Flexible arm space robot torque double-domain constraint control method based on improved FWN

Abstract

The invention discloses an improved FWN-based torque double-domain constraint control method for a flexible arm space robot, and the method comprises the steps: constructing a dynamic model of a flexible arm space robot system, and constructing a torque double-domain constraint control system mathematical model in combination with the posture of a body and the constraint of the amplitude and the amplitude change rate of the control torque of a joint hinge; a torque double-domain constraint controller based on the improved fuzzy wavelet network is constructed, the torque double-domain constraint controller meets a torque double-domain constraint control system mathematical model, and corresponding control laws comprise an improved fuzzy wavelet network control law and an improved fuzzy wavelet network parameter self-adjusting control law; and the body attitude angle and the joint angle of the robot and the deformation coordinate and the vibration amplitude of the flexible arm are adjusted through the control torque which is generated by the torque double-domain constraint controller and subjected to double-domain constraint till the control target is achieved. According to the method, the problems of saturation effect and dynamic performance degradation of the actuator caused by insufficient consideration of double-domain constraints of the torque amplitude and the change rate of the joint driving mechanism are solved.

Description

Technical Field

[0001] The present invention relates to the field of robotic arm control, and particularly to a flexible arm space robot torque dual-domain constraint control method for improving FWN. Background Art

[0002] In the long-term on-orbit service scenario of flexible arm space robots, the physical characteristics of the actuators impose strict dual constraints on the output torque of the joint actuators: one is the saturation limit of the torque amplitude, and the other is the dynamic threshold of the torque amplitude change rate. Torque amplitude saturation will significantly reduce the system control performance, leading to trajectory tracking deviation and even instability risks; while the sudden increase in the torque amplitude change rate will induce high-frequency tremors, exacerbate mechanical losses and threaten the mission reliability. How to coordinate the dual-domain dynamic constraints of torque amplitude - change rate has become the core challenge of high-precision anti-saturation control for space robots.

[0003] In current engineering practices, there are still the following technical bottlenecks for the above problems:

[0004] (1) Insufficient dynamic decoupling of dual-domain constraints: Existing anti-saturation control studies mostly focus on the single torque amplitude limited scenario, suppressing the problem of amplitude overrun through saturation compensation or amplitude limiting design, but lack the ability to dynamically model the time-varying boundaries of the torque amplitude change rate. The control strategy under the coupled constraints of amplitude - change rate is prone to cause high-frequency oscillations or response lags, and it is difficult to achieve adaptive coordinated regulation of the dual-domain boundaries.

[0005] (2) Limited real-time performance of intelligent methods: Although the nonlinear approximation method based on RBF neural network can partially compensate for system uncertainties, its multi-layer feedforward structure and gradient backpropagation mechanism lead to high computational complexity, making it difficult to meet the real-time calculation requirements of on-board computers and restricting the engineering feasibility of its on-orbit application.

[0006] (3) Poor time-frequency adaptability of fuzzy systems: Traditional fuzzy control systems rely on fixed membership functions and linear consequent structures, and cannot effectively capture the transient time-frequency characteristics of torque constraints; the lack of static characteristics of its rule base and wavelet multi-scale analysis ability further weakens the dynamic response accuracy and control robustness of the system to non-stationary mutation signals.

[0007] The above problems seriously restrict the control performance of space robots under complex dynamic working conditions. There is an urgent need for a new intelligent control architecture with high real-time performance, strong nonlinear approximation ability and multi-scale analysis characteristics to break through the dual technical barriers of coordinated optimization of dual-domain constraints and limited on-board computing resources. Summary of the Invention

[0008] The purpose of this application is to propose a flexible arm space robot torque dual-domain constraint control method based on improved FWN for the above-mentioned technical problems.

[0009] In a first aspect, the present invention provides a flexible arm space robot force and torque dual-domain constraint control method based on an improved FWN, comprising the following steps:

[0010] Construct a dynamic model of a flexible arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints. Under the condition that the attitude of the body and the control torque of the joint hinge in the flexible arm space robot system are restricted by the amplitude and the rate of change of the amplitude, combine the dynamic model of the flexible arm space robot system to construct a mathematical model of a torque dual-domain constraint control system;

[0011] Construct a torque dual-domain constraint controller based on an improved fuzzy wavelet network. The torque dual-domain constraint controller satisfies the mathematical model of the torque dual-domain constraint control system, and its corresponding control law includes an improved fuzzy wavelet network control law and a parameter self-adjusting control law of the improved fuzzy wavelet network;

[0012] Obtain the rotation angle vector formed by the body attitude angle and two joint angles at the current moment and the vibration mode coordinates of the flexible manipulator calculated based on the coordinates and amplitudes of the flexible manipulator, and construct an input vector; input the input vector into the torque dual-domain constraint controller. First, use the parameter self-adjusting control law of the improved fuzzy wavelet network to update the parameters of the improved fuzzy wavelet network, and then use the improved fuzzy wavelet network control law with the updated parameters of the improved fuzzy wavelet network to predict the controlled torque after dual-domain constraint at the next moment, and apply the controlled torque after dual-domain constraint at the next moment to the flexible arm space robot system to adjust the body attitude angle, two joint angles, and the coordinates and amplitudes of the flexible manipulator at the next moment.

[0013] Preferably, constructing a dynamic model of a flexible arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints specifically includes:

[0014] The flexible arm space robot system includes a freely floating body, a rigid manipulator, and a flexible manipulator. There are joints between the body and the rigid manipulator and between the rigid manipulator and the flexible manipulator, and each joint is hinged through a joint hinge;

[0015] Adopt the assumed mode method to reduce the order of the flexible manipulator, intercept the first two dominant modes to characterize the elastic deformation characteristics of the flexible manipulator, and based on the law of conservation of momentum and the second Lagrangian equation, establish a dynamic model of the flexible arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints, as shown in the following formula

[0016] ;

[0017] Wherein, represents a symmetric positive definite inertia matrix, represents the angular variable, 、 and respectively represent the body attitude angle, the joint angle between the body and the rigid manipulator, and the joint angle between the rigid manipulator and the flexible manipulator, and respectively represent the first derivative and the second derivative of the angular vector with respect to time, represents the mode coordinate of the flexible manipulator, and respectively represent the mode coordinates of the first mode and the second mode. T represents the transpose, and respectively represent the first derivative and the second derivative of the mode coordinate of the flexible manipulator with respect to time, represents the column vector containing the centrifugal force and the Coriolis force, is the flexible stiffness matrix, represents the attitude of the unconstrained body and the control torque of the joint hinge; represents the set of real numbers.

