Sensitivity analysis method for time-varying output of movement mechanism under interval uncertainty input

Abstract

The invention provides a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input, which comprises 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; the interval variable input Xi is fixed at the nominal value xi in the interval, and the condition interval process of time-varying output of the movement mechanism is obtained; inputting the interval variable into Xi, and traversing in the interval to obtain different condition interval processes corresponding to different nominal values; accumulating the interval process models to form an interval field model; and defining a global sensitivity index lambda i based on a volume ratio between interval fields in the interval field model, and obtaining a contribution degree sequence of interval uncertainty input to time-varying output of the motion mechanism. According to the method, how the uncertainty of the interval input influences the uncertainty of the time-varying output of the movement mechanism is quantified, and the contribution degree sequence of the uncertainty of the interval input is obtained, so that the input which has significant influence on the uncertainty of the time-varying output of the movement mechanism is screened out.

Description

Technical Field

[0001] The present invention belongs to the technical field of global sensitivity analysis, and in particular relates to a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input. Background Art

[0002] As modern equipment develops towards high precision and high reliability, higher requirements are placed on the motion accuracy of mechanisms in the aerospace field. In engineering practice, due to the widespread uncertainty in input factors, such as component size, joint clearance, manufacturing and assembly tolerances, material properties, loads, etc., these uncertainties will be transformed into uncertainty in performance response, causing the actual time-varying output of the motion mechanism to deviate from its designed time-varying output.

[0003] The deviation between the actual time-varying output and the designed time-varying output may cause vibration or even failure of the motion mechanism. Investigating the main uncertain factors that may cause vibration or even failure of the air-bleed motion mechanism and sorting the input factors according to the relative importance of the main uncertain factors is called "fault source identification", which can be achieved using sensitivity analysis technology.

[0004] Sensitivity analysis technology aims to explore the allocation of output uncertainty to input uncertainty and reflect 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 that cause uncertainty in the actual time-varying output of the motion mechanism. The results can effectively guide the motion accuracy analysis of the motion mechanism and help improve the performance of the motion mechanism.

[0005] At present, most of the sensitivity analysis methods for time-varying outputs of motion mechanisms are based on probability models. In the field of probability, random variables can better describe the randomness of parameters, but they require a large amount of sample statistical information. However, in the uncertainty problems faced in the field of engineering applications, we usually can only obtain limited and incomplete data resources, which are often insufficient to build a complete probability model that accurately describes the true model parameter values. Imprecise probability technology provides an effective solution for this. The most basic imprecise probability technology is the interval model, which only considers the lower and upper limits of the data. The interval model only needs to determine the boundary information of the input and is considered to be one of the practical and effective tools for quantifying non-probabilistic uncertain inputs.

[0006] At present, there is almost no research on the sensitivity analysis of the time-varying output of the motion mechanism under interval uncertainty input. In view of this, it is urgent to study the sensitivity analysis method of the time-varying output of the motion mechanism under interval variable input. Summary of the invention

[0007] The purpose of the present invention is to solve the shortcomings of the sensitivity analysis under interval uncertainty input in the prior art, and to provide a sensitivity analysis method for the time-varying output of a motion mechanism under interval uncertainty input, to quantify how the uncertainty of the interval input affects the uncertainty of the time-varying output of the motion mechanism, and to obtain the contribution ranking of the interval input uncertainty, so as to screen out the inputs that have a significant impact on the uncertainty of the time-varying output of the motion mechanism.

[0008] To achieve the above purpose, the technical solution provided by the present invention is:

[0009] The sensitivity analysis method of the time-varying output of the motion mechanism under interval uncertainty input includes:

[0010] Step 1: According to interval analysis theory, a time-varying output interval process model of a motion mechanism under interval input is constructed, wherein the time-varying output interval process model is a total interval process of obtaining a time-varying output of the motion mechanism under interval vector X input;

[0011] Step 2: Input the interval variable into X i The nominal value x is fixed within its interval i At the interval vector X, except for the interval variable input X i The other input variables are still interval variables, and the conditional interval process of the time-varying output of the motion mechanism is obtained;

[0012] Step 3: Repeat step 2 to input the interval variable into X i Traversing within the interval, fixing at different nominal values, obtaining different conditional interval processes corresponding to different nominal values; constructing different interval process models according to the total interval process and different conditional interval processes, and accumulating the geometric surfaces of the different interval process models to form an interval field model;

[0013] 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 Obtain the contribution ranking of interval uncertainty input to the time-varying output of the motion mechanism.

