CN113836697B - Spatial gravitational wave detection configuration stability evolution method based on dynamic Gaussian mixture - Google Patents
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Abstract
The invention discloses a dynamic Gaussian mixture-based spatial gravitational wave detection configuration stability evolution method, and belongs to the technical field of space. The invention realizes the evolution analysis of the stability of the space gravitational wave detection configuration based on the dynamic Gaussian mixture model, predicts the uncertainty distribution of the stability index of the space gravitational wave detection configuration, avoids an unstable region in the construction process of the space gravitational wave detection configuration, and improves the detection precision and efficiency of the space gravitational wave. The dynamic evolution method is used for recursion of the orbit error of the detector in the space gravitational wave detection configuration, different numbers of sub-Gaussian distributions are used for approximation according to the non-Gaussian degree of uncertainty distribution of the orbit error, and compared with the Gaussian mixture method, the prediction efficiency is remarkably improved; meanwhile, based on the track error distribution, the uncertainty distribution of the structural stability index is represented by using weighted sub-Gaussian distribution, and the non-Gaussian representation precision is high. When the evolution condition of the configuration stability index is predicted, the prediction accuracy is high and the efficiency is high.
Description
Technical Field
The invention relates to a method for predicting the uncertainty evolution of the stability of a spatial gravitational wave detection configuration, and belongs to the technical field of space.
Background
The gravitational wave detection is helpful for knowing the origins of black holes and neutron stars, promotes the development of relativity theory, and has important significance for the development of physics and exploration of universe. Because of the influence of atmospheric shielding, ground vibration noise and ground surface gravity gradient, the application range of the ground gravitational wave detection method is narrow, so that the space gravitational wave detector becomes a feasible scheme for future gravitational wave detection. For space gravitational wave detection, on one hand, the stability of the inspection quality needs to be controlled, and on the other hand, the measurement accuracy of laser interference and space inertial sensing needs to be ensured, so that higher requirements are placed on the stability of the detection configuration. On the one hand, the initial tracking error will shift the configuration, thus leading to instability of the configuration and further failure of the detection device. On the other hand, the evolution of the configuration stability also puts demands on the navigation accuracy and the navigation interval. In addition to the design of the nominal trajectory, an evolutionary analysis of the stability of the configuration corresponding to the nominal trajectory is therefore required in the task design. Because the configuration needs to be stably maintained for a long time, is sensitive to various error sources and needs to be iterated continuously in the design process, a long-period, high-precision and high-efficiency configuration stability evolution method needs to be studied.
Disclosure of Invention
Aiming at the long-period stability evolution problem of the three-star configuration of the space gravitational wave detection, the invention discloses a space gravitational wave detection configuration stability evolution method based on dynamic mixed Gaussian, which mainly solves the technical problems that: based on the dynamic Gaussian mixture model, the stability evolution analysis of the space gravitational wave detection configuration is realized, and the uncertainty distribution of the stability index of the space gravitational wave detection configuration is predicted, so that an unstable region is avoided in the construction process of the space gravitational wave detection configuration, and the space gravitational wave detection precision and efficiency are improved. The dynamic evolution method is used for recursion of the track errors of the detectors in the space gravitational wave detection configuration, different numbers of sub-Gaussian distributions are used for approximation according to the non-Gaussian degree of uncertainty distribution of the track errors, and compared with the mixed Gaussian method, the prediction efficiency is remarkably improved; meanwhile, based on the track error distribution, the uncertainty distribution of the structural stability index is represented by using weighted sub-Gaussian distribution, and the non-Gaussian representation precision existing in the actual engineering is high. By applying the invention to the evolution analysis of the stability of the space gravitational wave detection configuration, the engineering problems related to the space gravitational wave detection field are solved. The related engineering problems in the field of space gravitational wave detection comprise optimization of space gravitational wave detection configuration, determination of configuration stability domain and detector navigation.
The aim of the invention is achieved by the following technical scheme.