[0018] Preferably, under the condition that the control torque of the attitude of the body and the joint hinge in the flexible arm space robot system is subject to the constraints of the amplitude and the amplitude change rate, a mathematical model of the torque double-domain constrained control system is constructed in combination with the dynamic model of the flexible arm space robot system, specifically including:

[0019] Assume that at the unconstrained control torque is subject to the constraints of the amplitude and the amplitude change rate, that is:

[0020] , ;

[0021] wherein, respectively represent the known lower amplitude saturation limit value and the upper amplitude saturation limit value, respectively represent the known lower amplitude change rate saturation limit value and the upper amplitude change rate saturation limit value;

[0022] To process the constraints of the amplitude and the amplitude change rate, the following functions are defined:

[0023] ;

[0024] wherein, represents the control torque after double-domain constraint output by the torque double-domain constraint controller, represents the joint constraint upper limit, and its expression is: , Denote the control torque at a moment, denote the sampling time interval, denote taking the minimum value among them, denote the lower bound of the joint constraint, and its expression is: , denote taking the maximum value among them;

[0025] Design the mathematical model of the torque dual-domain constraint control system in combination with the dynamic model of the flexible-arm space robot system, as shown in the following formula:

[0026] .

[0027] Preferably, in the improved fuzzy wavelet network, the improved fuzzy wavelet network includes an input layer, a reconstructed fuzzification layer, a reconstructed rule layer, and a reconstructed wavelet function layer;

[0028] In the reconstructed fuzzification layer, replace the antecedent membership function in the fuzzification layer of the traditional fuzzy wavelet network with a wavelet basis function, as shown in the following formula:

[0029] ;

[0030] Among them, is the mother wavelet function, and multi-scale wavelet basis functions are generated through stretching and translation operations , denote the membership function; is the stretching parameter, is the translation parameter, denote the set of real numbers; denote the th input variable, , is the input dimension, denote the index of the fuzzy subset, , is the number of fuzzy subsets of each input variable, denote the index of the translation parameter of the wavelet basis function, , is the total number of indices of the translation parameter;

[0031] In the reconstructed rule layer, define the th fuzzy rule as:

[0032] If is and is ... and is , then ;

[0033] Among them, is the rule excitation intensity, represents the th rule, and is the weight of the th input variable on the , is the total number of rules, represents the th output variable, , is the output dimension, represents the index of the wavelet decomposition scale, , is the total number of wavelet decomposition scales;

[0034] Combining the rule excitation intensity with the wavelet basis function, the fuzzy wavelet basis function is defined as follows:

[0035] ;

[0036] Among them, represents the input vector of the improved fuzzy wavelet network, including input variables;

[0037] In the reconstructed wavelet function layer, through the matrix multiplication of the weight matrix and the fuzzy wavelet basis function vector , the output variables of all rules are weighted and combined, and the output vector of the improved fuzzy wavelet network is represented in vector form as follows:

[0038] ;

[0039] Among them, represents the weight matrix, represents the th rule's weight on the th output variable, represents the output vector of the improved fuzzy wavelet network, including output variables.

[0040] Preferably, a torque dual-domain constraint controller based on the improved fuzzy wavelet network is constructed, specifically including:

[0041] Using the output variables of the improved fuzzy wavelet network to establish an ideal control law:

[0042] ;

[0043] Among them, represents the ideal control torque after double-domain constraint, which is a non-linear uncertainty term; , is an unknown ideal weight matrix, represents the fuzzy wavelet basis function vector corresponding to the input vector at time represents a small error vector with boundedness. Respectively, is estimated as , and further, the control torque after double-domain constraint output by the torque double-domain constraint controller is used as the estimated value of, then the ideal control law is modified to the improved fuzzy wavelet network control law as shown in the following equation:

[0044] ;

[0045] Therefore, the control torque after double-domain constraint output by the improved fuzzy wavelet network predictive torque double-domain constraint controller can be obtained.

[0046] Preferably, the parameter self-adjusting control law of the improved fuzzy wavelet network is as follows:

[0047] When , or and , or and , based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as:

[0048] ;

[0049] Among them, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, , corresponding to the attitude of the body and two joint hinges respectively, represents the first derivative of with respect to time, represents the adaptive gain coefficient, ; represents the adjustment gain, ;

[0050] When and or and When, based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as:

[0051] ;

[0052] wherein, represents the robustness compensation coefficient, .

[0053] Compared with the prior art, the present invention has the following beneficial effects:

[0054] (1) The method for dual-domain constraint control of the flexible-arm space robot force and torque proposed by the present invention constructs a mathematical model of the torque dual-domain constraint control system on the basis of the dynamic model of the flexible-arm space robot system and combines the constraint conditions of the amplitude and the amplitude change rate. The torque dual-domain constraint is modeled as a time-varying boundary condition, and the constraint weight distribution is optimized in real time through a fuzzy rule base, so as to achieve the coordinated balance of trajectory tracking and elastic deformation while ensuring the physical limit of the actuator.

