A hybrid uncertainty propagation method, system, device, and medium for complex equipment response

Abstract

The application discloses a hybrid uncertainty propagation method, system, equipment and medium for complex equipment response, and the method comprises the following steps: finding samples determining the CDF boundary in an iterative manner through a sequence simulation strategy, so as to gradually approach the upper and lower boundaries of the structural response P-box. In the above sequence solving process, the CDF solving of each step is used to convert the high-dimensional probability propagation problem into the moment analysis problem of a single-variable sub-system through a single-variable dimension reduction integral method. After the interval of the statistical moment of the system response is obtained, the statistical moment of the response is fitted in the interval range to obtain the CDF through Johnson fitting distribution. The sequence simulation strategy of the application is more effective in the probability box analysis of complex equipment response than the direct simulation method, and more accurate system response is obtained only by using less sample point information under the condition of ensuring the same calculation accuracy. The application combines the advantages of interval sequence simulation and statistical moment analysis, and solves the sampling efficiency problem in a high-dimensional problem.

Description

A method, system, device, and medium for propagating hybrid uncertainty in complex equipment responses. Technical Field

[0001] This invention belongs to the field of uncertainty propagation analysis methods, specifically relating to a hybrid uncertainty propagation method, system, device, and medium for complex equipment response. Background Technology

[0002] With the development of science and technology and the advancement of industrial technology, engineering technicians have increasingly higher demands for equipment reliability. However, engineering equipment contains a large number of uncertainties, such as load, size, materials, and processing errors, which greatly hinder the improvement of equipment reliability. Therefore, it is urgent to conduct propagation analysis of uncertainties in engineering equipment to lay the foundation for improving equipment reliability.

[0003] Uncertainty is generally classified into random uncertainty and cognitive uncertainty based on differences in cognitive level and parameter information. Random uncertainty, also known as objective uncertainty, has been the subject of extensive and long-term research. For probabilistic uncertainty propagation problems with sufficient parameter information, probability density functions and cumulative distribution functions from probability theory can be used to quantify and analyze the propagation of random uncertainty. A series of uncertainty propagation methods have emerged for probabilistic propagation models, including Monte Carlo simulation methods, uncertainty propagation methods based on perturbation theory, uncertainty propagation methods based on sequence reliability, uncertainty propagation methods based on function expansion, and uncertainty propagation methods based on moment propagation. For cognitive uncertainty problems, related uncertainty propagation methods have also emerged, including interval models, evidence theory, fuzzy theory, and probability theory. Due to the small amount of relevant parameter sample information in engineering practice, interval models have been widely used. This has led to the development of uncertainty propagation methods based on interval analysis, interval perturbation, interval sampling, interval optimization and surrogate models, and interval sequence simulation.

[0004] Based on the above analysis, probabilistic models and interval models are two common methods for measuring random and cognitive uncertainties. However, in practical engineering, due to differences in the acquisition methods, measurement difficulties, and measurement methods of input parameters, mixed uncertainties often arise where random variables and interval numbers coexist in the system structure. In this case, the uncertainty propagation problem transforms into a probability-interval mixed uncertainty propagation problem. Based on the location of interval variables, probability-interval mixed uncertainty propagation is divided into separate mixed uncertainty propagation and coupled mixed uncertainty propagation. This invention primarily focuses on the research of separate mixed uncertainty propagation problems. Many uncertainty propagation methods have emerged for separate mixed uncertainty problems, including the double-layer nested sampling (DLS) uncertainty propagation method evolved from probability sampling and interval sampling methods, and optimization-based mixed uncertainty propagation methods. Currently, a relatively efficient strategy is lacking for separate probability-interval mixed uncertainty propagation problems, becoming a pressing issue in handling mixed uncertainty propagation in engineering problems. Summary of the Invention

[0005] This invention addresses the shortcomings of existing technologies by providing a method, system, device, and medium for propagating hybrid uncertainty in the response of complex equipment. Through a sequential simulation (SS) strategy, it iteratively finds samples that determine the boundaries of the cumulative distribution function (CDF), thereby progressively approximating the upper and lower boundaries of the structural response probability box (P-box). During the sequential solution process, at each step of the CDF calculation, this invention transforms the high-dimensional probability propagation problem into a moment analysis problem of a single-variable subsystem using a univariate dimensionality reduction method (UDRM). After obtaining the interval of the system response statistical moments, a Johnson fit distribution is used to fit the response statistical moments within its interval to obtain the CDF. Compared with existing methods, this invention achieves a more accurate system response with fewer sample points while maintaining the same computational accuracy. This invention combines the advantages of interval-based sequential simulation and statistical moment analysis, solving the sampling efficiency problem in high-dimensional problems through a high-precision approximation strategy.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A hybrid uncertainty propagation method for complex equipment response is applied to a coordination mechanism, which is a rigid-flexible coupling system including a motor, a reducer, a coordination robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The method includes the following steps:

[0008] Step 1: Establish a rigid-flexible coupling dynamic model of the coordination mechanism; the input parameters of the rigid-flexible coupling dynamic model of the coordination mechanism include probability uncertainty parameters and interval uncertainty parameters;

[0009] Step 2: Characterize the uncertainty of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism based on P-box;

[0010] Step 3: According to the distribution type of the input parameters, the input parameters of the rigid-flexible coupling dynamic model are divided into layers: the outer layer is the interval uncertainty parameter and the inner layer is the probability uncertainty parameter.

[0011] Step 4: Select the interval uncertainty parameters and sample space of the rigid-flexible coupling dynamic model, and perform global and local sampling in the sample space to obtain multiple sets of interval samples, including global and local samples.

[0012] Step 5: Take any set of interval samples from the outer layer and use the dimension reduction integral method to simulate and calculate the probability uncertainty parameters of the inner layer to obtain the P-box response CDF; determine whether all interval samples of the outer layer have been calculated. If yes, obtain a series of CDF results and proceed to step 6; otherwise, continue to step 5.

[0013] Step 6: Based on the series of CDF results obtained in Step 5, obtain the lower boundary of the P-box calculated in this step, determine the composition of the contributing CDF corresponding to the lower boundary, find the contributing samples corresponding to the contributing CDF, and complete one sequence simulation.

[0014] Step 7: Determine whether the area difference between the lower bounds of CDF obtained from the two most recent sequence simulations meets the preset standard. If yes, the calculation is considered to have converged, and proceed to step 8; otherwise, proceed to step 4 to generate new global and local samples.

