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

School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China Department of Control Science and Engineering, Hubei Normal University, Huangshi 435002, China School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan 411201, China Center for Polymer Studies, Boston University, Boston, MA 02215, USA Department of Physics, Boston University, Boston, MA 02215, USA School of Electrical and Electronic Engineering, Hubei University of Technology, Wuhan 430068, China School of Science, Hubei University of Technology, Wuhan 430068, China

erefore, more and more researchers embark on the cooperative control of NRSs [3,6,29]. However, the aforementioned literature studies mainly focused on the joint-space control of NRSs. In real-world applications, the task-space algorithms are indeed more practical due to the consideration of the kinematics analysis, which thus leads to a few researches studying the task-space control of NRSs [30,31]. It noteworthy that the heterogeneity of NRSs has not been taken into account in the above researches. Besides, when performing tasks, redundant robots are generally more functional than the nonredundant robots for their extra degrees of freedom (DOFs) and usually utilized to perform some corresponding subtasks [32]. us, in a system and an application view, it is of great significance to investigate the task-space cooperative control of networked heterogeneous robotic systems (NHRSs).
With the development of consensus theory, there is a tendency for researchers to embark on the studies of formation problems. To be specific, Dong et al. have studied the time-varying formation tracking problem of multiagent systems under switching topologies in [18]. e time-invariant formation problems of multiagent systems with time delays have been solved in [33]. A distributed position estimation method is employed in [34] to realize the formation of multiagent systems. Ge et al. [35] have proposed a finitetime algorithm to achieve a switching type of time-varying formation for NRSs in the joint space. It should be pointed out that most of the aforementioned works focused on the time-invariant or time-varying formation in the case of a single leader, namely, all agents inside form a single formation. However, in many practical circumstances, the main results of the above literature studies cannot be directly applied if the networked systems are required to execute multiple tasks concurrently. A few works centralized on multiple tracking [7,36,37], multiple formation [16,17], and group time-varying formation [19] are thus activated. To the best of our knowledge, the time-varying formation tracking problem of NHRSs in the task space remains unsolved, let alone the multiple time-varying formation tracking problem.
On the contrary, in the practical robotic application, inexact physical parameters and external disturbances are ineluctable and usually have negative impacts on the performance of the NHRS. It thus becomes extremely challenging to study the time-varying formation tracking problem of NHRSs in the task space with parameter uncertainties and external disturbances.
Based on the above discussions, this paper investigates both the time-varying formation and multiple time-varying formation problems of NHRSs in the task space with parameter uncertainties and external disturbances. e main contributions of this paper can be summed up threefold: (1) e time-varying formation and multiple time-varying formation tracking problems of NHRSs in the task space are firstly studied and successfully addressed by designing several novel estimator-based hierarchical cooperative (EBHC) algorithms. (2) e proposed finite-time distributed estimator algorithm can guarantee the accurate estimation of the states of the virtual leaders, namely, the availability of leaders' information for each robot is assured. (3) Without interfering with the time-varying formation tracking task, the possible preset subtasks for redundant robots can also be accomplished.
Notations: let R n equal the n-dimensional Euclidean space; especially, R represents the real number field. e Kronecker product is denoted by ⊗ . I m is the m-order identity matrix, and 1 m (0 m ) symbolizes an m-dimensional column vector with all its elements being 1 (0). L 2 and L ∞ denote the L 2 space and L ∞ space. ‖·‖ and sup‖·‖ denote the Euclidean norm and its supremum, respectively. For given ℓ > 0 and I � [I 1 , I 2 , . . . , I n ] T , one has sig(I) ℓ � [sgn(I 1 ) |I| ℓ , sgn(I 2 )|I| ℓ , . . . , sgn(I n )|I| ℓ ] T .

Preliminaries and Problem Formulation
Taking the virtual leader into consideration, a pinning matrix is introduced to indicate the interactions between the robots and the leader. Specifically, b i > 0 if the information of the leader is accessible for the robot i; otherwise, b i � 0. It is noteworthy that the leader will not receive any information from the robots.
en, an usual assumption on the interaction topology is given as follows.

Assumption 1.
e communication topology between the robots is undirected. Besides, for each robot, there exists at least one path such that the considered robot can receive the information of the leader, namely, the information of the leader is globally reachable for all robots inside.