The method for evolving stability of the spatial gravitational wave detection configuration based on the dynamic Gaussian mixture is used for establishing a dynamic model of a detector in the spatial gravitational wave detection configuration, and setting initial conditions of the detector according to the performance of the spatial gravitational wave detection detector. And then carrying out track error distribution forecast on each detector in the gravitational wave detection configuration by using dynamic Gaussian mixture, and for each sub Gaussian distribution, using the maximum eigenvalue of a covariance matrix to represent the non-Gaussian degree so as to judge whether splitting is needed or not, namely dynamically adjusting the number of Gaussian distribution of neutrons in the Gaussian mixture. The track error distribution is converted to a configuration stability index distribution based on deterministic sampling criteria. In addition, when the method is applied to predicting the evolution condition of the configuration stability index, the prediction precision is high and the efficiency is high.
The invention discloses a space gravitational wave detection configuration stability evolution method based on dynamic Gaussian mixture, which comprises the following steps:
step 1: and establishing a dynamic model of the detector in the space gravitational wave detection configuration, and setting initial conditions of the detector according to the performance of the space gravitational wave detection detector. The initial condition of the detector comprises an initial state x 0 Mean value m of error distribution 0 Covariance P of error distribution 0 Task end time t f A number of discrete segments M and a non-gaussian splitting threshold epsilon.
The dynamic model of the detector in the space gravitational wave detection configuration is established as shown in a formula (1):
wherein x (t) k )＝[r(t k ) T ,v(t k ) T ] T Indicating that the detector is at t k The state of the moment, wherein the state comprises position and speed, F (·, ·) represents a state transition equation, and the state transition equation can be obtained by integrating a detector dynamics equation F (·, ·) with time as shown in a formula (2).
x 0 At t 0 The initial state is given at the moment. The average value m of initial error distribution is given according to the detector performance of space gravitation wave detection 0 And covariance P 0 Setting the moment of the recursion tail end as t f And the number of discrete segments is M, with a deliveryPush time interval dt= (t) f -t 0 ) and/M. The initial error distribution is Gaussian distribution, the Gaussian number N≡1 in the mixed Gaussian distribution is initialized, the subscript k≡0 at the current time and the time t at the current time k And c, a non-Gaussian splitting threshold epsilon. And epsilon is a preset value.
Step 2: the uncertainty distribution of the detector orbit error in the configuration is characterized using a dynamically weighted sub-gaussian distribution. And (3) representing the non-Gaussian degree through the maximum eigenvalue of the covariance matrix P of the detector track error, and judging whether to dynamically adjust or not according to the non-Gaussian splitting threshold epsilon preset in the step (1), namely judging whether to split or not according to the non-Gaussian degree of the uncertainty distribution of the detector track error in the configuration. If the maximum eigenvalue of the covariance matrix P is larger than a preset non-Gaussian splitting threshold epsilon, splitting corresponding sub-Gaussian distribution along the eigenvector of the covariance matrix P, namely dynamically adjusting the number of the sub-Gaussian distribution, reducing the calculated amount and improving the approximation precision of the non-Gaussian distribution. Recursion of each sub-Gaussian distribution using deterministic sampling criteria to obtain t k+1 Moment sub-Gaussian distribution i-mean m i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k )。
Step 2.1: covariance matrix P through detector track error 0 Is characterized by a non-gaussian degree. Computing a sub-Gaussian distribution i covariance matrix P i (t k ) Has a characteristic value of [ V ] i (t k ),Λ i (t k )]＝eig(P i (t k ) V), where V i (t k ) As eigenvectors, lambda i (t k ) Is a matrix with diagonal lines composed of eigenvalues, letRepresenting covariance matrix P i (t k ) Maximum eigenvalue, corresponding eigenvector +.>If->And if the resolution threshold epsilon is larger than the non-Gaussian resolution threshold epsilon, turning to the step 2.2, otherwise turning to the step 2.3.