[0014] As a further limitation of the present invention, the step 1 is specifically:

[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 represents the initial moment of the time observation region, t fIndicates the end time of the time observation area;

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

[0019]

[0020] In the formula, I represents the interval, Represents interval variable input X i The lower bound of Represents interval variable input X i The upper bound of

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

[0022]

[0023] Interval variable input X i The median value of X c Defined as:

[0024]

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

[0026]

[0027] At any time t j , the output of the motion mechanism is also an interval, expressed as:

[0028]

[0029] In the formula, Indicates the output interval The lower bound of Indicates the output interval The upper bound of

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

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

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

[0033]

[0034] Where Y l (t) represents the lower boundary 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, the step 2 is specifically as follows:

[0036] (21) Input the interval variable into X i The nominal value x is fixed within its interval i At, it is expressed as:

[0037]

[0038] In the formula, Represents interval variable input X i The interval of

[0039] (22) The interval vector X is divided by the interval variable input X i Internal division of nominal value x i Other input variables are still interval variables, expressed 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 3:

[0044] (31) Input the interval variable into X i Traversing within its interval is expressed as:

[0045]

[0046] In the formula, Represents interval variable input X i The kth nominal value of Represents interval variable input X i The interval of

[0047] (32) Obtain different conditional interval processes corresponding to different nominal values

[0048] (33) Different interval process models are constructed based on the total interval process and different conditional interval processes, including:

[0049] The interval process model is expanded to include the total interval process YI (t) and conditional interval process

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

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

[0052] According to 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 processes 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 (t) corresponds to the total interval field, the total interval process Y I (t) The corresponding geometric surface is input from the interval variable X i The lower bound To the interval variable input X i The upper bound of The stack accumulates to form a geometry defined as:

[0054]

[0055] In the formula, represents the lower boundary surface of the total interval field, Represents the upper boundary surface of the total interval field; conditional interval field Y I (X i , t), conditional interval process under different nominal values The corresponding geometric surface is input from interval variables X i The lower bound To the interval variable input X i The upper bound of The stack accumulates to form a geometry defined as:

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

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

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

[0059] As a further limitation of the present invention, the step 4 specifically includes:

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

[0061] (411) is 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

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

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

[0064]

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

[0066] (42) Define the global sensitivity index λ based on the volume ratio of the interval field i , the global sensitivity index λ iThe size of measures the influence of interval uncertainty input on the uncertainty of time-varying output 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] Where Y u (t) represents the upper boundary of the interval process, Y l (t) represents the lower boundary line of the interval process, Represents interval variable input X i The upper bound of Represents interval variable input X i The lower bound of

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

[0071]

[0072] Global sensitivity index λ i , and its calculation formula is defined as:

[0073]

[0074] As a further limitation of the present invention, in the step 1, the step 2 and the step 4, 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 uses a discretization method to handle the time transformation problem;

[0075] Discretize the time variable corresponding to the time transformation into:

[0076] T=[t1,t2,…,t N ]=[t s , t f ]

[0077] Where t1 = t s , t N =t f .

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

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

[0080] Calculate the lower bound Y of the unconditional output by optimizing the modell (t) and the lower bound of the conditional output The optimization model is expressed as:

[0081]

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

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

[0084] The numerical integration method is used to solve the integral calculation, and the approximation based on the Newton-Cotes formula has high accuracy.

[0085] The advantages of the present invention are:

[0086] 1. The present invention uses a global sensitivity index to screen inputs that have an important impact on the time-varying output of the motion mechanism. The volume ratio 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 the sensitivity index to not only obtain the contribution ranking of the input to the time-varying output of the motion mechanism, but also obtain the change in 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 present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0090] Figure 1 : A flow chart of a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input provided by the present invention;

[0091] Figure 2 : The interval process model of the time-varying output of the motion mechanism provided by the present invention;

[0092] Figure 3 : The conditional interval process and unconditional interval process of the motion mechanism with time-varying output provided by the present invention are shown;

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

[0094] Figure 5: A diagram of the total interval field and the conditional interval field provided by the present invention;

[0095] Figure 6 : The essential illustration of the definition of the volume and sensitivity index of the interval field in the interval field model provided by the present invention;

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

[0097] Figure 8 : A motion diagram of the four-bar linkage mechanism provided by the present invention;

[0098] Fig. 9 : A diagram of the total interval process of the time-varying output of the four-bar linkage example provided by the present invention;

