In this paper, we study the circular formation problem for the second-order multi-agent systems in a plane, in which the agents maintain a circular formation based on a probabilistic position. A distributed hybrid control protocol based on a probabilistic position is designed to achieve circular formation stabilization and consensus. In the current framework, the mobile agents follow the following rules: 1) the agent must follow a circular trajectory; 2) all the agents in the same circular trajectory must have the same direction. The formation control objective includes two parts: 1) drive all the agents to the circular formation; 2) avoid a collision. Based on Lyapunov methods, convergence and stability of the proposed circular formation protocol are provided. Due to limitations in collision avoidance, we extend the results to LaSalle’s invariance principle. Some theoretical examples and numerical simulations show the effectiveness of the proposed scheme.
Owing to the current technological development/progress in computation and communication, distributed cooperative control of multi-agent systems (MASs) has received enormous attention from research communities in recent years. Its practical applications include search mechanisms, navigation, map manipulation, target interception, and tracking [1–4]. The main objective of cooperative control theory is to develop and design a protocol that guarantees the synchronization of a group of neighboring agents via local information exchange. Literature studies present different phenomena in cooperative control, such as consensus [5,6], formation control [7], and containment control [8].
Formation control is one of the fundamental research problems in distributed cooperative control of MASs that has presented wide application prospects, such as unmanned aerial vehicles (UAVs) [9], unmanned ground vehicles (UGVs) [10], autonomous underwater vehicles (AUVs) [11], coordination control of satellites [12], etc. In general, formation control aims to design a distributed control protocol that leads the state/output of the agent to maintain an expected shape. The formation control problem has been investigated using different control techniques in recent years. Based on [13], the techniques are classified into three strategies, namely, distance-based [14], displacement-based [15], and bearing-based [16] strategies. The consensus-based techniques are used to address the formation control problems in [17,18]. In [19], the adaptive control design is introduced to address the formation tracking problem of multiple mobile robots with unknown skidding and slipping environments. The distributed formation control problems are addressed in [20] by using an event-triggered mechanism. Based on multiple Euler-Lagrange functions, the authors in [21] investigated H∞ formation control design. In [22], a novel fault-tolerant control strategy is developed by utilizing decentralized state observers for multi-quadrotor systems. Based on the leader-follower method, a distributed adaptive-based sliding mode formation control scheme is designed for a class of second-order nonlinear MASs with unknown dynamics [23].
Among the challenging problems proposed in the formation control of MASs, the circular formation becomes a hot subject of interest because of its manifold applications, such as source-seeking exploration [24], surveillance [25], and sensor networks [26]. Circular formation control is to drive a group of agents to converge to or move on a defined circular trajectory with spacing adjustment between the neighboring agents [27]. Up to now, scholars are progressively adopting innovative research strategies and measurements to investigate the circular formation control problem of MASs. Several methodologies have been proposed to deal with the circular formation control problem, including the leader-follower technique [28], the cyclic pursuit technique [29], the behavioral technique [30], and the virtual structure technique [31]. In [32], authors developed a circular motion control law and phase-distributed protocol to achieve a circular formation for any preset relative phase. In [33,34], distributed control protocols are designed to solve the circle-forming problem of a group of anonymous agents. The circular formation stabilization of networked dynamic unicycles is considered in [35], where a distributed dynamic protocol is developed for each unicycle. Interested readers are referred to the survey paper [36] for a comprehensive review of the techniques and methodologies in circular formation control of MASs.
From a practical perspective, collision avoidance is one of the fundamental and challenging problems in formation control. A collision avoidance strategy is developed for multiple UAVs in formation flight to avoid collisions and obstacles [37]. In [38], formation tracking control with collision avoidance problem is addressed for nonlinear MASs by adopting the artificial potential approach with the neural networks technique. A novel control scheme based on adaptive neural networks is designed for a class of second-order nonlinear MASs to solve the formation control problem with multiple tasks, including obstacle avoidance, collision avoidance, and connectivity maintenance [39].
Although considerable research efforts devoted to the circular formation control problem, most of the existing results consider the single-integrator model [28,33,40], and only a few works have considered collision avoidance. Thus, it is of practical significance to study more realistic models, such as models that capture UAV systems. In this work, the circular formation problem has a wide array of practical potential applications in engineering. It has applications in the defense industry to provide surveillance and navigation of a particular area within a defined radius. It has applications in escorting and patrolling tasks of multi-robots, such as UAVs patrolling borders [41]. These facts motivate us to develop a novel distributed control scheme to achieve circular formation and meet practical challenges.
