A multi-robot prescribed time event-triggered sliding mode control method

Abstract

The application discloses a kind of multi-robot specified time event trigger sliding mode control method, it is related to robot processing technical field, including the following steps: S1, the leader and follower of multi-robot are established kinematics model and dynamics model, and specified time control criterion is determined;S2, the consistency error of robot is determined;S3, error conversion function is determined;S4, non-singular terminal sliding mode surface is determined;S5, constructs first self-constructing neural network, and specified time self-constructing neural network disturbance observer is determined;S6, constructs second self-constructing neural network, and specified time controller is determined, and the consistency tracking of multi-robot is completed.The application improves the formation control precision in continuous time when disturbance exists by the introduction of error function and the proper adjustment of sliding mode gain parameter;Event trigger mechanism avoids unnecessary continuous monitoring by periodic monitoring, reduces repeated periodic calculation, reduces the system hardware requirement, further improves the physical realizability of the method.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical arm processing, and particularly relates to a multi-mechanical arm prescribed time event triggering sliding mode control method. BACKGROUND

[0002] In the past few decades, the distributed cooperative control of multi-agent systems has been widely applied due to its stronger robustness, stability and high fault tolerance. In particular, as the most important branch of distributed cooperative control, consensus control plays a crucial role in practical engineering applications such as ship formation control, underwater robot cooperative control and unmanned aerial vehicle formation flight. In view of the fact that the dynamic characteristics of complex mechanical systems are often difficult to describe by simple linear system models in practical engineering applications, the research focus of the consensus problem has gradually shifted to more complex nonlinear systems. At the same time, as a special nonlinear case, the multi-agent system with Lagrange dynamics has attracted the attention of researchers due to its ability to model some practical industrial systems, such as the cooperative control of multi-robot arms.

[0003] Compared with traditional linear or nonlinear systems, multi-robot arm systems exhibit higher uncertainty and nonlinear characteristics, especially in complex scenarios involving unmodeled dynamics and strong parameter coupling. Some existing researches use adaptive control, neural network control and fuzzy control methods to deal with this model uncertainty. However, when considering the prescribed time consensus problem, these three methods have certain deficiencies. As an alternative, a prescribed time self-constructing neural network disturbance observer can be introduced to estimate the model uncertainty and external disturbance.

[0004] For these systems with high model uncertainty, the current control method shows obvious limitations, therefore, it is particularly important to develop an innovative control algorithm. Considering the excellent effect of sliding mode control strategy in weakening the negative impact of model uncertainty and unknown disturbance in the control process, it becomes a strong candidate for our exploration of new control schemes. Therefore, how to apply sliding mode control to solve the problem of multi-robot system cooperative control is a very interesting topic. On the one hand, the fixed time stability theory has been widely used in the field of sliding mode control due to its many advantages. However, the determination process of the upper bound of its convergence involves complex system parameter adjustment, which may lead to the upper bound set being too conservative. In order to overcome this limitation, the pre-defined time stability theory is introduced in the subsequent research, and the upper bound of its convergence time can be more closely related to the user-specified adjustment parameters, but still has some ambiguity. Therefore, the prescribed time stability research has attracted widespread attention and has been applied to sliding mode control, adaptive control and optimal control, and its convergence time can be specified in advance and is independent of the initial condition and other parameters, ensuring the timeliness of the control process. On the other hand, compared with the traditional linear sliding mode control, the terminal sliding mode control can ensure that the system state reaches the sliding surface and converges to zero within a specified time, with faster response speed. However, the tracking error is usually unstable and the upper and lower fluctuations are obvious, and a conversion function based on performance function is introduced to limit the range of error.

[0005] In order to effectively reduce the load pressure of the communication network of the closed-loop system and maintain a high level of control performance satisfaction, a distributed event-triggered control strategy is developed to efficiently utilize resources. The strategy realizes the optimal allocation and utilization of resources by reasonably managing the timing of the triggering event. It is worth noting that the important idea of event-triggered control is that when the constant quantity of measurement error exceeds the preset allowable threshold or the norm of system state reaches a certain condition, the event triggering mechanism will be activated to update the controller. This process ensures that the system maintains a certain performance level while ensuring that the internal execution time has a positive minimum limit. Therefore, it is necessary to study the multi-robot distributed event-triggered formation control. SUMMARY

[0006] The present application proposes a multi-robot prescribed time event-triggered sliding mode control method to solve the above problems.

[0007] The technical scheme of the present application is: a multi-robot prescribed time event-triggered sliding mode control method comprising the following steps:

[0008] S1, kinematic and dynamic models are established for the leader and follower of the multi-robot, and a prescribed time control criterion is determined;

[0009] S2, determining the consistency error of the robot arm according to the specified time control criterion, the leader's kinematic model and dynamic model, and the follower's kinematic model and dynamic model;

[0010] S3, determining the error conversion function according to the consistency error of the robot arm and the performance function;

[0011] S4, determining the non-singular terminal sliding mode surface according to the error conversion function;

[0012] S5, constructing the first self-constructing neural network according to the non-singular terminal sliding mode surface, and determining the specified time self-constructing neural network observer;

[0013] S6, constructing the second self-constructing neural network according to the specified time self-constructing neural network observer, and determining the specified time controller, completing the consistency tracking of the multi-robot arm.

[0014] Further, in S1, the expression of the kinematic model of the follower is:

[0015]

[0016] In the formula, M i,11 represents the first symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,12 represents the second symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,21 represents the third symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,22 represents the fourth symmetric positive definite inertia matrix of the i th double-joint robot arm, represents the first velocity of the i th double-joint robot arm, represents the second velocity of the i th double-joint robot arm, represents the first acceleration of the i th double-joint robot arm, represents the second acceleration of the i th double-joint robot arm, C i,11 represents the first Coriolis force of the i th double-joint robot arm, C i,12 represents the first centrifugal force of the i th double-joint robot arm, C i,21 represents the second Coriolis force of the i th double-joint robot arm, C i,22 represents the second centrifugal force of the i th double-joint robot arm, G i,1 represents the first gravitational force of the i th double-joint robot arm, G i,2 represents the second gravitational force of the i th double-joint robot arm, w i,1 represents the first control signal of the i th double-joint robot arm after event-triggered processing, w i,2 represents the second control signal of the i th double-joint robot arm after event-triggered processing, τ di,1denotes the external disturbance experienced by the ith dual-joint robot arm, τ di,2 denotes the second external disturbance experienced by the ith dual-joint robot arm;

[0017] In S1, the expression of the follower's kinematics model is:

[0018]

[0019] where x i,λ denotes the position of the ith robot arm's λth joint, v i,λ denotes the velocity of the ith robot arm's λth joint, denotes the first derivative of the velocity of the ith robot arm's λth joint, denotes the first derivative of the position of the ith robot arm's λth joint, f i,λ (x i,λ ) denotes the set of model uncertainties of the ith robot arm's λth joint, w i,λ denotes the control signal of the ith robot arm's λth joint after event-triggered processing, τ di,λ denotes the external time-varying disturbance experienced by the ith robot arm's λth joint, M i,λ denotes the moment of inertia of the motor and link of the ith robot arm's λth joint;

[0020] In S1, the expression of the leader's kinematics model is:

[0021]

[0022] where denotes the first derivative of the position of the leader robot arm, v 0,λ denotes the velocity of the leader robot arm's λth joint;

[0023] In S1, the expression of the leader's dynamics model is:

[0024]

[0025] where denotes the first derivative of the velocity of the leader robot arm, u 0,λ denotes the control signal of the leader robot arm's λth joint;

[0026] In S1, the expression of the prescribed-time control criterion is:

[0027]

[0028] where V i,λ (t) denotes a continuously differentiable function, μ(t) denotes a time-varying function, is a first-order derivative of a time-varying function, t represents time of a control process, κ1 represents a first positive real constant, κ2 represents a second positive real constant, and T represents a specified time.