System Formulation.
Consider the NHRS with the virtual leader. e dynamics and kinematics of the robot i are given as follows: where q i ∈ R p i denotes the generalized coordinates in the joint space and _ q i and € q i are its velocity and acceleration, respectively; x i ∈ R d represents the generalized coordinates in the task space with _ x i being its velocity; symbolize the positive-definite inertia matrix, the Coriolis-centrifugal matrix, and the gravitational torque, respectively; μ i denotes the input torque; d i (t) is the external disturbance; and J i (q i ) � zψ(q i )/zq i ∈ g i (q i ) ∈ R d×p i represents the Jacobian matrix, which is supposed to be bounded and nonsingular throughout the full text.
Besides, some properties associated with system (1) are presented as follows for the later analysis.
is skew symmetric, namely, for any 2 Complexity where σ H , σ H , σ C , and σ g are positive constants.

Property 3.
e involved dynamic terms can be linearly parameterized for any given vectors υ and ρ with proper dimensions, i.e., is the dynamical regressor and ϑ i is the unknown constant parameter vector. Remark 1. Property 3 is obtained because the inertia matrix H i (q i ) can be linearly represented by a set of properly chosen combinations of dynamic parameters. e specific evolutionary process is omitted here for the space limit. Refer to [38] for more details.
On the contrary, the virtual leader is described as where x r , v r , and a r , respectively, denote the reference position, velocity, and acceleration of the leader. In practical robotic applications, NHRSs are always expected to track the certain desired trajectory with respect to the practical task. en, a reasonable assumption on the acceleration of the leader is given as follows.
Assumption 2. It is assumed that the jerk of the virtual leader is bounded, namely, ‖ _ a r ‖ ≤ σ _ a , where σ _ a is a positive constant.

Problem Formulation.
e control objective is to design an appropriate control algorithm aiming at solving the timevarying formation tracking problem defined hereinafter. Firstly, the time-varying formation offset of the i-th robot is specified by h i (t) ∈ R d , and the tracking errors are predefined as Some useful lemmas that will be invoked in the later analysis are presented as follows.

Analysis for Time-Varying
Formation Tracking

Estimator-Based Hierarchical Cooperative Algorithm I.
In this section, EBHC algorithm I is proposed for addressing the time-varying formation tracking problem of NHRSs. For the sake of simplification, the redundant robot set and nonredundant robot set are severally defined as Let x i , v i , a i , and ϑ i , respectively, be the estimators of x i , v i , a i , and a i , ∀i ∈ ], where v i and a i are, respectively, the velocity and acceleration in the task space. An auxiliary joint-space velocity is properly defined as where ω i is a positive scalar and φ i ∈ R p i denotes the negative gradients of some performance indices with respect to the redundant robots. It follows that the auxiliary acceleration can be presented as ereupon, the sliding vector is defined as en, EBHC algorithm I is presented as follows: where i ∈ ], κ oi ∈ R d×d , and κ si , κ di ∈ R p i ×p i are the feedback gain matrix, Λ i is the adaptive positive-definite matrix with appropriate dimensions, and α, β, c, ρ 1 , and ρ 2 are positive constants. To be specific, (9) is the adaptive smooth control layer and (10) is the distributed estimator layer. (9) and (10) into (1) yields the following cascade closed-loop system:

Coordination Analysis and Boundedness Analysis for the
en, the closed-loop system of the estimator layer can be further rewritten in the following compact form: Theorem 1. Suppose that Assumptions 1 and 2 hold. e distributed estimator algorithm (10) guarantees that there Besides, the states q i and _ q i remain bounded when t 0 ≤ t < T f for any given bounded initial values, where t 0 denotes the initial time.
Proof. For the first presentation, we are going to prove that x i and v i , ∀i ∈ ], converge to the origin in finite time T f ≥ t 0 .

