Formation method supporting multiple robots to have target formation and have no target point
By combining ICP algorithm and auction algorithm, a multi-robot formation method is designed, which solves the problem that traditional methods are difficult to adjust formation in real time in dynamic environments, and achieves efficient and flexible formation control and task allocation.
Patent Information
- Application Number
- CN202411992869.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-31
- Publication Date
- 2025-05-16
AI Technical Summary
In multi-robot systems, traditional formation control methods are difficult to adjust in real time in dynamic environments, and it is difficult to maintain stability and effective communication between robots, resulting in inefficiency and collision risks.
Combining the ICP algorithm and auction algorithm, a formation method that supports multiple robots in the case of target formation without target points is designed. By iteratively optimizing position alignment, the robot can dynamically respond to environmental changes and efficiently allocate tasks through the auction mechanism to achieve real-time formation adjustment.
This approach provides greater flexibility and adaptability, enables efficient formation control and task allocation in complex environments, reduces collision risks, and improves system responsiveness and overall efficiency.
Smart Images
Figure CN120010467A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the field of multi-robot formation, and in particular relates to a method for supporting multi-robot formation in the case of having a target formation but no target point. Background Art
[0002] In recent years, formation control in multi-robot systems has received extensive attention as it plays a vital role in achieving collaborative behaviors among robots. In applications such as self-driving cars, drone swarms, and robot teams for industrial automation, maintaining a specific formation can enhance overall system efficiency, improve situational awareness, and facilitate effective communication between agents. Coordinated mobility enables robots to complete tasks that are impossible or inefficient for a single robot, such as exploring hazardous environments or performing complex assembly tasks. Despite the many advantages of formation control, there are still significant challenges in achieving effective control. One of the main issues is ensuring that the robots can adapt to dynamic environments, as obstacles may suddenly appear. In addition, it is also crucial to maintain stability during movement and ensure that all robots can contribute to maintaining the formation without collisions or interference. Traditional methods often rely on fixed position assignments, which can lead to inefficiencies and difficulty in real-time adjustments when conditions change.
[0003] To address these challenges, this paper combines the ICP (Iterative Closest Point) algorithm with an auction algorithm. The ICP algorithm is known for its robustness in point cloud matching and is well suited to ensure that robots can accurately assess their relative positions and adjust accordingly. By iteratively optimizing the alignment of positions, robots can dynamically respond to changes in the environment and each other. On the other hand, the auction algorithm promotes efficient task allocation among robots. In a multi-robot system, each robot usually has different capabilities and task preferences. The auction mechanism allows robots to bid based on their current state, distance to the task, and priority. This not only optimizes resource utilization, but also enhances the system's responsiveness to the emergence of new tasks. The method proposed in this paper combines these algorithms, allowing robots to achieve the desired formation while allocating tasks in real time. By not requiring predefined positions, the method provides greater flexibility and adaptability, which is critical for operating in unpredictable environments.
[0004] The impact of this research is wide-ranging. For example, in search and rescue operations, teams of robots can quickly adjust their formation to cover different areas while efficiently managing tasks such as identifying survivors or mapping terrain. In agricultural applications, robots can coordinate their movements to optimize the planting or harvesting process, significantly improving efficiency.
[0005] In summary, this invention aims to promote the development of multi-robot systems and provide a powerful, flexible and efficient formation control and task allocation framework. The combination of ICP algorithm and auction algorithm not only solves existing challenges, but also opens up new ways for innovative applications in complex environments. Summary of the invention
[0006] Based on the above-mentioned background technology problems, the purpose of the present invention is to design a formation method that supports multiple robots in the case of a target formation but no target point, and to verify the designed formation method on the Matlab R2018a simulation platform.