[0055] (2) The core innovation of the method for dual-domain constraint control of the flexible-arm space robot force and torque proposed by the present invention is achieved through the three-level collaborative reconstruction of the improved fuzzy wavelet network: First, the wavelet basis function is used to replace the traditional Gaussian function as the membership function, and the adaptive mechanism of the scaling parameter and the translation parameter is introduced, breaking through the limitation of the traditional method in time-frequency resolution, so as to accurately capture the transient components in the non-stationary signal of the system; Second, a real-coefficient product-type consequent architecture is created, and the output of the rule layer is reconstructed by non-linearly weighting and combining the input vectors, significantly enhancing the modeling ability of the strong coupling complex dynamics of the flexible-arm space robot joint torque-pose; Finally, a fuzzy wavelet basis function fusion mechanism is constructed, the rule excitation intensity is fused with the multi-scale time-frequency features, and the weight vector is used to realize the cross-level interaction between the global fuzzy inference and the local time-frequency features, and finally an intelligent control kernel with time-frequency analysis ability is formed. The entire reconstruction process establishes a collaborative optimization mechanism of the dynamic scale factor and the parameter space through dual-mode fusion, providing a general solution for high-precision control under complex working conditions while maintaining the good interpretability of the fuzzy system.

[0056] (3) The flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN proposed by the present invention designs a torque dual-domain constraint controller based on the improved fuzzy wavelet network through the mathematical model of the torque dual-domain constraint control system, and uses its corresponding improved fuzzy wavelet network control law and the parameter self-regulation control law of the improved fuzzy wavelet network to generate the controlled torque after dual-domain constraint. By means of the cooperation of wavelet network frequency domain feature extraction and fuzzy inference dynamic constraint, the anti-saturation ability and dynamic response robustness of the system are significantly improved. Description of the Drawings

[0057] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0058] Figure 1 It is a schematic flowchart of the flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN in the embodiments of the present application;

[0059] Figure 2 It is a schematic structural diagram of the flexible arm space robot system of the flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN in the embodiments of the present application;

[0060] Figure 3 It is a schematic structural diagram of the fuzzy wavelet network of the flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN in the embodiments of the present application;

[0061] Figure 4 It is a schematic structural principle diagram of the torque dual-domain constraint controller of the flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN in the embodiments of the present application;

[0062] Figure 5 It is the trajectory tracking diagram of the body attitude angle of Simulation 1 in the embodiments of the present application ;

[0063] Figure 6 It is the trajectory tracking diagram of the joint angle of Simulation 1 in the embodiments of the present application ;

[0064] Figure 7 It is the time-varying characteristic curve diagram of the vibration mode coordinates of the flexible manipulator of Simulation 1 in the embodiments of the present application;

[0065] Figure 8 It is the result diagram of the control torque of the body attitude angle of Simulation 1 in the embodiments of the present application ;

[0066] Figure 9 The joint angle of Simulation 1 of the embodiment of the present application Result graph of the control torque;

[0067] Figure 10 The body attitude angle of Simulation 1 of the embodiment of the present application Result graph of the amplitude change rate of the control torque;

[0068] Figure 11 The joint angle of Simulation 1 of the embodiment of the present application Result graph of the amplitude change rate of the control torque;

[0069] Figure 12 Topological graph of the motion trajectory of the flexible arm space robot in Simulation 1 of the embodiment of the present application

[0070] Figure 13 The body attitude angle of Simulation 2 of the embodiment of the present application Trajectory tracking graph;

[0071] Figure 14 The joint angle of Simulation 2 of the embodiment of the present application Trajectory tracking graph;

[0072] Figure 15 Time-varying characteristic curve graph of the vibration mode coordinates of the flexible manipulator in Simulation 2 of the embodiment of the present application

[0073] Figure 16 The body attitude angle of Simulation 2 of the embodiment of the present application Result graph of the control torque;

[0074] Figure 17 The joint angle of Simulation 2 of the embodiment of the present application Result graph of the control torque;

[0075] Figure 18 The body attitude angle of Simulation 2 of the embodiment of the present application Result graph of the amplitude change rate of the control torque;

[0076] Figure 19 The joint angle of Simulation 2 of the embodiment of the present application Result graph of the amplitude change rate of the control torque. Detailed implementation manner

[0077] To make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0078] Figure 1 An improved FWN-based flexible arm space robot force and torque dual-domain constraint control method provided by an embodiment of the present application is shown, including the following steps:

[0079] S1. Construct a dynamic model of a flexible arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints. Combine the dynamic model of the flexible arm space robot system to construct a mathematical model of a torque dual-domain constraint control system under the condition that the attitude of the body and the control torque of the joint hinge in the flexible arm space robot system are constrained by the amplitude and the amplitude change rate.

[0080] In a specific embodiment, constructing a dynamic model of a flexible arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints specifically includes:

[0081] The flexible arm space robot system includes a freely floating body, a rigid manipulator arm, and a flexible manipulator arm. There are joints between the body and the rigid manipulator arm and between the rigid manipulator arm and the flexible manipulator arm, and each joint is hinged through a joint hinge;

[0082] The assumed mode method is used to reduce the order of the flexible manipulator arm, and the first two dominant modes are intercepted to characterize the elastic deformation characteristics of the flexible manipulator arm. Based on the law of conservation of momentum and the second Lagrange equation, a dynamic model of the flexible arm space robot system under the conditions that no active control is applied to the translational degrees of freedom and the attitude motion has servo constraints is established as shown in the following formula

[0083] ;

[0084] Wherein, represents a symmetric positive definite inertia matrix, represents the angular displacement variable, , and respectively represent the body attitude angle, the joint angle between the body and the rigid manipulator arm, and the joint angle between the rigid manipulator arm and the flexible manipulator arm, and respectively represent the first-order derivative and the second-order derivative of the angular displacement vector with respect to time, represents the vibration mode coordinate of the flexible manipulator arm, and respectively represent the modal coordinates of the first-order mode and the modal coordinates of the second-order mode, T represents the transpose, and respectively represent the modal coordinates of the flexible manipulator the first derivative and the second derivative with respect to time, represents the column vector containing the centrifugal force and the Coriolis force, is the flexible stiffness matrix, represents the attitude of the unconstrained body and the control torque of the joint hinge; represents the set of real numbers.

