CN110175391A - One kind being based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos - Google Patents

One kind being based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos Download PDF

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CN110175391A
CN110175391A CN201910428449.0A CN201910428449A CN110175391A CN 110175391 A CN110175391 A CN 110175391A CN 201910428449 A CN201910428449 A CN 201910428449A CN 110175391 A CN110175391 A CN 110175391A
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accelerometer
formula
polynomial
analysis
chaos
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CN110175391B (en
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滕霖
姜潮
陈亮
姚中洋
叶坤
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Hunan University
Xian Flight Automatic Control Research Institute of AVIC
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Xian Flight Automatic Control Research Institute of AVIC
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Abstract

The invention belongs to technical field of inertial, disclose a kind of based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos.The present invention includes uncertain factor present in analysis accelerometer assembling process；Establish accelerometer analysis of uncertainty model；According to existing experimental data and engineering experience, statistical analysis obtains the statistical moment information of uncertain parameters, acquires accelerometer bias output mean value, variance, kurtosis, the degree of bias based on any type chaos Polynomial Method；The probability distribution of accelerometer bias output is fitted using principle of maximum entropy.The influence that the uncertain factor in accelerometer assembling process exports its bias is quantitatively evaluated out in the present invention, theoretical direction is provided for subsequent optimization accelerometer assembly technology and structure, to promote accelerometer qualification rate.

