CN112560343B - J2 perturbation Lambert problem solving method based on deep neural network and targeting algorithm - Google Patents
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Abstract
The invention discloses a J2 perturbation Lambert problem solving method based on a deep neural network and a targeting algorithm, which comprises the following steps: obtaining a starting end speed initial value according to the starting end position vector and the flight time, and performing track recursion based on the obtained starting end speed initial value to obtain an end position error under J2 perturbation interference; according to the end position error, the start and end positions in initial conditions and the flight time, utilizing a depth neural network obtained through training to estimate the error of the initial value of the start speed, and correcting the initial value of the start speed by taking the error as a correction amount to obtain a corrected initial guess value of the start speed; and taking the initial guess value of the initial end speed as an initial value, and performing targeting correction on the initial guess value of the initial end speed by utilizing a Newton iterative targeting algorithm based on differential approximation until the precision of the end position meets the requirement. The method solves the problems of low calculation efficiency, insufficient convergence stability and poor multi-circle Lambert problem solving effect when the J2 perturbation Lambert problem is solved in the prior art.
Description
Technical Field
The invention belongs to the technical field of orbit dynamics, and particularly relates to a J2 perturbation Lambert problem solving method based on a deep neural network and a targeting algorithm.
Background
The Lambert problem is to solve for the velocity of the beginning and end given the position of the beginning and end and the time of flight, and is the fundamental problem in the field of orbit dynamics. The classical Lambert problem is proposed based on a two-body dynamics model, but the actual motion of a spacecraft is interfered by various perturbation, so that the kepler solution of the classical Lambert problem cannot meet the accuracy requirement of an actual task. Thus, considering that J2 perturbation is the main perturbation term of the middle and low orbits, the J2 perturbation Lambert problem is presented on the basis of the classical Lambert problem.
According to the solution principle, the existing J2 perturbation Lambert problem solving algorithm can be summarized into two categories: analytical methods and targeting methods. The analysis method is used for deducing the analysis form of the J2 perturbation Lambert problem, converting the problem into a series of parameter algebraic equations and solving the parameter algebraic equations. The targeting method is to utilize various targeting algorithms to carry out iterative correction on the initial velocity vector according to the error feedback of the tail end state. Feng Haoyang et al, novel solution to the largely convergent perturbation Lambert problem: the method has the advantages that the initial speed and transfer track of the perturbation Lambert problem can be rapidly and accurately obtained under larger time and space scale by utilizing the large-scale convergence characteristic of the linearization and the rapid convergence and high-precision characteristic of the local variation iteration method. However, for the multi-turn Lambert problem with longer transfer time, the error of the linearization operation will be amplified due to the enhanced nonlinear characteristics, resulting in poor convergence. Yang Z, luo Y Z, zhang J, et al, homotopic perturbed Lambert algorithm for long-duration rendezvous optimization [ J ]. Journal of Guidance, control, and Dynamics,2015,38 (11): 2215-2223. A solution to the J2 perturbation multi-turn Lambert problem is proposed based on the homotopy technique, which has the advantages that the homotopy technique is introduced to effectively improve the convergence stability of the algorithm, but the introduction of homotopy parameters also leads to a substantial increase in the iteration times and calculation time.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a J2 perturbation Lambert problem solving method based on a deep neural network and a targeting algorithm, so as to solve the problems of low calculation efficiency, insufficient convergence stability and poor multi-turn Lambert problem solving effect in the prior art when the J2 perturbation Lambert problem is solved.
In order to achieve the above purpose, the invention adopts the following technical scheme:
the invention discloses a J2 perturbation Lambert problem solving method based on a deep neural network and a targeting algorithm, which comprises the following steps:
1) According to the initial and final position vectors and the flight time, solving a two-body Lambert problem by using a universal variable method to obtain an initial value of initial speed, and performing orbit recursion based on the initial value of the initial speed to obtain a final position error under J2 perturbation interference;
2) According to the end position error obtained in the step 1) and the start and end positions and the flight time in the initial conditions, the error of the initial value of the start speed is estimated by using the trained deep neural network, and the initial value of the start speed obtained in the step 1) is corrected by taking the error as a correction amount, so that a corrected initial guess value of the start speed is obtained;
3) And 2) taking the initial guess value of the initial speed obtained in the step 2) as an initial value, and performing targeting correction on the initial guess value of the initial speed by utilizing a Newton iterative targeting algorithm based on differential approximation until the precision of the position of the tail end meets the requirement.
