CN108959182A - The small feature loss gravitational field modeling method returned based on Gaussian process - Google Patents
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Abstract
The small feature loss gravitational field modeling method disclosed by the invention that GPR is returned based on Gaussian process, belongs to field of deep space exploration.Implementation method of the present invention is as follows: using the spherical coordinates of site near small feature loss, the training set of gravitation field data is obtained by polyhedron method；Gaussian process, which is established, using the training set of acquisition returns GPR model.The gravitational field at check point is predicted again, obtain the mapping relations between site and gravitational acceleration, realize that returning GPR method using Gaussian process quickly and accurately carries out Modeling Calculation to the gravitational field near small feature loss, and it can reduce calculation amount, modeling speed is improved, the requirement of on-line operation is met.The present invention can be applied to deep-space detection field, and the establishment for dynamics environment around small feature loss in small celestial body exploration task provides technical support and reference, and solves the problems, such as correlation engineering.
Description
Technical field
The present invention relates to a kind of small feature loss gravitational field modeling methods that GPR is returned based on Gaussian process, belong to deep space exploration
Technical field.
Background technique
There are many requirements for the detection mission of small feature loss, wherein determine that orbital environment is a vital ring, it can be accurate
The gravitational field for obtaining small feature loss will directly influence the progress of entire detection mission.Therefore efficient gravitational field modeling is not only
The main scientific goal of the matter of utmost importance and small celestial body exploration task that solve required for researching and designing small feature loss satellite orbit it
One.
There are mainly three types of traditional small feature loss modeling methods.First method is spheric-harmonic method, and spheric-harmonic method is main
Gravitational potential energy is directly approached with series expansion.Such methods are unable to get accurate solution in Brillouin's ball domain.Second method
It is polyhedron method, main thought is that the volume in gravitational potential point is finally turned to polyhedron rib using Gauss formula and green theorem
The line integral on side.The third method is particle group's method, and particle group's method principle is simple, mainly by replacing multi-panel using particle group
Body Model, by calculating the gravitational field that each particle generates to model to the gravitational field of small feature loss, but particle group with
And polyhedral techniques can not all evade a large amount of complicated calculations.
Summary of the invention
The problems such as the complexity of calculating process existing for traditional gravitational field modeling method and accurate solution can not be obtained, this hair
The bright disclosed small feature loss gravitational field modeling method technical problems to be solved for returning GPR based on Gaussian process are: utilizing Gauss
Process returns GPR method and quickly and accurately carries out Modeling Calculation to the gravitational field near small feature loss, and can reduce calculation amount, mentions
High modeling speed.The present invention can be applied to deep-space detection field, for dynamics ring around small feature loss in small celestial body exploration task
Establishing for border provides technical support and reference, and solves the problems, such as correlation engineering.
Object of the present invention is to what is be achieved through the following technical solutions.
The small feature loss gravitational field modeling method disclosed by the invention that GPR is returned based on Gaussian process, near small feature loss
The spherical coordinates of site obtains the training set of gravitation field data by polyhedron method.Gaussian process is established using the training set of acquisition
Return GPR model.The gravitational field at check point is predicted again, obtains the mapping relations between site and gravitational acceleration,
Realize that returning GPR method using Gaussian process quickly and accurately carries out Modeling Calculation, and energy to the gravitational field near small feature loss
Calculation amount is enough reduced, modeling speed is improved.
The small feature loss gravitational field modeling method disclosed by the invention that GPR is returned based on Gaussian process, is included the following steps:
Step 1: the training set of gravitation field data is obtained by polyhedron method.
In the training process to training set, take a little at random in preset range around the small feature loss first, by site
Spherical coordinates position data λ,Input vector of the r as training set, the output of training set are the gravitational acceleration g of site,
Gravitational acceleration g is acquired by the calculating of polyhedron method.
