Sensitivity analysis method for time-varying output of motion mechanism under interval uncertain input

Abstract

The application provides a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input, comprising the following steps: constructing a time-varying output interval process model of the motion mechanism to obtain a total interval process of the time-varying output of the motion mechanism; inputting an interval variable X i at a nominal value x i fixed in its interval to obtain a conditional interval process of the time-varying output of the motion mechanism; traversing the interval variable X i in its interval to obtain different conditional interval processes corresponding to different nominal values; accumulating the interval process models to form an interval field model; defining a global sensitivity index λ i based on the volume ratio between interval fields in the interval field model to obtain a contribution degree ranking of the interval uncertainty input to the time-varying output of the motion mechanism. The application quantifies how the uncertainty of the interval input affects the uncertainty of the time-varying output of the motion mechanism, and obtains the contribution degree ranking of the interval input uncertainty, so as to screen out the input which has a significant influence on the uncertainty of the time-varying output of the motion mechanism.

Description

Technical Field

[0001] This invention belongs to the field of global sensitivity analysis technology, specifically relating to a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input. Background Technology

[0002] As modern equipment develops towards higher precision and reliability, the aerospace field places higher demands on the motion accuracy of mechanisms. In engineering practice, due to the inherent uncertainties in input factors, such as component dimensions, joint clearances, manufacturing and assembly tolerances, material properties, and loads, these uncertainties translate into uncertainties in performance response, causing the actual time-varying output of the motion mechanism to deviate from its designed time-varying output.

[0003] Deviation between actual time-varying output and design time-varying output may cause vibration or even failure of the motion mechanism. Investigating the main uncertainties that may cause vibration or even failure of the air-assisted motion mechanism and ranking the input factors according to the relative importance of the main uncertainties is called "fault source identification," which can be achieved using sensitivity analysis techniques.

[0004] Sensitivity analysis aims to explore the allocation of output uncertainty to input uncertainty, reflecting the contribution of input uncertainty to output response uncertainty. Sensitivity analysis of the time-varying output of a motion mechanism is used to identify the main inputs causing uncertainty in the actual time-varying output. The results can effectively guide the motion accuracy analysis of the motion mechanism, thus improving its performance.

[0005] Currently, most sensitivity analysis methods for time-varying outputs of motion mechanisms are based on probabilistic models. In the realm of probability, random variables can effectively describe the randomness of parameters, but require a large amount of sample statistical information. However, in the uncertainties faced in engineering applications, we typically only have limited and incomplete data resources, which are often insufficient to construct a complete probabilistic model that accurately describes the true parameter values. Inexact probabilistic techniques offer an effective solution to this problem, with the most basic technique being the interval model, which only considers the lower and upper bounds of the data. Interval models only need to determine the boundary information of the input and are considered one of the practical and effective tools for quantifying non-probabilistic uncertain inputs.

[0006] Currently, there is a lack of research on the sensitivity analysis of time-varying outputs of motion mechanisms under interval uncertainty inputs. Therefore, there is an urgent need to study sensitivity analysis methods for the time-varying outputs of motion mechanisms under interval variable inputs. Summary of the Invention

[0007] The purpose of this invention is to address the shortcomings of existing sensitivity analysis methods for inputs with interval uncertainty, and to provide a sensitivity analysis method for time-varying outputs of motion mechanisms under inputs with interval uncertainty. This method quantifies how the uncertainty of the interval input affects the uncertainty of the time-varying output of the motion mechanism, and obtains a ranking of the contribution of the interval input uncertainty, thereby screening out inputs that have a significant impact on the uncertainty of the time-varying output of the motion mechanism.

[0008] To achieve the above objectives, the technical solution provided by this invention is:

[0009] Sensitivity analysis methods for time-varying outputs of motion mechanisms under interval uncertainty inputs include:

[0010] Step 1: Based on interval analysis theory, construct a time-varying output interval process model of the motion mechanism under interval input. The time-varying output interval process model is the total interval process of obtaining the time-varying output of the motion mechanism under interval vector X input.

[0011] Step 2: Input the interval variable X i The nominal value x is fixed within its interval i At, within the interval vector X, excluding the interval variable input X i Other input variables remain interval variables, resulting in a conditional interval process for obtaining the time-varying output of the motion mechanism;

[0012] Step 3: Repeat step 2 to input the interval variable X. i Traverse within its interval, fix at different nominal values, and obtain different conditional interval processes corresponding to different nominal values; construct different interval process models based on the total interval process and different conditional interval processes, and accumulate the geometric surfaces of the different interval process models to form an interval field model;

[0013] Step 4: Define a global sensitivity index λ based on the volume ratio between the interval fields in the interval field model. i According to the global sensitivity index λ i The contribution of interval uncertainty input to the time-varying output of the motion mechanism is ranked.

