Abstract : This paper briefly introduces the two main components of the NuBot robot: omnidirectional vision and omnidirectional motion system, and provides kinematic analysis. Based on this robot platform, two tracking algorithms, DA and DD control, are proposed. Distributed control of multi-robot formation is achieved through relative positioning and local communication between robots. Simultaneously, the algorithm can independently control the robot orientation. For the formation obstacle avoidance problem under different conditions, two methods, formation deformation and formation transformation, are proposed. Simulation and actual robot experiments show that the DA control method can achieve smooth formation transformation; the formation deformation method can ensure smooth obstacle avoidance while maintaining the original formation as much as possible. Keywords : Multi-robot; Formation control ; Omnidirectional vision; Omnidirectional motion Based on relative positioning and local communication, the proposed algorithms can implement distributed formation control of multiple robots and control the orientation of the robots separately. Two methods, formation distortion and formation switching, are proposed for the obstacle avoidance problem of robot formation. Real robot experiments and simulation results show that the DA controller can smoothly switch the robot formation, and the formation distortion method can avoid obstacles while minimizing changes to the original formation. Keywords : multi-robot; formation control; omni-vision; omnidirectional robot 1 Introduction Multi-robot formation control refers to the control technology that allows a team of multiple robots to maintain a specific geometric shape (i.e., formation) while adapting to environmental constraints (such as obstacle avoidance) as they move towards a specific target or direction. Multi-robot formation control is a common collaborative problem in the field of multi-robot system research. Using multiple robots to form a formation has many advantages, such as being able to obtain more environmental information in military reconnaissance, search, mine clearance and other applications, and being able to complete more and more complex tasks in aircraft and satellite formation flight applications. In recent years, with the development of network technology, communication technology and information technology, the research on multi-robot formation control has made great progress, especially in the field of military and space exploration, where practical research has become increasingly mature, making the research of this technology have certain strategic significance. From the perspective of the multi-robot system control framework, multi-robot formation control can be mainly divided into three types: centralized control, distributed control and monitoring control. From the perspective of specific technical implementation, there are mainly Leader-Follower method, behavior-based control method and virtual structure method. Reference [2] uses the virtual structure method to realize the precise formation control of multiple robots. However, this method requires knowing the absolute position of each robot and implementing it through centralized control. References [3-6] all use distributed control to achieve formation control. Reference [3] uses feedback linearization to achieve robot formation tracking control for a class of nonholonomic constrained robots. Reference [4] studies the cooperative problem of multi-robot formation surrounding and capturing "intruders" from the perspective of formation vector. Reference [5] uses the method of tracking specific robots to achieve formation. This method does not use global sensors and ensures a small amount of communication. However, because it avoids the motion control problem of the robot itself, the robot motion is not smooth and the speed is very slow in the experiment. Reference [6] studies the information flow control of multi-robot formation under a distributed framework, which has good robustness to changes in the formation diagram. Based on the NuBot omnidirectional vision-omnidirectional mobile robot platform independently developed by the Robotics Laboratory of the National University of Defense Technology, this paper improves the two tracking algorithms in Reference [3] and realizes the formation control of multiple robots. The characteristics of the method presented in this paper are: 1) fully distributed control; 2) no global localization required; 3) the robot uses local communication; 4) the robot's orientation can be controlled independently; 5) obstacle avoidance is achieved through formation deformation or transformation. 2 Omni-vision and omnidirectional robot 2.1 NuBot robot As shown in Figure 1, the NuBot robot uses an omni-vision system as a sensor to obtain the relative position information of surrounding robots and obstacles, and can be positioned and oriented in a specific environment. In terms of movement, the robot is equipped with four self-developed NuBot omnidirectional wheels at a certain angle to form an omnidirectional motion mechanism, which enables the robot to move and rotate in any direction in the plane. 1) Omni-vision system The omni-vision system consists of an omnidirectional reflecting mirror and a camera, as shown in Figure 1(a). The convex mirror is the omnidirectional reflecting mirror, which plays the role of reflecting light, while the camera collects the light reflected by the omnidirectional reflecting mirror through lens refraction to obtain a panoramic image of the surrounding environment. Meanwhile, each robot is equipped with a specific color mark. The robot identifies each other by analyzing the color marks in the acquired panoramic image and calculates their relative coordinates to itself. 