Further, the step 1) specifically includes: according to the start-end position vector (r) 0 ,r f ) And flight time tof, solving a two-body Lambert problem by using a universal variable method to obtain initial value v of initial end speed 0 ；
v 0 ＝L(r 0 ,r f ,tof)
Where L represents the Lambert solution process.
Further, the step 1) specifically further includes: according to the initial value v of the obtained initial end speed 0 And a known start position r 0 Performing orbit recursion, wherein the recursion time is equal to the flight time, and the recursion dynamics model adopts a J2 perturbation dynamics model to obtain the actual arrival end position and velocity vector (r) a ,v a ) In combination with known ideal end position r f Obtaining the end position error Deltar f ＝r f -r a 。
Further, the training samples of the deep neural network in the step 2) are represented by a start position vector r 0 And a starting end speed initial value v obtained based on two-body Lambert problem solving 0 Time of flight tof and simplicityTerminal position error vector Deltar of Kepler solution under two-body dynamics f For input, the initial velocity error vector Deltav of Kepler solution under simple two-body dynamics 0 For output, all the position and speed vectors are described by spherical coordinates; initial guess value of initial end speed v d ＝v 0 +Δv 0 。
Further, the training sample obtaining step of the deep neural network in the step 2) specifically includes the following steps:
21 Randomly generating an initial orbit state and a time of flight r 0 ；v s0 ；tof]；
22 To the initial state [ r ] 0 ；v s0 ]Recursion under J2 perturbation kinetic model to obtain end state [ r ] f ；v f ]；
23 Based on the start-end state and the flight time parameters, calculating the initial value v of the start-end speed by solving the two-body Lambert problem 0 ；
24 Initial state [ r ] with initial end position and initial end speed 0 ；v 0 ]Recursion under J2 perturbation dynamics model to obtain the actual end position r a ；
25 Calculating initial value v of initial end speed 0 And end position r a Error of Deltav 0 ＝v s0 -v 0 ，Δr f ＝r f -r a ；
26 With a start position vector r 0 And initial value v of initial end speed 0 Time of flight tof and end position error vector Δr f For input, an error vector Deltav of initial value of initial end speed 0 For output, training samples are formed, and all vectors are described in a spherical coordinate system.
Further, the specific form of the training sample S in the step 2) is as follows:
S＝{[r 00 ,α r0 ,β r0 ,v 00 ,α v0 ,β v0 ,Δr ff ,Δα f ,Δβ f ,tof]，[Δv 00 ,Δα 0 ,Δβ 0 ]}
wherein α and β represent the azimuth angles of the vectors, respectivelyAnd polar angle, subscripts 0 and f are the start and end flags, deltav, respectively 00 ＝||Δv 0 The modulus length of the initial velocity correction amount, r 00 ＝||r 0 The || represents the module length of the start end position, Δr ff ＝||Δr f And I is the modular length of the end position error, then:
further, the step 3) specifically includes: calculating a jacobian matrix using a differential approximation method, and applying a small disturbance δv=10 to each component of the initial velocity vector -6 km/s; integrating to the end state and recording the deviation δr of the end position; the resulting jacobian matrix is:
then, the start-end speed correction amount is:
Δv i ＝J -1 Δr fi
wherein Deltar fi Representing the end position error at the ith iteration.
The invention discloses a J2 perturbation Lambert problem solver based on a deep neural network and a targeting algorithm, which comprises the following components:
one or more processors;
a memory for storing one or more programs;
the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the solution method described above.
The invention has the beneficial effects that:
according to the invention, the kepler solution of the two-body Lambert problem is corrected by introducing the deep neural network, so that the accuracy of an initial value is greatly improved, the iteration times and the calculation time of an iterative targeting algorithm are effectively reduced, and the convergence stability of the algorithm is ensured. In addition, the transfer time of the training sample is not limited, so that the invention can still maintain high calculation efficiency while ensuring convergence stability when processing the problem of multi-circle J2 perturbation lambert; the whole idea is novel, and the method has wide application prospect.
Drawings
Fig. 1 is a flow chart of the algorithm of the present invention.
FIG. 2 is a schematic diagram of the principle of the target practice correction in the present invention.
FIG. 3a is a graph showing the error distribution of initial velocity components before and after correction of the deep neural network in the present invention.
