Bidirectional formation-involved consensus for uncertain multi-Lagrange systems under directed signed topology networks

Regarding complex behaviors of multi-agent systems (MAS), this paper proposes control protocols to reach a bidirectional formation-involved (FI) consensus for MAS. It primarily considers the multiple Lagrange systems (MLS) with uncertain parameters. And bipartite topology communication networks. Then, the validity of control protocols and the stability of the MLS are confirmed by determining the consistency convergence of the Lyapunov functions. With the stabilized MLS, the first-order linear system is adopted as a leader to provide trajectory guidance. With bipartite topology, MLS performs the FI complex behavior with pairwise tracking motion. The experiment part provides sufficient simulation examples that are selected to be consistent with the systems in the theoretical part.


Introduction
In the robotics field, the Lagrange dynamics analysis method has become a hot-spot for researchers because of its feasibility of constructing all robotic systems. The Lagrange dynamics equation portrays the agent system with noble expressions, which is also called the Lagrange agent. In the current era of "unmanned intelligence," Lagrange agents exist in the form of manipulators for manufacturing, rehabilitation robots for health-care, unmanned surface vehicles for marine, and other physical forms. But in recent tasks, the complexity, difficulty, and dangerous (CDD) are increasing. Thus, it is urgent to improve stability, security and simplicity (SSS) for unmanned control. There are two feasible methods to enhance SSS. One is to propose new agent mechanisms, which are almost carried out around Lagrange systems with higher precision and a higher degree of freedom. The second solution emerges from the control theory for multiple agent systems (MAS), which produces flexible algorithms for CDD problems. And this work will discuss the control issues of multiple Lagrange systems (MLS). Those control issues for MAS are concerned as swarm control, consensus control, and formation control. The swarm control has a large number of controlled agents with simple systems, which is generally solved by graph-theory algorithms. However, global control may not be achieved due to a lack of interactions among agents. For reaching a global agreement of states, such as velocities and accelerators, consensus control has been a new hot point. The following concern will undoubtedly be the consensus control problem.
Regardless of the control process, positional constraints are always problems to be overcome. Thus, a desired formation is required for MAS. Regarding the existence of the desired formation in the control process, the formation control problems are divided into formation reaching problems and formation keeping problems. Formation-keeping problems are mainly described as formation-tracking problems. According to tracking schemes, the formation tracking problems can be solved by leader-follower and virtual structure approaches [1][2][3][4]. Moreover, it should be noted that formation control focuses on the alignment of position coordinates. It implies that the form control is meaningful in position-constrained consensus problems. There are matured works to solve formation problems by using consensus protocols. These use a typical formation control scheme to realize a global agreement about relative positions. Nevertheless, there are demands to carry out formation and consensus control independently. Based on this, the complex behaviors of MAS have been introduced. In control works of complex behaviors, cohesion, collision avoidance, and obstacle avoidance may be included. However, any complex action should be based on consensus. Additionally, complex behavior should be considered based on consensus. The following are mainly about formationinvolved consensus control.
For the evolution of complex behaviors, sensing ability and interaction topology are required for MAS. The former depends on the realization of the terminal. For instance, the orientations of the coordinates can be aligned to the global coordinates by using magnetic sensors [5]; the positions and the velocities of quadrotors were obtained based on GPS sensors and accelerometers [6]. The latter reflects the consensus among agents, thus determining the feasibility of the control scheme. There are three primary control algorithms as usage information interaction is regarded. The Decentralized control of complex behaviors for MAS has been unreliable due to the lack of information. Centralized control requires excessive computation owing to massive inputs. Distributed control bal-ances the number of computing centers with the flow of information exchange, which achieves global control by partial control using local information. Distributed control has thus been used in coping with complex control problems [7]. At this point, this work takes one effort to produce distributed control protocols for reaching the formation-involved consensus first. In addition to these control algorithms, there are also research works explored in interaction topology. A type of cluster consensus protocol was given based on the topology with acyclic partition [8]; Group consensus states of MAS with single integrator dynamics are characterized by a large enough coupling strength [9]; the sampled-data consensus of MAS was recently investigated with switching jointly connected topologies [10]. But the aforementioned works only worked with cooperative interactions. Indeed, the competitive topology has received attention since Altafini, C. put forward the concept of bipartite consensus [11].
In recent research, the interaction networks discussed have been extended from cooperative to competitive-cooperative networks. Such as, the states of MLS demonstrated final symmetry in symmetric progresses [12][13][14]. In view of these, the bipartite topology tends to be used to solve problems of group consensus, symmetric consensus, etc. There is little literature on such works. The distributed cooperative formation control was extended to bipartite formation control for MAS using a first-order integrator [15]; Group bipartite coherent formation was examined for second-order MAS [16]. Accordingly, this paper will discuss formation control and consensus problems by adopting the bipartite topology. Different from the existing works, this work will be distinguished by controlled MLS. As multiple Lagrange dynamics as concerned, the protocols are to submit the torque to the controllers. Since the protocols cope with the torque controller, the controller presented in this paper is not only for industrial robot arms but also can be applied to agents with flexible attachments, such as small lunar probes and small landers. In short, this literature will study the behaviors of MLS regarding consensus control of formation under the bipartite topology. The highlights are listed below 1. Multiple Lagrangian systems are controlled objects which are expressed by higher-order differential equations. It will obviously fill the gap in the work of formation control in nonlinear systems.
Specially, the protocols are provided with torque controllers by adopting slider vectors, which will promise the SSS of controlled systems. 2. Bipartite topology, which is described by the directed signed graph, is chosen as an interaction network. At first, distributed protocols are introduced to make a consensus on formation control for MLS. Then, protocols for formation tracking are given after employing a virtual leader. Based on these, the bipartite consensus of formation is induced. 3. FI complex behaviors are presented in the results.
The bipartite topology contributes to making MLS motion pairwise back-to-back or pairwise motion face-to-face; Among these movements, the desired formation can be symmetric or asymmetric.
The rest of this study will be organized as follows. In Sect. 2, the mathematical, theoretical concepts, and dynamics analytical methods will be reviewed. In Sect. 2, the main results will be given and verified. It is arranged as follows: Firstly, the distributive consensus protocols for formation control are proposed; Secondly, the virtual leader is introduced to make global agreements of complex behaviors in cases of formation reaching and formation track. Then, simulation results will be given in Sect. 4. Finally, a summary and discussions are provided in Sect. 5.

