Distributed Formation Control Method and Computer Equipment for Multi-Agent Systems Based on Distance Ratio Constraints

Abstract

This invention discloses a distributed formation control method for multi-agent systems based on range ratio constraints. It establishes the definition of graph rigidity, graph topology, and kinematic model of the multi-agent system. Based on the geometric constraints of the range ratio, a range ratio rigid frame and a range ratio stiffness matrix are constructed. Combining the target formation configuration, the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame is solved. A control input criterion function is designed to minimize formation error, and the controller design and characteristics of the formation system are explored. For triangular formations in two-dimensional space and tetrahedral formations in three-dimensional space, the undesired equilibrium points of the formation system are calculated, and the conditions for global convergence are solved. This invention introduces range ratio constraints and designs a distributed formation control algorithm based on range ratio and direction information measured by low-cost sensors, thereby reducing dependence on global information and high-cost sensors.

Description

Technical Field

[0001] This invention belongs to the field of cooperative control technology for multi-agent systems, and specifically relates to a distributed formation control method for multi-agent systems based on distance ratio constraints. Background Technology

[0002] Distributed formation control is an important and challenging research area in multi-agent systems. It involves a group of agents cooperating and coordinating to achieve specific formation patterns or tasks. These tasks span multiple domains, including robotic transportation, motion sensing, UAV optical displays, and satellite formation. To achieve effective distributed formation control, the relationships and constraints between agents must be considered. In this context, rigid graph theory has been widely applied to distributed formation control. A rigid graph is a mathematical tool used to describe the connections and constraints between agents. It treats agents as nodes in a graph and uses edges to represent the connections between nodes. Past research has proposed many distributed formation control algorithms and theories based on rigid graphs. Common distributed formation constraints include displacement constraints, distance constraints, orientation constraints, angle constraints, and distance ratio constraints. Distributed formation control methods offer several advantages. First, they do not rely on a central node or global information, thus exhibiting high robustness and scalability. Second, they are adaptable and flexible, allowing for adjustments to formation patterns and task objectives as needed. Furthermore, this method is low-cost and efficient because it does not require a large number of sensors and communication devices. Most importantly, the distributed formation control method enables coordination and cooperation among multiple agents, thus exhibiting diversity and synergy.

[0003] However, despite some progress, several challenges and problems remain to be addressed. For example, displacement information is often difficult to obtain in the absence of a global positioning system; distance constraints may limit the scaling degrees of freedom of the formation system; orientation constraint methods rely on the unification of global coordinate axes; and angle constraints are difficult to implement in spatial coordinates. Furthermore, rigid graphical models exhibit nonlinearity and strong multivariable coupling, posing new challenges to distributed formation control research. Therefore, this study proposes a distributed formation control method for multi-agent systems based on distance ratio constraints, enabling multi-agent systems to form formations with translational, rotational, and scaling degrees of freedom in two-dimensional or three-dimensional space. This provides a new solution to the control problem of distributed formation in multi-agent systems and offers potential opportunities for further research and application of distributed formation control.

[0004] Summary of the Invention

[0005] The purpose of this invention is to provide a distributed formation control method for multi-agent systems based on distance ratio constraints.

[0006] The technical solution to achieve the purpose of this invention is: a distributed formation control method for a multi-agent system based on distance ratio constraints, comprising the following steps:

[0007] Step 1: Considering the formation size and sensor network, establish the definition of graph rigidity, graph topology, and kinematic model of multi-agent system;

[0008] Step 2: Based on the geometric constraints of the distance ratio, construct the distance ratio rigid frame and the distance ratio stiffness matrix;

[0009] Step 3: Based on the target formation configuration, solve the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame;

[0010] Step 4: Design the control input criterion function, minimize the formation error, and explore the controller design and characteristics of the formation system.

[0011] Step 5: For triangular formations in two-dimensional space and tetrahedral formations in three-dimensional space, calculate the undesirable equilibrium points of the formation system and solve for the conditions for global convergence of the system.

