A directed formation control method based on distance and angle in a local coordinate framework

Abstract

The present invention discloses a directed formation control method based on distance and angles in a local coordinate framework, comprising: 1) designing communication relationships between multiple agents; 2) establishing a dynamic model of the multi-agent system and analyzing the objectives required for the system to form a desired formation; 3) for n non-particle single-integral agent models, transforming the distances through a conversion function ensures signal connectivity and collision avoidance between adjacent agents, while simultaneously avoiding the non-uniqueness of the formation equilibrium point by imposing constraints on the angles of the kth agent; 4) defining the errors of the distances and angles converted in step 3) and designing a controller; 5) after completing the design of the agent controller, the controller is physically verified in a visual positioning experimental platform. By introducing a state-dependent time-varying control gain, the disadvantage of existing methods that require repeated adjustment of control parameters according to the desired formation pattern is avoided.

Description

Technical Field

[0001] The present invention relates to the field of multi-agent collaborative control, and in particular to a distance- and angle-based directed formation control method in a local coordinate framework. Background Art

[0002] In recent years, the application prospects of multi-agent formation control technology in military and aerospace fields have continued to expand. Formation control methods have been widely applied and deeply studied, with a trend of diversification, including leader-following methods, behavior-based methods, virtual structure methods, and graph theory methods. With the continuous deepening of formation control research, involving aspects such as formation generation, formation maintenance, formation switching, and self-adaptation, these studies have promoted the development and application of formation control technology. At the same time, the challenges faced by multi-agent systems, such as heterogeneity and network optimization, have pointed the way for further research in the theory and application of multi-agent systems. In addition, the integration of new technologies, especially the development of visual sensors, has provided new perception and navigation methods for formation control technology, enhancing the system's adaptability and flexibility. These factors have jointly promoted the development and advancement of multi-agent formation control technology.

[0003] Based on the types of sensing and control variables used in the controller, existing formation control methods can be categorized as position-based, displacement-based, distance-based, and orientation- and angle-based. Furthermore, formation control methods based on the relative positions of agents are also widely adopted. However, these methods are similar to distance- and orientation-based formation methods in that they require agents to have a common global position or a shared reference frame orientation, which places high demands on the agents' perception capabilities. Summary of the Invention

[0004] A distance- and angle-based directed formation control method in a local coordinate framework ensures the uniqueness of equilibrium points in multi-agent formation control by introducing angle constraints and combining distance and angle variables. A 2-D formation control scheme is provided for acyclic triangulated directed graphs, ensuring strong consistency between actual and desired formations. The introduction of state-dependent time-varying control gains avoids the need to repeatedly adjust control parameters based on the desired formation pattern. Furthermore, for a model of n non-particle single-integrator agents, signal connectivity maintenance and collision avoidance between adjacent agents are addressed through corresponding distance conversion. The agents are mobile robots, and inter-agent communication ensures that the mobile robots always maintain a safe distance and avoid collisions during motion. The method specifically includes the following steps:

[0005] (1) Step 1: According to the connection relationship and information communication between the leader and followers in the multi-agent system, the leader is set as agent 1, which has a time-varying bounded reference speed and leads the movement of the entire formation; agent 2 is set as the first follower in the formation architecture, and agent 1 is its only neighbor, that is, agent 2 can uniquely obtain the distance between itself and agent 1; the remaining agents k are set as followers, k ≥ 3, each follower agent follows its two neighbors, and each follower agent k can obtain the distance between itself and its two neighbors, and with itself as the vertex, measure the size of the angle formed by its two neighbors [0, 2π); the underlying directed graph of the entire formation is generated by a series of Henneberg vertex additions, which is applicable to any two-dimensional formation configuration, ensuring the flexibility and scalability of the formation;

[0006] (2) Step 2: Based on the connection relationship of multiple agents, a multi-agent system dynamics model is established, and the expected distance and expected angle indicators that need to be met by the control system to form the expected formation are analyzed;

[0007] (3) Step 3: After establishing the multi-agent dynamics model and index analysis, for the n non-mass single integrator agent model, the distance is converted through the conversion function to maintain the connectivity between adjacent agents, while avoiding the non-uniqueness of the equilibrium point during the formation process, and the expected edge angle of the kth agent is constrained accordingly;

[0008] (4) Step 4: Define the distance and angle errors converted in step 3 and design a controller to achieve strong consistency between the actual formation and the expected formation by combining the weighted fusion of distance and angle variables and imposing strict constraints on the angle;

[0009] (5) Step 5: After completing the controller design of the intelligent agent, the controller is physically verified in the visual positioning experimental platform.

