A flexible interaction topology design method for affine formation control
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2026-02-24
- Publication Date
- 2026-07-10
AI Technical Summary
Existing affine formation control methods, while ensuring stability, cannot effectively adapt to non-affine deformations, leading to a sharp increase in internal stress and control energy requirements, making it difficult to maneuver flexibly in complex environments.
Using stress energy as a performance indicator, a flexible interactive topology is designed. By optimizing the edge weights through a semidefinite programming model, a sparse interactive topology is generated, which reduces internal stress and control costs, and enables flexible deformation of the formation.
While ensuring global stability, it automatically generates the optimal interaction topology, optimizes communication costs, and enables flexible maneuverability and efficient transformation of the formation to adapt to dynamically changing environments.
Smart Images
Figure CN122363335A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of collaborative control technology, specifically relating to a flexible interactive topology design method for affine formation control. Background Technology
[0002] With the widespread application of multi-agent systems (MAS) in environmental monitoring, target encirclement, and sensor network localization, formation control has become a crucial cooperative operation mode. Traditional formation control methods, such as those based on displacement, distance, or orientation constraints, while maintaining formation configuration, limit flexibility during formation maneuvers due to inherent constant constraints. To improve formation maneuverability, affine formation control has emerged. The core of this method is the use of a stress matrix to describe the target configuration, allowing the formation to reconstruct its shape in arbitrary-dimensional space through affine transformations (such as translation, rotation, scaling, and shearing), thus achieving higher degrees of freedom. The stress matrix not only defines the interaction relationships between agents but also determines the interaction weights; its design and optimization directly affect the overall performance of the entire system.
[0003] Existing methods for constructing stress matrices primarily focus on ensuring the stability and rigidity of the formation. For example, semidefinite programming (SDP) or mixed-integer semidefinite programming (MISDP) is used to balance the closed-loop performance and communication costs of the system. However, these methods typically sacrifice dynamic maneuverability for high stability. In complex environments, due to limitations such as safety distances, formations cannot achieve obstacle avoidance through affine deformations like scaling; they often need to perform non-affine deformations to respond to dynamic changes, such as locally altering their shape to avoid obstacles. In this case, traditional rigid interactive topologies inherently resist these non-affine deformations, leading to a sharp increase in internal stress. This not only requires enormous control energy to overcome but may even affect system performance. Therefore, designing a flexible interactive topology that can adapt to non-affine deformations and reduce internal stress and control costs while ensuring global formation stability is a critical technical problem that urgently needs to be solved in the field of affine formation control. Summary of the Invention
[0004] In view of this, this invention proposes a flexible interactive topology design method for affine formation control to address the multi-agent cooperative control problem. Using stress energy as a performance index, it can automatically generate the optimal sparse interactive topology and achieve optimization of omnidirectional and directional formation flexibility.
[0005] A flexible interactive topology design method for affine formation control includes:
[0006] Step 1: Introduce stress energy as a performance metric, establish a mapping relationship between structural mechanics and virtual formation, and analogize the interaction links between agents to a spring with zero rest length. Measure the work done to overcome internal forces required for formation deformation, i.e., the control cost, specifically including: Consider by A formation composed of intelligent agents Operating in 3D space, the dynamic model of each agent is a single integrator, i.e.:
[0007] in, Represents intelligent agents Location coordinates, To control the input, an affine control law based on the stress matrix is adopted:
[0008] in, For intelligent agents , The edge weights between them; the compact form of the closed-loop system is ,in, The position vector formed by the position coordinates of all agents constitutes the formation configuration; express An identity matrix of 3D; weight of edge The stress matrix formed; Represents intelligent agents The neighbor set; Apply external control input to the leader in the intelligent agent. ;set up ,in It is a given formation nominal configuration. The constant control gain matrix of the leader, the complete system dynamics are:
[0009] The internal controller for formation control originates from a strain potential function, defined as the sum of the energies of all virtual springs in the system, i.e.:
[0010] in, Represents an edge set; For virtual spring nodes and The spring constant between Represents virtual spring nodes and The current length between, Represents virtual spring nodes and The static length between; In affine formation control, the virtual connections between agents are treated as springs of zero natural length, that is, ... When the formation changes configuration Along a certain deformation direction A small perturbation occurs, forming a new configuration. At that time, the total elastic potential energy of the system, i.e., the stress energy, is:
[0011] In the above formula, and They represent the configurations respectively. and The total elastic potential energy under the condition; , Representing virtual spring nodes respectively and Deformation vector; scalar This represents the magnitude of the disturbance and is used to scale the deformation vector. Total elastic potential energy about The second derivative in The value at that location is:
[0012] use As a performance indicator for measuring flexibility; Step 2: Construct a flexible topology design optimization framework. Addressing the flexibility requirements of directional and omnidirectional formations, and using formation stability and equilibrium conditions as hard constraints, minimize the stress energy and the factor characterizing communication costs. Norm penalty term, establish a semidefinite programming model, specifically: The basic form of the objective function in the optimization model is:
[0013] in, It is weighted. Norm penalty terms are used to promote topological sparsity, thereby reducing the number of communication links. It is a regularization parameter that balances flexibility and communication cost; It is a diagonal matrix, whose diagonal elements are... Indicates the first The communication cost coefficient of a link. Indicates the number of links, set to the level of the agent. , Distance Proportional, of which , Representing intelligent agents respectively , The given nominal configuration; To maintain formation shape, internal interaction forces must be present in the nominal configuration. In equilibrium; given an affine tension nominal configuration The configuration matrix is defined as The equilibrium condition is expressed as: ; According to rigidity theory, to ensure the affine formation is stable, the frame must be universally rigid; this is equivalent to the stress matrix needing to satisfy... and To transform non-convex rank constraints into convex constraints, an augmented configuration matrix is used. The singular value decomposition, i.e. ,in, , Let them represent matrices composed of left and right singular vectors, respectively; let It is by A matrix whose null space is a set of orthonormal bases, i.e., whose column vectors are orthogonal matrices. The end In this case, the stability constraint is equivalently transformed into a linear matrix inequality:
[0014] in, Represents the identity matrix; This is to provide a stiffness margin to ensure robust stability; The following constraints are applied to the equilibrium stress:
[0015] in, express A 1-dimensional row vector; This represents the number of edges in a fully connected graph. This represents the given total stress budget, which is a positive constant. For directional flexible topology design, considering that the formation needs to deform along a preset direction, the optimization model is established as follows:
[0016] in, For the unit deformation vector related to the task; the goal of this optimization model is to design a vector that... The most flexible topology in a particular direction, that is, a directional flexible topology designed for a specific task, whose link and weight distribution is designed to minimize deformation resistance in a specific direction; For omnidirectional flexible topology design, when the deformation task is unknown, or when it is desired that the formation has universal flexibility in all directions, the stress energy index is configured as the trace of the stress matrix, and the optimization model is established as follows:
[0017] Step 3: Model Solving and Simulation Verification Solve the optimization model constructed above for directional and omnidirectional flexible topology design to obtain the optimal edge weight vector and realize the flexible interactive topology design of affine formation control.
[0018] Preferably, when solving the optimization model constructed above for directional and omnidirectional flexible topology design, the global optimal solution can be obtained in polynomial time using the CVX toolkit in conjunction with the MOSEK solver.
[0019] Preferably, the diagonal elements of a diagonal matrix The calculation is as follows: The predefined agent spacing range Linear mapping to a preset cost weight range The specific calculation formula is as follows: .
[0020] The present invention has the following beneficial effects: 1. This invention is an intelligent topology construction method. Given a nominal configuration and desired flexibility direction, it can automatically generate a globally optimal interactive topology, while optimizing communication costs and overall system performance, thereby realizing affine formation maneuver control of multi-agent systems.
[0021] 2. This invention changes the existing affine formation's topology design mode that simply pursues rigidity. It models the flexible topology design problem as a convex optimization model that can be solved efficiently, enabling complex maneuvers such as contraction and stretching to be completed with less deformation cost. This allows it to flexibly adapt to dynamically changing environments, such as traversing narrow passages and avoiding dynamic obstacles. Attached Figure Description
[0022] Figure 1 This is the optimal topology for directional flexibility.
[0023] Figure 2 This is the optimal topology diagram for omnidirectional flexibility.
[0024] Figure 3 A simulated trajectory diagram for a formation performing a mission to traverse a narrow passage.
[0025] Figure 4 The diagram shows the relationship between formation tracking error and velocity.
[0026] Figure 5 This is a comparison diagram of the internal stresses of flexible and rigid topologies. Detailed Implementation
[0027] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.