Motivated by the aforegoing observation, the problem of circular formation for the second-order MASs based on a probabilistic position is addressed in this paper. The mobile agents are required to follow a circular trajectory such that the agents in the circular trajectory must have the same direction. Compared to the existing circular formation control techniques [28,42], the leader-follower strategy and probabilistic position are combined to solve the circular formation problem, which significantly enhances the flexibility and stability of the system. The main difficulty in this paper is caused by the fact that the agents may get a tangential path after getting in a circular trajectory. By using the Lyapunov methods, convergence analysis of the designed circular formation control protocol is provided. The main contributions of this work are as follows. First, unlike [33,40], this paper considers the circular formation of second-order MASs, which makes this work more application-oriented. The second-order systems can be used to model many real systems, such as unicycle dynamics (after dynamic feedback linearization) or quadrotor UAV simplified dynamics. Second, by introducing a probabilistic position control law, a novel distributed control protocol is proposed to achieve circular formation, which is different from [43]. The probabilistic position law is proposed to represent the probabilistic position of each agent in the circular trajectory. It is shown that under the proposed control scheme the agents move along the circular trajectory of the desired radius and also avoid the tangential path. Third, the proposed control strategy guarantees inter-agent collision avoidance.
The rest of the paper is organized as follows. Section 2 represents notations and preliminaries. Section 3 formulates the circular formation problem and presents the controller design. Section 4 discusses the main results. Finally, Section 5 presents simulation results and Section 6 summarizes the conclusions of the study.
Notations and PreliminariesNotations
Throughout this paper, let Rn, Rn×n, and Z represent the sets of nth dimensional space, (n×n) real matrices, and integer numbers, respectively In denotes the identity matrix of dimension . denotes the transpose of a matrix or vector M. diag{A1,…,An} denotes a block diagonal matrix. ∥x∥ is the Euclidean norm of a vector x. The notation ⊗ stands for the Kronecker product. The subscript 𝒪 represents the orbits. 𝒞=1,2 represents the orientation of agents such that 𝒞=1 for the clockwise direction and 𝒞=2 for the counterclockwise direction.
Graph Theory
In MASs, the interaction topology is represented by a graph [44]. A graph 𝒢(𝒱,ℰ)isapairofavertexset𝒱={v1,v2,…,vn} and an edge set ℰ={eij=(vi,vj)}⊂𝒱×𝒱. The weighted adjacency matrix of graph 𝒢 is given by 𝒜=[aij]∈Rn×n where aij=1 if (vi,vj)∈ℰ and aij=0 otherwise. Neighbors set of the vertex is denoted by a set 𝒩i={j|aij>0}. The graph 𝒢 is undirected if aij=aji. The (weighted) graph Laplacian matrix ℒ∈Rn×n is represented as ℒ=𝒟−𝒜 where 𝒟=diag{di} is the degree matrix with di=∑j∈𝒩iaij. A graph 𝒢 contains a spanning tree if there exists a node from vr such that there is a path vr to any node in the graph. A graph 𝒢 is strongly connected if every pair vi, vj are connected.
Definition 1(Laplacian). The Laplacian matrix is given asℒ=[lij𝒪𝒞]where[lij]𝒪𝒞={∑𝒪,j=1n[aij]𝒪𝒞ifj≠i,𝒞=1,2,0otherwise.where the adjacency matrix is defined by 𝒜ij=[aij]𝒪𝒞fori,j=1,…,n,
[aij]𝒪𝒞={1if(i,j)∈𝒪i∪𝒪i+1.0otherwise.
Control Law DesignProblem Formulation
We consider a group of n(n≥2) agents initially located in a plane, each agent must maintain circular formation as shown in Fig. 1. The dynamics of each agent i is defined as
p˙(t)=vi(t),v˙(t)=ui(t),i=1,…,n,where pi∈Rn, vi∈Rn , and ui∈Rn represent the position, velocity, and control input of the agent i to be designed, respectively.
The desired circular formation in a counterclockwise direction, maintaining circular distance and avoiding tangential movement
Definition 2. (Circular path). The agents move in a circular formation if
∥pi−p0∥=r,i=1,…,n.
For a given desired radius vector of different circular trajectoriesri,i=1,…,nthe agents maintain circular formations with different radii if∥pi−p0∥=ri,i=1,…,n.
Ifiri=jrjthen the agents are moving in the same circular trajectory otherwise they are in a different circular trajectory.
Definition 3.(Agents direction). The agent must have the same direction of motionθi∈[0,2π],i=1,…,ninthesameorbit/circularpathsuchthat∑iθi=2πbut for different circular trajectories, the agent direction may be different.
Definition 4.(Collision avoidance condition). For given desiredαi∈θ⊂[0,2πn]. The agents in the same circular trajectory have the property of collision avoidance if∥αi−αi±1∥>0,∀i,and constant.
Definition 5.(Problem definition). For a given n(n≥2)agents,designadistributedcontrollawuifori=1,…,nsuch that each agent maintains the circular formation with collision avoidance.
