Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots

Abstract

This paper proposes a coupled-error-based formation control algorithm for the rapid formation completion of multi-robot systems. We consider a multi-robot system with omni-directional robots with swerve-driving mechanisms and a communication system with minimized constraints. This paper introduces a coupled error that links the distance errors with the leading robot and the following robot through a coupling ratio. We propose a controller using the coupled error to achieve the control objectives of this paper. Unlike existing results that only use the information of the preceding robot, this algorithm couples the information of both the preceding robot and one’s follower. Using the proposed error-coupling-based formation control algorithm, multi-robot systems can quickly establish formations for collaboration, allowing tasks to commence swiftly and reducing deformations in formations due to speed variations. With stability analysis and simulation results for the practical application of the proposed algorithm, the approach has been verified to improve the speed of both the completion of the formation and overall system trajectory tracking, balancing trade-offs between them.

1. Introduction

The cooperation of multiple robots has been an interesting research topic in the robotics field, as it expands the scope of applications in which robots can be employed and enhances the performance of the target system. Recently, the application of multi-robot systems has increased across various fields, demonstrating commercial capabilities, and prompting more practical products and services with multi-robot systems. This trend has accelerated with advancements in control, networking, and AI technologies. Examples include multi-drone entertainment such as drone shows [1,2], disinfection and pest control systems using multiple robots, goods movement within logistics centers [3], and military applications such as Manned–Unmanned Teaming (MUM-T) [4,5]. These innovations represent significant progress in the utilization of multi-robot systems across diverse fields.

Successful multi-robot collaboration fundamentally requires formation control, a method that involves controlling multiple robots to form and maintain a specified formation. Since the collaboration of multiple robots typically occurs after the formation has been established, the rapid formation and maintenance of this structure are critical to the success of the mission. As illustrated in commercial applications such as drone shows or the collaborative movement of objects, as well as docking among unmanned vehicles during transit (e.g., AUV (Autonomous Underwater Vehicles)/USV (Unmanned Surface Vehicles) deployment/retrieval and UAV (Unmanned Aerial Vehicles) aerial refueling), maintaining formation between entities is directly linked to the success or failure of the mission.

Various multi-agent system-based commercial applications have been supported by the following research outcomes. Extensive studies have been conducted on the leader–follower strategy for building and keeping multiple robot formations. This method involves controlling followers that move in a formation dictated by a leader, as depicted in Figure 1, when the desired position of the followers is generated from the preceding robot. Alternatively, when a follower acts as a leader for the next, it can be called an inter-connected system [6]. In this interconnected system, the configuration of the information flow may lead to temporary disruptions in maintaining the formation.

If a control algorithm utilizing each follower’s tracking control law is applied, the ability to maintain the formation depends on the control performance of individual followers. This strategy can derive the ‘slinky-type effect’, where errors generated during the movement of a preceding robot are transferred and accumulate among subsequent followers [6,7,8,9].

To prevent such phenomena, a leader formation strategy has been proposed [10], where, as depicted in Figure 2a, followers moving based on a preceding agent are provided with the leader’s information to avoid the transmission and accumulation of errors. However, even if the entire multi-robot system is maintained, for this control strategy to be applied in real systems, a network system capable of consistently transmitting the leader’s information to all followers should be guaranteed.

In contrast to the leader–follower technique, a virtual-structure approach has been conducted. This method involves setting a virtual leader [11]. Each follower moves the formation based on the virtual leader. Unlike the leader–follower strategy, the virtual-structure strategy generates the formation of information from a virtual leader, thus preventing the transmission and accumulation of errors. However, this strategy faces the challenge of requiring all followers to share synchronized virtual leader information. With the advent of ultra-low latency and broadband communications such as 5G and Wi-Fi 6, this drawback is being mitigated. Yet, in such cases, a centralized control method using a supervisor that collects and controls information from all agents might be more efficient.