[0029] Further, in S2, the consistency error includes a position consistency error of the λth joint of the ith robot arm, a velocity consistency error of the λth joint of the ith robot arm, and an acceleration consistency error of the λth joint of the ith robot arm.

[0030] The position consistency error e xi,λ of the λth joint of the ith robot arm has an expression as follows:

[0031]

[0032] where N represents a number of robot arms, a ij represents a communication link between the ith follower robot arm and the jth follower robot arm, b i represents a communication link between the leader robot arm and the ith follower robot arm, x i,λ represents a position of the λth joint of the ith robot arm, x j,λ represents a position of the λth joint of the jth follower robot arm, x 0,λ represents a position of the leader robot arm;

[0033] The velocity consistency error e vi,λ of the λth joint of the ith robot arm has an expression as follows:

[0034]

[0035] where v i,λ represents a velocity of the λth joint of the ith robot arm, v j,λ represents a velocity of the λth joint of the jth robot arm, v 0,λ represents a velocity of the leader robot arm;

[0036] The acceleration consistency error e ai,λ of the λth joint of the ith robot arm has an expression as follows:

[0037]

[0038] where a i,λ represents an acceleration of the λth joint of the ith robot arm, a j,λ represents an acceleration of the λth joint of the jth robot arm, a 0,λ represents an acceleration of the leader robot arm.

[0039] Further, in S3, the error conversion function δ i,λ has an expression as follows:

[0040]

[0041] where e xi,λ represents the position consistency error, Φ a represents the performance function.

[0042] Further, in S4, the nonsingular terminal sliding mode surface s i,λ is expressed as:

[0043]

[0044] where δ i,λ represents the error conversion function, represents the first derivative of the error conversion function, κ 1i,λ represents the first positive real constant of the i-th robot arm λ-th joint, κ 2i,λ represents the second positive real constant of the i-th robot arm λ-th joint, μ(t) represents the time-varying function, represents the first derivative of the time-varying function.

[0045] Further, in S5, the self-constructing neural network f i,λ (σ i,λ ) is expressed as:

[0046] f i,λ (σ i,λ ) = W i,λ h i,λ (σ i,λ ) + ε i,λ ;

[0047] where σ i,λ represents the input of the self-constructing neural network, ε i,λ represents the estimation error of the self-constructing neural network, W i,λ represents the optimal weight, h i,λ (σ i,λ ) represents the Gaussian kernel function.

[0048] In S5, the expression of the prescribed time self-constructing neural network disturbance observer is:

[0049]

[0050] where represents the estimated value of the neural network approximation error, represents the estimated value of the uncertainty, M i,λ represents the moment of inertia of the motor and the connecting rod of the i-th robot arm λ-th joint, u ai,λ represents the control signal after input saturation, κ 7i,λdenotes the seventh positive real constant of the i-th robot arm λ-th joint, κ 8i,λ denotes the eighth positive real constant of the i-th robot arm λ-th joint, μ(t) denotes a time-varying function, denotes the first derivative of a time-varying function, ψ i,λ denotes a state variable, denotes an adjustable parameter.

[0051] Further, S6 comprises the following sub-steps:

[0052] S61, constructing a second self-constructing neural network according to a prescribed time, and determining a prescribed time controller;

[0053] S62, determining an auxiliary system and an event-triggered mechanism;

[0054] S63, using the prescribed time controller, the auxiliary system and the event-triggered mechanism to make the consistency error converge to the zero domain within a prescribed time, and completing the consistency tracking of the multiple robot arms.

[0055] Further, in S61, the prescribed time controller u i,λ has the expression:

[0056]

[0057] In the formula, M i,λ denotes the moment of inertia of the motor and the connecting rod of the i-th robot arm λ-th joint, denotes a conversion error function, denotes the derivative of the conversion error function, N denotes the number of robot arms, a ij denotes the communication link between the i-th follower robot arm and the j-th follower robot arm, b i denotes the communication link between the leader robot arm and the i-th follower robot arm, Φ a denotes a performance function, denotes the first derivative of the performance function, denotes the second derivative of the performance function, denotes the first derivative of the velocity of the i-th robot arm λ-th joint, denotes the first derivative of the velocity of the leader robot arm, e xi,λ denotes a position consistency error, denotes the first derivative of the position consistency error, κ 1i,λ denotes the first positive real constant of the i-th robot arm λ-th joint, κ 2i,λ denotes the second positive real constant of the i-th robot arm λ-th joint, κ 11i,λ denotes the eleventh positive real constant of the i-th robot arm λ-th joint, κ 12i,λLet μ(t) represent the twelfth positive real constant of the λ-th joint of the i-th robotic arm, and let μ(t) represent the time-varying function. s represents the first derivative of the time-varying function. i,λ Indicates a non-singular terminal sliding surface. h represents the estimated value of the first optimal weight. ai,λ Let ξ represent the first Gaussian function. i,λ This represents the first approximation error. Let represent the estimated external disturbance value of the λ-th joint of the i-th robotic arm.

[0058] Furthermore, in S62, the expression for the auxiliary system update rate is:

[0059]

[0060] In the formula, M i,λ Let represent the moment of inertia of the motor and link at the λ-th joint of the i-th robotic arm. Indicates conversion error. The derivative of the conversion error is represented by N, where N represents the number of robotic arms, and a ij This represents the communication between the i-th follower robotic arm and the j-th follower robotic arm, b. i Δu represents the communication between the leader robotic arm and the i-th follower robotic arm. i,λ ξ represents the control input difference after input saturation. i,λ θ represents the first approximation error. α Indicates the auxiliary system, κ 13i,λ κ represents the thirteenth positive real constant of the λ-th joint of the i-th robotic arm. 14i,λ Let represent the fourteenth positive real constant of the λ-th joint of the i-th robotic arm;

[0061] In S62, the event triggering mechanism w i,λ The expression for (t) is:

[0062]

[0063] In the formula, μ c Indicates the first positive design parameter. Indicates the second positive design parameter, m a Indicates the third positive design parameter. This represents the fourth positive design parameter, t represents the control process time, and u... ai,λ This indicates the control signal after input saturation. Let Γ denote the transpose of the non-singular terminal sliding surface, θ denote the natural number, and w denote the monitoring period. i,λ (t k-1 ) represents the control signal for the next trigger time after the event has been triggered, wi,λ (t k ) represents the current trigger time control signal after the event trigger, t k represents the current trigger time, t k-1 represents the next trigger time, t k+1 represents the last trigger time, E i,λ (t) represents the trigger condition defined by the measurement error.

