 1 and ω > 0. According to equation (2.7), the decaying function ζ(·) monotonically decays to ζ ∞ as time goes to infinity. Combing the decaying function with the boundaries q uk and q lk , we can obtain the boundaries ρ uk and ρ lk of the error e i . ζ ∞ is a constant we select in advance and it is invariable in the process. In other words, ζ ∞ is bounded, and the actual value of boundary of ζ ∞ equals to the initial value of ζ ∞ .

Problem formulation
The bipartite tracking control problem of the IRSs is solved if the position states q i (t) for all the robots finally reach two opposite directions of the leader q 0 (t).
For i ∈ V 1 , the position tracking error is defined as e i (t) = q i (t) − q 0 (t), and the velocity tracking error is defined asė For i ∈ V 2 , the position tracking error is defined as e i (t) = q i (t) + q 0 (t), and the velocity tracking error is defined asė The main control objective is to design the control input τ i such that the following goals can be achieved.
• The robots of the IRSs track the target trajectory of the leader within the prescribed boundaries, namely, where δ i is a selectable positive constant. • The state constraints are never violated during the convergence process, namely,

Estimator-based bipartite tracking algorithm
In this subsection, the estimator-based bipartite tracking algorithm is proposed to solve the tracking problem of the IRSs with the aforementioned state constraints.
Considering that only partial robots can directly communicate with the leader, the following estimator is designed in a distributed way to estimate the states of the leader.
where α 1 , α 2 > 0 is the control gain,q i andq j are the estimated states of the robots i and j,v i andv j are the estimated states of the leader velocity. The objective of the estimator is to drive the estimated statesq i and v i to track the leader states q 0 and v 0 , respectively. The design of the finite-time estimator can also be referred to the sliding mode estimators presented in [39].
Then, define the following error statesê i =q i −h i q 0 andε i =v i −h i v 0 . We can further obtain the following error system.
In the following analysis, we will use equation (3.1) to design the state-constrained estimator-based control algorithm and use equation (3.2) to analyze the convergence ofq i andv i , ∀i ∈ V.

Tracking error boundary
Before moving on, the tracking error transformation methods are developed. To obtain the prescribed performance, the decaying function is used to establish the prescribed limits of the tracking errors. For i ∈ V 1 , the prescribed limits ρ uk and ρ lk are presented as follows.
where ρ lk and ρ uk are presented to establish the constraints of e i (t),q ik is the k-th element ofq i . Then we firstly prove thatq i (t) is bounded. Based on Assumption 2, the boundedness of q uk , q lk andq ik implies that the error between q uk andq ik is less than the constant a u and the error between q lk andq ik is greater than the constant −a u . Thus, we can conclude that From what will be proved later, we can establish that q i (t) is bounded. Based on Assumption 2, the boundedness of q uk , q lk andq ik implies that the error between q uk andq ik is less than the constant a u and the error between q lk andq ik is greater than the constant −a u . It then follows from the property of the decaying function (2.8) and equation (3.2) that where Δ i = a u ζ ∞ is the boundary of the tracking error as the closed-loop system becomes stable. Similarly, when i ∈ V 2 , the boundary of the tracking error can be obtained by replacingq ik with −q ik . Furthermore, we establish the state-constrained estimator-based control algorithm to force q i track the trajectories of the estimated stateq i .

State-constrained estimator-based control algorithm
For designing the controller of the local control layer, the local layer error is designed asẽ i = q i −q i .ẽ ik is the k-th element ofẽ i , ∀k ∈ {1, 2, . . . , n}, being the local layer error of each joint. The auxiliary virtual controller is presented as follows.
where c i > 0 represents the control gain and ϕ ik (t) is designed to ensure ρ lk <ẽ ik (t) < ρ uk . Moreover, the velocity error containing the virtual controller is presented as follows.
where v ik is the k-th element of the velocity v i . To stabilize ε ik (t), the actual controller is designed as follows.
The generalized control frame of the state-constr ained estimator-based control algorithm is presented below (Fig. 1).

Remark 3
In this paper, a hierarchical control method based on the state estimator is proposed to advance the research of prescribed performance control of multiagent systems with state constraints. Due to the high Fig. 1 The proposed hierarchical control frame nonlinearity, complex constructor and strong coupling of multi-robot systems, the hierarchical control method proposed in this paper can be used to divide the complex tracking problem into the subproblems of state estimation and local control. We propose the bipartite tracking algorithm in which the estimator layer is used to estimate the information of the leader. Besides, we design the decaying function to guarantee that the tracking errors do not exceed a given boundary and the actual state of each robot can track the estimated states by using the proposed decaying function. Compared with the traditional tracking control for a single robot aiming to achieve prescribed performance, we present the bipartite tracking control methods to guarantee state constraints and prescribed performance for IRSs.