Preliminaries
To describe the problems of MLS, this section will recall some basic notations of graph theory, the Lagrange dynamic equation and the control theory.

Graph Theory
Generally, G = (V, E) represents a graph involving a node set V = {v i |i ∈ I} and an edge set E = {e i j |v i (directed)link to v j }, where I denotes the index set. Moreover, G is called a directed graph if e i j represents the directed link. There exists a (directed) path for a pair of nodes v i to v j if and only if there exists edges e i 1 , e 1 2 , . . . , e i w , e w j ∈ E. In addition, the graph G has a (directed) spanning tree if and only if there exists a (directed) path for each pair of nodes v i , v j . The adjacency matrix A = a i j I×I is used to present the edges weights for graph G. If there are both positive and negative edge weights, then the graph G is called a bi-graph. The properties of a directed bi-graph G will be discussed in this paper. For these cases, the Laplace matrix L(A) = i j I×I of this paper is given below: Lemma 2.1 [11] If G is structurally balanced, then there exists a matrix = diag(φ i ) i∈I such that A has no negative entries, where {φ i ∈ {1, −1}|i ∈ I}.
Lemma 2.2 [12] If G further has a directed spanning tree, then L has 1 I as the right eigenvector associated with its simple zero eigenvalue and the other eigenvalues have positive real parts. Moreover, there are non-negative numbers π i (i ∈ I) that satisfy i∈I π i = 1 such that L has π = diag(π i ) as the left eigenvector π of eigenvalue 0.
In this sense, it is said that agent-i and agentj are in competitive relationship if v i and v j are linked by a negative path. Otherwise, agent-i and agentj have cooperative relationship. Furthermore, notice that = T = −1 and L( A ) = L(A) . This means that the bipartite control problem of MAS can be transformed into a complete control problem of MAS by using the matrix .
For a given matrix Obviously, i is invertible. By taking a further calculation, one has that Moreover, is Hurwitz stable [17].