[0010] Furthermore, in step one, the following steps are included:

[0011] In the present invention, a directed graph is used To represent the perception topology between agents, is a set of perception topology vertices consisting of n agents, and is a set of directed edges describing directed perception between agents. If (j,i)∈ε, Then i is called j’s neighbor, and agent j can measure the relative position of its neighbor agent i in its local coordinate system. Conforms to the "leader - first follower" paradigm of the minimum persistent directed graph. Specifically, agent 1 is the leader with a time - varying bounded reference speed; agent 2 is the first follower in the formation architecture, and agent 1 is its only neighbor, that is, agent 2 can uniquely obtain the distance from agent 1; agent k (k≥3) acts as a follower, and each follower agent follows two agents with smaller indices, that is, the two neighbor agents of agent k. Each follower agent k (k≥3) can obtain the distances from itself to its two neighbors, and with itself as the vertex, measure the size of the included angle formed by its two neighbors in the range [0, 2π). Such a graph can be generated by a series of Henneberg vertex additions, applicable to any two - dimensional formation, and has the following two properties:

[0012] 1) out(1) = 0, out(2) = 1, and out(i) = 2, (i≥3) (where out(i) represents the out - degree of vertex i);

[0013] 2) If there is an edge between agent i and j, where i < j, then the direction of this edge must be j→i.

[0014] Furthermore, in step two, it includes the following steps:

[0015] The dynamic models of n agents on a plane are as follows:

[0016]

[0017] where pand u i respectively represent the position and velocity (control input) of agent i relative to the fixed coordinate system, and δ i (t) represents an unknown, bounded, piece - wise continuous external disturbance. The relative position vector corresponding to the directed edge (j, i) is defined as:

[0018] p ji = p i - p j ∈R 2 , (j, i) ∈ ε

[0019] where represents the relative position vector between agent i and agent j, and and respectively represent the positions of agent i and agent j relative to the fixed coordinate system, R represents the set of real numbers, and represents the set of directed edges between agents.

[0020] The included angle α kij between each follower k and its two neighbors i and j (i < j < k) ranges from [0, 2π), and is formed counter - clockwise from the edge (k, i) to the edge (k, j), as Figure 1 shown, and can be obtained through the modulo algorithm:​

[0021] α kij =mod{α kj -α ki ,2π}

[0022] α ki and α kj and are the orientation angles of agents i and j in the local coordinate system relative to agent k, respectively.

[0023] Based on the description of step 1, you can and expected corners Uniquely define the desired formation, thus avoiding reflection, flipping and bending ambiguities, e.g. Figure 2 shown.

[0024] Based on the collision avoidance and signal connectivity maintenance of adjacent intelligent agents in the non-particle model, in the present invention, each intelligent agent i is defined as having a radius r si The circular safety range and radius are known as r ci The circular perception range, where r si <r ci In order to avoid collision between adjacent agents i and j, define d ji =d ji =||p i (t)-p j (t)||>r sji :=r si +r ci (t≥0). At the same time, in order to ensure that the information between adjacent agents can be smoothly transmitted when the agents move to the desired formation, define d ji > d ji :=r sji +2r sk Therefore, in order to ensure that the agents can smoothly converge to the desired formation during the movement and to ensure that there is no collision between adjacent agents, it is further required that d ji <r cji :=mib(r ci +r si ,r cj +r si Based on the above analysis, the goals that need to be accomplished to achieve the desired formation are as follows:

[0025]

[0026] d ji <d ji <r cji ,t>0

[0027] Among them, (j,i)∈ε and (k,i), (k,j)∈ε\{(2,1)}, i <j<k。

[0028] Furthermore, in step three, the following steps are included:

