Vertex-Weighted Consensus-Based Formation Control with Area Constraints and Collision Avoidance

Abstract

In this paper, a consensus-based formation control strategy is presented, subject to area constraints and collision avoidance. To achieve a desired formation pattern, a control law is proposed that incorporates a vertex-tension function along with signed area constraints. The vertex-tension function provides the capabilities of collision avoidance among agents. Moreover, signed area constraints avoid local minimum stagnation and mitigate ambiguities within the formation shape. Additionally, the proposed approach can be implemented considering a group of differential-drive mobile robots in both centralized and decentralized settings. Simulation and real-world experiments are performed to validate the effectiveness of the proposed approach, where the experimental setup includes a multi-robot system composed of Turtlebot3^® Waffle Pi robots.

1. Introduction

Formation control is a very well-studied problem due to its potential applications in collaborative and cooperative tasks [1]. Formation control aims to drive a multi-agent system to achieve prescribed patterns. Commonly, formation control is solved as a consensus problem [2,3,4], where the state variables of all agents converge to a common value that defines the centroid of the formation. A remarkable advantage of consensus algorithms is that a consensus agreement is reached even in the presence of delays in communication [5,6]. Moreover, the various problems for consensus formation control have been addressed, like formation control with collision avoidance [7,8,9], formation control with communication delays [10,11], noise tolerance formation control [12], fuzzy formation control with input saturation [13], PID formation control with disturbances [14], and formation control with leader–follower networks [15,16,17].

Since collision among agents can occur during the formation control task, artificial potential field approaches are commonly used to deal with collisions [18,19,20]. Repulsive potential fields are combined with consensus-based formation control to include collision avoidance. However, incorporating repulsive fields does not guarantee system stability in the presence of delays [21].

Edge-weighted consensus-based formation control guarantees collision avoidance among agents [22,23]. These approaches exploit the properties of weighted graphs, allowing the formation to rotate and adapt its shape to avoid collision among robots. However, the desired formation cannot always be preserved due to local minima. Local minima may occur for some particular agent positions, especially when agents are aligned. Moreover, a connection between agents is required to avoid a collision between agents. Then, when more agents are involved in a network, more connections are required to avoid collisions. Thus, to guarantee formation and collision capabilities, fully consensus-based communication is always required. In [6], a virtual relabeling strategy, along with a broadcasting algorithm, is proposed to avoid local minima while ensuring that agents converge to the desired formation. The virtual relabeling strategy assigns a virtual label to each agent. Local minimum stagnation is avoided by modifying agents’ labels following an anti-clockwise order concerning the formation’s center of mass. However, the broadcasting algorithm assigns the appropriate label to an agent based on its own position and the position of all other agents, losing the capability of performing a decentralized formation control. Moreover, some particular initial agents’ positions still provoke local minima stagnation. Additionally, the virtual relabeling strategy required fully consensus-based communication.

In [24,25], formation control is achieved based on a gradient descent control law, which incorporates a signed area term to mitigate ambiguities within the formation. When attempting to attain a specific formation, ambiguities may arise in the absence of sufficient inter-agent connections. In the context of n agents that are not fully connected, multiple formations may exist that meet the desired inter-agent distance criteria [25]. This approach enables formation to remain translation- and rotation-independent. However, the proposed control law does not inherently ensure collision avoidance among agents. Additionally, the methodology in [25] is structured around a hierarchical approach, implying that control strategies employed differ among agents within the information.

In this work, vertex-weighted consensus-based formation control is addressed, where a control law is proposed that incorporates a vertex-tension function along with signed area constraints. Inspired by the edge-tension function from [6], a vertex-tension function is proposed, which maintains the capabilities of collision avoidance among agents. Moreover, inspired by the area constraints from [24,25], a signed triangle area term is considered to avoid local minimum stagnation and mitigate formation ambiguities. Thanks to area constraints, collision among agents is guaranteed even if not performing a fully consensus-based communication. The advantage of this approach is that virtual relabeling and broadcasting algorithms are not required, given the capacity to perform formation control in a completely decentralized manner. Plus, the desired formation pattern is achieved for any initial agent positions. Finally, a multi-robot system composed of four Turtlebot3^® (Turtlebot3 is a registered trademark of ROBOTIS Inc., Corona, CA, USA) Waffle Pi systems is considered for real-world experimentation, which is a non-holonomic robot.