[0028] This invention proposes a flexible interactive topology design method for affine formation control, which mainly includes the following steps: Step 1: Introduce stress energy as a performance metric, establish a mapping relationship between structural mechanics and virtual formation, and compare the interaction links between intelligent agents to a spring with zero rest length to measure the work done to overcome internal forces required for formation deformation, i.e., control cost.
[0029] Step 2: Construct a flexible topology design optimization framework. Addressing the flexibility requirements of directional and omnidirectional formations, and using formation stability and equilibrium conditions as hard constraints, minimize the stress energy and the factor characterizing communication costs. Norm penalty terms are used to establish a semidefinite programming (SDP) model.
[0030] Step 3: Solve the model and generate the topology. Use a standard convex optimization solver, such as the polynomial time point method, to efficiently solve the SDP model established in Step 2 and obtain the optimal edge weight vector. This vector defines the sparse interaction topology and weights between agents.
[0031] The detailed implementation method is as follows: Step 1: Establish stress energy performance indicators This invention is considered to be by A formation composed of intelligent agents Operating in 3D space, the dynamic model of each agent is a single integrator, i.e.
[0032] in Represents intelligent agents Location coordinates, To control the input, an affine control law based on the stress matrix is adopted:
[0033] in For intelligent agents , The edge weights between them. The compact form of the closed-loop system is: ,in, The position vector formed by the position coordinates of all agents constitutes the formation configuration; express An identity matrix of 3D; weight of edge The stress matrix formed. Represents intelligent agents The neighbor set.
[0034] To achieve formation maneuvers to the desired geometry, external control input needs to be applied to the leader of the agent. .set up ,in It is a given formation nominal configuration. This is the constant control gain matrix of the leader. Therefore, the complete system dynamics are:
[0035] The overall goal is to construct the optimal stress matrix. This invention innovatively transforms the problem of quantifying the control cost of formation deformation into a structural mechanics problem. Traditional distance-based formation control typically uses an internal controller derived from a strain potential function, defined as the sum of the energies of all virtual springs in the system.
[0036] in, Represents an edge set; For virtual spring nodes and The spring constant between Represents virtual spring nodes and The current length between, Represents virtual spring nodes and The static length between them. In this model, the controller focuses on the distance between the agent and the target. Driven to nonzero natural length .
[0037] In affine formation control, this invention treats the virtual connections between intelligent agents as zero-natural-length springs, that is, it makes... Therefore, the above strain energy function This is transformed into a quadratic form that depends only on the current configuration and edge weights, called stress energy. When the formation changes from configuration... Along a certain deformation direction A small perturbation occurs, forming a new configuration. At that time, the total elastic potential energy of the system, i.e., the stress energy, is:
[0038] In the above formula, and They represent the configurations respectively. and The total elastic potential energy under the condition; , Representing virtual spring nodes respectively and Deformation vector; scalar This represents the magnitude of the disturbance and is used to scale the deformation vector.
[0039] Furthermore, total elastic potential energy about The second derivative in The value at that location is:
[0040] This quadratic form quantifies the formation's resistance direction. The greater the stiffness of the deformation, the greater the control cost required to force the deformation. Therefore, this invention employs... As a performance indicator for measuring flexibility.
[0041] Step 2: Construct a flexible topology design optimization model To obtain the optimal flexible topology, this invention constructs a unified convex optimization framework aimed at solving for the edge weight vector that defines the topology. The goal of this optimization problem is to minimize a composite function consisting of stress energy and communication cost.
[0042] The basic form of the objective function in the optimization model is:
[0043] in, It is the stress energy index defined in step one. It is weighted. Norm penalty terms are used to promote topological sparsity, thereby reducing the number of communication links. It is a regularization parameter that balances flexibility performance with communication costs. It is a diagonal matrix, whose diagonal elements are... Indicates the first The communication cost coefficient of a link. Indicates the number of links, usually set to the number of links between the agent and the network. , Distance Proportional, of which , Representing intelligent agents respectively , The given nominal configuration. This proportional relationship is achieved through a mapping function, which maps the predefined agent spacing range. Linear mapping to a preset cost weight range The specific calculation formula is as follows:
[0044] To ensure the edge weight vector obtained is accurate To define a physically valid stable formation, the vector must satisfy three fundamental constraints: balance, stability, and nontriviality.
