Control of 2D Minimally Persistent Formations with Three Co-Leaders in a Cycle

This paper mainly solves the problem of 2D minimally persistent formation control in which three co-leaders are in a cycle, which raises a great challenge in the area of persistent formation control. First, a novel control law is proposed for this problem. The fundamental moving principles of the agents are well-designed based on the property of persistence, and the non-square rigidity matrix is converted into the square one for the design of the control law. Then, the method of leading principal minor is utilized to prove that the formation with the above control law can be stabilized. Finally, simulation results show that the proposed controllers are able to stabilize the group formation to a rigid shape, while keeping the distances between the agents to the desired value.

The underlying graph of the directed formations may be either cyclic or acyclic. Formation control with an acyclic graph is easy dealt with because of its particular structure, where the follower agents cannot influence the leader agents. However, formation control with a cyclic graph has well-known difficulties, since the control of the leader agents may be affected by the follower [18]. In the area of distance maintenance control with a cyclic graph, some authors have made some contributions [15,[23][24][25]. Baillieul [19] proposed a method of formation control considering the distance measurements in a cyclic structure, but he did not research how to maintain the distance of all the agents. Hendrix [15] then discussed the possibility of keeping the distance between every pair of agents constant in a cycle graph and raised the concept of minimally persistent formation, but the specific control of the whole formation was not analysed. Furthermore, Yu [17] researched the special control law in a minimally persistent formation with a leader-first-follower variety in the plane and tried to design the control law using a rigidity matrix. However, he only considered the oneleader situation in which the followers are in a cycle. Moreover, in the research of leaders in a cycle, Anderson [25] considered a control law for a formation with only three agents named three co-leaders, but he did not explain the topology relationship of the followers.
To the best of our knowledge, all the aforementioned papers share the following common drawbacks: oneleader model for minimally persistent formation cannot complete the complex mission very soon and it is difficult for one leader to find out the goal in a complicated environment as soon as possible. Consequently, the one-leader model is an important factor limiting the performance of the whole formation. In addition, all the algorithms based on the one-leader model seem to have poor scalability and lack adaptability and flexibility to both tasks and environment.
To overcome the aforementioned drawbacks, a multileader architecture is necessary. Yu [17] mentioned that only the minimally persistent formation systems with three leaders can be stabilized. Therefore, the problem of minimally persistent formation control with three co-leaders is discussed in this paper. In the three co-leader model, the three co-leaders are equivalent. In addition, the non-square rigidity matrix, which indicates the distance of all the agents, is introduced in the design of the control law to deal with the timeliness problem.
The primary contribution of this paper is the proposing of a novel control law using a non-square rigidity matrix for minimally persistent formations, in particular, under the condition that the three co-leaders are in the cycle. Firstly, since the leaders and followers have different types of the moving principle, we devise different control laws for them respectively. Secondly, during the course of designing the control laws, the novel method of converting the non-square rigidity matrix into the square one is discussed in detail. Thirdly, it is necessary to consider the situation where the followers are in a cycle and when they are not. Finally, we do the simulations in which the followers are in the acyclic and cyclic graphs to prove the efficiency of the proposed control law.

Graph rigidity
Before introducing persistent formations in a directed graph, it is necessary to know the underlying rigid formations in an undirected graph. Let ( , ) G V E  be an undirected graph with n vertices; denote the composite vector 2  i j p p . In addition, the introduction of the rigidity matrix in [11] claims that: Assume that the trajectory is smooth, then we could get the following expressions from (1): where i p  is the velocity at point i p . Then, we obtain the following homogeneous equation by combining (2) at different points: In addition, the rigidity function is described as follows for another definition of rigidity matrix:   Lemma 2.1 tells us that some frameworks are rigid but not infinitesimally rigid. However, if the framework is infinitesimally rigid, then it is sure to be rigid. Fig. 1 illustrates these properties with two examples. It is easy to compute that ����� � � � ��� � �� � � in Fig. 1 (a) and ����� � � � ��� � �� � � in Fig. 1 (b), so Fig. 1 (a) is rigid and infinitesimally rigid; Fig. 1 (b) is rigid but not infinitesimally rigid as p is not a regular point. In general, the rigid graphs which fail to be infinitesimally rigid almost have parallel or collinear edges. In this paper, "rigid" almost always means "infinitesimally rigid". Lemma 2.2 [28] Consider a formation F in � � , with agents in generic positions and with defined agent pairs having the inter-agent distances maintained. Let � � ��� �� be the undirected graph. Then F is rigid if and only if there is a subset � � � � satisfying the following two conditions: 1. |� � | � �|�| � �� 2. For all � �� � � � � � �� � �� |� �� | � �|��� �� �| � �, where |��� �� �| is the number of vertices which are the end vertices of the edges in � �� .

