Distance and corner-based directed formation control method under local coordinate framework

Abstract

The invention discloses a directed formation control method based on distance and corners under a local coordinate framework. The method comprises the following steps: 1) designing a communication relationship among multiple agents; 2) establishing a dynamic model of a multi-agent system, and analyzing a target required to be completed by the system to form an expected formation; 3) for n non-particle type single-integral agent models, converting the distance through a conversion function to ensure the signal connectivity and collision avoidance between adjacent agents, and meanwhile, restraining the corner of the kth agent to avoid the non-uniqueness of a formation balance point; (4) error definition is conducted on the distance and the corners converted in the step (3), and a controller is designed; and 5) after the design of the intelligent agent controller is completed, carrying out physical verification on the controller in a visual positioning experiment platform. By introducing state-related time-varying control gains, the defect that control parameters need to be repeatedly adjusted according to an expected formation mode in an existing method 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, aerospace and other fields have been continuously expanded. Formation control methods have been widely used and deeply studied, and the research methods have shown a diversified trend, including leader following method, behavior-based method, virtual structure method and graph theory method. With the continuous deepening of the content of formation control research, involving formation generation, formation maintenance, formation switching and adaptation, these studies have promoted the development and application of formation control technology. At the same time, the problems faced by multi-agent systems, such as heterogeneity and network optimization, have pointed out the direction for further research on the theory and application of multi-agent systems. In addition, the integration of new technologies, especially the development of visual sensor technology, has provided new perception and navigation means for formation control technology, enhancing the adaptability and flexibility of the system. These factors have jointly promoted the development and progress of multi-agent formation control technology.

[0003] According to the types of sensing and control variables used in the controller, the existing formation control methods can be divided into position-based, displacement-based, distance-based, orientation-based and angle-based methods. In addition, the formation control method based on the relative position between agents is also widely adopted. However, this method is similar to the distance-based and orientation-based formation method because it requires the agents to have common global orientation information or have a shared reference frame direction, which places high demands on the perception ability of the agents. Summary of the invention

[0004] A directed formation control method based on distance and angle in a local coordinate framework ensures the uniqueness of the equilibrium point in the multi-agent formation control by introducing angle constraints and combining distance and angle variables; provides a 2-D formation control scheme for acyclic triangulated directed graphs to ensure strong consistency between actual and expected formations; introduces state-dependent time-varying control gains to avoid the disadvantage of repeatedly adjusting control parameters according to the expected formation mode; in addition, for n non-particle single integrator agent models, the signal connectivity maintenance and collision avoidance between adjacent agents are solved through corresponding distance conversion. The agent is a mobile robot, and communication between agents is used to ensure that the mobile robot always maintains a safe distance and avoids collision during movement; specifically, the following steps are included:

[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 its two neighbors and measure the size of the angle formed by its two neighbors with itself as the vertex [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-particle 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 in the formation process, and the expected 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. It 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 to 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 to its two neighbors and measure the included angle size [0, 2π) formed by its two neighbors with itself as the vertex. Such a graph can be generated by a series of Henneberg vertex additions, is 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, the following steps are included:

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

[0016]

[0017] where p i and 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 edge (k, i) to 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 azimuths of agents i and j in the local coordinate system relative to agent k, respectively.

[0023] Based on the description of step 1, the expected distance and the expected edge 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 to be r ci The circular sensing range of 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 completed 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, the following two types of angles α are considered. 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 ) = π or 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 the expected edge Perform the following transformation to apply constraints

[0034]

[0035] where s kij and are the transformed corners and expected corners of the agent, respectively, (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 the 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 the agent conversion and the expected distance, 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 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 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. The state constraint control problem is effectively transformed 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 can maintain collision avoidance and connectivity maintenance between adjacent agents, while all signals in the closed-loop system are bounded; μ(t) involves the controller's adjustment of distance and corner weights, effectively overcoming the disadvantage of repeated adjustment of control gains when multiple agents form different desired formation modes.

[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 carried out using five mobile robots (Turt leBot3).

[0053] The experiment used a processor (NUC11TNK-i7) equipped with Ubuntu18.04 and Robot Operating System (ROS) to receive and process the perception information from the industrial camera (MV-LD-4-4M-G). Note that the relative distance and corner size between the agents do not change with the change of the coordinate system, so we use the position data in the fixed camera coordinate frame to indirectly simulate the local coordinate frame of each agent. Subsequently, the control commands are calculated using a decentralized control protocol and broadcast using a Zigbee wireless communication module. Finally, each agent subscribes to the corresponding control signal, enabling them to perform movements.