Complexity
Based on Assumption 1, it can be derived that J is positive definite. en, the Lyapunov function is constructed for the third differential system in (12) as Equation (13) It then follows that . us, we can derive that a converges to zero in finite time J f1 . en, when t ≥ J f1 , (12) equals Taking the linear transformation for en, according to Lemma 1 and (13), system (16) is globally finite-time stable with the origin being its stable equilibrium, i.e., there exists a settling time T f2 > 0 such that X and V converge to the origin on t ∈ [J f1 , J f2 ]. Note that e second presentation focuses on the boundedness analysis. Note that Besides, Assumption 2 implies that x r , v r , and a r are all globally bounded on t ∈ [t 0 , T f ], namely, x r , v r , a r ∈ L 2 ∩ L ∞ as t 0 ≤ t < T f , and the formation offsets of robots are usually set to be bounded in practical robotic en, substituting the aforementioned variables into (6)- (8) yields that In this section, we will show whether the time-varying formation tracking problem can be addressed under the proposed EBHC algorithm I or not. Firstly, the norm variables associated with ψ · i , € ψ i , and s i are given as Based on eorem 1, it can be figured out that when t ≥ T f , the following closedloop system arises based on (11): namely, it can be obtained that the task-space coordinates x i and v i can eventually form the time-varying formations, i.e., Proof. Consider the following storage function as the Lyapunov function candidate for system (21): Complexity 5 Differentiating (23) along (21) yields that where Property 1 and λ min (κ di ) ≥ sup t≥t 0 ‖d i (t)‖ in (22) have been invoked to make the above inequality hold. ereupon, λ min (κ oi ) > 0 and λ min (κ si ) > 0 imply that _ V(s i ) < 0. Furthermore, (22) and (24) provide that s i ∈ L 2 ∩ L ∞ . en, one obtains that _ s i ∈ L ∞ through the closed-loop system (21). us, € V(s i ) ∈ L ∞ is derived. According to Barbalat's lemma [41] and the above stability analysis, it thus can be concluded that s i ⟶ 0 as t ⟶ ∞.
According to (18) and (20), J i (q i )s i is presented as follows when t ≥ T f : which can be further rewritten in the following form: Note that J i (q i )s i ⟶ 0 as t ⟶ ∞ for the boundedness of J i (q i ). According to Lemma 2, it thus follows that the kinematics loop (26) is input-to-state stable with J i (q i )s i being the input and e i and _ e i being the states. en, ∀i ∈ ], we draw the conclusion that e i ⟶ 0 and _ e i ⟶ 0 as t ⟶ ∞ through the fact that J i (q i )s i ⟶ 0 as t ⟶ ∞.
is completes the proof.
□ Remark 2. Selecting appropriate φ i for a redundant robot can execute some subtasks but does not conflict with the main task (i.e., the time-varying formation tracking task). For instants, denote q i (1) as the first element of q i , i ∈ ℸ, and choose the performance index as H i (q i ) � 0.4(12 − q i (1)) 2 , then one has φ i � − (zH i (q i )/zq i ) � [0.8(12 − q i (1)), 0, . . . , 0] T . Such a setting mainly aims at forcing the first joint of the i-th robot toward 12 rad. A corresponding simulation experiment on redundant robots will be conducted to support the above presentation.

Remark 3.
According to the study in [32], the subtask for the redundant robots is called completed if lim t⟶∞ e si � lim en, it will be discussed whether the subtasks can be addressed or not under the proposed EBHC algorithm I. Note that when t ≥ T f , where (18) and (20) have been invoked to obtain the above equation. It has been already obtained from eorem 2 that s i ⟶ 0 as t ⟶ ∞.
en, one can easily derive that namely, lim t⟶∞ e si � 0. is completes the proof.

Further Results on Multiple Time-Varying Formation Tracking
In this section, a similar controller will be designed to address the multiple time-varying formation tracking problem of NHRSs. Firstly, some different definitions on graph theory should be further presented here. e adjacency matrix W is no longer nonnegatively weighted here. If the robot i cooperates with the robot j, then w ij � w ji > 0; if the robot i competes with the robot j, then w ij � w ji < 0. Besides, the considered NHRS can be divided into M subnetworks. e undirected graph g k � ] k , E k , W k , k ∈ 1, 2, . . . , M { }, which is nonnegatively weighted, is employed to describe the interaction of the k-th subnetwork. Besides, each robot can only belong to one of these subnetworks, namely, ] k ∩ ] p � ∅ for k ≠ p ∈ 1, 2, . . . , M { } and ∪ M i�1 ] i � ]. n k is utilized to denote the number of robots in the k-th subnetwork. Each subnetwork corresponds to a single virtual leader. Taking the M virtual leaders into consideration, the pinning matrix B k is employed to illustrate the interactions between the robots in the subnetwork k and their corresponding leader. en, 6 Complexity ereupon, two different assumptions on the interaction topology are given as follows.
Assumption 3. For each subnetwork with the corresponding leader, there exists at least one path, through which the information of the leader can be obtained by the robots inside.

Assumption 4.
e total information that each robot receives from any other different subnetworks is zero, i.e., r k j�r k l ij � 0, ∀i ∈ ] p , k ∈ 1, 2, . . . , M { } − p , where r k � 1 + k i�1 n i− 1 and r k � k i�1 n i . Note that n 0 � 0. e virtual leader associated with the k-th subnetwork is described as where x r,k , v r,k , and a r,k , respectively, denote the position, velocity, and acceleration of the k-th leader, k ∈ 1, 2, . . . , M { }.

