Continuous adaptive integral sliding mode leader-following multi-robot affine formation control method based on event trigger mechanism

Abstract

According to the continuous self-adaptive integral sliding mode leader-following multi-robot affine formation control method based on the event triggering mechanism, distributed control and affine formation strategies are fused, a formation controller with high maneuverability and high robustness is constructed, the event triggering mechanism is introduced to reduce the communication load and the control frequency, and the control efficiency is improved. The method belongs to the technical field of multi-robot intelligent cooperative control. Aiming at the common problem of insufficient formation rigidity and flexibility in a multi-robot formation control task, the invention provides a cooperative control method capable of realizing various affine transformations such as zooming, rotating, shearing, coplanar and collinear, and the cooperative control method has good environmental adaptability and system performance.

Description

Technical Field

[0001] The present invention belongs to the field of multi-robot system control technology, and specifically relates to a distributed affine formation control method that integrates continuous adaptive integral sliding mode control and event triggering mechanism. The method is suitable for multi-agent systems with dynamic formation requirements, such as underwater robots, ground mobile robots, and drone clusters. Background Art

[0002] In recent years, multi-robot systems have been widely used in disaster relief, cargo handling, patrolling, and dynamic target search, owing to their greater flexibility, adaptability, and robustness compared to individual robots. By collaborating as a team, these systems achieve overall efficiency exceeding the sum of the individual robots. To enhance multi-robot collaboration and improve efficiency, research on multi-robot teaming has become a hot topic.

[0003] Multi-robot formations require control methods. Since the successful application of self-consistency theory to formation control, extensive research has been conducted on convergence-guaranteed formation control methods, such as displacement-based methods, distance-based methods, and orientation-based methods. These three traditional formation control methods control the robot formation by imposing constant constraints on the displacement, distance, and orientation between robots. The invariance of the constant constraints on the robot formation has a significant impact on the formation's maneuverability. For example, formation control methods based on displacement constraints can track robot formations with time-varying translations, but have difficulty controlling the formation's scale or orientation. This is because changing the scale or orientation requires changing the displacement constraints, while formation control methods based on displacement constraints are invariant to formation displacement. Formation control methods based on distance constraints can track robot formations with time-varying translations and orientations, but have difficulty tracking formations with time-varying scales. Formation control methods based on orientation constraints can track formations with time-varying translations and scales, but have difficulty tracking formations with time-varying orientations.

[0004] In order to realize the dynamic transformation of the nominal formation of a multi-robot affine formation, the present invention proposes a continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on an event trigger mechanism. Summary of the Invention

[0005] The purpose of the present invention is to provide a continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on an event trigger mechanism, which can realize flexible collaborative formation of multiple robots and has good robustness, adaptability and communication efficiency.

[0006] The present invention discloses a continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on an event trigger mechanism, and the specific steps are as follows:

[0007] The steps are as follows:

[0008] S1. Affine hierarchical control framework construction: Based on the leader-follower model, the leader and followers in the multi-robot formation are determined, and the nominal formation configuration and the communication topology of the cluster are established. The stress matrix of the nominal formation is calculated to support the subsequent affine localizability estimation;

[0009] S2. Distributed Estimator Design: By constructing a distributed estimator to obtain the formation's time-varying maneuver information, we can estimate the desired trajectory of each follower. Followers only need to calculate and track their affine localized target trajectory based on the position information of their local neighbors and the leader.

[0010] S3. Continuous Adaptive Integral Sliding Mode Controller Design: Combining an adaptive sliding mode control strategy with an integral sliding mode surface, this controller improves formation control accuracy and robustness, mitigating chattering. The controller utilizes a σ-modification technique to design a nonmonotonic adaptive gain law and incorporates an event-triggered mechanism to reduce unnecessary control updates.