Assumption 2.4
The graph G is structurally balanced and has a directed spanning tree.
In this paper, the main result is based on this assumption.

Lagrange dynamics
For a Lagrange agent system with the generalized coordinate q ∈ R p , the dynamic is formulated as where M(q) ∈ R p× p is the inertia matrix, C(q,q) ∈ R p× p is the matrix related to the Coriolis force and centrifugal force, g(q) ∈ R p is the gravitational torque, and τ ∈ R p is the control torque. Regarding the kinetic coefficients involved in Eq. 4, in addition to the fact that M(q) is a symmetric positive definite matrix, there are the following three properties that are often mentioned.

Proposition 2.5
There exist positive constants m 1 , m 2 , (4) is linear with respect to a certain constant parameter theta. That is, for arbitrary differentiable vector u, v ∈ R p , the following equation holds.

Proposition 2.7 The Lagrange dynamic equation
where Y (q,q, u, v) is called the regressor matrix with respect to the dynamic parameter theta. In fact, θ is generally the column vector of the moment of the rigid body about the orthogonal basis.
Remark 2.8 It is worth noting that only the estimates of M(q), C(q), g(q) can be used to design the controller for such a Lagrange system. Therefore, the control algorithms have to eliminate the uncertainty of the parameters. Nevertheless, the dynamics equation of the Lagrangian system has a parametric linearization property. Therefore, the problem of parameter uncertainty is transformed into the problem of the estimation error of θ . For this point, the controller will take the estimation error of the parameter into account.
Since the multi-Lagrange systems with topology G are considered in this article, denote q i as the generalized coordination of agent i for all i ∈ I. Then the dynamics of the MLS are expressed by

Bidirectional formation consensus
The desired formation is specified by the column stack vector as

Definition 2.9
For the MAS, it is said to form the formation F if the state of systems satisfy lim then the MAS is said to reach bidirectional formation consensus.
The distributed adaptive laws will be introduced in the next section.

Main results
Before giving two main results of this paper, we provide some notations for further illustration.

Reaching Formation Consensus
The notationq r i (t) is the reference velocity of the ith agent and is given aṡ Then one has thaẗ In addition, sliding vectors s i (t) are added to enhance the robustness of the system in this article.
Under these notations, the control torques for systems (6) are designed as with the positive definite coupling matrix K i ∈ R p× p , i ∈ I. In (10), each termθ i is the estimation of parameter θ i . Each estimation errorθ i =θ i − θ i is required to obey the laẇ where the constant matrix i ∈ R p× p is symmetric positive definite. From Proposition 2.7, there are linear parameters regression expressions for systems (6), By adopting the control laws to (6), with the equation (12), then there are closed loop systems.
Next, q is denoted as the column stack vector arranged in order for naturally is the column stack vector arranged in order for {g(q i )|i ∈ I}. θ is also such a column stack vector of {θ i |i ∈ I}. Following these notations, the above variables and corresponding equations can be resorted to compact expressions aṡ where s is the column stack vector of s 1 , s 2 , . . . , s N , and L ⊗ I p is the Kronecker-product of L , I p . The dynamics equations (13) is rewritten as equation (17). And, it is sufficient and necessary to analysis the following equations (16) and (17) to complete the discussions.
Remark 3.1 Note that compact form (16) is the inputto-state equation corresponding to systems. Under the proposed control law, the systems can be described by estimated dynamics (17). Additionally, matrix L exists in the input-to-state equation. The systems analysis will be thus rely on topology. Regarding the bipartite topology, the matrix is used for symbol switching.
After left applying i ⊗ I p on both sides of equation (16), the equation (18) is thus deduced.
From Lemma 2.2, it is worth noting matrices expressed as (19) In this sense, denote that L i = i L( i ) −1 . As a result, the following equation (20) are obtained.
One finally obtains that Based the above preliminaries, the next step is to approach the main result of this paper. Proof First, consider the following Lyapunov function, Immediately, one obtains that Combining (23) with (24), V (t) has bounded limitation andV (t) exists. Meanwhile, one has the claim as follows: Claim:V (t) is bounded. From Barbalat theorem [18], this Claim ensures thaṫ V (t) → 0 as t → ∞. Accordingly, there is that s(t) → 0 when t → ∞. As a result, the stability of the closedloop systems (13) is verified. Based on the aforementioned discussion, the followings are to give states an analysis of systems. As discussed before this theorem, one can obtain that the multi-Lagrange systems achieve bidirectional formation defined in 2.9.
It is left to verify the Claim that has been given, i.e. to clarify thatV Observe the fact that both (23) and (24) hold implies that s i ∈ L 2 ∩ L ∞ , which leads toθ i ∈ L ∞ . Thus, both s i andθ i are bounded. In this situation,q r i is bounded for all i ∈ I since the equation (18) is inputto-state stable. Moreover, from Proposition 2.5 and Proposition 2.7, Y (q i ,q i ,q r i ,q r i ) is bounded. All over the closed-loop system (17),ṡ i is bounded for all i ∈ I. Therefore,V (t) = − i∈Iṡ T i (t)K i s i (t) is bounded. Consequently, the above discussion gives the self-contained proof for this theorem.