[0029] The expected edge angle of the kth (k≥3) agent The range is [0,π) or [π,2π). In order to avoid non-unique equilibrium points in the formation process, that is, to avoid ambiguity in the movement of multiple agents, for the following two types of edge angles α kij Introduce the corresponding constraint function:

[0030] 1) For Defining β rk (t) is a continuous non-increasing function, β rk (0) = 2π, β rk (T β )=πor β lk =-ε0;

[0031] 2) For Defining β lk As a continuous non-decreasing function, β lk =0,β lk (T β )=π-ε0 or β rk =2π.

[0032] Where 0<ε0≤1 is a very small constant, T β represents the transition time of the constraint function, β rk and β lk Respectively represent about α kij upper and lower constraints.

[0033] Next, the opposite angle α kij and expected corners Perform the following transformation to apply constraints

[0034]

[0035] where s kij and are the corners after transformation and the expected corners of the agent, (k,i), (k,j)∈ε\{(2,1)}, i <j<k,λ> 0 is the design parameter. lk (0) = -ε0, β rk (0)=2π,α kij , The initial conditions for corner conversion described above always hold.

[0036] In order to ensure collision avoidance and connectivity maintenance between adjacent agents i and j, the distance is transformed as follows:

[0037]

[0038] where s ji and are the distance after transformation and the expected distance of the agent, respectively. d ji <d ji (0)<r cji The initial conditions ensure that the formation meets the conditions of collision avoidance and connectivity at the initial moment.

[0039] For any d ji (0)Satisfy d ji <d ji (0)<r cji , if s kij and s ji Keeping bounded, the uniqueness of the formation and the collision avoidance and connectivity of adjacent agents are guaranteed. Therefore, as long as the distance variable d in the initial distribution of agents is ji satisfy d ji <d ji (0)<r cji , that is, each agent satisfies the conditions of collision avoidance and connectivity at the initial moment, which means that s ji The boundedness of ensures the communication between agent i and agent j. It effectively transforms the state constraint control problem into a bounded control problem of new variables, which greatly facilitates the design of the controller in the subsequent steps.

[0040] Furthermore, in step 4, the following steps are included:

[0041] In order to quantify the control target, the agent distance error e is defined as ji and the agent's edge error e kij as follows:

[0042]

[0043] At the same time, the agent conversion distance error z is introduced ji and the agent transformation angle error z kij as follows:

[0044]

[0045] When e ji = 0, z ji =0, when e kij = 0, z kij = 0. In addition, According to the above steps, the decentralized formation control protocol is designed as follows:

[0046] u1=u L (t)

[0047] u2=c2μ(t)z 21 p 21

[0048]

[0049]

[0050] where c2, c k , v k , k μ , T μ and μ0 are positive design parameters, and J represents the counterclockwise rotation matrix. The above decentralized control protocol maintains collision avoidance and connectivity between adjacent agents, while also ensuring that all signals within the closed-loop system are bounded. μ(t) involves the controller's adjustment of distance and corner weights, effectively overcoming the drawback of repeated control gain adjustments required by multiple agents to form different desired formation patterns.

[0051] In step five, the following steps are included:

[0052] To further verify the effectiveness of the proposed decentralized control protocol in an environment with sparse obstacles, experiments were conducted using five mobile robots (Turt leBot3).

[0053] The experiment used a processor (NUC11TNK-i7) running Ubuntu 18.04 and the Robot Operating System (ROS) to receive and process sensory information from an industrial camera (MV-LD-4-4M-G). Noting that the relative distances and corner sizes between agents do not change with coordinate systems, we indirectly simulated each agent's local coordinate frame using position data within a fixed camera coordinate frame. Control commands were then calculated using a decentralized control protocol and broadcast using a Zigbee wireless communication module. Ultimately, each agent subscribed to the corresponding control signals, enabling them to execute movements.

