1. Introduction

Over the past few decades, multi-agent system has attracted great interest from many fields due to its successful utilization for variety of practical applications. Flocking is an instance of self-organized entities of large number of mobile agents with a common objective. Like multi-agent system, flocking has also attracted many researchers of rather diverse disciplines such as biology, physics, computer science, animal behavior ⁽²⁾.

In 1986, Reynold ⁽¹⁾ introduced three rules known as funda- mental rules of flocking as follows: 1) Cohesion: attempt to stay close to flockmates 2) Separation: avoid collisions with nearby flockmates 3) Alignment: attempt to match velocity with nearby flockmates. For decades, many control law was released to achieve the flock behavior of multi-agent system. Up to this point, most of researches for flocking control were the implement of the Reynold rules as control objectives. For instance, consensus pro- blems are

addressed by graph theory ⁽³⁾. The author in ⁽⁴⁾ provided basic theories and two distributed algorithms for flocking control based on potential field approach to avoid collision and obstacle. The leader - follower approach was used in ⁽⁶⁾ where only few members are selected as leader who knows desired trajectory. Further, the method in research ⁽⁵⁾ figured out a framework for flocking control of double-integrator modeled agent based on virtual leaders and multiple non-homolonomic agents was consi- dered in ⁽¹¹⁾.

The aforementioned literatures just considered agents as a single point. There are several recent studies tackling the problem of circular shape agents in ⁽⁷⁾⁽⁸⁾. K D Do in ⁽⁹⁾ pointed out that in many situations in which, the circular shape results in large conservative areas. Therefore, the author considered agent shape as ellipses in order to be more appropriate for many practical cases. However, these agents fitted in ellipses may have large redundant areas so the research in ⁽¹⁰⁾ deal with the pro- blem of rectangle agents. Although problem of rectangle agents was handled in the research, the distances between agents are not always differentiable at all position of agents, the control signals were discontinuous. Hence, the algorithm in ⁽¹⁰⁾ is impracticable and not able to improve the algorithm for higher order systems. In Fig. 1, it is clearly to see that if agents have diamond shape, there exist large conservative areas and even intersect areas when agents are close together. The collision is considered to happen when there exist intersect areas although there are not really exist collision among the agents.

The main contribution of this paper is to create the flocking control law for multi-agents having polygon shapes with limited communication range. The study on polygon shape agents not only is theoretical interest but also provide the framework for arbitrary shape based on the ideal that there always exist a polygon to approximate any 2-D real entities. The control design is based on a collision avoidance condition to achieve all the flocking properties. The control signals has become differentiable despite of limitation on communication range.

The organization of the paper is presented as follow. In the next section, we put forward the description of the arbitrary poly- gonal agent and avoidance collision condition. The agent dynamic and control objective is given in section 3. In section 4, we design the control input. The simulation of agents are implemented in section 5 to demonstrate the performance of proposed control law.

Notation : $\|\cdot\|_{2}$ denotes the Euclidean norm.

2. Preliminaries

Let us consider a multi-agent system including N agents, of which the th agent is characterized by $q_{i}(t)=\left[p_{i}^{T}(t), \phi_{i}(t)\right]^{T}$, where $p_{i}(t)=\left[x_{i}(t), y_{i}(t)\right]^{T}$ and $\phi_{i}(t)$ stand for position in Earth-fixed frame $O X Y$ and its heading angle, respectively. Here, each agent is assumed to be bounded by a polygon $\Phi_{i}$ with $M$ vertices $\left\{p_{i}^{m} | m=1,2, \ldots, M\right\}$ in Earth-fixed frame $O X Y$ (see Figure 2), and has a body-fixed frame $O_{i} X_{i} Y_{i}$ with $O_{i}$. Especially, let the position of vertex m of the $i$-th agent be given as $\hat{p}_{i}^{m}$ in the frame $O_{i} X_{i} Y_{i}$. Then, the relationship between $\hat{p}_{i}^{m}$ and $P_{i}^{m}$ becomes

where the rotation matrix is formed as

$$ R_{i}(t)=R_{i}\left(\phi_{i}(t)\right)=\left[\begin{array}{cc}{\cos \left(\phi_{i}(t)\right)} & {-\sin \left(\phi_{i}(t)\right)} \\ {\sin \left(\phi_{i}(t)\right)} & {\cos \left(\phi_{i}(t)\right)}\end{array}\right] $$

$$ \dot{R}_{i}(t)=S R_{i}(t) \dot{\phi}_{i}, S=\left[\begin{array}{cc}{0} & {1} \\ {-1}& {0}\end{array}\right] $$