[0014] As a further limitation of the present invention, step one specifically includes:

[0015] (11) Define the time variable t of the motion mechanism, where, during the motion of the motion mechanism, the time variable is defined as:

[0016] t∈[t s , t f ]

[0017] In the formula, t s t represents the initial time of the time observation region. fIndicates the end time of the time observation region;

[0018] (12) Define the interval vector of the motion mechanism as X = [X1, X2, ..., X...]. n ], where X is a single interval variable input to the interval vector X. i Defined as:

[0019]

[0020] In the formula, I represents the interval. Input X represents a range variable. i The lower bound, Input X represents a range variable. i The upper bound;

[0021] (13) According to the interval analysis theory, the interval variable input X i radius X r Defined as:

[0022]

[0023] Input range variable X i median X c Defined as:

[0024]

[0025] (14) Define the output Y(t) of the motion mechanism. During the total time interval of the time-varying output of the motion mechanism, the time interval is discretized as follows:

[0026]

[0027] At any time t j Output of the motion mechanism Also a range, represented as:

[0028]

[0029] In the formula, Indicates the output range The lower bound, Indicates the output range The upper bound;

[0030] (15) Define the relationship between the input and output of the motion mechanism as follows:

[0031] Y(t) = f(X, t);

[0032] The time-varying output of the motion mechanism is an interval process model, represented by Y. I (t), the expression is:

[0033]

[0034] In the formula, Y l (t) represents the lower boundary line of the interval process, Y u (t) represents the upper boundary line of the interval process.

[0035] As a further limitation of the present invention, step two specifically includes:

[0036] (21) Input the interval variable into X i The nominal value x is fixed within its interval i The location is represented as:

[0037]

[0038] In the formula, Input X represents a range variable. i The interval;

[0039] (22) The interval vector X excluding the interval variable input X i Internal division of nominal value x i Other input variables besides these are still range variables, represented as:

[0040]

[0041] (23) The conditional interval process of the time-varying output of the motion mechanism is expressed as:

[0042]

[0043] As a further limitation of the present invention, in step three:

[0044] (31) Input the interval variable into X i Traversing within its interval can be represented as:

[0045]

[0046] In the formula, Input X represents a range variable. i The kth nominal value, Input X represents a range variable. i The interval;

[0047] (32) Obtaining different condition intervals corresponding to different nominal values

[0048] (33) Construct different interval process models based on the total interval process and interval processes with different conditions, including:

[0049] The interval process model is extended to include the total interval process Y.I (t) and conditional interval process

[0050] The extension stacks the geometric surfaces corresponding to the interval process model into a geometric body along a certain direction, and the geometric body is the interval field model of the time-varying output of the motion mechanism.

[0051] (34) Accumulating the geometric surfaces of different interval process models to form an interval field model, including:

[0052] Based on the construction principle of the interval field model, the interval field model includes the total interval process Y. I (t) The total interval field formed by accumulation and conditional interval process The accumulated conditional interval field Y I (X i ,t).

[0053] As a further limitation of the present invention, the total interval process Y I In the total interval field corresponding to (t), the total interval process Y I The geometric surface corresponding to (t) is input from the interval variable X. i lower bound Input X to the interval variable i The upper realm Stacking and accumulating to form a geometry, which is defined as:

[0054]

[0055] In the formula, This represents the lower boundary surface of the total interval field. The upper boundary surface of the total interval field is represented; the conditional interval field Y I (X i In (t), the conditional interval process under different nominal values The corresponding geometric surface is input from the interval variable X. i lower bound Input X to the interval variable i The upper realm Stacking and accumulating to form a geometry, which is defined as:

[0056] Y I (X i ,t)=[Y l (X i ,t),Y u (X i ,t)]

[0057] In the formula, Y l (X i ,t) represents the lower boundary surface of the conditional interval field, Yu (X i ,t) denotes the upper boundary surface of the conditional interval field;

[0058] Conditional interval field represents the interval variable input X i Under different nominal values, the remaining interval variable X in the interval vector X ～i The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism.