[0007] The technical solution adopted by the present invention is a formation method that supports multi-robots with target formations but no target points. A network consisting of N mobile robots is built, and the positions of the robots in the network are represented by a set X, where: The i-th element in the set X represents the position x of robot i i , expressed as two-dimensional coordinates. is a mathematical symbol for a two-dimensional real number space, where represents the set of real numbers, and represents the set of all ordered pairs of two real numbers. Multiple robots start from an arbitrary initial configuration X 0 Initially, the goal is to make them form a specific formation that is invariant under scaling, translation, and rotation. The required formation model is Indicates that the i-th element in the set P represents the position p of the i-th point in the formation i , element p i are two-dimensional coordinates, and these points are chosen so that their centroid is at the origin, expressed as The set of assignments S is any bijection of P such that robot x i is assigned to element y i , Element y i is a two-dimensional coordinate, representing the two-dimensional coordinate of the i-th position in the transformed formation. The assignment set S assigns the robots to the points in the transformed formation in a one-to-one correspondence. Each robot corresponds to a point in the target formation, and each point in the target formation corresponds to only one robot. At the same time, Represents the translation vector of the robot formation, and τ is a two-dimensional coordinate. θ represents the rotation angle of the robot formation, indicating that the rotation of the robot formation is relative to the center of the formation. The local coordinate system is used to describe the rotation and adjust the orientation of the formation. It is expressed in radians. A positive rotation angle indicates a counterclockwise rotation, while a negative rotation angle indicates a clockwise rotation. K represents the scaling factor of the robot formation, and the scaling factor K scales the coordinates of all robots in the formation proportionally. If K>1, the formation will be enlarged; if K<1, the formation will be reduced; if K=1, the size of the formation remains unchanged, and the scaling operation is performed relative to the center of the formation. The goal is to determine S,τ,θ, and K to minimize the cost function L defined by the following formula.
[0008]
[0009] The rotation matrix R(θ) is defined as:
[0010]
[0011] where KR(θ)y i Represents the source point cloud y i The rotation (via the rotation matrix R(θ)) and scale adjustment (via the scaling factor K) are performed, and +τ is a translation operation used to transform the rotated and scaled point cloud KR(θ)y i Move to the appropriate position so that it matches the point x in the robot position point cloud X i Align. || || 2 is the square of the Euclidean distance, which is used to calculate the difference between the scaled, rotated and translated point and the multi-robot point. For each point i, this difference is KR(θ)y i +τ vs x i The entire cost function L is the sum of the errors of all N points, aiming to minimize the overall error of all point alignments.
[0012] The specific implementation steps of the whole method are as follows:
[0013] Step 1: Initialize the robot's related variables, using the initial estimated robot formation's rotation angle θ, the robot formation's translation vector τ, and the robot formation's scaling factor K, τ[0] = τ 0 , θ[0]=θ 0 , K[0]=K 0 At this time, the target formation will be transformed according to the given initial τ, θ, K, that is, KR(θ)y i +τ.
[0014] Step 2: Increase the iteration variable k. According to the rotation angle θ of the robot formation in the previous iteration, the translation vector τ of the robot formation and the scaling factor K of the robot formation, select the assignment set S to minimize the cost function:
[0015]
[0016] This formula means that the allocation result S obtained through the auction algorithm minimizes the total distance or time to complete the task, that is, minimizes the cost function L, and then calculates S, which is the allocation result.
[0017] Step 3: Based on the assignment set S determined in step 2, that is, the robot allocation result, calculate the optimal robot formation rotation angle θ, the robot formation translation vector τ and the robot formation scaling factor K to minimize the cost function:
[0018]
[0019] This formula represents updating τ, θ and K by transforming the derivative to minimize the cost function L.
[0020] Step 3.1: The rotation angle θ of the optimal robot formation can be derived by the following formula:
[0021]
[0022] in
[0023]
[0024] and
[0025]
[0026] Here, μ x and μ y Represents the centroid of point sets X and Y.
[0027] The derivation process is as follows:
[0028]
[0029] Substitute τ from step 3.2 to obtain
[0030]
[0031] Step 3.2: The translation vector τ of the optimal robot formation is calculated by aligning the centroids of the two point sets:
[0032]
[0033] Step 3.3: At the same time, the scaling factor K of the robot formation is calculated as follows:
[0034]
[0035] The derivation process is as follows:
[0036]
[0037] Available
[0038]
[0039] This step is to update various parameters to facilitate the next formation transformation calculation.
