A preset time formation control method based on distance rigidity matrix under performance constraint

Abstract

This invention discloses a pre-defined time-based multi-agent formation control method based on a rigid matrix under performance constraints, comprising: 1) establishing the communication topology of the multi-agent system based on rigid graph theory, and defining the minimum and infinitesimal rigid frame corresponding to the desired formation; 2) applying upper and lower bound constraints to the squared distance error, transforming collision avoidance, connectivity maintenance, and infinitesimal rigidity preservation into a unified performance boundary; 3) introducing a time-varying scaling function to construct a time scale transformation; 4) designing a distributed pre-defined time-based performance controller; and 5) conducting experimental verification on a motion capture system and a mobile robot platform. By introducing a time-varying scaling function and a double power feedback term, a user-defined preset convergence time is achieved, avoiding the shortcomings of existing methods where the convergence time depends on initial conditions or cannot strictly guarantee transient performance, while ensuring collision avoidance and connectivity during the formation process.

Description

Technical Field

[0001] This invention relates to the field of multi-agent cooperative control technology, specifically to a preset-time formation control method based on a distance rigid matrix under performance constraints. Background Technology

[0002] In recent years, multi-agent formation control has become a research hotspot due to its widespread application in autonomous robots, vehicle networks, and UAV formations. Achieving and maintaining rigid formations—a geometric configuration where the distances between all agents remain constant—is crucial for tasks requiring precise relative positioning, such as cooperative payload transport and synthetic aperture imaging. This need has driven in-depth research into rigidity theory and its applications in distributed control. Early foundational work established the application of graph rigidity in distributed formation stability, proving that if the desired formation is infinitesimally rigid, a formation can be globally stabilized using only local relative measurement information. Subsequent research extended these ideas to more complex scenarios, proposing a leaderless, distributed, coordinate-free method that achieved global stability of rigid formations with a single integrator and a unicycle agent. Furthermore, it combined rigidity-based control with adaptive techniques to compensate for unknown system parameters, further extending rigidity theory to handle formation shape control in three-dimensional space, overcoming the challenge of configuration fuzziness in low-dimensional controllers.

[0003] However, while current research on rigid formation control is extensive, it primarily focuses on linear systems and asymptotic convergence, leaving the crucial aspect of state constraints relatively underexplored. Researchers recognize that unconstrained controllers can lead to collisions or connection loss in practice, prompting some to pioneer research on performance constraints. For example, some studies have introduced pre-defined performance functions to impose transient and steady-state boundaries on distance errors, ensuring collision avoidance and connection maintenance. Further improvements have been made to this method, achieving predefined performance in both centralized and distributed settings. Similarly, some studies employ barrier Lyapunov functions to constrain system states, ensuring safety specifications are not violated. However, while these methods provide crucial safety guarantees, their convergence rates are inherently asymptotic, and adjusting the performance function only provides a single and limited degree of freedom for controlling transient performance, which is far from sufficient for applications requiring precise time control.

[0004] To overcome the limitations of asymptotic convergence and achieve more predictable temporal performance, the research community has turned its attention to finite-time and fixed-time control schemes. Some studies have incorporated finite-time convergence into rigid-based control, achieving formation stability within a finite timeframe dependent on initial conditions. To further decouple convergence time from initial conditions, a pre-defined time control paradigm has been proposed, allowing users to arbitrarily pre-specify the convergence time. Subsequent research has successfully applied these time-critical strategies to multi-agent systems, demonstrating excellent convergence performance. However, a significant drawback remains: these finite-time and pre-defined time controllers primarily prioritize convergence speed, often neglecting the integration of strict safety constraints during transient processes. These methods may lead to unsafe trajectories or connection interruptions before convergence. Therefore, to achieve safe and time-critical formation maneuvers, this invention aims to bridge this gap by researching a control scheme that embeds performance constraints into a pre-defined time convergence framework, thereby ensuring agent safety while achieving strict time objectives. Summary of the Invention

[0005] The present invention aims to provide a pre-defined time-based formation control method based on a distance rigid matrix under performance constraints. Its key features include: defining the squared distance error between multiple agents using a rigid matrix, and introducing a pre-defined performance function to apply time-varying constraints to the squared distance error, ensuring collision avoidance and connectivity maintenance between adjacent agents during formation; and achieving convergence of the formation error to a given bound within a user-defined pre-defined time by introducing a time-varying scaling function and a double-power feedback term containing positive integer and fractional power terms. Specifically, the method includes the following steps:

[0006] Step 1: Based on rigid graph theory, establish the communication topology of the multi-agent system, define the minimum and infinitesimal rigid frame corresponding to the desired formation, and calculate the expected distance and squared distance error of each edge;

[0007] Step 2: Design a preset performance function for each edge, apply upper and lower bound constraints to the squared distance error, and transform constraints such as collision avoidance, connectivity maintenance, and infinitesimal rigidity preservation into performance boundaries of the error variable;

[0008] Step 3: Convert the constrained error variables into unconstrained variables using a general barrier function, and introduce a time-varying scaling function to construct a time scale transformation to achieve the preset time convergence;

[0009] Step 4: Design a distributed preset time preset performance controller. The controller includes a time-varying scaling function, a double power feedback term including positive integer power terms and fractional power terms, and a desired formation speed feedforward term to ensure that all distance errors converge to the desired value within a preset time, while ensuring that all signals in the closed-loop system are bounded.

[0010] Step 5: Conduct experimental verification on the motion capture system and mobile robot platform to verify the effectiveness of the controller in achieving formation maneuvering within a preset time and meeting performance constraints.

[0011] Step 1 includes the following steps:

[0012] Step 1.1: Use an undirected graph To represent the perceptual topology between intelligent agents, where It is by A set of vertices consisting of intelligent agents. It is the set of edges describing the perceptual relationships between agents. The desired formation consists of a minimum and infinitesimal rigid frame. Unique definition, in which For an infinitesimal rigid graph, For the desired position, express The overall position column vector space of an intelligent agent in a two-dimensional plane. Let be the desired position vector of the first agent. For the first Each agent has a desired location.

[0013] Step 1.2: For each edge , Indicates the current intelligent agent, It represents its neighboring intelligent agents, and the expected distance is defined as follows: The actual distance is ,in and For the first The expected position vector and the actual position vector of each agent. and For the first The distance error between the target position vector and the actual position vector of each agent is... To facilitate controller design, the squared distance error is further defined. .when hour, The two are equivalent.

[0014] Step 2 includes the following steps:

[0015] Step 2.1: In this invention, the first... An intelligent agent has a radius known to be... The circular safety range and radius are known to be The circular sensing range, in which To avoid neighboring agents and A collision requires the actual distance. To ensure smooth information exchange between adjacent agents during their movement to the desired formation, the actual distance is required to be... Furthermore, according to the infinitesimal rigidity preservation lemma, as long as the distance error between the actual formation and the desired formation is small enough, the actual formation can maintain infinitesimal rigidity.

[0016] Step 2.2: Unify the handling of constraints and design a preset performance function for each edge. ,in , The decay rate is then used. The following time-varying constraint is applied to the squared distance error:

[0017]

[0018] in The distance is determined by the collision avoidance distance, the sensing distance, and the desired distance. The physical meaning of this constraint is that as long as the squared distance error remains within its upper and lower bounds, the actual distance must be greater than the collision avoidance distance (ensuring no collision) and less than the sensing distance (ensuring communication continuity). Simultaneously, the deviation between the actual formation and the desired formation must be sufficiently small to ensure that the infinitesimal rigidity is not violated. In this way, multiple safety constraints that previously required separate handling are unified into a performance constraint on the squared distance error, greatly simplifying the control design.

[0019] Step 3 includes the following steps:

[0020] Step 3.1: To facilitate controller design, this step first defines the conversion error. As can be seen from the constraints in step two, Subject to upper and lower bounds of constants: Then, a general barrier function is introduced to address the issue of constant upper and lower bounds. Mapping to unconstrained variables :

[0021]

[0022] The characteristic of this function is: when Approaching the lower bound or upper boundary hour, It tends towards negative infinity or positive infinity, thus naturally ensuring that constraints are not violated in control design.

[0023] Step 3.2: Introduce the time-varying scaling function ,in The function sets a user-defined upper bound for the convergence time. The time tends towards infinity. This can be achieved by introducing this function and performing a time-scale transformation. It can make a requirement that needs to be set within a limited time. The control problem of internal convergence is transformed into an asymptotically convergent problem on an infinite time scale.