To achieve this objective, let us assume that:
All the agents must move in one direction with the same constant angular velocity in the same orbit.
There exist a constant distance between agents to avoid collisions, the distance between agents is rotational in the same circle as well as different circles.
Each agent knows its initial velocities ωi(t0).
The position of each agent is presumed, because of its circular trajectory.
The dynamics of followers (trajectory and uncertainty) are stabilizable which means that the pair (A,B) is stabilizable.
Probabilistic Position
The probabilistic position law of each dynamic agent along circular trajectories modeled by the system Eq. (1) can be designed as
P(r)=n−∫02πr(1(n−1)t)ndt
Main Results
Lemma 1. [44] Let ℒ∈Rn×nbetheLaplacianmatrixofanundirectedgraph𝒢, then
If 𝒢isconnected,then0isasimpleeigenvalueofℒ,andalltheothern−1 eigenvalues are positive.
Lemma 2. (Young’s Inequality, [45]). If aandbarenon−negativerealnumbersandp,qarepositiverealnumberssuchthat1/p+1/q=1,thenab≤app+bqq.
Lemma 3. [46] Under a time-invariant information exchange topology, the continuous-time protocol achieves consensus asymptotically if and only if the information exchange topology has a spanning tree.
Lemma 4.Consider the multi-agent system (1). The system uncertainties are supposed as:
Agents not maintaining a circular path,∥pi−p0∥≠ri,∀i,
Agents may get a tangential path after getting in a circular trajectory, getting ap˙islopewhen∥pi−p0∥∥p˙i−p0∥suchthat∥pi−p0∥∈riand∥p˙i−p0∥≠ri.
Proof.
Agents not maintaining a circular path is a contradiction to Definition 2, which will affect formation. Let’s consider ri∈ZsuchthatZ=±1,±2,±3,…circleswithareaA=πr2 . Let’s assume that pi is the position of agent i and pj is the position of agent . If (pi−pj)∈ri (circular path), we have d(pi−pj)dt because the derivative across any point/line/plane is a slope A=π,4π,9π,16π,…. Let’s assume the distance covered by pi is π, then the distance covered by pj≃π, which means: if (pi−pj)≃0 is achieved it means that the consensus is achieved for i and j, pi and pj are following the same trajectory (circular).
While getting a tangential trajectory is a contradiction to Definition 2. We have d(pi−pj)dt=0, which means that the value for (pi−pj)≃0, implies that agent i is the leader following the circular path, and j follows the same trajectory.
Remark 1.Each dynamic agent must follow the circular trajectory in the same radius and avoid the tangential path.
Theorem 1. Consider the multi-agent system (1). The control law generated by the probabilistic position is designed asui=𝒞ij𝒪𝒞+k∑i∈𝒩jaij(vi−vj)such that𝒞ij𝒪𝒞=f(P),will make sure that the agents remain in the same radius and avoids the tangential path where𝒞ij𝒪𝒞represents the probabilistic position of each agent in each circle,𝒪𝒞represents the circular orbit, andkcontrol gain.
Proof. For the multi-agent system (1), the controller is designed as follows [6]:
ui=∑i∈𝒩j(pi−pj)+∑i∈𝒩j(vi−vj),in our paper, ∑i=1i≠jn(pi−pj) converges to a constant value it will make sure that agents are apart from each other while maintaining strong communication links. For general formation control design, we construct the Lyapunov function as
V(e)=eT(p,v)Pe(p,v)such that e(p,v)=[pT,vT], where e(p,v) is error function e(p,v)→0, as T→∞. Then, the derivative of V(e) is given by
V(e)=eT(p,v)Pe˙(p,v)+e˙T(p,v)Pe(p,v)
The desired trajectory is described as follows
p=cos(θ)+jsin(θ),p˙=cos(θ)−jsin(θ),θ˙=12(x32+xn−1).
The error can be calculated as e(θ)=I3−[p,p˙,θ˙]T. We have e˙T=eTAT. Thus
A=[sin(θ)−jcos(θ)1−(cos(θ)+jsin(θ))000sin(θ)+jcos(θ)1−(cos(θ)−jsin(θ)000−12(32p12+(n−1)pn−2)1−12(p32+pn−1)]
By solving the Lyapunov equation ATP+PA=−Q for any Q>0, the matrix is obtained as
P=−12×A−1=12×[1−(jsin(θ)+cos(θ))jcos(θ)−sin(θ)000−jsin(θ)+cos(θ)−1jcos(θ)+sin(θ)00012(32p12+(n−1)pn−2)1−12(p32+pn−1)]
The matrix P is positive semi-definite if and only if
{P11=1−(cos(θ)+jsin(θ))−sin(θ)+jcos(θ)>0forθ=0;P22=−jsin(θ)+cos(θ)−1jcos(θ)+sin(θ)>0forθ=(π6,15π36);P33=(32p12+(n−1)pn−2)1+12(p32+pn−1)>0forθ=(0,π1.59);
Remark 2.Theorem 1 leads to the contradiction that∑i∈𝒩j(pi−pj)convergesto𝒞ij𝒪𝒞only in the static case. Thus, Lemma 4 is true only for a static case. The results are extended to the dynamic case by applying the virtual leader-follower strategy in the next subsection.