[0040] Step 4: Determine whether the algorithm has converged. If the algorithm has not converged, return to step 2 and use the updated τ, θ, and K. If the algorithm converges, then τ, θ, and K at this time are the final robot formation transformation parameters. The target formation given at this time is transformed according to τ, θ, and K, which is the final formation position of the robot.
[0041] The present invention provides a formation method supporting multiple robots with target formations but without target points. Compared with the prior art, the present invention has the following beneficial effects:
[0042] 1. Compared with many existing methods for solving multi-robot formation problems, most of them provide target points for multiple robots, while the algorithm we designed is a formation method with target formation but no target points.
[0043] 2. The designed formation algorithm adds a scaling ratio K, which can be scaled on the existing formation to achieve the scaling of the formation. BRIEF DESCRIPTION OF THE DRAWINGS
[0044] Figure 1 Flowchart of the designed formation method.
[0045] Figure 2 Schematic diagram of eight groups of robots in numerical simulation, where the black × represents the initial position of the robot, the red dot represents the target formation, the blue dot represents the final formation, and the straight line represents the robot path. Figure 3 to Figure 4 The black ×, red dots, blue dots and straight lines in numerical simulation represent the same meanings as Figure 2 same.
[0046] Figure 3 Schematic diagram of the numerical simulation of fourteen groups of robots.
[0047] Figure 4 Schematic diagram of the numerical simulation of twenty-five groups of robots. DETAILED DESCRIPTION
[0048] The present invention is described in detail below with reference to the accompanying drawings and embodiments.
[0049] A formation method that supports multiple robots with target formations but no target points, the specific contents are as follows:
[0050] Consider a network consisting of N mobile robots, whose positions are represented by The i-th element in the set X represents the position x of robot i. i , expressed as two-dimensional coordinates. is a mathematical symbol for a two-dimensional real number space, where represents the set of real numbers, and represents the set of all ordered pairs of two real numbers. The multi-robots start from an arbitrary initial configuration X 0 Initially, the goal is to make them form a specific formation that is invariant under scaling, translation, and rotation. The required formation model is Indicates that the i-th element in the set P represents the position p of the i-th point in the formation i , element p i are two-dimensional coordinates, and these points should be chosen so that their centroid is at the origin, expressed as Furthermore, the set of assignments S is any bijection of P such that robot x i is assigned to element y i , Element y i is a two-dimensional coordinate, representing the two-dimensional coordinate of the i-th position in the transformed formation. The assignment set S assigns the robots to the points in the transformed formation in a one-to-one correspondence. Each robot corresponds to a point in the target formation, and each point in the target formation corresponds to only one robot. At the same time, Represents the translation vector of the robot formation, and τ is a two-dimensional coordinate. θ represents the rotation angle of the robot formation, indicating that the rotation of the robot formation is relative to the center of the formation. The local coordinate system is used to describe the rotation and adjust the orientation of the formation. It is expressed in radians. A positive rotation angle indicates a counterclockwise rotation, while a negative rotation angle indicates a clockwise rotation. K represents the scaling factor of the robot formation, and the scaling factor K scales the coordinates of all robots in the formation proportionally. If K>1, the formation will be enlarged; if K<1, the formation will be reduced; if K=1, the size of the formation remains unchanged, and the scaling operation is performed relative to the center of the formation. Our goal is to determine S,τ,θ, and K to minimize the cost function L defined by the following formula.
[0051]
[0052] The rotation matrix R(θ) is defined as:
[0053]
[0054] where KR(θ)yi Represents the source point cloud y i The rotation (via the rotation matrix R(θ)) and scale adjustment (via the scaling factor K) are performed, and +τ is a translation operation used to transform the rotated and scaled point cloud KR(θ)y i Move to the appropriate position so that it matches the point x in the robot position point cloud X i Align. || || 2 is the square of the Euclidean distance, which is used to calculate the difference between the scaled, rotated and translated point and the multi-robot point. For each point i, this difference is KR(θ)y i +τ vs x i The entire cost function L is the sum of the errors of all N points, aiming to minimize the overall error of all point alignments.