These results emphasize the importance of a network configuration capable of sharing information among all agents to address the issues arising during formation control. With recent advancements in communication technology, the practical implementation of such proposals is becoming feasible in manufacturing sites, indoor services, and urban areas. Nonetheless, extensive and broadband network deployment remains unfeasible in many environments such as underwater settings, disaster zones, military operations, and remote or extreme locations like polar regions or space. Therefore, a method that rapidly generates and maintains swarm formations using only the information available from surrounding agents is essential.

To overcome the slinky-type effect that occurs when controlling followers’ movement along with surrounding agents, a platoon control method based on coupled sliding mode utilizes the information of nearby followers to form and maintain the entire system. By coupling the information of leading and following agents in the controller, the entire system can be rapidly formed, mitigating the slinky-type effect and maintaining the structural integrity of the system. In this scenario, although the overall trajectory convergence of the system may slow down, the advantage lies in the rapid formation of the entire system. However, this approach, which primarily targets a one-dimensional platoon system, has limitations when applied to mobile robots moving in a two-dimensional plane, and the coupling ratio fixed as a constant renders the system rigid.

Therefore, this paper proposes a formation control algorithm based on error-coupling that allows for rapid formation and maintenance of a multi-robot system with minimal communication constraints. Furthermore, moving beyond the static coupling ratio proposed in previous works, this paper adopts a flexible coupling ratio that enhances the speed of swarm formation and the trajectory tracking performance of the entire system based on the completeness of the formation. To ensure the stability of the entire system, this paper shows that the coupled error (CE), is bounded and converges to zero, which is the error coupled between the position error from the desired position generated by its leader and the following robot’s position error. Additionally, the coupling ratio between the forward and backward errors is represented using the distance between the desired position and the agent. This approach ensures that as the position error of the following agent increases, the formation aspect is more significantly reflected over the trajectory tracking performance, and as the error decreases, the focus shifts back to enhancing the trajectory tracking performance.

The contributions of this paper are as follows. Firstly, by coupling the information of surrounding followers, the proposed algorithm enables rapid formation and maintenance of formations, surpassing traditional methods that control each follower independently. Unlike previous approaches where the leader merely follows the trajectory and the followers track the leader—making the completion time of the entire formation dependent on the performance of the followers—the algorithm presented here allows the leader to consider the positions of the followers and move towards them to quickly establish the formation before proceeding along the trajectory. This reduces the formation time, which is crucial for applications requiring swift collaboration start, such as aerial refueling of UAVs and deployment/retrieval of USVs/AUVs, where a slight delay in trajectory following is acceptable for faster mission initiation.

The paper is organized as follows: Section 2 describes the system under study, followed by an explanation of the coupled-error-based swarm control in Section 3. Section 4 shares the results of simulations and experimental findings, and Section 5 re-analyzes the performance of the proposed algorithm based on the insights gained from Section 3 and Section 4, concluding with final remarks.

2. Target Multi-Robot System

This paper proposes an algorithm considering robots equipped with a swerve drive mechanism [12,13], where all four wheels are capable of steering. As depicted in Figure 3, the swerve drive allows each wheel to independently determine its direction, enabling omni-directional movement. The configuration of the swerve drive robot is illustrated in Figure 3.

In Figure 3, L represents the distance between the front and rear wheels, W denotes the distance between the left and right wheels, v_x and v_y are the robot linear velocities in x and y directions, respectively, relative to the robot’s coordinate system, and ω is the angular velocity. v_fl, v_fr, v_rl, and v_rr are linear velocities of wheels and the steering angles ξ_fl, ξ_fr, ξ_rl, and ξ_rr represent the steering angles of the wheels, where the subscripts fl, fr, rl, and rr denote the front left, front right, rear left, and rear right wheels, respectively.

$$a=-v_y+\omega L/2, b=-v_y-\omega L/2, c=v_x+\omega W/2, \mathrm{and} d=v_x-\omega W/2,$$

