Leader following multi-robot affine formation control method based on nonsingular terminal sliding mode

Abstract

The invention discloses a leadership following multi-robot affine formation control method based on a nonsingular terminal sliding mode, relates to a formation controller combining distributed control and an affine formation method, a nonsingular terminal sliding mode controller and a dynamic event triggering mechanism, and belongs to the technical field of multi-robot control. In order to solve the problems of fixed formation and lack of flexibility in a multi-robot formation task, the method comprises the steps of S1, determining a navigator and a follower in a multi-robot formation based on a navigation follower mode, and determining the configuration of a nominal formation and a communication topological structure of a cluster; and S2, estimating formation maneuvering information through a designed distributed estimator. The follower only needs to track the expected trajectory which is affine localized by the leader; s3, designing a controller by using a nonsingular terminal sliding mode control theory so as to improve the accuracy and robustness of formation control and reduce a chattering phenomenon; the method has the advantages of high flexibility, high environmental adaptability, high control performance and the like.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-robot formation control, and in particular to a multi-robot affine formation control and a non-singular terminal sliding membrane controller. 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 leader-follower multi-robot affine formation control method based on a non-singular terminal sliding mode. Summary of the Invention

[0005] The present invention addresses the shortcomings of the prior art by providing a method for controlling affine maneuverable formations in a multi-robot system. Leveraging nonsingular terminal sliding mode control theory, the controller is designed to improve the accuracy and robustness of formation control and mitigate chattering. Incorporating rapid adaptive control technology, control parameters are adjusted in real time to accommodate external disturbances and system state changes, effectively handling uncertainties and disturbances in the multi-robot formation and improving system stability and robustness.

[0006] The present invention discloses a leader-follower multi-robot affine formation control method based on a non-singular terminal sliding mode, and the specific steps are as follows:

[0007] The steps are as follows:

[0008] S1. Construction of an affine hierarchical control framework: Based on the leader-follower model, the leader and followers in a multi-robot formation are determined, the nominal formation configuration and the cluster communication topology are determined, and the stress matrix of the nominal formation is calculated;

[0009] S2. Distributed Estimator: The formation maneuver information is estimated through a designed distributed estimator to obtain its desired reference trajectory. Followers only need to track the desired trajectory affine-localized by the leader.

[0010] S3. Non-singular terminal sliding film controller: Design a controller to improve the accuracy and robustness of formation control, reduce chattering, and combine it with fast adaptive control technology to adjust control parameters in real time.

[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 Represented. A multi-robot system consists of n robots. The communication between n 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, the stress matrix of the nominal formation is calculated using a linear matrix inequality solution method in S1. Furthermore, the specific steps of calculating the stress matrix of the nominal formation using a linear matrix inequality solution method are as follows:

[0016] 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:

[0017]

[0018] Step S1-2: yes The i-th column of

[0019] definition:

[0020]

[0021] in, Defined as:

[0022]

[0023] Let z1,…,z q ∈R m yes A set of bases;

[0024] Step S1-3: Perform singular value decomposition to obtain

[0025] Let U = [U1, U2], where U1 contains the first d+1 columns of U;

[0026] definition:

[0027]

[0028] Step S1-4, solve the following linear matrix inequality to obtain the equilibrium stress of the nominal formation

[0029]

[0030] where c1,…,c q Satisfies the linear matrix inequality:

[0031]

[0032] Step S1-5: Using the equilibrium stress of the nominal formation obtained Solve for the equilibrium stress matrix:

[0033]

[0034] Where Ω is the equilibrium stress matrix.

[0035] Compared with the prior art, the advantages of the present invention are:

[0036] 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.

[0037] 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.

[0038] 3. A distributed leader-follow multi-robot control strategy based on a non-singular terminal 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 both the robot's own model uncertainty 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.

[0039] Furthermore, this method employs a leader-follow multi-robot affine formation control strategy based on a nonsingular terminal 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

[0040] Figure 1 This is a schematic diagram of the implementation flow of the leader-follower multi-robot affine formation control method based on non-singular terminal sliding mode in this embodiment.

[0041] Figure 2 Schematic diagram of the communication topology of seven robot clusters in a specific application embodiment.

[0042] Figure 3 1 is a motion trajectory diagram of seven multi-robot systems on a plane used in a specific application embodiment.

[0043] 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

[0044] 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.

[0045] like Figure 1 As shown, the steps of the leader-follower multi-robot affine formation control strategy based on the non-singular terminal sliding mode in this embodiment include:

[0046] S1. Construction of an affine hierarchical control framework: Based on the leader-follower model, the leader and followers in a multi-robot formation are determined, the nominal formation configuration and the cluster communication topology are determined, and the stress matrix of the nominal formation is calculated;

[0047] S2. Distributed Estimator: The formation maneuver information is estimated through a designed distributed estimator to obtain its desired reference trajectory. Followers only need to track the desired trajectory affine-localized by the leader.

[0048] S3. Non-singular terminal sliding film controller: Design a controller to improve the accuracy and robustness of formation control, reduce chattering, and combine it with fast adaptive control technology to adjust control parameters in real time.

[0049] 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.

[0050] 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:

[0051]

[0052] Among them I n represents 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 3×3-dimensional matrices. The affine transformation is recorded as (A, b). The matrices A(t) and b(t) are used to realize the robot's geometric transformation actions such as translation, rotation, scaling, and shearing.

[0053] 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:

[0054]

[0055] To make the expression more compact, ij}(i,j)∈ε Written as the following stress matrix:

[0056]

[0057] Rewrite the above formula as:

[0058]

[0059] 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.

[0060] The distributed error expression for the i-th item is as follows:

[0061]

[0062] 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.