[0011] Furthermore, in a preferred embodiment, the multi-robot formation and stress matrix to be formed in S1 are defined as:

[0012] There are n robots to be formed. Two coordinate systems are introduced to describe the motion of the robots. The nonlinear kinematic and dynamic models of each robot are simplified as follows:

[0013]

[0014] Among them, x i and y i represents the coordinates, θ i represents the orientation of the ith robot relative to the inertial frame; the linear velocity and angular velocity of the ith robot in its body fixed frame are represented by v i and ω i The multi-robot system consists of N robots and a dynamic virtual navigator. The communication between N+1 robots can be described by an undirected graph G = (N, ε, A), where N = {0, 1, n} is the set of robots. Edge Set, is the adjacency matrix. When a ij >0, otherwise a ij = 0. The configuration of the nominal formation is constructed as follows: l and f represent the leader and follower, and r represents the position of the N robots in the nominal formation. Based on the communication topology and the configuration of the nominal formation, the nominal formation of the multi-robot system is obtained as (G, r).

[0015] Furthermore, in S1, a linear matrix inequality solution method is used to calculate the stress matrix of the nominal formation.

[0016] Furthermore, the specific steps of calculating the stress matrix of the nominal formation using the linear matrix inequality solution method are as follows:

[0017] Step S1-1, assign any direction to each edge of the undirected graph G, using B∈R n×m represents the incidence matrix of the undirected graph G, and the incidence matrix is defined as follows:

[0018]

[0019] Step S1-2: yes The i-th column of

[0020] definition:

[0021]

[0022] in, Defined as:

[0023]

[0024] Let z1,…,z q ∈R m yes A set of bases;

[0025] Step S1-3: Perform singular value decomposition to obtain

[0026] Let U = [U1, U2], where U1 contains the first d+1 columns of U;

[0027] definition:

[0028]

[0029] Step S1-4, solve the following linear matrix inequality to obtain the equilibrium stress of the nominal formation

[0030]

[0031] where c1,…,c q Satisfies the linear matrix inequality:

[0032]

[0033] Step S1-5: Using the equilibrium stress of the nominal formation obtained Solve for the equilibrium stress matrix:

[0034]

[0035] Where Ω is the equilibrium stress matrix.

[0036] Compared with the prior art, the advantages of the present invention are:

[0037] 1. This technology implements affine formation control for a multi-robot system based on a leader-follower control model. By constructing a follower-tracking multi-robot affine formation control strategy, followers track the leader's movements and automatically adjust their relative positions within the formation based on the leader's cluster configuration. This approach enables a multi-robot system to manage the entire cluster through the coordinated control of a small number of key pilot nodes, enabling the formation to generate, track, and transform while moving along different trajectories.

[0038] 2. This multi-robot affine formation control method based on the leader-follower model has good scalability and applicability in large-scale multi-robot formation control tasks, in which the overall shape of the formation can be determined by the configuration of the leader subsystem.

[0039] 3. By designing a distributed estimation mechanism, followers can locally compute their own reference trajectories based on the leader's position information. The reference trajectory is derived by time-transforming the nominal configuration, generating the desired position and attitude (especially yaw angle) in real time without the need for a centralized path planner. Each follower relies solely on information from its neighbors to track the reference trajectory, achieving fully distributed control.

[0040] 4. A distributed leader-follow multi-robot control strategy based on a continuous adaptive integral sliding mode for followers achieves the following key benefits: This strategy ensures that the follower robots accurately converge to their desired relative position or trajectory with respect to the leader within a finite time. Thanks to the inherent robustness of sliding mode control, this method is highly robust against model uncertainties within the robots themselves and external environmental disturbances, significantly improving formation tracking accuracy and stability. The non-singular terminal sliding mode ensures that the tracking error not only converges to zero within a finite time but also avoids the singularity issues that can arise with traditional terminal sliding modes, ensuring the boundedness of the control input and the good performance of the entire closed-loop system.

[0041] Furthermore, this method employs a leader-follow multi-robot affine formation control strategy based on a continuous adaptive integral sliding mode for the followers. This enables fast, accurate, and robust distributed formation tracking control for the follower robots in dynamically changing environments or when maneuvering with the leader. Furthermore, this control method inherently and conveniently implements various affine transformations, such as translation, rotation, scaling, and shearing, for the entire robot formation, significantly enhancing the flexibility of the formation's maneuvers. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic diagram of the implementation flow of the continuous adaptive integral sliding mode leading-following multi-robot affine formation control method based on the event triggering mechanism of this embodiment.