Step 2.2: and dynamically adjusting the number of sub-Gaussian distributions according to the non-Gaussian degree of uncertainty distribution of the detector orbit error in the space gravitational wave detection configuration, reducing the calculated amount and improving the approximation precision of the non-Gaussian distribution. Split molecular gaussian distribution i along eigenvectorSplitting the sub-Gaussian distribution i, wherein the corresponding new average value of the sub-Gaussian distribution is as follows:
the covariance matrix is:
P i,q (t k )＝V i (t k )Λ i,q (t k )V i (t k ) T (4)
the weight is as follows:
wherein:
at this time N++N q -1. Parameters (parameters)And +.>Can be obtained by solving the optimization problem (7).
Wherein:
the parameter β is a preset parameter, and β=2 is usually taken. p is the true distribution of the light,is a distribution approximated by a mixture of gaussians, as shown in equation (9).
Step 2.3: each sub-gaussian distribution is recursively computed. Using deterministic sampling criteria to deliver a clipper Gaussian distribution i to get t k+1 Time average value m i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ). I.e. the uncertainty distribution of the detector orbit error in the configuration is characterized using a weighted sub-gaussian distribution, as shown in equation (10).
Step 3: according to the sub Gaussian distribution i mean value m obtained in the step 2 i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ) And the uncertainty distribution of the stability index of the spatial gravitational wave detection configuration is characterized by using the weighted sub-Gaussian distribution, so that the characterization precision of the uncertainty distribution of the stability index of the actual configuration is improved.
Each detector in the gravitational wave detection configuration obtained according to step 2 is at t k+1 Sub-gaussian distribution p of orbit error at time instant i (x,t k+1 ) Where i=1, 2,3 denotes detector 1,2,3, n j The number of the neutron Gaussian distribution is corresponding to the Gaussian mixture distribution. In this case, the distribution of the configuration stability is represented by N 1 ×N 2 ×N 3 The individual sub-gaussian distributions are approximated by an approximation formula such as formula (11):
wherein Σ= { x 1 ,x 2 ,x 3 The K (Σ) represents a configuration stability index including arm length, respiratory angle, and arm length change rate, corresponding to one configuration.And->Is the mean and covariance of the sub-gaussian distribution of the configuration stability distribution. Using equation (10), by N 1 ×N 2 ×N 3 The sub Gaussian distribution characterizes the uncertainty distribution of the stability index of the space gravitational wave detection configuration, and the characterization precision of the uncertainty distribution of the stability index of the actual configuration is improved.
Step 4: judging whether the corresponding evolution time reaches the task end time t in the step 1 f If the end time t of the task is not reached f Let k +1 and return to step 2, the uncertainty distribution of the configuration stability index at the next moment is continuously predicted. If reaching the end time t of the task f And outputting an uncertainty distribution result of the spatial gravitational wave detection configuration stability index, namely realizing the spatial gravitational wave detection configuration stability evolution analysis based on the dynamic Gaussian mixture model.
Further comprising the step 5: and (3) avoiding an unstable region in the construction process of the space gravitational wave detection configuration according to the uncertainty distribution of the stability index of the space gravitational wave detection configuration predicted in the steps (2) to (4), and improving the detection precision and efficiency of the space gravitational wave.
The beneficial effects are that:
1. according to the dynamic Gaussian mixture-based spatial gravitational wave detection configuration stability evolution method disclosed by the invention, spatial gravitational wave detection configuration stability evolution analysis is realized based on a dynamic Gaussian mixture model, and the uncertainty distribution of spatial gravitational wave detection configuration stability indexes is predicted, so that an unstable region is avoided in the construction process of the spatial gravitational wave detection configuration, and the spatial gravitational wave detection precision and efficiency are improved.
2. The spatial gravitational wave detection configuration stability evolution method based on the dynamic Gaussian mixture, disclosed by the invention, uses the weighted sub-Gaussian distribution to represent the uncertainty distribution of the configuration stability index, and has high non-Gaussian representation precision in actual engineering.
3. According to the spatial gravitational wave detection configuration stability evolution method based on the dynamic Gaussian mixture, whether the separation is carried out or not is judged according to the magnitude relation between the maximum eigenvalue of the covariance matrix P and the preset non-Gaussian separation threshold epsilon, so that the non-Gaussian degree of uncertainty distribution is approximated by using different numbers of sub-Gaussian distributions, and the prediction efficiency is remarkably improved compared with that of the Gaussian mixture method.