Assumption 5.
e jerks of all the leaders are bounded, namely, there exists a uniform upper bound σ _ a r such that e control objective in this section mainly focuses on the multiple time-varying formation tracking problem of the NHRS defined hereinafter.

Definition 2.
e multiple time-varying formation tracking problem of the NHRS is addressed if the tracking errors satisfy

Estimator-Based Hierarchical Cooperative Algorithm II.
In this section, a similar controller is designed to address the multiple time-varying tracking problem of NHRSs. e definitions of ℸ, ℶ, and s i are the same as in Section 3.

Theorem 3. Suppose Assumptions 3-5 hold, then the multiple time-varying formation tracking problem can be solved under the designed EBHC algorithm II
namely, Proof. Note the adaptive smooth control layer is the same as in the previous part. us, we just need to redo the cooperative analysis and boundedness analysis of the distributed estimator layer. Similarly, redefine the following compact vectors: � col a 1,1 , . . . , a n 1 ,1 , . . . , a 1+ It thus follows that 8 Complexity Remark 5). e fact that g is undirected guarantees the symmetry of L. It thus follows from Assumption 3 and (36) that J � L + B is positive definite. en, by similar analysis in eorem 1, it can be derived that x i and v i converge to x r,k and v r,k in finite time, namely, there exists T f > t 0 such that e remaining convergence analysis of the adaptive smooth control layer is the same as that in eorem 2 and is omitted here. Besides, the subtasks for the redundant robots can be fulfilled by the analysis in Remark 3. is completes the proof.   Figure 2(a). To be specific, the node L denotes the leader, node 1 represents the redundant robot, nodes 2-4 are the nonredundant robots, and robots 1 and 2 can directly receive the information from the leader. It thus follows that B � diag(2, 0, 0, 3) and Let the external disturbances be d i (t) � − 0.1[cos(t), sin(t)] T for 2-DOF robots and d i (t) � − 0.1[sin(t), cos(t), cos(t)] T for 3-DOF robots. e control parameters are selected and shown in Table 1, which guaranteed the establishment of (13) and (22).     Figure 7 provides the trajectories of all the robots inside and the leader, from which we can observe that the four robots form a square, and the square is rotating itself with a certain angular velocity while moving along the trajectory of the leader. It is noted that the tracking errors e i and _ e i are bounded by [− 0.08, 0.08], which are tiny enough to support our main results.

Robot 4 Leader
(d) Figure 4: On the contrary, we choose (redundant) robot 1 to verify that the subtask can also be finished but does not conflict with the tracking task. e negative gradient of the performance index corresponding to robot 1 is designed as

Robot 4 Leader
(d) Figure 6: e external disturbances, the control parameters, leader 1, and the formation offsets of robots 1 − 4 are the same as in Example 1. Besides, leader 2 is described as e second subnetwork is expected to form a timevarying regular triangle, which is specified by Finally, the initial values are randomly chosen in the same corresponding ranges given in Example 1.
e simulation results are shown in Figures 10-14. Similarly, Figures 10 and 11 indicate that x i can eventually track the desired position, and the convergence of x i − h i is completed in finite time. Besides, Figure 12 depicts the tracking performance of v i and v i . Figure 13 indicates that the NHRS can achieve the multiple time-varying formation in the XY plane under the proposed EBHC algorithm II, which is also depicted in Figure 14. In detail, the robots belonging to the first subnetwork form the time-varying square as same as in Example 1, and the robots belonging to the second subnetwork form a time-varying regular triangle with its size changing periodically. Note that the tracking errors in Figures 10 and 12 do not converge to zero.
is is mainly because although it has been theoretically proved that the states converge to zero asymptotically, the final simulation results converge to the neighbourhood of the origin due to the presence of external disturbances,     parametric uncertainties, and kinematic redundancy and the employment of the sign function. is phenomenon is very general in the applications of the sliding mode control technique on regulating complex systems with various uncertainties. us, it is theoretically correct to obtain the simulation results shown in Figures 10 and 12. It thus can be concluded that the time-varying formation tracking problems of NHRSs with both a single leader and multiple leaders can be solved under the proposed EBHC algorithms.

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

Data Availability
is paper mainly brings some theoretical perspectives and methodologic approaches. All the figures are obtained through simulations based on the presented approaches. No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.