The contributions of this paper are summarized below:

• The proposed signed triangle area term avoids local minimum stagnation and mitigates ambiguities within the formation.
• The proposed vertex-tension function gives the capabilities of collision avoidance among agents, which is an inconvenience in formation control with distance and area constraints [24,25].
• The proposed approach does not require virtual relabeling and broadcasting algorithms. Then, formation control can be performed in a completely decentralized manner.
• The desired formation pattern is achieved for any initial agent positions, which is a drawback of edge-weighted formation [6].

This paper is organized as follows: The next section presents the kinematic model formulation of a non-holonomic differential-drive robot, along with the proposed formation control design. Then, simulation and real-world are presented in Section 3. An analysis of the reported results and future research directions is given in Section 4. Finally, the conclusions are provided in Section 5.

2. Problem Formulation

2.1. Kinematic Model Formulation

Our case study considers a set of N non-holonomic differential-drive robots, which are modeled as unicycle kinematics (1). A differential-drive model is illustrated in Figure 1. For each mobile platform i, the robot pose is given by position $(x_i,y_i)$ and orientation ${\theta }_i$ with respect to world frame $\left\{F_W\right\}$. Moreover, a base frame $\left\{F_B\right\}$ is attached to middle point of the two wheels, and ${\theta }_i$ is measured from $(x_i,y_i)$ pointing to $(x_i^L,y_i^L)$ relative to the x-axis of the frame $\left\{F_W\right\}$. The point $(x_i^L,y_i^L)$ is proposed for control purposes to deal with non-holonomic constraints [26]. Parameter $L_i$ represents a distance from $(x_i,y_i)$ to $(x_i^L,y_i^L)$. Finally, $v_i$ and ${\omega }_i$ are the linear and angular velocities, respectively.

The unicycle kinematics model is given by

$$\begin{array}{cc}\hfill {\dot{x}}_i& =v_{i}cos\left({\theta }_i\right)\hfill \\ \hfill {\dot{y}}_i& =v_{i}sin\left({\theta }_i\right)\hfill \\ \hfill {\dot{\theta }}_i& ={\omega }_i.\hfill \end{array}$$

(1)

Non-holonomic constraints make the design of $v_i$ and ${\omega }_i$ difficult, such as $(x_i,y_i)$ approaching a zero state. For this, we define

$$\begin{array}{cc}\hfill x_i^L& \hfill =x_i+L_{i}cos\left({\theta }_i\right)\hfill \\ \hfill y_i^L& \hfill =y_i+L_{i}sin\left({\theta }_i\right),\hfill \end{array}$$

(2)

where $L_i\ne 0$. After evaluating (1) in the time derivative of (2), we obtain

$$\left[\begin{array}{c}{\dot{x}}_i^L\\ {\dot{y}}_i^L\end{array}\right]=\left[\begin{array}{cc}cos\left({\theta }_i\right)& -L_{i}sin\left({\theta }_i\right)\\ sin\left({\theta }_i\right)& L_{i}cos\left({\theta }_i\right)\end{array}\right]\left[\begin{array}{c}{v}_i\\ {\omega }_i\end{array}\right].$$

(3)

The following input transformation can be used to drive the robot

$$\left[\begin{array}{c}{v}_i\\ {\omega }_i\end{array}\right]={\left[\begin{array}{cc}cos\left({\theta }_i\right)& -L_{i}sin\left({\theta }_i\right)\\ sin\left({\theta }_i\right)& L_{i}cos\left({\theta }_i\right)\end{array}\right]}^{-1}\left[\begin{array}{c}{\dot{x}}_i^L\\ {\dot{y}}_i^L\end{array}\right],$$

(4)

where $({\dot{x}}_i^L,{\dot{x}}_i^L)$ will be designed for formation control purposes.