[0099] Fig.10 : Illustration of the interval field model of the interval input of the four-bar linkage example provided by the present invention;

[0100] Fig.11 : A graphical representation of the ranking results of the sensitivity index of the input variables of the four-bar linkage example provided by the present invention;

[0101] Fig.12 : The calculation result table of sensitivity index of the four-bar linkage motion mechanism provided by the present invention. DETAILED DESCRIPTION

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

[0103] See also Figure 1 The embodiment of the present invention discloses a sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input, including steps 1 to 4, specifically:

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

[0105] The above step 1 of the embodiment of the present invention is specifically as follows:

[0106] (11) Time variable: Define the time variable t of the motion mechanism. 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 srepresents the initial moment of the time observation region, t f Indicates the end moment of the time observation area; the time input usually includes angle, displacement, time and other time-varying variables related to the time-varying output of the motion mechanism.

[0109] (12) Interval uncertainty input: Define the interval vector X of the motion mechanism = [X1, X2, …, X n ], in which the experimental data obtained in engineering practice is often limited. Therefore, it is difficult to establish an accurate probability model for the input variables. The interval model only needs to obtain the boundary information of the input variables, so describing the input variables as interval variables is an effective means to quantify the uncertainty input. i Defined as:

[0110]

[0111] In the formula, I represents the interval, Represents interval variable input X i The lower bound of Represents interval variable input X i The upper bound of

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

[0113]

[0114] Interval variable input X i The median value of X c Defined as:

[0115]

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

[0117]

[0118] At any time t j In this case, due to the existence of interval variables, the output of the mechanism must also be an interval, and the output of the motion mechanism is also an interval, expressed as:

[0119]

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

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

[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, expressed as Y I (t), the expression is:

[0124]

[0125] Where Y l (t) represents the lower boundary 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 area enclosed by the upper and lower boundary lines.

[0126] Step 2: Input the interval variable into X i The nominal value x is fixed within its interval i At the interval vector X, except for the interval variable input X i The other input variables are still interval variables, and the conditional interval process of the time-varying output of the motion mechanism is obtained.

[0127] The above-mentioned step 2 of the embodiment 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 At, it is expressed as:

[0129]

[0130] In the formula, Represents interval variable input X i The interval of

[0131] (22) The interval vector X is divided by the interval variable input X i Other input variables are still interval variables, expressed as:

[0132]

[0133] The time-varying output of the motion mechanism under this condition It is still an interval process like Figure 3 As shown. In order to I (t) naming distinction, It 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 Procedure When the interval variable X i =x i When , the time-varying output of the motion mechanism is still an interval process and is defined as:

[0137]

[0138] in, and are the upper and lower boundaries of the conditional interval process. i is fixed at its nominal value (X i The uncertainty of the output may be reduced, so the conditional interval process is the total interval process Y I A subset of (t).

[0139] Step 3: Repeat step 2 to input the interval variable into X i Traverse within its interval and fix at different nominal values ​​to obtain different conditional interval processes corresponding to different nominal values; construct different interval process models according to 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 the above step 3 of the embodiment of the present invention:

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

[0142]

[0143] In the formula, Represents interval variable input X i The kth nominal value of Represents interval variable input X i The interval of

[0144] (32) Obtain different conditional interval processes corresponding to different nominal values Next, the interval process model is expanded (the interval process model includes the total interval process Y I(t) and conditional interval process ).

[0145] (33) Different interval process models are constructed based on the total interval process and different conditional interval processes, including:

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

[0147] The geometric surfaces corresponding to the interval process model are stacked in a certain direction into a geometric body, which is the interval field model of the time-varying output of the motion mechanism, such as Figure 4 shown.

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

[0149] According to 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 processes The accumulated conditional interval field Y I (X i ,t).