[0045] First, to maintain formation shape, internal interaction forces must be within the nominal configuration. The state is in equilibrium. Given an affine tension... nominal configuration The configuration matrix is defined as The equilibrium condition can be expressed as:
[0046] Secondly, according to rigidity theory, to ensure the affine formation's stability, the frame must be universally rigid. This is equivalent to the stress matrix needing to satisfy... and To transform non-convex rank constraints into convex constraints, we utilize the augmented configuration matrix. The singular value decomposition, i.e. ,in, , Let represent matrices composed of the left and right singular vectors, respectively. It is by A matrix whose null space is a set of orthonormal bases, i.e., whose column vectors are orthogonal matrices. The end In this case, the stability constraint is equivalently transformed into a linear matrix inequality (LMI):
[0047] in, Represents the identity matrix; This is a stiffness margin to ensure robust stability.
[0048] Furthermore, to establish a consistent scale for stress values and avoid trivial solutions, the following constraints are imposed on the equilibrium stress:
[0049] in, express A 1-dimensional row vector; This represents the number of edges in a fully connected graph. Let represent the given total stress budget, which is a positive constant. This constraint transforms the problem from finding the absolute magnitude of the stress to finding the stress within a given total stress budget. Then, determine its optimal distribution scheme.
[0050] Within this unified framework, by configuring stress energy It can be designed to meet different flexibility requirements.
[0051] For directional flexible topology design, considering that the formation needs to deform along a preset direction, such as compression to pass through narrow passages, the optimization model is established as follows:
[0052] in, Let be the unit deformation vector relevant to the task. The goal of this optimization model is to design a vector that... The most flexible topology in a particular direction is a directional flexible topology designed for a specific task, in which the link and weight distribution is designed to minimize deformation resistance in a particular direction.
[0053] For omnidirectional flexible topology design, when the deformation task is unknown, or when it is desired that the formation has universal flexibility in all directions, the stress energy index is configured as the trace of the stress matrix, and an optimization model is established as follows:
[0054] Due to the trace function yes Minimizing the trace function, which is the sum of all eigenvalues, is equivalent to reducing the average stiffness across all deformation modes. This topology is relatively more uniform, accommodating deformation demands from any direction.
[0055] Step 3: Model Solving and Simulation Verification Since the objective function in the above optimization models for directional and omnidirectional flexible topology design is linear and the constraints are linear or positive semidefinite, they are all standard positive semidefinite programming (SDP) problems. The global optimal solution can be obtained efficiently in polynomial time using toolkits such as CVX and solvers such as MOSEK.
[0056] Example: This embodiment presents a simulation experiment of the proposed affine formation flexible topology construction and control method. Consider a formation consisting of 5 agents in a 2D plane, with the following nominal configuration:
[0057] For directional flexible topology design, a non-affine deformation task needs to be performed, where an agent 3 moves downward by 1 unit. The sparsity weights in the directional and omnidirectional topology optimization models are set as follows: and This results in a topological graph containing 9 edges. Let the stiffness margin be... Total stress Optimal topologies for directional and omnidirectional flexibility, such as... Figure 1 and Figure 2 As shown.
[0058] Furthermore, considering a simulation scenario of a multi-agent system performing a cooperative emergency obstacle avoidance task, the formation needs to traverse a narrow passage. During the task execution phase, a standard LQR controller is used for trajectory tracking, guiding the formation to move forward at a constant speed and dynamically adjusting its shape to pass through the narrow passage. After the external control input is removed, the formation automatically returns to a nearby stable configuration. The affine formation system's trajectory, tracking error, and velocity are shown below. Figure 3 and Figure 4 As shown, by employing the directional flexible topology of the present invention, the formation can smoothly complete the transformation from the initial configuration to the target configuration, and finally stabilize within the affine set of the nominal configuration.
[0059] Furthermore, the directional flexible topology designed in this invention and a method aimed at maximizing stiffness, i.e., maximizing the stress matrix, are utilized. The traditional rigid topology with the smallest non-zero eigenvalue is compared. The comparison diagram shows the internal stresses that the controller needs to overcome throughout the deformation process. Figure 5 As shown, the method of the present invention can reduce the total internal stress by 49.1% and the peak internal stress by 46.1% while maintaining almost the same tracking accuracy, which verifies the significant advantages of the present invention in reducing control costs and improving formation maneuverability.