[0085] In a specific embodiment, under the constraint conditions of the amplitude and the amplitude change rate of the attitude of the body and the control torque of the joint hinge in the flexible arm space robot system, a mathematical model of the torque double-domain constraint control system is constructed in combination with the dynamic model of the flexible arm space robot system, specifically including:

[0086] Assume that at the unconstrained control torque at time is subject to the constraints of the amplitude and the amplitude change rate, that is:

[0087] , ;

[0088] Among them, respectively represent the known lower amplitude saturation limit value and the upper amplitude saturation limit value, respectively represent the known lower amplitude change rate saturation limit value and the upper amplitude change rate saturation limit value;

[0089] To process the constraints of the amplitude and the amplitude change rate, the following functions are defined:

[0090] ;

[0091] Among them, represents the control torque after the double-domain constraint, represents the joint constraint upper limit, and its expression is: , represents the control torque at time represents the sampling time interval, represents taking the minimum value among them, represents the joint constraint lower limit, and its expression is: , represents taking the maximum value among them;

[0092] Combined with the dynamic model of the flexible arm space robot system, a mathematical model of the torque double-domain constraint control system is designed as shown in the following formula:

[0093] 。

[0094] Specifically, referring to Figure 2 , the flexible arm space robot system in the embodiments of the present application consists of a freely floating body , a rigid robotic arm and a flexible robotic arm . The flexible robotic arm can be regarded as a cantilever beam installed on a joint hinge connected to the rigid robotic arm . is the flexible robotic arm. During the movement, it will inevitably deform. However, since it is a slender rod, the influence of its axial deformation and shear deformation can be ignored, and it can be regarded as an Euler - Bernoulli beam for processing. Establish the principal axis body - fixed right - hand coordinate system of each component , where coincides with the centroid of . and are respectively the centers of the joint hinges connecting and , and and . is the axis of symmetry of the rigid robotic arm . passes through and is consistent with the axis of symmetry of when it is undeformed. Let the distance between and be . The centroid of and the distance between be . The lengths of and are respectively and . At time , The mass and the central inertia tensor of and ; The mass and the central inertia tensor of and ; The mass per unit length of is , and the uniform bending stiffness is is the total centroid of the system, is the total mass of the system.

[0095] The assumed mode method is used to reduce the order of the flexible manipulator, and its first two dominant modes are selected to characterize the elastic deformation characteristics. The complete motion state of the flexible arm space robot system is described by a set of generalized coordinates, which includes three types of independent variables: the body attitude angle, two joint angles and and the mode coordinates of the flexible manipulator ; among them, the body attitude angle is used to define the basic attitude of the system, and the two joint angles and characterize the relative rotation of the two joints of the manipulator. The mode coordinates of the flexible manipulator describe its dynamic behavior under elastic vibration by reflecting the natural vibration modes of each order of the flexible manipulator. Based on the law of conservation of momentum and the second Lagrange equation, the dynamic model of the flexible arm space robot system is established. This dynamic model considers that the translational degrees of freedom are not actively controlled, the attitude motion has servo constraint conditions, and fully reflects the strong coupling effect brought by the flexible manipulator.

[0096] By means of frequency-domain constraint equivalent mapping, the amplitude change rate constraint is converted into an amplitude constraint, as shown in the following formula:

[0097] ;

[0098] For a relatively small sampling time interval , .

[0099] From the constraint functions of the amplitude and the amplitude change rate, it can be seen that if the condition is satisfied, the amplitude and the change rate of the control torque of the joint actuator can be strictly limited within the preset saturation range. Assuming that the dynamic range of the control torque set of the ideal joint actuator is limited within the amplitude-change rate joint feasible region , then the compatibility criterion can be derived based on this.

[0100] Combined with the above dynamic model of the flexible arm space robot system, the mathematical model of the torque double-domain constraint control system is designed. Therefore, the torque double-domain constraint controller corresponding to the mathematical model of the torque double-domain constraint control system can be designed, and It is the control torque after double-domain constraint output by the torque double-domain constraint controller corresponding to the mathematical model of the torque double-domain constraint control system. This torque double-domain constraint controller captures the flexible vibration characteristics through the time-frequency analysis of the wavelet network, embeds the fuzzy logic into the double-domain constraint rules, and combines the adaptive learning to compensate for the uncertainty of the model. Finally, it realizes the constraint-feasible and robustly stable control in the strongly coupled and underactuated flexible-arm space robot system. Its core innovation lies in the deep integration of the frequency-domain constraint equivalence and the intelligent control architecture, breaking through the conservatism of the traditional saturation control.

[0101] S2. Construct a torque double-domain constraint controller based on the improved fuzzy wavelet network. The torque double-domain constraint controller satisfies the mathematical model of the torque double-domain constraint control system, and its corresponding control law includes the improved fuzzy wavelet network control law and the parameter self-regulation control law of the improved fuzzy wavelet network.

[0102] In a specific embodiment, in the improved fuzzy wavelet network, the improved fuzzy wavelet network includes an input layer, a reconstructed fuzzification layer, a reconstructed rule layer, and a reconstructed wavelet function layer;

[0103] In the reconstructed fuzzification layer, the antecedent membership function in the fuzzification layer of the traditional fuzzy wavelet network is replaced by a wavelet basis function, as shown in the following formula:

[0104] ;

[0105] Where, is the mother wavelet function, and multi-scale wavelet basis functions are generated through stretching and translation operations , represents the membership function; is the stretching parameter, is the translation parameter, represents the set of real numbers; represents the th input variable, , is the input dimension, represents the index of the fuzzy subset, , is the number of fuzzy subsets of each input variable, represents the index of the translation parameter of the wavelet basis function, , is the total number of indices of the translation parameter;

[0106] In the reconstructed rule layer, define the th fuzzy rule as:

[0107] If is and is ... and is then ;

[0108] wherein, is the rule excitation strength, represents the weight of the th input variable to the th output variable in the th rule; , is the total number of rules, represents the th output variable, , is the output dimension, represents the index of the wavelet decomposition scale, , is the total number of wavelet decomposition scales;

[0109] Combining the rule excitation strength with the wavelet basis function, the fuzzy wavelet basis function is defined as follows:

[0110] ;

[0111] wherein, represents the input vector of the improved fuzzy wavelet network, including input variables;

[0112] In the reconstructed wavelet function layer, through the matrix multiplication of the weight matrix and the fuzzy wavelet basis function vector , the output variables of all rules are weighted and combined, and the output vector of the improved fuzzy wavelet network is represented in vector form as follows:

[0113] ;

[0114] wherein, represents the weight matrix, represents the weight of the th rule to the th output variable, represents the output vector of the improved fuzzy wavelet network, including output variables.