[0064] The beneficial effects of the present application are:

[0065] (1) The present application determines a specified time disturbance observer based on a self-constructed neural network, which is used to approximate uncertain external disturbances. Compared with the existing technology, the proposed observer can realize more accurate disturbance estimation within a specified time, and further improve the convergence speed of the observer. The introduction of adaptive structure neural network can continuously adjust the number of neurons according to the segmentation strategy under the premise of ensuring good approximation accuracy, and finally obtain the optimal number;

[0066] (2) The present application determines a multi-robot system consistency control method based on a disturbance observer, which is a multi-robot specified time event triggered sliding mode control method. Compared with the existing technology, the proposed sliding mode controller realizes specified time convergence, and improves the formation control accuracy in continuous time when there is disturbance by introducing error function and appropriately adjusting sliding mode gain parameter; The event triggered mechanism avoids unnecessary continuous monitoring through periodic monitoring, reduces repeated periodic calculation, reduces the hardware requirements of the system, and further improves the physical realizability of the method. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 Flow chart of multi-robot specified time event triggered sliding mode control method;

[0068] Figure 2 Structure diagram of multi-robot specified time event triggered sliding mode control system;

[0069] Figure 3 Schematic diagram of topological communication relationship of multi-robot system;

[0070] Figure 4 First schematic diagram of the change of position tracking error of joint 1 and joint 2 in the multi-robot system with time;

[0071] Figure 5 Second schematic diagram of the change of position tracking error of joint 1 and joint 2 in the multi-robot system with time;

[0072] Figure 6A first schematic diagram of the variation of the velocity tracking error of joint 1 and joint 2 in the multi-robot system over time;

[0073] Figure 7 A second schematic diagram of the variation of the velocity tracking error of joint 1 and joint 2 in the multi-robot system over time;

[0074] Figure 8 A schematic diagram of the variation of the error of joint 1 and joint 2 of all robots during disturbance observation;

[0075] Figure 9 A schematic diagram of the distribution of the event-triggering time of joint 1 and joint 2 of all robots;

[0076] Figure 10 A schematic diagram of the number of event-triggering of joint 1 and joint 2 of all robots;

[0077] Figure 11 A first schematic diagram of the variation of the number of neurons of the self-constructing neural network;

[0078] Figure 12 A second schematic diagram of the variation of the number of neurons of the self-constructing neural network;

[0079] Figure 13 A first schematic diagram of the saturated control signal of joint 1 and joint 2 of all robots after being processed by the event-triggering mechanism;

[0080] Figure 14 A second schematic diagram of the saturated control signal of joint 1 and joint 2 of all robots after being processed by the event-triggering mechanism. DETAILED DESCRIPTION

[0081] The embodiments of the present application will be further described below with reference to the accompanying drawings.

[0082] As shown in the drawings, Figure 1 the present application provides a multi-robot specified time event-triggering sliding mode control method, comprising the following steps:

[0083] S1, establishing kinematic models and dynamic models for the leader and the follower of the multi-robot, and determining a specified time control criterion;

[0084] S2, determining a consistency error of the robot according to the specified time control criterion, the kinematic model and the dynamic model of the leader, and the kinematic model and the dynamic model of the follower;

[0085] S3, determining an error conversion function according to the consistency error of the robot and a performance function;

[0086] S4, determining a non-singular terminal sliding mode surface according to the error conversion function;

[0087] S5, constructing a first self-constructing neural network according to the non-singular terminal sliding mode surface, and determining a prescribed time self-constructing neural network observer;

[0088] S6, constructing a second self-constructing neural network according to the prescribed time self-constructing neural network observer, and determining a prescribed time controller, and completing the consistency tracking of the multiple robot arms.

[0089] In the embodiment of the application, in S1, the expression of the kinematics model of the follower is:

[0090]

[0091] In the formula, M i,11 represents the first symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,12 represents the second symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,21 represents the third symmetric positive definite inertia matrix of the i th double-joint robot arm, M i,22 represents the fourth symmetric positive definite inertia matrix of the i th double-joint robot arm, represents the first velocity of the i th double-joint robot arm, represents the second velocity of the i th double-joint robot arm, represents the first acceleration of the i th double-joint robot arm, represents the second acceleration of the i th double-joint robot arm, C i,11 represents the first Coriolis force of the i th double-joint robot arm, C i,12 represents the first centrifugal force of the i th double-joint robot arm, C i,21 represents the second Coriolis force of the i th double-joint robot arm, C i,22 represents the second centrifugal force of the i th double-joint robot arm, G i,1 represents the first gravitational force of the i th double-joint robot arm, G i,2 represents the second gravitational force of the i th double-joint robot arm, w i,1 represents the first control signal of the i th double-joint robot arm after event-triggered processing, w i,2 represents the second control signal of the i th double-joint robot arm after event-triggered processing, τ di,1 represents the external disturbance suffered by the i th double-joint robot arm, τ di,2 represents the second external disturbance suffered by the i th double-joint robot arm;

[0092] In S1, the expression of the dynamics model of the follower is:

[0093]

[0094] where x i,λ represents the position of the i-th robot arm at the λ-th joint, v i,λ represents the velocity of the i-th robot arm at the λ-th joint, represents the first derivative of the velocity of the i-th robot arm at the λ-th joint, represents the first derivative of the position of the i-th robot arm at the λ-th joint, f i,λ (x i,λ ) represents a set of model uncertainties of the i-th robot arm at the λ-th joint, w i,λ represents the control signal of the i-th robot arm at the λ-th joint after event-triggered processing, τ di,λ represents the external time-varying disturbance on the i-th robot arm at the λ-th joint, M i,λ represents the moment of inertia of the motor and the link of the i-th robot arm at the λ-th joint;

[0095] In S1, the expression of the kinematic model of the leader is:

[0096]

[0097] where represents the first derivative of the position of the leader robot arm, v 0,λ represents the velocity of the leader robot arm at the λ-th joint;

[0098] In S1, the expression of the dynamic model of the leader is:

[0099]

[0100] where represents the first derivative of the velocity of the leader robot arm, u 0,λ represents the control signal of the leader robot arm at the λ-th joint;

[0101] In S1, specifically, the specified time control criterion is determined by a time-varying function μ(t) = (T / T-t) ω , t ∈ [t0, t0+T); μ(t) = 1, t ∈ [t0+T, ∞).

[0102]

[0103] where V i,λ (t) represents a continuously differentiable function, μ(t) represents a time-varying function, is the first derivative of the time-varying function, t represents the time of the control process, κ1 represents a first positive real constant, κ2 represents a second positive real constant, and T represents the specified time.