Description

One kind being based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos
Technical field
The present invention relates to technical field of inertial, more particularly to one kind to be based on any polynomial accelerometer of type chaos Uncertainty Analysis Method.
Background technique
Inertial navigation is the precise navigation technology that a kind of independence is strong, precision is high, safe and reliable, is widely used at present Numerous key areas such as Aeronautics and Astronautics, navigation, high-speed rail.
Accelerometer is as the core element in inertial navigation, guidance system, for real-time measurement sensitive carrier in space In line movement, thus for carrier navigation, guide the information such as Position And Velocity are provided.The performance of accelerometer is to determine inertia Navigation, guidance system precision most direct and most important factor, therefore modern high-precision inertial navigation, guidance system are to adding Speedometer performance (bias output accuracy) proposes higher requirement.
The multi-sources such as number of bubbles in accelerometer there are jelly bond areas during practical set inconsistent, adhesive Uncertain factor, cause accelerometer bias export dispersion it is bigger, to its precision improve approaches and methods and It the application such as guides in subsequent inertial navigation and proposes challenge.
Summary of the invention
In view of the above-mentioned problems, the present invention more considers the uncertain factor for influencing the output of accelerometer bias；It is right For accelerometer practical set process, after obtaining uncertain factor and exporting affecting laws to accelerometer bias, Targetedly certain uncertain factors of assembling process can be controlled according to this rule, be subsequent optimization acceleration It counts assembly technology and theoretical direction is provided, promote accelerometer qualification rate.
It is uncertain based on any polynomial accelerometer of type chaos that the technical problem to be solved in the present invention is to provide one kind Property analysis method, this method are all suitable for arbitrary structures, there is good versatility.
In order to realize the purpose, the technical solution adopted by the present invention is that:
One kind being based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos, includes the following steps:
Uncertain factor present in step 1, analysis accelerometer assembling process；
In conjunction with practical set technique and engineer experience, analysis obtain in accelerometer assembling process it is uncertain because Element, respectively x1,x2…,xn。
Step 2 establishes accelerometer analysis of uncertainty model:
In formula:
F is the output of accelerometer bias；
P=(n+d)！/(n！d！) be chaos polynomial expansion item number, d be the polynomial order of chaos；X={ x1, x2,…,xnIt is that n ties up uncertain variable (factor), ψ in accelerometer assembling processkIt (x) is with uncertain variable for x ginseng The multi-dimensional orthogonal multinomial of amount；ckFor chaos multinomial coefficient.
Step 3, according to existing experimental data and engineer experience, statistical analysis obtains the statistical moment of each uncertain parameters Information acquires accelerometer bias output mean value, variance, kurtosis, the degree of bias based on chaos Polynomial Method, specifically includes following step It is rapid:
(1) solving the polynomial expansion of chaos in accelerometer analysis of uncertainty model using the golden projecting method of gal the Liao Dynasty is Number is shown below:
In formula:
The polynomial expansion coefficient of chaos in accelerometer analysis of uncertainty model will be solved by formula (2) and formula (3) It is converted into and solves multidimensional Gauss integration problem.
(2) in conjunction with irrelevance between each uncertain factor of accelerometer assembling process, by multi-dimensional orthogonal polynomial transformation For the polynomial product of multiple one-dimensional orthogonals, it is shown below:
In formula: P is the item number of chaos polynomial expansion；
It is with stochastic variable xiFor parameterRank one-dimensional orthogonal multinomial,
And
It can be expressed as:
In formula:
It can be acquired by formula (6)；
At this point, formula (6) need to meet formula (7) condition；
When using traditional tensor product method, with the increase of uncertain variable (factor) dimension, Integration Solving can go out Existing " dimension explosion " problem, calculation amount are exponentially increased；For the analysis of uncertainty problem of accelerometer, calculation amount is excessively huge Greatly；The present invention solves the problems, such as this using the sparse grid method based on sparse tensor product.
(3) according to existing experimental data and engineering experience, the statistical moment of 2d rank before uncertain parameter is obtained, calculates and adds The preceding Fourth-order moment of speedometer bias output；
Using the sparse grid method based on sparse tensor product, i.e. Smolyak algorithm, as shown in formula (8)；
The important uncertain factor in assembling process is filtered out by 1 norm, ignores secondary cause, reduces calculation amount；
By formula (7) it is found that Metzler matrix is nonsingular, M=R can be obtained using Cholesky decompositionTR；
R is shown below:
By the explicit analytical expressions of Cholesky Matrix Solving orthogonal polynomial coefficient:
In formula:
r0,0=1, r0,1=0.
The allocation optimum point and weight of Orthogonal Polynomial are obtained by the Jacobian matrix J of following formula, wherein Jacobi The characteristic value of matrix J is lkThe root of rank multinomial, i.e. allocation optimum point, weight be corresponding ith feature value standard feature to Square of first element of amount, as shown in formula (13):
The Smolyak integral of different random distribution input is calculated by formula (14)；
In formula (14) and (16) | i | indicate vector i={ i1,i2,…,inNorm, while also referring to matrixJth row Sum.
Calculate the preceding Fourth-order moment of accelerometer bias output, i.e. mean value, variance, the degree of bias, kurtosis.
(4) the preceding Fourth square obtained is fitted to the probability density of accelerometer bias output according to principle of maximum entropy Function.
The beneficial effects of the present invention are:
(1) present invention considers the uncertain factor in accelerometer assembling process, is consistent with actual conditions, obtains Accelerometer bias output probability accurately objectively characterize accelerometer existing uncertainty during practical set, Theoretical reference is provided for subsequent optimization accelerometer assembly technology, fail-safe analysis and Optimal Structure Designing；
(2) for traditional Deterministic Methods, based on uncertainty propagation analysis method can quantitatively characterizing not Deterministic parameter propagation characteristic in accelerometer, so as to carry out analysis of uncertainty to accelerometer, to accelerometer Uncertainty optimization is carried out, its qualification rate is improved；
(3) selection of traditional Gauss integral node and weight is all based on specific continuous probability distribution, in the present invention not The specific Unknown Distribution for determining parameter solves Gauss integration node and weight using based on sample moment information approach；