4. The method for evolving the stability of the spatial gravitational wave detection configuration based on the dynamic Gaussian mixture disclosed by the invention is applied to the evolution analysis of the stability of the spatial gravitational wave detection configuration, so that the problem of engineering related to the field of spatial gravitational wave detection is solved. The related engineering problems in the field of space gravitational wave detection comprise optimization of space gravitational wave detection configuration, determination of configuration stability domain and detector navigation.
Drawings
FIG. 1 is a flow chart of a dynamic Gaussian mixture-based spatial gravitational wave detection configuration stability evolution method disclosed by the invention;
fig. 2 shows the distribution of the obtained structural stability index, wherein fig. 2 (a) shows the distribution of the structural arm length, fig. 2 (b) shows the distribution of the structural respiratory angle, and fig. 2 (c) shows the distribution of the structural arm length change rate.
Detailed Description
For a better description of the objects and advantages of the present invention, the following description of the invention refers to the accompanying drawings and examples.
According to the method proposed in the present project and as shown in fig. 1, the present example predicts the uncertainty distribution of the configuration stability index by using the spatial gravitational wave detection configuration stability evolution method based on dynamic mixed gauss disclosed in the present invention, for the configuration of the constellation of the pian.
The embodiment discloses a space gravitation wave detection configuration stability evolution method based on dynamic Gaussian mixture, which comprises the following specific implementation steps:
step 1: establishing a problem model and setting parameters:
the pian constellation consists of three detectors and the corresponding orbit parameters are shown in table 1. In this example, the kinetic model is implemented by modulating GMAT software, and the high-precision kinetic model used includes the earth 10 x 10 order gravitational field, the moon, and the sun's central attraction. The stability indicators considered include arm length, respiratory angle, and rate of change of arm length. Evolution time t f For 3 months. The initial error is set as: the position error mean value is 0m, the standard deviation is 10m, the speed error mean value is 0mm/s, the standard deviation is 1 mm/s, and the initial error distribution is represented as formula (12).
The number of discrete segments is m=100. The initial error distribution is Gaussian distribution, the Gaussian number N≡1 in the mixed Gaussian distribution is initialized, the subscript k≡0 at the current time and t at the current time k ζ0, non-gaussian splitting threshold ε=10.
TABLE 1 Piano constellation track parameters
Step 2: the uncertainty distribution of the detector orbit error in the configuration is characterized using a dynamically weighted sub-gaussian distribution. And (3) representing the non-Gaussian degree through the maximum eigenvalue of the covariance matrix P of the detector track error, and judging whether to dynamically adjust or not according to the non-Gaussian splitting threshold epsilon preset in the step (1). If the maximum eigenvalue of covariance matrix P is greater than 10, splitting the corresponding sub-gaussian distribution along the eigenvector of covariance matrix P, where β=2, n q Solving problem (7) =5 gives split parameters as shown in table 2. Recursion of each sub-Gaussian distribution using deterministic sampling criteria to obtain t k+1 Moment sub-Gaussian distribution i-mean m i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ). The deterministic sampling criteria in this example is a fifth order volume criteria.
TABLE 2 sub-Gaussian distribution splitting parameters
Step 3: according to the sub-Gaussian distribution i mean value m obtained in the step 2 i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ) The uncertainty distribution of the stability index of the spatial gravitational wave detection configuration is characterized using a weighted sub-gaussian distribution.
Step 4: judging whether the corresponding evolution time reaches the task end time t in the step 1 f If the end time t of the task is not reached f I.e. t k+1 ＜t f Let k +1 and return to step 2, and continuously predicting the uncertainty distribution of the configuration stability index at the next moment. If reaching the end time t of the task f I.e. t k+1 ＝t f And outputting an uncertainty distribution result of the spatial gravitational wave detection configuration stability index, namely realizing the spatial gravitational wave detection configuration stability evolution analysis based on the dynamic Gaussian mixture model, as shown in figure 2.