Virtual Leader-Follower Strategy
In this subsection, the virtual leader-follower strategy is designed to deal with the system uncertainties where each follower agent tracks the virtual leader dynamics. A distributed hybrid control law is designed to ensure the formation control stabilization and consensus in presence of the uncertain trajectory with tracking error converging to zero.
Lemma 5.The dynamic of the virtual leader follows a desired circular trajectory defined asθ˙0=ω0,ω˙0=u0,where θ0∈[0,2π],ω0,andu0are respectively the rotation angle, angular velocity, and control input of the virtual leader given asu0=k0∑i=1nbi0(vi−v0),where k0isthecontrolgain,bi0=diag(B1,B2,…,Bn)denotestheconnectivityanddirectaccessofthefolloweragentito the virtual leader.
Proof. The distributed formation stabilization controller can be designed as
uform=ui+u0,and the consensus controller is proposed as follows
ucons=∑𝒪=1𝒞=1,2n∑i∈𝒩jaij𝒪𝒞((pi+pip)−(p0+p0p)).
By adding the two controllers Eqs. (8) and (9), a distributed hybrid controller is given by
ui=uform+ucons,which is equivalent to
ui=𝒞ij𝒪𝒞+k0∑i∈𝒩jbi0(vi−v0)+k∑i∈𝒩jaij𝒪𝒞(vi−vj)+∑𝒪=1𝒞=1,2n∑i∈𝒩jaij𝒪𝒞((pi+pip)−(p0+p0p)).
For 𝒞=2, we obtain
ui=∑i∈𝒩j[aij𝒪2((pi+pip)−(p0+p0p)+k(vi−vj))+k0∑i∈𝒩jbi0(vi−v0)+Cij𝒪2].
Eq. (11) can be also written in a vector form as
ui=∑i∈𝒩jℒijpij+kℒijvij+k0bi0ℒi0vi0+𝒞ij02,where pij=((pi+pip)−(p0+p0p)). Furthermore, for θ∈[0,2π], by substituting p=r(cos(θ)+jsin(θ)) and v=p˙ into Eq. (11), we obtain
ui=∑i∈𝒩jaij𝒪2[ri(cos(θi)+jsin(θi))+rip(cos(θip)+jsin(θip))−r0(cos(θ0)+jsin(θ0))+r0p(cos(θ0p)+jsin(θ0p))+k(ri(−sin(θi)+jcos(θi))−rj(−sin(θj)+jcos(θj)))]+k0∑i∈𝒩jbi0[ri(−sin(θi)+jcos(θi))−r0(−sin(θ0)+jcos(θ0))]+Cij𝒪2.
Remark 3. Under the proposed control scheme, inter-agent collision avoidance is guaranteed under the following assumptions:
All the agents move in a counterclockwise direction, i.e., orientation with constant velocity.
There exist a positive or constant relative distance between the agents, i.e.,‖pi(t)−pj(t)‖>0andconstant,∀i,j∈{1,2,…,n},i≠j,atallt≥0.
Lemma 6. Consider a graph𝒢with a directed spanning tree. The consensus is reached for the multi-agent system Eq. (1) and the control law Eq. (12) with a complex polynomial of the following formi(s)=s2+(a1+ib1)s+a0+ib0,wherea1,b1,a0,b0∈R.Thus,i(s)is stable ifa1>0and
a1b1b0+a12a0−b02>0.
Proof. Let A=(−1000.5), B=(0010), and hi=(pi,vi)T for i=1,…,n. We have
h˙i=Ahi+((ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+𝒞ij02B)h,for h=(h1T,h2T,…,hnT)T. Then, it can be written as
h˙=[(In⊗A)+(ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+d)]h,for ℒ=PJP−1, we have
h˙=(P−1⊗I2)[(In⊗A)+(ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+d]=[(P−1⊗A)+(JP−1⊗B)+k(JP−1⊗B)+k0bi0(JP−1⊗B)+P−1d]=[(I2⊗A)+(J⊗B)+k(J⊗B)+k0bi0(J⊗B)+P−1d]h,where J is the Jordan matrix given as
Jl=(ul0001⋱0000⋱0⋱10ul)nl×nl,such that ul are complex eigenvalues. By using the Leonhard-Mikhailov theorem, the condition of spanning (directed) is true only if 0<bi0(k+1)<1 and for every i=1,…,n,
f(k,bi0,ul)=(bi0/(k+1))21−bi0(k+1)[sin2(θi)−sin2(ϕi)]2×[cos2(ϕi)−cos2(θi)]2−4sin2(θi)sin2(ϕi)>0,where
ϕi=(|k+1|(∥ul∥)+|d|−sign(k+1)(Re(ul))2)1/2,θi=(|k+1|(∥ul∥)+|d|+sign(k+1)(Re(ul))2)1/2,which completes the case for a directed spanning tree.