(4) accelerometer Uncertainty Analysis Method proposed by the present invention is suitable for the analysis of uncertainty of arbitrary structures, There is no particular/special requirement to structure, it is applied widely.
Detailed description of the invention
Fig. 1 is provided in an embodiment of the present invention based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos Flow chart.
Fig. 2 is a kind of arrangements of accelerometers schematic diagram provided in an embodiment of the present invention.
Fig. 3 is uncertain factor (arrow directed section) in the accelerometer assembling process provided in the embodiment of the present invention Location diagram.
Fig. 4 (a-c) is the histogram of each uncertain variable in accelerometer assembling process in the embodiment of the present invention；
A is epoxy glue elasticity modulus histogram；
B is that epoxy glue is bonded angular histogram at aluminium cushion block；
C is tungsten cushion block angular histogram Nian Jie with epoxy glue at tungsten counterweight.
Fig. 5 is displacement cloud atlas of the accelerometer under real load operating condition in the embodiment of the present invention.
Fig. 6 is the probability density function schematic diagram that the output of accelerometer bias is acquired according to method proposed by the invention.
Specific embodiment
Fig. 1 shows according to the present invention based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos Process.
As shown in Figure 1, provided by the invention a kind of based on the polynomial accelerometer analysis of uncertainty of any type chaos Method includes the following steps:
Uncertain factor present in step 1, analysis accelerometer assembling process shown in Fig. 2；
In conjunction with practical set technique and engineer experience, analysis obtain in accelerometer assembling process it is uncertain because Element, as shown in figure 3, being respectively as follows: the bonding angle of the concrete moduli of adhesive, aluminum material and copper product, aluminum material and silicon materials Bonding angle, the bonding angle of tungsten material and copper product, the bonding angle of tungsten material and silicon materials, counterweight and silicon materials it is viscous Connect angle.
Above-mentioned 6 uncertain factors are successively denoted as x1,x2,x3,x4,x5,x6。
Step 2 establishes accelerometer analysis of uncertainty model using chaos multinomial；
In formula:
F is the output of accelerometer bias；
P=(6+d)！/(6！d！) be chaos polynomial expansion item number, d be the polynomial order of chaos；
X={ x1,x2,…,x6It is the uncertain variable (factor) of 6 dimensions in accelerometer assembling process；
ψk(x) be with uncertain variable be X parameter multi-dimensional orthogonal multinomial；
ckFor chaos multinomial coefficient.
Step 3, according to existing experimental data and engineer's engineering experience, statistical analysis obtains the system of each uncertain parameters Count square information, based on chaos Polynomial Method acquire accelerometer bias output mean value, variance, kurtosis, the degree of bias, specifically include as Lower step
(1) solving the polynomial expansion of chaos in accelerometer analysis of uncertainty model using the golden projecting method of gal the Liao Dynasty is Number is shown below:
In formula:
(2) in conjunction with irrelevance between each uncertain factor of accelerometer assembling process, by multi-dimensional orthogonal polynomial transformation For the polynomial product of multiple one-dimensional orthogonals, it is shown below:
In formula: P is the item number of chaos polynomial expansion；
It is with stochastic variable xiFor parameterRank one-dimensional orthogonal multinomial, and It can be expressed as:
In formula:
It can be acquired by formula (6), at this point, formula (6) need to meet formula (7) condition:
(3) according to existing experimental data and engineering experience, the statistical moment of 2d rank before uncertain parameter is obtained, calculates and adds The preceding Fourth-order moment of speedometer bias output；
According to existing experimental data and engineering experience, the statistical moment of 2d rank before uncertain parameter is obtained, d is that chaos is multinomial Order is unfolded in formula；
Using the sparse grid method of sparse tensor product, i.e. Smolyak algorithm, as shown in formula (8), screened by 1 norm Important uncertain factor in assembling process out ignores secondary cause, reduces calculation amount；
By formula (7) it is found that Metzler matrix is nonsingular, M=R can be obtained using Cholesky decompositionTR, R are as follows:
By the explicit analytical expressions of Cholesky Matrix Solving orthogonal polynomial coefficient:
ξφj-1(ξ)=bj-1φj-2(ξ)+ajφj-1(ξ)+bjφj(ξ) (10)
In formula:
r0,0=1, r0,1=0.
The allocation optimum point and weight that Orthogonal Polynomial is obtained by the Jacobian matrix J of (12) formula, wherein refined can Characteristic value than matrix J is lkThe root of rank multinomial, i.e. allocation optimum point, weight are the standard feature of corresponding ith feature value Square of first element of vector, as shown in formula (13):
The Smolyak integral of uncertain factor is calculated by formula (14)；
In formula (14) and (16) | i | indicate vector i={ i1,i2,…,inNorm, while also referring to matrixJth row Sum, calculate the preceding Fourth-order moment of accelerometer bias output, i.e. mean value, variance, the degree of bias, kurtosis.
(4) probability density function of accelerometer bias output is fitted based on principle of maximum entropy, as shown in Figure 6.
The above is only specific steps of the invention, are not limited in any way to protection scope of the present invention；All use is equal Transformation or equivalence replacement and the technical solution that is formed, all fall within rights protection scope of the present invention；The present invention does not explain in detail State the well-known technique for partly belonging to those skilled in the art.
The method of the present invention is relative to traditional Uncertainty Analysis Method beneficial effect:
(1) present invention quantitatively calculates the shadow that the uncertain factor in accelerometer assembling process exports its bias It rings, provides theoretical direction for subsequent optimization accelerometer assembly technology, fail-safe analysis and Optimal Structure Designing, promote acceleration Count qualification rate.
(2) selection of traditional Gauss integral node and weight is all based on specific continuous probability distribution, however acceleration The specific Probability Distributed Unknown for counting the uncertain parameter during practical set can obtain the system of uncertain parameter by counting The method of meter square, traditional solution Gauss integration node and weight is no longer applicable in；The present invention, which uses, is based on sample moment information approach Gauss integration node and weight are solved, the deficiency that tradition solves the method for Gauss integration node and weight is compensated for.
(3) present invention uses the sparse grid method based on sparse tensor product, is solved in sparse tensor product by 1 norm Important element, reduce calculation amount, improve computational efficiency.