[0055] The specific steps of the algorithm are as follows:
[0056] Step 1: Initialize the robot's related variables, using the initial estimated robot formation's rotation angle θ, the robot formation's translation vector τ, and the robot formation's scaling factor K, τ[0] = τ 0 , θ[0]=θ 0 , K[0]=K 0 At this time, the target formation will be transformed according to the given initial τ, θ, K, that is, KR(θ)y i +τ.
[0057] Step 2: Increase the iteration variable k. According to the rotation angle θ of the robot formation in the previous iteration, the translation vector τ of the robot formation and the scaling factor K of the robot formation, select the assignment set S to minimize the cost function:
[0058]
[0059] This formula means that the allocation result S obtained by the auction algorithm minimizes the total distance or time to complete the task, that is, minimizes the cost function L. This step can be achieved by using an established method, that is, by an auction algorithm in the present invention, to calculate S, that is, the allocation result.
[0060] Step 3: Based on the assignment set S determined in step 2, that is, the robot allocation result, calculate the optimal robot formation rotation angle θ, the robot formation translation vector τ and the robot formation scaling factor K to minimize the cost function:
[0061]
[0062] This formula represents updating τ, θ and K by transforming the derivative to minimize the cost function L.
[0063] Step 3.1: The rotation angle θ of the optimal robot formation can be derived by the following formula:
[0064]
[0065] in
[0066]
[0067] and
[0068] Here, μ x and μ y Represents the centroid of point sets X and Y.
[0069] The derivation process is as follows:
[0070]
[0071] Substituting τ in step 3.2, we can get
[0072]
[0073] Step 3.2: The translation vector τ of the optimal robot formation is calculated by aligning the centroids of the two point sets:
[0074]
[0075] Step 3.3: At the same time, the scaling factor K of the robot formation is calculated as follows:
[0076]
[0077] The derivation process is as follows:
[0078]
[0079] Available
[0080]
[0081] This step is to update various parameters to facilitate the next formation transformation calculation.
[0082] Step 4: Determine whether the algorithm has converged. If the algorithm has not yet converged, return to step 2 and use the updated τ, θ, and K. If the algorithm converges, then τ, θ, and K at this time are the final robot formation transformation parameters. The target formation given at this time is transformed according to τ, θ, and K, which is the final formation position of the robot. The above are the specific steps of the algorithm and the calculation method of parameter iteration in each step.
[0083] Next, the convergence of the algorithm is proved.
[0084] The proof steps are as follows:
[0085] The algorithm converges in finite time. To prove this, we will analyze the behavior of the cost function at each iteration k. The post-assignment error in step 2 is defined as:
[0086]
[0087] Where S represents the allocation set. In step 3, the error after translation and rotation can be expressed as:
[0088]
[0089] It is important to note that d[k]≤e[k]. The reason this holds is that both e[k] and d[k] represent cost function values for a fixed allocation Y[k], but d[k] takes into account the optimal translation and rotation based on S[k]. If d[k]>e[k], this indicates that the rotation and translation calculated in step 3 were not optimal. Therefore, the algorithm returns to step 2 to calculate a new optimal allocation given the translation. Since τ and θ remain unchanged, we have e[k+1]≤d[k]. If e[k+1]>d[k], this means that the new allocation is also suboptimal. Therefore, we can conclude that:
[0090] 0≤d[k+1]≤e[k+1]≤d[k]≤e[k]
[0091] Eventually, given that the number of assignment sets S that can be paired one-to-one with the set P is finite, the algorithm will converge in finite time.
[0092] The algorithm reaches a Nash equilibrium. In the context of game theory, the two "players" are the morphological posture update (step 3) and the role allocation update (step 2). The goal of each "player" is to minimize a utility function, namely the cost function L. Therefore, reaching a Nash equilibrium means that the final allocation is the optimal choice considering the final posture, and the final posture is the optimal choice given the final allocation:
[0093]
[0094] Proof: At iteration k f When converged, we have;
[0095] S[k f ]=S[k f -1]=S *
[0096] (τ[k f ],θ[k f ],K[k f])=(τ[k f -1],θ[k f -1],K[k f -1])=(τ * ,θ * ,K * )
[0097] Substituting the above equation into steps 2 and 3, we get:
[0098]
[0099] The overall algorithm flow chart is as follows Figure 1 shown.