[0054] The present invention provides a distance- and angle-based directed formation control method in a local coordinate framework. Compared with existing technologies, the present invention has the following advantages:

[0055] 1. Achieving strong formation consistency by leveraging distance and angles essentially involves dynamic weight adjustment. The distance and angle control variables introduced in this invention overcome the drawback of requiring users to repeatedly adjust control gains based on different desired formation patterns.

[0056] 2. The present invention solves the collision avoidance and connectivity maintenance problems of n non-particle intelligent agent models affected by external interference in an acyclic triangulated directed graph framework by directly converting distance constraints.

[0057] 3. The present invention was experimentally verified using a micro mobile robot to verify the applicability of the decentralized control protocol. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 is the corner graph in the triangular subgraph.

[0059] Figure 2 (a) and (b) are the expected formation diagrams.

[0060] Figure 3 Snapshots of the agent at different time points, where gray dots represent obstacles. (a) t = 0s, (b) t = 50s, (c) t = 95s, (d) t = 200s.

[0061] Figure 4 (a) is the edge angle α kij The evolution diagram of (b) is the distance d ji 's evolution diagram. DETAILED DESCRIPTION

[0062] The following is a further detailed description of the distance and angle-based directed formation control method in the local coordinate framework of the present invention in conjunction with the accompanying drawings and embodiments.

[0063] 1. The present invention uses five mobile robots to form a multi-agent system for experimental verification, where robot 1 is the leader agent and the remaining robots are follower agents, numbered 1, 2, 3, 4, and 5 respectively.

[0064] 2. The desired formation and underlying perception topology are as follows Figure 2 As shown in (b), the expected directed edge length is as follows: The expected corner sizes are as follows: For i=1,…,5, the collision avoidance radius and the perception radius are set to r si =0.06m,r ci =5m. Select function β rk and β lk as follows:

[0065]

[0066] Edge angle α kij and expected corners Perform the following conversion:

[0067]

[0068] Distance d ji and expected distance Perform the following conversion:

[0069]

[0070] The decentralized control protocol is designed as follows:

[0071] u1=u L (t)

[0072] u2=0.12μ(t)z 21 p 21

[0073]

[0074]

[0075] 3. In a cluttered environment with obstacles, the control protocol is combined with the artificial potential field method to obtain the changes of the intelligent agent at different time points. Figure 3 The distance and angle between adjacent agents change as shown in Figure 4 As shown in the figure, it is worth noting that the initial positions of agents 3, 4, and 5 are roughly opposite to their desired positions, with obstacles in between, which can easily lead to formation ambiguity. From the above results, it can be seen that the robots not only maintain connectivity and collision avoidance between adjacent agents during movement, but also achieve and maintain the desired formation.

Claims

1. A distance and angle-based directed formation control method in a local coordinate framework, characterized by: The uniqueness of the equilibrium point in multi-agent formation control is ensured by introducing angle constraints and combining distance and angle variables. A 2-D formation control scheme is proposed for acyclic triangulated directed graphs, ensuring strong consistency between the actual and desired formations. By introducing state-dependent time-varying control gains, the need to repeatedly adjust control parameters according to the desired formation pattern is avoided. In addition, for a model of n non-particle single-integrator agents, signal connectivity maintenance and collision avoidance between adjacent agents are addressed through corresponding distance conversion. The agents are mobile robots, and communication between the agents ensures that the mobile robots always maintain a safe distance and avoid collisions during movement. The specific steps include: Step 1: Based on the connection relationship and information communication between leaders and followers in a multi-agent system, the leader is set as agent 1, which has a time-varying and bounded reference velocity and leads the movement of the entire formation. Agent 2 is set as the first follower in the formation architecture, and agent 1 is its only neighbor, that is, agent 2 can uniquely obtain the distance between itself and agent 1. The remaining agents k are set as followers, k ≥ 3, and each follower agent follows its two neighbors. Each follower agent k can obtain the distance between itself and its two neighbors and measure the size of the angle formed by its two neighbors [0, 2π). The underlying directed graph of the entire formation is generated by a series of Henneberg vertex additions, which is applicable to any two-dimensional formation configuration and ensures the flexibility and scalability of the formation. Step 2: Based on the connection relationship of multiple agents, a multi-agent system dynamics model is established, and the expected distance and expected angle indicators that need to be met by the control system to form the desired formation are analyzed; Step 3: After establishing the multi-agent dynamics model and indicator analysis, for the n non-mass single integrator agent model, the distance is converted through a conversion function to maintain the connectivity between adjacent agents, while avoiding the non-uniqueness of the equilibrium point during the formation process, and the desired angle of the kth agent is constrained accordingly; Step 4: Define the distance and angle errors converted in step 3 and design a controller to achieve strong consistency between the actual and expected formations by combining weighted fusion of distance and angle variables and imposing strict constraints on angles; Step 5: After completing the controller design of the intelligent agent, the controller will be physically verified in the visual positioning experimental platform.

