Formation Control of Multi-Agent Systems with Region Constraint

In this paper, the formation problem for multi-agent systems with region constraint is studied while few researchers consider this problem. e goal is to control all multi-agents to enter the constraint area while reaching formation. Each agent is constrained by a common convex set. A formation control law is presented based on local information of the neighborhood. It is proved that the positions of all the agents would converge to the set constraint while reaching formation. Finally, two numerical examples are presented to illustrate the validity of the theoretical results.


Introduction
In recent years, more and more researchers have discussed the formation control of multi-agent systems (MASs).e formation of MASs is widely used in load transportation, satellite formation ying, and unmanned aerial vehicle formation [1]. e purpose of multi-agent formation control is to control all agents to realize and maintain a prespeci ed geometric shape.
Many control strategies have been used for formation control, such as distance-based, displacement-based, and position-based strategies [2,3].e consensus theory has also been widely used to solve the formation control [4][5][6][7].Lin et al. [8] studied the time-varying formation having shape constraints, which lie in the null space of a complex Laplacian.e necessary conditions for the formation of a multi-agent under xed and undirected graphs are given in [9].e leader-following formation control problem of second-order MASs with time-varying delay and nonlinear dynamics was investigated [10].Xiao et al. [11] developed a new control framework to solve the nite-time formation of MASs.Sun et al. [12] gave a modi ed gradient control algorithm which can achieve nite time formation stabilization of rst-order MASs. Lee and Ahn [13] proposed a formation control method which includes a combination of global orientation estimation and formation control law.In Ref. [14], adaptive formation control of MASs is proposed, which has an unknown leader.e time-varying formation problem was researched in previous studies [15][16][17].Xia et al. [18] proposed an optimal formation control strategy using the estimated position information.In Ref. [19], a combination of attractive-repulsive arti cial potential eld and a control term related to the angular information between robots was used to reduce undesired local minima.In Ref. [20], an adaptive control approach was developed to solve such a problem by using the volume condition constraints.Inspired by distance rigidity theory and bearing rigidity theory, Jing [21] developed an angle rigidity theory to study whether the shape of a planar graph can be uniquely determined by angles only.
e above research into formation control relies on formation maintenance or stabilization.Recently, formation control with constraints has been studied by some researchers.Ref. [22] designed a space constrained controller based on neural network, which provides a tool for tracking control or path planning in nonomniscient constrained space.Hernandez-Martinez et al. [23] considered the desired formation speci cation with signed area constraints, and designed a tracking control strategy using distance and area terms.A leader-follower formation control method for under-actuated autonomous underwater vehicles with lineof-sight and angle constraints is proposed in Ref. [24].Liu [25] proposed a formation control algorithm which uses distance and signed area information to ensure convergence to the required formation shape.
To the best of our knowledge, there has not been a formal article to systematically address the formation protocols in the presence of region constraints.But in real control systems, we need to control agents to the desired area.at is, to consider situations where the only concern of the nal state is a constraint rather than some speci c geometry.We focus on the formation control problem of second-order MASs with region constraints.We know that the leaderfollowing formation can solve this problem by imposing that one robot (leader) converges to the goal position in the plane (within a desired area).However, the main advantages of this scheme are that: the controller does not depend on the leader, and has good robustness; the agent can converge to the constraint region quickly; all agents can enter the constrained area accurately.Compared with previous results, the features of this paper are as follows: First, we propose a new control law with region constraints, which has the sum of a formation part and a projection part.Second, an asymptotic convergence is proved via a common Lyapunov function method.e remainder of this paper is organized as follows: Section 2 introduces the basic principles and problem statement of graph theory.In Section 3, a formation protocol with region constraint is presented.In Section 4, numerical examples are given.Section 5 contains the conclusion.