[0015] Step 8: Output the final P-box response CDF lower boundary;

[0016] Step 9: Perform sequence simulation to obtain the upper boundary of the P-box response CDF. Based on the upper and lower boundaries of the P-box response CDF, obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism, and adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

[0017] To optimize the above technical solution, the specific measures also include:

[0018] Furthermore, in step 1, the rigid-flexible coupling dynamic model of the coordination mechanism is specifically as follows:

[0019]

[0020] In the formula, M(q,t) is the mass matrix of the coordinating mechanism, t is the motion time of the coordination process, and q, and These represent the relative displacement, relative velocity, and relative acceleration of the rotating hinge in the coordinating mechanism. For the generalized damping matrix of the coordinating mechanism, Let Φ(q,t) = 0 be the generalized stiffness matrix of the coordinating mechanism, and let Φ(q,t) = 0 be the constraint equation of the coordinating mechanism. q For Φ(q,t) differentiated with respect to q, (g) T This represents the transpose of the matrix, where l is the Lagrange multiplier and Q is the external force of the coordinating mechanism.

[0021] Further, in step 2, the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism is the angular displacement angle by which the coordinated robotic arm rotates from its initial -3° position to a position making an angle of 45° with the horizontal direction, and the CDF of the output response is expressed as F. z (Z) indicates that F z (Z) is enveloped by the lower boundary of CDF. F z (Z) and upper boundary Within the formed area, The P-box of the output response Z of the rigid-flexible coupling dynamic model of the coordinating mechanism is written as:

[0022]

[0023] In the formula, In response to Z's P-box.

[0024] Furthermore, in step 4, the global samples are generated from the entire bounded sample space, and the local samples are generated in the neighborhood of the contributing samples according to the contribution index of the contributing samples; the contributing samples refer to the samples that contribute to the upper and lower boundaries of the P-box.

[0025] The contribution index is calculated as follows:

[0026] For each contribution sample The CDF of the P-box is denoted as Its contribution is evaluated using the span of the ordinate where the CDF is maximized:

[0027]

[0028] In the formula, Contributing samples Contribution index, Indicates the contribution sample The ordinate span, F, that affects the upper or lower boundary of the CDF. z (Z(y)) represents the total vertical coordinate span of the upper and lower boundaries of the P-box.

[0029] Furthermore, the neighborhood of the contributing sample is defined by a circle centered at... radius r * The circular region or multidimensional spherical region; the formula for calculating the radius is as follows:

[0030]

[0031] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, V(W) α W is the bounded sample space. α The volume or area of ​​the sample space is m, which is the number of interval uncertainty parameters and also the dimension of the bounded sample space.

[0032] Furthermore, the neighborhood of the contributing sample is centered at... The side length is 2r * Square or multidimensional cubic region; r * The calculation formula is as follows:

[0033]

[0034] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, V(W) α W is the bounded sample space. α of

[0035] Volume or area, where m is the number of interval uncertainty parameters and also the dimension of the bounded sample space.

[0036] Furthermore, the neighborhood of the contributing sample is centered at... The side length is 2r i * A rectangular or multidimensional rectangular solid region, i = 1, 2, ..., m, where m is the number of interval uncertainty parameters, and r... i * The calculation formula is as follows:

[0037]

[0038] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, For interval The width.

[0039] Furthermore, the value of b is set between 1 / 12 and 1 / 4.

[0040] Furthermore, in step 5, the simulation calculation of the inner layer probability uncertainty parameter using the dimension reduction integral method to obtain the P-box response CDF is specifically as follows:

[0041] Step 5.1: Apply a univariate dimensionality reduction integration method to the probability uncertainty parameters, expanding the dimensionality reduction around the mean, thus transforming the multivariate probability uncertainty parameter system into a subsystem containing only a single variable, expressed as:

[0042]

[0043] Among them, E l (Z) represents the l-th moment of the response, l = 1, 2, 3, 4; m i It is the mean of the i-th probability uncertainty parameter. yes The abbreviation for X represents the i-th univariate subsystem after decomposition. i Y represents the i-th probability uncertainty parameter. (d) Indicates the interval uncertainty parameter. For binomial expansion, j is the order index. Represents the function Find the j-th moment, where m represents the number of probability uncertainty parameters, g0(m1,m2,L,m) m ,Y (d) ) indicates that all probability uncertainty parameters are initialized to their mean values;

[0044] Step 5.2: Calculate the first four moments of the single-variable subsystem using Gauss-Hermite integration, expressed as:

[0045]

[0046] In the formula, E{·} represents the function The l-th moment, r represents the number of Gaussian integral nodes, U i,k and w k These are the weights of the Gaussian integral node and the node, respectively;

[0047] Step 5.3: Iteratively calculate the first four central moments using the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, expressed as:

[0048]

[0049] Among them, E l (Y (d) ), l=1,2,3,4 are the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, M l (Y (d) ), l=1,2,3,4 are the first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism;

[0050] Step 5.4: Obtain the P-box response CDF by using the Johnson fitting method to obtain the first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism. The Johnson fitting method is expressed as:

[0051]

[0052] Where g and δ are shape parameters, is the size parameter, x is the position parameter, w ~ N(0,1) follows a standard normal distribution, Y is the transformation function, w represents the Johnson distribution, and h is the Johnson variable of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism.

[0053] Further, in step 7, determining whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard specifically involves:

[0054] Will As the lower boundary of the CDF obtained from the k-th sequence simulation, The lower boundary of the CDF obtained from the (k-1)th sequence simulation is used as the area difference between the lower boundaries of the CDF obtained from two adjacent sequence simulations as the convergence criterion, as follows:

[0055]

[0056] in, The lower boundary of the CDF obtained from the k-th sequence simulation. The lower boundary of the CDF obtained from the (k-1)th sequence simulation The Minkowski-L1 distance between them Indicates the contribution sample The differential;

[0057] Distance of Minkowski-L1 The absolute error of the simulation of the k-th sequence The convergence criterion is:

[0058]

[0059] In the formula, e0 is the preset value for error control.

[0060] Furthermore, in step 5.2, the number r of the Gaussian integration nodes is 6.

[0061] Furthermore, the preset value for error control is e0 = 1′10 -3 .

[0062] This invention also proposes a hybrid uncertainty propagation system for complex equipment response, applied to a coordination mechanism. The coordination mechanism is a rigid-flexible coupling system, including a motor, a reducer, a coordination robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The hybrid uncertainty propagation system includes:

[0063] The modeling module is used to establish a rigid-flexible coupling dynamic model of the coordination mechanism; the input parameters of the rigid-flexible coupling dynamic model of the coordination mechanism include probability uncertainty parameters and interval uncertainty parameters.

[0064] The uncertainty characterization module is used to characterize the output response of the rigid-flexible coupling dynamic model of the coordination mechanism based on P-box.

[0065] The layering module is used to stratify the input parameters of the rigid-flexible coupling dynamics model according to the distribution type of the input parameters. The outer layer consists of parameters with interval uncertainty, and the inner layer consists of parameters with probability uncertainty.