Thus, the time derivative of offers

Let us define as a set of all consecutive vertices of the a agent : $U=\{(1,2),(2,3), \ldots,(M, 1)\}$. The following lemma pro- poses a valuable anti-collision condition for two polygonal agents, which is used for our main approach.

Lemma 1 (Anti-collision condition) : Given two polygonal agents $i$ and $j$ that are outside of each other at the initial time $t_{0}$, there is no collision between both two agents and if and only if the relative distance $∆_{ij}>0 $ holds, for all $t>t_{0}$, where

Proof : It is worth noting that the triangle inequalities $z_{i j}^{m k}>0$ implies non-negative $\Delta_{i j}$. From definition (3), the relative distance $\Delta_{i j}$ equals zero if at least one value of $z_{\ddot{y}}^{m n k}$ is zero. In addition, two equalities $z_{i j}^{m k}=0$ and $z_{j i}^{m k}=0$ hold if and only if there exist a vertex of polygon $\Phi_{i}$ lay on an edge of agent $\Phi_{j}$ or vice versa. Hence, the condition $\Delta_{i j}>0$ holds for all $t>t_{0},(i, j) \in\left(S_{N} \times S_{N}\right), S_{N}=\{1, \ldots, N\}$, there are no collisions among the group of agents.

3. Problem statement

Like the aforementioned problems in section 1, the main work of this paper focuses on the difficulty in designing flocking control law for multi polygonal agents. Hence, for simplicity, let us consider the group of N agents, each agent is modeled a first order dynamics as follows

where $u_{i}=\left[u_{i x}, u_{i j}, u_{i \phi}\right]^{T}$ is control input of agent. The control input vectors of all agents is designed to achieve the following properties: 1) p-times differentiable, 2) no collision between all agents, 3) the velocity of each agent converge to rendezvous velocity, 4) the flocking trajectory error $e(t)=\sum_{i=1}^{N}\left\|q_{o d}-q_{i}\right\|_{2}$ is bounded. To achieve the control objects, Let us assume that each pair of agents $i$ and $j$ has the circular communication area with radius $R_{i}$ and $R_{j}$ centered at $O_{i}$ and $O_{j}$, respectively, where these agents broadcast their position and heading angular. The radius $R_{i}$,$R_{j}$ satisfy the following condition

for all $1 \leq i, j \leq N$ where $d\left(O_{i}, p_{i}^{m}\right)$ is a distance between center $O_{i}$ and vertex $p_{i}^{m}$. Moreover, agent $i$ can receive the position and the heading angle of agent $j$ if the centered point $O_{j}$ is in the circular communication of agent $i$. At the initial time $t_{0}$, the agent $i$ and $j$ are out of each other for all $(i, j) \in \square$ and $i \neq j$. In addition, all agents are able to know the information of the flocking rendezvous trajectory $q_{o d}=\left[\begin{array}{lll}{x_{o d}} & {y_{o d}} & {\phi_{o d}}\end{array}\right]^{T}$. In addition, $q_{o d}$ and its derivative $\dot{q}_{o d}$ are bounded all time.

4. Control design

In this section, a kind of pairwise potential function $\varphi_{i j}\left(\Delta_{i j}, \alpha_{i j}, \beta_{i j}\right), \Delta_{i j}, \alpha_{i j}, \beta_{i j} \in \square$ is proposed to, represent the impact of agent $i$ on agent $j$, which has following properties

1) $\varphi_{i j}$ is -times differentiable function respect to

$\Delta_{i j}, \forall \Delta_{i j} \in(0,+\infty)$,

2) $\varphi_{i j} \geq 0, \forall \Delta_{i j} \in(0,+\infty), \lim _{\Delta_{j} \rightarrow 0^{+}} \varphi_{i j}=+\infty$,

3) $\partial \varphi_{i j} / \partial \Delta_{i j}=0, \forall \Delta_{i j} \in\left[\beta_{i j},+\infty\right)$,

4) $\varphi_{i j}$ has unique minimum point at $\alpha_{i j}$ and $\varphi_{i j}=0 \Leftrightarrow \Delta_{i j}=\alpha_{i j}$.