[0059] As a further limitation of the present invention, step four specifically includes:

[0060] (41) Based on the interval field model obtained in step three, the conditional interval field Y I (X i The upper boundary surface Y of t) u (X i ,t) and the lower boundary surface Y l (X i ,t) will be the total interval field It is divided into the following 3 parts:

[0061] (411) The upper boundary surface of the total interval field and the upper boundary surface Y of the conditional interval field u (X i The volume enclosed by t)

[0062] (412) The upper boundary surface Y of the conditional interval field u (X i The lower boundary surface Y of the conditional interval field (t) and t) l (X i The volume enclosed by t)

[0063] (413) The lower boundary surface Y of the conditional interval field l (X i ,t) and the lower boundary surface of the total interval field The volume enclosed Total Interval Field volume Represented as:

[0064]

[0065] In the formula, Representing the interval variable X i The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism;

[0066] (42) Define the global sensitivity index λ based on the volume ratio of the interval field. i Global sensitivity index λ iThe magnitude of the value measures the degree of influence of the interval uncertainty input on the time-varying output uncertainty of the motion mechanism.

[0067] As a further limitation of the present invention, the total interval field volume Its calculation formula is defined as:

[0068]

[0069] In the formula, Y u (t) represents the upper boundary line of the interval process, Y l (t) represents the lower boundary line of the interval process. Input X represents a range variable. i The upper realm, Input X represents a range variable. i The lower bound;

[0070] Conditional interval field Y I (X i volume of t) Its calculation formula is defined as:

[0071]

[0072] Global sensitivity index λ i Its calculation formula is defined as:

[0073]

[0074] As a further limitation of the present invention, in steps one, two and four, the time-varying output of the motion mechanism is a continuous process that changes with time, and the time-varying output of the motion mechanism is processed by a discretization method to handle the time transformation problem.

[0075] Discretize the time variable corresponding to the time transformation as follows:

[0076] T = [t1, t2, ..., t] N ] = [t s , t f ]

[0077] In the formula, t1 = t s , t N =t f .

[0078] As a further limitation of the present invention, the volume of the interval field and the total volume of the geometry are calculated when calculating the sensitivity index, wherein:

[0079] The upper and lower bounds of the inner layer calculation output, specifically:

[0080] The lower bound Y of the unconditional output is calculated by optimizing the model.l (t) and the lower bound of conditional output The optimization model is represented as:

[0081]

[0082] The upper bound Y of the unconditional output is calculated by optimizing the model. u (t) and the upper bound of conditional output Transformed into: maxf(·)=min(-f(·));

[0083] The outer layer calculates the integral, specifically:

[0084] The integral calculation is solved by numerical integration, and the approximation accuracy is high based on the Newton-Cotes formula.

[0085] The advantages of this invention are:

[0086] 1. This invention uses a global sensitivity index to screen inputs that have a significant impact on the time-varying output of the motion mechanism, and adopts a volume ratio form that is simple and intuitive; the calculation process is clear and can better reflect the impact of interval inputs on the uncertainty of the time-varying output of the motion mechanism.

[0087] 2. The present invention uses a sensitivity index to obtain not only the ranking of the contribution of the input to the time-varying output of the motion mechanism, but also the change of the influence of the interval variable on the time-varying output of the motion mechanism when it changes within its interval.

[0088] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0089] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0090] Figure 1 The present invention provides a flowchart of a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input;

[0091] Figure 2 The present invention provides an interval process model for the time-varying output of a motion mechanism;

[0092] Figure 3 The present invention provides illustrations of the conditional interval process and the unconditional interval process of the time-varying output of the motion mechanism;

[0093] Figure 4 : A diagram illustrating the construction of the interval field model for the time-varying output of the motion mechanism provided by this invention;

[0094] Figure 5The present invention provides an illustration of the total interval field and the conditional interval field;

[0095] Figure 6 The essential diagram illustrating the definition of the volume and sensitivity index of the interval field in the interval field model provided by this invention;

[0096] Figure 7 Illustrations of two extreme cases of conditional interval fields in the interval field model provided by this invention;

[0097] Figure 8 A simplified kinematic diagram of the four-bar linkage provided by this invention;

[0098] Figure 9 The diagram illustrates the total interval process of the time-varying output of the four-bar linkage provided by this invention.

[0099] Figure 10 The diagram shows the interval field model of the four-bar linkage example provided by this invention, with interval input.

[0100] Figure 11 The following is a diagram showing the ranking results of the sensitivity indices of the input variables in the four-bar linkage example provided by this invention.

[0101] Figure 12 Table of calculation results for sensitivity index of the four-bar linkage provided by this invention. Detailed Implementation

[0102] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0103] Please see Figure 1 This invention discloses a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input, comprising steps one to four, specifically:

[0104] Step 1: Based on interval analysis theory, construct a time-varying output interval process model of the motion mechanism under interval input. The time-varying output interval process model is the total interval process of obtaining the time-varying output of the motion mechanism under interval vector X input.