[0012] Step 6: Use the formation stabilization controller designed in Step 4 to perform distributed formation of the multi-agent system.

[0013] Preferably, in step 1, a class is first defined by... strip and An undirected graph consisting of vertices Its point set is edge set is Intelligent agents The neighbor set is defined as The matrix that associates nodes with edges is called the incidence matrix. Specifically, it is expressed as follows:

[0014]

[0015] Secondly, the definition of rigid graph theory is given, which is a framework in three-dimensional space. It is considered rigid when the following conditions are met: There exists a neighborhood satisfy ,in It is a with The same complete graph. And when At that time, it is called a framework. It is globally rigid. If translation, rotation, scaling, or combinations thereof are defined as infinitesimal motions, then infinitesimal motion is defined as follows: if the motion of the graph vertices ensures the invariance of the graph, that is... If this motion is such that it is called an infinitesimal motion, then the motion is called an infinitesimal motion. , As vertices The speed. And the framework. When all of a frame’s motions are infinitesimal motions, the frame is said to have infinitesimal stiffness.

[0016] Consider another set of... A single integrator The multi-agent system built by Wiene:

[0017]

[0018] in and They are the first The position and control input (i.e., velocity) of an agent in the global coordinate system. Define vectors. Representation diagram exist Implementation in spatial form. Introducing matrices. A relative position vector can be constructed into a position vector. Mapping:

[0019]

[0020] in , It is the first The edge determines the relative position vector formed by the fixed point pair.

[0021] Preferably, in step 2, based on the geometric constraints, a distance-rigid frame is first constructed, and the frame is defined. Distance stiffness function for:

[0022]

[0023] Define the distance stiffness matrix Then each row of the distance stiffness matrix can be represented as:

[0024]

[0025] Redefining the distance stiffness matrix Represented as The following simplified form of the distance stiffness matrix can be obtained:

[0026]

[0027] in Secondly, a distance ratio rigid frame is constructed. Unlike a distance-rigid frame, distance ratio constraints in real-world systems are often redundant. Taking a triangle as an example, it has three pairs of distance ratio constraints, but only any two pairs are needed to uniquely determine the triangle's frame. Definition Then the distance ratio stiffness function It can be represented as:

[0028]

[0029] in This represents the number of distance ratio constraints. Redundant constraints are removed, and a new distance ratio stiffness function is defined. ,in . merely a subset of It is the minimum set of constraints that can realize both minimum and infinitesimal rigid frames. Combining this with the distance stiffness matrix, the following relationship can be obtained:

[0030]

[0031] in , It is the distance stiffness matrix. Defined as the distance-to-stiffness matrix.

[0032] Preferably, in step 3, the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame is first solved based on the target formation configuration. The following two-dimensional rotation matrix is ​​defined:

[0033]

[0034] and three-dimensional rotation matrix :

[0035]

[0036] The frame motion in two-dimensional space can be defined by the following set. The corresponding frame motion in three-dimensional space is caused by Indicated. Based on and It is possible to obtain an infinitesimal distance ratio that satisfies the rigid frame. Therefore, the distance ratio stiffness matrix satisfies... Therefore, the following conclusion can be drawn: For framework If and only if The system is infinitesimally rigid in terms of distance ratio. Based on this conclusion, the objective constraints for achieving a formation with minimum and infinitesimally rigid distance ratios can be solved.

[0037] Preferably, in step 4, a set of equilibrium points for the target formation is defined; a distributed formation control algorithm is designed for the constructed target set to minimize the formation error, and the characteristics of the controller relying only on local information and the fact that the entire formation system has degrees of freedom of translation, rotation, and scaling are explored. First, the desired equilibrium point is defined as follows:

[0038]

[0039] This leads to the control objective of distributed formation based on distance ratio: The desired formation under this control objective satisfies the conditions of minimum and infinitesimal distance-to-stiffness. To address this, a formation stabilization controller is designed:

[0040]

[0041] in:

[0042]

[0043] and It controls the gain; and It is a unit direction vector; This is distance ratio information. Preferably, the formation system satisfies invariance of center of mass, total angular momentum, and overall scaling:

[0044]

[0045] And when the initial state of the multi-agent system satisfies

[0046]

[0047] At that time, the formation system will not collide.