[0043] Figure 2 Schematic diagram of the communication topology of seven robot clusters in a specific application embodiment.

[0044] Figure 3 It is a diagram of the release and interval time of the event trigger mechanism used in a specific application embodiment.

[0045] Figure 4 1 is a graph showing error variation of four follower multi-robots used in a specific application embodiment on a two-dimensional plane. DETAILED DESCRIPTION

[0046] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0047] like Figure 1 As shown, the steps of the continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on the event trigger mechanism of this embodiment include:

[0048] S1. Affine hierarchical control framework construction: Based on the leader-follower model, the leader and followers in the multi-robot formation are determined, and the nominal formation configuration and the communication topology of the cluster are established. The stress matrix of the nominal formation is calculated to support the subsequent affine localizability estimation;

[0049] S2. Distributed Estimator Design: By constructing a distributed estimator to obtain the formation's time-varying maneuver information, we can estimate the desired trajectory of each follower. Followers only need to calculate and track their affine localized target trajectory based on the position information of their local neighbors and the leader.

[0050] S3. Continuous Adaptive Integral Sliding Mode Controller Design: Combining an adaptive sliding mode control strategy with an integral sliding mode surface, this controller improves formation control accuracy and robustness, mitigating chattering. The controller utilizes a σ-modification technique to design a nonmonotonic adaptive gain law and incorporates an event-triggered mechanism to reduce unnecessary control updates.

[0051] The design concept for forming robots is divided into a leader-follower system, employing a follower-leader architecture to achieve affine formation. The leader determines the formation's formation, while the followers use the leader's position and control methods to determine their desired positions. They then move from their current positions to the desired positions, bringing the entire formation into the target formation. By changing the relative position of the leader, the leader can implement affine formation transformations such as shearing, scaling, and rotation.

[0052] In this embodiment, it is assumed that there are n robots in a plane, and the interactions between the robots are described by a fixed graph G∈(V,ε). is an affine formation. Among them, the leader's formation is The robots in the follower formation are affine formations. They include leaders and followers. The goal is to keep the followers and leaders at the same speed and in a stable formation. Assume there are n robots in a plane. The interactions between the robots are described by a fixed graph G∈(V,ε). This graph consists of a vertex set and an edge set ε∈V×V. An edge (i,j)∈ε indicates that robot i can receive information from robot j, and robot j is a neighbor of i. The neighbor set of vertex i is N i ={j∈V:(j,i)∈ε}. This invention only considers undirected graphs, that is, assuming that the first n l The robot is the leader, and the remaining n f The robots are followers, where n f =nn l , then V l ={1,...,n l} and V f =V / V l are the sets of leaders and followers respectively. Let it be an affine formation. Among them, the leader's formation is and the follower's formation is a formation (G,p) represents a one-to-one mapping between the formation of the i-th robot and the i-th node in the graph G. A nominal formation corresponding to the graph G can be defined as (G,r), where the nominal formation is a constant. Affine transformation is a conventional linear transformation form, such as translation, rotation, reduction, shearing, and combinations of these transformations. Therefore, the affine mapping of the nominal formation r can be defined as:

[0053]

[0054] Among them I nrepresents the n×n identity matrix, represents the Kronecker product, 1 n is an n-dimensional column vector and its value is 1. A(t) and b(t) both represent 3-dimensional column vectors, representing a 3×3-dimensional matrix. The affine transformation is recorded as (A, b). The matrices A(t) and b(t) are used to realize the geometric transformation actions of the unmanned boat, such as translation, rotation, scaling, and shearing.

[0055] For the formation, the stress corresponding to each side (i, j)∈ε{ω ij} (i,j)∈ε is a set of scalars. When there is attraction between i and j, ω ij >0, when there is repulsive force between i and j ω ij <0, in other cases ω ij =0, the relationship between stress and formation is described as:

[0056]

[0057] To make the expression more compact, ij} (i,j)∈E Written as the following stress matrix:

[0058]

[0059] Rewrite the above formula as:

[0060]

[0061] in, represents the stress matrix of the navigator, represents the stress matrix of the navigator, represents the stress matrix between the leader and the follower, represents the stress matrix between the follower and the leader.