2. The distance- and angle-based directed formation control method in a local coordinate framework according to claim 1, characterized in that: Using directed graphs To represent the perception topology between agents, is a set of perception topology vertices consisting of n agents, and j≠i} is a set of directed edges describing directed perception between agents. If (j,i)∈ε, Then i is called j's neighbor, and agent j measures the relative position of its neighbor agent i in its local coordinate system; directed graph It conforms to the "leader-first follower" paradigm of a minimal persistent directed graph; agent 1 is the leader, with a time-varying bounded reference velocity; agent 2 is the first follower in the formation architecture, and agent 1 is its only neighbor, meaning that agent 2 can uniquely determine its distance from agent 1; agent k acts as a follower, each following two agents with smaller indices, namely agent k's two neighbors; each follower agent k can determine the distance between itself and its two neighbors, and, with itself as a vertex, measure the size of the angle formed by its two neighbors ([0, 2π)); this type of graph is generated by a series of Henneberg vertex additions, is applicable to any two-dimensional formation, and has the following two properties: 1) out(1) = 0, out(2) = 1, and out(i*) = 2, i ≥ 3, where out(i*) represents the out-degree of vertex i*; 2) If there is an edge between agents i and j, where i<j, then the direction of the edge must be j→i.

3. The distance- and angle-based directed formation control method in a local coordinate framework according to claim 1, characterized in that: Based on the connection relationship of multiple agents, a multi-agent system dynamics model is established, and the expected distance and expected angle indicators that need to be met for the control system to form the desired formation are analyzed; The dynamic model of n agents on the plane is as follows: where p i and u i Represent the position and velocity of agent i relative to the fixed coordinate system, δ i (t) represents an unknown, bounded, piecewise continuous external disturbance; The relative position vector corresponding to the directed edge (j,i) is defined as: p ji =p i -p j ∈R 2 ,(j,i)∈ε where p ji represents the relative position vector of agent i and agent j, p i and p j Respectively represent the positions of agent i and agent j relative to the fixed coordinate system, R represents the set of real numbers, and ε represents the set of directed edges between agents; The corner angle α between each follower k and its two neighbors i and j kij ranges from [0, 2π), where i < j < k, and is formed counterclockwise from edge (k, i) to edge (k, j), obtained through modulo arithmetic: a kij =mod{a kj -a ki ,2π} α ki and α kj and are the orientation angles of agents i and j in the local coordinate system relative to agent k, respectively; Based on the description of step 1, according to the expected distance and expected corners Uniquely define the desired formation, thus avoiding reflection, flipping and bending ambiguities; Based on the collision avoidance and signal connectivity maintenance of adjacent intelligent agents in the non-mass point model, each intelligent agent i is defined as having a radius r si The circular safety range and radius are known as r ci The circular perception range, where r si <r ci ; To avoid collision between adjacent agents i and j, define d ji =d ji =||p i (t)-p j (t)||>r sji :=r si +r ci (t≥0); At the same time, in order to ensure smooth information transmission between adjacent agents during the process of moving to the desired formation, define d ji > d ji :=r sji +2r sk In order to ensure that the agents can smoothly converge to the desired formation during the movement and to ensure that there is no collision between adjacent agents, it is required that d ji <r cji :=min(r ci +r si , r cj +r si ); The goals that need to be completed to achieve the desired formation are as follows: d ji <d ji <r cji ,t>0 Among them, (j, i)∈ε and (k, i), (k, j)∈ε\{(2, 1)}, i<j<k.