[0115] In a specific embodiment, a torque double-domain constraint controller based on the improved fuzzy wavelet network is constructed, specifically including:

[0116] Establishing an ideal control law using the output variables of the improved fuzzy wavelet network:

[0117] ;

[0118] wherein, represents the ideal control torque after double-domain constraint, and is a non-linear uncertainty term; , is an unknown ideal weight matrix, represents the fuzzy wavelet basis function vector corresponding to the input vector at time represents a small error vector with boundedness. Respectively, is estimated as , and further, the control torque after double-domain constraint output by the torque double-domain constraint controller is used as the estimated value of, then the ideal control law is modified to the improved fuzzy wavelet network control law as shown in the following formula:

[0119] ;

[0120] Therefore, the control torque after double-domain constraint can be predicted by the improved fuzzy wavelet network.

[0121] In a specific embodiment, the parameter self-adjusting control law of the improved fuzzy wavelet network is as follows:

[0122] When , or and , or and , based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as:

[0123] ;

[0124] wherein, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, represents the sub-vector of, , corresponding to the attitude of the body and two joint hinges respectively, represents the first derivative of with respect to time, represents the adaptive gain coefficient, ; represents the adjustment gain, ;

[0125] When and or and At this time, based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as:

[0126] ;

[0127] Wherein, represents the robustness compensation coefficient, .

[0128] In the embodiment of the present application, the improved fuzzy wavelet network (improved FWN) retains the basic four-layer architecture (input layer → fuzzification layer → rule layer → wavelet function layer), and realizes a breakthrough innovation by reconstructing the coupling mechanism among the three layers (fuzzification layer, rule layer, wavelet function layer):

[0129] (1) Fuzzification layer innovation: Using wavelet basis functions to replace traditional Gaussian functions as membership functions, and through the dynamic adaptive adjustment of parameters, effectively breaking through the time-frequency resolution limitation of Gaussian functions, so as to accurately extract the transient characteristics in the non-stationary signals of the system;

[0130] (2) Rule layer reconstruction, establishing a real coefficient product-type nonlinear consequent structure, which significantly enhances the accurate modeling ability for strongly coupled complex dynamics such as flexible manipulators;

[0131] (3) Wavelet function layer collaborative innovation: Constructing fuzzy wavelet basis functions (FWBF), this mechanism effectively integrates the excitation intensity information of the rule layer and the multi-scale time-frequency characteristics of the wavelet function layer, and with the help of the weight matrix, realizes the deep coupling of global fuzzy reasoning and local time-frequency characteristics in a multi-input multi-output (MIMO) system, and finally comprehensively improves the control accuracy and dynamic response performance of the entire control system.

[0132] The specific description is as follows:

[0133] First of all, the embodiment of the present application reconstructs the fuzzification layer, establishes a reconstructed fuzzification layer, and replaces the antecedent membership function from a Gaussian function with a wavelet basis function with excellent time-frequency localization characteristics, so as to overcome the problem that in traditional fuzzy systems, Gaussian-type functions are usually used as membership functions, and the energy dispersion characteristics of Gaussian-type functions in the time-frequency domain make it difficult to accurately capture the rapidly changing local characteristics in non-stationary signals, resulting in insufficient feature extraction ability. The embodiment of the present application utilizes the time-frequency localization characteristics of wavelet basis functions to accurately capture the transient characteristics in signals. And by adaptively adjusting the scaling parameter and the translation parameter , enabling the membership function to have the ability to dynamically respond to changes in the input signal.

[0134] Secondly, in the reconstructed rule layer of this application, by introducing a non-linear multiplicative consequent structure, the modeling and expression ability for complex non-linear mapping relationships (such as the strong coupling between torque and position in the dynamics of a flexible-arm space robot system) is significantly enhanced. While the output in the rule layer of the traditional Mandani fuzzy system is calculated through center-average defuzzification (or similar linear weighting methods), its ability to express complex non-linear mapping relationships is limited.

[0135] Finally, in the reconstructed wavelet function layer of this application, the excitation intensity output by the rule layer is combined with the wavelet basis function to define the fuzzy wavelet basis function (FWBF). Finally, the FWN structure with inputs and outputs is as shown in Figure 3 . For the convenience of designing the torque double-domain constraint controller, the final output of the improved FWN is represented in vector form: the wavelet function layer performs a matrix multiplication of the weight matrix and the fuzzy wavelet basis function vector to perform a weighted combination of the outputs of all rules. Compared with neural networks, wavelet networks can achieve the same approximation quality with a smaller network scale. Therefore, the improved FWN is used as an approximator in the design of the torque double-domain constraint controller.

[0136] Specifically, the basic idea of designing a torque double-domain constraint controller based on an improved fuzzy wavelet network is: first, using the dynamic model of the flexible-arm space robot system and the physical constraints of the actuator, combined with the frequency-domain constraint equivalent mapping principle and the joint constraint function, to establish a mathematical model of the torque double-domain constraint control system; furthermore, based on this mathematical model of the torque double-domain constraint control system, a torque double-domain constraint controller based on an improved fuzzy wavelet network is designed, and its principle structure is as shown in Figure 3 . The ultimate goal is to achieve safe and feasible tracking control of the desired trajectory of the flexible-arm space robot under strict physical constraints and effectively suppress the vibration of the flexible arm.