[0024] Step 4 includes the following steps:

[0025] Step 4.1: This step involves designing a distributed preset time preset performance controller. Considerations include... The dynamic model of an agent is as follows ,in To control the input, This is a bounded external disturbance. The specific form of the controller is:

[0026]

[0027] in, For the first The neighbor set of an agent, It is a relative position vector. This is the derivative term of the general barrier function; , Similarly, where is the fractional exponent. satisfy , It is a positive integer exponent and satisfies ; To control the gain; For the Kronecker function (when (1 if it is 1, otherwise 0). To achieve the desired formation speed, the controller comprises three core parts. The first part is the time-varying scaling function. The product of the double-power feedback term and the first part is the weighting term associated with the rigid matrix. The second part is the weighting term associated with the rigid matrix, which maps the error information of each edge to the control input of each agent through the rigid matrix. The third part is the feedforward term for the desired formation speed, which is applied only to the leader agent (agent 1) and transmitted to the entire formation through the directed communication topology, ensuring that all agents can synchronously track the desired trajectory. The double-power feedback term consists of a fractional power term and a positive integer power term. The fractional power term provides a large gain when the error is small, accelerating the convergence of small errors; the positive integer power term provides a fast response when the error is large, accelerating the convergence of large errors. The two work together to ensure that the system remains efficient throughout the convergence process.

[0028] Step 5 includes the following steps:

[0029] Step 5.1: Physical verification on the motion capture system and mobile robot platform. The experimental platform included the NOKOV motion capture system (equipped with 12 Mars 2H infrared cameras, sampling frequency of 380 Hz, positioning accuracy of ±0.15 mm), a central processing unit (NUC11TNK-i7, running Ubuntu 18.04 and the robot operating system), and five TurtleBot3 mobile robots. The robots received control commands via a wireless network at a control frequency of 10 Hz.

[0030] Step 5.2: At the start of the experiment, the five robots are in random initial positions. After the controller is activated, the robots begin to converge toward the desired formation. The robot pose information is acquired in real time through a motion capture system. The central processing unit calculates control commands according to the control algorithm and broadcasts them to each robot wirelessly. The distance error of each edge is recorded as a function of time, and it is observed whether any collisions or communication interruptions occur between the robots.

[0031] This invention provides a preset-time formation control method based on a distance rigidity matrix under performance constraints. Compared with the prior art, this invention has the following advantages:

[0032] 1. By introducing a time-varying scaling function and a double-power feedback term containing both positive integer and fractional exponent terms, precise convergence within a user-defined preset time is achieved for the first time in rigid formation control. The upper bound of the convergence time can be arbitrarily specified and is independent of the initial conditions, overcoming the shortcomings of traditional asymptotic convergence and finite-time convergence being sensitive to initial conditions, and providing predictable convergence performance for time-critical tasks.

[0033] 2. By applying time-varying constraints to the squared distance error through a preset performance function, it is ensured that the overshoot, convergence rate, and steady-state accuracy of the distance error are all within the preset range. Satisfying this constraint simultaneously guarantees collision avoidance, connectivity maintenance, and infinitesimal rigidity preservation, avoiding the cumbersome process of designing collision avoidance potential functions, connectivity preservation mechanisms, and rigidity preservation conditions separately in traditional methods.

[0034] 3. The controller incorporates both fractional exponent terms and positive integer exponent terms, accelerating convergence for small and large errors respectively. Users can independently adjust the parameters of the fractional and positive integer exponents, thus flexibly configuring the convergence rate and steady-state accuracy, offering far greater flexibility than existing fixed-gain or single-power controllers.

[0035] 4. The present invention was experimentally verified using a micro mobile robot, which confirmed the applicability of the distributed control protocol. Attached Figure Description

[0036] Figure 1 This is a topology diagram of a five-robot formation.

[0037] Figure 2 The following are robot trajectory diagrams taken at different time snapshots: (a) t=0s, (b) t=50s, (c) t=95s, (d) t=200s.

[0038] Figure 3 This is a graph showing how distance error changes over time. Detailed Implementation

[0039] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0040] This embodiment uses five TurtleBot3 Burger mobile robots for experimental verification. The desired formation is a regular pentagon, with the following topology: Figure 1 As shown, the edge set is The desired distance is calculated based on the formation geometry. rice, Meters. Collision avoidance distance set to [number]. Meters, sensing distance set to Meters. The parameters of the preset performance function are set as follows: , , The controller parameters are set as follows: , , , The preset time is set to... Seconds. Expected formation speed is set to... meters per second.