[0066] The sampling module is used to select the interval uncertainty parameters and sample space of the rigid-flexible coupling dynamic model, and to perform global and local sampling within the sample space to obtain multiple sets of interval samples, including global and local samples.

[0067] The lower boundary generation module is used to randomly select a set of interval samples from the outer layer, and simulate the inner layer probability uncertainty parameters using a dimensionality reduction integral method to obtain the P-box response CDF. It then checks whether all interval samples in the outer layer have been calculated; if so, it obtains a series of CDF results. Otherwise, it continues to select uncalculated interval samples for simulation to obtain the P-box response CDF, until all interval samples in the outer layer have been calculated. Based on the series of CDF results, it obtains the lower boundary of the P-box for this calculation, determines the composition of the contributing CDF corresponding to the lower boundary, identifies the contributing samples corresponding to the contributing CDF, and completes one sequence simulation. Finally, it checks whether the area difference between the lower boundaries of the CDF obtained from the two most recent sequence simulations meets a preset standard. If so, it considers the calculation to have converged and outputs the final lower boundary of the P-box response CDF; otherwise, it controls the sampling module to generate new global and local samples.

[0068] The upper boundary generation module is used to perform sequence simulation to obtain the upper boundary of the P-box response CDF;

[0069] The adjustment module is used to obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism based on the upper and lower boundaries of the P-box response CDF, and to adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

[0070] To optimize the above technical solution, the specific measures also include:

[0071] Furthermore, the rigid-flexible coupling dynamic model of the coordination mechanism is specifically as follows:

[0072]

[0073] In the formula, M(q,t) is the mass matrix of the coordinating mechanism, t is the motion time of the coordination process, and q, and These represent the relative displacement, relative velocity, and relative acceleration of the rotating hinge in the coordinating mechanism. For the generalized damping matrix of the coordinating mechanism, Let Φ(q,t) = 0 be the generalized stiffness matrix of the coordinating mechanism, and let Φ(q,t) = 0 be the constraint equation of the coordinating mechanism. q For Φ(q,t) differentiated with respect to q, (g) T This represents the transpose of the matrix, where l is the Lagrange multiplier and Q is the external force of the coordinating mechanism.

[0074] Furthermore, the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism is the angular displacement angle by which the coordinated robotic arm rotates from an initial -3° position to a position making an angle of 45° with the horizontal direction, and the CDF of the output response is expressed as F. z (Z) indicates that F z (Z) is enveloped by the lower boundary of CDF. F z (Z) and upper boundary Within the formed area, The P-box of the output response Z of the rigid-flexible coupling dynamic model of the coordinating mechanism is written as:

[0075]

[0076] In the formula, In response to Z's P-box.

[0077] Furthermore, the global samples are generated from the entire bounded sample space, and the local samples are generated in the neighborhood of the contributing samples according to the contribution index of the contributing samples; the contributing samples refer to the samples that contribute to the upper and lower boundaries of the P-box.

[0078] The contribution index is calculated as follows:

[0079] For each contribution sample The CDF of the P-box is denoted as Its contribution is evaluated using the span of the ordinate where the CDF is maximized:

[0080]

[0081] In the formula, Contributing samples Contribution index, Indicates the contribution sample The ordinate span, F, that affects the upper or lower boundary of the CDF. z (Z(y)) represents the total vertical coordinate span of the upper and lower boundaries of the P-box.

[0082] Furthermore, the neighborhood of the contributing sample is defined by a circle centered at... radius r * The circular region or multidimensional spherical region; the formula for calculating the radius is as follows:

[0083]

[0084] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, V(W) α W is the bounded sample space. α The volume or area of ​​the sample space is m, which is the number of interval uncertainty parameters and also the dimension of the bounded sample space.

[0085] Furthermore, the neighborhood of the contributing sample is centered at... The side length is 2r * Square or multidimensional cubic region; r * The calculation formula is as follows:

[0086]

[0087] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, V(W) α W is the bounded sample space. α of

[0088] Volume or area, where m is the number of interval uncertainty parameters and also the dimension of the bounded sample space.

[0089] Furthermore, the neighborhood of the contributing sample is centered at... The side length is 2r i * A rectangular or multidimensional rectangular solid region, i = 1, 2, ..., m, where m is the number of interval uncertainty parameters, and r... i * The calculation formula is as follows:

[0090]

[0091] In the formula, b represents the neighborhood relative to the entire bounded sample space W. α Size, For interval The width.

[0092] Furthermore, the value of b is set between 1 / 12 and 1 / 4.

[0093] Furthermore, the P-box response CDF is specifically calculated by simulating the inner layer probability uncertainty parameters using a dimension reduction integral method:

[0094] A univariate dimensionality reduction integral method is used to expand the probabilistic uncertainty parameter around the mean, transforming the multivariate probabilistic uncertainty parameter system into a subsystem containing only a single variable, as follows:

[0095]

[0096] Among them, E l (Z) represents the l-th moment of the response, l = 1, 2, 3, 4; m i It is the mean of the i-th probability uncertainty parameter. yes The abbreviation for X represents the i-th univariate subsystem after decomposition. i Y represents the i-th probability uncertainty parameter. (d) Indicates the interval uncertainty parameter. For binomial expansion, j is the order index. Represents the function Find the j-th moment, where m represents the number of probability uncertainty parameters, g0(m1,m2,L,m) m ,Y (d) ) indicates that all probability uncertainty parameters are initialized to their mean values;

[0097] The first four moments of the univariate subsystem are calculated using Gauss-Hermite integrals and are expressed as follows:

[0098]

[0099] In the formula, E{·} represents the function The l-th moment, r represents the number of Gaussian integral nodes, U i,k and w k These are the weights of the Gaussian integral node and the node, respectively;

[0100] The first four central moments are calculated iteratively from the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, and are expressed as follows:

[0101]

[0102] Among them, E l (Y (d) ), l=1,2,3,4 are the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, M l (Y (d) ), l=1,2,3,4 are the first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism;

[0103] The first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism are used to obtain the P-box response CDF using the Johnson fitting method. The Johnson fitting method is expressed as follows:

[0104]

[0105] Where g and δ are shape parameters, is the size parameter, x is the position parameter, w ~ N(0,1) follows a standard normal distribution, Y is the transformation function, w represents the Johnson distribution, and h is the Johnson variable of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism.