[0054] The present invention provides a directed formation control method based on distance and angle in a local coordinate frame. Compared with the prior art, the present invention has the following beneficial effects:

[0055] 1. Achieving strong consistency of formation by using distance and angle essentially involves dynamic adjustment of weights. The distance and angle control variables introduced in the present invention overcome the disadvantage that the user needs to repeatedly adjust the control gain according to different desired formation modes.

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

[0057] 3. The present invention has been 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 side 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 leading agent and the remaining robots are follower agents, numbered 1, 2, 3, 4, and 5 respectively.

[0064] 2. The expected formation and underlying perception topology are as follows Figure 2 As shown in (b), the expected length of the directed edge 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 respectively. si =0.06m,r ci =5m. Select function β rk and β lk as follows:

[0065]

[0066] Edge Angle α kij and the expected edge 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 agent at different time points. Figure 3 The distance and angle between adjacent agents change as shown in Figure 4 As shown. It is worth noting that the initial positions of agents 3, 4 and 5 are roughly opposite to their expected positions, and there are obstacles in the middle, which can easily lead to formation ambiguity. From the above results, it can be seen that the robot not only maintains connectivity and collision avoidance between adjacent agents during movement, but also achieves and maintains the expected formation.

Claims

1. A directed formation control method based on distance and angle in a local coordinate frame, characterized by: The uniqueness of the equilibrium point in the multi-agent formation control is ensured by introducing angle constraints and combining distance and angle variables; a 2-D formation control scheme is provided for the acyclic triangulated directed graph to ensure strong consistency between the actual and expected formations; the disadvantage of repeatedly adjusting the control parameters according to the expected formation mode is avoided by introducing state-dependent time-varying control gains; in addition, for n non-particle single integrator agent models, the signal connectivity maintenance and collision avoidance between adjacent agents are solved by corresponding distance conversion. The agent is a mobile robot, and the communication between the agents ensures that the mobile robot always maintains a safe distance and avoids collision during movement; the specific steps include: Step 1: According to the connection relationship and information communication between the leader and the follower 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 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; 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 for the control system to form the expected formation are analyzed; Step 3: After establishing the multi-agent dynamics model and index analysis, for the n non-particle single integrator agent models, the distance is converted through the conversion function to maintain the connectivity between adjacent agents, while avoiding the non-uniqueness of the equilibrium point in the formation process, and the expected 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 formation and the expected formation by combining the weighted fusion of distance and angle variables and imposing strict constraints on the angle; 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 method for controlling a directed formation based on distance and angle in a local coordinate frame 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 the minimum continuous directed graph; 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 between itself and agent 1; agent k acts as a follower, and each follower agent follows two agents with smaller indexes, that is, the two neighbor agents of agent k; each follower agent k can obtain the distance between 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 the local coordinate frame 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 expected formation are analyzed; The dynamic model of n agents on the plane is as follows: where p i and u i Respectively 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 included 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 modular arithmetic: a kij =mod{a kj -a ki ,2π} α ki and α kj and are the azimuths 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 the expected edge Uniquely define the desired formation, thus avoiding reflection, flipping and bending ambiguities; Based on the non-particle model, the collision avoidance and signal connectivity maintenance of adjacent agents are defined as follows: each agent i has a radius r si The circular safety range and radius are known to be r ci The circular sensing range of 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 method for controlling a directed formation based on distance and angle in a local coordinate frame according to claim 1, characterized in that: After establishing the multi-agent dynamics model and index analysis, for the n non-particle 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 in the formation process, and the expected 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 non-unique equilibrium points in the formation process, that is, to avoid ambiguity in the movement of multiple agents, the following two types of angles α kij Introduce the corresponding constraint function: 1) For Defining β rk (t) is a continuous non-increasing function, β rk (0) = 2π, β rk (T β ) = π or β lk = -ε0; 2) For Defining 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 about α kij The upper and lower constraints of Next, the opposite angle α kij and the expected edge Perform the following transformation to apply constraints where s kij and are the transformed corners and 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 the 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 the agent conversion and the expected distance, 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 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 the 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 method for controlling a directed formation based on distance and angle in a local coordinate frame 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 formation and the expected formation by combining the weighted fusion of distance and angle variables and imposing strict constraints on the 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 corner weights.

6. The method for controlling a directed formation based on distance and angle in a local coordinate frame according to claim 1, characterized in that: After completing the controller design of the intelligent agent, the controller is 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 were conducted using five mobile robots; A processor equipped with Ubuntul8.04 and robot operating system is used to receive and process the perception information from the industrial camera. Noting that the relative distance and corner size between the agents do not change with the change of the coordinate system, the position data in the fixed camera coordinate frame is used to indirectly simulate the local coordinate frame of each agent. Subsequently, the control commands are calculated using a decentralized control protocol and broadcast using a Zigbee wireless communication module. Finally, each agent subscribes to the corresponding control signal, enabling them to perform movements.

Images