Preliminaries
Notations.‖ ‖ denotes the Euclidean norm of the vector .e 1-norm of the vector is denoted by ‖ ‖ 1 .( ) denotes the projection of the vector onto the closed convex set .
In this paper, an undirected graph G is considered with node set V = {1, 2, . . ., } and edges set E . is called a neigh- bor of node if , ∈ E .e neighbors of vertex are given by = :∈ : , ∈ E .e adjacency matrix of the graph In this paper, a system that is made up of agents (indexed by 1, 2, . . ., ) is considered.e state of agent is denoted by ∈ 2 ( = 1, . . ., ). e dynamics of each agent is described by the integrators where ∈ 2 is the velocity vector, and ∈ 2 is the control input acting on agent .
A framework is de ned as a pair (G , ) where = 1 , 2 , . . .∈ 2 .Ordering edges of E in some way, the rigidity function ( ) : 2 → |E | associated with the framework (G , ) is de ned as: (1) De nition 2.1 (see [26]).A framework (G , ) is rigid if there exists a neighborhood U of 2 such that where K is the complete graph on -vertices.
In this paper, we consider the region constraint on the formation control of multi-agents.at is, all agents are given a region constraint ∈ 2 , which limits all multi-agents to enter the constraint area while reaching formation.To do that, give a point * = * 1 , * 2 , . . ., * , * ∈ 2 such that (G , ) is in nitesimally rigid, the target formation is de ned as at is, is the set of all formations congruent with * .

Main Results
In this section, we propose a new controller to make multi-agent realize asymptotic formation.e controller has a projection part and the sum of the formation part.e new distributed formation controller is as follows: where > 0 and > 0 are two constants, * = * − * .We now propose the important result of this paper.
Theorem 1.For all > 0, and > 0, for the system (1) with the control algorithm ( 5), all agents achieve the asymptotical convergence of the formation shape, and ∈ as → ∞.

Proof. Consider the Lyapunov function of system (1) as
Its derivative along the system ( 1) is (5) .

Complexity
Because = = 1, we have Equation ( 7) can be rewritten as erefore, based on Lasalle's invariance principle, we have, → ∞, It then follows from ( 1) and ( 5) that for all ∈ V , en, we get us, we have ( 7) shape is a regular hexagon which is shown in Figure 1.In the simulation, the projection ( ) is implemented as follows: Find the shortest distance min ∈ ‖ ( ) − ‖ from ( ) to the constraint set , and the point e initial positions and velocities of all multi-agents are shown in Figure 2. e initial positions are set randomly, and the initial velocities are chosen randomly in the unit square.Figure 3 shows the motion trajectory of the multi-agent.Figure 4 depicts the nal formation shape of the multi-agent.
e convergence of the errors of the position and the region constraint are plotted in Figure 5.So we can see that the sixagent group moves into the constraint set and then achieves the desired formation shape.e errors converge to zero, but they jump to other values in a discontinuous manner.So, the convergence of the errors is asymptotic.e control inputs of the simulation are plotted in Figure 6.We can see that the control input converges to zero asymptotically.
Example 2. Simulation is performed with 155 agents moving in a 2D plane.In simulation, the parameters in algorithm (4) are also = 1, = 1.e initial positions are set randomly and the initial velocities are chosen randomly in the unit square.We take the random network [28] as the initial con guration of the MAS.e construction method of the random network is as follows: starting with a set of isolated vertices and adding successive edges between them according to probability .We take the adjacency matrix of the random network as the matrix = . Here, we choose = 155, = 0.05.

Conclusion
is paper studied the formation problem for MASs with region constraint.A formation controller law was proposed to enable all multi-agents to enter the constraint area while reaching formation.Each agent was constrained by a common convex set.We proved that the positions of all the agents would converge to the set constraint while reaching formation.Finally, two numerical simulations were carried out to illustrate the validity of the theoretical results.

F 3 :
e motion trajectory of a six-agent.

F 4 :
e nal formation shape of a six-agent.

F
of the position and the area constraint of a six-agent.

F 6 :
e control inputs of a six-agent.

10 F 7 :
e motion trajectory of a 155-agent.

F 8 :
e nal formation shape of a 155-agent.

F 9 :
e control inputs of a 155-agent.