[0062] The distributed error expression for the i-th item is as follows:

[0063]

[0064] in, represents the position error, which reflects the local deviation of robot i in the formation position configuration. The introduction of the stress matrix ensures the rigid constraint of the formation against affine transformations (rotation, scaling, translation, shear). θ represents the posture error, which reflects the local deviation of robot i in the formation posture configuration. This error is used to ensure that the posture of all robots is consistent with the expected movement direction of the formation. i represents the posture of the i-th robot.

[0065] The virtual control rates of linear velocity and angular velocity are designed as follows:

[0066]

[0067] Among them, J i J represents the position transformation matrix of the i-th robot, describing the velocity transformation relationship from the body coordinate system to the earth-fixed coordinate system. i v j K represents the velocity vector of the jth neighbor robot in the earth-fixed coordinate system. i1p With K i2p is a diagonal positive definite matrix, K i1θ Represents the proportional gain of the yaw angle error, which is a positive constant used to adjust the convergence speed of the yaw angle error. K i2θ The integral gain of the yaw angle error is a positive constant used to eliminate the steady-state error of the yaw angle error. represents the time derivative of the desired yaw angle of the ith robot.

[0068] To make the expression more compact, iv =(α ip ,α iθ ) T , then the velocity tracking error vector expression is as follows:

[0069] v ie =e i -α iv .

[0070] The non-singular terminal sliding mode manifold of the velocity tracking error vector can be described as:

[0071]

[0072] Among them, K i3 is a positive definite matrix.

[0073] The event trigger mechanism is designed as follows:

[0074]

[0075] The adaptive continuous sliding mode control law is designed as follows:

[0076]

[0077] in, represents the nominal inertia matrix of the ith robot, represents the nominal Coriolis and centripetal force matrices of the ith robot, ξ i represents the velocity vector of the i-th robot, represents the nominal damping matrix of the ith robot.

[0078] The non-monotonic adaptive gain law is designed as:

[0079]

[0080] Among them, γ ij (j=0,1,2) represents the adaptive gain, a positive constant used to adjust the amplitude of the adaptive rate. The adaptive law dynamically updates the upper bound parameters of uncertainty, allowing the estimated value to gradually approach the true value. This design, based on the norms of the sliding surface vector and the velocity vector, can adjust the control strategy according to real-time operating conditions, effectively offsetting the effects of disturbances. The advantage of the adaptive law is that it does not require prior knowledge of the specific characteristics of the disturbance, reducing the control law's reliance on prior knowledge. This dynamic adjustment mechanism significantly improves the flexibility and environmental adaptability of the control law, enabling multi-robot formations to maintain stable performance in complex environments.

[0081] An adaptive, fast-converging continuous integral sliding mode control law is proposed, specifically for flexible affine formation tasks for multi-robot systems in two-dimensional space. This control law, consisting of a nominal control term, a nonlinear sliding mode feedback term, and a robust compensation term, can rapidly compress the tracking error to zero within a finite time and significantly enhance the system's adaptability to modeling uncertainties, external disturbances, and dynamic obstacles. In this control structure, the nominal control term generates basic control inputs based on the known system dynamics model and target trajectory changes, ensuring the system's tracking stability under ideal conditions. The nonlinear feedback term, by constructing an integral sliding mode surface and a nonmonotonic gain law, provides the control system with rapid convergence and robust dynamic performance. The robust control term compensates for uncertainties and interference signals, while incorporating boundary layer technology to achieve continuity in the control law and suppress common chattering issues.

[0082] The adaptive gain component dynamically adjusts the control gain based on the sliding mode error, automatically adapting to varying error scales and system disturbance levels, avoiding the limitations of manual parameter adjustment. Furthermore, the event triggering mechanism, a key component of this control strategy, effectively suppresses the frequent updates of redundant control instructions. It triggers the controller to execute calculations only when the error exceeds a set threshold, effectively reducing network communication pressure and the execution burden on the robot.