[0106] Furthermore, the determination of whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard specifically involves:

[0107] Will As the lower boundary of the CDF obtained from the k-th sequence simulation, The lower boundary of the CDF obtained from the (k-1)th sequence simulation is used as the area difference between the lower boundaries of the CDF obtained from two adjacent sequence simulations as the convergence criterion, as follows:

[0108]

[0109] in, The lower boundary of the CDF obtained from the k-th sequence simulation. The lower boundary of the CDF obtained from the (k-1)th sequence simulation The Minkowski-L1 distance between them Indicates the contribution sample The differential;

[0110] Distance of Minkowski-L1 The absolute error of the simulation of the k-th sequence The convergence criterion is:

[0111]

[0112] In the formula, ε0 is the preset value for error control.

[0113] Furthermore, the number r of the Gaussian integration nodes is 6.

[0114] Furthermore, the preset value for error control is ε0 = 1 × 10⁻⁶. -3 .

[0115] The present invention also proposes an electronic device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the hybrid uncertainty propagation method for complex equipment response as described above.

[0116] The present invention also proposes a computer-readable storage medium storing a computer program that causes a computer to execute the hybrid uncertainty propagation method for complex equipment response as described above.

[0117] The beneficial effects of this invention are:

[0118] 1. For the probability-interval mixed uncertainty propagation analysis problem, the outer layer uses a sequential simulation method to quickly find samples that define the CDF boundary iteratively. Compared with the MCS (Monet Carlo Method), this significantly improves computational efficiency with a smaller error. In the inner layer, for each step of CDF solving, this invention transforms the high-dimensional probability propagation problem into a moment analysis problem of a single-variable subsystem using a univariate dimensionality reduction integration method. After obtaining the interval of the system response statistical moments, a Johnson fit distribution is used to fit the response statistical moments within its interval to obtain the CDF. Compared with the MCS method, this requires only a very small number of sample points to ensure high accuracy.

[0119] 2. This method is a non-embedded method, which is generally applicable to complex engineering problems such as nonlinear, high-input-parameter, and non-monotonic dynamic responses of mechanisms and structures. Attached Figure Description

[0120] Figure 1 is a flowchart of the present invention.

[0121] Figure 2 is a schematic diagram of the rigid-flexible coupling system of the present invention.

[0122] Figure 3 shows the contribution samples and their CDFs of the present invention.

[0123] Figure 4 shows the neighborhood of the contributing sample of the present invention.

[0124] Figure 5 is a schematic diagram of the convergence criterion of the present invention. Detailed Implementation

[0125] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of this application.

[0126] Example 1

[0127] This invention proposes a hybrid uncertainty propagation method for complex equipment response, applied to a coordination mechanism. This coordination mechanism is a rigid-flexible coupling system, as shown in Figure 2, comprising a motor, a reducer, a coordination robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The main function is to coordinate the conveying mechanism to a specific position using the coordination robotic arm and attitude-adjusting electric cylinder, ensuring a certain level of positioning accuracy. The specific actions are as follows: The robotic arm is connected to a rotating shaft via the reducer and can rotate around the shaft. An attitude-adjusting limit block and an attitude-adjusting electric cylinder support are fixedly connected to the robotic arm. The front end of the attitude-adjusting electric cylinder is hinged to the conveying mechanism, and the rear end is hinged to the attitude-adjusting electric cylinder support. The conveying mechanism is mounted at the end of the robotic arm via an attitude-adjusting shaft and, under the action of the attitude-adjusting electric cylinder, performs a small-angle rotation around the robotic arm. The extreme positions of the movement are mechanically limited by the limit block and the upper and lower limit blocks of the conveying mechanism. Under the combined action of the motor and attitude-adjusting electric cylinder assembly, the axis of the conveying mechanism can be aligned with the transmission and receiving device at any angle, thereby achieving accurate object transmission. The presence of mixed uncertainties in geometric dimensions, physical properties, and other parameters can affect the positioning accuracy of a robotic arm. Therefore, it is necessary to conduct mixed uncertainty propagation analysis of the rigid-flexible coupled dynamic model.

[0128] The proposed hybrid uncertainty propagation method for complex equipment response, as shown in Figure 1, includes the following steps:

[0129] Step 1: Establish a rigid-flexible coupling dynamic model of the coordination mechanism; the input parameters of the rigid-flexible coupling dynamic model of the coordination mechanism include probability uncertainty parameters and interval uncertainty parameters.

[0130] During the motion, there will be contact collisions between the shaft hole and the limiting block, resulting in contact force and damping torque. The contact force model is based on the Lankarani-Nikravesh normal contact force model F. n and the improved Coulomb friction model F t Composition. Their calculations are as follows:

[0131]

[0132] In the formula, F n Where K is the actual collision force, K is the stiffness coefficient of the two contacting objects, δ is the penetration depth of the contacting body, n is the collision index, and C is the contact collision damping coefficient. Let a1 be the relative collision velocity at the point of contact, and c be the coefficient of sliding friction. d v is a dynamic correction coefficient. t This represents the relative tangential velocity. M c Let a be the rotational damping torque, and a2 be the coefficient of friction of the damping torque. Let be the rotational angular velocity, and sign be the sign function. hour, when hour, when hour, a3 is the damping coefficient of the damping torque.

[0133] Since the robotic arm is a thin-plate structure, it will deform during movement. Therefore, in the modeling process, the robotic arm is considered as a flexible body. Combining the kinematic relationships, forces and constraints of the mechanism, and based on the principle of virtual power, a rigid-flexible coupling dynamic model of the coordinating mechanism is established as follows:

[0134]

[0135] In the formula, M(q,t) is the mass matrix of the coordinating mechanism, t is the motion time of the coordination process, and q, and These represent the relative displacement, relative velocity, and relative acceleration of the rotating hinge in the coordinating mechanism. For the generalized damping matrix of the coordinating mechanism, Let Φ(q,t) = 0 be the generalized stiffness matrix of the coordinating mechanism, and let Φ(q,t) = 0 be the constraint equation of the coordinating mechanism. q For Φ(q,t) differentiated with respect to q, (g) TThis represents the transpose of the matrix, where l is the Lagrange multiplier and Q is the external force of the coordinating mechanism.

[0136] In the rigid-flexible coupling dynamic model of the coordinated mechanism, material parameters such as elastic modulus and density, initial position of the mechanism, Coulomb friction coefficient, torque friction coefficient, damping coefficient, and load mass all exhibit uncertainties. The values ​​and types of these uncertainties are shown in Table 1. The input parameters of the rigid-flexible coupling dynamic model of the coordinated mechanism include probabilistic uncertainty parameters and interval uncertainty parameters. Probabilistic uncertainty parameters include elastic modulus, density, initial position, load mass, and moment of inertia. Interval uncertainty parameters include Coulomb friction coefficient, dry friction coefficient, damping coefficient, and eccentricity of the center of mass along the x-direction, y-direction, and z-direction.

