A Quantized Control Method for Multi-Robots Based on State Observer and Prescribed Finite-Time Control

Abstract

The present invention discloses a multi-robot prescribed finite time quantitative control method based on a state observer, which belongs to the technical essentials of consistency control of a multi-robot system. A model of a leader and a follower is established based on the multi-robot system, and a prescribed finite time control criterion is determined; a state observer is determined, and new leader information is reconstructed for each follower using switching topology graph theory; consistency error is determined by switching topological relationships, and an obstacle Lyapunov function is constructed according to the designed performance constraints; a prescribed finite time virtual controller is constructed, a prescribed finite time adaptive command filter is determined, and an adaptive rate is established; a prescribed finite time H∞ theorem is proposed, and a prescribed finite time H∞ controller and a hysteresis quantizer are constructed. The method solves the problems of limited control accuracy, slow convergence speed, uncertainty in model parameters, external disturbance influence, and inability to accurately estimate convergence time when the current control method is applied to a multi-robot system.

Description

Technical Field

[0001] The invention relates to the technical essentials of consistency control of a multi-robot system, and in particular to a multi-robot prescribed finite time quantitative control method based on a state observer. Background Art

[0002] With the increase in actual working conditions, it is difficult for a single robot to meet the growing work tasks, and the research on multi-robot collaborative control has received widespread attention. Therefore, many methods and technologies have been proposed for the consistency tracking control of multi-robot systems. Some of these methods are based on traditional control theories, such as PID control and adaptive control, while others use modern control theories, such as fuzzy control and neural network control. The goal of consistency tracking control is to require multiple robots to achieve consistency in state quantities such as position, velocity, and acceleration through information sharing and interaction during the control process. The ultimate goal of this control method is to promote the robot team as a whole to perform tasks in the most optimized way, improve overall efficiency and accuracy of task completion. The inverse design method of asymmetric obstacle Lyapunov function is a simple and efficient controller construction tool. It is not only easy to implement itself, but also can be flexibly integrated with many other advanced algorithms. For example, the introduction of performance function in asymmetric obstacle Lyapunov function is widely used, which limits the trajectory tracking errors of multiple robots within a predetermined boundary, so that the dynamic response of multiple robots meets the expected control performance.

[0003] In the past, most formation consistency control strategies required obtaining the leader's acceleration information and other states. However, in some cases, it is relatively easier to obtain accurate velocity information than to obtain accurate acceleration information, and in practical applications, the controlled object may be disturbed by external disturbances. Undoubtedly, inaccurate leader acceleration measurements will affect the final formation control effect. In order to solve the above problems, the present invention proposes a finite-time leader state estimation observer, which reconstructs new leader information and assigns it to each follower, and no longer relies on uncertain acceleration measurements.

[0004] At present, with the continuous improvement of control performance requirements, the backstepping design method has been widely promoted and applied as an important control design tool. However, the existing backstepping method is designed to calculate the virtual control law in a differential manner, which leads to repeated derivation of the virtual control rate, and as the system order increases, computational complexity problems will inevitably arise, and even become difficult to solve. Dynamic surface control technology is generally used in existing research to solve this problem. However, this method usually requires the assumption of the upper bound of the second-order derivative of the desired signal, which undoubtedly increases the uncertainty of the system. Therefore, the introduction of adaptive rate is used to estimate the unknown fuzzy upper bound. The present invention proposes an adaptive command filter based on a specified finite time, which can well limit the filtering error within the constraint boundary within the specified finite time.

[0005] In order to achieve high-performance tracking control of multi-robot systems, nonlinear terms and external environmental disturbances are important factors affecting control performance. Previous studies have explored the use of neural networks as a solution, including self-constructed neural networks and fuzzy neural networks. However, the use of pure neural networks will increase the computational burden. Predefined time H ∞ The control method is widely used in various control systems due to its anti-interference characteristics. This method can not only effectively handle model uncertainty and external disturbances, ensure that the system output remains within the set upper limit, but also achieve rapid convergence of the system internal state within a preset time. ∞ In the controller, since the L2 gain of the system itself is less than or equal to the given γ value, it has good robustness and good disturbance suppression effect. However, the convergence time of the predefined time theory still has some ambiguity. Therefore, a predefined finite time H based on the time scale change strategy is designed. ∞ Control method. The purpose of this method is to achieve a stable state of the system within a specified finite time without being affected by the initial conditions.

[0006] Therefore, through the above analysis, the present invention combines the state observer, dynamic surface control technology and the specified finite time H ∞ A multi-robot finite-time adaptive H ∞ Quantitative control method. The controller designed by this method is relatively simple, and the system has high stability under the premise of ensuring that the tracking error converges quickly within a specified finite time. The convergence time of the system can directly give an accurate time without tedious parameter calculation. Summary of the invention

[0007] In view of the above-mentioned deficiencies in the prior art, the present invention provides a multi-robot finite-time quantitative control method based on a state observer.

[0008] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0009] A multi-robot finite-time quantitative control method based on a state observer comprises the following steps:

[0010] S1. Establish the model of leader and follower based on the multi-robot system, including kinematic and dynamic models, and determine the finite time control criteria;

[0011] S2, determining the state observers of the leader and the follower respectively, and reconstructing new leader information for each follower using switching topology graph theory without requiring a leading measurement of acceleration information;

[0012] S3, determine the consistency error by switching the topological relationship, design the performance constraint function and construct the obstacle Lyapunov function;

[0013] S4, constructing a prescribed finite time virtual controller, determining a prescribed finite time adaptive command filter and establishing an adaptive rate;

[0014] S5, based on limited time H ∞ Theorem construction stipulates that the finite time H ∞ The controller and the hysteresis quantizer make the position error and the velocity error converge to the neighborhood of zero within a specified finite time, thus achieving consistent tracking of the multi-robot system.