[0060] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A flexible interactive topology design method for affine formation control, characterized in that, include: Step 1: Introduce stress energy as a performance metric, establish a mapping relationship between structural mechanics and virtual formation, and analogize the interaction links between agents to a spring with zero rest length. Measure the work done to overcome internal forces required for formation deformation, i.e., the control cost, specifically including: Consider by A formation composed of intelligent agents Operating in 3D space, the dynamic model of each agent is a single integrator, i.e.: in, Represents intelligent agents Location coordinates, To control the input, an affine control law based on the stress matrix is adopted: in, For intelligent agents , The edge weights between them; the compact form of the closed-loop system is ,in, The position vector formed by the position coordinates of all agents constitutes the formation configuration; express An identity matrix of dimensionality; weight of edge The stress matrix formed; Represents intelligent agents The neighbor set; Apply external control input to the leader in the intelligent agent. ;set up ,in It is a given formation nominal configuration. The constant control gain matrix of the leader, the complete system dynamics are: The internal controller for formation control originates from a strain potential function, defined as the sum of the energies of all virtual springs in the system, i.e.: in, Represents an edge set; For virtual spring nodes and The spring constant between Represents virtual spring nodes and The current length between, Represents virtual spring nodes and The static length between; In affine formation control, the virtual connections between agents are treated as springs of zero natural length, that is, let When the formation changes configuration Along a certain deformation direction A small perturbation occurs, forming a new configuration. At that time, the total elastic potential energy of the system, i.e., the stress energy, is: In the above formula, and They represent the configurations respectively. and The total elastic potential energy under the condition; , These represent virtual spring nodes. and Deformation vector; scalar This represents the magnitude of the disturbance and is used to scale the deformation vector. Total elastic potential energy about The second derivative in The value at that location is: use As a performance indicator for measuring flexibility; Step 2: Construct a flexible topology design optimization framework. Addressing the flexibility requirements of directional and omnidirectional formations, and using formation stability and equilibrium conditions as hard constraints, minimize the stress energy and the factor characterizing communication costs. Norm penalty term, establish a semidefinite programming model, specifically: The basic form of the objective function in the optimization model is: in, It is weighted. Norm penalty terms are used to promote topological sparsity, thereby reducing the number of communication links. It is a regularization parameter that balances flexibility and communication costs; It is a diagonal matrix, and its diagonal elements are... Indicates the first The communication cost coefficient of a link. Indicates the number of links, set to the level of the agent. , Distance Proportional, of which , Representing intelligent agents respectively , The given nominal configuration; To maintain formation shape, internal interaction forces must be present in the nominal configuration. In equilibrium; given an affine tension nominal configuration The configuration matrix is defined as The equilibrium condition is expressed as: ； According to rigidity theory, to ensure the affine formation is stable, the frame must be universally rigid; this is equivalent to the stress matrix needing to satisfy... and To transform non-convex rank constraints into convex constraints, an augmented configuration matrix is used. The singular value decomposition, i.e. ,in, , Let them represent matrices composed of left and right singular vectors, respectively; let It is by A matrix whose null space is a set of orthonormal bases, i.e., whose column vectors are orthogonal matrices. The end In this case, the stability constraint is equivalently transformed into a linear matrix inequality: in, Represents the identity matrix; This is to provide a stiffness margin to ensure robust stability; The following constraints are applied to the equilibrium stress: in, express A 1-dimensional row vector; This represents the number of edges in a fully connected graph. This represents the given total stress budget, which is a positive constant. For directional flexible topology design, considering that the formation needs to deform along a preset direction, the optimization model is established as follows: in, For the unit deformation vector related to the task; the goal of this optimization model is to design a vector that... The most flexible topology in a particular direction, that is, a directional flexible topology designed for a specific task, whose link and weight distribution is designed to minimize deformation resistance in a specific direction; For omnidirectional flexible topology design, when the deformation task is unknown, or when it is desired that the formation has universal flexibility in all directions, the stress energy index is configured as the trace of the stress matrix, and the optimization model is established as follows: Step 3: Model Solving and Simulation Verification Solve the optimization model constructed above for directional and omnidirectional flexible topology design to obtain the optimal edge weight vector and realize the flexible interactive topology design of affine formation control.
2. The flexible interactive topology design method for affine formation control as described in claim 1, characterized in that, When solving the optimization model constructed for the above-mentioned directional and omnidirectional flexible topology design, the global optimal solution is obtained in polynomial time using the CVX toolkit in conjunction with the MOSEK solver.
3. The flexible interactive topology design method for affine formation control as described in claim 1, characterized in that, diagonal elements of a diagonal matrix The calculation is as follows: The predefined agent spacing range Linear mapping to a preset cost weight range The specific calculation formula is as follows: 。