[0104] In this embodiment of the invention, in S2, it is assumed that graph theory θ(L,Π) is used to describe the communication topology between the follower robotic arms, Y=[y1,K,y n ] represents the vertex set, and Π∈y×y represents the edge set (y i ,y j ), where i represents the number of robotic arms, and the directed graph A = [a ij ]∈R n The adjacency matrix is ​​defined as Ω, where a ij =1. a ij =1, j≠i indicates that the i-th follower robotic arm communicates with the j-th follower robotic arm; otherwise, a ij =0. Define the exponential correlation matrix as C = diag(c i )∈R n ,in Let node i be an example. Define Y' = CA as the Laplace matrix. The adjacency matrix of the leader robotic arm is defined as B = diag(b). i )∈R n , where b i =1 indicates that the leader robot arm and the i-th follower robot arm have a communication connection; otherwise, the leader robot arm and the i-th follower robot arm do not have a communication connection, defined as b. i =0.

[0105] In this embodiment of the invention, in S2, the consistency error includes the position consistency error of the ith joint of the λth robotic arm, the velocity consistency error of the ith joint of the λth robotic arm, and the acceleration consistency error of the ith joint of the λth robotic arm.

[0106] The position consistency error of the i-th robotic arm and the λ-th joint is e xi,λ The expression is:

[0107]

[0108] In the formula, N represents the number of robotic arms, a ij This represents the communication between the i-th follower robotic arm and the j-th follower robotic arm, b. i x represents the communication between the leader robotic arm and the i-th follower robotic arm. i,λ Let x represent the position of the λ-th joint of the i-th robotic arm. j,λ Let x represent the position of the λ-th joint of the j-th follower robotic arm. 0,λ Indicates the position of the leader's robotic arm;

[0109] The speed consistency error of the i-th robotic arm at the λ-th joint is e vi,λ The expression is:

[0110]

[0111] wherein v i,λ represents the velocity of the i-th robot arm at the λ-th joint, v j,λ represents the velocity of the j-th robot arm at the λ-th joint, v 0,λ represents the velocity of the leader robot arm.

[0112] The acceleration consistency error e ai,λ of the i-th robot arm at the λ-th joint is expressed as:

[0113]

[0114] wherein a i,λ represents the acceleration of the i-th robot arm at the λ-th joint, a j,λ represents the acceleration of the j-th robot arm at the λ-th joint, a 0,λ represents the acceleration of the leader robot arm.

[0115] In the embodiment of the present application, in S3, a performance function Φ a∞ is determined, and the error conversion function δ i,λ is determined:

[0116]

[0117] wherein e xi,λ represents the position consistency error, Φ a represents the performance function.

[0118] In S3, the first derivative of the error conversion function is expressed as:

[0119]

[0120] wherein e xi,λ represents the position consistency error, represents the first derivative of the position consistency error, Φ a represents the performance function, represents the first derivative of the performance function,

[0121] In S3, the second derivative of the error conversion function is expressed as:

[0122]

[0123] wherein, represents the conversion error, represents the derivative of the conversion error, Second derivative of position consistency error, Second derivative of performance function.

[0124] In embodiments of the present application, in S4, the nonsingular terminal sliding mode surface s i,λ is expressed as:

[0125]

[0126] wherein δ i,λ represents error conversion function, represents first derivative of error conversion function, κ 1i,λ represents first positive real constant of the i-th robot arm, κ 2i,λ represents second positive real constant of the i-th robot arm, μ(t) represents time-varying function, represents first derivative of time-varying function.

[0127] In S4, the first derivative of nonsingular terminal sliding mode surface s is expressed as:

[0128]

[0129] wherein, represents first derivative of error conversion function, represents second derivative of error conversion function.

[0130] In S4, the first derivative of nonsingular terminal sliding mode surface s can be further expressed as:

[0131]

[0132] wherein, represents conversion error, represents derivative of conversion error, e xi,λ represents position consistency error, represents first derivative of position consistency error, represents second derivative of position consistency error, Φ a represents performance function, represents first derivative of performance function, represents second derivative of performance function.

[0133] In embodiments of the present application, in S5, the self-constructed neural network f i,λ (σ i,λ ) is expressed as:

[0134] f i,λ (σ i,λ ) = W i,λ hi,λ (σ i,λ )+ε i,λ ;

[0135] In the formula, σ i,λ represents the input of the self-constructing neural network, ε i,λ represents the estimation error of the self-constructing neural network, W i,λ represents the optimal weight, h i,λ (σ i,λ ) represents the Gaussian basis function;

[0136] In S5, specifically, the first self-constructing neural network update rate and the adaptive rate The self-constructing neural network disturbance observer is determined within a specified time:

[0137]

[0138] In the formula, σ represents the estimation value of the neural network approximation error, W represents the estimation value of the first optimal weight, h ai,λ represents the Gaussian basis function of the first self-constructing neural network, ξ i,λ represents the first approximation error, M i,λ represents the moment of inertia of the motor and connecting rod of the i-th manipulator and the λ-th joint, u ai,λ represents the control signal after input saturation, κ 7i,λ represents the seventh positive real constant of the i-th manipulator and the λ-th joint, κ 8i,λ represents the eighth positive real constant of the i-th manipulator and the λ-th joint, μ(t) represents a time-varying function, represents the first derivative of the time-varying function, ψ i,λ represents the state variable, represents an adjustable parameter.

[0139] In the embodiment of the present application, S6 includes the following sub-steps:

[0140] S61, according to the specified time self-constructing neural network observer, a second self-constructing neural network is constructed, and a specified time controller is determined;

[0141] S62, an auxiliary system and an event triggering mechanism are determined;

[0142] S63, the specified time controller, the auxiliary system and the event triggering mechanism are used to make the consistency error converge to the zero domain within a specified time, and the consistency tracking of the multi-manipulator is completed.

[0143] In the embodiment of the present application, in S61, specifically, the second self-constructed neural network update rate The external disturbance estimation value of the i th robot arm λ th joint is determined Further, the time controller u i,λ The expression is:

[0144]

[0145] In the formula, M i,λ Indicates the moment of inertia of the motor and connecting rod of the i th robot arm λ th joint, Indicates the conversion error function, Indicates the derivative of the conversion error function, N indicates the number of robots, a ij Indicates the communication link between the i th follower robot arm and the j th follower robot arm, b i Indicates the communication link between the leader robot arm and the i th follower robot arm, Φ a Indicates the performance function, Indicates the first derivative of the performance function, Indicates the second derivative of the performance function, Indicates the first derivative of the velocity of the i th robot arm λ th joint, Indicates the first derivative of the velocity of the leader robot arm, e xi,λ Indicates the position consistency error, Indicates the first derivative of the position consistency error, κ 1i,λ Indicates the first positive real constant of the i th robot arm λ th joint, κ 2i,λ Indicates the second positive real constant of the i th robot arm λ th joint, κ 11i,λ Indicates the eleventh positive real constant of the i th robot arm λ th joint, κ 12i,λ Indicates the twelfth positive real constant of the i th robot arm λ th joint, μ(t) indicates a time-varying function, Indicates the first derivative of the time-varying function, s i,λ Indicates a non-singular terminal sliding mode surface, Indicates the estimation value of the first optimal weight, h ai,λ Indicates the first Gauss base function, ξ i,λ Indicates the first approximation error, Indicates the external disturbance estimation value of the i th robot arm λ th joint.