[0063] The virtual control rates of linear velocity and angular velocity are designed as follows:

[0064]

[0065] Where J represents the position transformation matrix of the i-th robot, which describes 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.

[0066] To make the expression more compact, iv =(αip ,α iθ ) T , then the velocity tracking error vector expression is as follows:

[0067] v ie =e i -α iv .

[0068] The non-singular terminal sliding mode manifold of the velocity tracking error vector can be described as:

[0069]

[0070] Among them, K i3 With K i4 is a positive definite matrix, m i ,n i ,a i ,b i is a positive odd number and satisfies

[0071] The adaptive non-singular terminal sliding mode control rate is designed as:

[0072]

[0073] in, represents the pseudo-inverse matrix of the thruster configuration matrix of the i-th robot, 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.

[0074] The adaptive law is designed as:

[0075]

[0076] 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.

[0077] The adaptive nonsingular terminal sliding mode control rate and the adaptive rate jointly construct an adaptive fast nonsingular integral terminal sliding mode control law for affine formation control of multiple robots in two-dimensional space. This control law generates precise thrust control inputs to drive the velocity tracking error to converge to zero within a finite time, while compensating for system uncertainties and external disturbances. Its core advantages lie in high precision, fast response, and strong robustness, enabling robots to perform complex formation maneuvers such as rotation, scaling, and shearing, and adapt to flexible task requirements in complex environments. The synergistic effect of nominal control terms, nonlinear feedback terms, and robust control terms, combined with the dynamic adjustment of the adaptive law, ensures the stability and adaptability of the control law, providing strong technical support for the collaborative operation of multiple robots.

[0078] 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.

[0079] Figure 3 Figure 2 shows the motion trajectories of seven robots during the simulation. The red color represents the leader robot, the blue represents the follower robots, and the black rectangles represent obstacles in the environment. As can be seen from the figure, the robots, guided by the leader robot, gradually evolve from a scattered state to a stable formation. Throughout this process, the formation undergoes multiple orderly transformations, including adjustments from a compact to an expanded formation and dynamic switching between formations at different time points (e.g., t = 17.2 s, t = 39.8 s, t = 66.0 s, and t = 89.8 s) based on mission requirements. During this process, the affine formation control strategy enables each robot to autonomously adjust its position and velocity based on the formation's changing needs, achieving coordinated movement of the entire structure. In particular, between t = 66.0 s and t = 125.0 s, the formation undergoes a gradual adjustment from an intermediate transition state to the final target formation. The robot trajectories exhibit distinct structural changes in space, demonstrating excellent formation maintenance and control capabilities. Simulation results show that the control method can effectively guide the multi-robot system to complete complex formation transformation tasks, ensure the orderliness and consistency of the formation structure, and demonstrate good coordination and dynamic adaptability.

[0080] Figure 4Figure 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.

[0081] To achieve the transformation of a multi-robot affine formation from its nominal formation, this paper proposes a leader-follower multi-robot affine formation control method based on a nonsingular terminal sliding mode. Leveraging nonsingular terminal sliding mode control theory, a controller is designed to improve the accuracy and robustness of formation control and mitigate chattering. Incorporating fast adaptive control techniques, control parameters are adjusted in real time to adapt to external disturbances and system state changes, effectively handling uncertainties and disturbances in the multi-robot formation and improving system stability and robustness.

[0082] 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 leader-follower multi-robot affine formation control method based on non-singular terminal sliding mode, characterized in that the steps include: S1. Construction of an affine hierarchical control framework: Based on the leader-follower model, the leader and followers in a multi-robot formation are determined, the nominal formation configuration and the cluster communication topology are determined, and the stress matrix of the nominal formation is calculated; S2. Distributed Estimator: The formation maneuver information is estimated through a designed distributed estimator to obtain its desired reference trajectory. Followers only need to track the desired trajectory affine-localized by the leader. S3. Non-singular terminal sliding film controller: Design a controller to improve the accuracy and robustness of formation control, reduce chattering, and combine it with fast adaptive control technology to adjust control parameters in real time.

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 Directed edges Refers to the path from node i to node j. The neighbor set of vertex i is composed of Therefore, the adjacency matrix is ​​defined as Among them if but otherwise Since there is no self-loop, Applicable. The Laplace matrix is ​​defined as The expression is as follows: The affine mapping of the nominal formation is expressed as: The affine representation of the position set of a multi-robot formation based on the leader-follower method is: The required position stack p can be obtained by affine mapping * , the expression is as follows: in, Represents the nominal configuration. The matrix are the gains associated with the rotation, scaling, and shear transformations, which control the entire formation to perform the desired maneuver with respect to r. The affine image includes all affine transformations of the nominal configuration. In addition, the desired position stack can be extracted from the affine image. 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 non-singular terminal sliding mode 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 non-singular terminal sliding mode according to claim 3, characterized in that: In the S3: The non-singular terminal sliding mode manifold of the velocity tracking error vector can be described as: Among them, K i3 With K i4 is a positive definite matrix, m i ,n i ,a i ,b i is a positive odd number and satisfies The adaptive non-singular terminal sliding mode control rate is designed as: The adaptive law is designed as:

5. A leader-follower multi-robot affine formation control method based on non-singular terminal sliding mode, 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 working multi-robot formation controller. The specific process of the control method is: the multi-robot formation controller transmits the signal to the controller of the adjacent robot through a distributed communication network; the controller synchronously fuses the local state and the received neighborhood information, uses the stress matrix to solve the affine geometric constraint relationship of the formation, updates the non-singular terminal sliding mode control law to generate an anti-interference control signal, drives the multiple robots to complete affine formation actions such as scaling, rotation or shearing, and ensures the convergence and consistency of the overall formation under disturbance and communication delay.

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