Step 1 concrete methods of realizing is as follows:
In polyhedron method, arbitrary point P coordinate is P (x, y, z), in addition, coordinate of the P under spherical coordinate system:
Training set will will use spherical coordinates position data and be trained, the transformational relation of spherical coordinates and cartesian coordinate are as follows:
The mistake of the gravitational acceleration function g (x, y, z) at the P of gravitational field midpoint is substantially sought in the modeling of small feature loss gravitational field
Journey.And gravitational acceleration g is obtained by gravitational potential energy V (x, y, z), the relationship of the two are as follows:
During seeking the gravitational acceleration g in training set, small feature loss is divided into several differential element of volumes, and S is small feature loss
The differential element of volume that an internal quality is dm, r are the distance of S to check point P.Gravitational potential energy at check point is determined by triple integral
Justice:
Finally, using Gauss formula and green theorem, gravitational acceleration g is derived to obtain are as follows:
In formula, R is position vector of the check point P in the case where asteroid is connected coordinate system, eedgeIndicate side, reFor polyhedron side e
Upper any point to check point vector, and
Wherein EeFor 3 × 3 matrixes；For the exterior normal direction vector of the side e in the A of face；For the exterior normal side of face A
To vector；Re1, re2 be check point to side two endpoints distance；E12 is the length of side e；For the outer normal direction side of face f
To vector.
In addition, can directly judge check point or not outside celestial body, in above formulaFor gravitational field Laplce calculation
Son judges the position of check point, and criterion is as follows:
It is calculated by polyhedron method, the input point spherical coordinates that combined training is concentrated obtains the output valve of training set, that is, inputs
Gravitational acceleration g at point.
The input and output of training set are as follows:
Wherein, giFor the gravitational acceleration g in input vector at i-th of data point.
Formula (12) is the training set of the gravitation field data obtained by polyhedron method.
Step 2: establishing Gaussian process using the training set that step 1 obtains and return GPR model.
Step 2 concrete methods of realizing are as follows:
Training set D={ (Xi,Yi) | i=1,2 ..., n }=(X, Y), input vectorI.e. described in step 1
Spherical coordinates data.Export scalar YiFor the gravitational acceleration g at i-th of point.The task of recurrence is exactly according to given training set
Learnt, obtains output scalar YiWith input vector XiBetween mapping relations, finally according to test point X*Calculate possibility
Maximum output valve Y*。
When establishing Gaussian process recurrence GPR model using the training set that step 1 obtains, the observation Y of n trained functioni
∈ R, i=1,2 ... n.
Given training set, since observation is with noise, noise ε~N (0, σ2), then there is observation:
Y=f (X)+ε (13)
Wherein, X is input vector, and Y is noise-containing observation, and f (X) is functional value.Give f (X) Gaussian process elder generation
It tests, it may be assumed that
F (X)~GP (m (X), k (X, X')) (14)
Wherein, in order to succinct on symbol, definition mean function m (X)=0, and covariance function k has different selections.
So far it completes to establish Gaussian process recurrence GPR model, GPR problem is returned for Gaussian process, target is for new
Test point X*, corresponding Y can be obtained*。
Step 3: GPR model is returned using Gaussian process described in step 2, and the gravitational acceleration at check point is predicted,
Realize that returning GPR method using Gaussian process quickly and accurately carries out Modeling Calculation, and energy to the gravitational field near small feature loss
Calculation amount is enough reduced, modeling speed is improved.
Step 3 concrete methods of realizing are as follows:
Due to test data (X*,Y*) and training data (X, Y) all derive from same distribution, obtain training data (X, Y) with
Test data (X*,Y*) Joint Distribution are as follows:
Wherein, m is mean function, and K is covariance function, also referred to as kernel function.
Using kernel function, the input/output relation that Gaussian process returns GPR prediction model just can be obtained, in conjunction with formula
(15), it will be able to obtain Y*Predicted value.
After obtaining training set shown in formula (12) as step 1, chooses square exponential form (EQ) and be used as kernel function shape
Formula, zero-mean is as mean function, expression formula are as follows:
After noise is added, k (X, X') are as follows:
Y*Condition distribution:
The mean value of distribution is chosen as Y*Estimated value, it may be assumed that
Then to get arrive Y*Predicted value, in the kernel function k (X, X') shown in formula (18), the parameter θ that is related to=
[l,σf,σn], l is variance measure,For signal variance, σnFor the variance of noise.The variance measure l, signal variance
The variance parameter σ of noisenReferred to as hyper parameter.Hyper parameter is sought using maximum-likelihood method herein, obtains priori first
The negative log-likelihood function of probability distribution then seeks partial derivative to likelihood function and seeks likelihood function by conjugate gradient method
Minimum value obtains the optimal solution of hyper parameter in turn.