[0015] Furthermore, the kinematic model of the follower robot in S1 is expressed as:

[0016]

[0017] In the formula, q i,1 ,q i,2 represents the joint position information of the i-th dual-joint robot, represents the joint velocity information of the i-th dual-joint robot, represents the joint acceleration information of the i-th dual-joint robot, C i,1a , C i,2a represents the Coriolis force of the i-th dual-joint robot, C i,1b , C i,2b represents the centrifugal force of the i-th dual-joint robot; M i,1a , M i,1b , M i,2a and M i,2b represents the symmetric positive definite inertia matrix of the i-th dual-joint robot; G i,1 and G i,2 represents the gravity of the i-th dual-joint robot; Q(u i,1 ) and Q(u i,2) is the quantitative control input signal of the i-th dual-joint robot; τ di.1 and τ di,2 is the external disturbance suffered by the i-th dual-joint robot; u i,1 and u i,2 is the control input of the i-th dual-joint robot; subscripts 1 and 2 are the joint numbers of the i-th dual-joint robot, and subscripts a and b are the ordinal numbers of the i-th dual-joint robot;

[0018] The dynamic model of the follower robot is expressed as:

[0019]

[0020] In the formula, Expressed as the velocity of the kth joint of the ith robot, x n,k It is expressed as the velocity of the kth joint of the ith robot; is the acceleration of the kth joint of the ith robot; f σi,k (x i,k ) is represented as the set of model uncertainties of the kth joint of the ith robot; Q(u i,k ) is represented by the control signal after input quantization; τ di,k It is represented by the external time-varying disturbance on the kth joint of the ith robot; M i,k It is represented as the moment of inertia of the motor and connecting rod of the kth joint of the ith robot; u i,k Expressed as a specified finite time H ∞ Control signal;

[0021] The leader robot dynamics and kinematics model is expressed as:

[0022]

[0023] In the formula, is the velocity of the kth joint of the leader robot; v 0,k is the velocity of the kth joint of the leader robot, is the acceleration of the kth joint of the leader robot; u 0,k is the control input of the kth joint of the leader robot.

[0024] Furthermore, the S2 specifically includes the following steps:

[0025] S21. Designing graph theory based on the communication relationship of the leader's state observer and the subsequent leader's state observer;

[0026] S22, based on the graph theory designed in S21, use the differential equations to solve the estimated value of the leader's acceleration information;

[0027] S23. Determine a state observer of a finite-time cascade leader according to the estimated value of the leader acceleration information obtained in S22, and redesign state information for each follower.

[0028] Furthermore, the specific method of using the differential equation group to solve the estimated value of the leader's acceleration information in S22 is:

[0029]

[0030] In the formula, β 1i,k , β 2i,k , β 3i,k , β 4i,k , Ξ is the constant term in the finite-time leader observer, x 0,k is the position of the kth joint of the leader robot; z xi,k , z vi,k , z ai,k is the observer adaptation term for the kth joint of the i-th prescribed finite-time leader, First-order derivative of the adaptive term; z ai,k (t0) is z ai,k The value at the initial moment, a 0,k (t0) is the initial acceleration of the kth joint of the leader robot, μ is the time-varying function, and t0 is the initial moment.

[0031] Furthermore, the state information of each follower is redesigned in S23 as follows:

[0032]

[0033] In the formula, β 5i,k , β 6i,k , β 7i,k , β 8i,k , β 9i,k and β 10i,k is a constant term, μ is a time-varying function; i≠j is an element in a directed graph and a ij represents the communication relationship between the ith follower robot and the jth follower robot; b i represents the communication relationship between the ith follower robot and the leader robot; is the position estimation information of the i-th and j-th follower robots, is the velocity estimation information of the i-th and j-th follower robots, is the acceleration estimation information of the i-th and j-th follower robots, Its first-order differential, x 0,k , v 0,k is the position and velocity of the leader robot.

[0034] Furthermore, the consistency error in S3 includes:

[0035] The position consistency error of the kth joint of the ith robot is expressed as:

[0036]

[0037] In the formula, e xi,k Expressed as position consistency error, x i,k Represented as the position of the ith follower robot, x j,k Represented as the position of the j-th follower robot;

[0038] The velocity consistency error of the kth joint of the ith robot is expressed as:

[0039]

[0040] In the formula, e vi,k Expressed as velocity consistency error, v i,k Denotes the velocity of the ith follower robot, v j,k Denotes the speed of the j-th follower robot;

[0041] The acceleration consistency error of the kth joint of the ith robot is expressed as:

[0042]

[0043] In the formula, e ai,k Expressed as acceleration consistency error, a i,k Expressed as the acceleration of the ith follower robot, a j,k Expressed as the acceleration of the jth follower robot;

[0044] The performance constraint function expression is:

[0045]

[0046] In the formula, e xi,k is the position consistency error, is the lower bound constraint of the performance function of the kth joint of the ith robot; is the initial value of the lower bound of the performance function of the kth joint of the ith robot; is the final value of the lower bound of the performance function of the kth joint of the ith robot; The upper boundary constraint of the performance function of the kth joint of the ith robot; is the initial value of the upper boundary of the performance function of the kth joint of the ith robot; is the final value of the upper boundary of the performance function of the kth joint of the ith robot; γ is the adjustment parameter of the performance function; t is the time of the control process; T is the convergence time;

[0047] The barrier Lyapunov function expression in S3 is:

[0048]

[0049] In the formula, q(e xi,k ) is the judgment function, that is, e xi,k >0, q(e xi,k )=1;e xi,k When ≤0, q(e xi,k )=0.

[0050] Furthermore, the finite time virtual controller specified in S4 is expressed as:

[0051]

[0052] In the formula, e xi,k is the position consistency error, β 11i.k , β 12i.k , κ0 is the constant term in the virtual control rate, μ is the time-varying function; The upper boundary constraint of the performance function of the kth joint of the ith robot, is its first-order differential; is the lower bound constraint of the performance function of the kth joint of the ith robot, is its first-order differential; Γ ai,k , Γ bi,k is the virtual control rate error term.

[0053] Furthermore, the S5 specifically includes the following steps:

[0054] S51, determining the specified finite time H according to the model of the leader and the follower in step S1 and the specified finite time control criterion ∞ theorem;

[0055] S52, in order to make the position error and the velocity error converge to the neighborhood of zero within a specified finite time, based on the specified finite time H obtained in S51 ∞ Theorem construction stipulates that the finite time H ∞ Controller;

[0056] S53, the constructed specified finite time H ∞ The output signal of the controller is processed by a hysteresis quantizer to achieve consistent tracking of the multi-robot system.