4. The distance- and angle-based directed formation control method in a local coordinate framework according to claim 1, characterized in that: After establishing a multi-agent dynamics model and indicator analysis, for a model of n non-mass single integrator agents, a conversion function is used to transform the distance to maintain the connectivity between adjacent agents while avoiding the non-uniqueness of the equilibrium point during the formation process. The desired angle of the kth agent is constrained accordingly. The expected edge angle of the kth (k≥3) agent The range is [0, π) or [π, 2π). In order to avoid the occurrence of non-unique equilibrium points in the formation process, that is, to avoid the ambiguity of multi-agents in the movement process, for the following two types of edge angles α kij Introduce the corresponding constraint function: 1) For Define β rk (t) is a continuous non-increasing function, β rk (0) = 2π, β rk (T β )=πor β lk = -ε0; 2) For Define β lk As a continuous non-decreasing function, β lk (0) = -ε0, β lk (T β )=π-ε0 or β rk =2π; Where 0<ε0≤1 is a very small constant, T β represents the transition time of the constraint function, β rk and β lk Respectively represent the kij Upper and lower constraints; Next, the opposite angle α kij and expected corners Perform the following transformation to apply constraints where s kij and are the corners after transformation and the expected corners of the agent respectively, (k, i), (k, j)∈ε\{(2, 1)}, i<j<k, λ>0 are design parameters; lk (0) = -ε0, β rk (0) = 2π and The initial conditions for corner conversion described above always hold; In order to ensure collision avoidance and connectivity maintenance between adjacent agents i and j, the distance is transformed as follows: where s ji and are the distance after transformation and the expected distance of the agent, respectively. d ji <d ji (0) <r cji ; The initial conditions ensure that the formation meets the conditions of collision avoidance and connectivity during the multi-agent movement at the initial moment; For any d ji (0)Satisfaction d ji <d ji (0)<r cji , if s kij and s ji Keeping bounded, the uniqueness of the formation and the collision avoidance and connectivity of adjacent agents are guaranteed; as long as the distance variable d in the initial distribution of agents ji satisfy d ji <d ji (0) <r cji , that is, each agent satisfies the conditions of collision avoidance and connectivity at the initial moment, which means that s ji The boundedness of ensures the communication between agent i and agent j; it effectively transforms the state constraint control problem into a bounded control problem of new variables, which greatly facilitates the design of the controller in the subsequent steps.

5. The distance and angle-based directed formation control method in a local coordinate framework according to claim 1, characterized in that: Define the distance and angle errors converted in step 3 and design a controller to achieve strong consistency between the actual and expected formations by combining weighted fusion of distance and angle variables and imposing strict constraints on angles; In order to quantify the control target, the agent distance error e is defined as ji and the agent's edge error e kij as follows: At the same time, the agent conversion distance error z is introduced ji and the agent transformation angle error z kij as follows: When e ji = 0, z ji =0, when e kij = 0, z kij =0; In addition, The decentralized formation control protocol is designed as follows: u1=u L (t) u2=c2μ(t)z 21 p 21 where c2, c k , v k , k μ , T μ and μ0 are positive design parameters, J represents the counterclockwise rotation matrix; μ(t) involves the controller's adjustment of distance and edge weights.

6. The distance- and angle-based directed formation control method in a local coordinate framework according to claim 1, wherein: After completing the controller design of the intelligent agent, the controller will be physically verified in the visual positioning experimental platform; To verify the effectiveness of the proposed decentralized control protocol in an environment with sparse obstacles, experiments are conducted using five mobile robots; A processor equipped with Ubuntu l8.04 and a robot operating system is used to receive and process perception information from an industrial camera. Noting that the relative distances and corner sizes between agents do not change with the coordinate system, the position data within the fixed camera coordinate frame is used to indirectly simulate the local coordinate frame of each agent. Subsequently, control commands are calculated using a decentralized control protocol and broadcast using a Zigbee wireless communication module. Ultimately, each agent subscribes to the corresponding control signals, enabling them to perform movements.

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