Likelihood function L (θ) are as follows:
Local derviation is asked to obtain likelihood function:
After obtaining the partial derivative of likelihood function, optimal hyper parameter is acquired by conjugate gradient method, then super by what is obtained
Parameter substitution formula (18) and formula (20) can acquire the predicted value of gravitational acceleration g, that is, realize and return GPR using Gaussian process
Method quickly and accurately carries out Modeling Calculation to the gravitational field near small feature loss, and can reduce calculation amount, improves modeling speed.
Further include applying step 4: step 1 to step 3 the method is applied to deep-space detection field, is small celestial body exploration
Establishing for dynamics environment provides technical support and reference around small feature loss in task, and solves the problems, such as correlation engineering.
The utility model has the advantages that
1, a kind of gravitational field modeling method that GPR algorithm is returned based on Gaussian process disclosed by the invention, utilizes small feature loss
The spherical coordinates of neighbouring site, the training set of gravitation field data is obtained by polyhedron method, establishes Gauss using the training set of acquisition
Process returns GPR model, and it is a kind of machine learning method that Gaussian process, which returns GPR, and returning GPR model using Gaussian process can
The complicated calculations process for avoiding traditional gravitational field modeling, acquires check point spherical coordinates λ from data statistics angle,R adds with gravitation
Mapping relations between speed g are realized and return GPR method quickly and accurately to the gravitation near small feature loss using Gaussian process
Field carries out Modeling Calculation, and can reduce calculation amount, improves modeling speed, meets the requirement of on-line operation.
2, a kind of gravitational field modeling method that GPR algorithm is returned based on Gaussian process disclosed by the invention, can be applied to
Deep-space detection field, the establishment for dynamics environment around small feature loss in small celestial body exploration task provide technical support and reference,
And solve the problems, such as correlation engineering.
Detailed description of the invention
Fig. 1 is a kind of gravitational field modeling method flow chart that GPR algorithm is returned based on Gaussian process disclosed in the present embodiment.
Fig. 2 is 433Eros gravitational acceleration calculated result in embodiment；
Fig. 3 is the error distribution of 433Eros 10000 check points within the scope of 20km.
Specific embodiment
Objects and advantages in order to better illustrate the present invention with reference to the accompanying drawing do further summary of the invention with example
Explanation.
This example is counted for small feature loss 433Eros away from the gravitational acceleration g of 10000 check points in mass center 20km
It calculates, 433Eros density is 2.67 × 1012kg/km3, small feature loss gravitational constant is 0.4401 × 10-3km3/s2.In order to prove this
The applicability of method, check point are random acquirement, and result that modeling result and polyhedron method calculate and time are carried out pair
Than.
As shown in Figure 1, the disclosed small feature loss gravitational field modeling method for returning GPR based on Gaussian process of the present embodiment, tool
Body implementation method is as follows:
Step 1: the training set of gravitation field data is obtained by polyhedron method.
In the training process to training set, carrying out taking 800 at random within the scope of small feature loss mass center 20km first
Site is to generate training set, by the spherical coordinates position data λ of site,Input vector of the r as training set, the output of training set
It is acquired for the gravitational acceleration g of site, gravitational acceleration g by the calculating of polyhedron method.
Step 1 concrete methods of realizing is as follows:
In polyhedron method, arbitrary point P coordinate is P (x, y, z), in addition, coordinate of the P under spherical coordinate system:
Training set will will use spherical coordinates position data and be trained, the transformational relation of spherical coordinates and cartesian coordinate are as follows:
The mistake of the gravitational acceleration function g (x, y, z) at the P of gravitational field midpoint is substantially sought in the modeling of small feature loss gravitational field
Journey.And gravitational acceleration g is obtained by gravitational potential energy V (x, y, z), the relationship of the two are as follows:
During seeking the gravitational acceleration g in training set, small feature loss is divided into several differential element of volumes, and S is small feature loss
The differential element of volume that an internal quality is dm, r are the distance of S to check point P.Gravitational potential energy at check point is determined by triple integral
Justice:
Wherein, G=6.672x10-11Nm2/kg2For universal gravitational constant.Finally, public using Gauss formula and Green
Formula derives to obtain gravitational acceleration g are as follows:
In formula, R is position vector of the check point P in the case where asteroid is connected coordinate system, eedgeIndicate side, reFor polyhedron side e
Upper any point to check point vector, and
Wherein EeFor 3 × 3 matrixes；For the exterior normal direction vector of the side e in the A of face；For the exterior normal side of face A
To vector；Re1, re2 be check point to side two endpoints distance；E12 is the length of side e；For the outer normal orientation of face f
Vector.