[0150] In the embodiment of the present invention, the total interval process Y I (t) corresponds to the total interval field, the total interval process Y I (t) The corresponding geometric surface is input from the interval variable X i The lower bound To the interval variable input X i The upper bound of Stacking accumulates to form geometric shapes, such as Figure 5 (A) is defined as:

[0151]

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

[0153] Conditional interval field Y I (X i ,t), conditional interval process under different nominal values The corresponding geometric surface is input from interval variables X i The lower bound To the interval variable input X i The upper bound of Stacking accumulates to form geometric shapes, such as Figure 5(B) is defined as:

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

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

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

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

[0158] Step 4 of the embodiment of the present invention specifically includes:

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

[0160] (411) is 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

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

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

[0163]

[0164] In the formula, Represents interval variable input X i The contribution of uncertainty to the uncertainty of the time-varying output of the motion mechanism;

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

[0166] Total interval field of the embodiment of the present invention Volume Its calculation formula is defined as:

[0167]

[0168] Where Y u (t) represents the upper boundary of the interval process, Y l (t) represents the lower boundary line of the interval process, Represents interval variable input X i The upper bound of Represents interval variable input X i The lower bound of

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

[0170]

[0171] Global sensitivity index λ i , and its calculation formula is defined as:

[0172]

[0173] In the above-mentioned calculation of the sensitivity index in the embodiment of the present invention, the volume of the interval field and the total volume of the geometric body are solved, wherein:

[0174] According to the above formula, it can be seen that the calculation of the sensitivity index requires solving the volume of the interval field and the total volume of the geometric body, that is, solving the above formula. Obviously, its calculation is a nested process.

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

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

[0177]

[0178] Calculate the upper bound Y of the unconditional output by optimizing the model u (t) and the upper bound of the conditional output Transformed into: maxf(·)=min(-f(·)); In the embodiment of the present invention, the optimization problem is solved by the "particleswarm" function (particle swarm optimization algorithm (PSO)) in MATLAB software.

[0179] The outer layer of the embodiment of the present invention needs to calculate the integral, specifically:

[0180] The embodiment of the present invention adopts the method of numerical integration to solve the integral calculation, and the approximation accuracy based on the Newton-Cotes formula is high. The formula is also relatively simple, and it is a relatively common engineering numerical integration calculation method. The trapezoidal integration of the embodiment of the present invention is implemented by the "trapz" function in the MATLAB software.

[0181] Furthermore, in step 1, step 2 and step 4 of the embodiment of the present 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 uses a discretization method to handle the time transformation problem;

[0182] Discretize the time variable corresponding to the time transformation into:

[0183] T=[t1,t2,…,t N ]=[t s ,t f ]

[0184] Where t1 = t s , t N =t f .

[0185] The indicator calculation steps of the embodiment of the present invention include:

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

[0187] (11) Generate a set of discrete time parameters T corresponding to the time variable t, that is, 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 Next, that is: t = t j . Calculate the upper bound Y of the unconditional output at this moment u (t) and the 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 the lower bound Y l (t).

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

[0191]

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

[0193]

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

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

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

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

[0198] (222) from t1 to t M Fix the time variable t and repeat step (21) to calculate the upper bound of the conditional output at all times With the lower bound

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

[0200]

[0201] (23) From arrive Enter the interval variable into X i Fixed, and repeatedly execute 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 interval variables i The lower bound To the upper bound Integrate to obtain the volume of the conditional interval field, that is:

[0203] (3)λ i Calculation: It can be calculated according to the following formula, namely:

[0204] The embodiment of the present invention further studies the value range and characteristics of the indicator. A detailed description is given below:

[0205] (1) The proposed global sensitivity index λ i The formula is defined as the ratio of volumes, namely: 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 ,t) of the upper boundary surface Y u (X i ,t) and the total interval field The upper boundary of Always coincident, and the conditional interval field Y I (X i ,t) of the lower boundary surface Y l (X i ,t) and the total interval field The lower boundary of When the conditional interval field Y always coincides I (X i ,t) and the total interval field are equal when their volumes are equal, that is: In this case λ i =0, indicating that the input variable X i There is 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 ,t) of the upper boundary surface Y u (X i ,t) and its lower boundary surface Y u (X i ,t) always overlap, other intervals input X ～i The variable X of the uncertainty of the time-varying output of the motion mechanism i It is eliminated, resulting in the volume of the conditional interval field being 0, that is: In this case λ i =1, indicating 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 i The greater the impact on the uncertainty of the time-varying output of the motion mechanism; conversely, i The smaller the variable X is, i The smaller the impact on the uncertainty of the time-varying output of the motion mechanism.

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

[0210] In order to verify the feasibility of the proposed method, the proposed index is applied to the numerical example of a typical four-bar linkage mechanism in engineering.