The method is compared with other methods in terms of approximation accuracy using a criterion called likelihood consistency. The likelihood consistency criterion is shown in formula (13):
LAM(p,q)＝∫ ∞ p(x)q(x)dx (13)
wherein LAM (p, q) represents the proximity degree of two distributions p (x) and q (x), wherein q (x) is a real distribution, p (x) is a distribution obtained by approximation of a dynamic mixture gaussian model, and the larger the LAM (p, q) value is, the higher the approximation accuracy is. In this example comparison, the true distribution was tabulated using a distribution of 10000 Monte Carlo sampling points. The dynamic Gaussian mixture model is compared with a conventional covariance function method, and the obtained result is shown in Table 3. In table 3, in order to highlight the comparison result, the LAM value of the dynamic gaussian mixture model is normalized, and it is seen from the table that the LAM value of the covariance function method is smaller than the method of the present invention, which shows that the method of the present invention has advantages over the conventional precision.
TABLE 3 approximation accuracy vs. results
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (4)
1. The evolution method for the stability of the spatial gravitational wave detection configuration based on the dynamic mixing Gaussian is characterized by comprising the following steps of: comprises the following steps of the method,
step 1: establishing a dynamic model of the detector in a space gravitational wave detection configuration, and setting initial conditions of the detector according to the performance of the space gravitational wave detection detector; the initial condition of the detector comprises an initial state x 0 Mean value m of error distribution 0 Covariance P of error distribution 0 Task end time t f A discrete segment number M and a non-Gaussian splitting threshold epsilon;
step 2: characterizing an uncertainty distribution of detector orbit errors in the configuration using a dynamically weighted sub-gaussian distribution; the non-Gaussian degree is represented by the maximum eigenvalue of the covariance matrix P of the detector track error, and whether dynamic adjustment is carried out is judged according to the non-Gaussian splitting threshold epsilon preset in the step 1, namely according to the detector track in the configurationJudging whether the track error is split or not according to the non-Gaussian degree of uncertainty distribution of the track error; if the maximum eigenvalue of the covariance matrix P is larger than a preset non-Gaussian splitting threshold epsilon, splitting corresponding sub-Gaussian distribution along the eigenvector of the covariance matrix P, namely dynamically adjusting the number of the sub-Gaussian distribution, reducing the calculated amount and improving the approximate precision of the non-Gaussian distribution; recursion of each sub-Gaussian distribution using deterministic sampling criteria to obtain t k+1 Moment sub-Gaussian distribution i-mean m i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k )；
The implementation method of the step 2 is that,
step 2.1: covariance matrix P through detector track error 0 The maximum eigenvalue of (2) characterizes the non-gaussian degree; computing a sub-Gaussian distribution i covariance matrix P i (t k ) Has a characteristic value of [ V ] i (t k ),Λ i (t k )]＝eig(P i (t k ) V), where V i (t k ) As eigenvectors, lambda i (t k ) Is a matrix with diagonal lines composed of eigenvalues, letRepresenting covariance matrix P i (t k ) Maximum eigenvalue, corresponding eigenvector +.>If->If the resolution threshold epsilon is larger than the non-Gaussian resolution threshold epsilon, turning to step 2.2, otherwise turning to step 2.3;
step 2.2: according to the non-Gaussian degree of uncertainty distribution of detector orbit errors in a space gravitational wave detection configuration, the number of sub-Gaussian distributions is dynamically adjusted, the calculated amount is reduced, and the approximation precision of the non-Gaussian distribution is improved; split molecular gaussian distribution i along eigenvectorSplitting the sub-Gaussian distribution i, wherein the corresponding new average value of the sub-Gaussian distribution is as follows:
the covariance matrix is:
P i,q (t k )＝V i (t k )Λ i,q (t k )V i (t k ) T (4)
the weight is as follows:
wherein:
at this time N++N q -1; parameters (parameters)And +.>Can be obtained by solving an optimization problem (7);
wherein:
the parameter beta is the pre-preparationSetting parameters, β=2; p is the true distribution of the light,is a distribution approximated by a mixture of gaussians, as shown in equation (9);
step 2.3: recursion of each sub-Gaussian distribution; using deterministic sampling criteria to deliver a clipper Gaussian distribution i to get t k+1 Time mean value m i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ) The method comprises the steps of carrying out a first treatment on the surface of the I.e., using a weighted sub-gaussian distribution to characterize the uncertainty distribution of detector orbit errors in the configuration, as shown in equation (10);
step 3: according to the sub Gaussian distribution i mean value m obtained in the step 2 i (t k+1 ) Covariance matrix P i (t k+1 ) Weight alpha i (t k+1 )＝α i (t k ) The uncertainty distribution of the stability index of the spatial gravitational wave detection configuration is represented by using the weighted sub Gaussian distribution, and the representation precision of the uncertainty distribution of the stability index of the actual configuration is improved;
step 4: judging whether the corresponding evolution time reaches the task end time t in the step 1 f If the end time t of the task is not reached f Let k +1 and return to step 2, continuously predicting the uncertainty distribution of the configuration stability index at the next moment; if reaching the end time t of the task f And outputting an uncertainty distribution result of the spatial gravitational wave detection configuration stability index, namely realizing the spatial gravitational wave detection configuration stability evolution analysis based on the dynamic Gaussian mixture model.