In addition, can directly judge check point or not outside celestial body, in above formulaFor gravitational field Laplce calculation
Son judges the position of check point, and criterion is as follows:
It is calculated by polyhedron method, the input point spherical coordinates that combined training is concentrated obtains the output valve of training set, that is, inputs
Gravitational acceleration g at point.
The input and output of training set are as follows:
Wherein, giFor the gravitational acceleration in input vector at i-th of data point.
Formula (12) is the training set of the gravitation field data obtained by polyhedron method.
Step 2: establishing Gaussian process using the training set that step 1 obtains and return GPR model.
Training set D={ (Xi,Yi) | i=1,2 ..., n }=(X, Y), input vectorI.e. described in step 1
Spherical coordinates data.Export scalar YiFor the gravitational acceleration g at i-th of pointi.The task of recurrence is exactly according to given training set
Learnt, obtains output scalar YiWith input vector XiBetween mapping relations, finally according to test point X*Calculate possibility
Maximum output valve Y*。
When establishing Gaussian process recurrence GPR model using the training set that step 1 obtains, the observation Y of n trained functioni
∈ R, i=1,2 ... n.
Given training set, since observation is with noise, noise ε~N (0, σ2), then there is observation:
Y=f (X)+ε (13)
Wherein, X is input vector, and Y is noise-containing observation, and f (X) is functional value.Give f (X) Gaussian process elder generation
It tests, it may be assumed that
F (X)~GP (m (X), k (X, X')) (14)
Wherein, in order to succinct on symbol, definition mean function m (X)=0, and covariance function k has different selections.
So far it completes to establish Gaussian process recurrence GPR model, GPR problem is returned for Gaussian process, target is for new
Test point X*, corresponding Y can be obtained*。
Step 3: returning GPR model to the gravity assist at 10000 randomized test points using Gaussian process described in step 2
Degree is predicted, that is, realizes that returning GPR method using Gaussian process quickly and accurately builds the gravitational field near small feature loss
Mould calculates, and can reduce calculation amount, improves modeling speed.
Due to test data (X*,Y*) and training data (X, Y) all derive from same distribution, obtain training data (X, Y) with
Test data (X*,Y*) Joint Distribution are as follows:
Wherein, m is mean function, and K is covariance function, also referred to as kernel function.
Using kernel function, the input/output relation that Gaussian process returns GPR prediction model just can be obtained, in conjunction with formula
(15), it will be able to obtain Y*Predicted value.
After obtaining training set shown in formula (12) as step 1, chooses square exponential form (EQ) and be used as kernel function shape
Formula, zero-mean is as mean function, expression formula are as follows:
After noise is added, k (X, X') are as follows:
Y*Condition distribution:
The mean value of distribution is chosen as Y*Estimated value, it may be assumed that
Then to get arrive Y*Predicted value, in the kernel function k (X, X') shown in formula (18), the parameter θ that is related to=
[l,σf,σn], l is variance measure,For signal variance, σnFor the variance of noise.The variance measure l, signal variance
The variance parameter σ of noisenReferred to as hyper parameter.Hyper parameter is sought using maximum-likelihood method herein, obtains priori first
The negative log-likelihood function of probability distribution then seeks partial derivative to likelihood function and seeks likelihood function by conjugate gradient method
Minimum value obtains the optimal solution of hyper parameter in turn.
Likelihood function L (θ) are as follows:
Local derviation is asked to obtain likelihood function:
After obtaining the partial derivative of likelihood function, optimal hyper parameter is acquired by conjugate gradient method, then super by what is obtained
Parameter substitution formula (18) and formula (20) can acquire the predicted value of gravitational acceleration g, that is, realize and return the side GPR using Gaussian process
Method quickly and accurately carries out Modeling Calculation to the gravitational field near small feature loss, and can reduce calculation amount, improves modeling speed.
Further include applying step 4: step 1 to step 3 the method is applied to deep-space detection field, is small celestial body exploration
Establishing for dynamics environment provides technical support and reference around small feature loss in task, and solves the problems, such as correlation engineering.