Remark 4
In practice, the target trajectory generally cannot be previously known and is possibly attainable at each time instant. Therefore, the estimator layer can be used to estimate the information of the target trajectory through the interconnection of IRSs at each time instant.

Stability analysis for state-constrained estimator-based control algorithm
For the designed control algorithm, it is necessary to ensure the stability of the estimator layer. Furthermore, the target trajectory at each time instant is acquired, and meanwhile the boundedness of the convergence procedure can be guaranteed.
Proof Based on (3.2), the compact form of the derivative of the velocity error is obtained as follows. , if there exists a proper constant α 2 satisfying α 2 > v 0 , then ε ≡ 0 as t ≥ t 1 , namely,ε i ≡ 0 as t ≥ t 1 , and t 1 can be calculated following [40]. Thus, lim  [41], if G is heterogeneous and structurally balanced, then the matrix M D is positive stable. Therefore, the systemē is asymptotically stable. Obviously, the systemê is also asymptotically stable. Then we can conclude that lim Remark 5 According to above analysis of the estimator errorê i , it can be obtained thatq i =ê i +q 0 for ∀i ∈ V 1 andq i =ê i − q 0 for ∀i ∈ V 2 . Due to the fact that lim t→+∞ ê i = 0 and q 0 is bounded, we then obtain that q i is bounded.
Furthermore, the decaying rate of the position error e i can be further acquired. The decaying rate can be described as the following equation.
where μ is a positive constant. This ends the proof. Then, the following two lemmas are firstly presented to demonstrate Theorem 2.

Lemma 2 For the k-th joint of the i-th robot, if
Based on (3.4), it can be obtained that 0 ≤ q uk (t) − q 0k (t) ≤ a u and −a u ≤ q lk (t) − q 0k (t) < 0. It then follows that ρ uk ≤ a u ζ 1 (t) and ρ lk ≥ −a u ζ 1 (t). Therefore, the tracking error satisfies that |ẽ ik (t)| < a u ζ 1 (t).
Since Δ i =a u ζ ∞ , the error can be limited in the set (−Δ i , Δ i ) ultimately. The proof is completed.
Before proceeding, the auxiliary variable is presented as the following form. . (3.14)

Theorem 2
Considering the initial conditions presented in (2.5) and right after (3.9), the proposed controller can guarantee that for all k ∈ {1, 2, . . . , n}, Proof The proof contains two steps including the stability of the position error and the velocity error. In addition, the position errors are demonstrated to stay within the constraint sets presented as in (3.16).
Then the reduction to absurdity is used to carry out the following analysis. We firstly assume that there exist time instants t v , v ∈ Z + such that one or more statements of the following conditions holds.
where t v is regarded as the first time to violate the restricted conditions. Therefore, ρ lk <ẽ ik (t) < ρ uk and |ε ik (t)| < ζ 2 (t) hold for 0 < t < t v .
According to the continuity of the errorsẽ ik and ε ik , we can further conclude that one or more items of the following statements hold.
It then follows the following analysis.
Step 1: The position error is firstly analyzed. The Lyapunov function candidate is selected as follows.

(3.18)
Due to the definition ofẽ i , one obtains thaṫ By substituting (3.19) into the (3.18), we conclude thaṫ In the following, we demonstrate the boundedness of ω i during t < t v . Due to the boundedness ofq lk (t), q uk (t) andq i (t), we obtain thatρ lk ∈ L ∞ andρ uk ∈ L ∞ . From (2.6) and (2.7), we can obtain that That is, Therefore, it can be concluded that ω i is bounded. There exists a constant o i such that (3.20) Based on (3.3) and(3.14), we obtain that Considering (3.20) and (3.21), it follows thaṫ It can be further obtained thatV i,1 < 0 as |ϕ ik | > o i c i . On account of (3.17), we conclude thatV i,1 < 0 as . Based on the properties of the Lyapunov function and the aforementioned statements, one can conclude that (3.22) We then obtain that ϕ ik (t) is bounded due to (3.22). According to (3.6)-(3.9), the velocity error ε ik (t), the auxiliary virtual controller β ik (t) and ϕ ik (t), as well as the control input τ i are all bounded. By (3.22), there exists a contradictory deduction between the derived result and the three hypothesis. Therefore, the hypothesis that position errors violate the specified rules is invalid, namely, (3.16) is true.
Step 2: The boundedness of the velocity error is also analyzed. Consider the Lyapunov function candidate as the following form.
Subsequently, we can derive the boundedness of Γ . Firstly, the boundedness of χ i for t < t v is obtained according to (2.3) on account of the boundedness of each term. Based on Lemma 3, it follows thatβ i ∈ L ∞ for t < t v . Then, γ for t < t v is bounded on the basis of (3.7). Therefore, there exists a positive constant κ such that Furthermore, it follows thaṫ On the basis of the above analysis, we obtainV i,2 < 0 for i (t)ψ i (t) 2 > κ λm 1 . By means of (3.9), we conclude that . (3.32) According to the property of ζ 2 and (3.19), we can obtain that Furthermore, it can be concluded thatV i,2 < 0 for Based on the properties of the Lyapunov function and the three hypothesis, it can be obtained that (3.33) By (3.33), there exists a contradictory deduction between the derived result and the hypothesis. Therefore, the hypothesis that the velocity errors violate the specified rules is invalid, namely, (3.16) is true. Consequently, the proposed control algorithm can establish that the states follows the desired trajectories. Thus, we ensure the state constraints simultaneously. According to Lemma 2, it is concluded that the proposed control algorithm can be used to achieve bipartite tracking with prescribed performance. The proof is completed.
Remark 6 Motivated by the above proof, it follows that the tracking errors are constrained in the boundary. That is,