2.2. Formation Control Design

Since communication topology among a set of robots can be represented with an undirected graph, some basic graph theory concepts are summarized ahead. An undirected graph with N elements is defined as a pair $G=(\mathcal{V},\mathcal{E})$, where $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the edge set of cardinality M and $\mathcal{V}=\{{\mathcal{v}}_i,i=1,2,\dots ,N\}$ stands for the vertex set of cardinality N. An edge ${\mathcal{E}}_k=({\mathcal{v}}_i,{\mathcal{v}}_j)$ represents bidirectional communication between agents i and j, where ${\mathcal{E}}_k$ is the k-th edge of G.

Inspired by the edge-tension function proposed in [6], we define a vertex-tension function for every vertex ${\mathcal{v}}_i\in \mathcal{V}$ instead of every edge in the graph. The vertex-tension is defined as

$$V_i=\sum _{j\in {\mathcal{N}}_i}{\alpha }_{ij}\left(coth\left({\displaystyle \frac{\parallel {\mathit{l}}_{ij}\parallel -\delta }{k_{ij}}}\right)+{\displaystyle \frac{1}{2}}{\parallel {\mathit{l}}_{ij}\parallel }^2-V_{ij}^{min}\right),$$

(5)

where ${\mathcal{N}}_i=\{\forall {\mathcal{v}}_j\in \mathcal{V}\mid (i,j)\in \mathcal{E}\}$ is the subset of neighbors of the i-th robot, and $K_{ij}$ is a constant that defines a related position of the minimum of the edge-tension function, i.e., the desired distance for each couple of agents.

Moreover, ${\alpha }_{ij}$ defines the intensity of the inter-robot influence, $\delta$ is a safety distance to avoid collision between agents, and $V_{ij}^{min}$ is defined as follows

$$\underset{\parallel {\mathit{l}}_{ij}\parallel >\delta }{min}{V}_i=0$$

(6)

Finally, ${\mathit{l}}_{ij}={\mathit{p}}_i-{\mathit{p}}_j$ is the edge vector between agents i and j, where ${\mathit{p}}_i$ is the state of the i-th agent. In the case of differential-drive robots, the state of the i-th agent is given by

$${\mathit{p}}_i=\left[\begin{array}{c}{x}_i^L\\ y_i^L\end{array}\right]$$

(7)

Inspired by [24,25], we proposed creating a connected graph such that every vertex ${\mathcal{v}}_i\in \mathcal{V}$ is part of at least one triplet $(i,j,k)$, where $\left\{\right(i,j),(j,k),(k,i\left)\right\}\in \mathcal{E}$ (see Figure 2). Every triplet is related to a signed area function that can be useful to avoid ambiguities and local minima.

The set of all triplets $(i,j,k)$ in the graph is denoted by $\mathcal{C}$. Notice that if we permute the triplet in a cyclical way, we will be referring to the same triplet, that is, $(i,j,k):=(k,i,j):=(j,k,i)$. Then, we can define the subset ${\mathcal{C}}_i=\left\{(j,k)\right|(i,j,k)\in \mathcal{C}\}$ that denotes all triplets where the vertex ${\mathcal{v}}_i$ is involved.

As an example, we can see that the set $\mathcal{C}$ for Figure 2a is defined as

$$\mathcal{C}=\left\{\right(0,1,5),(1,2,3),(3,4,5),(1,3,5\left)\right\},$$

where we have the next subsets

$$\begin{array}{cc}\hfill {\mathcal{C}}_0& =\left\{\right(1,5\left)\right\}\hfill \\ \hfill {\mathcal{C}}_1& =\left\{\right(5,0),(2,3),(3,5\left)\right\}\hfill \\ \hfill \hfill & \dots \hfill \\ \hfill {\mathcal{C}}_5& =\left\{\right(0,1),(3,4),(1,3\left)\right\}.\hfill \end{array}$$

We proposed the triplet-tension function for every vertex $v_i$

$$V_{{\mathcal{C}}_i}=\sum _{(j,k)\in {\mathcal{C}}_i}{\displaystyle \frac{1}{2}}{K}_{{\mathcal{C}}_{ijk}}{\left(Z_{ijk}-Z_{ijk}^{*}\right)}^2,$$

(8)

where $K_{{\mathcal{C}}_{ijk}}\in \mathbb{R}$ is a constant greater than zero that allows for avoiding ambiguities and local minima. Moreover, $Z_{ijk}$ and $Z_{ijk}^{*}$ are the currently signed area and desired signed area of the triplet $(i,j,k)$, respectively.