Simulation parameter is as shown in table 1:
1 simulation parameter of table
| Learning sample quantity | 800 |
| Test sample quantity | 10000 |
| Small feature loss density (kg/km3) | 2.67×1012 |
| Take point range (km) | 20 |
| Small feature loss gravitational constant (km3/s2) | 0.4401×10-3 |
Modeling accuracy and time are as shown in table 2:
2 modeling accuracy of table and time
| Mean error (%) | The polyhedron method time (s) | The GPR method time (s) |
| 1.22 | 611.81 | 0.71 |
It can be seen that the small feature loss gravitational field modeling method energy that GPR is returned based on Gaussian process from table 2 and Fig. 2-Fig. 3
Enough quick Accurate Models for realizing the gravitational field near small feature loss, have larger mention compared to conventional method especially on operation time
It rises, this efficient Modeling Calculation method can greatly reduce the modeling time of gravitational field, help to shorten small celestial body exploration times
It is engaged in the design cycle.
Above-described specific descriptions have carried out further specifically the purpose of invention, technical scheme and beneficial effects
It is bright, it should be understood that the above is only a specific embodiment of the present invention, the protection model being not intended to limit the present invention
It encloses, all within the spirits and principles of the present invention, any modification, equivalent substitution, improvement and etc. done should be included in the present invention
Protection scope within.
Claims (5)
1. returning the small feature loss gravitational field modeling method of GPR based on Gaussian process, characterized by the following steps:
Step 1: the training set of gravitation field data is obtained by polyhedron method；
In the training process to training set, take a little at random in preset range around the small feature loss first, by the ball of site
Coordinate position data λ,Input vector of the r as training set, the output of training set are the gravitational acceleration g of site, gravitation
Acceleration g is acquired by the calculating of polyhedron method；
Step 2: establishing Gaussian process using the training set that step 1 obtains and return GPR model；
Step 3: returning GPR model using Gaussian process described in step 2 and the gravitational acceleration at check point is predicted, i.e., in fact
GPR method now is returned using Gaussian process, Modeling Calculation quickly and accurately is carried out to the gravitational field near small feature loss, and can drop
Low calculation amount improves modeling speed.
2. the small feature loss gravitational field modeling method of GPR is returned based on Gaussian process as described in claim 1, it is characterised in that:
Further include applying step 4, step 1 to step 3 the method is applied to deep-space detection field, is small in small celestial body exploration task
Establishing for dynamics environment provides technical support and reference around celestial body, and solves the problems, such as correlation engineering.
3. returning the small feature loss gravitational field modeling method of GPR based on Gaussian process as claimed in claim 1 or 2, feature exists
In: step 1 concrete methods of realizing is as follows:
In polyhedron method, arbitrary point P coordinate is P (x, y, z), in addition, coordinate of the P under spherical coordinate system:Training
Collection will will use spherical coordinates position data and be trained, the transformational relation of spherical coordinates and cartesian coordinate are as follows:
The process of the gravitational acceleration function g (x, y, z) at the P of gravitational field midpoint is substantially sought in the modeling of small feature loss gravitational field；And
Gravitational acceleration g is obtained by gravitational potential energy V (x, y, z), the relationship of the two are as follows:
During seeking the gravitational acceleration g in training set, small feature loss is divided into several differential element of volumes, and S is inside small feature loss
The differential element of volume that one quality is dm, r are the distance of S to check point P；Gravitational potential energy at check point is defined by triple integral:
Finally, using Gauss formula and green theorem, gravitational acceleration g is derived to obtain are as follows:
In formula, R is position vector of the check point P in the case where asteroid is connected coordinate system, eedgeIndicate side, reTake up an official post for polyhedron side e
Meaning a little arrives the vector of check point, and
Wherein EeFor 3 × 3 matrixes；For the exterior normal direction vector of the side e in the A of face；It is sweared for the exterior normal direction of face A
Amount；re1,re2For the distance of two endpoints of check point to side；e12For the length of side e；For the outer normal orientation vector of face f；
In addition, can directly judge check point or not outside celestial body, in above formulaFor gravitational field Laplace operator, sentence
The position of disconnected check point, criterion are as follows:
It is calculated by polyhedron method, the input point spherical coordinates that combined training is concentrated obtains the output valve of training set, i.e., at input point
Gravitational acceleration g；
The input and output of training set are as follows:
Wherein, giFor the gravitational acceleration g in input vector at i-th of data point；
Formula (12) is the training set of the gravitation field data obtained by polyhedron method.