[0100] To validate and enhance our theoretical framework, we implemented the proposed allocation algorithm in Matlab R2018a and simulated it with a virtual robot network. These robots are modeled as point entities in space, with no inertia and can respond to velocity commands instantly. This full dynamic property allows the robots to move instantly in any direction without any constraints.
[0101] Initially, we organized the robot network into a complete graph, where each robot can sense the relative position of all other robots in the network. The initial position of the robot is randomly generated in a 4*4 square area, thus ensuring the diversity of the initial configuration. To fully evaluate the performance of the algorithm, we used networks of different sizes for three independent tests, with network sizes of N=8, N=14, and N=25. In each test, all robots executed the same algorithm at the same time. Figure 2 , Figure 3 and Figure 4 Provides a visual representation of the robots' initial and final positions, as well as the paths they traveled during their movement.
[0102] The algorithm exhibits impressive scalability, with the transition from N=8 to N=25 occurring smoothly without any modifications to the underlying framework. Notably, each robot follows a straight trajectory toward its assigned goal location, which is in the desired formation. This observation suggests that the assignment and formation pose remain consistent as the robots progress toward their respective goals. Furthermore, we find that no robot needs to travel an excessively long distance to reach its destination, which is highly consistent with our initial expectations for efficiency and performance.
[0103] In addition, it is worth mentioning that the centroid of the final configuration coincides with the centroid of the initial configuration, which further verifies the algorithm’s ability to maintain the overall spatial integrity of the robot network. Importantly, in all three test scenarios, the robots’ trajectories did not cross, even when the network size was increased to N = 25 robots. This result is significant because it verifies the algorithm’s effectiveness in preventing collisions, which is a critical issue in multi-robot systems.
[0104] The following is a further detailed description of the formation method of supporting multiple robots with target formations but without target points in conjunction with the accompanying drawings and embodiments.
[0105] In order to better test the effectiveness of the formation method that supports multi-robots with target formations but without target points, numerical simulations are performed for different numbers of robots. The numerical simulation verifies the correctness of the design algorithm by comparing the number of robots.
[0106] The numerical simulation experiment platform uses Matlab R2018a, and the specific implementation method of numerical simulation is as follows:
[0107] Step 1: Enter the program code of the designed formation method in the software platform Matlab R2018.
[0108] Step 2: Create a new file and input the program code of the algorithm we designed in the software platform Matlab R2018. The algorithm data inputs the initial position of each robot and the target formation of the robot. The difference is that the number of robots continues to increase.
[0109] Step 3: Continue to increase the number of robots from 8 to 14, and then to 25, run the algorithm, and obtain simulation results.
[0110] Step 4: Compare the simulation results of different robot operation algorithms. The specific operation results of the formation method we designed are as follows: Figure 2 , Figure 3 , Figure 4 .
[0111] These simulation results are highly consistent with our theoretical expectations, confirming that formation synthesis can always be successfully achieved in the context of a complete graph. Through this study, we not only verified the effectiveness of the proposed algorithm, but also laid the foundation for further exploration of optimizing robot formation control in different applications.
[0112] Experimental results show that the algorithm can effectively achieve the desired formation even when only the target formation is provided without specifying the specific target point. The robot is able to autonomously explore unoccupied target positions, allowing flexible target allocation and demonstrating the adaptability and efficiency of the algorithm in complex environments. The smooth transition between networks of different sizes shows that the algorithm maintains good performance at different scales, further emphasizing its practicality in large-scale robotic systems. This finding further verifies the effectiveness of the algorithm and shows that the algorithm can achieve efficient formation control even in the absence of a clear target point.