Simulation results
In this subsection, several simulation examples are presented to testify the effectiveness of the control algo- Fig. 2 The communication topology rithm. The simulation parts can be divided into three parts including the estimator layer, the local control layer and the total control algorithm. The estimator layer is used to satisfy the demands on the interconnections of the IRSs. The interconnection topology includes one leader and eight followers. The eight followers can be divided into two groups. In the same group, the followers cooperate with each other. In different groups, the followers compete with each other.
The leader only sends its information to partial followers. Then the eight followers can be accessible to the leader information directly or indirectly. The topology is displayed in Fig. 2.
The control object of the local control layer is to force the eight followers to track the leader. The Euler-Lagrange individual is chosen as two-link robot manipulator. The detailed physical parameters of the eight followers are listed in Table 1, where m is the quality of the link, l stands for the length of the link, r is described as barycenter of the link, and J represents the moment of inertia of the link, ∀ ∈ {1, 2}. Then, the physical parameters are chosen for i ∈ {1, 2, 3, 4, 5, 6, 7, 8} as follows.
In the simulation part, the initial conditions are determined in a random way. The control objective is to track the trajectory of the leader. The target trajectory of the leader is given as follows.
; cos(t) + 1 5 . (4.1) Besides, the position constraints obeys the following items. (4.2) Furthermore, for the aforementioned controller, we choose the parameters as α 1 = 1, α 2 = 2, c i = 8, κ = 8, μ = 0.25 and λ = 8. Then, the decaying func- In order to verify the effectiveness of the proposed control algorithm, we design the numerical simulation to illustrate the theoretical result. The followers exchange their information in a distributed away. Besides, only partial robots can receive the information from the leader. Based on the proposed control algorithm, we can testify the effectiveness of the presented control algorithm for the eight robots. The simulation results Figs. 3, 4, 5, 6, 7, 8, 9 and 10 are presented to testify the effectiveness of the control frame and the control algorithms. Therefore, we can achieve the control objective that the followers track the target trajectory of the leader with the prescribed boundary. Meanwhile, Fig. 5 shows that the velocity states of the four followers belonging to one group can reach agreement and the velocity states of the four followers belonging to another group reach agreement on the exact opposite state of the former group. Figure 6 illustrates thatẽ i between the actual states and the estimated states are bounded between the upper boundary ρ u1 , ρ u2 , and the lower boundary ρ l1 , ρ l2 under the local controller (3.18). Figure 7 illustrates thatê i between the estimated states and the leader states are bounded between the upper boundary and the lower boundary under the estimator (3.9). From Figs. 6 and 7, the presented state-constrained estimator-based control algorithm can guarantee that the performance of results (i.e., the errors between the actual states and the estimated states, or the errors between the estimated states and the leader states) is limited to the prescribed performance (i.e., the upper boundary and the lower boundary). That is, the prescribed performance control is effective. In Figs. 8 and 9,˙ i between   the actual velocity and the estimated velocity andˆ i between the estimated velocity and the leader velocity respectively converge to the origin under the stateconstrained estimator-based control algorithm. Finally, Fig. 10 illustrates that e i is also controlled to be located within the prescribed performance. In other words, the actual states of each robot can track the leader states within the range determined according to the prescribed performance.

Conclusion
This paper has designed a new state-constrained estim ator-based bipartite tracking algorithm for the IRSs with prescribed performance and state constraints. It also has demonstrated that the performance of the estimator-based control algorithm can be guaranteed if the following items hold. (1) The tracking problem with a prescribed accuracy is solved. (2) The position constraint is never violated during the convergence process.
(3) The estimator-based control algorithm reduces the complexity of algorithm without using the pre-known tracking trajectory. It has provided the theoretical analysis and simulation results to prove the effectiveness of the control algorithm. Besides, the results can be easily extended to solve the formation tracking problems of networked complex systems and the multi-target tracking problems of other complex systems. In future work, we will focus on controlling the IRSs with both state constraints and prescribed convergence time.
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