The triplet area $Z_{ijk}$ is defined as

$$Z_{ijk}:={\displaystyle \frac{1}{2}}\det \left[\begin{array}{ccc}1& 1& 1\\ {\mathbf{p}}_i& {\mathbf{p}}_j& {\mathbf{p}}_k\end{array}\right].$$

(9)

As can be seen in (9), $Z_{ijk}=Z_{kij}=Z_{jki}$ because of the switching property of the determinant. Moreover, the triplet could be permuted in a cyclical way.

Based on (5) and (8), a total potential function can be defined as

$$V=\sum _{i=1}^N\left(V_i+V_{{\mathcal{C}}_i}\right).$$

(10)

The total potential function V in (10) exploits the collision avoidance capacities of (5), plus the match for all the signed areas formed by the triplets (8), which is useful to avoid ambiguities such as a flipped version of the desired formation and local minima. It is a fact that (10), including (8), causes each area to be counted three times since each area is considered in three of the subsets ${\mathcal{C}}_0,{\mathcal{C}}_1,\dots ,{\mathcal{C}}_N$. In other words, vertex j or k of the triplet in a different iteration could be the value of i, but that is covered with the gain $K_{{\mathcal{C}}_{ijk}}>0$. Actually, if $K_{{\mathcal{C}}_{ijk}}$ were equal to $1/3$, the result obtained would be as if each area had only been counted once. What we obtain by doing it this way is unifying the potential energy Equations (5) and (8) in terms of the vertices instead of having one in terms of bonds [6] and another in terms of triplets [24,25]. Using $d^{*}=2a\Rightarrow a=d^{*}/2>0$ according to (6), it is possible to ensure the Anderson [24,25] and Falconi [6] convergence and stability conditions at $d^{*}=\delta$, with $d^{*}$ as the desired distance between two defined agents (see Appendix A). Then, the value of $d^{*}$ is calculated according to

$$d^{*}=\parallel {\mathit{l}}_{ij}\parallel$$

(11)

For control purposes, we can use the next gradient descent law

$${\dot{\mathbf{p}}}_i=-{\displaystyle \frac{\partial V}{\partial {\mathbf{p}}_i}},$$

(12)

which can be used as control input for agent i. Then, we have

$${\dot{\mathbf{p}}}_i={\mathbf{U}}_{{\mathcal{N}}_i}+{\mathbf{U}}_{{\mathcal{C}}_i},$$

(13)

where

$${\mathbf{U}}_{{\mathcal{N}}_i}=-\sum _{j\in {\mathcal{N}}_i}{\alpha }_{ij}\left(-{\displaystyle \frac{1}{k_{ij}\parallel {\mathit{l}}_{ij}\parallel }}{\mathrm{csch}}^2\left({\displaystyle \frac{\parallel {\mathit{l}}_{ij}\parallel -\delta }{k_{ij}}}\right)+1\right){\mathit{l}}_{ij},$$

(14)

$${\mathbf{U}}_{{\mathcal{C}}_i}=-\sum _{(j,k)\in {\mathcal{C}}_i}{K}_{{\mathcal{C}}_{ijk}}\left(Z_{ijk}-Z_{ijk}^{*}\right)\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]{\mathit{l}}_{jk}.$$

(15)

Since control law (13) is only based on agent i and its neighbors’ $(j,k)$ positions, it could be implemented as a decentralized control as long as every vertex in the graph is part of at least one triplet. This is very useful because every 2D shape could be represented as a 2D triangulated mesh; this allows us to reduce the number of connections between agents.

The signed area function defined in (9) requires the global positions ${\mathbf{p}}_i$, ${\mathbf{p}}_j$, and ${\mathbf{p}}_k$ for control. However, this function can be expressed in terms of the relative position of ${\mathbf{p}}_j$ and ${\mathbf{p}}_k$ with respect to ${\mathbf{p}}_i$ for decentralized control purposes. Then, (9) can be rewritten as

$$Z_{ijk}={\displaystyle \frac{1}{2}}{\mathit{l}}_{ij}^T\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]{\mathit{l}}_{ik}.$$

(16)

The use of a signed area function (16) allows for the control of any number of agents in a decentralized way because we only need the neighbor’s relative position.