(1)

$$v_{fl}=\sqrt{b^2+d^2}, v_{fr}=\sqrt{b^2+c^2}, v_{rl}=\sqrt{a^2+d^2}, \mathrm{and} v_{rr}=\sqrt{a^2+c^2}$$

(2)

$${\xi }_{fl}=a\mathit{tan}2\left(b,d\right), {\xi }_{fr}=a\mathit{tan}2\left(b,c\right), {\xi }_{rl}=a\mathit{tan}2\left(a,d\right), \mathrm{and} {\xi }_{rr}=a\mathit{tan}2\left(a,c\right),$$

(3)

where −π ≤ atan2(·) < π. By Equations (1)–(3), control input v_x, v_y, and ω can be realized. Also, the omni-directional robot with the swerve drive mechanism can be described using the following kinematic model:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{ccc}cos\left(\theta \right)& sin\left(\theta \right)& 0\\ -sin\left(\theta \right)& cos\left(\theta \right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]$$

(4)

In this paper, we assume that the orientation angle is controlled to zero degrees. This assumption can be realized by Equations (1)–(3). If θ→0, Equation (4) can be simplified as follows:

$$\dot{x}=v_x, \dot{y}=v_y, \mathrm{and} \dot{\theta }=\omega .$$

(5)

Based on the assumption of Equation (5), a multi-robot system of the omni-directional robot system is configured as shown in Figure 4. In Figure 4, (x_i, y_i) represents the position of the i-th robot, and θ_i is the orientation angle. In this paper, the 0th robot is used as the trajectory of the entire system, and the 1st robot is designated as the leader. Through the assumptions stated, θ_i is controlled to be θ_i = 0. The target positions ($x_i^d$, $y_i^d$) for the i-th robot are generated from the robot i − 1 as follows.

$$x_i^d=x_{i-1}+L\cos \left(\psi \right), y_i^d=y_{i-1}+L\sin \left(\psi \right).$$

(6)

Here, in the case of i = 1, the leader robot which tracks the trajectory, and the desired position ($x_i^d$, $y_i^d$) is (x₀, y₀). The position errors $e_i^x$ and $e_i^y$ are defined as follows.

$$e_i^x=x_i^d-x_i, e_i^y=y_i^d-y_i.$$

(7)

As can be seen in Figure 4 and Equations (6) and (7), the i-th robot follows the desired position generated by the (i − 1)-th robot and simultaneously becomes the leader for the (i + 1)-th robot. The time derivatives of error variables in (7) are described in the following.

$$\begin{array}{cc}\hfill {\dot{e}}_i^x& ={\dot{x}}_i^d-{\dot{x}}_i\hfill \\ & =\frac{d}{dt}\left\{x_{i-1}+Lcos\left(\psi \right)\right\}-v_i^x\hfill \\ & =v_{i-1}^x-v_i^x\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill {\dot{e}}_i^y& ={\dot{y}}_i^d-{\dot{y}}_i\hfill \\ & =\frac{d}{dt}\left\{y_{i-1}+Lsin\left(\psi \right)\right\}-v_i^y\hfill \\ & =v_{i-1}^y-v_i^y\hfill \end{array}$$

(8b)

where d{Lcos(ψ)}/dt = 0 and d{Lsin(ψ)}/dt = 0.

The formation control algorithm proposed in this paper designs and suggests a swarm control algorithm that utilizes the information of both the leader, which is the (i − 1)-th robot, and the follower, which is the (i + 1)-th robot, for the i-th robot.