[0105] Step one of the embodiments of the present invention is specifically as follows:

[0106] (11) Time variable: Define the time variable t of the motion mechanism, where, during the motion of the motion mechanism, the time variable t is defined as:

[0107] t∈[t s ,t f ]

[0108] In the formula, t st represents the initial time of the time observation region. f This indicates the end time of the time observation area; time inputs typically include time-varying variables related to the time-varying output of the motion mechanism, such as angle, displacement, and time.

[0109] (12) Interval uncertainty input: Define the interval vector X = [X1, X2, ..., X...] of the motion mechanism. n In practical engineering, the experimental data obtained is often limited. Therefore, it is difficult to establish accurate probabilistic models for the input variables. Interval models only need to obtain boundary information of the input variables, so describing the input variables as interval variables is an effective means of quantifying uncertain inputs. A single interval variable input X in the interval vector X. i Defined as:

[0110]

[0111] In the formula, I represents the interval. Input X represents a range variable. i The lower bound, Input X represents a range variable. i The upper bound;

[0112] (13) According to the interval analysis theory, the interval variable input X i radius X r Defined as:

[0113]

[0114] Input range variable X i median X c Defined as:

[0115]

[0116] (14) Define the output Y(t) of the motion mechanism. During the total time interval of the time-varying output of the motion mechanism, the time interval is discretized as follows:

[0117]

[0118] At any time t j Under the following conditions, due to the existence of interval variables, the output of the mechanism must also be an interval, and the output of the motion mechanism... Also a range, represented as:

[0119]

[0120] In the formula, Indicates the output range The lower bound, Indicates the output range The upper bound; throughout the entire motion process, the time-varying output of the motion mechanism is an interval process model, such as... Figure 2 As shown.

[0121] (15) Define the relationship between the input and output of the motion mechanism as follows:

[0122] Y(t) = f(X,t);

[0123] Due to the existence of interval variables, the time-varying output of the motion mechanism is an interval process model, represented as Y. I (t), the expression is:

[0124]

[0125] In the formula, Y l (t) represents the lower boundary line of the interval process, Y u (t) represents the upper boundary line of the interval process. The uncertainty of the time-varying output of the motion mechanism is represented by the region enclosed by the upper and lower boundary lines.

[0126] Step 2: Input the interval variable X i The nominal value x is fixed within its interval i At, within the interval vector X, excluding the interval variable input X i Other input variables remain interval variables, resulting in a conditional interval process for the time-varying output of the motion mechanism.

[0127] Step two in the embodiments of the present invention is specifically as follows:

[0128] (21) Input the interval variable into X i The nominal value x is fixed within its interval i The location is represented as:

[0129]

[0130] In the formula, Input X represents a range variable. i The interval;

[0131] (22) The interval vector X excluding the interval variable input X i Other input variables besides these are still range variables, represented as:

[0132]

[0133] The time-varying output of the motion mechanism under this condition It is still a range process like Figure 3 As shown. In order to connect with Y I (t) Naming distinction, This is called a conditional interval process, YI (t) is called the total interval process.

[0134] (23) The conditional interval process of the time-varying output of the motion mechanism is expressed as:

[0135]

[0136] Conditional interval process When the interval variable X i =x i At that time, the time-varying output of the motion mechanism is still an interval process, and is defined as:

[0137]

[0138] in, and These are the upper and lower boundary lines of the conditional interval process, respectively. When the interval variable is input X... i Fixed at its nominal value (X) i (If the uncertainty is eliminated), the uncertainty of the output may be reduced, so the conditional interval process... It is the total interval process Y I A subset of (t).

[0139] Step 3: Repeat step 2 and input the interval variable X. i Traverse within its interval, fix at different nominal values, and obtain different conditional interval processes corresponding to different nominal values; construct different interval process models based on the total interval process and different conditional interval processes, and accumulate the geometric surfaces of different interval process models (each of these interval process models includes the total interval process and the corresponding conditional interval process) to form an interval field model.

[0140] In step three of the above embodiments of the present invention:

[0141] (31) Input the interval variable into X i Traversing within its interval (with fixed, different nominal values) is represented as:

[0142]

[0143] In the formula, Input X represents a range variable. i The kth nominal value, Input X represents a range variable. i The interval;

[0144] (32) Obtaining different condition intervals corresponding to different nominal values Next, the interval process model is extended (the interval process model includes the total interval process Y). I(t) and conditional interval process ).

[0145] (33) Construct different interval process models based on the total interval process and interval processes with different conditions, including:

[0146] The interval process model is extended to include the total interval process Y. I (t) and conditional interval process

[0147] The extension involves stacking the geometric surfaces corresponding to the interval process model along a certain direction to form a geometric body, which is the interval field model of the time-varying output of the motion mechanism, such as... Figure 4 As shown.