[0057] Furthermore, the limited time H is specified in S52. ∞Control signal u i,k It is expressed as:

[0058]

[0059] In the formula, β 15i,k , β 16i,k is a constant term, is the virtual control rate derivative after filtering, M i,k is a symmetric positive definite matrix, μ is a time-varying function; i≠j is an element in a directed graph and a ij represents the communication relationship between the ith follower robot and the jth follower robot, N is the number of follower robots; b i represents the communication relationship between the ith follower robot and the leader robot, Φ i,k is the performance representation vector; is the acceleration of the leader robot, is the acceleration of the jth follower robot; e xi,k is the position consistency error; Γ ai,k , Γ bi,k is the virtual control rate error term.

[0060] Furthermore, in S53, the specified limited time H constructed ∞ Control signal u i,k The specific way of processing through the hysteresis quantizer is:

[0061]

[0062] In the formula, u i,k To specify a limited time H ∞ Control signal, is its first-order differential; sgn() represents a step function; a di,k Expressed as the basic parameter of the quantization process; u ni,k Input quantization parameters and satisfy the conditions n=1,…,12,i=1,…,4,k=1,2,θ i,k is the transmission rate of the communication channel from the controller to the multi-robot system, t - Indicates the previous moment of the control process.

[0063] The present invention has the following beneficial effects:

[0064] The present invention determines a cascade leader state estimation observer based on a switching communication topology with a specified finite time, which is used to approximate and transmit the leader's state information without leader acceleration measurement. Compared with the existing technology, the observer can converge within the specified finite time, has strong robustness, and obtains a faster convergence speed.

[0065] A multi-robot prescribed finite-time adaptive H-based state observer is proposed for the consensus control of multi-robot systems. ∞ Quantized control method. Compared with the existing technology, the filter error of the filter can achieve fast convergence within a specified finite time. The introduction of adaptive rate solves the fuzzy problem of unknown upper bound of the derivative of the virtual control law. The H obtained by the specified finite time control criterion proposed by the present invention ∞ The controller is obviously simpler. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 The present invention is a schematic flow chart of a multi-robot finite time quantitative control method based on a state observer.

[0067] Figure 2 This is a switching topology communication relationship diagram of a multi-robot system according to an embodiment of the present invention.

[0068] Figure 3 Schematic diagram of position error of robot joint 1 according to an embodiment of the present invention.

[0069] Figure 4 Schematic diagram of position error of robot joint 2 according to an embodiment of the present invention.

[0070] Figure 5 Schematic diagram of the velocity error of the robot joint 1 according to an embodiment of the present invention.

[0071] Figure 6 Schematic diagram of the velocity error of the robot joint 2 according to an embodiment of the present invention.

[0072] Figure 7 Schematic diagram of position observation error of robot joint 1 according to an embodiment of the present invention.

[0073] Figure 8 Schematic diagram of position observation error of robot joint 2 according to an embodiment of the present invention.

[0074] Fig. 9 Schematic diagram of the speed observation error of the robot joint 1 according to an embodiment of the present invention.

[0075] Fig.10 Schematic diagram of the speed observation error of the robot joint 2 according to an embodiment of the present invention.

[0076] Fig.11 Schematic diagram of acceleration observation error of robot joint 1 according to an embodiment of the present invention.

[0077] Fig.12 Schematic diagram of acceleration observation error of robot joint 2 according to an embodiment of the present invention.

[0078] Fig.13Schematic diagram of filtering error of robot joint 1 according to an embodiment of the present invention.

[0079] Fig.14 Schematic diagram of filtering error of robot joint 2 according to an embodiment of the present invention.

[0080] Fig.15 Schematic diagram of control input of robot joint 1 according to an embodiment of the present invention.

[0081] Fig.16 Schematic diagram of control input of robot joint 2 according to an embodiment of the present invention. DETAILED DESCRIPTION

[0082] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0083] A multi-robot finite-time quantitative control method based on state observer, such as Figure 1 As shown, the following steps are included:

[0084] S1. Establish the model of leader and follower based on the multi-robot system, including kinematic and dynamic models, and determine the finite time control criteria;

[0085] Consider a multi-robot system consisting of a leader robot and N follower robots. The kinematic equation of the i-th robot under disturbance is expressed as:

[0086]

[0087] Where:

[0088]

[0089] In the formula, q i,1 , q i,2 , and Respectively represent the joint position, velocity and acceleration information of the i-th dual-joint robot. i,1a , C i,1b , C i,2a and C i,2b represents the Coriolis force and centrifugal force of the i-th dual-joint robot; M i,1a , M i,1b , M i,2a and M i,2brepresents the symmetric positive definite inertia matrix of the i-th dual-joint robot; G i,1 and G i,2 represents the gravity of the i-th dual-joint robot; Q(u i,1 ) and Q(u i,2 ) is the quantitative control input signal of the i-th dual-joint robot; τ di.1 and τ di,2 is the external disturbance suffered by the i-th dual-joint robot; u i,1 and u i,2 is the control input of the i-th dual-joint robot; subscripts 1 and 2 are the joint numbers of the i-th dual-joint robot, and subscripts a and b are the ordinal numbers of the i-th dual-joint robot.

[0090] The dynamic equation of the ith follower robot is expressed as an uncertain second-order nonlinear system, which is obtained by the following differential equation:

[0091]

[0092] In the formula, Expressed as the velocity of the kth joint of the ith robot, x n,k It is expressed as the velocity of the kth joint of the ith robot; Its first-order differential, f σi,k (x i,k ) is represented as the set of model uncertainties of the kth joint of the ith robot; Q(u i,k ) is represented by the control signal after input quantization; τ di,k It is represented by the external time-varying disturbance on the kth joint of the ith robot; M i,k It is represented as the moment of inertia of the motor and connecting rod of the kth joint of the ith robot; u i,k Expressed as a specified finite time H ∞ Control signal.