2) Omnidirectional motion mechanism Four NuBot omnidirectional wheels are installed as shown in Figure 1(b). The adjacent wheels are perpendicular to each other to form the robot's omnidirectional motion mechanism. It should be noted that since three points can determine a plane, ensuring that all four wheels touch the ground at the same time during the movement has a significant impact on the robot's motion performance. In this paper, the four omnidirectional wheel system is divided into two groups and fixed on two base plates. The two base plates are connected by a hinge, which effectively solves the problem of the robot touching the ground at four points. 2.2 Kinematic analysis The robot's workspace is a plane. An absolute coordinate system X[sub]w[/sub]-Y[sub]w[/sub] is established on this plane, and a local coordinate system X[sub]m[/sub]-Y[sub]m[/sub] is established on the vehicle body with its origin coinciding with the robot's center, as shown in Figure 2. Where θ is the angle between X[sub]m[/sub] and X[sub]w[/sub], and δ [/font] is the angle between the wheel and Y[sub]m[/sub]. The robot kinematic equations are as follows: 3 Formation-control algorithm 3.1 Tracking control Controlling one robot to track another or several robots at a certain distance and angle is the basis of the Leader-Follower multi-robot formation control method. This paper studies two tracking algorithms: distance-angle (DA) control and distance-distance (DD) control. (1) Distance-angle (DA) tracking control As shown in Figure 3, assume that robot R[sub]i[/sub] is the Leader and robot R[sub]j[/sub] is the Follower. R[sub]j[/sub] needs to track robot R[sub]i[/sub] at a certain distance and angle. In the figure, X_mi and X_mj represent the positive directions of the two robots, respectively, and and are the expected velocity directions of the two robots, respectively. is the actual velocity, and and are the angles observed by R_i to the direction of R_j, respectively, from its positive direction, expected velocity direction, and actual velocity direction. is the angle observed by R_j to the direction of R_i. Note 1: During formation changes, the direction of R_i's velocity may change significantly. If the direction is used as the reference direction for R_j, R_j and its subsequent robots may experience large oscillations, leading to formation control failure. Therefore, this paper uses the expected velocity as the reference direction, which reflects the overall movement direction of the formation. System State Equation Let equation (1) be written as: Control Law Using the input-output linearization method, we get: Where P is the auxiliary control input: Select an appropriate control gain k, k > 0, and the corresponding closed-loop linear system is: The control law of R[sub]j[/sub] is: Note 2: From the above process, it can be seen that the formation tracking control of R[sub]j[/sub] only involves and is independent of its angular velocity, which means that the orientation angle of R[sub]j[/sub] can be controlled independently. This feature of the algorithm can ensure that the orientation of the platform can be adjusted arbitrarily as needed during the formation operation and formation change of the robot. Implementation Method Since robots R[sub]i[/sub] and R[sub]j[/sub] can observe each other's relative positions, and each robot knows its own motion direction and orientation, they can obtain the input parameters required for tracking control through certain communication rules. ① Robot R[sub]i[/sub] sends the following data to robot R[sub]j[/sub]: ② Robot R[sub]j[/sub] calculates: ③ Robot R[sub]j[/sub] uses the new parameters for DA control. Note 3: Robot R[sub]j[/sub] tracks R[sub]i[/sub], which only requires R[sub]i[/sub] to send it speed and angle information. R[sub]j[/sub] obtains all the data required for DA tracking control based on this information and its own observation data. It can be seen that this method does not require global positioning and only requires less communication in the local area. (2) Distance-Distance (DD) Tracking Control As shown in Figure 4, assume that the robot tracks robots R[sub]i[/sub] and R[sub]j[/sub] at a certain distance. Let and be the positive directions of the three robots, and and be the actual speeds of R[sub]i[/sub] and R[sub]j[/sub], respectively, and and be the distances of R[sub]i[/sub] and R[sub]j[/sub] to. The system state equation (3) can be written as: Control Law Using the input-output linearization method, we get: Where p is the auxiliary control input: Selecting appropriate control gain k[sub]1[/sub], k[sub]2[/sub]>0, the corresponding closed-loop linear system is: The obtained control law is: Note 4: Unlike the DA tracking control method, here and are the angles between the current velocity directions of robots R[sub]i[/sub] and R[sub]j[/sub] and the direction where the robot is located. Implementation Method Compared with the DA tracking control method, since it does not involve maintaining a certain angle with the Leader robot in a specific direction, the implementation of this method is simpler. R[sub]i[/sub] and R[sub]j[/sub] send data to, and combined with their own observations, perform DD control. 