[0137] For the mathematical model of the torque double-domain constraint control system, first define the expected angular displacement variables of the body and joints and their tracking errors . The derivatives of all orders of the expected angular displacement variables are continuously differentiable and satisfy and exist and are continuous, and the velocity error , then the error dynamic equation is: . Under the constraint conditions of the known dynamic model of the flexible-arm space robot system, by constructing a positive definite diagonal sliding mode surface , is a positive definite diagonal gain matrix, a quantization benchmark for the motion state deviation is established, and then its time derivative operation is performed to reveal the dynamic characteristics of the system. When the mechanical system model is accurate and satisfies the mathematical conditions, a globally stable linear state feedback controller can be designed based on the stability theory, and this method is listed as a standard solution in authoritative international journals.

[0138] Taking the first derivative with respect to time gives: Taking the first derivative with respect to time gives:

[0139] ;

[0140] Combining the mathematical model of the torque double-domain constraint control system with the above equation, we get:

[0141] ;

[0142] Transforming it gives the following equation:

[0143] ;

[0144] Therefore, we can obtain: ;

[0145] When the mechanical system model is accurate and satisfies the mathematical conditions, based on the stability theory, a linear state feedback controller is designed, and its corresponding control law is:

[0146] ;

[0147] where is a symmetric positive definite gain matrix, represents a diagonal matrix, represents submatrix of, , ;

[0148] From this, we can obtain:

[0149] Define the Lyapunov function :

[0150] ;

[0151] Calculate the total derivative of , and from and , we get: .

[0152] Therefore, according to the Lyapunov stability theory, on the premise that the system model is accurately known, through the linear state feedback controller, if the symmetric positive definite gain matrix and can ensure the asymptotic stable convergence of the tracking error of the closed-loop control system. Among them, the feedforward term of the linear state feedback controller can accurately compensate for the system dynamics, while the feedback term can ensure the stability of the error.

[0153] However, in the actual application of space robots, the high complexity of its system structure often makes it difficult to accurately identify the dynamic parameters, and it has become the norm that the parameters are unknown or have uncertainties. In this context, if the control law corresponding to the linear state feedback controller is directly used for design, aiming to control the flexible-arm space robot system with unknown parameters while meeting the torque double-domain constraints, the control effect usually fails to meet the expectations.

[0154] Referring to Figure 4 , the embodiments of this application utilize the improved FWN to efficiently approximate the nonlinear uncertainties . Compared with traditional neural networks, the improved FWN can significantly improve the approximation efficiency while streamlining the network structure by integrating the semantic interpretability of fuzzy logic and the time-frequency localization characteristics of wavelet transform: the antecedent of its fuzzy rules uses adaptively adjustable wavelet basis functions to capture the multi-scale dynamic characteristics of the uncertainties, and the consequent structure realizes the accurate modeling of the nonlinear coupling relationship through parameter optimization, and finally achieves the efficient online estimation of complex uncertainties with a low-dimensional rule base.

[0155] Furthermore, aiming at the actuator saturation effect and the deterioration of control accuracy caused by the double-domain constraints of the control torque amplitude and its change rate in the flexible-arm space robot system, a parameter adaptive compensation mechanism is constructed based on the Lyapunov stability theory, that is, a parameter self-adjusting control law of the improved fuzzy wavelet network is proposed. The introduction of in the parameter self-adjusting control law of the improved fuzzy wavelet network is to improve the robustness of the controller and ensure is bounded.

[0156] The proof process of Lyapunov stability is as follows:

[0157] Theorem 1: Based on the synergistic effect of the control law of the improved fuzzy wavelet network and the parameter self-adjusting control law of the improved fuzzy wavelet network, it is proved by the stability theory that the state of the flexible-arm space robot system has global convergence, while ensuring that the tracking error converges within the preset accuracy range and strictly satisfying the dynamic constraint conditions of the control torque amplitude and its change rate: ;

[0158] Proof: Add and subtract on the right side of

[0159] ;

[0160] Substitute , and into the above formula, we can get:

[0161] ;

[0162] Construct the Lyapunov function as shown in the following formula:

[0163] ;

[0164] where represents the Lyapunov function, is the estimated value, and the estimation error is defined as .

[0165] Take the first derivative of the above formula with respect to time, we can get:

[0166] ;

[0167] where represents the first derivative of with respect to time, represents

[0168] 1) When the control torque is within the dynamic amplitude range, , that is, no saturation constraint occurs, there is . Substitute into , we can deduce:

[0169] ;

[0170] 2) When and , or and , substitute into , we can deduce:

[0171] ;

[0172] 3) When and , or and , substitute into , we can deduce:

[0173] ;

[0174] Due to the assumption , there is , so the above formula becomes:

[0175] ;

[0176] It is observed that the formulas derived in 1), 2), and 3) above have a consistent parameter coupling structure. To improve the writing efficiency, the formula derived in 1) will be used as a representative case in the subsequent analysis to systematically elaborate its dynamic characteristics and convergence conditions.

[0177] From the inequalities and , the formula derived in 1) can be rewritten as:

[0178] ;

[0179] where is the adjustment gain, is the adjustment gain, is the weight coefficient, , so it can be obtained that:

[0180] ;

[0181] where , the Lyapunov function satisfies the following conditions:

[0182] ;

[0183] It can be seen from the above formula that and are uniformly bounded, converges to the interval .

[0184] S3. Obtain the rotation angle vector formed by the body attitude angle and two joint angles at the current moment, and the vibration mode coordinates of the flexible manipulator calculated based on the coordinates and amplitudes of the flexible manipulator, and construct the input vector; input the input vector into the torque double-domain constraint controller. First, update the parameters of the improved fuzzy wavelet network using the parameter self-adjustment control law of the improved fuzzy wavelet network, and then predict the control torque after double-domain constraint at the next moment using the improved fuzzy wavelet network control law with the updated parameters of the improved fuzzy wavelet network. Apply the control torque after double-domain constraint at the next moment to the flexible arm space robot system to adjust the body attitude angle, two joint angles, and the coordinates and amplitudes of the flexible manipulator at the next moment.