[0093] Kinematic and dynamic models of leaders based on multi-robot systems:

[0094]

[0095] In the formula, is the velocity of the kth joint of the leader robot; v 0,k is the velocity of the kth joint of the leader robot, is its first-order differential; u 0,k is the control input of the kth joint of the leader robot.

[0096] Determine the specified finite time control criterion:

[0097] Consider a monotonically increasing function:

[0098]

[0099] Where h is an adjustable parameter, T>0, t∈R n If a continuously differentiable function V i,k (t) where c1 and c2 are both positive constants and their first-order partial derivatives are in [0,T]→[0,+∞) satisfy:

[0100]

[0101] Inequality (6) can be transformed into the following inequality:

[0102]

[0103] The solution of differential inequality (7) is given as follows:

[0104]

[0105] Where V i,k (0) is its initial value.

[0106] From (5), we can get the following inequality:

[0107]

[0108] According to the properties of the function ln(Tt), we have:

[0109]

[0110] S2, determining the state observers of the leader and the follower respectively, and reconstructing new leader information for each follower using switching topology graph theory without requiring a leading measurement of acceleration information;

[0111] In this embodiment, the following steps are specifically included:

[0112] S21. Designing graph theory based on the communication relationship of the leader's state observer and the subsequent leader's state observer;

[0113] In multi-robot systems, information transmission mechanisms generally rely on directed and undirected graphs in graph theory as theoretical foundations. Directed graphs represent the key characteristics of unidirectional information flow between systems, while undirected graphs represent the important theoretical representation of bidirectional information flow. In the above formula, Z=[z1,…,z n ] represents the vertex set, ξ∈Z×Z represents the edge set (z i ,z j ), N represents the number of robots, the directed graph A = [a ij ]∈Rn The incident adjacency matrix of is defined as G, where a ij =1. ij =1, j≠i means that the i-th follower robot can receive messages from the j-th follower robot; otherwise, a ij = 0. The exponential correlation matrix is ​​defined as E = diag (e i )∈R n ,in Denotes the communication relationship of the follower robots of node i. Define L'=EA as the Laplace matrix. The adjacency matrix of the leader robot is defined as B=diag(b i )∈R n , where b i =1 indicates that the leader robot can connect to the ith follower robot; otherwise, there is no communication between the leader robot and the ith follower robot, defined as b i =0.

[0114] S22, based on the graph theory designed in S21, use the system of differential equations to solve the estimated value of the leader's acceleration;

[0115] Based on the graph theory knowledge of the multi-robot system mentioned above, the following differential equation is proposed to solve the estimated value of the leader's acceleration information:

[0116]

[0117] In the formula, β 1i,k ,β 2i,k ,β 3i,k ,β 4i,k ,Ξis a constant term in the finite time cascade leader observer, and forms a matrix and Satisfies Hurwitz's theorem.

[0118] S23, determining the state observer of the leader of the specified finite time cascade according to the estimated value of the leader acceleration information obtained in S22, and redesigning the state information for each follower, expressed as:

[0119]

[0120] In the formula, x 0,k =[x 0,1 ,…,x 0,r ] T , v 0,k =[x 0,1 ,…,x 0,r ] T , a 0,k =[a 0,1,…,a 0,r ] T and i,j=1,…,4,,r=1,2. In the case of establishing a direct communication link with the leader, z ai,k is considered a 0,k The estimated value of . 5i,k , β 6i,k , β 7i,k , β 8i,k , β 9i,k and β 10i,k Satisfies the condition of being greater than 0.

[0121] S3, determine the consistency error by switching the topological relationship, design the performance constraint function and construct the obstacle Lyapunov function;

[0122] In this embodiment, the consistency error of each robot is:

[0123] The position consistency error of the kth joint of the ith robot is expressed as:

[0124]

[0125] In the formula, e xi,k Expressed as position consistency error, x i,k Represented as the position of the ith follower robot, x j,k Denoted as the position of the j-th follower robot.

[0126] The velocity consistency error of the kth joint of the ith robot is expressed as:

[0127]

[0128] In the formula, e vi,k Expressed as velocity consistency error, v i,k Denotes the velocity of the ith follower robot, v j,k Denotes the speed of the j-th follower robot;

[0129] The acceleration consistency error of the kth joint of the ith robot is expressed as:

[0130]

[0131] In the formula, e ai,k Expressed as acceleration consistency error, a i,k Expressed as the acceleration of the ith follower robot, a j,k Denoted as the acceleration of the j-th follower robot.

[0132] Since the consistency error of the kth joint of the ith robot is introduced, the present invention designs the following performance constraints to limit it:

[0133]

[0134] To ensure that the constructed barrier Lyapunov function is well - constrained, the performance function is expressed as:

[0135]

[0136] According to the consensus error of multi - robots and the designed performance constraints, the expression of the log - type barrier Lyapunov function in S3 is:

[0137]

[0138] In the formula, q(e xi,k ) is defined as follows:

[0139]

[0140] S4. Construct a prescribed finite - time virtual controller, determine a prescribed finite - time adaptive command filter and establish an adaptation rate;

[0141] In this embodiment, first, combine the log - type barrier Lyapunov function proposed in S3, take its derivative and introduce an error variable to provide a theoretical basis for the subsequent virtual control rate design:

[0142] Lemma 1: For any performance boundary parameter, if there exists a steady - state error E in the compact set {Ω: - ω < E < ω}, then

[0143]

[0144] The expression of the first - order differential of the Lyapunov function (19) in S3 is:

[0145]

[0146] The following error variables are introduced into the virtual controller:

[0147]

[0148] Substitute into formula (14) to obtain the following formula:

[0149]

[0150] Substitute (24) into (22) to get:

[0151]

[0152] According to the above formula (25) and the specified finite time control criterion, the virtual control rate expression in S4 is:

[0153]

[0154] In the formula, κ0, β 11i,k , β 12i,k Both greater than 0 and

[0155] Since the virtual control rate is introduced, the Lyapunov function can be further determined as:

[0156] Substituting (26) into (25), we can obtain, according to Lemma 1:

[0157]

[0158] In the formula,

[0159] Considering the theoretical basis of the adaptive command filter with a finite time specification, the following lemma and error variables are given:

[0160] Lemma 2: If the condition If it holds, then there must exist constants δ>0 and γ∈R, where w=0.2785. Lemma 3: For constants ζ>1, (uX) ζ ≤v ζ -x ζ Or for a constant ζ>0, The premise is that the following conditions are met: χ>0,υ≤χ.