[0137] Table 1 Uncertainty parameters and their distribution types

[0138]

[0139]

[0140] Step 2: Characterize the uncertainty of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism based on P-box; the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism is the angular displacement angle of the coordinated robotic arm rotating from the initial -3° position to the position with an angle of 45° with the horizontal direction, and the CDF of the output response is expressed as F. z (Z) indicates that F z (Z) is enveloped by the lower boundary of CDF. F z (Z) and upper boundary Within the formed area, The P-box of the output response Z of the rigid-flexible coupling dynamic model of the coordinating mechanism is written as:

[0141]

[0142] In the formula, In response to Z's P-box.

[0143] Step 3: According to the distribution type of the input parameters in Table 1, the input parameters of the rigid-flexible coupling dynamic model are divided into layers: the outer layer consists of interval uncertainty parameters and the inner layer consists of probability uncertainty parameters.

[0144] Step 4: Select the interval uncertainty parameters and sample space of the rigid-flexible coupling dynamic model, and perform global and local sampling within the sample space (the first local sampling is 0) to obtain multiple sets of interval samples. Where ns-initial =50 is the initial sample size. At this point, all interval variables degenerate into scalars, and the system is transformed into a system containing only random variables. Interval samples include global samples and local samples (there are no local samples in the initial sequence). Global samples are generated from the entire bounded sample space, and local samples are generated in the neighborhood of contributing samples based on their contribution index. Contributing samples refer to samples that contribute to the upper and lower boundaries of the P-box. In each sequence simulation of the outer interval, samples that contribute to the upper and lower boundaries of the P-box need to be identified. The upper and lower boundaries obtained in each sequence simulation must be composed of some parts of the existing CDF response. Therefore, these input samples that can constitute the boundary components of the CDF response are called contributing samples.

[0145] The contribution index is calculated as follows:

[0146] As shown in Figure 3, for each contribution sample The CDF of the P-box is denoted as Its contribution is evaluated using the span of the ordinate where the CDF is maximized:

[0147]

[0148] In the formula, Contributing samples Contribution index, Indicates the contribution sample The ordinate span, F, that affects the upper or lower boundary of the CDF. z (Z(y)) represents the total vertical coordinate span of the upper and lower boundaries of the P-box.

[0149] As shown in Figure 4, the contributing sample points neighborhood It can be defined in different forms or shapes. The circular neighborhood shown in Figure 4(a) is the center of the circle. radius r * The circular region (for two interval uncertainty parameters) or the multidimensional spherical region (for more than two interval uncertainty parameters) shown in Figure 4(b) is a region centered at... The side length is 2r * A square or multi-dimensional cubic region. The rectangular neighborhood shown in Figure 4(c) is centered at... The side length is 2r i * The neighborhood is a rectangular or multidimensional rectangular solid region, i = 1, 2, ..., m, where m is the number of interval parameters. An appropriately sized set helps accelerate the convergence of the sequence simulation. For ease of description, a ratio b is introduced to represent the neighborhood relative to the entire bounded space W. αThe size of the neighborhood. For circular and square neighborhoods, it is defined as... Where V(W) α W is a bounded space. α The volume or area, m is W α The dimension. For a rectangular region, by Confirmed, among which For interval The width of b. Based on calculations of interval uncertainty problems, this invention suggests setting b between 1 / 12 and 1 / 4.

[0150] Using the rectangular neighborhood form shown in Figure 4(c), with a relative neighborhood radius b = 0.1, the global sampling number for each sequence simulation is 20, and the local sampling number is determined based on the contributing samples. The number of simulations for the sequence simulation is shown in Table 2.

[0151] Table 2 Number of simulations in the sequence simulation process

[0152]

[0153]

[0154] Step 5: Randomly select one set of interval samples from the outer layer, and simulate the inner layer probability uncertainty parameters using the dimension reduction integral method to obtain the P-box response CDF; determine whether all interval samples in the outer layer have been calculated. If yes, obtain a series of CDF results and proceed to Step 6; otherwise, continue to Step 5. Specifically, the simulation calculation of the inner layer probability uncertainty parameters using the dimension reduction integral method to obtain the P-box response CDF is as follows:

[0155] Step 5.1: Apply a univariate dimensionality reduction integration method to the probability uncertainty parameters, expanding the dimensionality reduction around the mean, thus transforming the multivariate probability uncertainty parameter system into a subsystem containing only a single variable, expressed as:

[0156]

[0157] Among them, E l (Z) represents the l-th moment of the response, l = 1, 2, 3, 4; m i It is the mean of the i-th probability uncertainty parameter. yes The abbreviation for X represents the i-th univariate subsystem after decomposition. i Y represents the i-th probability uncertainty parameter. (d) Indicates the interval uncertainty parameter. For binomial expansion, j is the order index. Represents the function Find the j-th moment, where m represents the number of probability uncertainty parameters, g0(m1,m2,L,m) m ,Y (d) ) indicates that all probability uncertainty parameters are initialized to their mean values;

[0158] Step 5.2: Calculate the first four moments of the single-variable subsystem using Gauss-Hermite integration, expressed as:

[0159]

[0160] In the formula, E{·} represents the function The l-th moment, where r represents the number of Gaussian integral nodes, is set to r = 6 in this embodiment. i,k and w k These are the weights of the Gaussian integral node and the node, respectively;

[0161] Step 5.3: Iteratively calculate the first four central moments using the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, expressed as:

[0162]

[0163] Among them, E l (Y (d) ), l=1,2,3,4 are the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, M l (Y (d) ), l=1,2,3,4 are the first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism;

[0164] Step 5.4: Obtain the P-box response CDF by using the Johnson fitting method to obtain the first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism. The Johnson fitting method is expressed as:

[0165]

[0166] Where g and δ are shape parameters, is the size parameter, x is the position parameter, w ~ N(0,1) follows a standard normal distribution, Y is the transformation function, w represents the Johnson distribution, and h is the Johnson variable of the output response of the rigid-flexible coupling dynamic model of the coordination mechanism.

[0167] Step 6: Based on the series of CDF results obtained in Step 5, obtain the lower boundary of the P-box calculated in this step, determine the composition of the contributing CDF corresponding to the lower boundary, find the contributing samples corresponding to the contributing CDF, and complete one sequence simulation.

[0168] Step 7: Determine whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard. If yes, the calculation is considered to have converged, and proceed to step 8; otherwise, proceed to step 4 to generate new global and local samples. Specifically, determining whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard involves:

[0169] Will As the lower boundary of the CDF obtained from the k-th sequence simulation, The lower boundary of the CDF obtained from the (k-1)th sequence simulation is used as the area difference between the lower boundaries of the CDF obtained from two adjacent sequence simulations as the convergence criterion, as follows:

[0170]

[0171] in, The lower boundary of the CDF obtained from the k-th sequence simulation. The lower boundary of the CDF obtained from the (k-1)th sequence simulation The Minkowski-L1 distance between them Indicates the contribution sample The differential;

[0172] Distance of Minkowski-L1 The absolute error of the simulation of the k-th sequence The convergence criterion is:

[0173]

[0174] In the formula, e0 is the preset value for error control, e0 = 1′10 -3 .