Claims
1. A method for supporting multi-robot formation with target formation but without target point, characterized in that: Build a network consisting of N mobile robots. The positions of the robots in the network are represented by a set X, where The i-th element in the set X represents the position x of robot i i , expressed as two-dimensional coordinates; is a mathematical symbol for a two-dimensional real number space, where represents the set of real numbers, represents the set of all ordered pairs of two real numbers; multiple robots start from an arbitrary initial configuration X0, and the goal is to make them form a specific formation that is invariant under scaling, translation and rotation; the required formation model is denoted by Indicates that the i-th element in the set P represents the position p of the i-th point in the formation i , element p i are two-dimensional coordinates, and these points are chosen so that their centroid is at the origin, expressed as The set of assignments S is any bijection of P such that robot x i is assigned to element y i , Element y i is a two-dimensional coordinate, representing the two-dimensional coordinate of the i-th position in the transformed formation; the assignment set S assigns the robots to the points in the transformed formation in a one-to-one correspondence; Each robot corresponds to a point in the target formation, and each point in the target formation corresponds to only one robot; at the same time, Represents the translation vector of the robot formation, τ is a two-dimensional coordinate; θ represents the rotation angle of the robot formation, indicating that the rotation of the robot formation is relative to the center of the formation, and the local coordinate system is used to describe the rotation and adjust the orientation of the formation; it is expressed in radians; a positive rotation angle represents a counterclockwise rotation, while a negative rotation angle represents a clockwise rotation; K represents the scaling factor of the robot formation, and the scaling factor K will scale the coordinates of all robots in the formation proportionally; if K>1, the formation will be enlarged; if K<1, the formation will be reduced; if K=1, the size of the formation remains unchanged, and the scaling operation is performed relative to the center of the formation; the goal is to determine S, τ, θ and K to minimize the cost function L defined by the following formula; The rotation matrix R(θ) is defined as: Where KR(θ)y i Represents the source point cloud y i Perform rotation and scale adjustment, +τ is a translation operation, which is used to transform the rotated and scaled point cloud KR(θ)y i Move to the appropriate position so that it matches the point x in the robot position point cloud X i Align;|| || 2 is the square of the Euclidean distance, which is used to calculate the difference between the scaled, rotated and translated point and the multi-robot point; for each point i, this difference is KR(θ)y i +τ vs x i The entire cost function L is the sum of the errors of all N points, aiming to minimize the overall error of all point alignments.
2. A method for supporting multi-robot formation with target formation but without target point according to claim 1, characterized in that: The specific implementation steps of the whole method are as follows: Step 1: Initialize the relevant variables of the robot, using the initial estimated rotation angle θ of the robot formation, the translation vector τ of the robot formation and the scaling factor K of the robot formation, τ[0] = τ0, θ[0] = θ0, K[0] = K0; at this time, the given target formation will be transformed according to the given initial τ, θ, K, that is, KR(θ)y i +τ; Step 2: Add the iteration variable k; select the assignment set S to minimize the cost function based on the rotation angle θ of the robot formation in the previous iteration, the translation vector τ of the robot formation, and the scaling factor K of the robot formation: This formula means that the allocation result S obtained by the auction algorithm minimizes the total distance or time to complete the task, that is, minimizes the cost function L, and then calculates S, which is the allocation result; Step 3: Based on the assignment set S determined in step 2, that is, the robot allocation result, calculate the optimal robot formation rotation angle θ, the robot formation translation vector τ and the robot formation scaling factor K to minimize the cost function: This formula indicates that τ, θ and K are updated by transforming derivatives to minimize the cost function L; Step 3.1: The rotation angle θ of the optimal robot formation can be derived by the following formula: in and Here, μ x and μ y represents the centroid of point sets X and Y; The derivation process is as follows: Substitute τ from step 3.2 to obtain Step 3.2: The translation vector τ of the optimal robot formation is calculated by aligning the centroids of the two point sets: Step 3.3: At the same time, the scaling factor K of the robot formation is calculated as follows: The derivation process is as follows: Available This step updates various parameters to facilitate the next formation transformation calculation; Step 4: Determine whether the algorithm has converged; if the algorithm has not yet converged, return to step 2 and use the updated τ, θ and K; if the algorithm converges, then the τ, θ, K at this time are the final robot formation transformation parameters. The target formation given at this time is transformed according to τ, θ, K, which is the final formation position of the robot.