[0161] A new error variable is determined and the filtering error is expressed as follows:

[0162]

[0163] In the formula, Defined as the signal before filtering, is defined as the filtered signal. In addition, the initial value satisfies the condition Determine the estimated error as follows:

[0164]

[0165] In the formula, It is considered to be i,k The estimated value of .

[0166] According to the above lemma and error variables, the expression of the finite time adaptive command filter specified in S4 is:

[0167]

[0168] In the formula, Its first-order differential, β 13i.k , β 14i.k , σ c is a constant term. Due to the introduction of the specified finite time adaptive command filter, the Lyapunov function is determined as follows:

[0169]

[0170] By taking the derivative of formula (31), we can get:

[0171]

[0172] Substituting formula (30) into formula (32) and according to Lemma 2, we can obtain:

[0173]

[0174] According to Lemma 3, we can get Substituting into formula (33):

[0175]

[0176] Where p1 = min{β 11i,k ,β 13i,k}, p2 = min{β 12i,k ,β 14i,k}and

[0177] S5, based on limited time H ∞ Theorem construction stipulates that the finite time H ∞ The controller and the hysteresis quantizer make the position error and the velocity error converge to the neighborhood of zero within a specified finite time, thus achieving consistent tracking of the multi-robot system.

[0178] In this embodiment, the following steps are specifically included:

[0179] S51, determining the specified finite time H according to the model of the leader and the follower in step S1 and the specified finite time control criterion ∞ theorem;

[0180] According to the dynamic equation of the multi-robot system, the following finite time H is determined: ∞ Theorem 1:

[0181] For multi-robot systems:

[0182]

[0183] In the formula, x i,k ∈Rn is the state vector, Q(u i,k )∈R n is the quantized control input vector, Φ i,k ∈R n is the performance characterization vector, f σi,k (x i,k )∈R n is an uncertain set vector, g(x i,k ) is the inertia matrix and h i,k (x i,k ) is the error vector. The dynamic compensator is:

[0184]

[0185] The controller is stable for a specified time if the following conditions are met:

[0186] 1. If the multi-robot systems in (35) and (36) are globally finite-time stable, they must satisfy the condition f σi,k (x i,k )=0;

[0187] 2. Given conditions If the output Φ i,k By σi,k (x i,k ) is generated and the initial value x i,k (T) satisfies, then the system gain L2 in (35) and (36) does not exceed

[0188]

[0189] For all t1>t0 and all uncertain perturbations f σi,k (t) is established.

[0190] According to the specified finite time control criterion, determine the following specified finite time H ∞ Theorem 2:

[0191] If in the neighborhood H∈R of the nonlinear system (35) n There exists a condition that satisfies B>0,β ai,k >0 and β bi,k Lyapunov function V > 0 i,k (x), then it follows that:

[0192] 1. V i,k (x) in the positive region H;

[0193] 2.

[0194] If the gain L2 of the system (35) does not exceed or H∈R n and V i,k (x) is radially unbounded and satisfies the condition V when ||x||→+∞ i,k (x)→+∞, the system is considered to be partially finite-time stable at the origin.

[0195] Proof: If the above conditions are met, σi,k (x i,k )=0, then Inequality 2 in Theorem 2 can be expressed as:

[0196]

[0197] If the condition f is met σi,k (x i,k )≠0,V i,k (x)>0, then Inequality 2 in Theorem 2 can be expressed as:

[0198]

[0199] In short, the system gain L2 satisfies the condition that it does not exceed And the system is stable within a specified limited time.

[0200] S52, in order to make the position error and the velocity error converge to the neighborhood of zero within a specified finite time, based on the specified finite time H obtained in S51 ∞ Theorem construction stipulates that the finite time H ∞ Controller:

[0201]

[0202] Proof: The proof that the stable phase is reached within a specified finite time is as follows.

[0203] Determine the Lyapunov function as follows:

[0204]

[0205] By taking the derivative of formula (41), we can get:

[0206]

[0207] Substituting formula (24) into the equation, we can obtain:

[0208]

[0209] Substituting formula (40) into the equation, we can obtain:

[0210]

[0211] Substitute formula (44) into the following inequality:

[0212]

[0213] Where η1=min{p1,β 15i,k} and η2=min{p2,β 16i,k}.

[0214] S53, the constructed specified finite time H ∞ The output signal of the controller is processed by a hysteresis quantizer to achieve consistent tracking of the multi-robot system.

[0215] Due to the introduction of the above-mentioned finite time actual controller, the finite time H is specified ∞ Control signal u i,k It needs to be processed by the following hysteresis quantizer:

[0216]

[0217] In the formula, u i,k To specify a limited time H ∞ Control signal, is its first-order differential; sgn() represents a step function; a di,k Expressed as the basic parameter of the quantization process; u ni,k Input quantization parameters and satisfy the conditions n=1,…,12,i=1,…,4,k=1,2,θ i,k is the transmission rate of the communication channel from the controller to the multi-robot system, t - Indicates the previous moment of the control process.

[0218] Based on the above method, a multi-robot system module, a specified finite time leader state observer module, a multi-robot formation relative position constraint module, a virtual control rate module, a specified finite time adaptive command filter module, a specified finite time H ∞ A control system of a controller module and an input quantization mechanism module, wherein:

[0219] The multi-robot system module is used to obtain the quantized input control signal, and calculate the position, velocity and acceleration of the follower according to the actual control signal, and then adjust the follower's own state according to the obtained position, velocity and acceleration information.

[0220] The finite-time leader state observer module is used to obtain the position and velocity information input by the leader, and reconstruct the position, velocity and acceleration of each follower according to the obtained position and velocity information.