Based on above properties, a scalar functionis given

where

$\lambda_{i j}$ is positive constant and $\alpha_{t j}, \beta_{i j}$ is selected as

in which, $\Delta_{j d}$ is desired relative distance between agents, $h\left(\Delta_{i j}, \alpha_{i j}, \beta_{i j}\right)$ is a bounded scalar smooth step function in ⁽¹¹⁾. Fig. 3 presents an example of the potential function. The partial derivative of $\varphi_{i j}$ with respect to $\Delta_{i j}$ is:

For calculating, let take a time derivative $\dot{z}_{i j}^{m n q}$ and note that $\left\|p_{j}^{n}-p_{j}^{k}\right\|_{2}$ is the length of a edge of agent $j$ and S is skew matrix $\left(x^{T} S x=0, \quad \forall x\right)$

where

The combination of and offers first time derivative of $\Delta_{i j}$ as:

where

Because of $\varphi_{i j}=\varphi_{j i}$, the potential function $\varphi$ is the sum of all pairwise potential function $\varphi_{i j}$ with $1 \leq i, j \leq N$ as following form:

From (7), (13) and using the following property

the time derivative both side of is given as follows

This section uses Lyapunov directed method to design a control law. The control Lyapunov function is constructed from the proposed potential function to guarantee no collision in the group of agents and adds sum of square error to obtain the boundedness of the flocking trajectory error

where $k_{i}^{p}, k_{i}^{\phi} \quad i=1,2, \ldots, N$ are positive constant, $Q_{i}$ is invertible matrix depend on $\phi_{i}$ such that

By selecting

time differentiation of both sides of is given by

where

The control law $u_{p i}, u_{\phi i}$ is selected such that the right side of is negative define function. In other word, to reduce the magnitude of control inputs when two agents are too closed, let us use the scalar smooth saturation function $\sigma(x)$ such that:

1) $\sigma(x)$ is a smooth function,

2) $\frac{\partial^{(k)} \sigma}{(\partial x)^{k}}$ is bounded , $\forall k \in \square, \partial \sigma / \partial x>0 \quad \forall x \in \square$,

3) $x \sigma(x)>0 \forall x \neq 0$ and $\sigma(x)=0 \Leftrightarrow x=0$,

by which, the control inputs are chosen as the following

where $\gamma_{i x}, \gamma_{i y}, \gamma_{i \phi}$ are positive constants, from (22), we can obtain that

Theorem 1 : The group of $N$ agent , each agent is steered by the control input , achieves the control objective in section 3. In particular, the following results hold:

1) Each agent in the group is capable of avoiding the others in the group.

2) The Euclid distances between agent and the rendezvous trajectory is bounded $\left\|q_{i}(t)-q_{\alpha d}(t)\right\|<\infty$.

3) The generalized velocity of agent $i$ asymptotically approaches the one of the others as well as the generalized flocking rendezvous velocity

Proof : From , it established that $\dot{V} \leq 0$. Hence, $V(t)$ is a non-increasing function respect to $t$. By integrating both sides of from initial time $t_{0}$ to $t$, we derive that $V(t) \leq V\left(t_{0}\right)$. It is worth noting that $q_{od}$ and $\dot q_{od}$ are bounded, we can drive the boundedness of $V(t)$. Further, it implies that $\| \phi_{i}-\phi_{o d} |$ is bounded for all $i$. The boundedness of $\varphi_{i j}$ considering for all $(i, j)$, $i \neq j$ it implies that there are no collisions among the group of agents. Thus, statement 1 and 2 of theorem 1 is proven.