Notice that the proposed control law in (13) is designed for holonomic robots. In the case of non-holonomic systems, such as differential-drive mobile platforms, the transformation in (4) can be useful to drive the non-holonomic with ($v_i,{\omega }_i$) inputs. Then, the agents converge in the position $(x_i^L,y_i^L)$ rather than $(x_i,y_i,{\theta }_i)$.

3. Experimental Results

The applicability of the proposed approach is validated through simulations and real-world experiments. The performance of the proposed approach is compared against edge-weighted formation control with virtual relabeling [6] in both simulations and real-world experiments. In this work, the compared formation control is called a conventional approach.

In the simulation experiments, a network of six holonomic mobile robots was considered to achieve the formation patterns shown in Figure 2a,b. Moreover, a network of four holonomic mobile robots was considered to achieve the formation pattern shown in Figure 2c. For real-world experimentation, four non-holonomic Turtlebot3^® Waffle Pi mobile robots were considered to achieve the formation pattern shown in Figure 2c.

Formation control for non-holonomic mobile robots is achieved by considering the use of coordinate transformations, as defined in (2). Instead of employing the conventional ${\mathbf{p}}_i={[x_i,y_i]}^T$ coordinates, we adopt ${\mathbf{p}}_i^L={[x_i^L,y_i^L]}^T$ and utilize the input transformation outlined in (4).

The $k_{ij}$ and $Z_{ijk}^{*}$ values for the hexagonal, rectangular, and square formation patterns are given in

Table 1. Values used for the hexagonal formation.

${\mathit{k}}_{01},{\mathit{k}}_{12},{\mathit{k}}_{34},{\mathit{k}}_{45}$	${\mathit{k}}_{13},{\mathit{k}}_{15}$	${\mathit{k}}_{05},{\mathit{k}}_{23}$	${\mathit{k}}_{35}$
$0.1123$	$0.4858$	$0.1493$	$0.4764$
$Z_{015}^{*}$	$Z_{123}^{*}$	$Z_{135}^{*}$	$Z_{345}^{*}$
$0.1$	$0.1$	$0.28$	$0.08$

,

Table 2. Values used for the rectangular formation.

${\mathit{k}}_{01},{\mathit{k}}_{12},{\mathit{k}}_{34},{\mathit{k}}_{45}$	${\mathit{k}}_{05},{\mathit{k}}_{14},{\mathit{k}}_{23}$	${\mathit{k}}_{04},{\mathit{k}}_{13}$	${\mathit{Z}}_{014}^{*},{\mathit{Z}}_{045}^{*},{\mathit{Z}}_{123}^{*},{\mathit{Z}}_{134}^{*}$
$0.08322$	$0.0832$	$0.2027$	$0.08$

, and

Table 3. Values used for the square formation.

${\mathit{k}}_{01},{\mathit{k}}_{12},{\mathit{k}}_{23},{\mathit{k}}_{03}$	${\mathit{k}}_{02}$	${\mathit{Z}}_{012}^{*},{\mathit{Z}}_{023}^{*}$
$0.8517$	$2.2881$	$0.5$

, respectively. The value of $k_{ij}$ is tuned to define the distance between agents ${\mathbf{p}}_i$ and ${\mathbf{p}}_j$; indeed, $k_{ij}$ define the formation patterns. Moreover, $Z_{ijk}^{*}$ is a desired signed area, which can be computed with (9).

With respect to formation control settings, the safety distance was set to $\delta =0.2$, the inter-robot influence was fixed to $\alpha =0.7$, and the signed area control gain for all triplets was fixed to $K_{{\mathcal{C}}_{ijk}}=10$. These parameters were experimentally selected.

For simplicity, the physical parameter $L_i$ is set to $0.05$ meters for all Turtlebot3^® systems. Additionally, the linear and angular velocities $(v_i,{\omega }_i)$ are bounded to $-0.2\le v_i\le 0.2$ and $-0.9\le {\omega }_i\le 0.9$. This saturation is imposed to protect the equipment, but only for real-world implementations.