[0146] In the embodiment of the present application, in S62, the auxiliary system update rate expression is:

[0147]

[0148] In the formula, M i,λdenotes the moment of inertia of the motor and link of the i th manipulator at the λ th joint, denotes the conversion error, denotes the derivative of the conversion error, N denotes the number of manipulators, a ij denotes the communication link between the i th follower manipulator and the j th follower manipulator, b i denotes the communication link between the leader manipulator and the i th follower manipulator, Δu i,λ denotes the control input difference after input saturation, ξ i,λ denotes the first approximation error, θ α denotes the auxiliary system, κ 13i,λ denotes the thirteenth positive real constant of the i th manipulator at the λ th joint, κ 14i,λ denotes the fourteenth positive real constant of the i th manipulator at the λ th joint;

[0149] In S62, the event-triggered mechanism w i,λ The expression of (t) is:

[0150]

[0151] In the formula, μ c denotes the first positive design parameter, denotes the second positive design parameter, m a denotes the third positive design parameter, denotes the fourth positive design parameter, t denotes the time of the control process, u ai,λ denotes the control signal after input saturation, denotes the transpose of the nonsingular terminal sliding surface, Γ denotes a natural number, θ denotes a monitoring period, w i,λ (t k-1 ) denotes the control signal at the next triggering time after event triggering, w i,λ (t k ) denotes the control signal at the current triggering time after event triggering, t k denotes the current triggering time, t k-1 denotes the next triggering time, t k+1 denotes the last triggering time, E i,λ (t) denotes the triggering condition defined by the measurement error.

[0152] In the embodiment of the present application, as shown in Figure 2 The multi-manipulator prescribed time event-triggered sliding mode control system based on the disturbance observer should include a leader trajectory generator module, an error conversion module, a performance constraint module, a prescribed time controller module, an auxiliary system module, an event-triggered mechanism module, a multi-manipulator system module, a prescribed time self-constructed neural network disturbance observer module, a self-constructed neural network module, and an adaptive rate module.

[0153] The leader trajectory generator module is used to generate the desired position, desired velocity and desired acceleration of the control system, and input the desired position, desired velocity and desired acceleration of the control system into the error conversion module and the self-constructing neural network module respectively.

[0154] The error conversion module is used to obtain the desired trajectory input by the leader trajectory generator module, the follower state information input by the multi-robot system module, and the performance function input by the performance constraint module, and calculate the errors between each follower and the follower and between the leader and each follower according to the communication topology relationship, and then combine the performance function input by the performance constraint module with the errors between each follower and the follower and between the leader and each follower to solve the error conversion function, and solve the prescribed time non-singular terminal sliding mode through the calculated error conversion function.

[0155] The performance constraint module is used to generate a performance function applied to the multi-robot system and input into the error conversion module.

[0156] The prescribed time controller module is used to obtain the non-singular terminal sliding mode input by the error conversion module, the errors between each follower and the follower and between the leader and each follower, and the performance function, and further obtain the sum of the product of the optimal weight and the Gaussian basis function input by the self-constructing neural network module and the approximation error (i.e. the estimated value of the model uncertainty) and the external disturbance estimate value input by the prescribed time self-constructing neural network disturbance observer module, and solve the prescribed time controller output signal according to the non-singular terminal sliding mode, the errors between each follower and the follower and between the leader and each follower, the performance function, the sum of the product of the optimal weight and the Gaussian basis function and the approximation error, and the external disturbance estimate value.

[0157] The auxiliary system module is used to obtain the output signal of the prescribed time controller module, and according to the output signal of the prescribed time controller module, to perform saturation suppression processing on the control signal and calculate the processed control signal.

[0158] The event-triggered mechanism module is used to obtain the control signal of the auxiliary system module, and according to the control signal, to design an event-triggered mechanism, which is used to limit the transmission frequency of the continuous control signal sent from the auxiliary system to the multi-robot system.

[0159] The multi-robot system module is used to obtain the control signal of the event-triggered mechanism module, and according to the obtained actual control signal, to solve the position, velocity and acceleration of the follower by differentiation, and further to adjust the follower state information according to the obtained position, velocity and acceleration information.

[0160] The prescribed time self-constructing neural network disturbance observer module is used to obtain the position, velocity and acceleration information of the follower of the multi-robot system module, the control signal of the auxiliary system module, the product of the optimal weight and Gaussian basis function input by the self-constructing neural network module and the sum of the approximation error (i.e. the estimated value of the model uncertainty) and the product of the optimal weight and Gaussian basis function and the sum of the approximation error (i.e. the estimated value of the external disturbance) and the adaptive rate of the adaptive rate module.

[0161] The self-constructing neural network module is used to obtain the desired position, desired velocity and desired acceleration of the leader trajectory generator module, the position, velocity and acceleration of the follower of the multi-robot system module and the state variables of the prescribed time self-constructing neural network disturbance observer module, automatically adjusts and optimizes its internal structure according to the characteristics of the desired position, desired velocity and desired acceleration of the leader trajectory generator module, the position, velocity and acceleration of the follower of the multi-robot system module and the state variables of the prescribed time self-constructing neural network disturbance observer module data, automatically removes the neural network neurons with poor effect, and splits the neural network neurons with good effect.

[0162] The adaptive rate module is used to obtain the state variables of the self-constructing neural network module, and differentiates to obtain the adaptive rate.

[0163] The input end of the error conversion module is connected with the output end of the leader trajectory generator module, the input end of the error conversion module is connected with the output end of the multi-robot system, the input end of the error conversion module is connected with the output end of the performance constraint module, the input end of the specified time controller module is connected with the output end of the error conversion module, the input end of the specified time controller module is connected with the output end of the self-constructing neural network module, the input end of the specified time controller module is connected with the output end of the specified time self-constructing neural network disturbance observer module, the input end of the auxiliary system module is connected with the output end of the specified time controller module, the input end of the event trigger mechanism module is connected with the output end of the auxiliary system module, the input end of the multi-robot system module is connected with the output end of the event trigger mechanism module, the input end of the specified time self-constructing neural network disturbance observer module is connected with the output end of the multi-robot system module, the input end of the specified time self-constructing neural network disturbance observer module is connected with the output end of the auxiliary system module, the input end of the specified time self-constructing neural network disturbance observer module is connected with the output end of the self-constructing neural network module, the input end of the specified time self-constructing neural network disturbance observer module is connected with the output end of the adaptive rate module, the input end of the self-constructing neural network module is connected with the output end of the leader trajectory generator module, the input end of the self-constructing neural network module is connected with the output end of the multi-robot system, the input end of the self-constructing neural network module is connected with the output end of the specified time self-constructing neural network disturbance observer module, and the input end of the adaptive rate module is connected with the output end of the self-constructing neural network module.

[0164] In order to verify the effectiveness of the method of the application, simulation experiments were carried out using the following model, as follows.