In this paper, we make the following two assumptions: Assumption 1: Communication between robots is minimal, and near real-time communication can be ensured. Assumption 2: The positions of the robots are obtained to reduce signal noise and compensate for error accumulation. These two assumptions allow this paper to focus on proposing a coupled-error-based formation control algorithm. Although real-time communication is assumed in this paper, the realistic implementation of this assumption is feasible due to the existing technologies such as 5G communication, WIFI-6, mesh local networks, and low-altitude satellite communication, which enable communication with nearby robots. Furthermore, the assumption regarding position acquisition is also practically achievable, given that position information can be obtained with issues such as signal noise and error accumulation resolved through technologies like D-GPS (Differential Global Positioning System), sensor fusion, and SLAM (Simultaneous Localization And Mapping).

3. Coupled-Error-Based Formation Control Method

This paper focuses on rapidly forming formations, even if the overall system’s trajectory tracking velocity is reduced. Nonetheless, as the formation is completed, the entire system should move along the trajectory. Therefore, as in [6], this paper employs a coupled error that integrates the information from its own trajectory (or the desired position generated by its own leader) and the followers as in the following.

$$E_i^x=e_i^x-q{e}_{i+1}^x, E_i^y=e_i^y-q{e}_{i+1}^y.$$

(9)

where q > 0. The time derivatives of the coupled error in (9) are

$$\begin{array}{cc}\hfill {\dot{E}}_i^x& =\frac{d}{dt}\left\{e_i^x-q{e}_{i+1}^x\right\}\hfill \\ & =\left(v_{i-1}^x-v_i^x\right)-q\left(v_i^x-v_{i+1}^x\right)\hfill \\ & =-\left(q+1\right)v_i^x+v_{i-1}^x+q{v}_{i+i}^x\hfill \end{array},$$

(10a)

$$\begin{array}{cc}\hfill {\dot{E}}_i^y& =\frac{d}{dt}\left\{e_i^y-q{e}_{i+1}^y\right\}\hfill \\ & =\left(v_{i-1}^y-v_i^y\right)-q\left(v_i^y-v_{i+1}^y\right)\hfill \\ & =-\left(q+1\right)v_i^y+v_{i-1}^y+q{v}_{i+1}^y\hfill \end{array}$$

(10b)

Here, since the coupled error in (9) and (10) use the information of the following robot, the last robot’s coupled error is described as

$$E_n^x=e_n^x, E_n^y=e_n^y.$$

(11)

where n is the index of the last follower.

To ensure the relationship between the position error and coupled error, we refer to Lemma 1 in [6]. According to Lemma 1, it can be shown that if the coupled error defined in Equations (9)–(11) is bounded, then the position errors can also be bounded, and if the coupled error becomes zero, then the position errors will also be zero.

Theorem 1.

(Equivalence of the Boundedness and Convergences of the Coupled Error and Each Position Error Toward Zero): If the coupled error is bounded and becomes zero then, position error of each robot is bounded and becomes zero for all i = 1, 2, …, n at the same time.

Proof. 

To show that coupled error $E_i^x$ and $E_i^y$ are bounded and become zero if and only if $e_i^x$ and $e_i^y$ are bounded and become zero for all i = 1, 2, …, n at the same time, we can describe the coupled error $E_i^x$ and $E_i^y$, and $e_i^x$ and $e_i^y$ as $E=Qe$, where

$$\begin{gathered} e_x={\left[\begin{array}{cccc}{e}_1^x& e_2^x& \cdots & e_n^x\end{array}\right]}^T, e_y={\left[\begin{array}{cccc}{e}_1^y& e_2^y& \cdots & e_n^y\end{array}\right]}^T \\ {E}_x={\left[\begin{array}{cccc}{E}_1^x& E_2^x& \cdots & E_n^x\end{array}\right]}^T, E_y={\left[\begin{array}{cccc}{E}_1^y& E_2^y& \cdots & E_n^y\end{array}\right]}^T \\ Q=\left[\begin{array}{ccccc}1& -q& 0& \cdots & 0\\ 0& 1& -q& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& \cdots & 0& 1& -q\\ 0& 0& \cdots & 0& 1\end{array}\right]. \end{gathered}$$