3.1. Simulation Experiments

In Figure 3, the hexagonal formation results using the conventional approach are reported. We can see how the virtual relabeling algorithm works, which can be beneficial in avoiding local stagnation. However, it is clear that we are not treating every agent as unique since the final formation has no order. Starting from robot 0 in the anti-clockwise direction, we have the robots 3, 1, 5, 4, and 2, which is not the desired order. In contrast, the hexagonal formation results using the proposed approach are reported in Figure 4. In this case, we achieved the same formation without the need for virtual relabeling. The final positions give us the desired order. Here, we have the following robot order—0, 1, 2, 3, 4, and 5—in anti-clockwise direction. As mentioned, the conventional approach suffers from local stagnation for some initial conditions of the robots. The performance of both the conventional and proposed approaches are illustrated in Figure 5 and Figure 6, where the drawback of [conventional approach arose. When starting in a formation where all robots are lined up, the control in the conventional approach only actuates along this line, causing all agents to remain stuck in this region, as depicted in Figure 3b and Figure 4b. In contrast, the proposed approach does not have this local stagnation, thanks to the signed triangle area term. We can see, in Figure 5c and Figure 6c, that the proposed approach achieved the formations depicted in Figure 2b and Figure 2a, respectively.

One of the benefits of adding the desired signed area is that the agents now have more knowledge about the final formation. In the proposed approach, agents need fewer connections to achieve their objective, which is in contrast to the conventional approach in which consensus-based communication is often required, especially in the case of collision avoidance and formation ambiguities. As can be seen in Figure 7, the conventional approach was unable to achieve the final formation, despite all robots being in their correct order, as suggested by the virtual relabeling algorithm. However, the proposed approach successfully achieved the desired formation. Figure 8 shows the square formation results starting from a line and using four non-holonomic robots. The drawbacks of the conventional approach arose due to the robots being initially arranged in a line formation. However, this inconvenience occurs when all of them are oriented in the same direction as the line. In contrast, the proposed approach successfully achieved the desired formation pattern.

Figure 9 shows the formation results of two equilateral triangles using four non-holonomic robots. In this case, the performance of both approaches is compared under situations where ambiguities arise. The formation produced by conventional control meets prescribed edge lengths but deviates from the desired configuration. To realize the desired formation with conventional control, additional inter-agent connections would be needed. For this assessment, we utilized the following formation values: $k_{01}=k_{02}=k_{12}=k_{13}=k_{23}=0.852$ and $z_{012}=z_{132}=0.433$.

3.2. Real-World Experiments

In real-world experimentation, four Turtlebot3^® Waffle Pi robots are considered. The Turtlebot3^® Waffle Pi is an ROS-based differential-drive mobile robot used for education, research, and product development applications [27] (see Figure 10). The ROS (ROS is a registered trademark of Open Robotics, Mountain View, CA, USA) packages of Turtlebot3^® provide access to send velocity control inputs and read its odometry.

In this section, we perform two experiments, which we call Experiment 1 and Experiment 2. In Experiment 1, we will address the initial line position to achieve the square formation depicted in Figure 2c with the values listed in Table 3. In Experiment 2, we will tackle a specific ambiguity case where the desired formation is two equilateral triangles. Thus, the formation patterns are defined as $k_{01}=k_{02}=k_{12}=k_{13}=k_{23}=0.852$, and the desired signed areas are $Z_{012}^{*}=Z_{132}^{*}=0.433$.

In the case of Experiment 1, Figure 11 illustrates the performance of the conventional approach without virtual relabeling. We have intentionally omitted the use of a virtual relabeling algorithm to highlight its significance within this control law. To assess the performance of the proposed approach, please refer to Figure 12. It is reported that the line starting position does not pose a challenge for the proposed approach, regardless of whether it is applied to holonomic or non-holonomic robots. The achieved formation patterns are depicted in Figure 11b and Figure 12b.