Remark 4.In the circular orbit, we have an infinite set of points for the tangential trajectory. Let’s definehias the tangential path, for eachi=1,…,n,hi∈(−1,1). Thus,riwill no longer be the radius as the agent not following the circular path, it will be changed tosi, wheresiis the tangent on any point in the circle. The tangential trajectorysi(−sin(hi)+jcos(hi))will vary in the intervalsi∈(−1,1).
From Remark 4, for every i=1,…,n, there exists δi which defines the signal of the actual trajectory to be followed as
{δ0(hi)=r0(cos(h0)+jsin(h0))δi(hi)=r0−si(cos(hi)+jsin(hi))+ri,for i=1,…,n. Then, differentiating Eq. (14) yields
{δ˙0(θ0)=r0(−sin(θ0)+jcos(θ0))δ˙i(θi)=−si(−sin(θi)+jcos(θi))
To show the stability of the system, consider the Lyapunov function
V1(δi)=12∑i=1n(δi−δ0)2.
It is easy to check that V1 is differentiable and positive definite. By taking the derivative of V1 along the trajectories of the system, we obtain
V˙1(δi)=∑i=1n(δi−δ0)(δ˙i−δ˙0),by using Young’s inequality, for each i=1,…,n and ξ1,ξ2,ξ3,μ1,μ2,μ3,μ4>0, we have
V˙1(δi)=∑i=1n(δiδ˙i−δiδ˙0−δ0δ˙i+δ0δ˙0)≤∑i=1n((δi22ξ1+δ˙i22μ1)−(δi22ξ1+δ˙022μ2)−(δ022ξ3+δ˙i22μ3)+(δ022ξ3+δ˙022μ4)),
We take
d=δ˙i22μ1−δ˙022μ2−δ˙i22μ3+δ˙022μ4
Thus, for ξ2≃ξ3, we have V˙1≤d, which implies that V˙1 is a negative definite. Consequently, for every i=1,…,n, δi is asymptotically stable. Further, the system can achieve both formation stabilization with collision avoidance and consensus.
For every i=1,…,n, we define the error ei=(δi−δ0δ˙i−δ˙0). We discuss two cases.
Case 1: when e1=e2=…=ei,… We consider the following Lyapunov function candidate
V2(δ)=12e2,
Differentiating V2 yields
V˙2(δ)=ee˙.
Case 2: when e1≠e2≠…≠ei…, we define the error sets as e={ei,i=1,…,n}. We choose a Lyapunov function as follows
V3(δi)=12∑i=1nei2,
Taking the derivative of V3 yields
V˙3(e)=∑i=1neie˙i=ei(Aei+Bx)=ei(Aei+B(ℒijpij+kℒijvij+k0bi0+ℒi0vi0+𝒞ij02))=eiAei+eiBℒijpij+eiBkℒijvij+eiBk0bi0ℒi0vi0+eiB𝒞ij02⏟dby using Young’s inequality, we have
V˙3(e)≤ei22ξ1+(Aei)22μ1+ei22ξ2+(Bℒijpij)22μ2+ei22ξ3+(Bℒijvij)22μ3+ei22ξ4+(Bk0bi0ℒi0vi0)22μ4+ei22ξ5+d22μ5,for ξ1,ξ2,ξ3,ξ4,ξ5,μ1,μ2,μ3,μ4,μ5>0. If ξ1=−ξ2 , ξ3=−ξ4 and ξ5<<<<<; we have
V˙3(ei)≤(Aei)22μ1+(Bℒijpij)22μ2+(Bℒijvij)22μ3+(Bk0bi0ℒi0vi0)22μ4+d22μ5=d¯^≈h˙.
We conclude that the system is stable, and the error will converge to zero. We write the system Eq. (1) in generalized form as
{e˙i=Aei+Buiyi=Cei,where ei∈Rn, ui∈Rm, and yi∈Rq are the state, control input, and output of agent i, respectively. A,B, and C are constant matrices with appropriate dimensions given as follows
A=[∂δ0∂θ0∂δ0∂θi∂δi∂θ0∂δi∂θi]=[r0(−sin(θ0)+jcos(θ0))00−si(−sin(θi)+jcos(θi))],B=[0010],C=[10]
The feedback control gain k is given by k=μBBTP, such that P is the positive definite solution of the following algebraic Riccati inequality (ARI)
ATP+PA−12BBTP+2I2≤0.