[0165] Through the simulation experiment of the selected model parameters on the five double-joint robot systems, the effectiveness of the designed multi-robot specified time event trigger sliding mode control method based on the disturbance observer was verified. The parameters of the multi-robot system are shown in Table 1, the parameters of the controller are shown in Table 2, and other parameters are shown in Table 3.

[0166] Table 1

[0167] Parameter name Parameter value Parameter name Parameter value Parameter name Parameter value J 1,1 ]]> 0.21 m 4,1 ]]> 0.98 l 3,2 ]]> 0.88 J 2,1 ]]> 0.28 m 1,2 ]]> 1.06 l 4,2 ]]> 0.97 J 3,1 ]]> 0.19 m 2,2 ]]> 1.66 d1,1 ]]> ​ 2 + 1.6 sin(0.5t) J 4,1 ]]> 0.41 m 3,2 ]]> 1.86 d2,1 ]]> ​ 1 - 0.5 sin(0.3t) J 1,2 ]]> 0.42 m 4,2 ]]> 2.16 d3,1 ]]> ​ 2 - 0.6 sin(0.5t) J 2,2 ]]> 0.4 l 1,1 ]]> 1.54 d4,1 ]]> ​ 2 + 0.5 sin(0.3t) J 3,2 ]]> 0.33 l 2,1 ]]> 1.34 d1,2 ]]> ​ 1.5 - 0.5 cos(0.4t) J 4,2 ]]> 0.21 l 3,1 ]]> 1.39 d2,2 ]]> ​ 1.8 + 0.3 cos(0.2t) m 1,1 ]]> 0.86 l 4,1 ]]> 1.59 d3,2 ]]> ​ 0.9 - 0.8 cos(0.6t) m 2,1 ]]> 0.91 l 1,2 ]]> 0.94 d4,2 ]]> ​ 1.2 + 0.6 cos(0.7t) m 3,1 ]]> 1.34 l 2,2 ]]> 0.78 g 9.81

[0168] 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 2,2J2 represents the moment of inertia of the second joint of the second robot arm 3,2 J3 represents the moment of inertia of the second joint of the third robot arm 4,2 J4 represents the moment of inertia of the second joint of the fourth robot arm 1,1 m1 represents the weight of the link of the first joint of the first robot arm 2,1 m2 represents the weight of the link of the first joint of the second robot arm 3,1 m3 represents the weight of the link of the first joint of the third robot arm 4,1 m4 represents the weight of the link of the first joint of the fourth robot arm 1,2 m5 represents the weight of the link of the second joint of the first robot arm 2,2 m6 represents the weight of the link of the second joint of the second robot arm 3,2 m7 represents the weight of the link of the second joint of the third robot arm 4,2 m8 represents the weight of the link of the second joint of the fourth robot arm 1,1 l1 represents the length of the link of the first joint of the first robot arm 2,1 l2 represents the length of the link of the first joint of the second robot arm 3,1 l3 represents the length of the link of the first joint of the third robot arm 4,1 l4 represents the length of the link of the first joint of the fourth robot arm 1,2 l5 represents the length of the link of the second joint of the first robot arm 2,2 l6 represents the length of the link of the second joint of the second robot arm 3,2 l7 represents the length of the link of the second joint of the third robot arm 4,2 l8 represents the length of the link of the second joint of the fourth robot arm d1,1 τ1 represents the disturbance variable of the first joint of the first robot arm d2,1 τ2 represents the disturbance variable of the first joint of the second robot arm d3,1 τ3 represents the disturbance variable of the first joint of the third robot arm d4,1 τ4 represents the disturbance variable of the first joint of the fourth robot arm d1,2 τ5 represents the disturbance variable of the second joint of the first robot arm d2,2 τ6 represents the disturbance variable of the second joint of the second robot arm d3,2 τ7 represents the disturbance variable of the second joint of the third robot arm d4,2 τ8 represents the disturbance variable of the second joint of the fourth robot arm g represents the gravitational acceleration.

[0169] Table 2

[0170] Parameter name Parameter value Parameter name Parameter value Parameter name Parameter value T 1.5 Kappa 1i,λ ]] 1.2 Kappa 12i,λ ]] 0.75 [t0] 0 Kappa 2i,λ ]] 1.5 ω 1 Kappa 11i,λ ]] 1.35

[0171] In Table 2, T denotes a time of controlling a process, t0denotes an initial time, ω denotes an adjustable parameter set by a time-varying function, κ 1i,λ denotes a first positive real constant of the i-th robot arm at the λ-th joint, κ 2i,λ denotes a second positive real constant of the i-th robot arm at the λ-th joint, κ 11i,λ denotes an eleventh positive real constant of the i-th robot arm at the λ-th joint, κ 12i,λ denotes a twelfth positive real constant of the i-th robot arm at the λ-th joint.

[0172] Table 3

[0173]

[0174]

[0175] In Table 3, κ 3i,λ denotes a third positive real constant of the i-th robot arm at the λ-th joint, κ 4i,λ denotes a fourth positive real constant of the i-th robot arm at the λ-th joint, Γ ai,λ , Ξ i,λ and denotes an adjustable parameter, κ 5i,λ denotes a fifth positive real constant of the i-th robot arm at the λ-th joint, κ 6i,λ denotes a sixth positive real constant of the i-th robot arm at the λ-th joint, κ 7i,λ denotes a seventh positive real constant of the i-th robot arm at the λ-th joint, κ 8i,λ denotes an eighth positive real constant of the i-th robot arm at the λ-th joint, Γ bi,λ denotes an adjustable parameter, κ 9i,λ denotes a ninth positive real constant of the i-th robot arm at the λ-th joint, κ 10i,λ denotes a tenth positive real constant of the i-th robot arm at the λ-th joint, κ 13i,λ denotes a thirteenth positive real constant of the i-th robot arm at the λ-th joint, κ 14i,λ denotes a fourteenth positive real constant of the i-th robot arm at the λ-th joint, c ai,λ denotes a Gaussian kernel center range of a first self-constructing neural network, b ai,λ denotes a Gaussian kernel width of the first self-constructing neural network, Ψ a,s denotes a split threshold value for the first self-constructing neural network, Ψ a,d denotes a deletion threshold value for the first self-constructing neural network, h a,e denotes a boundary value for the first self-constructing neural network, c bi,λ denotes a Gaussian kernel center range of a second self-constructing neural network, b bi,λ denotes a Gaussian kernel width for the second self-constructing neural network, Ψ b,sSegmentation threshold represented as a second self-constructing neural network, Ψ b,d Deletion threshold represented as a second self-constructing neural network, h b,e Boundary value represented as a second self-constructing neural network, u i,1max Maximum value that the control signal of all robot joints 1 is allowed to reach, u i,1min Minimum value that the control signal of all robot joints 1 is allowed to reach, u i,2max Maximum value that the control signal of all robot joints 2 is allowed to reach, u i,2min Minimum value that the control signal of all robot joints 2 is allowed to reach, u a0 Initial value of the performance function, u a∞ Final value of the performance function.