3.2 Formation Description Robot formations are typically categorized into several types: rows, columns, triangles, rhombuses, etc., as shown in Figure 5. The description of robot formations is fundamental to formation control. Generally, a graph structure is used to represent formations, and matrices are employed for description. Nodes in the graph represent robots, and the lines connecting nodes represent the relationships between robots. Definition 1 (Robot Number): A unique identifier for each robot, denoted by R[sub]i[/sub]. Definition 2 (formation number J: used to identify different formations, denoted by F[sub]i[/sub]. Definition 3 (position number): used to distinguish different positions in the formation, denoted by P[sub]i[/sub]. After determining the current formation F[sub]i[/sub], robot R[sub]i[/sub] obtains its position P[sub]i[/sub] in the current formation F[sub]i[/sub] based on its own and the surrounding robots' situation, thereby determining its Leader and the corresponding expected distance and angle parameters, and performing DA or DD tracking control. 3.3 Formation obstacle avoidance When encountering obstacles, it may be necessary to adjust the formation to ensure that the formation passes smoothly. This adjustment can be divided into two cases: formation deformation and formation transformation. (1) Formation deformation obstacle avoidance When the obstacle has little impact on the entire formation, it is only necessary to deform the formation in a certain way (such as compressing it in a certain direction) without completely changing the formation, so as to ensure that the formation passes smoothly through the obstacle area. Definition 4 (Safety Distance): The minimum distance that ensures no collision between robots or between a robot and an obstacle during formation movement, denoted by D. Principles of Formation Transformation: ① When robot Rj is tracking Ri, if the distance between Rj and the nearest obstacle point O is less than the safety distance D, Rj uses the D-D control algorithm to track Ri and O, with the tracking parameters being the expected distance to Ri and D, respectively. ② When robot Rj is tracking Ri, if a robot's distance to Rj is less than the safety distance D due to obstacle avoidance, Ri will use the D-D control algorithm to track Ri and O, with the tracking parameters being the expected distance to Ri and D, respectively. (2) Obstacle Avoidance by Formation Change When an obstacle has a significant impact on the entire formation, and the formation change method cannot smoothly pass through the obstacle area (e.g., a formation passing through a narrow area with obstacles on both sides), it is necessary to change the formation to avoid the obstacle. If the robot in the formation (usually the formation leader) finds that a formation change is needed to pass through the obstacle area, the robot determines whether to change the formation and what new formation to use through commands or negotiation. Since the process of the robot negotiating to determine the formation is not the focus of this study, this paper uses the method of the leader robot directly issuing commands to realize the formation change and the determination of the new formation. 4 Experiments and Result Analysis The formation control method proposed in this paper is verified by combining simulation and actual robot formation experiments. This paper only gives some experimental scenarios. 4.1 In the DA formation control, the three robots initially move forward in a line formation. R1 is the leader of the entire formation, R2 tracks R2 at a distance of 80cm and an angle of 1.5π, and R3 tracks R2 at the same distance of 80cm and an angle of 1.5π. After a period of time, the formation changes to a column formation. At this time, R1 maintains its original speed, R2 changes to tracking R1 at a distance of 80cm and an angle of π, and R3 tracks R2 at the same distance of 80cm and an angle of π. The formation control in this experiment uses the DA control algorithm proposed in this paper. Figure 6 shows three scenarios in this experiment, where (A) is maintaining row formation, (C) is changing to column formation, and (B) is a moment in the formation transformation process. As can be seen from Figure (B), using the DA control algorithm in this paper, even during the formation transformation process, R3 can track R2 well, and the three robots still maintain a straight line, and the entire process is very smooth. It should be noted that if the control method in [3] is directly used, as shown in Figure 6 (B), during the formation transformation process, R3 will first move to point P behind the current movement direction of R2, and only after R2 completes the formation transformation will R3 change the target point to realize the formation transformation. In this way, not only is the formation transformation time prolonged, but the formation formation will also oscillate. 