Since, A−Bk is Hurwitz the pair (A, B) is controllable.
Lemma 7. Lyapunov candidate (potential function) is the error in the dynamical system Eq. (14) given as
V=eiTPei,∀i=1,…,n,whereei=(δi−δ0δ˙i−δ˙0).
Proof.
We have e˙i=(δ˙i−δ˙0δ¨i−δ¨0). The derivative of V along the trajectories leads to
V˙=2eiTPe˙i=2eiTe˙i(k11k)e˙i=2(k(δi−δ0)(δ˙i−δ˙0)+(δ˙i−δ˙0)2+(δi−δ0)(δ¨i−δ¨0)+k(δ˙i−δ˙0)(δ¨i−δ¨0)).
Applying Young’s inequality, we obtain
V˙≤2((δ˙i−δ˙0)22ξ1+k2(δi−δ0)22μ1+(δ˙i−δ˙0)22ξ2+k2(δi−δ0)22μ2+(δ¨i−δ¨0)22ξ3+(δi−δ0)22μ3+(δ¨i−δ¨0)22ξ4+k2(δ˙i−δ˙0)22μ4).
If ξ1=−ξ2 and ξ3=−ξ4, we have
V˙≤2(k2(δi−δ0)22μ1+k2(δi−δ0)22μ2+(δi−δ0)22μ3+k2(δ˙i−δ˙0)22μ4)≤d~.
The system error is given as e=(pi−p0p¨i−p¨0), and its derivative is e˙=(vi−v0ui−v˙0).
The system can be written as
e˙i=Aei+Bui=Aei+BK∑j(ej−ei)−BKbisuch that bi=D, the system can be written in compact form as
e˙i=(In⊗A)ei−(In⊗B)(ℒ⊗Im)(In⊗K)ei−(In⊗B)(D⊗Im)(In⊗K)ei=[(In⊗A)−(ℒ+D)⊗BK]ei=[(In⊗A)−H⊗BK]ei
Consider candidate Lyapunov function as
V(e)=eT(In⊗P)e.
Calculating the derivative of V, we get
V˙(e)=eT[(In⊗AT−ℋ⊗KTBT)(In⊗P)]e+eT[(In⊗P)(In⊗A−ℋ⊗BK]e=eT[(In⊗(ATP+PA)−ℋ⊗(KTBTP+PKB))]e=eT[In⊗(ATP+PA)−ℋ⊗(2PBBTP)]e=eT[In⊗(ATP+PA)−Oλ⊗(2PBBTP)]e=eT[In⊗(ATP+PA)−λi⊗(2PBBTP)]e=eT[In⊗(ATP+PA)−δ⊗(2PBBTP)]e.
Thus, we have
V˙≤−δ∑ieiTei,for e≠0, we have V˙<0, which implies that the system is globally asymptotically stable. The pair (A,B) is stabilizable. Hence, the consensus is reached, i.e., limt→∞∥pi−p0∥=εp, limt→∞∥vi−v0∥=εv.
Since the graph 𝒢 is connected, we have
ui=K∑j(pi−pj)+Kbi(pi−p0),,for a connected graph to solve the Riccati equation. For ℋ to be symmetric there exists an orthogonal matrix Oℋ such that OℋℋOℋT=Oλ=diag(λ1,λ2,…,λn).
Theorem 2.Consider the multi-agent system (1). The dynamic control law Eq. (12) under assumptions solves the circular formation control problem.
Proof. Appendix A.
Remark 5.Theorem 2 leads to a contradiction with the Assumptions proposed. Since the virtual leader velocity may not be constant (changing velocity) the agents may maintain circular trajectories but with weak collision avoidance between interacting agents.
An Extension to Lasalle’s Invariance Principle
In this subsection, the results are extended to LaSalle’s invariance principle due to the limitations of the virtual leader-follower strategy Theorem 2. Consider a function
L=f(θ), L=r(cos(θ)+jsin(θ)) for 0≤θ≤2πn, we have −r≤L≤r, these function limits can be divided as
For 0≤L≤r , the function L is Lyapunov stable;
For −r≤L≤0, the function L is unstable.
For this Lyapunov strict function, we use the LaSalle principle. LaSalle will make sure that the function with −r≤L≤0 is globally stable. For i=1,2,…, ei=r0+ri(cos(θi)+jsin(θi)), the function is Lyapunov stable for positive ei. We have ei negative for θ<3π4+nπ. We choose the Lyapunov function as V∗(ei)=V(ei)+u(ei) where V(ei)=eiTθei. The derivative of V∗ along the trajectories is given by
V˙∗(e)=V˙(ei)+u˙(ei)=δeTθδ˙e+δ˙eTθδe+u˙(δe).