[0185] Specifically, the control method takes the system state at the current moment as the input: the rotation angle vector composed of the body attitude angle and two joint angles at the current moment is collected, and the vibration mode coordinates of the flexible manipulator are calculated based on the coordinates and amplitude of the flexible manipulator. The two jointly construct the input vector. After inputting the input vector into the torque double-domain constraint controller, the torque double-domain constraint controller first uses the parameter self-adjusting control law of the improved fuzzy wavelet network to update the parameters of the improved fuzzy wavelet network online, and then uses the control law of the improved fuzzy wavelet network with adjusted parameters to predict the controlled torque after double-domain constraint at the next moment; finally, the predicted controlled torque after double-domain constraint is applied to the flexible arm space robot system to adjust the body attitude angle, two joint angles, and the coordinates and amplitude of the flexible manipulator at the next moment.

[0186] The embodiments of the present application will be described below through specific simulation cases.

[0187] Taking Figure 2 the flexible arm space robot system as an example, the structural parameters of the flexible arm space robot system are taken as , , ; the body and the rigid manipulator arm have a mass of ; the uniform bending stiffness of the flexible manipulator is , and its mass per unit length is ; the body and the rigid manipulator have a moment of inertia about the centroid of .

[0188] During the simulation, the expected motion laws of the body attitude angle and two joint angles of the flexible arm space robot are set as:

[0189] , , , with the unit of rad.

[0190] The initial values of the motion are taken as: ; the tracking time from the start to the end is .

[0191] The constraint range of the output torque amplitude of the joint actuator is set as: , , with the unit of .

[0192] The constraint range of the change rate of the output torque amplitude of the joint actuator is set as:

[0193] , , in the unit of .

[0194] Simulation scheme design: To verify the adaptability of the control method proposed in the embodiments of this application to the double-domain constraints of the joint output control torque, two sets of comparative experiments are set up:

[0195] Simulation 1: Method verification experiment

[0196] Calculation example simulation content and result analysis: Using the improved fuzzy wavelet network control law and the parameter self-adjusting control law of the improved fuzzy wavelet network designed in the embodiments of this application, a control experiment is carried out for the flexible-arm space robot system with unknown parameters and subject to double-domain torque constraints. The simulation results are as Figures 5 - 12 shown. The simulation verification shows that the method proposed in the embodiments of this application realizes the high-precision motion control of the flexible-arm space robot system: the tracking errors of the body attitude angle and the two joint angles quickly converge to zero, and the time-varying characteristics of the flexible-arm vibration mode confirm that the elastic vibration is actively suppressed; the amplitude and its change rate of the control torque output by the actuator are strictly limited to the preset threshold, meeting the requirements of double-domain constraints, and the multi-body system motion trajectory topological map further reveals the global dynamic coordination law of the body motion, joint rotation and flexible vibration. This scheme is verified through multi-dimensional experiments, providing a solution path with both theoretical depth and engineering feasibility for the control of strong-constraint flexible-arm space robots.

[0197] Simulation 2: Comparative verification experiment

[0198] To further verify the comprehensive performance of the method proposed in the embodiments of this application under the double-domain constraints of the torque amplitude and change rate, under the same system parameters and torque double-domain constraint conditions, taking the "Neural Network L2-Gain Robust Control Based on Virtual Force for Flexible-Arm Space Robots" published in the 23rd issue of the 48th volume of the Chinese Journal of Mechanical Engineering in 2012, pages 23-29 as the benchmark control method, a comparative simulation study on control accuracy and robustness is carried out.

[0199] The simulation comparison results show that, as Figures 13 - 19 shown, although the benchmark control method can constrain the actuator torque amplitude and its change rate within the preset range, there are significant steady-state deviations and insufficient convergence in the trajectory tracking of its body attitude angle and the two joint angles, and the flexible-arm vibration suppression efficiency decreases synchronously; in contrast, the method proposed in the embodiments of this application realizes a significant improvement in trajectory tracking accuracy, rapid dissipation of vibration energy and effective suppression of control torque fluctuations within the same constraint range through the dynamic boundary coordination and multi-scale compensation mechanism, verifying the comprehensive improvement of its control performance and robustness in complex constraint scenarios.

[0200] Therefore, the simulation experiments show that the method proposed in the embodiments of the present application, without relying on the accurate model of the system, can not only real-time constrain the control torque output by the joint actuator and its change rate within the preset double-domain boundary, but also significantly improve the trajectory tracking accuracy and elastic vibration suppression ability of the system under restricted working conditions through the efficient cooperation of dynamic fuzzy rules and adaptive wavelet basis functions, while effectively smoothing the torque fluctuation amplitude. Compared with the traditional control strategy, the proposed method has the characteristics of high computational efficiency, strong parameter self-adaptability and robustness to uncertain dynamics, providing a more practical solution for the engineering control of space robot systems in complex restricted environments.

[0201] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A flexible-arm space robot force-moment double-domain constraint control method based on improved FWN, characterized in that Including the following steps: Construct the dynamic model of a flexible-arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints. Under the conditions that the control torques of the body attitude and joint hinges in the flexible-arm space robot system are subject to constraints on the amplitude and the amplitude change rate, combine the dynamic model of the flexible-arm space robot system to construct the mathematical model of the torque double-domain constraint control system; Construct a torque double-domain constraint controller based on an improved fuzzy wavelet network. The torque double-domain constraint controller satisfies the mathematical model of the torque double-domain constraint control system, and its corresponding control law includes an improved fuzzy wavelet network control law and a parameter self-regulation control law of the improved fuzzy wavelet network; Obtain the rotation angle vector composed of the body attitude angle and two joint angles at the current moment and the vibration mode coordinates of the flexible robotic arm calculated based on the coordinates and amplitudes of the flexible robotic arm, and construct an input vector; input the input vector into the torque double-domain constraint controller. First, use the parameter self-regulation control law of the improved fuzzy wavelet network to update the parameters of the improved fuzzy wavelet network, and then use the improved fuzzy wavelet network control law with the updated parameters of the improved fuzzy wavelet network to predict the controlled torque after double-domain constraint at the next moment, and apply the controlled torque after double-domain constraint at the next moment to the flexible-arm space robot system to adjust the body attitude angle, two joint angles, and the coordinates and amplitudes of the flexible robotic arm at the next moment.