Remark 3.3
Under the networked topology of the bipartite graph, the control protocol (14) uses ⊗ I p F as the formation information to complete the formation of F, while the control protocol defined with F makes MLS reach the formation of ⊗ I p F. Remark 3.4 The above proof procedure demonstrates the uniform convergence of the MLS by using the Kronecker Product. Referring to this procedure, it can be derived that the generalized relative coordinate of any two agents eventually converge in each dimension. That is, (φ i q i −φ j q j ) → ( f i − f j ), i, j ∈ I can be deduced. Additionally, there are also (φ iqi − φ jq j ) → 0, i, j ∈ I. Furthermore, these results imply that agents i and j finally maintain generalized distance f i − f j if the information flow between agents i and j is positive. However, only the midpoints of agent i with agent j will be determined when the information flow between agent i and agent j is negative. In the case of positive information flow, the relative position between agent i and agent j is not available. However, the division vector between the two sides can be identified by any p midpoints.

FI consensus in bidirectional tracking
In the following discussion, a virtual leader is introduced to provide the tracking path. It is given that the virtual leader the notation with respect to the graph index and generalized coordinate as 0 and q 0 , respectively, and assumed thatq 0 ,q 0 and ... q 0 are all bounded. In the following discussion, . . , f T N T and f 0 will be the general origin. The relating adjacency matrixĀ will be in block form as describes the interaction from the leader to the follower agent in G. In addition, the graphḠ is the extension of topology graph G by adding the leader, which also possess the extended index setĪ = I ∪ {0}. Furthermore, the corresponding extension forms are noted by givinḡ L,π .