4. the small feature loss gravitational field modeling method of GPR is returned based on Gaussian process as claimed in claim 3, it is characterised in that:
Step 2 concrete methods of realizing are as follows:
Training set D={ (Xi,Yi) | i=1,2 ..., n }=(X, Y), input vectorI.e. ball described in step 1 is sat
Mark data；Export scalar YiFor the gravitational acceleration g at i-th of point；The task of recurrence is exactly to be carried out according to given training set
Study obtains output scalar YiWith input vector XiBetween mapping relations, finally according to test point X*Calculate possibility maximum
Output valve Y*；
When establishing Gaussian process recurrence GPR model using the training set that step 1 obtains, the observation Y of n trained functioni∈ R, i
=1,2 ... n；
Given training set, since observation is with noise, noise ε~N (0, σ2), then there is observation:
Y=f (X)+ε (13)
Wherein, X is input vector, and Y is noise-containing observation, and f (X) is functional value；F (X) Gaussian process priori is given,
That is:
F (X)~GP (m (X), k (X, X')) (14)
Wherein, in order to succinct on symbol, definition mean function m (X)=0, and covariance function k has different selections；
So far it completes to establish Gaussian process recurrence GPR model, GPR problem is returned for Gaussian process, target is for new inspection
Measuring point X*, corresponding Y can be obtained*。
5. the small feature loss gravitational field modeling method of GPR is returned based on Gaussian process as claimed in claim 4, it is characterised in that:
Step 3 concrete methods of realizing are as follows:
Due to test data (X*,Y*) and training data (X, Y) all derive from same distribution, obtain training data (X, Y) and test
Data (X*,Y*) Joint Distribution are as follows:
Wherein, m is mean function, and K is covariance function, also referred to as kernel function；
Using kernel function, the input/output relation that Gaussian process returns GPR prediction model just can be obtained, in conjunction with formula (15), just
It can obtain Y*Predicted value；
After obtaining shown in formula (12) training set as step 1, chooses square exponential form (EQ) and be used as kernel function form, zero
Mean value is as mean function, expression formula are as follows:
After noise is added, k (X, X') are as follows:
Y*Condition distribution:
The mean value of distribution is chosen as Y*Estimated value, it may be assumed that
Then to get arrive Y*Predicted value, in the kernel function k (X, X') shown in formula (18), the parameter θ that is related to=[l,
σf,σn], l is variance measure,For signal variance, σnFor the variance of noise；The variance measure l, signal varianceNoise
Variance parameter σnReferred to as hyper parameter；Hyper parameter is sought using maximum-likelihood method herein, obtains prior probability first
The negative log-likelihood function of distribution then seeks partial derivative to likelihood function and seeks likelihood function minimum by conjugate gradient method
Value obtains the optimal solution of hyper parameter in turn；
Likelihood function L (θ) are as follows:
Local derviation is asked to obtain likelihood function:
After obtaining the partial derivative of likelihood function, optimal hyper parameter, then the hyper parameter that will be obtained are acquired by conjugate gradient method
Substitution formula (18) and formula (20) can acquire the predicted value of gravitational acceleration g, that is, realize and return GPR method using Gaussian process
Modeling Calculation quickly and accurately is carried out to the gravitational field near small feature loss, and can reduce calculation amount, improves modeling speed.