[0048] Preferably, in step 5, the set of equilibrium points in the two-dimensional space under the controller (12) is first calculated, and the set is defined as follows:

[0049]

[0050] in and Let represent the target formation and the undesired equilibrium point, respectively. Without loss of generality, taking agent 3 as an example, let ,and The following equation was calculated:

[0051]

[0052] It can be concluded that the three agents are collinear. The same logic applies to agents 1 and 2. Linearize the formation error system and solve for the system. The Hessian matrix is ​​calculated as follows:

[0053]

[0054] in For coefficients, satisfy

[0055]

[0056] Substitute into the balanced set The points in the middle are calculated to obtain If a negative root exists, then the set of undesirable equilibrium points is unstable.

[0057] Preferably, the equilibrium point is determined for the tetrahedral formation in three-dimensional space:

[0058]

[0059] Then the four agents are coplanar. Similarly, solve the system... The Hessian matrix is ​​calculated to obtain If there is a negative root, then the set of undesirable equilibrium points is unstable. In conclusion, for triangular formations in the plane and tetrahedral formations in space, if the initial states of the multi-agent system do not satisfy the collinear or coplanar conditions, then the entire system can achieve global stability based on the controller (12).

[0060] The present invention also discloses a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the distributed formation control method for a multi-agent system based on distance ratio constraints as described in the present invention.

[0061] Beneficial effects of the present invention

[0062] By introducing distance ratio constraints, a distributed formation control algorithm is designed based on distance ratio information and direction information measured by low-cost sensors. This reduces the dependence on global information and high-cost sensors, ultimately achieving the goal of distributed formation with translation, scaling and rotation degrees of freedom. Attached Figure Description

[0063] Figure 1This is a flowchart of the distributed formation control method for multi-agent systems based on distance ratio constraints according to the present invention.

[0064] Figure 2 This is a schematic diagram of the spatial formation of six intelligent agents in an embodiment of the method of the present invention.

[0065] Figure 3 This is a graph showing the spatial formation error curves of six intelligent agents in an embodiment of the method of the present invention.

[0066] Figure 4 This is a diagram showing the planar triangle formation result in an embodiment of the method of the present invention.

[0067] Figure 5 This is a diagram showing the spatial tetrahedral formation result in an embodiment of the method of the present invention.

[0068] Figure 6 This is a diagram of a quadcopter UAV verification device in an embodiment of the method of the present invention.

[0069] Figure 7 This is a graph showing the experimental results of a quadcopter UAV in an embodiment of the method of the present invention.

[0070] Detailed Implementation

[0071] The present invention will be further described below with reference to embodiments, but the scope of protection of the present invention is not limited thereto:

[0072] A distributed formation control method for multi-agent systems based on distance ratio constraints, combined with Figure 1 This includes the following steps:

[0073] Step 1, considering formation size and sensor network, establish the definition of graph rigidity, graph topology, and kinematic model of multi-agent system, specifically:

[0074] Step 1-1, define a... strip and An undirected graph consisting of vertices Its point set is edge set is Intelligent agents The neighbor set is defined as The matrix that associates nodes with edges is called the incidence matrix. Specifically, it is expressed as follows:

[0075]

[0076] Steps 1-2 give the definition of rigid graph theory, a frame in three-dimensional space. It is considered rigid when the following conditions are met: There exists a neighborhood satisfy ,in It is a with The same complete graph. And when At that time, it is called a framework. It is globally rigid. If translation, rotation, scaling, or combinations thereof are defined as infinitesimal motions, then infinitesimal motion is defined as follows: if the motion of the graph vertices ensures the invariance of the graph, that is... If this motion is such that it is called an infinitesimal motion, then the motion is called an infinitesimal motion. , As vertices The speed. And the framework. When all of a frame’s motions are infinitesimal motions, the frame is said to have infinitesimal stiffness.