[0041] The experimental platform includes a NOKOV motion capture system (equipped with 12 Mars 2H infrared cameras, a sampling frequency of 380 Hz, and a positioning accuracy of ±0.15 mm), a central processing unit (NUC11TNK-i7, running Ubuntu 18.04 and the robot operating system), and five TurtleBot3 robots. The robots receive control commands via a wireless network at a control frequency of 10 Hz.

[0042] At the start of the experiment, the five robots were in random initial positions. After the controller was activated, the robots began to converge toward the desired formation. Figure 2 As shown, in The robots were dispersed within seconds; In seconds, the robots had basically formed a regular pentagonal formation; Seconds and At the specified time, the formation remains stable and moves in the desired direction. For example... Figure 3 As shown, the distance error of each edge converges to near zero within a preset time of 10 seconds, and remains within the preset performance boundary (dashed line in the figure), indicating that no collision or connection loss has occurred.

Claims

1. A pre-set time formation control method based on a distance rigidity matrix under performance constraints, characterized in that: The squared distance error between multiple agents is defined by a rigid matrix, and a preset performance function is introduced to apply time-varying constraints to the squared distance error, so as to ensure collision avoidance and connectivity maintenance between adjacent agents during the formation process. By introducing a time-varying scaling function and a double-power feedback term containing both positive integer and fractional exponent terms, the formation error is made to converge to a given bound within a user-defined preset time. Specifically, the steps include: Step 1: Based on rigid graph theory, establish the communication topology of the multi-agent system, define the minimum and infinitesimal rigid frame corresponding to the desired formation, and calculate the expected distance and squared distance error of each edge; Step 2: Design a preset performance function for each edge, apply upper and lower bound constraints to the squared distance error, and transform the collision avoidance, connectivity maintenance and infinitesimal rigidity preservation constraints into the performance boundary of the error variable; Step 3: Convert the constrained error variables into unconstrained variables using a general barrier function, and introduce a time-varying scaling function to construct a time scale transformation to achieve the preset time convergence; Step 4: Design a distributed preset time preset performance controller. The distributed preset time preset performance controller includes a time-varying scaling function, a double power feedback term including positive integer power terms and fractional power terms, and a desired formation speed feedforward term to ensure that all distance errors converge to the desired value within a preset time, while ensuring that all signals of the closed-loop system are bounded. Step 5: Conduct experimental verification on the motion capture system and mobile robot platform to verify the effectiveness of the distributed preset time preset performance controller in achieving formation maneuvering and meeting performance constraints within a preset time.

2. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: In step 1, an undirected graph is used to represent the perceptual topology between agents. The vertex set consists of multiple agents, and the edge set describes the perceptual relationships between agents. The expected formation is uniquely defined by a minimum and infinitesimal rigid frame, including the expected infinitesimal rigid graph and the expected position. For each edge, the expected distance and the actual distance are defined, and the squared distance error is further defined.

3. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: In step 2, each agent has a circular safety range and a circular perception range with a known radius, and the safety radius is smaller than the perception radius; to avoid collisions, the actual distance is required to be greater than the safety distance; to ensure communication connectivity, the actual distance is required to be less than the perception distance; a preset performance function is designed for each edge, and a time-varying constraint is applied to the squared distance error accordingly. The upper and lower bounds of the time-varying constraint are jointly determined by the collision avoidance distance, the perception distance, and the expected distance.

4. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: In step 3, the transformation error is first defined, which is constrained by constant upper and lower bounds. Then, a general barrier function is introduced to map the transformation error constrained by constant upper and lower bounds into an unconstrained variable. This general barrier function tends to infinity as the transformation error approaches the boundary, thus naturally ensuring that the constraints are not violated in the control design. Next, a time-varying scaling function is introduced, which tends to infinity at the upper bound of the user-preset convergence time. Through time scale transformation, the convergence problem within a preset finite time is transformed into an asymptotic convergence problem on an infinite time scale.