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Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20060129331A1 (en) * | 2003-04-29 | 2006-06-15 | The Jackson Laboratory | Expression data analysis systems and methods |
| CN104048675A (en) * | 2014-06-26 | 2014-09-17 | 东南大学 | Integrated navigation system fault diagnosis method based on Gaussian process regression |
| CN104699894A (en) * | 2015-01-26 | 2015-06-10 | 江南大学 | JITL (just-in-time learning) based multi-model fusion modeling method adopting GPR (Gaussian process regression) |
| CN106778012A (en) * | 2016-12-29 | 2017-05-31 | 北京理工大学 | A kind of small feature loss attachment detection descending trajectory optimization method |
| CN107323692A (en) * | 2017-07-04 | 2017-11-07 | 北京理工大学 | A kind of energy optimizing method of small feature loss soft landing avoidance |
| CN107341581A (en) * | 2017-08-08 | 2017-11-10 | 国网江苏省电力公司盐城供电公司 | A kind of new energy output short term prediction method returned based on experience wavelet transformation and Gaussian process |
| CN107367942A (en) * | 2017-08-31 | 2017-11-21 | 北京理工大学 | A kind of normal thrust control method of small feature loss spot hover |
| CN107451102A (en) * | 2017-07-28 | 2017-12-08 | 江南大学 | A kind of semi-supervised Gaussian process for improving self-training algorithm returns soft-measuring modeling method |
-
2018
- 2018-05-21 CN CN201810485427.3A patent/CN108959182B/en active Active
Patent Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20060129331A1 (en) * | 2003-04-29 | 2006-06-15 | The Jackson Laboratory | Expression data analysis systems and methods |
| CN104048675A (en) * | 2014-06-26 | 2014-09-17 | 东南大学 | Integrated navigation system fault diagnosis method based on Gaussian process regression |
| CN104699894A (en) * | 2015-01-26 | 2015-06-10 | 江南大学 | JITL (just-in-time learning) based multi-model fusion modeling method adopting GPR (Gaussian process regression) |
| CN106778012A (en) * | 2016-12-29 | 2017-05-31 | 北京理工大学 | A kind of small feature loss attachment detection descending trajectory optimization method |
| CN107323692A (en) * | 2017-07-04 | 2017-11-07 | 北京理工大学 | A kind of energy optimizing method of small feature loss soft landing avoidance |
| CN107451102A (en) * | 2017-07-28 | 2017-12-08 | 江南大学 | A kind of semi-supervised Gaussian process for improving self-training algorithm returns soft-measuring modeling method |
| CN107341581A (en) * | 2017-08-08 | 2017-11-10 | 国网江苏省电力公司盐城供电公司 | A kind of new energy output short term prediction method returned based on experience wavelet transformation and Gaussian process |
| CN107367942A (en) * | 2017-08-31 | 2017-11-21 | 北京理工大学 | A kind of normal thrust control method of small feature loss spot hover |
Non-Patent Citations (5)
| Title |
|---|
| HAIBIN SHANG等: "Assessing Accessibilityof Main-Belt Asteroids Basedon Gaussian Process Regression", 《JOURNAL OF GUIDANCE, CONTROL, ANDDYNAMICS》 * |
| XIAOKE YANG 等: "Aircraft Centre-of-Gravity Estimation using Gaussian Process", 《2016 IEEE/CSAA INTERNATIONAL CONFERENCE ON AIRCRAFT UTILITY SYSTEM》 * |
| 刘国海 等: "基于多准则和高斯过程回归的动态软测量建模方法", 《东 南 大 学 学 报 ( 自 然 科 学 版 )》 * |
| 刘宇鑫: "地球同步卫星转移轨道设计与在轨保持方法研究", 《中国优秀硕士学位论文全文数据库（工程科技Ⅱ辑）》 * |
| 崔祜涛 等: "多面体模型的 Eros433 引力场计算与分析", 《哈尔滨工业大学学报》 * |
Cited By (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109978025A (en) * | 2019-03-11 | 2019-07-05 | 浙江工业大学 | A kind of intelligent network connection vehicle front truck acceleration prediction technique returned based on Gaussian process |
| CN109978025B (en) * | 2019-03-11 | 2022-03-01 | 浙江工业大学 | Intelligent internet vehicle front vehicle acceleration prediction method based on Gaussian process regression |
| CN110490112A (en) * | 2019-08-13 | 2019-11-22 | 新华智云科技有限公司 | Football video segment detection method, device, system and storage medium |
| CN110910972A (en) * | 2019-11-20 | 2020-03-24 | 长沙理工大学 | Fatigue stress concentration coefficient prediction method based on Gaussian process |
| CN111552003A (en) * | 2020-05-11 | 2020-08-18 | 中国人民解放军军事科学院国防科技创新研究院 | Asteroid gravitational field full-autonomous measurement system and method based on ball satellite formation |
| CN113065287A (en) * | 2021-04-14 | 2021-07-02 | 北京理工大学 | Small celestial body gravitational field rapid prediction method based on implicit characteristics |
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| CN116047448B (en) * | 2022-12-30 | 2024-03-12 | 西安电子科技大学 | Method for predicting conductor target RCS |
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