[0077] Steps 1-3, consider a set of... A single integrator The multi-agent system built by Wiene:

[0078]

[0079] in and They are the first The position and control input (i.e., velocity) of an agent in the global coordinate system. Define vectors. Representation diagram exist Implementation in spatial form. Introducing matrices. A relative position vector can be constructed into a position vector. Mapping:

[0080]

[0081] in , It is the first The edge determines the relative position vector formed by the fixed point pair.

[0082] Step 2: Based on the geometric constraints of the distance ratio, construct the distance ratio rigid frame and the distance ratio stiffness matrix, specifically as follows:

[0083] Based on geometric constraints, a distance-rigid frame is first constructed, and the frame is defined. Distance stiffness function for:

[0084]

[0085] Define the distance stiffness matrix Then each row of the distance stiffness matrix can be represented as:

[0086]

[0087] Redefining the distance stiffness matrix Represented as The following simplified form of the distance stiffness matrix can be obtained:

[0088]

[0089] in Secondly, a distance ratio rigid frame is constructed. Unlike a distance-rigid frame, distance ratio constraints in real-world systems are often redundant. Taking a triangle as an example, it has three pairs of distance ratio constraints, but only any two pairs are needed to uniquely determine the triangle's frame. Definition Then the distance ratio stiffness function It can be represented as:

[0090]

[0091] in This represents the number of distance ratio constraints. Redundant constraints are removed, and a new distance ratio stiffness function is defined. ,in . merely a subset of It is the minimum set of constraints that can realize both minimum and infinitesimal rigid frames. Combining this with the distance stiffness matrix, the following relationship can be obtained:

[0092]

[0093] in , It is the distance stiffness matrix. Defined as the distance-to-stiffness matrix.

[0094] Step 3, combining the target formation configuration, solve the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame, specifically:

[0095] Based on the target formation configuration, the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame is solved. The following two-dimensional rotation matrix is ​​defined:

[0096]

[0097] and three-dimensional rotation matrix :

[0098]

[0099] The frame motion in two-dimensional space can be defined by the following set. The corresponding frame motion in three-dimensional space is caused by Indicated. Based on and It is possible to obtain an infinitesimal distance ratio that satisfies the rigid frame. Therefore, the distance ratio stiffness matrix satisfies... Therefore, the following conclusion can be drawn: For framework If and only if The system is infinitesimally rigid in terms of distance ratio. Based on this conclusion, the objective constraints for achieving a formation with minimum and infinitesimally rigid distance ratios can be solved.

[0100] Step 4: Design the control input criterion function to minimize the formation error, and explore the controller design and characteristics of the formation system. Specifically:

[0101] Step 4-1: Define the set of equilibrium points for the target formation; design a distributed formation control algorithm for the constructed target set to minimize formation error, and explore the characteristic that the controller relies only on local information and that the entire formation system has degrees of freedom of translation, rotation, and scaling. First, the desired equilibrium point is defined as follows:

[0102]

[0103] This leads to the control objective of distributed formation based on distance ratio: The desired formation under this control objective satisfies the conditions of minimum and infinitesimal distance-to-stiffness. To address this, a formation stabilization controller is designed:

[0104]

[0105] in:

[0106]

[0107] and It controls the gain; and It is a unit direction vector; It is distance ratio information.

[0108] Step 4-2, the formation system satisfies the invariance of center of mass, total angular momentum, and overall scaling:

[0109]

[0110] And when the initial state of the multi-agent system satisfies

[0111]

[0112] At that time, the formation system will not collide.

[0113] Step 5: For triangular formations in two-dimensional space and tetrahedral formations in three-dimensional space, calculate the undesirable equilibrium points of the formation system and solve for the conditions for global convergence of the system. Specifically:

[0114] Step 5-1: Calculate the set of equilibrium points in the two-dimensional space under the controller (12), and define the following set:

[0115]

[0116] in and Let represent the target formation and the undesired equilibrium point, respectively. Without loss of generality, taking agent 3 as an example, let ,and The following equation was calculated:

[0117]

[0118] It can be concluded that the three agents are collinear. The same logic applies to agents 1 and 2.