[0211] The embodiment of the present invention analyzes a typical four-bar linkage mechanism in engineering. Figure 8is a simplified diagram of a four-bar linkage. The angle between the crank R1 and the frame R4 is θ, the angle between the connecting rod R2 and the horizontal axis x is δ, and the angle between the rocker arm R3 and the frame R4 is ψ. Consider the lengths of the four rods X = [X1, X2, X3, X4] = [R1, R2, R3, R4] as interval variables, where the value and deviation are X 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 relationship of the four-bar linkage during the entire motion process can be expressed as:

[0213]

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

[0215]

[0216] in:

[0217] D=-2R1R2sinθ

[0218] E=2R3(R4-R1cosθ)

[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 the embodiment of the present invention is to study the variation of the error of the actual output angle ψ of the four-bar linkage with θ, that is, Y(θ)=ψ(θ)-ψ d (θ). Next, a sensitivity analysis is performed on the change in the actual output angle error of the four-bar linkage.

[0223] Fig. 9 It is the interval process diagram of the actual output angle error change of the four-bar linkage mechanism. Fig.10 It is an interval field model diagram of the time-varying output corresponding to each interval input variable. By observing it, we can clearly see how each interval input affects the change of the actual output angle error of the four-bar linkage within its interval. Fig.12 The table gives the calculation results of the interval input sensitivity index. Fig.11 The order of the uncertainty of the interval input on the actual output angle error of the four-bar linkage is: X1>X2>X3>X4. The interval variable connecting rod R2 has the greatest influence; the interval variable frame R4 has the second largest influence; the influence of crank R1 is smaller than that of frame R4; the influence of 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 connecting rod R2, frame R4, and crank R1.

[0224] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should be included in the protection scope 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: According to interval analysis theory, a time-varying output interval process model of a motion mechanism under interval input is constructed, wherein the time-varying output interval process model is a total interval process of obtaining a time-varying output of the motion mechanism under interval vector X input; Step 2: Input the interval variable into X i The nominal value x is fixed within its interval i At the interval vector X, except for the interval variable input X i The other input variables are still interval variables, and the conditional interval process of the time-varying output of the motion mechanism is obtained; Step 3: Repeat step 2 to input the interval variable into X i Traversing within its interval, fixing at different nominal values, different conditional interval processes corresponding to different nominal values ​​are obtained; Construct different interval process models according to the total interval process and the interval processes under different conditions, and accumulate the geometric surfaces of the different interval process models to form an interval field model; 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 Obtain the contribution ranking of interval uncertainty input to the time-varying output of the motion mechanism.

2. The sensitivity analysis method of time-varying output of a motion mechanism under interval uncertainty input according to claim 1 is characterized in that: The step 1 is specifically as follows: (11) Define the time variable t of the motion mechanism. During the motion of the motion mechanism, the time variable t is defined as: t∈[t s ,t f ] In the formula, t s represents the initial moment of the time observation region, t f Indicates the end time of the time observation area; (12) Define the interval vector X of the motion mechanism = [X1, X2, …, X n ], where the single interval variable input X in the interval vector X i Defined as: In the formula, I represents the interval, Represents interval variable input X i The lower bound of Represents interval variable input X i The upper bound of (13) According to interval analysis theory, the interval variable input X i Radius X r Defined as: Interval variable input X i The median value of X c Defined as: (14) Define the output Y(t) of the motion mechanism. In the total interval of the time-varying output of the motion mechanism, the time interval is discretely expressed as: At any time t j , the output of the motion mechanism is also an interval, expressed as: In the formula, Indicates the output interval The lower bound of Indicates the output interval The upper bound of (15) Define the relationship between the input and output of the motion mechanism, expressed as: Y(t)=f(X,t); The time-varying output of the motion mechanism is an interval process model, expressed as Y I (t), the expression is: Where Y l (t) represents the lower boundary of the interval process, Y u (t) represents the upper boundary line of the interval process.

3. The sensitivity analysis method of time-varying output of a motion mechanism under interval uncertainty input according to claim 1 is characterized in that: The step 2 is specifically as follows: (21) Input the interval variable into X i The nominal value x is fixed within its interval i At, it is expressed as: In the formula, Represents interval variable input X i The interval of (22) The interval vector X is divided by the interval variable input X i Other input variables are still interval variables, expressed as: (23) The conditional interval process of the time-varying output of the motion mechanism is expressed as:

4. The sensitivity analysis method of time-varying output of a motion mechanism under interval uncertainty input according to claim 1 is characterized in that: In the step three: (31) Input the interval variable into X i Traversing within its interval is expressed as: In the formula, Represents interval variable input X i The kth nominal value of Represents interval variable input X i The interval of (32) Obtain different conditional interval processes corresponding to different nominal values (33) Different interval process models are constructed based on the total interval process and different conditional interval processes, including: The interval process model is expanded to include the total interval process Y I (t) and conditional interval process The expansion stacks the geometric surfaces corresponding to the interval process model into a geometric body along a certain direction, and the geometric body is an interval field model of the time-varying output of the motion mechanism; (34) accumulating the geometric surfaces of the different interval process models to form an interval field model, including: According to 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 processes The accumulated conditional interval field Y I (X i ,t).