Assumption 3.5
The graphḠ is structurally balanced and has a directed spanning tree.
To avoid increasing additional computation, it will introduce the leader as a first-order linear dynamics to provide tracking trajectories q 0 . In this sense, the system of the leader obeys the equationq 0 (t) = v 0 (t).
Note that e i = q i − φ i q 0 is the position coordinate error. And i = e i − φ i f i is the error about the relative position of the state between the ith follower agent and leader agent with respect to the desired relative position. Under the topologyḠ, the reference velocity of agent i(i ∈ I) will be given aṡ (25) Furthermore, one obtains thaṫ For the ith agent, the expected velocity is introduced aṡ Let the tolerable error for the ith velocity be defined by the following sliding variablē Then, further calculations deduce that Remark 3.6 From the assumption 3.5, the virtual leader has direct paths to every node in G. Moreover, Under the leader-follower network topology, Eq. (25) is based on the addition of each error vector i with leader, which is also extended from the definition in Eq. (7). Meanwhile, Eq. (27) further defines the reference velocity considering the bidirectional consensus with the velocity vector of the leader. If the velocity with trajectory tracking set by (27) is not considered, then the reference velocity given by (25) is used to define the control torque of formation coordinated control based on the virtual-frame method. Moreover, different from [19], the introduced protocols do not use the estimation of the velocity for the ith agent. Thus, the control protocols of this part are not distributed laws. But such kind of centralized control laws absolutely reduces computations in practical application.
Next, the bidirectional formation control protocols are given as with the adaptive protocols for the estimated parameter ϑ i as follows, There are also closed-loop systems.
Similarly, the above equations (29) and (32) can be resorted to compact expressions as Since 2 = I N , one can obtain that It should be noticed thats contains distributed sliding vector s. This implies that one part of the following discussions will consist of Theorem 3.2. Nevertheless, there are still problems caused by adding a leader agent to the origin systems. The lemmas given below are used to deal with precisely this problem.
Lemma 3.7 [20,21] For μ, ν > 0, denote that W = Consider the MLS with any given tracking error bound , which is also adopted with the control protocols provided in Equations (30) and (31). It is said to reach the bidirectional formation consensus. This result is clarified by the following theorem [22]. Proof Choose the following Lyapunov-like function, As a result, there iṡ Considering that symmetric matrix, the above equation can be deduced tȱ From the conditions of this theorem, one obtains thaṫ

Fig. 1 Topology examples used in simulation experiment
Referring to analysis of proof for Theorem 3.2, it similarly discuss q i ,q i withs.
Remark 3.9 Similar to Remark 3.4, there are also In this sense, the MLS will reach the consensus on the desired formation. Moreover, this result will lead to (φ iqi −q 0 ) → 0, i, j ∈ I. In straightforward terms, the MLS will make bidirectional movements with reference to the trajectory of the leader.

Simulation
In this section, simulation examples of formation control on MLS under bipartite topology networks are given. The following simulation experiments consider a multi-agent team composed of 8 Lagrange agents, one leader with a first-order linear system, using a total of 3 communication topology graphs for networking without and with a leader. The formation control objective is the formation of the desired form F and its trajectory tracking under formation keeping.

Simulation requirement
Before proceeding to the main work in this section, an introduction to the requirements of the simulation environment is necessary.

The setup of agent body
Immediately after, 8 Lagrange agents with uncertain parameters are selected, and q = (q i1 , q i2 ) T ∈ R 2 denotes the position coordinates of the ith agent in space. Therefore,q = (q i1 ,q i2 ) T ∈ R 2 denotes the velocity vector in space of the ith agent. For the ith agent, the inertia of rotation of each of its pth link is chosen as J i p = 1 12 m i p 2 i p (kg · m 2 ), ( p = 1, 2), where the link mass m i p and link length i p are taken as shown in Tab. 1.
In addition, for the agent system with uncertain parameters currently used, taking the constants u i1 = m i1 to linearize the coefficients of the dynamic equation, the inertia matrix, Coriolis and centrifugal matrix, and the gravity vector of each rigid body system is obtained as indicated by the following equation, where x and denote that s( * ) . = sin( * ), c( * ) . = cos( * ).

Simulation procedure
Based on the above preparations, the procedure for the formation control of the networked MLS is now described. Basically, a networked MLS of eight agents based on Fig. 1 is prepared, noting q(t) as its global position coordinate vector. After initializing the system and setting up the formation task F, the coupling coefficient and the control parameter K are given. agent-i obtains the position information q j (t) and the velocity informationq j (t) of all neighbor nodes in response to the network topology. The ith agent obtains the reference velocity vector by performing the calculation as defined in Equation (7) with this information. Simultaneously, MLS gains the distributed reference velocity vectorq r (t). Regarding the reference velocity, the ith agent obtains the velocity error s i , and the MLS obtains the input to state equation defined by Eq. (16); For the ith local controller, the control torque τ i is designed by Eq. (10). In a global view, MLS is a closed-loop control system. Then the agent i adjusts its own acceleration under that control protocol, conducting adaptive adjustments locally with respect to velocity and position. Thus, MLS achieves global control through distributed local adaptation, with an output containing the position coordinates q(t + 1) and the velocity vectorq(t + 1). See the algorithm flowchart Fig. 2 for the above process.