[0148] (34) Accumulate the geometric surfaces of different interval process models to form an interval field model, including:

[0149] Based on the construction principle of the interval field model, the interval field model includes the total interval process Y. I (t) The total interval field formed by accumulation and conditional interval process The accumulated conditional interval field Y I (X i ,t).

[0150] In this embodiment of the invention, the total interval process Y I In the total interval field corresponding to (t), the total interval process Y I The geometric surface corresponding to (t) is input from the interval variable X. i lower bound Input X to the interval variable i The upper realm Stacking and accumulating forms geometry, such as Figure 5 As shown in (A), it is defined as:

[0151]

[0152] In the formula, This represents the lower boundary surface of the total interval field. This represents the upper boundary surface of the total interval field;

[0153] Conditional interval field Y I (X i In (t), conditional interval processes under different nominal values The corresponding geometric surface is input from the interval variable X. i lower bound Input X to the interval variable i The upper realm Stacking and accumulating forms geometry, such as Figure 5As shown in (B), it is defined as:

[0154] Y I (X i ,t)=[Y l (X i ,t),Y u (X i ,t)]

[0155] In the formula, Y l (X i ,t) represents the lower boundary surface of the conditional interval field, Y u (X i ,t) denotes the upper boundary surface of the conditional interval field;

[0156] Conditional interval field represents the interval variable input X i Under different nominal values, the remaining interval variable X in the interval vector X ～i The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism.

[0157] Step 4: Define the global sensitivity index λ based on the volume ratio between the interval fields in the interval field model. i According to the global sensitivity index λ i The contribution of interval uncertainty input to the time-varying output of the motion mechanism is ranked.

[0158] Step four of this embodiment of the invention specifically includes:

[0159] (41) Based on the interval field model obtained in step three, the conditional interval field Y I (X i The upper boundary surface Y of t) u (X i ,t) and the lower boundary surface Y l (X i ,t) will be the total interval field It is divided into the following 3 parts, such as Figure 6 As shown:

[0160] (411) The upper boundary surface of the total interval field The upper boundary surface Y of the conditional interval field u (X i The volume enclosed by t)

[0161] (412) The upper boundary surface Y of the conditional interval field u (X i The lower boundary surface Y of the conditional interval field (t) and t) l (X i The volume enclosed by t)

[0162] (413) The lower boundary surface Y of the conditional interval field l (X i ,t) and the lower boundary surface of the total interval field The volume enclosed Total Interval Field volume Represented as:

[0163]

[0164] In the formula, Input X represents a range variable. i The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism;

[0165] (42) Define the global sensitivity index λ based on the volume ratio of the interval field. i Global sensitivity index λ i The magnitude of the value measures the degree of influence of the interval uncertainty input on the time-varying output uncertainty of the motion mechanism.

[0166] Total Interval Field of the Embodiment of the Invention volume Its calculation formula is defined as:

[0167]

[0168] In the formula, Y u (t) represents the upper boundary line of the interval process, Y l (t) represents the lower boundary line of the interval process. Input X represents a range variable. i The upper realm, Input X represents a range variable. i The lower bound;

[0169] Conditional interval field Y I (X i volume of t) Its calculation formula is defined as:

[0170]

[0171] Global sensitivity index λ i Its calculation formula is defined as:

[0172]

[0173] In the above embodiments of the present invention, when calculating the sensitivity index, the volume of the solution interval field and the total volume of the geometry are determined, wherein:

[0174] As can be seen from the above formula, calculating the sensitivity index requires solving for the volume of the interval field and the total volume of the geometry, i.e., solving the above formula. Clearly, the calculation is a nested process.

[0175] The inner layer needs to calculate the upper and lower bounds of the output, specifically:

[0176] The lower bound Y of the unconditional output is calculated by optimizing the model. l (t) and the lower bound of conditional output The optimization model is represented as:

[0177]

[0178] The upper bound Y of the unconditional output is calculated by optimizing the model. u (t) and the upper bound of conditional output This is transformed into: maxf(·)=min(-f(·)); In this embodiment of the invention, the optimization problem is solved using the “particleswarm” function (Particle Swarm Optimization Algorithm (PSO)) in MATLAB software.

[0179] The outer layer of this invention requires integral calculation, specifically:

[0180] This invention employs a numerical integration method to solve integral calculations, based on the Newton-Cotes formula, which offers high approximation accuracy. The formula is also relatively simple and is a commonly used numerical integration method in engineering. The trapezoidal integral in this invention is implemented using the "trapz" function in MATLAB software.

[0181] Furthermore, in steps one, two, and four of this embodiment of the invention, the time-varying output of the motion mechanism is a continuous process that changes with time, and the time-varying output of the motion mechanism is processed by a discretization method to handle the time transformation problem.