[0119] Step 5-2: Linearize the formation error system and solve the system. The Hessian matrix is ​​calculated as follows:

[0120]

[0121] in For coefficients, satisfy

[0122]

[0123] Substitute into the balanced set The points in the middle are calculated to obtain If a negative root exists, then the set of undesirable equilibrium points is unstable.

[0124] Step 5-3: Solve for the equilibrium point of the tetrahedral formation in three-dimensional space:

[0125]

[0126] Then the four agents are coplanar. Similarly, solve the system... The Hessian matrix is ​​calculated to obtain If there is a negative root, then the set of undesirable equilibrium points is unstable. In conclusion, for triangular formations in the plane and tetrahedral formations in space, if the initial states of the multi-agent system do not satisfy the collinear or coplanar conditions, then the entire system can achieve global stability based on the controller (12).

[0127] Step 6: Use the formation stabilization controller designed in Step 4 to perform distributed formation of the multi-agent system.

[0128] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, performs the following steps:

[0129] Step 1: Considering the formation size and sensor network, establish the definition of graph rigidity, graph topology, and kinematic model of multi-agent system;

[0130] Step 2: Based on the geometric constraints of the distance ratio, construct the distance ratio rigid frame and the distance ratio stiffness matrix;

[0131] Step 3: Based on the target formation configuration, solve the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame;

[0132] Step 4: Design the control input criterion function, minimize the formation error, and explore the controller design and characteristics of the formation system.

[0133] Step 5: For triangular formations in two-dimensional space and tetrahedral formations in three-dimensional space, calculate the undesirable equilibrium points of the formation system and solve for the conditions for global convergence of the system.

[0134] The present invention will be further described below with reference to the embodiments. Example

[0135] To verify the performance of the method proposed in this paper, a system simulation environment was built, and a formation experiment was conducted using three quadcopter UAVs. First, a formation experiment was performed using six agents modeled with single integrators, and the target formation constraints were defined as follows:

[0136]

[0137] Select controller parameters Formation Figure 2 As shown, the formation error is as follows Figure 3 As shown. Its formation structure satisfies:

[0138]

[0139] Next, we perform triangular formation on the plane, defining the target formation constraints as follows:

[0140]

[0141] Select controller parameters The three agents were initially almost collinear, with agent number 3 in... An offset of less than 0.01m on the axis yields the formation and formation error as follows: Figure 4 As shown. Next, perform tetrahedral formation in space, defining the target formation constraints as follows:

[0142]

[0143] Select controller parameters The initial positions of the four agents are almost coplanar, with agent number 4 in... An offset of less than 0.01m on the axis yields the formation and formation error as follows: Figure 5 As shown. (Through) Figure 4 , 5 It can be concluded that for triangular formations in a plane and tetrahedral formations in space, if the initial states of the multi-agent system do not satisfy the collinearity or coplanarity condition, the entire system can achieve global stability based on the controller (12). Finally, three quadcopter UAVs with a wheelbase of 600mm were used for formation testing. Figure 6 As shown, select target formation constraints. Controller parameters The three drones first ascended to a height of 3 meters, then formed a formation, and the result was as follows. Figure 7 As shown. The method of this invention introduces a range ratio constraint and designs a distributed formation control algorithm based on the range ratio information and direction information measured by low-cost sensors, thereby reducing the dependence on global information and high-cost sensors.