2. The dynamic mixed Gaussian-based spatial gravitational wave detection configuration stability evolution method according to claim 1, wherein the method is characterized by comprising the following steps: and 5, avoiding an unstable region in the construction process of the space gravitational wave detection configuration according to the uncertainty distribution of the space gravitational wave detection configuration stability indexes predicted in the steps 2 to 4, and improving the space gravitational wave detection precision and efficiency.
3. The dynamic mixed gaussian based spatial gravitational wave detection configuration stability evolution method according to claim 1 or 2, wherein the method is characterized in that: the implementation method of the step 1 is that,
the dynamic model of the detector in the space gravitational wave detection configuration is established as shown in a formula (1):
wherein x (t) k )＝[r(t k ) T ,v(t k ) T ] T Indicating that the detector is at t k The state at the moment, wherein the state comprises the position and the speed, F (·, ·) represents a state transition equation, and the state transition equation can be obtained by integrating a detector dynamics equation F (·, ·) with time as shown in a formula (2);
x 0 at t 0 An initial state given at a moment; the average value m of initial error distribution is given according to the detector performance of space gravitation wave detection 0 And covariance P 0 Setting the moment of the recursion tail end as t f And the number of discrete segments is M, with recursive time interval dt= (t) f -t 0 ) M; the initial error distribution is Gaussian distribution, the Gaussian number N≡1 in the mixed Gaussian distribution is initialized, the subscript k≡0 at the current time and t at the current time k The non-Gaussian splitting threshold epsilon is described as (C) 0; and epsilon is a preset value.
4. The method for evolution of spatial gravitational wave detection configuration stability based on dynamic mixing gauss as claimed in claim 3, wherein the method is characterized in that: the implementation method of the step 3 is that,
each detector in the gravitational wave detection configuration obtained according to step 2 is at t k+1 Sub-gaussian distribution p of orbit error at time instant i (x,t k+1 ) Where i=1, 2,3 denotes detector 1,2,3, n j The number of the neutron Gaussian distributions in the corresponding Gaussian mixture distribution is the number of the neutron Gaussian distributions; in this case, the distribution of the configuration stability is represented by N 1 ×N 2 ×N 3 The individual sub-gaussian distributions are approximated as shown in equation (11):
wherein Σ= { x 1 ,x 2 ,x 3 A configuration, K (Σ) represents a configuration stability index including arm length, respiratory angle, and arm length change rate;and->Is the mean and covariance of the sub-gaussian distribution of the configuration stability distribution; using equation (10), by N 1 ×N 2 ×N 3 The sub Gaussian distribution characterizes the uncertainty distribution of the stability index of the space gravitational wave detection configuration, and the characterization precision of the uncertainty distribution of the stability index of the actual configuration is improved.
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| 航天器轨迹偏差的非线性非高斯预报方法;孙振江;罗亚中;张进;唐国金;;宇航学报(第11期);全文 * |
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