Claims (1)

1. one kind is based on the polynomial accelerometer Uncertainty Analysis Method of any type chaos, which is characterized in that specifically include Following steps:
Uncertain factor present in step 1, analysis accelerometer assembling process；
In conjunction with practical set technique and engineering experience, analysis obtains the uncertain factor in accelerometer assembling process, respectively For x1,x2…,xn；
Step 2 establishes accelerometer analysis of uncertainty model:
In formula:
F is the output of accelerometer bias；
P=(n+d)！/(n！d！) be chaos polynomial expansion item number, d be the polynomial order of chaos；
X={ x1,x2,…,xnIt is that n ties up uncertain variable (factor) in accelerometer assembling process；
ψk(x) be with uncertain variable be x parameter multi-dimensional orthogonal multinomial；
ckFor chaos multinomial coefficient；
Step 3, according to existing experimental data and engineering experience, statistical analysis obtains the statistical moment information of uncertain parameters, base Accelerometer bias output mean value, variance, kurtosis, the degree of bias are acquired in any type chaos Polynomial Method, specifically includes following step It is rapid:
(1) the polynomial expansion coefficient of chaos in accelerometer analysis of uncertainty model is solved using the golden projecting method of gal the Liao Dynasty, It is shown below:
In formula:
The polynomial expansion coefficient of chaos in accelerometer analysis of uncertainty model will be solved by formula (2) and formula (3) to convert To solve multidimensional Gauss integration problem；
It (2) is more by multi-dimensional orthogonal polynomial transformation in conjunction with irrelevance between each uncertain factor of accelerometer assembling process A polynomial product of one-dimensional orthogonal, is shown below；
In formula:
P is the item number of chaos polynomial expansion；
It is with stochastic variable xiFor parameterRank one-dimensional orthogonal multinomial, and
It can be write as:
In formula:
It can be acquired by following formula；
At this point, formula (6) need to meet formula (7) condition:
In formula: M is matrix；
(3) according to existing experimental data and engineering experience, the statistical moment of 2d rank before uncertain parameter is obtained, acceleration is calculated Count the preceding Fourth-order moment of bias output；
Using the sparse grid method of sparse tensor product, i.e. Smolyak algorithm, it is shown below:
The important uncertain element in assembling process is filtered out by 1 norm, ignores minor element, reduces calculation amount；
By formula (7) it is found that Metzler matrix is nonsingular, M=R can be obtained using Cholesky decompositionTR, R are shown below:
By the explicit analytical expressions of Cholesky Matrix Solving orthogonal polynomial coefficient:
ξφj-1(ξ)=bj-1φj-2(ξ)+ajφj-1(ξ)+bjφj(ξ) (10)
In formula:
r0,0=1, r0,1=0；
The allocation optimum point and weight of Orthogonal Polynomial are obtained by formula (12) Jacobian matrix J, wherein Jacobian matrix J Characteristic value be lkThe root of rank multinomial, i.e. allocation optimum point, weight are the of the standard feature vector of corresponding ith feature value Square of one element, as shown in formula (13):
The Smolyak integral of random distribution input is calculated by formula (14)；
In formula (14) and (16) | i | indicate vector i={ i1,i2,…,inNorm, while also referring to matrixJth row With；
(4) the preceding Fourth square obtained is fitted to the probability density letter of accelerometer bias output based on principle of maximum entropy Number.
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