Since q > 0 is constant and Q is invertible, $‖e_x‖=‖Q^{-1}‖‖E_x‖$. If $‖E_x‖\le \overline{E}$, $‖e_x‖\le ‖Q^{-1}‖\overline{E}$. Also, if $\overline{E}=0$, then it follows the equivalence of E_x = 0 and e_x = 0, and E_y = 0 and e_y = 0. □

As shown in Theorem 1 we should design the control law to make the CE converge to zero. The objective of the control law proposed in this paper is to make the coupled error zero. From Figure 4 and Equations (5)–(10), the following control algorithm is proposed:

$$v_i^x=X_i+k{E}_i^x, v_i^y=Y_i+k{E}_i^y,$$

(12)

where $X_i=\frac{1}{\left(1+q\right)}\left(v_{i-1}^x+q{v}_{i+1}^x\right)$, $Y_i=\frac{1}{\left(1+q\right)}\left(v_{i-1}^y+q{v}_{i+1}^y\right)$, and k > 0 constant. In the case of i = n, the formation control law can be expressed as

$$v_i^x=v_{i-1}^x+k{e}_i^x, v_i^y=v_{i-1}^y+k{e}_i^y.$$

(13)

By using the proposed formation control law in (12) and (13), the coupled error can be bounded and converge to zero as time goes on, as in the following theorem.

Theorem 2.

(The Stability of the Formation Control Law Based on Coupled Error): When the proposed formation control law in (12)–(13) is employed in the multi-robot system in Figure 4, the stability of each robot can be guaranteed in the sense that coupled error $E_i^x$ and $E_i^y$, and the distance error $e_i^x$ and $e_i^y$ for i = 1, …, n are bounded and converge to zero asymptotically.

Proof. 

To show the stability of the multi-robot system with the proposed formation control algorithm, we choose the Lyapunov function candidate as

$$V={\displaystyle \sum _{i=1}^n{V}_i},$$

(14)

where

$$V_i=\frac{1}{2}\frac{1}{\left(1+q\right)}\left\{{\left(E_i^x\right)}^2+{\left(E_i^y\right)}^2\right\}$$

(15)

Using (10), the time derivative of the Lyapunov function candidate in (15) for each robot can be described as

$$\begin{array}{cc}\hfill {\dot{V}}_i& =\frac{1}{\left(1+q\right)}{E}_i^x{\dot{E}}_i^x+\frac{1}{\left(1+q\right)}{E}_i^y{\dot{E}}_i^y\hfill \\ & =\frac{1}{\left(1+q\right)}{E}_i^x\left\{-\left(1+q\right)v_i^x+v_{i-1}^x+q{v}_{i+1}^x\right\}+\frac{1}{\left(1+q\right)}{E}_i^y\left\{-\left(1+q\right)v_i^y+v_{i-1}^y+q{v}_{i+1}^y\right\}\hfill \\ & =E_i^x\left\{-v_i^x+\frac{1}{\left(1+q\right)}\left(v_{i-1}^x+q{v}_{i+1}^x\right)\right\}+E_i^y\left\{-v_i^y+\frac{1}{\left(1+q\right)}\left(v_{i-1}^y+q{v}_{i+1}^y\right)\right\}\hfill \end{array}$$

(16)

Substituting (12) and (13) into (16) gives

$${\dot{V}}_i=-k{\left(E_i^x\right)}^2-k{\left(E_i^y\right)}^2\le 0$$

(17)

Since V_i(0) is bounded and V_i(t) is non-increasing and bounded, it can be shown by Barbalat’s Lemma [14,15,16] that the coupled errors $E_i^x$ and $E_i^y$ converge to zero.

The Lyapunov function of the whole system in (14) can be arranged as

$$\dot{V}={\displaystyle \sum _{i=1}^n{\dot{V}}_i}\le 0$$

(18)

In (14)–(18), it is shown that V(t) is bounded and converges to zero asymptotically for all times. This, in turn, implies that E_i and e_i in Theorem 1 are bounded for all i. □

Remark 1.