5. The sensitivity analysis method of time-varying output of a motion mechanism under interval uncertainty input according to claim 4 is characterized in that: Total interval process Y I (t) corresponds to the total interval field, the total interval process Y I (t) The corresponding geometric surface is input from the interval variable X i The lower bound To the interval variable input X i The upper bound of The stack accumulates to form a geometry defined as: In the formula, represents the lower boundary surface of the total interval field, represents the upper boundary surface of the total interval field; Conditional interval field Y I (X i ,t), conditional interval process under different nominal values The corresponding geometric surface is input from interval variables X i The lower bound To the interval variable input X i The upper bound of The stack accumulates to form a geometry defined as: Y I (X i ,t)=[Y l (X i ,t),Y u (X i ,t)] Where Y l (X i ,t) represents the lower boundary of the conditional interval field, Y u (X i ,t) represents the upper boundary surface of the conditional interval field; Conditional interval fields represent interval variable input X i Under different nominal values, the remaining interval variables X in the interval vector X ～i The contribution of uncertainty to the uncertainty of the time-varying output of the motion mechanism.

6. The sensitivity analysis method of time-varying output of a motion mechanism under interval uncertainty input according to claim 1 is characterized in that: The step 4 specifically includes: (41) Based on the interval field model obtained in step 3, the conditional interval field Y I (X i ,t) of the upper boundary surface Y u (X i ,t) and the lower boundary surface Y l (X i ,t) the total interval field Divided into the following 3 parts: (411) is the upper boundary surface of the total interval field and the upper boundary surface Y of the conditional interval field u (X i ,t) enclosed volume (412) The upper boundary surface Y of the conditional interval field u (X i ,t) and the lower boundary surface Y of the conditional interval field l (X i ,t) enclosed volume (413) The lower boundary surface Y of the conditional interval field l (X i ,t) and the lower boundary of the total interval field The volume enclosed Total interval field Volume It is expressed as: In the formula, represents the interval variable X i The contribution of uncertainty to the uncertainty of the time-varying output of the motion mechanism; (42) Define the global sensitivity index λ based on the volume ratio of the interval field i , the global sensitivity index λ i The size of measures the influence of interval uncertainty input on the uncertainty of time-varying output of the motion mechanism.

7. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input according to claim 6 is characterized in that: Total interval field Volume Its calculation formula is defined as: Where Y u (t) represents the upper boundary of the interval process, Y l (t) represents the lower boundary line of the interval process, Represents interval variable input X i The upper bound of Represents interval variable input X i The lower bound of Conditional interval field Y I (X i ,t) volume Its calculation formula is defined as: Global sensitivity index λ i , and its calculation formula is defined as:

8. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input according to claim 1 is characterized in that: In the step 1, the step 2 and the step 4, 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 uses a discretization method to handle the time transformation problem; Discretize the time variable corresponding to the time transformation into: T=[t1,t2,…,t N ]=[t s ,t f ] Where t1 = t s , t N =t f .

9. The sensitivity analysis method for time-varying output of a motion mechanism under interval uncertainty input according to claim 7 is characterized in that: When calculating the sensitivity index, the volume of the interval field and the total volume of the geometric body are solved, where: The upper and lower bounds of the inner layer calculation output, specifically: Calculate the lower bound Y of the unconditional output by optimizing the model l (t) and the lower bound Y of the conditional output x l i (t), the optimization model is expressed as: Find X′={X1′,X2′,…,X n ′}Find X ～i * ={X1 * ,…,X i-1 * ,X i+1 * ,…X n * } minf(X,t);minf(x i ,X ～i ,t) subject to X l ≤X≤X u Calculate the upper bound Y of the unconditional output by optimizing the model u (t) and the upper bound of the conditional output Transformed into: max f(·) = min(-f(·)); The outer layer computes the integral, specifically: The numerical integration method is used to solve the integral calculation, and the approximation based on the Newton-Cotes formula has high accuracy.

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