[0182] Discretize the time variable corresponding to the time transformation as follows:

[0183] T = [t1, t2, ..., t N ] = [t s ,t f ]

[0184] In the formula, t1 = t s , t N =t f .

[0185] The calculation steps for the indicators in this embodiment of the invention include:

[0186] (1) The calculation steps are as follows:

[0187] (11) Generate a set of discrete time parameter sets T corresponding to the time variable t, i.e.: T = {t1, t2, ..., t j ,…t M}=[t s ,t f ], where t1 = t s , t M =t f .

[0188] (12) Fix the time variable t at a certain time t j Below, that is: t = t j Calculate the upper bound Y of the unconditional output at this moment. u (t) and lower bound Y l (t).

[0189] (13) From t1 to t M Fix the time variable t and repeat step 2 to calculate the upper bound Y of the unconditional output at all times. u (t) and lower bound Y l (t).

[0190] (14) From the initial time t s At the final moment t f Integrate the upper and lower bounds of the unconditional output, i.e.:

[0191]

[0192] (15) Multiply the integral result by the interval variable X i The volume of the total interval field is obtained by determining the interval length, i.e.:

[0193]

[0194] (2) The calculation steps are as follows:

[0195] (21) Generate a set of nominal values ​​for the interval vector X, i.e.: in,

[0196] (22) Input variable X in the interval i A nominal value fixed within its interval Place, that is:

[0197] (221) Fix the time variable t at a certain time t. j Below, that is: t = t j Calculate the upper bound of the conditional output at this moment. and the lower realm

[0198] (222) From t1 to t2 in sequence M Fix the time variable t and repeat step (21) to calculate the upper bound of the conditional output at all times. and the lower world

[0199] (223) From the initial time t s At the final moment t f Integrate the upper and lower bounds of the conditional output, i.e.:

[0200]

[0201] (23) sequentially from arrive Input the interval variable X i Fix and repeat steps (221) to (222), and perform the integral calculation in step (223) on the upper and lower bounds of the conditional output under all nominal values.

[0202] (24) Input X from the interval variable i lower bound To the Upper Realm Integrating, we obtain the volume of the field over the conditional interval, i.e.:

[0203] (3)λ i The calculation can be obtained according to the following formula:

[0204] This invention further investigates the value range and characteristics of the indicators. Detailed explanations are provided below:

[0205] (1) The proposed global sensitivity index λ i The formula is defined as the ratio of volumes, that is: Therefore, its lower limit is 0 and its upper limit is 1, that is: 0≤λ i ≤1.

[0206] (2) Figure 7 (A), when the conditional interval field Y I (X i The upper boundary surface Y of t) u (X i ,t) and the total interval field upper boundary surface The fields always coincide, and the condition interval field Y I (X i The lower boundary surface Y of ,t) l (X i ,t) and the total interval field lower boundary surface When they always coincide, the condition interval field Y I (X i ,t) and the total interval field The volumes are equal, that is: In this case, λ i =0 indicates that the input variable X i It has no effect on the uncertainty of the time-varying output of the motion mechanism.

[0207] (3) Figure 7 (B), when the conditional interval field Y I (X i The upper boundary surface Y of t) u (X i ,t) and its lower boundary surface Y u (X i When ,t) always coincide, other intervals input X ～i The dependent variable X for the time-varying output uncertainty of the motion mechanism i This is eliminated, resulting in the volume of the field in the conditional interval being 0, i.e.: In this case, λ i =1 indicates that the input variable X i It has an absolute impact on the uncertainty of the time-varying output of the motion mechanism.

[0208] (4) When 0≤λ i When ≤1, λ i The larger the variable X is, the more likely it is to be affected. i The greater the impact on the time-varying output uncertainty of the motion mechanism, the greater the influence; conversely, the smaller the impact, the greater the influence. i The smaller the value of variable X, the better. i The smaller the impact on the time-varying output uncertainty of the motion mechanism.

[0209] In summary, the global sensitivity index proposed in this embodiment of the invention can not only obtain the ranking of the contribution of input variables to the time-varying output uncertainty of the motion mechanism, but more importantly, it can obtain the changes in the influence of interval input variables on the time-varying output uncertainty of the motion mechanism when they change within their interval through the interval field model.

[0210] To verify the feasibility of the proposed method, the proposed indexes are applied to a numerical example of a typical four-bar linkage in engineering.