[0175] Step 8: Output the final P-box response CDF lower boundary;

[0176] Step 9: Perform sequence simulation to obtain the upper boundary of the P-box response CDF. The sequence simulation process of the upper boundary is similar to that of the lower boundary. Based on the upper and lower boundaries of the P-box response CDF, obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism, and adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

[0177] Based on the above calculation steps, the comparison tables 3 and 4 of the P-box calculated by this invention with DLS (Double Layer Sampling) and MCS-UDRM are shown.

[0178] Table 3 Comparison of P-box calculation results for several hybrid uncertainty propagation methods

[0179]

[0180] Table 4 Comparison of fourth moments of several mixed uncertainty propagation methods

[0181]

[0182]

[0183] As shown in Table 2, for this example, the outer upper bound sequence simulation reached convergence in 8 iterations, requiring 330 samples; the lower bound sequence simulation reached convergence in 10 iterations, requiring 424 samples; the total number of samples for the outer layer was 804. Combined with Table 3, it can be seen that the P-box result calculated by the method of this invention has a maximum error of 0.25% compared to DLS and MCS-UDRM, while the sampling number for DLS is 10. 4 '10 4 The number of samples for MCS-UDRM was 10. 4 The method of this invention achieves such high accuracy using only 804′30. Furthermore, according to Table 4, the error of the first four central moments calculated by the method of this invention is about 2% compared with the DLS results. Only the error of the third central moment is relatively large. However, according to the calculation results of MCS-UDRM, this error is mainly due to the inner layer dimension reduction integration. The outer layer sequence simulation still has high efficiency and accuracy.

[0184] Therefore, it can be seen that the accuracy of the method of the present invention can still be guaranteed for rigid-flexible coupled dynamic systems in engineering practice with high parameter dimensions, and the computational efficiency is greatly improved. This demonstrates the high universality of the method of the present invention.

[0185] Example 2

[0186] This invention proposes a hybrid uncertainty propagation system for complex equipment response, corresponding to the method in Embodiment 1, applied to a coordination mechanism. The coordination mechanism is a rigid-flexible coupling system, including a motor, a reducer, a coordination robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The hybrid uncertainty propagation system includes:

[0187] The modeling module is used to establish a rigid-flexible coupling dynamic model of the coordination mechanism; the input parameters of the rigid-flexible coupling dynamic model of the coordination mechanism include probability uncertainty parameters and interval uncertainty parameters.

[0188] The uncertainty characterization module is used to characterize the output response of the rigid-flexible coupling dynamic model of the coordination mechanism based on P-box.

[0189] The layering module is used to stratify the input parameters of the rigid-flexible coupling dynamics model according to the distribution type of the input parameters. The outer layer consists of parameters with interval uncertainty, and the inner layer consists of parameters with probability uncertainty.

[0190] The sampling module is used to select the interval uncertainty parameters and sample space of the rigid-flexible coupling dynamic model, and to perform global and local sampling within the sample space to obtain multiple sets of interval samples, including global and local samples.

[0191] The lower boundary generation module is used to randomly select a set of interval samples from the outer layer, and simulate the inner layer probability uncertainty parameters using a dimensionality reduction integral method to obtain the P-box response CDF. It then checks whether all interval samples in the outer layer have been calculated; if so, it obtains a series of CDF results. Otherwise, it continues to select uncalculated interval samples for simulation to obtain the P-box response CDF, until all interval samples in the outer layer have been calculated. Based on the series of CDF results, it obtains the lower boundary of the P-box for this calculation, determines the composition of the contributing CDF corresponding to the lower boundary, identifies the contributing samples corresponding to the contributing CDF, and completes one sequence simulation. Finally, it checks whether the area difference between the lower boundaries of the CDF obtained from the two most recent sequence simulations meets a preset standard. If so, it considers the calculation to have converged and outputs the final lower boundary of the P-box response CDF; otherwise, it controls the sampling module to generate new global and local samples.

[0192] The upper boundary generation module is used to perform sequence simulation to obtain the upper boundary of the P-box response CDF;

[0193] The adjustment module is used to obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism based on the upper and lower boundaries of the P-box response CDF, and to adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

[0194] The implementation methods of each module and its function in the system are completely consistent with the steps of the method in Implementation Example 1, so they will not be repeated here.

[0195] Example 3

[0196] This invention proposes an electronic device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the hybrid uncertainty propagation method for complex equipment response as described in Embodiment 1.

[0197] Example 4

[0198] The present invention proposes a computer-readable storage medium storing a computer program that causes a computer to execute a hybrid uncertainty propagation method for complex equipment response as described in Embodiment 1.

[0199] In the embodiments disclosed in this application, a computer storage medium may be a tangible medium that may contain or store programs for use by or in conjunction with an instruction execution system, apparatus, or device. The computer storage medium may include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of computer storage media include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0200] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed in this application can be implemented in electronic hardware or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0201] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principle of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A hybrid uncertainty propagation method for complex equipment response, applied to a coordination mechanism, characterized in that, The coordinating mechanism is a rigid-flexible coupling system, including a motor, a reducer, a coordinating robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The method includes the following steps: Step 1: Establish a rigid-flexible coupling dynamic model of the coordinating mechanism; the input parameters of the rigid-flexible coupling dynamic model include probabilistic uncertainty parameters and interval uncertainty parameters; Step 2: Characterize the uncertainty of the output response of the rigid-flexible coupling dynamic model based on the P-box; Step 3: According to the distribution type of the input parameters, the input parameters of the rigid-flexible coupling dynamic model are layered, with the outer layer being interval uncertainty parameters and the inner layer being probabilistic uncertainty parameters; Step 4: Select the interval uncertainty parameters of the rigid-flexible coupling dynamic model and its sample space, and perform global sampling and local sampling within the sample space to obtain multiple sets of interval samples, including global samples and local samples; In Step 4, the global samples are generated from the entire bounded sample space, and the local samples are generated in the neighborhood of the contributing samples according to the contribution index of the contributing samples; the contributing samples refer to the samples that contribute to the upper and lower boundaries of the P-box; the contribution index is calculated as follows: for each contributing sample... The CDF of the P-box is denoted as Its contribution is evaluated by the span of the ordinate where the CDF reaches its maximum value: In the formula, Contributing samples Contribution index, Indicates the contribution sample The span of the ordinate that affects the upper or lower boundary of the CDF. The total ordinate span of the upper and lower boundaries of the P-box is represented; Step 5: Take any set of interval samples from the outer layer and use the dimension reduction integral method to simulate and calculate the probability uncertainty parameters of the inner layer to obtain the P-box response CDF; Determine whether all interval samples of the outer layer have been calculated. If yes, obtain a series of CDF results and proceed to Step 6; otherwise, continue to Step 5; Step 6: Based on the series of CDF results obtained in Step 5, obtain the lower boundary of the P-box calculated this time, determine the composition of the contribution CDF corresponding to the lower boundary, find the contribution samples corresponding to the contribution CDF, and complete a sequence simulation; Step 7: Determine whether the area difference between the lower boundaries of the CDF obtained from the two most recent sequence simulations meets the preset standard. If yes, it is considered that the calculation has converged and proceed to Step 8; otherwise, proceed to Step 4 to generate new global and local samples; Step 8: Output the final lower boundary of the P-box response CDF; Step 9: Perform a sequence simulation to obtain the upper boundary of the P-box response CDF. Based on the upper and lower boundaries of the P-box response CDF, obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism, and adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

2. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, In step 1, the rigid-flexible coupling dynamic model of the coordination mechanism is specifically as follows: In the formula, The quality matrix of the coordinating body. For the motion time of the coordination process, 、 and These represent the relative displacement, relative velocity, and relative acceleration of the rotating hinge in the coordinating mechanism. For the generalized damping matrix of the coordinating mechanism, For the generalized stiffness matrix of the coordination mechanism, The constraint equations for the coordinating mechanism. for right Differentiate, This indicates the transpose of the matrix. For Lagrange multipliers, External forces that coordinate the organization.

3. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, In step 2, the output response of the rigid-flexible coupling dynamic model of the coordination mechanism is the angular displacement angle by which the coordinated robotic arm rotates from its initial -3° position to a position making an angle of 45° with the horizontal direction. The CDF of the output response is used... express, Enclosed by the lower boundary of CDF and upper boundary Within the formed area, Output response of the rigid-flexible coupling dynamic model of the coordinating mechanism The P-box is written as: In the formula, In response P-box.

4. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, The neighborhood of the contributing sample is defined by a circle centered at... , radius is The circular region or multidimensional spherical region; the formula for calculating the radius is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For bounded sample space The volume or area, denoted as the number of interval uncertainty parameters, and also as the dimension of the bounded sample space.

5. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, The neighborhood of the contributing sample is centered at... The side length is A square or multi-dimensional cubic region; The calculation formula is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For bounded sample space The volume or area, denoted as the number of interval uncertainty parameters, and also as the dimension of the bounded sample space.

6. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, The neighborhood of the contributing sample is centered at... The side length is A rectangular or multi-dimensional rectangular solid area Let be the number of interval uncertainty parameters. The calculation formula is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For interval The width.

7. The hybrid uncertainty propagation method for complex equipment response as described in any one of claims 4-6, characterized in that, The value is set between 1 / 12 and 1 / 4.

8. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, In step 5, the simulation calculation of the inner-layer probability uncertainty parameters using the dimensionality reduction integral method to obtain the P-box response CDF specifically involves: Step 5.1, applying a univariate dimensionality reduction integral method to the probability uncertainty parameters, expanding the dimensionality reduction near the mean, and transforming the multivariate probability uncertainty parameter system into a subsystem containing only a single variable, as shown below: in, The first response Step moment, ； It is the first The mean of a probability uncertainty parameter, yes The abbreviation for , indicating the th after decomposition A single-variable subsystem, Indicates the first One probability uncertainty parameter, Indicates the interval uncertainty parameter. This is a binomial expansion, where j is the order index. Represents the function Find the j-th moment, where m represents the number of probability uncertainty parameters. This indicates that all probability uncertainty parameters are initial values ​​of the mean; Step 5.2: Calculate the first four moments of the univariate subsystem using Gauss-Hermite integrals, expressed as: In the formula, Representation function The l-th moment, This represents the number of nodes in the Gaussian integral. and These are the weights of the Gaussian integral nodes and the node weights, respectively; Step 5.3: Iteratively calculate the first four central moments using the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, expressed as: in, The first four origin moments are the output response of the rigid-flexible coupling dynamic model of the coordination mechanism. The first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism are given; Step 5.4: The first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism are obtained by using the Johnson fitting method to obtain the P-box response CDF. The Johnson fitting method is expressed as: in, and For shape parameters, These are dimensional parameters. It is a position parameter. Follows a standard normal distribution. For transformation function, Represents the Johnson distribution. Johnson variables represent the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism.

9. The hybrid uncertainty propagation method for complex equipment response as described in claim 1, characterized in that, In step 7, determining whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard specifically involves: As the first The lower boundary of the CDF obtained by the subsequence simulation will be The lower boundary of the CDF obtained from the (k-1)th sequence simulation is used as the area difference between the lower boundaries of the CDF obtained from two adjacent sequence simulations as the convergence criterion, as follows: in, For the first The lower boundary of CDF obtained from subsequence simulation The lower boundary of the CDF obtained from the (k-1)th sequence simulation The Minkowski-L1 distance between them Indicates the contribution sample The derivative; the Minkowski-L1 distance As the first Absolute error of a sequence simulation The convergence criterion is: In the formula, This is the preset value for error control.

10. The hybrid uncertainty propagation method for complex equipment response as described in claim 8, characterized in that, In step 5.2, the number of Gaussian integration nodes r is 6.