Examples of simulation experiments with the control variable method
In accordance with the last algorithm, the following simulation experiments involve parameters with the following values: = 3 0 0 3 ⊗ I 8 , K = 26 0 0 12 ⊗ I 8 .
Two kind of trajectories are given as follows: Note that formation F and formation F 1 represent symmetric and asymmetric formations, respectively, where

Performances in reaching formation F
In the absence of a leader, the formation performance of MLS with respect to F can be seen in this example. As shown in Fig. 3, this example achieve the final state. Figure 6a and b show the evolution with respect to the position coordinates of the 8 agents, and MLS reaches the stabilization and stays in the steady state. Correspondingly, Figs. 6c and d give the convergence of the velocity components of each agent, respectively. Figure 7 shows the results of this example simulation experiment. In addition, only the agents v 1 , v 2 , v 3 , and v 4 belonging to the same side are considered. This simulation example gives their relative distances in reaching the formation as shown in Fig. 8a and b, which is consistent with the explication given by Remark 3.4 in Sect. 3. Furthermore, with Fig. 3, it can also be seen that the center positions of any two agents linked by negative edges can be determined and that their final formation is center-symmetric.

Performances in bidirectional tracking trajectory tr 1 with formation F
Control experiment with simulation experiment 4.3.1: the parameters used in MLS of simulation example 4.3.1 are not changed, and a virtual leader is added to MLS with the network topology G 1 is required. Set the trajectory of the virtual leader as tr 1 , which allows MLS to achieve the complex behavior of bidirectional tracking tr 1 with F 1 formation. From the graph 4, the position relationship between any two agents remains consistent with Remark 3.4. Likewise, the relative distance curves between any two of the agents v 1 , v 2 , v 3 , and v 4 are given by Fig. 9. However, the velocities of this system(see Fig. 9c and d) appeared disturbed after reaching stability and then returned to equilibrium. The process is illustrated visually in Fig. 10.   Fig. 5. The cooperative velocity of the system is shown in Fig. 12a and b, and the system is less sensitive to joining the reference trajectory tr 2 compared to joining the reference trajectory tr 1 . There is also the problem of the lack of adaptability of the interior coupling coefficients. These situations are reflected by the lack of adaptability of the interior coupling of the system after adding the reference trajectory tr 1 . This is caused by the fact that the kinetic parameters θ i are variably selected. Adjusting the appropriate control coefficients K , is an effective way to deal with this problem and make the convergence process stable. In general, different control coefficients K , bring different convergence processes with different stability. Further, estimating the control coefficients and adding their estimation errors to the control protocol is a more powerful way. As a result, the control coefficients are continuously adjusted during the control process to maintain the stability of the convergence of the systems.   bipartite network topology can be used to construct both symmetric and asymmetric forms in formation control problems. It was therefore demonstrated that, in addition to including complete consensus control, bidirectional formation control allows the systems to remain in formation and to track trajectories in a bidirectional way, either backwards or towards each other.

Conclusions
This paper introduced methods to realize the FI consensus control of MLS. Under a bipartite topology, MLS completed bidirectional tracking with formation control in a distributed and coordinated way in torque mode. These studies were conducted in two main topics of formation: formation reaching and trajectory tracking under formation keeping. The theory part included complete consensus and bipartite consensus, which resulted in a bidirectional formation control method. In parallel with the theoretical part, four examples of MLS were presented in a series of control experiments. The final results were accompanied by the achievements of tracking with both symmetric and asymmetric formations. After comparing the theoretical and experimental results, the effectiveness of the proposed bidirectional formation control method in this paper was verified in the two simulation experiments. It was found that although the two simulation experiments showed good performance in producing formations, they were prone to internal perturbations in the case of trajectory tracking with a virtual leader. Therefore, the research work on the bidirectional FI consensus control of MLS needs to continue to improve the anti-perturbation capability, and the adaptive coupling coefficients should be further developed to improve the stability of the system.