[0176] The initial values of the multi-robot system for this simulation experiment are [x d1,1 , x d1,2 , x d2,1 , x d2,2 , x d3,1 , x d3,2 , x d4,1 , x d4,2 ] T = [1, 1.7, 0.5, 2, -0.5, 0.5, -1, 1] T and [v d1,1 , v d1,2 , v d2,1 , v d2,2 , v d3,1 , v d3,2 , v d4,1 , v d4,2 ] T = [0.2, 1, -0.2, -1, 0.8, 7, 1, 8] T . The desired trajectories of the multi-robot system are set as 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 position of the first joint of the first robot, x d1,2 represents the initial position of the second joint of the first robot, x d2,1 represents the initial position of the first joint of the second robot, x d2,2 represents the initial position of the second joint of the second robot, xd3,1 represents the initial position of the first joint of the third robot arm, x d3,2 represents the initial position of the second joint of the third robot arm, x d4,1 represents the initial position of the first joint of the fourth robot arm, x d4,2 represents the initial position of the second joint of the fourth robot arm, v d1,1 represents the initial velocity of the first joint of the first robot arm, v d1,2 represents the initial velocity of the second joint of the first robot arm, v d2,1 represents the initial velocity of the first joint of the second robot arm, v d2,2 represents the initial velocity of the second joint of the second robot arm, v d3,1 represents the initial velocity of the first joint of the third robot arm, v d3,2 represents the initial velocity of the second joint of the third robot arm, v d4,1 represents the initial velocity of the first joint of the fourth robot arm, v d4,2 represents the initial velocity of the second joint of the fourth robot arm. s d1,1 represents the desired position of the first joint of all robot arms, s d1,2 represents the desired position of the second joint of all robot arms, s d2,1 represents the desired velocity of the first joint of all robot arms, s d2,2 represents the desired velocity of the second joint of all robot arms, s d3,1 represents the desired acceleration of the first joint of all robot arms, s d3,2 represents the desired acceleration of the second joint of all robot arms.

[0177] Figure 3 represents the topological communication relationship of the multi-robot arm system. The simulation results are shown in Figure 4 . Figure 5 The position tracking errors of joint 1 and joint 2 of all robot arms are shown. From the figure, it can be clearly observed that the performance constraint curve effectively defines the fluctuation range of the error, ensuring that the error remains within a controllable level. In particular, at the preset time t = 1.5 s, the positions of the four follower robot arms can accurately match the expected trajectory of the leader robot arm, fully verifying the high precision and rapid response capability of the system. Figure 6-7The velocity tracking error of joint 1 and joint 2 in the multi-robot system is presented. Before t = 1.5s, the error curve shows a certain degree of overshoot, which reflects the system's initial rapid response and adjustment process to the target velocity. However, it is worth noting that by t = 1.5s, the errors quickly converge and stabilize, and in the subsequent control process, the errors remain near zero and do not show significant fluctuations or jitter, which demonstrates the system's good stability and precise control performance. Figure 8 The error changes of joint 1 and joint 2 of all robots during the disturbance observation process are clearly shown. From the figure, we can find that before t = 1.5s, the error value experiences a gradual decay from high to low, which reflects the system's initial perception and response to external disturbances. When reaching the pre-set time point t = 1.5s, the error appears a short peak. However, after that, the error quickly and effectively converges to near zero, fully demonstrating the excellent fitting ability and control accuracy of the proposed time-regulated self-structured neural network observer in accurately capturing and compensating disturbances. Figure 9 The distribution of event-triggered time for joint 1 and joint 2 of all robots is shown in the form of a scatter plot. By using the proposed periodic monitoring event-triggering algorithm, the figure clearly shows that this algorithm can significantly reduce the number of communications between the actuators, thereby effectively reducing the consumption of communication resources. This result not only reflects the algorithm's advantage in optimizing the communication efficiency of the system, but also proves its practical application value in multi-robot collaborative control. Figure 10 The number of event-triggered times for joint 1 and joint 2 of all robots is shown. Figure 11-12 The number of neurons in the self-structured neural network is shown, where the number of neurons for joint 1 of the first robot, the second robot, the third robot, and the fourth robot decreases from 41 to 2, 4, 6, and 3, respectively, and the number of neurons for joint 2 of the first robot, the second robot, the third robot, and the fourth robot decreases from 41 to 2, 3, 4, and 5, respectively. Figure 13-14 The saturated control signals of joint 1 and joint 2 of all robots after the event-triggering mechanism are shown. From the figure, it is not difficult to see that before t = 1.5s, the control signal is limited in a certain range due to the constraint of the saturation function, and the introduction of the auxiliary system significantly improves the performance of the control system. At t = 1.5s, the control signal reaches its peak state, and then quickly and smoothly approaches the zero value region. In the subsequent control process, the control signal shows a segmented form and remains consistent, which indicates that the system has good stability and controllability in this stage.

[0178] Those skilled in the art will appreciate that the embodiments described herein are presented for purposes of illustration and that the inventive principles are not limited to these particular embodiments. Other variations and modifications can be made to the embodiments without departing from the spirit and scope of the inventive principles.

Claims

1. A multi-manipulator prescribed time event-triggered sliding mode control method, characterized in that, The method comprises the following steps: S1, establishing kinematic model and dynamic model for the leader and the follower of the multi-robot arm, and determining a prescribed time control criterion; S2, determining consistency error of the robot arm according to the prescribed time control criterion, the kinematic model and the dynamic model of the leader, and the kinematic model and the dynamic model of the follower; S3, determining error conversion function according to the consistency error of the robot arm and a performance function; S4, determining a non-singular terminal sliding mode surface according to the error conversion function; S5, constructing a first self-constructing neural network according to the non-singular terminal sliding mode surface, and determining a prescribed time self-constructing neural network observer; S6, constructing a second self-constructing neural network according to the prescribed time self-constructing neural network observer, and determining a prescribed time controller, thereby completing consistency tracking of the multi-robot arm; In the S1, the expression of the kinematic model of the follower is: ； wherein, denotes a first symmetric positive definite inertia matrix of the i-th dual-joint robot arm, denotes a second symmetric positive definite inertia matrix of the i-th dual-joint robot arm, denotes a third symmetric positive definite inertia matrix of the i-th dual-joint robot arm, denotes a fourth symmetric positive definite inertia matrix of the i-th dual-joint robot arm, denotes a first velocity of the i-th dual-joint robot arm, denotes a second velocity of the i-th dual-joint robot arm, denotes a first acceleration of the i-th dual-joint robot arm, denotes a second acceleration of the i-th dual-joint robot arm, denotes a first Coriolis force of the i-th dual-joint robot arm, denotes a first centrifugal force of the i-th dual-joint robot arm, denotes a second Coriolis force of the i-th dual-joint robot arm, denotes a second centrifugal force of the i-th dual-joint robot arm, denotes a first gravitational force of the i-th dual-joint robot arm, denotes a second gravitational force of the i-th dual-joint robot arm, denotes a first control signal of the i-th dual-joint robot arm after event-triggered processing, denotes a second control signal of the i-th dual-joint robot arm after event-triggered processing, denotes a first external disturbance to the i-th dual-joint robot arm, denotes a second external disturbance to the i-th dual-joint robot arm; In the S1, the expression of the dynamic model of the follower is: ； wherein, represents the position of the i-th robot arm at the λ-th joint, represents the velocity of the i-th robot arm at the λ-th joint, represents the first derivative of the velocity of the i-th robot arm at the λ-th joint, represents the first derivative of the position of the i-th robot arm at the λ-th joint, represents a set of model uncertainties of the i-th robot arm at the λ-th joint, represents the control signal of the i-th robot arm at the λ-th joint after event-triggered processing, represents the external time-varying disturbance on the i-th robot arm at the λ-th joint, represents the moment of inertia of the motor and link of the i-th robot arm at the λ-th joint; In the S1, the expression of the kinematic model of the leader is: ； wherein denotes the position first derivative of the leader robot arm, denotes the velocity of the λ-th joint of the leader robot arm; In the S1, the expression of the dynamic model of the leader is: ； wherein denotes the first derivative of the velocity of the leader robot arm, denotes the control signal of the λ-th joint of the leader robot arm; In the S1, the expression of the prescribed time control criterion is: ； wherein denotes a continuously differentiable function, denotes a time-varying function, is a first derivative of the time-varying function, denotes a time of the control process, denotes a first positive real constant, denotes a second positive real constant, denotes a prescribed time.