We have e˙i=(A⊗In)ei+(B⊗In)u. Suppose that (B⊗In)u=u(ei), we get
V˙(ei)=eiT(θ(A⊗In)+(AT⊗In)θ)ei≤0.
Since θ(A⊗In)+(AT⊗In)θ=−Q from algebraic Riccati equation, Q is positive definite if and only if ei is positive. In addition, we have
u˙(ei)=−r0(cos(θ0)−jsin(θ0))+k0r0(jsin(θ0)+cos(θ0)),
u˙(ei) is positive for every θ0. Thus V˙(ei)∗≤0 when |V˙(ei)|≥|u˙(ei)|. The results are presented in the following Theorem.
Theorem 3. The dynamic control law solves the circular formation control problem for max(R)=1,wheretheprobabilityvalueR=∑iaij02(rip(cos(θip+jsin(θip))+rop(cos(θop))+jsin(θop))+Cij02.
Proof. We consider the following system
e˙=Ae+Bu,where the vector e=[δi−δ0δ˙i−δ˙0], we have
ri(cos(θi)+jsin(θi))=ri(1+(cos(θ1)+jsin(θ1))+…+(cos(θn)+jsin(θn))+…=ri[∑θ=0n(cos(θi)+jsin(θi))]=ri[∑θ=0n[Θ1⋮Θn]]=ri[In⊗Θ].
In the same way, we get
{(cos(θ0)+jsin(θ0))=(Im⊗Ψ),(cos(θip)+jsin(θip))=(Il⊗Γ),(cos(θ0p)+jsin(θ0p))=(Im⊗Δ),(−sin(θj)+jcos(θj))=(Im⊗Φ),(−sin(θi)+jcos(θi))=(Im⊗Λ),(−sin(θ0)+jcos(θ0))=(Im⊗Ξ),
Thus, by substituting Eq. (18) into (18), we obtain
e˙=(In×n⊗A)e+In×1⊗(∑i∈Njaij01(ri(Im⊗Θ)−r0(In⊗Ψ)+k(ri(Im⊗Θ)−rj(Il⊗Φ))))+k0∑j∈Nibi0(ri(Im⊗Λ)−r0(In⊗Ξ))+R)where R=∑iaij02(rip(cos(θip+jsin(θip))+rop(cos(θop))+jsin(θop))+Cij02 is the probability value.
We analyze two cases: If max(R)=1, it means that the follower agent follows a circular trajectory with all predefined conditions. If min(R)=0, it means that the follower agent is not following any agents, or the agent is not following the desired formation with necessary conditions.
Remark 6.In the current system, the probability value has a fundamental rule in making the follower agents follow the circular trajectory.
Let P(s) is a probability function. The function will always result for 0≤P(s)≤1, P(s) for any other value is referred to as an error function
P(s)=]0,1[.
P(s) is not following the circular trajectory. Thus the system is unstable. We have to define an error threshold for Pf={0,1}. Let consider P0={1} and Pf=[0,1] with respect to circular trajectory. To prove that the error is not affecting the system. Consider the following Lyapunov function candidate
V(P)=12∑i=1n(P0−Pf)2.
The function V(P) is differentiable and positive definite. Then, the derivative of V(P) is given by
V˙(P)=∑i=1n(P˙0−P˙f)(P0−Pf)
Since (P0−Pf)≤0.05, we have that V˙≤0. Thus, if the probability is more than 0.95 then the system will remain stable. So the system should have 0.05 uncertainty.
Simulation Results
This section presents the simulation of three agents (n=3) modeled by Eq. (1), controlled by the proposed control techniques above. The communication topology of agents is shown in Fig. 2. The system matrices are given as follows
A=[010000000],B=[001],ℒ=[2−1−1−12−1−1−1−1],and the matrixH=(ℒ+D)=[3−1−1−13−1−1−13].
Graph interaction topology of two follower agents with a virtual leader in the same radius
Fig. 3 shows the trajectories of the agents uer control law (5). P33 depicts the trajectory of the agent in the static case, where the agent follows a circular trajectory. P11 and P22 depict the trajectories of the agents in the dynamic case, where the agents follow tangential paths. It shows that the agents follow the desired circular trajectory in the static case. It also shows that when the agents move and change position and velocity, they may follow a tangential path or follow a circular trajectory.