11. The hybrid uncertainty propagation method for complex equipment response as described in claim 9, characterized in that, The preset value of the error control 。 12. A hybrid uncertainty propagation system for complex equipment response, applied to a coordination mechanism, characterized in that, The coordinating mechanism is a rigid-flexible coupling system, including a motor, a reducer, a coordinating robotic arm, an attitude-adjusting electric cylinder assembly, and a conveying mechanism. The hybrid uncertainty propagation system includes: a modeling module for establishing a rigid-flexible coupling dynamic model of the coordinating mechanism; the input parameters of the rigid-flexible coupling dynamic model include probabilistic uncertainty parameters and interval uncertainty parameters; an uncertainty characterization module for characterizing the output response of the rigid-flexible coupling dynamic model based on the P-box; a hierarchical module for hierarchically classifying the input parameters of the rigid-flexible coupling dynamic model according to the distribution type of the input parameters, with the outer layer being interval uncertainty parameters and the inner layer being probabilistic uncertainty parameters; and a sampling module for selecting the interval uncertainty parameters of the rigid-flexible coupling dynamic model and its sample space, performing global and local sampling within the sample space to obtain multiple sets of interval samples, including global samples and local samples; the global samples are generated from the entire bounded sample space, and the local samples are generated in the neighborhood of the contributing samples based on the contribution index of the contributing samples; the contributing samples refer to samples that contribute to the upper and lower boundaries of the P-box; the contribution index is calculated as follows: for each contributing sample... The CDF of the P-box is denoted as Its contribution is evaluated by the span of the ordinate where the CDF reaches its maximum value: In the formula, Contributing samples Contribution index, Indicates the contribution sample The span of the ordinate that affects the upper or lower boundary of the CDF. The total ordinate span of the upper and lower boundaries of the P-box is represented. The lower boundary generation module is used to randomly select a set of interval samples from the outer layer and simulate the inner layer probability uncertainty parameters using a dimensionality reduction integral method to obtain the P-box response CDF. It then determines whether all interval samples in the outer layer have been calculated; if so, a series of CDF results are obtained. Otherwise, it continues to select uncalculated interval samples for simulation calculation to obtain the P-box response CDF, until all interval samples in the outer layer have been calculated. Based on the series of CDF results, the lower boundary of the P-box calculated in this instance is obtained, and the contribution CDF composition corresponding to the lower boundary is determined, identifying the contribution... The algorithm contributes samples corresponding to the CDF and completes a sequence simulation. It then determines whether the area difference between the lower boundaries of the CDF obtained from the two most recent sequence simulations meets the preset standard. If so, the calculation is considered to have converged, and the final P-box response CDF lower boundary is output. Otherwise, the algorithm controls the sampling module to generate new global and local samples. The upper boundary generation module is used to obtain the upper boundary of the P-box response CDF through sequence simulation. The adjustment module is used to obtain the P-box result of the mixed uncertainty propagation of the coordination mechanism based on the upper and lower boundaries of the P-box response CDF, and adjust the positioning accuracy of the coordination robot arm according to the actual working conditions.

13. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The specific rigid-flexible coupling dynamic model of the coordination mechanism is as follows: In the formula, The quality matrix of the coordinating body. For the motion time of the coordination process, 、 and These represent the relative displacement, relative velocity, and relative acceleration of the rotating hinge in the coordinating mechanism. For the generalized damping matrix of the coordinating mechanism, For the generalized stiffness matrix of the coordination mechanism, The constraint equations for the coordinating mechanism. for right Differentiate, This indicates the transpose of the matrix. For Lagrange multipliers, External forces that coordinate the organization.

14. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The output response of the rigid-flexible coupling dynamics model of the coordination mechanism is the angular displacement angle by which the coordinated robotic arm rotates from an initial -3° position to a position making an angle of 45° with the horizontal. The CDF of the output response is used as... express, Enclosed by the lower boundary of CDF and upper boundary Within the formed area, Output response of the rigid-flexible coupling dynamic model of the coordinating mechanism The P-box is written as: In the formula, In response P-box.

15. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The neighborhood of the contributing sample is defined by a circle centered at... , radius is The circular region or multidimensional spherical region; the formula for calculating the radius is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For bounded sample space The volume or area, denoted as the number of interval uncertainty parameters, and also as the dimension of the bounded sample space.

16. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The neighborhood of the contributing sample is centered at... The side length is A square or multi-dimensional cubic region; The calculation formula is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For bounded sample space The volume or area, denoted as the number of interval uncertainty parameters, and also as the dimension of the bounded sample space.

17. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The neighborhood of the contributing sample is centered at... The side length is A rectangular or multi-dimensional rectangular solid area Let be the number of interval uncertainty parameters. The calculation formula is as follows: In the formula, The neighborhood is represented relative to the entire bounded sample space. Size, For interval The width.

18. The hybrid uncertainty propagation system for complex equipment response as described in any one of claims 15-17, characterized in that, The value is set between 1 / 12 and 1 / 4.

19. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The simulation calculation of the inner-layer probability uncertainty parameters using a dimension reduction integral method to obtain the P-box response CDF specifically involves: applying a univariate dimension reduction integral method to the probability uncertainty parameters, expanding the dimension reduction near the mean, and transforming the multivariate probability uncertainty parameter system into a subsystem containing only a single variable, as shown below: in, The first response Step moment, ； It is the first The mean of a probability uncertainty parameter, yes The abbreviation for , indicating the th after decomposition A single-variable subsystem, Indicates the first One probability uncertainty parameter, Indicates the interval uncertainty parameter. This is a binomial expansion, where j is the order index. Represents the function Find the j-th moment, where m represents the number of probability uncertainty parameters. This indicates that all probability uncertainty parameters are initialized to their mean values; the first four moments of the univariate subsystem are calculated using the Gauss-Hermite integral, and are expressed as: In the formula, Representation function The l-th moment, This represents the number of nodes in the Gaussian integral. and Here, represents the weights of the Gaussian integral nodes and the node weights; the first four central moments are calculated iteratively using the first four origin moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism, and are expressed as: in, The first four origin moments are the output response of the rigid-flexible coupling dynamic model of the coordination mechanism. The first four central moments of the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism are used to obtain the P-box response CDF using the Johnson fitting method. The Johnson fitting method is expressed as follows: in, and For shape parameters, These are dimensional parameters. It is a position parameter. Follows a standard normal distribution. For transformation function, Represents the Johnson distribution. Johnson variables represent the output response of the rigid-flexible coupling dynamic model of the coordinating mechanism.

20. The hybrid uncertainty propagation system for complex equipment response as described in claim 12, characterized in that, The determination of whether the area difference between the lower bounds of the CDF obtained from the two most recent sequence simulations meets the preset standard specifically involves: As the first The lower boundary of the CDF obtained by the subsequence simulation will be The lower boundary of the CDF obtained from the (k-1)th sequence simulation is used as the area difference between the lower boundaries of the CDF obtained from two adjacent sequence simulations as the convergence criterion, as follows: in, For the first The lower boundary of CDF obtained from subsequence simulation The lower boundary of the CDF obtained from the (k-1)th sequence simulation The Minkowski-L1 distance between them Indicates the contribution sample The derivative; the Minkowski-L1 distance As the first Absolute error of a sequence simulation The convergence criterion is: In the formula, This is the preset value for error control.

21. The hybrid uncertainty propagation system for complex equipment response as described in claim 19, characterized in that, The number of Gaussian integration nodes, r, is 6.

22. The hybrid uncertainty propagation system for complex equipment response as described in claim 20, characterized in that, The preset value of the error control 。 23. An electronic device, characterized in that, include: The memory, the processor, and the computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the hybrid uncertainty propagation method for complex equipment response as described in any one of claims 1-11.

24. A computer-readable storage medium storing a computer program, characterized in that, The computer program causes the computer to execute the hybrid uncertainty propagation method for complex equipment response as described in any one of claims 1-11.

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