2. The flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN according to claim 1, wherein Construct the dynamic model of a flexible-arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints, specifically including: The flexible-arm space robot system includes a freely floating body, a rigid robotic arm, and a flexible robotic arm. There are joints between the body and the rigid robotic arm and between the rigid robotic arm and the flexible robotic arm, and each joint is hinged by a joint hinge; Use the assumed mode method to reduce the order of the flexible robotic arm, intercept the first two dominant modes to characterize the elastic deformation characteristics of the flexible robotic arm, and based on the law of conservation of momentum and the second Lagrange equation, establish the dynamic model of the flexible-arm space robot system under the conditions that the translational degrees of freedom are not actively controlled and the attitude motion has servo constraints, as shown in the following formula ； Among them, represents a symmetric positive definite inertia matrix, represents the angular displacement variable, , and respectively represent the body attitude angle, the joint angle between the body and the rigid manipulator, and the joint angle between the rigid manipulator and the flexible manipulator, and respectively represent the first-order derivative and the second-order derivative of the angular displacement vector with respect to time, represents the mode coordinate of the flexible manipulator, and respectively represent the mode coordinate of the first-order mode and the mode coordinate of the second-order mode. T represents the transpose, and respectively represent the first-order derivative and the second-order derivative of the mode coordinate of the flexible manipulator with respect to time, represents the column vector containing the centrifugal force and the Coriolis force, is the flexible stiffness matrix, represents the attitude of the unconstrained body and the control torque of the joint hinge; represents the set of real numbers.

3. The flexible-arm space robot force-moment dual-domain constraint control method based on the improved FWN according to claim 2, characterized in that Under the conditions that the control torques of the body attitude and joint hinges in the flexible-arm space robot system are subject to constraints on the amplitude and the amplitude change rate, combine the dynamic model of the flexible-arm space robot system to construct the mathematical model of the torque double-domain constraint control system, specifically including: Assume that at the unconstrained control torque at time is subject to constraints on the magnitude and the rate of change of the magnitude, i.e.: ， ； wherein, respectively represent a known lower amplitude saturation limit value and an upper amplitude saturation limit value, respectively represent a known lower amplitude change rate saturation limit value and an upper amplitude change rate saturation limit value; To handle the constraints on the amplitude and the amplitude change rate, define the following functions: ； Among them, represents the controlled torque after double-domain constraint output by the torque double-domain constraint controller, represents the upper limit of the combined constraint, and its expression is: , represents the controlled torque at time represents the sampling time interval, represents taking the minimum value among them, represents the lower limit of the combined constraint, and its expression is: , represents taking the maximum value among them; Combine the dynamic model of the flexible-arm space robot system to design the mathematical model of the torque double-domain constraint control system, as shown in the following formula: 。 4. The flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN according to claim 3, characterized in that, In the improved fuzzy wavelet network, the improved fuzzy wavelet network includes an input layer, a reconstructed fuzzification layer, a reconstructed rule layer, and a reconstructed wavelet function layer; In the reconstructed fuzzification layer, replace the antecedent membership function in the fuzzification layer of the traditional fuzzy wavelet network with a wavelet basis function, as shown in the following formula: ； wherein, is the mother wavelet function, and generates wavelet basis functions of multiple scales through dilation and translation operations , represents the membership function; is the dilation parameter, is the translation parameter, represents the set of real numbers; represents the th input variable, , is the input dimension, represents the index of the fuzzy subset, , is the number of fuzzy subsets of each input variable, represents the index of the translation parameter of the wavelet basis function, , is the total number of indices of the translation parameter; In the reconstructed rule layer, define the th fuzzy rule as follows: If is and is ... and is , then ; Among them, is the rule excitation intensity, represents the weight of the th input variable in the th rule for the th output variable; , is the total number of rules, represents the th output variable, , is the output dimension, represents the index of the wavelet decomposition scale, , is the total number of wavelet decomposition scales; Combine the rule excitation strength with the wavelet basis function to define a fuzzy wavelet basis function , as shown in the following formula: ； Among them, represents the input vector of the improved fuzzy wavelet network, including input variables; In the reconstructed wavelet function layer, through the weight matrix and the matrix multiplication with the fuzzy wavelet basis function vector weighted combination of the output variables of all rules is performed, and the output vector of the improved FWN is represented in vector form as shown in the following formula: ； Among them, represents the weight matrix, represents the weight of the th rule for the th output variable, represents the output vector of the improved FWN, including output variables.

5. The flexible-arm space robot force-moment dual-domain constraint control method based on the improved FWN according to claim 4, characterized in that Construct a torque double-domain constraint controller based on the improved FWN, specifically including: Establish an ideal control law using the output variables of the improved FWN: ； Among them, represents the ideal control torque after double-domain constraint, which is a non-linear uncertainty term; , is an unknown ideal weight matrix, represents the fuzzy wavelet basis function vector corresponding to the input vector at time represents a small error vector with boundedness. Respectively, is estimated as , and further, the control torque after double-domain constraint output by the torque double-domain constraint controller is used as the estimated value, then the ideal control law is modified to the improved fuzzy wavelet network control law as shown in the following formula: ； Therefore, the control torque after double-domain constraint output by the torque double-domain constraint controller can be predicted by the improved fuzzy wavelet network.

6. The flexible arm space robot force and torque dual-domain constraint control method based on the improved FWN according to claim 5, characterized in that, The parameter self-adjustment control law of the improved fuzzy wavelet network is as follows: When or and or and At this time, based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as: ； Among them, represents 's subvector, represents 's subvector, represents 's subvector, represents 's subvector, represents 's subvector, , corresponding to the posture of the body and two joint hinges respectively, represents the first derivative with respect to time, represents the adaptive gain coefficient, ; represents the adjustment gain, ; When and or and At this time, based on the Lyapunov stability criterion, the real-time dynamic adjustment control law of the weight matrix of the reconstructed wavelet function layer of the improved fuzzy wavelet network can be expressed as: ； Among them, represents the robustness compensation coefficient, .

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