[0211] This invention analyzes a typical four-bar linkage mechanism in engineering. Figure 8This is a simplified diagram of a four-bar linkage. The angle between crank R1 and frame R4 is θ, the angle between connecting rod R2 and the horizontal axis x is δ, and the angle between rocker arm R3 and frame R4 is ψ. Consider the lengths of the four links X = [X1, X2, X3, X4] = [R1, R2, R3, R4] as interval variables, with their midpoints and deviations X1 and X2 respectively. c = [53mm, 122mm, 66.5mm, 100mm] and X r = [0.3mm, 0.3mm, 0.3mm, 0.3mm]. The input angle θ is a time variable, and its range is [105deg, 205deg].

[0212] The angular relationships of a four-bar linkage throughout the entire motion process can be expressed as:

[0213]

[0214] In this engineering example, the output response is considered to be the angle ψ between the rocker arm R3 and the frame R4, which can be derived from the above formula as follows:

[0215]

[0216] in:

[0217] D=-2R1R2sinθ

[0218] E = 2R³(R⁴ - R⁁cosθ)

[0219]

[0220] Meanwhile, the design value of the output response of interest is expressed as:

[0221] ψ d (θ)=76deg+60degsin[3(θ-95.5deg) / 4]

[0222] The time-varying output of this invention studies the variation of the error of the actual output angle ψ of a four-bar linkage with θ, i.e.: Y(θ)=ψ(θ)-ψ d (θ). Next, a sensitivity analysis is performed on the variation of the actual output angle error of the four-bar linkage.

[0223] Figure 9 This is a process diagram showing the range of changes in the actual output angle error of a four-bar linkage. Figure 10 It is a time-varying output interval field model diagram corresponding to the input variables of each interval. By observation, it can be clearly seen how each interval input affects the change of the actual output angle error of the four-bar linkage within its interval. Figure 12 The table presents the calculation results for the interval input sensitivity index.Figure 11 The impact of interval inputs on the uncertainty of the actual output angle error variation of the four-bar linkage is ranked as follows: X1 > X2 > X3 > X4. The influence of the interval variable link R2 is the greatest; the influence of the interval variable frame R4 is the second greatest; the influence of crank R1 is less than that of frame R4; and the influence of the interval variable rocker arm R3 is very small. Therefore, when considering the actual output angle error of the four-bar linkage, designers should focus on link R2, frame R4, and crank R1.

[0224] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention.

Claims

1. A sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input, characterized in that, include: Step 1: Based on interval analysis theory, construct a time-varying output interval process model of the motion mechanism under interval input. The time-varying output interval process model is based on interval vectors. The total interval process of obtaining the time-varying output of the motion mechanism under input; The above interval process model only needs to obtain the boundary information of the input variables. The uncertainty of the time-varying output of the motion mechanism is represented by the region enclosed by the upper and lower boundary lines. Step 2: Input the interval variable The nominal value fixed within its range At, interval vector Inner interval variable input Other input variables remain interval variables, resulting in a conditional interval process for obtaining the time-varying output of the motion mechanism; Step 3: Repeat step 2 to input the interval variable. By traversing within its interval and fixing at different nominal values, we obtain different conditional interval processes corresponding to different nominal values; Each interval process model includes a total interval process and a corresponding conditional interval process; Different interval process models are constructed based on the total interval process and different conditional interval processes. The geometric surfaces of the different interval process models are accumulated to form an interval field model. Specifically, the conditional interval field represents the interval variable input. Interval vectors under different nominal values Remaining interval variable The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism; according to the construction principle of the interval field model, the interval field model includes the total interval process. The total interval field formed by accumulation and conditional interval process The accumulated conditional interval field ; Step 4: Define a global sensitivity index based on the volume ratio between the interval fields in the interval field model. According to the global sensitivity index The contribution ranking of interval uncertainty inputs to the time-varying output of the motion mechanism is obtained; specifically, a global sensitivity index is defined based on the volume ratio of the interval fields. Global sensitivity index The magnitude measures the degree of influence of the interval uncertainty input on the time-varying output uncertainty of the motion mechanism. Global sensitivity index Its lower limit is 0 and its upper limit is 1, that is: ,when hour, The larger the input variable The greater the impact on the time-varying output uncertainty of the motion mechanism, the less impact it has on the time-varying output uncertainty. The smaller the input variable The smaller the impact on the time-varying output uncertainty of the motion mechanism; Step four specifically includes: (41) Based on the interval field model obtained in step three, conditional interval field upper boundary surface With the lower boundary surface Total interval field It is divided into the following 3 parts: (411) The upper boundary surface of the total interval field and the upper boundary surface of the conditional interval field The volume enclosed ; (412) The upper boundary surface of the condition interval field and the lower boundary surface of the conditional interval field The volume enclosed ; (413) The lower boundary surface of the conditional interval field and the lower boundary surface of the total interval field The volume enclosed Total Interval Field volume Represented as: In the formula, Representing interval variables The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism; (42) Define the global sensitivity index based on the volume ratio of the interval field. Global sensitivity index The magnitude measures the degree of influence of the interval uncertainty input on the time-varying output uncertainty of the motion mechanism. Total Interval Field volume Its calculation formula is defined as: In the formula, Represents the upper boundary line of an interval process. This represents the lower boundary line of an interval process. Indicates interval variable input The upper realm, Indicates interval variable input The lower bound; Conditional interval field volume Its calculation formula is defined as: In the formula, Indicates the initial time of the time observation region. Indicates the end time of the time observation region; Global sensitivity index Its calculation formula is defined as: 。 2. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 1, characterized in that, Step one specifically involves: (11) Define the time variable of the motion mechanism In the process of motion of the motion mechanism, the time variable is... Defined as: In the formula, Indicates the initial time of the time observation region. Indicates the end time of the time observation region; (12) Define the interval vector of the motion mechanism. , where interval vector Input of a single interval variable Defined as: In the formula, Represents an interval, Indicates interval variable input The lower bound, Indicates interval variable input The upper bound; (13) According to interval analysis theory, interval variable input radius Defined as: ； Interval variable input median Defined as: ； (14) Define the output of the motion mechanism In the total time-varying output process of the motion mechanism, the time interval is discretized as follows: ； At any time Output of the motion mechanism Also a range, represented as: In the formula, Indicates the output range The lower bound, Indicates the output range The upper bound; (15) Define the relationship between the input and output of the motion mechanism as follows: ； The time-varying output of the motion mechanism is an interval process model, expressed as follows: The expression is: In the formula, This represents the lower boundary line of an interval process. This represents the upper boundary line of an interval process.

3. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 1, characterized in that, Step two specifically involves: (21) Input the interval variable The nominal value fixed within its range The location is represented as: In the formula, Indicates interval variable input The interval; (22) Interval vectors Inner interval variable input Other input variables besides these are still range variables, represented as: (23) The conditional interval process of the time-varying output of the motion mechanism is expressed as: in For the total interval process, It is a conditional interval process.

4. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 1, characterized in that, In step three: (31) Input the interval variable Traversing within its interval can be represented as: In the formula, Indicates interval variable input The A nominal value, Indicates interval variable input The interval; (32) Obtaining different condition intervals corresponding to different nominal values ; (33) Construct different interval process models based on the total interval process and interval processes with different conditions, including: The interval process model is extended to include the total interval process. and conditional interval process ; The extension stacks the geometric surfaces corresponding to the interval process model into a geometric body along a certain direction, and the geometric body is the interval field model of the time-varying output of the motion mechanism. (34) Accumulating the geometric surfaces of different interval process models to form an interval field model, including: Based on the construction principle of the interval field model, the interval field model includes a total interval process. The total interval field formed by accumulation and conditional interval process The accumulated conditional interval field .

5. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 4, characterized in that, Total interval process In the corresponding total interval field, the total interval process The corresponding geometric surface is input from the interval variable. lower bound Input to interval variable The upper realm Stacking and accumulating to form a geometry, which is defined as: In the formula, This represents the lower boundary surface of the total interval field. This represents the upper boundary surface of the total interval field; Conditional interval field In the context of conditional interval processes under different nominal values... The corresponding geometric surface is input from the interval variable. lower bound Input to interval variable The upper realm Stacking and accumulating to form a geometric shape, which is defined as: In the formula, This represents the lower boundary surface of the conditional interval field. This represents the upper boundary surface of the conditional interval field; Conditional interval field represents interval variable input Interval vectors under different nominal values Remaining interval variable The contribution of uncertainty to the time-varying output uncertainty of the motion mechanism.

6. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 1, characterized in that, In steps one, two, and four, the time-varying output of the motion mechanism is a continuous process that changes over time. The time-varying output of the motion mechanism is handled using a discretization method to address the time transformation problem; the time variable corresponding to the time transformation is discretized into: In the formula, , .

7. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input as described in claim 1, characterized in that, When calculating the sensitivity index, the volume of the interval field and the total volume of the geometry are solved, where: The upper and lower bounds of the inner layer calculation output, specifically: The lower bound of the unconditional output is calculated by optimizing the model. and the lower bound of conditional output The optimization model is expressed as: ； The upper bound of the unconditional output is calculated by optimizing the model. and the upper bound of conditional output Transformed into: ; The outer layer calculates the integral, specifically: The integral calculation is solved by numerical integration, and the approximation accuracy is high based on the Newton-Cotes formula.

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