2. The multi-manipulator prescribed time event-triggered sliding mode control method according to claim 1, wherein, In the S2, the consistency error comprises position consistency error of the i-th robot arm and the λ-th joint, velocity consistency error of the i-th robot arm and the λ-th joint, and acceleration consistency error of the i-th robot arm and the λ-th joint. a position consistency error of the ith mechanical arm λth joint is expressed as: ； wherein denotes the number of manipulators, denotes the communication link between the i-th follower manipulator and the j-th follower manipulator, denotes the communication link between the leader manipulator and the i-th follower manipulator, denotes the position of the λ-th joint of the i-th manipulator, denotes the position of the λ-th joint of the j-th follower manipulator, denotes the position of the leader manipulator; a velocity consistency error of the i-th robot arm λ-th joint is expressed as: ； wherein denotes the velocity of the i-th robot arm at the λ-th joint, denotes the velocity of the j-th robot arm at the λ-th joint, denotes the velocity of the leader robot arm; an acceleration consistency error of the i-th robot arm λ-th joint The expression is: ； In the formula, denotes the acceleration of the i-th robot arm at the λ-th joint, denotes the acceleration of the j-th robot arm at the λ-th joint, denotes the acceleration of the leader robot arm.

3. The multi-manipulator prescribed time event-triggered sliding mode control method of claim 1, wherein, In the S3, the error conversion function is expressed as: ； wherein denotes the positional consistency error, denotes the performance function.

4. The multi-manipulator prescribed time event-triggered sliding mode control method of claim 1, wherein, In the S4, the nonsingular terminal sliding mode surface The expression is: ； wherein denotes the error transfer function, denotes the first derivative of the error transfer function, denotes a first positive real constant of the i-th robot arm at the λ-th joint, denotes a second positive real constant of the i-th robot arm at the λ-th joint, denotes a time varying function, denotes the first derivative of the time varying function.

5. The multi-manipulator prescribed time event-triggered sliding mode control method of claim 1, wherein, In the S5, the neural network is constructed The expression is: ； wherein denotes an input to the self-constructing neural network, denotes an estimation error of the self-constructing neural network, denotes an optimal weight, denotes a Gaussian kernel; In the S5, the prescribed time is from constructing the neural network disturbance observer The expression is: ； In the formula, represents an estimate value of a neural network approximation error, represents an estimate value of uncertainty, represents the moment of inertia of the motor and the connecting rod of the i-th robot arm λ-th joint, represents the control signal after input saturation, represents the seventh positive real constant of the i-th robot arm λ-th joint, represents the eighth positive real constant of the i-th robot arm λ-th joint, represents a time-varying function, represents the first derivative of the time-varying function, represents a state variable, represents an adjustable parameter.

6. The multi-manipulator prescribed time event-triggered sliding mode control method according to claim 1, wherein, The S6 comprises the following sub-steps: S61, constructing a second self-constructing neural network according to the prescribed time self-constructing neural network observer, and determining a prescribed time controller; S62, determining an auxiliary system and an event triggering mechanism; S63, using the prescribed time controller, the auxiliary system and the event triggering mechanism to make the consistency error converge to the zero domain within the prescribed time, thereby completing consistency tracking of the multi-robot arm.

7. The multi-manipulator prescribed time event-triggered sliding mode control method of claim 6, wherein, In the S61, the prescribed time controller The expression is: ； wherein denotes the moment of inertia of the motor and link of the i-th manipulator's λ-th joint, denotes the conversion error function, denotes the derivative of the conversion error function, denotes the number of manipulators, denotes the communication link between the i-th follower manipulator and the j-th follower manipulator, denotes the communication link between the leader manipulator and the i-th follower manipulator, denotes the performance function, denotes the first derivative of the performance function, denotes the second derivative of the performance function, denotes the first derivative of the velocity of the i-th manipulator's λ-th joint, denotes the first derivative of the velocity of the leader manipulator, denotes the position consistency error, denotes the first derivative of the position consistency error, denotes the first positive real constant of the i-th manipulator's λ-th joint, denotes the second positive real constant of the i-th manipulator's λ-th joint, denotes the eleventh positive real constant of the i-th manipulator's λ-th joint, denotes the twelfth positive real constant of the i-th manipulator's λ-th joint, denotes the time varying function, denotes the first derivative of the time varying function, denotes the nonsingular terminal sliding mode surface, denotes the estimate of the first optimal weight, denotes the first Gaussian kernel function, denotes the first approximation error, denotes the estimate of the external disturbance of the i-th manipulator's λ-th joint.

8. The multi-manipulator prescribed time event-triggered sliding mode control method of claim 6, wherein, In the S62, the expression of the auxiliary system update rate is: ； wherein denotes the moment of inertia of the motor and the link of the i-th manipulator's λ-th joint, denotes the conversion error, denotes the derivative of the conversion error, denotes the number of manipulators, denotes the communication link between the i-th follower manipulator and the j-th follower manipulator, denotes the communication link between the leader manipulator and the i-th follower manipulator, denotes the input difference after input saturation, denotes the first approximation error, denotes the auxiliary system, denotes the thirteenth positive real constant of the i-th manipulator's λ-th joint, denotes the fourteenth positive real constant of the i-th manipulator's λ-th joint; In the S62, the event triggering mechanism The expression of the formula is: ； wherein denotes a first positive design parameter, denotes a second positive design parameter, denotes a third positive design parameter, denotes a fourth positive design parameter, denotes a time of a control process, denotes a control signal after input saturation, denotes a transpose of a nonsingular terminal sliding surface, denotes a natural number, denotes a monitoring period, denotes a control signal of a next triggering time after event triggering, denotes a control signal of a current triggering time after event triggering, denotes a current triggering time, denotes a next triggering time, denotes a last triggering time, denotes a triggering condition defined by a measurement error.

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