In this paper, it is shown that the coupled error converges to zero as t→∞. Since we do not show finite time convergence of coupled error $E_i^x$ and $E_i^y$ but asymptotic convergence, we cannot ensure $E_i^x$ and $E_i^y$ become zero. However, since we have shown the boundedness of $E_i^x$ and $E_i^y$ by Lyapunov’s theorem, we can achieve the fact that the distance error $e_i^x$ and $e_i^y$ for i = 1, …, n are bounded without divergence of error using Theorems 1 and 2. Also, since we showed that $E_i^x$ and $E_i^y$ converge to zero as t → ∞, and distance errors are bounded in the reducing boundary as t → ∞, distance errors outside the boundary converge within the boundary, and errors that have converged within the boundary remain inside the boundary.

Remark 2.

The finite time convergence of coupled error can ensure the string stability preventing error-propagation toward following robots as in [6]. However, since this paper focuses on fast formation completion rather than string stability, string stability is not guaranteed in this paper.

4. Simulation Results

For the numerical simulations, we consider the multi-robot system in Figure 4, each of which consists of two robots—a leader and follower. Here, the 0th robot represents the entire system’s trajectory, the 1st robot is the leader, and the 2nd robot is the follower. The formation is built using L = 1 and ψ = π/2. The initial positions of each robot, (x_i(0), y_i(0), θ_i(0)) (m, m, rad) (i = 0, 1, 2), are (0, 0, 0), (−4, 2, 0), and (−8, 2, 0), respectively. The trajectory moves following the velocity profile.

$$\begin{gathered} v_0^x=0.3 (\mathrm{m}/\mathrm{s}), v_0^y=0.0 (\mathrm{m}/\mathrm{s}), 0\le t<20 \left(\mathrm{s}\right) \\ {v}_0^x=0.0 (\mathrm{m}/\mathrm{s}), v_0^y=0.3 (\mathrm{m}/\mathrm{s}), 20\le t<40 \left(\mathrm{s}\right) \\ {v}_0^x=-0.3 (\mathrm{m}/\mathrm{s}), v_0^y=0.0 (\mathrm{m}/\mathrm{s}), 40\le t \left(\mathrm{s}\right) \end{gathered}$$

Additionally, considering the real system, the analysis applies velocity saturation constraints of −1 < $v_i^x$ < 1 and −1 < $v_i^y$ < 1.

In this paper, a system that only uses the information of the preceding robot without coupling ratio q is configured to serve as the control group for comparing and analyzing the performance of the proposed clustering control algorithm. The control algorithm of the system used as the control group representing previous research has the form of a linear controller, as follows.

$$v_i^x=v_{i-1}^x+k_x{e}_i^x, v_i^y=v_{i-1}^y+k_y{e}_i^y$$

(19)

where the control gain, k_x > 0 and k_y > 0 are constant values. The linear controller in Equation (19) is used to compare the performance of the controller that uses only the information of the preceding robot with that of the proposed controller.

First, after applying q = 5, the tracking performance of the proposed formation control method in Scenario 1 is presented in Figure 5 and Figure 6. Figure 5 shows the routes of the member robots. Figure 5a shows the route of a multi-robot system using a controller that only follows its own desired position. Since it does not consider other robots, it can be observed that the formation is created after the follower moves a long distance. Compared to the single robot-based formation control algorithm in Figure 5a, Figure 5b shows that the leader, with the proposed CE-based formation control algorithm applied, moves towards the follower to form a cluster formation and then follows the trajectory. As can be seen in Figure 5a, the formation is quickly established after the follower travels a short distance. Figure 6 shows the detailed performance of error convergence and boundedness. From Figure 6a–c, we can see the coupled error and position errors of each robot are bounded and converge to zero. In Figure 6c, it is evident that the leader moves towards the follower between t = 0 and about t = 8 s, even at the expense of an increased trajectory tracking error. This highlights that, despite the increase in error, the leader prioritizes moving towards the follower. Figure 6d shows the distance between the follower robot and its desired position, which means the completeness of the formation. As can be seen in Figure 6d, it can be ensured that the distances between robots within the formation converge more quickly in the system with the proposed CE-based formation control algorithm applied.