[0083] This control law is widely applicable to a variety of platforms performing dynamic formation missions, such as wheeled ground robots, drone formations, and underwater robots. When faced with complex maneuvering commands such as rotation, scaling, and shearing, the system maintains the geometric consistency of the overall formation and ensures global convergence of the states of each subsystem. Combined with path planning and obstacle avoidance mechanisms, this method can also adapt to application scenarios such as multi-objective scheduling, spatial reconstruction, and formation reconfiguration in complex mission environments, providing a strong control foundation for autonomous collaborative operations of multi-robot systems.

[0084] Figure 2 The communication topology of a multi-robot formation network is presented. This topology illustrates how information is exchanged between robots and is key to implementing distributed control strategies.

[0085] Figure 3 Figure 2 shows how the trigger intervals for robots 4 through 7 change over time under the event-triggered control strategy. Each subgraph plots time t (in seconds) on the abscissa and the time interval (in seconds) between the current control input update and the previous one, depicting the dynamic behavior of the event-triggered mechanism.

[0086] The blue vertical bars in the figure represent the time points and intervals of trigger events, and the orange dots mark the specific interval values of each trigger. Each sub-graph corresponds to a robot, and the details are as follows:

[0087] The overall distribution of control trigger intervals for robot 4 is relatively dense, and the interval time is stably maintained in a small range, indicating that the robot frequently updates the control input throughout the operation process and has a high trigger frequency. Only in the middle and late stages does a large trigger interval peak (about 0.06 seconds) appear, which may correspond to the stage of low error variation of the system.

[0088] The trigger interval of robot 5 showed obvious jitter in the initial stage, reaching a maximum of about 0.22 seconds. The interval then decreased rapidly and stabilized. This shows that the controller appropriately delayed the update frequency when the error was large in the early stage, and accelerated the trigger response in the later stage as the error converged.

[0089] Robot 6 is similar to Robot 5. The trigger interval of this robot fluctuates greatly in the first 20 seconds, reaching a maximum of 0.27 seconds. It then stabilizes at approximately 0.02 seconds, indicating that the system is gradually entering a stable operation trend.

[0090] The maximum trigger interval of robot 7 in the initial stage was about 0.33 seconds, which then quickly shortened and stabilized, indicating that the event trigger mechanism can adaptively adjust the control update frequency according to the error change, effectively reducing unnecessary calculations and communications.

[0091] Figure 3The event-triggered mechanism employed effectively regulates the frequency of control updates. Initially, a longer trigger interval is allowed to reduce communication pressure. As the system error converges, the trigger frequency increases to ensure control accuracy. All robots experience large intervals at the beginning of the mission, but then gradually stabilize, demonstrating the control system's excellent convergence, adaptability, and communication efficiency optimization capabilities.

[0092] Figure 4 Figure 2 shows the tracking error curves of the four followers during the simulation, including position and attitude errors. As can be seen from the figure, although the four robots experience a certain degree of disturbance in the initial stage, especially during the first 20 seconds when the errors briefly fluctuate, all errors converge rapidly to near zero in a very short time, demonstrating the system's good convergence speed. In particular, for Robots 5 and 7, even with relatively large initial errors, the system is able to maintain stable control within a short period of time, demonstrating the robustness of this control strategy in the face of initial disturbances and uncertainties. The attitude errors consistently fluctuate within a very small range, further verifying the control method's efficiency and stability in attitude adjustment. Overall, this formation control method offers the advantages of rapid convergence and strong robustness, ensuring that the multi-robot system maintains a high degree of stability and coordination during dynamic formation adjustments.

[0093] To achieve the transformation of the nominal formation of a multi-robot affine formation, this paper proposes a continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on an event-triggered mechanism. A continuous adaptive integral sliding mode controller is introduced to implement motion control of the followers. The controller constructs an integral sliding surface to guide the system state toward the sliding surface. By incorporating a non-monotonic adaptive gain law designed using σ-modification technology, the sliding mode gain is adjusted online to avoid control chattering caused by fixed or excessive gains in traditional sliding mode control. Furthermore, to further reduce the controller update frequency and communication burden, an event-triggered mechanism is embedded in the control scheme. This mechanism uses the sliding surface error as a trigger condition and updates the control instructions only when the error exceeds a set threshold, thereby conserving computational and communication resources while ensuring system stability.