[0221] The relative position constraint module of the multi-robot formation is used to obtain the position, velocity and acceleration information input by the leader state observer module and the position, velocity and acceleration information of the multi-robot system module within a specified finite time, design the relative position constraints between the leader and the follower and between the followers, set the expected position values ​​between the leader and the follower and between the followers, and calculate the position tracking errors between the leader and the follower and between the followers according to the expected position values ​​between the leader and the follower and between the followers.

[0222] The virtual controller module is used to obtain the position tracking error between the leader and follower and between followers of the relative position constraint module of the multi-robot formation, and design the virtual controller according to the relative position constraint conditions between the leader and follower and between followers of the relative position constraint module of the multi-robot formation.

[0223] It is specified that the finite time adaptive command filter module is used to obtain the control signal of the virtual controller, and calculate the estimated value of the adaptive rate of the adaptive command filter according to the control signal of the virtual controller.

[0224] Limited time limit ∞ The controller module is used to obtain the control signal of the specified finite time adaptive command filter module and the position, velocity and acceleration of the multi-robot system module, and to design the specified finite time H according to the control signal of the specified finite time adaptive command filter module and the position, velocity and acceleration of the multi-robot system module. ∞ Controller.

[0225] The input quantization mechanism module is used to obtain the specified finite time H ∞ The control signal of the controller module is ∞ The control signal is designed to input a quantization mechanism to convert the continuous control signal sent by the controller to the actuator into a piecewise step signal.

[0226] The input end of the multi-robot system module is connected to the output end of the input quantization mechanism module, and the input end of the input quantization mechanism module is connected to the specified finite time H ∞ The output terminal of the controller module is connected to the specified time H ∞ The input end of the controller module is connected to the output end of the multi-robot system module, and a limited time H is specified. ∞ The input end of the controller module is connected to the output end of the specified finite time adaptive command filter module. The specified finite time H ∞The input end of the controller module is connected to the output end of the specified finite time leader state observer module, the input end of the specified finite time adaptive command filter module is connected to the output end of the virtual controller module, the input end of the virtual controller module is connected to the output end of the relative position constraint module of the multi-robot formation, the input end of the relative position constraint module of the multi-robot formation is connected to the output end of the specified finite time leader state observer module, and the input end of the relative position constraint module of the multi-robot formation is connected to the output end of the multi-robot system module.

[0227] In order to verify the effectiveness of the method of the present invention, simulation experiments were carried out using the following model. Specifically, five dual-joint robot systems were simulated by selecting model parameters to verify the designed multi-robot finite-time adaptive H based on state observer. ∞ The effectiveness of the quantitative control method is shown in Table 1. The parameters of the multi-robot system are shown in Table 1, the controller parameters are shown in Table 2, and the other parameters are shown in Table 3.

[0228] Table 1 Multi-robot system model parameters

[0229]

[0230] In Table 1, J 1,1 represents the moment of inertia of the first joint of the first robot, J 2,1 represents the moment of inertia of the first joint of the second robot, J 3,1 represents the moment of inertia of the first joint of the third robot, J 4,1 represents the moment of inertia of the first joint of the fourth robot, J 1,2 represents the moment of inertia of the second joint of the first robot, J 2,2 represents the moment of inertia of the second joint of the second robot, J 3,2 represents the moment of inertia of the second joint of the third robot, J 4,2 represents the moment of inertia of the second joint of the fourth robot, m 1,1 Indicates the weight of the connecting rod of the first joint of the first robot, m 2,1 Indicates the weight of the connecting rod of the first joint of the second robot, m 3,1 Indicates the weight of the connecting rod of the first joint of the third robot, m 4,1 Indicates the weight of the connecting rod of the first joint of the fourth robot, m 1,2 Indicates the weight of the connecting rod of the second joint of the first robot, m 2,2 Indicates the weight of the connecting rod of the second joint of the second robot, m 3,2 Indicates the weight of the connecting rod of the second joint of the third robot, m 4,2 represents the weight of the connecting rod of the second joint of the fourth robot, l1,1 represents the weight of the connecting rod of the first joint of the first robot, l 2,1 represents the weight of the connecting rod of the first joint of the second robot, l 3,1 represents the weight of the connecting rod of the first joint of the third robot, l 4,1 represents the weight of the connecting rod of the first joint of the fourth robot, l 1,2 represents the weight of the connecting rod of the second joint of the first robot, l 2,2 represents the weight of the connecting rod of the second joint of the second robot, l 3,2 represents the weight of the connecting rod of the second joint of the third robot, l 4,2 represents the weight of the connecting rod of the second joint of the fourth robot, τ d1,1 represents the disturbance variable of the first joint of the first robot, τ d2,1 represents the disturbance variable of the first joint of the second robot, τ d3,1 represents the disturbance variable of the first joint of the third robot, τ d4,1 represents the disturbance variable of the first joint of the fourth robot, τ d1,2 represents the disturbance variable of the second joint of the first robot, τ d2,2 represents the disturbance variable of the second joint of the second robot, τ d3,2 represents the disturbance variable of the second joint of the third robot, τ d4,2 represents the disturbance variable of the second joint of the fourth robot, and g represents the gravity.

[0231] Table 2 Controller parameters

[0232]

[0233]

[0234] In Table 2, T represents the time of the time scale function setting, h represents the exponential power of the time scale function setting, t represents the time of the control process, κ0, β 11i,k and β 12i,k represents the constant term in the virtual control rate, and β 15i,k and β 16i,k A constant term representing the actual control rate.