In the case of Experiment 2, Figure 13 reports the performance of the conventional approach with virtual relabeling under the presence of ambiguities. Moreover, the performance of the proposed approach is illustrated in Figure 14. Notably, even with the use of the virtual relabeling algorithm, the conventional approach fails to attain the desired formation pattern, in stark contrast to the proposed approach, which correctly achieves the desired formation. The incorporation of the signed area term allows for achieving the desired formation pattern, even in the presence of ambiguities.

4. Discussion

Based on the obtained results, we have verified that the proposed control law (13) preserves the advantages of the formation control law outlined in [6]. These advantages include rotational and translation independence, as well as collision avoidance among connected agents. Notably, the proposed approach does not rely on the virtual relabeling algorithm or broadcasting process.

The proposed approach consistently achieves the desired formation under various conditions, regardless of the initial state, as evidenced by Figure 5, Figure 6, Figure 8, and Figure 9. This achievement sets it apart from the conventional approach. In Experiment 1, we can see that, in real-world applications, the conventional approach drives the robots out of the line formation after a few seconds. This is mainly due to a range of factors, including errors in odometry, signal noise, and surface conditions. In such scenarios, our dependence on these factors becomes essential to attaining the desired formation. Consequently, there is a continuous need for the application of the virtual relabeling algorithm because, in the initial moments, it is not possible to ascertain correct virtual labels until the line formation is disrupted. Regrettably, this ongoing process can lead to increased latency in network communication.

Moreover, the proposed approach effectively handles ambiguities in the formation control problem without necessitating additional connections between agents, as illustrated in Figure 7. This is in contrast to the conventional approach, which would require the creation of a connection between agent 1 and agent 4 to resolve the ambiguity, similar to Experiment 2.

By utilizing (8), our proposed approach operates in a fully decentralized manner, relying only on the relative positions of neighboring agents. It also accurately recognizes the role of each agent within the formation. For example, if the goal is to arrange all robots in a counter-clockwise direction, the proposed approach can achieve it, a feat not possible with the conventional approach for certain initial conditions, as demonstrated in Figure 3 and Figure 4.

The simulations and real-world experiments showed that the robots successfully achieved the desired formation with smooth trajectories. However, the input control signals in real-world implementations exhibited some noise, primarily due to non-modeled dynamics, packet loss, actuator limitations, and other real-world factors.

This work predominantly centers on the realm of 2D planes featuring differential mobile robots. For future endeavors, analogous concepts can be explored in the domain of 3D space, specifically with aerial robots or similar platforms. In this scenario, the study would incorporate the notion of the signed volume of a tetrahedron comprising four vertices. Consequently, a minimum of four connected agents would be essential to establish volumetric formations.

5. Conclusions

This paper delves into the xy-plane formation problem, aiming to create formations with a minimum of three agents. The results show that incorporating the signed area term, coupled with collision avoidance control, consistently enables us to achieve the desired formations under various conditions.

The incorporation of the signed area in the control law allows us to address the formation problem with fewer connections, a crucial advantage when dealing with substantial distances between agents. This is illustrated in Figure 7, where, as previously discussed, the compared formation control method would necessitate a connection between agent one and agent four. However, this link would be the longest in the formation, and in real-world scenarios, measuring such a distance might be impractical.

Regardless of the initial agent’s positions, the proposal shows that local minima stagnation is avoided using area constraints based on triplets, a drawback of conventional edge-weighted consensus-based formation control. Moreover, the proposal avoids collision among agents thanks to the vertex-tension function, which is not considered in conventional formation control with distance and area constraints.

The proposed approach requires providing formation patterns based on triples. Then, the minimum number of agents to consider in a formation problem is three. However, defining triplets in a network of a bunch of agents can take time and effort. The signed area can be extended to use other geometric figures, such as squares, pentagons, and hexagons, to easily represent connection among agents.

The proposed control approach offers a decentralized solution to the formation problem, applying the same control law for all agents without requiring knowledge of the full network, in contrast to the conventional control method with the virtual relabeling algorithm. This characteristic facilitates easier implementation in real-world experiments.

Looking ahead, our proposed control approach can be extended to the xyz-space, utilizing the concept of the signed volume of a tetrahedron. In this context, the analysis should focus on quartets rather than triplets.