[0094] The above specific embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to examples, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the scope of the technical solutions of the present invention, and all of these should be included in the scope of the claims of the present invention.

Claims

1. A continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on event triggering mechanism, characterized in that the steps include: S1. Affine hierarchical control framework construction: Based on the leader-follower model, the leader and followers in the multi-robot formation are determined, and the nominal formation configuration and the communication topology of the cluster are established. The stress matrix of the nominal formation is calculated to support the subsequent affine localizability estimation; S2. Distributed Estimator Design: By constructing a distributed estimator to obtain the formation's time-varying maneuver information, we can estimate the desired trajectory of each follower. Followers only need to calculate and track their affine localized target trajectory based on the position information of their local neighbors and the leader. S3. Continuous Adaptive Integral Sliding Mode Controller Design: Combining an adaptive sliding mode control strategy with an integral sliding mode surface, this controller improves formation control accuracy and robustness, mitigating chattering. The controller utilizes a σ-modification technique to design a nonmonotonic adaptive gain law and incorporates an event-triggered mechanism to reduce unnecessary control updates.

2. The multi-robot affine formation control method based on the pilot-follower mode according to claim 1, characterized in that: In S1: In a communication topology diagram Contains a node set and an edge set A directed edge (i, j)∈ε is a path from node i to node j. The neighbor set of vertex i is given by Therefore, the adjacency matrix is defined as If (i,j)∈ε then otherwise Since there is no self-loop, Applicable. The Laplacian matrix is defined as Where D is the out-degree matrix of the node. The affine mapping of the nominal formation is expressed as: in, For nominal configuration, is the time-varying affine transformation matrix, Affine transformation is a translation vector that can be used to perform rotation, scaling, shearing, and other operations on the formation. For the formation The stress is defined as {ω ij } (i,j)∈E A scalar set of Assign to all edges. If a stress satisfies Condition, then the stress is classified as equilibrium stress and can be expressed in matrix form in represents a stress matrix that satisfies the following conditions: Rewrite the above formula as: definition and Denote the leader and follower respectively, and we get:

3. The multi-robot distributed affine formation control method based on a continuous adaptive integral sliding mode controller according to claim 2, characterized in that: In S2: The distributed error expression for the i-th item is as follows: The virtual control rates of linear velocity and angular velocity are designed as follows: To make the expression more compact, iv =(α ip ,α iθ ) T , then the velocity tracking error vector expression is as follows: v ie =e i -α iv 4. The multi-robot distributed affine formation control method based on a continuous adaptive integral sliding mode controller according to claim 3, characterized in that: In the S3: The continuous adaptive integral sliding mode manifold of the velocity tracking error vector can be described as: Among them, K i3 is a positive definite matrix. The event trigger mechanism is designed as follows: The adaptive continuous sliding mode control law is designed as follows: The non-monotonic adaptive gain law is designed as:

5. A continuous adaptive integral sliding mode leader-follower multi-robot affine formation control method based on event triggering mechanism, characterized in that: The method adopts the multi-robot formation controller construction method described in any one of claims 1-3 to design a corresponding controller implementation, and each robot to be formed is configured with an independently running formation controller. The specific process of the control method includes: each robot controller exchanges information with its neighboring nodes through a distributed communication network; the controller fuses the local state and the received neighborhood state data in real time, and solves the affine geometric relationship and the expected trajectory based on the constructed stress matrix; then, based on the estimation result, the speed tracking error is constructed and the integral sliding mode surface is generated, combined with the adaptive gain adjustment and event triggering mechanism, the continuous sliding mode control law is updated to generate a control signal with anti-interference ability, thereby driving the robot to realize affine formation action (including scaling, rotation, shearing, etc.) under local information conditions, and ensuring that the entire formation system still has convergence and dynamic consistency in an environment with disturbances and communication delays.

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