[0235] Table 3 Other parameters

[0236]

[0237] In Table 3, β 1i,k , β 2i,k , β 3i,k , β 4i,k ,Ξ,β5i,k , β 6i,k , β 7i,k , β 8i,k , β 9i,k and β 10i,k represents the constant term in the finite-time cascade leader observer, γ represents the tuning parameter of the performance function, represents the initial value of the lower bound of the performance function, represents the final value of the lower bound of the performance function, represents the initial value of the upper bound of the performance function, represents the final value of the upper boundary of the performance function, β 13i,k , β 14i,k and σ c Represents the constant term that specifies the finite time adaptive command filter. 1,1 represents the quantized density measure of the first joint of the first robot, ∏ 1,2 represents the quantized density measure of the second joint of the first robot, ∏ 2,1 represents the quantized density measure of the first joint of the second robot, ∏ 2,2 represents the quantized density measure of the second joint of the second robot, ∏ 3,1 represents the quantized density measure of the first joint of the third robot, ∏ 3,2 represents the quantized density measure of the second joint of the third robot, ∏ 4,1 represents the quantized density measure of the first joint of the fourth robot, ∏ 4,2 represents the quantized density measure of the second joint of the fourth robot, ρ 1i,k , ρ 2i,k , ρ 3i,k , ρ 4i,k , ρ 5i,k , ρ 6i,k , ρ 7i,k , ρ 8i,k , ρ 9i,k , ρ 10i,k , ρ 11i,k and ρ 12i,k represents the number of iterations of the quantization process, α di,k Represents the basic parameters of the quantization process.

[0238] In this simulation experiment, the initial position of the multi-robot system is selected as [x d1,1 ,x d1,2 ,x d2,1 ,x d2,2 ,x d3,1 ,x d3,2 ,x d4,1 ,x d4,2 ]=[0.13,1.7,0.2,2,-0.1,-1.7,-0.2,-2]T , and the initial velocities are all set to 0. The expected trajectory of the multi-robot system is set to s d1,1 = sin(0.5t), s d1,2 =2cos(0.5t),s d2,1 =0.5cos(0.5t),s d2,2 = -sin(0.5t), s d3,1 =-0.25sin(0.5t) and s d3,2 =-0.5cos(0.5t). Where x d1,1 represents the initial value of the first joint of the first robot, x d1,2 represents the initial value of the second joint of the first robot, x d2,1 represents the initial value of the first joint of the second robot, x d2,2 represents the initial value of the second joint of the second robot, x d3,1 represents the initial value of the first joint of the third robot, x d3,2 represents the initial value of the second joint of the third robot, x d4,1 represents the initial value of the first joint of the fourth robot, x d4,2 Indicates the initial value of the second joint of the fourth robot. d1,1 represents the expected position of the first joint of all robots, s d1,2 represents the expected position of the second joint of all robots, s d2,1 represents the expected velocity of the first joint of all robots, s d2,2 represents the expected velocity of the second joint of all robots, s d3,1 represents the expected acceleration of the first joint of all robots, s d3,2 represents the expected acceleration of the second joint of all robots.

[0239] Figure 2 Represents the switching topology communication relationship of the multi-robot system. The simulation results are shown in Figure 3-16 shown. Figure 3-4 is the position tracking error value of joint 1 and joint 2 of all robots. It can be seen that the performance constraint curve limits the error range well. At the specified finite time t=1s, the positions of the four follower robots coincide with the expected trajectory of the leader robot, and the tracking effect is good. Figure 5-6 is the velocity tracking error value of joints 1 and 2 of the multi-robot. The error has a certain overshoot phenomenon before t=1s, but completes convergence at t=1s, and in the subsequent control process, the error tends to be close to zero without obvious jitter. Figure 7-12The specified finite-time state observer position, velocity and acceleration values ​​of different joints of each robot are respectively represented. During the simulation process, each observer position, velocity and acceleration value converges from the preset initial value of the expected trajectory. The convergence process is smooth without obvious jitter phenomenon. The error is stabilized at t=1s. In the subsequent control process, the error value keeps converging to near zero, which proves that the control method proposed in the present invention is effective. Figure 13-14 Represents the filtering error values ​​of all robot joints 1 and 2 that specify the finite-time adaptive command filter. It can be seen that during the simulation process, the initial filtering error value is large and overshoot occurs, and then it gradually shows a convergence trend. The convergence is most obvious at t=1s and the error value is near zero. The error value of the subsequent control process no longer fluctuates significantly, which proves that the proposed method is feasible. Figure 15-16 It is the control signal of all robot joints 1 and 2 after being processed by quantization technology. The original control signal communication frequency band is unmeasurable. By inputting quantization technology, the continuous signal is converted into a segmented measurable signal, which greatly reduces the communication burden and provides feasibility for realizing collaborative control between more robots.