The trajectories of <inline-formula id="ieqn-195"><mml:math id="mml-ieqn-195"><mml:msub><mml:mi mathvariant="bold-italic">P</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mi mathvariant="bold-italic">P</mml:mi><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>, and <inline-formula id="ieqn-196"><mml:math id="mml-ieqn-196"><mml:msub><mml:mi mathvariant="bold-italic">P</mml:mi><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
Fig. 4 shows the different positions of agent i along the same circular trajectory in the plane under the proposed control protocol. It is shown that when the agent moves the tangential path is controlled but the agents can have different circular trajectories. The designed controller protocol ensures that the agents follow desired circular trajectory and avoid the tangential paths.
The different positions of the agent in the same circular trajectory under the control law
In the simulation, we consider a system consisting of three agents (n=3), and we set the center point of the circular trajectory at (0, 0) on the plane. The initial positions of the agents are generated randomly. The design parameters are set as r=rip=2, 𝒞ij𝒪2≅1 demonstrated in Fig. 5. As shown in Fig. 5, the simulation results indicate that the group of mobile robots move in a counterclockwise direction to form and maintain the spaced circular formation under the proposed control law (13).
The circular formation trajectories of agents in the plane
The simulation results indicate clearly that the proposed control scheme solves the circular formation problem while no collision occurs among agents.
Conclusion
In this paper, a novel formation control scheme is introduced to study the circular formation problem for second-order MASs in the plane. The problem has two sub-objectives: circular formation control and collision avoidance. First, by combining probabilistic position law with a leader-follower strategy, a novel distributed control protocol is developed to achieve circular formation. It is proved that under the developed control scheme all the agents achieve a circular formation with the desired radius and also avoid the tangential path. Under the proposed control protocol, inter-agent collision avoidance is guaranteed by keeping the same counterclockwise direction of the agents with constant velocity and preserving a positive or constant distance between any two agents. Based on Lyapunov methods, the stability analysis of the designed circular formation scheme is provided. The effectiveness of the proposed control strategy is illustrated in the numerical simulations. Future works will focus on extending the proposed technique to MASs with more realistic dynamics. Moreover, how to achieve circular formation in different circular radii is an open and challenging research topic that needs investigation.
Funding Statement
The authors received no specific funding for this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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Appendix A
From Eq. (12), we have
u=δ¨=∑i∈𝒩jℒijpij+kℒijvij+k0bi0ℒi0vi0+𝒞ij02.
Let φ=(δ˙,δ¨)T. Let A=(−1000.5) and B=(0010), one can obtain
φ˙=Aφ+((ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+𝒞ij02B)φ.
For a system solution for a single agent, the system will be of the form
g˙=Ag
Let g=[g1,g2]T . Let φ=[φ1T,φ2T,…,φnT]T, it yields that
φ˙=[(In⊗A)+(ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+d)]φ.
Let J be the Jordan form of the matrix associated with laplacian ℒ. We have ℒ=PJP−1 for P invertible matrix. Consider h=(P−1⊗)I2)C. The derivative of h is given as
h˙=(P−1⊗I2)[(In⊗A)+(ℒij⊗B)+k(ℒij⊗B)+k0bi0(ℒi0⊗B)+d]h=[(I2⊗A)+(J⊗B)+k(J⊗B)+k0bi0(J⊗B)+P−1d]φ,such that h=(P−1⊗I2)φ .
Remark 6.For a directed graph𝒢such thatuPcomplex eigenvalues of Laplacian would be complex andJ=diag(J1,J2,…,JP)whereJp=(up0001⋱0000⋱0⋱10up)np×np, such thatuPcomplex eigenvalues.
Let P=(P1,P2,…,Pn), P−1=(q1,q2,…,qn), h=(h1T,h2T,…,hnT)T, and hi=(hi1,hi2), it yields that h˙i=Ah. For a directed graph with a spanning tree P=1nn and ℒP=PJ. To achieve the desired formation, h must satisfy limt→∞∥hi∥→θ,e. In fact, we have
limt→∞∥φ−1Nh+d∥=limt→∞∥(P−1⊗I2)h−1Nh+d∥+d=0,if ℒP=PJ, then one can obtain that ∥h∥=∥(qiT⊗I2)φ∥→∥(qiT1n)⊗φ∗∥=d, where φ∗ is complex. Thus, limt→∞∥h∥=d,e.
For an asymptotically stable system (n−1) agents are asymptotically stable. Consensus is reached for the second-order multi-agent system with constant position and velocity for φ∗∈R2 such that limt→∞∥φ−1n⊗φ∥=0, 0n=P−1ℒ1n=JP−11n, q1T1n=0, i.e., ∥pi−pj∥=d and ∥vi−vj∥=e.
Therefore, it implies that
r|cos(θi−θj)+jsin(θi−θj)|=d, where d∈[0,2π[∈φ∗ and r|sin(θi−θj)+jcos(θi−θj)|=e such that e∈w,w∈[θ˙],θ˙∈φ∗.