Persistent and minimally persistent graph
Rigidity is the property of the undirected graph, and persistence is the corresponding property of the directed graph.

Lemma 2.3 [15] A representation is persistent if and only
if it is rigid and constraint consistent. A graph is generically persistent if and only if it is generically rigid and generically constraint consistent.
From Lemma 2.3, we know that rigidity and constraint consistence are the crucial factors for the persistence of a directed graph. However, in the plane, the rigidity of a directed graph is actually the rigidity of the corresponding underlying undirected graph, and the constraint consistence means that every agent should satisfy all their own distance constraints. As a result, the graph in Fig.2 (a) is persistent and the graph in Fig.2 (b) is not persistent as node 2 is not constraint consistent (one node is able to satisfy two distance constraints at most in the plane). The arrows in the figures indicate the "leading" relationship, instead of the direction of the information flow which is commonly used in graph persistent theory [14]. As a result, the arrows indicating node 4 in Fig.2 (a) mean that node 4 is the "leader" of node 1, 2 and 3; node 1, 2 and 3 can only know the information of node 4.
The minimal rigid graph requires that the graph has the least edges to satisfy the rigid conditions, and if one edge is removed, the graph will not be rigid. A comparative relationship between the rigidity and persistence is that the conditions of the minimally persistent graph are more complicated than that of the rigidity graph.
Lemma 2.4 [15] Consider a directed graph with more than one vertex. Then it is minimally persistent if and only if the underlying undirected graph is minimally rigid and no vertex has more than two outgoing edges. Lemma 2.3 and Lemma 2.4 tell us that the number of the vertices in a minimally persistent graph is always 2n-3 (n represents the number of nodes).

Formation control laws
An example of minimally persistent formation with three co-leaders is shown in Fig 3 in which agent 4, 5 and 6 are the leaders in a cycle and each of them has one Degree of Freedom (DF). The DF of any of the followers 1, 2 and 3 is zero. All the followers are directly or indirectly controlled by the leaders to fulfil the distance constraint, i.e., the motion of agent 1 is constrained by agents 4 and 6; while the motion of agent 2 is constrained by agent 4 and agent 1. In this paper, the first order kinematic model is adopted for every agent:

Control laws for followers
For n agents, the index of all the followers is defined from 1 to n-3. When one agent � � has to maintain the constant distance ( � ���� and � ���� ) from two leader neighbours ( � � and � � ), it has two choices (shown in Fig. 4). Agent � � may choose � ��� or � ��� , as its targeted position since both of them have the same distance ( � ���� and � ���� ) to � � and � � . However, the position � ��� is more feasible than � ��� , due to a short moving distance. Therefore, � ��� is chosen as the target position to be reached for the follower agent � � . All the followers in this paper are guided by such a principle.
where i K is the gain here.
Suppose that the position of agent i p can be expressed as ) is the desired position which satisfies the dynamic constraint on the distance for the ith agent, and ( )  means a small variable. From Fig.4, by using the cosine law to the triangle _ i j i o p p p and neglecting the non-linear second order terms, a linear equation can be obtained as follows: As a result, the control law of the followers is represented by: is the sub-matrix of the rigidity matrix R. (Please see [17] for more details).
So, for all the followers, the control law can be obtained as follows: where both K and e R are 2 2  diagonal block matrices. e R partly represents the rigidity of the formation.

Control laws for co-leaders
There are three co-leaders indexed by n-2, n-1, and n, and they lead each other. The minimally persistent formation controllers for three co-leaders are proposed as: where n k , _ 2 e n R  is the sub-matrix of e R corresponding to node n-2. Refer to (10), the _ 2 e n R  becomes: Each row of _ 2 e n R  is derived based on (10). As a result, the control law for 2 n p  is: Similarly, we get:

Control laws for the whole formation
Without loss of generality, any 2D linear motion can be equivalent to a two-step motion consisting of translation along the x-axis and y-axis. To simplify the analysis process, a two-step translation along the x-axis and yaxis are used to substitute the linear motion in this paper.
For the first step of the translation along the x-axis, the leader who firstly discovers the goal is defined as node n. In order to realize the x-axis translation, let the values of , So we get that: Likewise, for the second step of the translation along the y-axis, we arrive at the following conclusion: Since the translation along the x-axis and y-axis is equivalent, we can only discuss the case of translation along the x-axis here.
With (11), (21), (22) and (23), we obtain a new control law: where  Then the set of all the follower nodes is denoted by ʹ V and a subset of the follower node set is ʹ Here, we suppose that the two outgoing edges from node i are { , } p p i j and { , } p p i k , so:

Stability analysis
In section 3, we propose the formation control laws using the minimally persistent directed graph. In this section the stability of the control laws will be discussed.
The control laws will be expressed by: , , , n P p p p   at generic positions, the three co-leaders are indexed by n, n-1 and n-2.  R is the matrix obtained by removing the (2n-4), (2n-2), (2n-1) and 2n columns, and (2n-2) row, from the rigidity matrix R . Then a diagonal block matrix  Re K exists, such that the formation control laws (31) with the minimally persistent are stable.
Proof. In Theorem 3.2 in [17], we obtain the conclusion that if every leading principal minor of � � is nonzero, then a diagonal Λ exists such that the real parts of the eigenvalues of Λ� � are all negative. The leading principal minor of � � is denoted by � � �� � ������ � � � � ��.On account of the sequence appearing in the leading principal minor, only the last appearing node needs to be considered here. Suppose that the last appearing node is � � , and its virtual leaders are � � � � � .The edges from � � are{� � � � � } and {� � � � � }.
If � � � and � � �, which means that the index of both � � and � � are smaller than that of � � in the sequence of the leading principal minor, then this leading principal minor is ʹ | ( )| f R V . As analysed in [17], it is known that ʹ ( ) f R V is non-singular, so the determinant here is nonzero.
When both of them are followers, then it is similar to step (1); when one of them is the follower and the other is a coleader, then it is similar to step (2); when both of them are co-leaders, then the leading principal minor is: (1,2,3,..., 1) where X is a don't-care vector and , x y are the x-axis and y-axis coordinates. So it is also nonzero.
To sum up, the leading principal minor of all the followers is nonzero. Then the leading principal minor of the co-leaders is provided as follows: Both of them can be transformed into the form as follows: (1,2,3,..., where k  is the real coefficient gain and x indicates the real number, such as 1 n x  , 2 n x  and n x . As a result, they are nonzero. Therefore, the leading principal minor of  R is nonzero. From Theorem 3.2 in reference [17], we know that a diagonal  Re K exists such that the real parts of the eigenvalues of (31) are all negative, Theorem 4.1 holds.

Simulations
In this section, simulations using the control laws designed in this paper are presented. Mobilesim software is used to conduct the simulation platforms.
In the simulations, the topology of the mobile robot system is the same as in Fig. 3, in which robots 4, 5 and 6 are defined as the leaders, and the others are the followers. (c) Time=7s (d) Time=12s Figure 5. Simulation snapshots for a three co-leader minimally persistent formation with followers in an acyclic graph.
As analysed in section 3.3, to simplify the analysis, the simulations consist of two stages which are moving along the x-axis and then along the y-axis. In addition, two kinds of follower topology are studied, which are cyclic and acyclic.
In the simulations, the gains applied in the controller are defined as follows: Node �1,2,3� � � � , and their topology is acyclic, so based on Theorem 5.2 in [17], we know that all the eigenvalues corresponding to node �1,2,3� are -1. So only the eigenvalues corresponding to the leader needed to be adjusted by  K .   Fig.5 (c) depicts that the robots have adjusted their orientation along the y-axis. In Fig.5 (d), all the robots finally arrive at the destination.
Then, another situation is considered where the topology of the follower set �1,2,3� is cyclic. The snapshots are shown in Fig 6. (a) Time=0s (d) Time=12s Figure 6. Simulation snapshots for a three co-leader minimally persistent formation with followers in a cyclic graph.
The initial positions of the robots are defined as follows:   So it can be proved that this situation is also stable.

Conclusions
This paper mainly dealt with the problem of minimally persistent formation control with three co-leaders, and the minimally persistent formation control law for the case where three co-leaders are in the cycle was proposed. The non-square matrix, which mostly represents the characteristic of the persistent formation, is successfully utilized to design the control law. In the process of designing the control laws, the straight line motion is substituted by a two-step translation motion along the xaxis and y-axis, respectively. The method of leading principal minor is adopted to prove that the system with the designed law can be stabilized. The simulation results demonstrate that whether or not the followers lie on a cyclic graph, the designed control law is effective.