[0240] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0241] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0242] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0243] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0244] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A multi-robot finite-time quantitative control method based on a state observer, characterized in that: The steps include: S1. Based on the multi-robot system, a model of the leader and the follower is established, including kinematic and dynamic models, and the specified finite time control criterion is determined. The kinematic model of the follower robot is expressed as: In the formula, q i,1 ,q i,2 represents the joint position information of the i-th dual-joint robot, represents the joint velocity information of the i-th dual-joint robot, represents the joint acceleration information of the i-th dual-joint robot, C i,1a , C i,2a represents the Coriolis force of the i-th dual-joint robot, C i,1b , C i,2b represents the centrifugal force of the i-th dual-joint robot; M i,1a , M i,1b , M i,2a and M i,2b represents the symmetric positive definite inertia matrix of the i-th dual-joint robot; G i,1 and G i,2 represents the gravity of the i-th dual-joint robot; Q(u i,1 ) and Q(u i,2 ) is the quantitative control input signal of the i-th dual-joint robot; τ di.1 and τ di,2 is the external disturbance suffered by the i-th dual-joint robot; u i,1 and u i,2 is the control input of the i-th dual-joint robot; subscripts 1 and 2 are the joint numbers of the i-th dual-joint robot, and subscripts a and b are the ordinal numbers of the i-th dual-joint robot; The dynamic model of the follower robot is expressed as: In the formula, Expressed as the velocity of the kth joint of the ith robot, x n,k It is expressed as the velocity of the kth joint of the ith robot; is the acceleration of the kth joint of the ith robot; f σi,k (x i,k ) is represented as the set of model uncertainties of the kth joint of the ith robot; Q(u i,k ) is represented by the control signal after input quantization; τ di,k It is represented by the external time-varying disturbance on the kth joint of the ith robot; M i,k It is represented as the moment of inertia of the motor and connecting rod of the kth joint of the ith robot; u i,k Expressed as a specified finite time H ∞ Control signal; The leader robot dynamics and kinematics model is expressed as: In the formula, is the velocity of the kth joint of the leader robot; v 0,k is the velocity of the kth joint of the leader robot, is the acceleration of the kth joint of the leader robot; u 0,k is the control input of the kth joint of the leader robot; S2, determining the state observers of the leader and the follower respectively, and reconstructing new leader information for each follower using switching topology graph theory without requiring a leading measurement of acceleration information; S3. Determine the consistency error by switching the topological relationship, design the performance constraint function and construct the obstacle Lyapunov function, where the consistency error includes: The position consistency error of the kth joint of the ith robot is expressed as: In the formula, e xi,k Expressed as position consistency error, x i,k Represented as the position of the ith follower robot, x j,k Represented as the position of the j-th follower robot; The velocity consistency error of the kth joint of the ith robot is expressed as: In the formula, e vi,k Expressed as velocity consistency error, v i,k Denotes the velocity of the ith follower robot, v j,k Denotes the speed of the j-th follower robot; The acceleration consistency error of the kth joint of the ith robot is expressed as: In the formula, e ai,k Expressed as acceleration consistency error, a i,k Expressed as the acceleration of the ith follower robot, a j,k Expressed as the acceleration of the jth follower robot; The performance constraint function expression is: In the formula, e xi,k is the position consistency error, ω i,k is the lower bound constraint of the performance function of the kth joint of the ith robot; ω 0i,k is the initial value of the lower bound of the performance function of the kth joint of the ith robot; ω ∞i,k is the final value of the lower bound of the performance function of the kth joint of the ith robot; The upper boundary constraint of the performance function of the kth joint of the ith robot; is the initial value of the upper boundary of the performance function of the kth joint of the ith robot; is the final value of the upper boundary of the performance function of the kth joint of the ith robot; γ is the adjustment parameter of the performance function; t is the time of the control process; T is the convergence time; The barrier Lyapunov function expression in S3 is: In the formula, q(e xi,k ) is the judgment function, that is, e xi,k >0, q(e xi,k )=1;e xi,k When ≤0, q(e xi,k )=0; S4. Construct a virtual controller with a specified finite time, expressed as: In the formula, e xi,k is the position consistency error, β 11i.k , β 12i.k , κ0 is the constant term in the virtual control rate, μ is the time-varying function; The upper boundary constraint of the performance function of the kth joint of the ith robot, is its first-order differential; ω i,k is the lower bound constraint of the performance function of the kth joint of the ith robot, is its first-order differential; Γ ai,k , Γ bi,k is the virtual control rate error term; determining a prescribed finite time adaptive command filter and establishing an adaptive rate; S5, based on limited time H ∞ Theorem construction stipulates that the finite time H ∞ The controller and hysteresis quantizer make the position error and velocity error converge to the neighborhood of zero within a specified finite time, and realize the consistency tracking of the multi-robot system. The specific method is as follows: S51, determining the specified finite time H according to the model of the leader and the follower in step S1 and the specified finite time control criterion ∞ theorem; S52, in order to make the position error and the velocity error converge to the neighborhood of zero within a specified finite time, based on the specified finite time H obtained in S51 ∞ Theorem construction stipulates that the finite time H ∞ Controller, specifies a limited time H ∞ Control signal u i,k It is expressed as: In the formula, β 15i,k , β 16i,k is a constant term, is the virtual control rate derivative after filtering, M i,k is a symmetric positive definite matrix, μ is a time-varying function; i≠j is an element in a directed graph and a ij represents the communication relationship between the ith follower robot and the jth follower robot, N is the number of follower robots; b i represents the communication relationship between the ith follower robot and the leader robot, Φ i,k is the performance representation vector; is the acceleration of the leader robot, is the acceleration of the jth follower robot; e xi,k is the position consistency error; Γ ai,k , Γ bi,k is the virtual control rate error term; S53, the constructed specified finite time H ∞ The output signal of the controller is processed by the hysteresis quantizer to achieve the consistency tracking of the multi-robot system. The specific method is as follows: In the formula, u i,k To specify a limited time H ∞ Control signal, is its first-order differential; sgn() represents a step function; a di,k Expressed as the basic parameter of the quantization process; u ni,k Input quantization parameters and satisfy the conditions n=1,…,12,i=1,…,4,k=1,2,θ i,k is the transmission rate of the communication channel from the controller to the multi-robot system, t - Indicates the previous moment of the control process.

2. The multi-robot finite time quantitative control method based on state observer according to claim 1 is characterized in that: The S2 specifically includes the following steps: S21. Designing graph theory based on the communication relationship of the leader's state observer and the subsequent leader's state observer; S22, based on the graph theory designed in S21, use the differential equations to solve the estimated value of the leader's acceleration information; S23. Determine a state observer of a finite-time cascade leader according to the estimated value of the leader acceleration information obtained in S22, and redesign state information for each follower.

3. The multi-robot finite time quantitative control method based on state observer according to claim 2 is characterized in that: The specific method of using the differential equation group to solve the estimated value of the leader's acceleration information in S22 is: In the formula, β 1i,k , β 2i,k , β 3i,k , β 4i,k , Ξ is the constant term in the finite-time leader observer, x 0,k is the position of the kth joint of the leader robot; z xi,k , z vi,k , z ai,k is the observer adaptation term for the kth joint of the i-th prescribed finite-time leader, First-order derivative of the adaptive term; z ai,k (t0) is z ai,k The value at the initial moment, a 0,k (t0) is the initial acceleration of the kth joint of the leader robot, μ is the time-varying function, and t0 is the initial moment.

4. The multi-robot finite time quantitative control method based on state observer according to claim 2 is characterized in that: The state information of each follower is redesigned in S23 as follows: In the formula, β 5i,k , β 6i,k , β 7i,k , β 8i,k , β 9i,k and β 10i,k is a constant term, μ is a time-varying function; i≠j is an element in a directed graph and a ij represents the communication relationship between the ith follower robot and the jth follower robot; b i represents the communication relationship between the ith follower robot and the leader robot; is the position estimation information of the i-th and j-th follower robots, is the velocity estimation information of the i-th and j-th follower robots, is the acceleration estimation information of the i-th and j-th follower robots, Its first-order differential, x 0,k , v 0,k is the position and velocity of the leader robot.

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