5. The preset-time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: In step 4, the dynamic model of multiple agents is considered to include control inputs and bounded external disturbances. The controller specifically includes the product of a time-varying scaling function and a double-power feedback term, a weighting term associated with a rigid matrix, and a feedforward term for the desired formation speed. The double-power feedback term includes a fractional exponent term and a positive integer exponent term. The fractional exponent term provides a larger gain when the error is small, while the positive integer exponent term provides a fast response when the error is large. The weighting term associated with the rigid matrix is ​​used to map the error information of each edge to the control input of each agent. The feedforward term for the desired formation speed is applied only to the leader agent and is transmitted to the entire formation through a directed communication topology.

6. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: In step 5, the experimental platform includes a motion capture system, a central processing unit, and multiple intelligent agents, each being a mobile robot. The mobile robots receive control commands via a wireless network. At the start of the experiment, the mobile robots are in random initial positions. After the controller is activated, the mobile robots converge toward the desired formation. The motion capture system acquires the robot's pose information in real time. The central processing unit calculates control commands based on the control algorithm and broadcasts them to each mobile robot via wireless communication. The distance error of each edge is recorded as a function of time, and it is observed whether a collision or communication interruption occurs.

7. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: Step 1 includes the following steps: Step 1.1: Use an undirected graph To represent the perceptual topology between intelligent agents, where It is by A set of vertices consisting of 10 agents. It is the set of edges describing the perceptual relationships between agents; the expected formation consists of a minimum and infinitesimal rigid frame. Unique definition, in which For an infinitesimal rigid graph, For the desired position, express The overall position column vector space of an intelligent agent in a two-dimensional plane. Let be the desired position vector of the first agent. For the first The desired location of each agent; Step 1.2: For each edge , Indicates the current intelligent agent, Represents its neighboring intelligent agents, and the expected distance is defined as follows: The actual distance is ,in and For the first The expected position vector and the actual position vector of each agent. and For the first The distance error between the target position vector and the actual position vector of each agent is... Define the squared distance error. ;when hour, The two are equivalent.

8. The method according to claim 1, characterized in that: Step 2 includes the following steps: Step 2.1: The An intelligent agent has a radius known to be... The circular safety range and radius are known to be The circular sensing range, in which To avoid neighboring agents and A collision occurred, and the actual distance was... To ensure information exchange between adjacent agents during their movement to the desired formation, the actual distance... ; Step 2.2: Unify the handling of constraints and design a preset performance function for each edge. ,in , The decay rate is given; then the following time-varying constraint is applied to the squared distance error: ； in It is determined by the collision avoidance distance, the perceived distance, and the expected distance.

9. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: Step 3 includes the following steps: Step 3.1: Define the conversion error As can be seen from the constraints in step 2, Subject to upper and lower bounds of constants: Then, a general barrier function is introduced to address the issue of constant upper and lower bounds. Mapping to unconstrained variables : ； The property of the general barrier function is: when Approaching the lower bound or upper boundary hour, It tends toward negative infinity or positive infinity; Step 3.2: Introduce the time-varying scaling function ,in The function has a user-defined upper bound on the convergence time; The time tends to infinity; by introducing this function and performing a time scale transformation... To make a task that needs to be completed within a preset time limit The control problem of internal convergence is transformed into an asymptotically convergent problem on an infinite time scale.

10. The pre-set time formation control method based on a distance rigidity matrix under performance constraints according to claim 1, characterized in that: Step 4 includes the following steps: Step 4.1: Design a distributed preset time preset performance controller; consider The dynamic model of an agent is as follows ,in To control the input, For bounded external disturbances; the specific form of the controller is: ； in, For the first The neighbor set of an agent, It is a relative position vector. This is the derivative term of the general barrier function; , Similarly, where is the fractional exponent. satisfy , It is a positive integer exponent and satisfies ; To control the gain; Let Kronecker function be used when The value is 1 if it is true, and 0 otherwise. To achieve the desired formation speed, the controller comprises three core components: The first part is the time-varying scaling function. The product of the double power feedback term; the double power feedback term contains a fractional power term and a positive integer power term: the fractional power term provides a large gain when the error is small, accelerating the convergence of small errors; the positive integer power term provides a fast response when the error is large, accelerating the convergence of large errors; The second part consists of weighting terms related to the rigid matrix, which are used to map the error information of each edge to the control input of each agent through the rigid matrix; The third part is the feedforward term for the desired formation speed, which is applied only to the leader agent and passed to the entire formation through a directed communication topology, ensuring that all agents can synchronously track the desired trajectory.

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