Because $q_{i}(t)$ is bounded, it find easy to see that $\dot{V}$ is also bounded. Taking derivative of $\dot{V}$ we have:

Where $\xi=\left[q_{1}^{T}, q_{2}^{T}, \ldots, q_{N}^{T}, q_{o d}^{T}\right]^{T}$. We now going to proof the boun- dedness of $d \dot{V}(t) / d t$. As above proof of no collision in the group of agent, the boundedness of $q_{i}(t)$ implies that both $\Gamma_{i j}$ and $\partial \Gamma_{\tilde{j}} / \partial q$ are bounded. From the properties of saturation function $\sigma(\bullet)$, we also have boundedness of $\sigma\left(\Gamma_{\eta}\right)$, $\partial \sigma / \partial \Gamma_{j}$ and bounded control input $u_{i}$ in (24). Hence, $\dot{\xi}$ and $d \dot{V}(t) / d t$ is bounded. The application of Barbalat’s lemma in ⁽¹⁴⁾ offers $\dot{V}(t) \rightarrow 0$ as $t \rightarrow \infty$. It implies that $\lim _{t \rightarrow \infty} \Gamma_{i j}=0$. Hence, the statement 3 of theorem 1 is proven. □

5. Simulation

In this section we illustrate the effectiveness of the proposed control law presented by implementing numerical simulation. All agents in the group are the same shape. To indicate that there is no collision among the group of agents, we represent variable $D_{i}$ as geometric mean of $N-1$ relative distance from agent $i$ to the other:

In this case, we consider the system has $N=20$ agents with these dimension as diamond shape with the diagonals are 3 and 1.5 respectively. The rendezvous trajectory in the simulation scenario is $x_{o d}(t)=R_{d} \cos (t)-R_{d} / 2$, $\quad y_{o d}(t)=R_{d} \sin (t)$, $\quad \phi_{o d}(t)=0.2 t+\pi / 2$ It is presented as black dash line in Fig. 4b, 4c and Fig. 4d. Let us denote $u_{x}=\left[u_{x 1}, \ldots ., u_{x N}\right]^{T}, u_{y}=\left[u_{y 1}, \ldots, u_{y N}\right]^{T}$ and $u_{\phi}=\left[u_{\phi 1}, \ldots ., u_{\phi N}\right]^{T}$ as control inputs in Fig. 5a, 5b, 5c and the saturation function $\sigma(\bullet)$ is chosen as $x / \sqrt{1+x^{2}}$. At the initial time $t=0$, these agents were uniformly located in two circular rings as in Fig. 4a with the initial heading angles were randomly chosen as: $\phi(0)=\operatorname{rand}(1) \pi / 2$, where rand $(1)$ is the random number between 0 and 1. The communication range of all agents are equal with $R_{i}=5$ and the control parameters are selected as: $k_{p}^{i}=0.2, \quad k_{p}^{i}=0.2, \quad \gamma_{i 1}=5, \quad \gamma_{i 2}=5, \quad \gamma_{i 3}=5 \quad$ and $\quad \alpha_{i j}=0.25$ for all $(i, j) i \neq j$. In the simulation time 30s the group of agents tracks the circle form of rendezvous trajectory.

As can be seen in the Fig. 4b, 4c, >4d group of agents attempt to track the rendezvous circular trajectory $q_{o d}$. The angular velocity $\mathcal{U}_{\phi}$ in Fig. 5c asymptotically converse to $\dot{\phi}_{c d}=0.2$. From the Fig. 5a, 5b, the linear velocity of these agents $u_{x}, u_{y}$ asymptotically converse to $\dot{x}_{o d}(t)-\left(y_{i}(t)-y_{o d}(t)\right) \dot{\phi}_{o d}(t)$ and $\dot{y}_{o d}(t)+\left(x_{i}(t)-x_{o d}(t)\right) \quad \dot{\phi}_{o d}(t)$ , respectively as Theorem 1. In addition, all control inputs $u_{x}, u_{y}, u_{\phi}$ are differentiable. The presentation $D_{i}$ in Fig. 5d is greater than zero, it implies that there are no collision in the group of agents.

6. Conclusion

This paper has addressed the issue of flocking control for multiple polygonal agents with limited communication range. The proposed control law is designed, based on the new avoidance condition for polygonal shape and Lyapunov directed method. The potential function has been introduced to achieve desired flocking behaviors and guarantee the control input differentiable in spite of agents' limited communication areas. The simulation result shows the effectiveness of the proposed control law. In the future work, our study will extend to higher-order systems such as double-integrator modeled agent.