To analyze the effect of the coupling ratio q, the performance of the entire system is observed by varying q (q = 1, 5, 10, and 50). Figure 7 shows the changes in system performance as q varies. In Figure 7, we can see that as the coupling ratio q increases, the leader moves closer to the follower. Conversely, as q increases, the performance of trajectory tracking decreases. Especially, Figure 7d shows the distance between the desired position and the follower. We can see that increasing q results in faster completion of the formation. As observed in Figure 5, Figure 6 and Figure 7, the speed of completion of formation and the leader’s trajectory tracking performance are determined by the coupling ratio, q. It can be seen that these two factors have a trade-off relationship.

Therefore, it can be seen that by not using q as a fixed constant, but determining it in real-time based on the completeness of the formation, both the speed of formation completion and trajectory tracking performance can be achieved. This will be discussed further in the following section.

5. Discussion

As mentioned in the previous sections, the performance of trajectory tracking and the speed of cluster formation are determined by the coupling ratio q, and these two elements have a trade-off relationship. Therefore, to enhance the performance of both factors, it is necessary to determine q according to the overall system’s situation. Replacing the constant q used earlier, q is determined according to the completeness of the next formation as follows:

$$q=k_{d}d$$

Here, d represents the distance between the follower and its desired position needed to form the formation, and k_d is a positive constant. As evidenced in Figure 6 and Figure 7, lower values of q enhance tracking performance, while higher values of q significantly influence the elements needed to complete the formation. Therefore, q is determined by d. The results, when k_d = 50, are illustrated in the following Figure 8. As can be seen in Figure 8a, at the beginning of the simulation, the leader moves towards the follower to maintain the formation at point A, after which the trajectory following performance improves, as seen at point B. Compared to Figure 7, it is evident that the robots complete a formation quickly with a large q applied, yet exhibit similar trajectory following characteristics as when q is small. In Figure 8c–e, it can be seen that the CE and the distance errors between each robot are bounded near zero even as q changes with distance. Additionally, as observed in Figure 8b, the distances converge rapidly.

6. Conclusions

This paper focuses on rapidly configuring formations to support the swift execution of tasks by multiple robots. Accordingly, a CE-based formation control algorithm that utilizes the information of both the leader and followers is proposed for a multi-robot system comprised of omni-directional robots, aiming to achieve rapid formation completion and effective trajectory tracking. Unlike other approaches that rely solely on the desired position information for formation and maintenance, which predominantly depends on the follower’s tracking performance, the algorithm proposed in this paper enables the leader to adjust its actions based on the state of its follower, thereby maintaining the overall cluster.

As demonstrated in simulation results, by applying the proposed coupled-error-based algorithm to the multi-robot system, the leader not only follows the trajectory but also moves towards the followers upon checking their states, thus quickly completing a formation. Furthermore, unlike conventional algorithms that require high-performance communication to use the entire system’s leader information or to simulate a virtual leader, the proposed method swiftly forms clusters by using only the information of nearby entities. Additionally, velocity constraints were implemented in the simulations to validate the proposed algorithm under realistic environmental conditions.

In this paper, to verify the versatility of the proposed algorithm, the error coupling ratio q was varied, and the performance of the algorithm was evaluated across various aspects by allowing q to be determined based on the degree of cluster formation completion. Moving forward, the coupled-error-based cluster control algorithm introduced in this paper will be applied to actual omni-directional drive systems in practical use cases, and it is also expected to be implemented in other robotic systems with different locomotion capabilities.