[0144] The specific embodiments described herein are merely illustrative of the spirit of the invention. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific embodiments or use similar methods to substitute them, without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

Claims

1. A method for distributed formation control of multi-agent systems based on distance ratio constraints, characterized in that, Includes the following steps: Step 1: Based on the formation size and sensor network, establish the definition of graph rigidity, graph topology, and kinematic model of the multi-agent system; specifically including: Establishing graph topology: define an undirected graph G = (V, E) consisting of n vertices and m edges, where V = {v1, v2,..., vn} represents the vertices of the undirected graph, and E = {e1, e2,..., em} represents the edges of the undirected graph.​ By establishing the definition of graph rigidity, and defining translation, rotation, scaling, or combinations thereof as infinitesimal motions, infinitesimal motions are defined as follows: if the motion of a graph vertex ensures the invariance of the graph, then the motion is called an infinitesimal motion. Establish a kinematic model of a multi-agent system: a set of... A single integrator The multi-agent system built by Wiene: (2) in and They are the first The position and control input of an agent in the global coordinate system Represent velocity; define a vector Representation diagram exist Implementation in space; introduction of matrices ,in Indicates definition, For Kronecker product, express A 3D identity matrix, representing the relative position vectors Construct as a position vector Mapping: (3) in , It is the first Each edge determines the relative position vector formed by the vertex pairs; Step 2: Based on the geometric constraints of the distance ratio, construct the distance ratio rigid frame and the distance ratio stiffness matrix; specifically including: Based on geometric constraints, construct a distance-ratio rigid frame and define the frame. Distance stiffness function for: (4) The following distance stiffness matrix is ​​obtained. The simplest form: (6) in Distance ratio stiffness function Represented as: (7) in The number of distance ratio constraints; remove redundant constraints and define a new distance ratio stiffness function. The new undirected graph ;definition , yes a subset of It is the minimum set of constraints that can realize the minimum and infinitesimal rigid frames; Distance-to-stiffness matrix: Combining the distance-to-stiffness matrix, the following relationship is obtained: (8) in , It is the distance stiffness matrix; Defined as the distance-to-stiffness matrix; Step 3: Based on the target formation configuration, solve the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame; Step 4: Design a formation stability controller and provide the characteristics of the formation system; Step 5: Solve for the conditions for global convergence of the system for triangle formation in two-dimensional space and tetrahedral formation in three-dimensional space; Step 6: Use the formation stabilization controller designed in Step 4 to perform distributed formation of the multi-agent system.

2. The method according to claim 1, characterized in that, In step 3, considering the target formation configuration, the algebraic relationship between the range ratio stiffness matrix and the infinitesimal range ratio rigid frame is solved: For framework If and only if The system is rigid with an infinitesimal distance ratio; based on this conclusion, the target constraints for achieving a formation with the minimum and infinitesimal distance ratio are solved.

3. The method according to claim 1, characterized in that, In step 4, the desired equilibrium point set is... Define it as follows: (11) in, , , , Representing intelligent agents respectively and The relative displacement vector between them and The relative displacement vector between them and The expected relative displacement vector between them and The expected relative displacement vector between them; This leads to the control objective of distributed formation based on distance ratio: The desired formation under this control objective satisfies the conditions of minimum and infinitesimal distance-to-stiffness ratio; therefore, a formation stabilization controller is designed: (12) in: and It controls the gain; , , , , and It is the distance ratio feedback error; and It is a unit direction vector; , , , and It is distance ratio information; , , , , and These respectively indicate the existence of objective constraints. , , , , and .

4. The method according to claim 3, characterized in that, In step 4, the formation system satisfies the centroid. Total angular momentum and overall scaling Invariance: (13) And when the initial state of the multi-agent system satisfies (14) At that time, the formation system will not collide, among which , respectively intelligent agents , Target formation position; This is the initial position of the cluster. as a unit 3D column vector; The target cluster formation position; Let be an arbitrarily small positive integer.

5. The method according to claim 1, characterized in that, In step 5, the condition for global convergence of the system is: For triangular formations in a plane and tetrahedral formations in space, if the initial states of the multi-agent system do not satisfy the collinearity or coplanarity condition, the entire system can achieve global stability based on the formation stability controller.

6. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that... When the processor executes the computer program, it implements the distributed formation control method for a multi-agent system based on distance ratio constraints as described in any one of claims 1-5.

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