4.2 Formation Deformation Obstacle Avoidance This experiment is used to verify that a multi-robot formation can pass through an obstacle area by using formation deformation. Six robots are used to form a triangular formation and pass through an area with an obstacle on one side of the formation's running direction. Due to the limited number of laboratory robot platforms, a simulation experiment is used for verification. Figure 7 shows several key scenarios in the simulation process. The obstacle is on the left, and the six robots form a triangular formation and move forward. The arrows between the robots represent the tracking control relationship of each robot in the formation. If a robot has only one outward arrow, it uses the DA control algorithm; if it has two outward arrows, it uses the DD control algorithm. (1) Figure 7 (A) shows the standard triangular formation movement of the robots when there are no obstacles. All robots use DA control. (2) When the closest distance between R2 and the obstacle is less than the safe distance (50cm), its control algorithm becomes DD control that tracks R1 and the closest obstacle point O, with expected distances of 80cm and 50cm, respectively, as shown in Figure 7 (B). (3) When R4 When the distance to O2 is less than 50cm, the DD control is changed to track R3 and point O2, with expected distances of 80cm and 50cm respectively. At the same time, since the distance from the wind to R is less than 50cm, when R5 is also changed to track R4 and R3, the DD control is also changed to track R4 and R3, with expected distances of 80mm and 50em respectively, as shown in Figure 7(C). (4) When the robot's closest distance to the obstacle is greater than the safe distance, the DA control is restored, as shown in Figure 7(D), (E), and (F). 4.3 Formation Change Obstacle Avoidance This experiment verifies that a multi-robot formation can pass through an obstacle area by changing its formation. As shown in Figure 8, the experiment uses three NuBot robots to form a row formation and pass through a narrow area with obstacles on both sides. Since the area is narrow, the formation needs to be adjusted to a column formation. After passing through the obstacle area, the formation automatically restores the initial row formation and continues to move forward. (1) Initially, the three robots form a row formation and move. R1 is the leader of R2 and R3. R2 and R3 are both controlled by DA, as shown in Figure 8 (A). (2) When the formation approaches the obstacle area, R1 sends a formation change command to R2 and R3, changing to a triangular formation. R2 and R3 still use DA control to track R1, as shown in Figure 8(B). (3) After completing the triangular formation, the formation changes again to a column formation. At this time, R2 tracks R1, and R3 tracks R2, still using DA control, as shown in Figure 8(C) and (D). (4) After entering the obstacle area, the robot detects the closest distance to the obstacle. If the distance is less than the safe distance (50cm), the control algorithm is changed to DD control to track the robot in front and the closest obstacle point, with tracking distances of 80cm and 50cm respectively; otherwise, DA control is still used. As shown in Figure 8(E) and (F). (5) After passing through the obstacle area, the formation is restored to triangle and row in sequence. At this time, R3 and R2 are also restored to tracking R1 using DA control, as shown in Figure 8 (G), (H), and (I). 5 Conclusion and future work This paper briefly introduces the two main components of the NuBot robot: omnidirectional vision and omnidirectional motion system, and gives the kinematic model of the robot. Based on the robot platform, two tracking control algorithms, DA and DD, are proposed to realize the formation control of multiple robots. This paper also proposes and implements two formation obstacle avoidance methods: formation deformation and formation transformation. The formation control method in this paper has the following characteristics: (1) fully distributed control; (2) no global positioning required; (3) the robot adopts local communication; (4) the robot orientation can be controlled independently; (5) the formation can effectively achieve obstacle avoidance. Simulation and actual robot experiments verified the effectiveness of the formation control method in this paper. Among them, the DA control method can ensure smooth formation transformation, avoid large fluctuations, and has high efficiency. The formation deformation method can achieve smooth obstacle avoidance while maintaining the original formation as much as possible. The negotiation and determination of formation and the formation obstacle avoidance processing in more complex obstacle situations have not been studied in depth in this paper, which will be the problem to be solved in further work. References [1] Ren Dehua, Lu Guizhang. Thoughts on formation control [J]. Control and Decision, 2005, 20(6): 601-606. [2] Lewis MA, Tan KH. High precision formation control of mobile robots using virtual structures [J]. Autonomous Robots, 1997, 4(4): 387-403. [3] Das AK, FietTo R, Kumar V, et al. A vision-based formation control framework [J]. IEEE Transactions on Robotics and Automation, 2002, 18(5): 813-825. [4] Han Xuedong, Hong Bingrong, Meng Wei. Research on distributed control of multiple robots in arbitrary formations[J]. Robot. 2003, 25(1): 66-72. [5] Fredslund J, Mataric MJ. A general algorithm for robot formations using local sensing and minimal communication[J]. IEEE Transactions on Robotics and Automation, 2002, 18(5): 837-845. [6] Fax JA, Murray R. Information flow and cooperative control of vehicle formations[J]. IEEE Transactions on Automatic Control, 2004, 49(9): 1465-1476. [7] Shen Jie, Fei Shumin, Liu Huai. Research on control of multiple mobile robots forming arbitrary formations[J]. Robotics, 2004, 26(4): 295-300, [8] Balcb T, Arkin RC. Behavior based formation control for multibot teams[J]. IEEE Transactions on Robotics and Automation, 1999, 14(6): 926-939. About the authors: Liu Lin (1977-), male, PhD student. Research field: multi-robot systems, intelligent robots and robot soccer. Zheng Zhiqiang (1965-), male, professor, PhD supervisor. Research field: multi-robot systems, intelligent robots, robot soccer and precision guidance and control.