FIG. 3b is a graph showing the error distribution of the end position components before and after correction by the deep neural network according to the present invention.
FIG. 4 is a schematic diagram of the convergence of the solution of the different turns J2 perturbation Lambert problem using the present invention.
FIG. 5 is a schematic diagram of the time consumption of the invention for solving the different turns J2 perturbing Lambert problem.
Detailed Description
The invention will be further described with reference to examples and drawings, to which reference is made, but which are not intended to limit the scope of the invention.
Referring to fig. 1, the J2 perturbation Lambert problem solving method based on the deep neural network and the targeting algorithm comprises the following steps:
1. the start-end position vector (r) of the spacecraft is known 0 ,r f ) And flight time tof, solving a two-body Lambert problem by using a universal variable method to obtain initial value v of initial end speed 0 ：
v 0 ＝L(r 0 ,r f ,tof)；
2. According to the initial value v of the initial end speed calculated in the step 1 0 And a known start position r 0 Performing orbit recursion, wherein the recursion time is equal to the flight time, and the recursion dynamics model adopts a J2 perturbation dynamics model to obtain the actual arrival end position and velocity vector (r) a ,v a ) Thereby combining the known ideal end position r f Obtaining the end position error Deltar f ＝r f -r a ；
3. According to the known starting position r 0 Time of flight tof, initial value of initial velocity v 0 And end position error Deltar f Describing the vector concerned with spherical coordinates and predicting the correction amount Deltav of the initial velocity by using the trained deep neural network 0 ：
[Δv 00 ,Δα 0 ,Δβ 0 ]＝DNN(r 00 ,α r0 ,β r0 ,v 00 ,α v0 ,β v0 ,Δr ff ,Δα f ,Δβ f ,tof)
Wherein Deltav 00 ＝||Δv 0 The modulus length of the initial velocity correction amount, r 00 ＝||r 0 The < I > represents the modular length of the start end position, v 00 ＝||v 0 The | is the modulo length of the initial velocity calculated in step 1, Δr ff ＝||Δr f The I is the modular length of the end position error, alpha and beta respectively represent the azimuth angle and the polar angle of the vector, and subscripts 0 and f respectively represent the start-end marks;
the vector represented by the spherical coordinates is converted into a Cartesian coordinate system description by using the following formula:
r 0 ＝[r 00 sin(β r0 )cos(α r0 )；r 00 sin(β r0 )sin(α r0 )；r 00 cos(β r0 )]
v 0 ＝[v 00 sin(β v0 )cos(α v0 )；v 00 sin(β v0 )sin(α v0 )；v 00 cos(β v0 )]
Δr f ＝[Δr ff sin(Δβ f )cos(Δα f )；Δr ff sin(Δβ f )sin(Δα f )；Δr ff cos(Δβ f )]
Δv 0 ＝[Δv 00 sin(Δβ 0 )cos(Δα 0 )；Δv 00 sin(Δβ 0 )sin(Δα 0 )；Δv 00 cos(Δβ 0 )]
the prediction effect of the neural network is mainly determined by the sample quality, and the rapid acquisition steps of the sample in the invention are as follows:
random generation of initial orbit states and time of flight [ r ] 0 ；v s0 ；tof]；
Will be the initial state [ r ] 0 ；v s0 ]Recursion under J2 perturbation kinetic model to obtain end state [ r ] f ；v f ]；
Based on the start-end state and the flight time parameters, obtaining a start-end speed initial value v by solving a two-body Lambert problem 0 ；
The initial state [ r ] is the initial value of the initial end position and the initial end speed 0 ；v 0 ]Recursion under J2 perturbation dynamics model to obtain the actual end position r a ；
Calculating initial value v of initial end speed 0 And end position r a Error of Deltav 0 ＝v s0 -v 0 ，Δr f ＝r f -r a ；
With a start state vector r 0 And v 0 Time of flight tof and end position error vector Δr f For input, an error vector Deltav of initial value of initial end speed 0 For output, training samples are formed, and all vectors are described in a spherical coordinate system.
4. According to the initial speed correction amount Deltav obtained in the step 3 0 Correcting the initial value v of the initial end speed obtained in the step 1 0 Calculating the corrected initial end speed initial guess vector v d ＝v 0 +Δv 0 ；
5. The corrected initial end speed initial guess value v calculated in the step 4 d And (3) as an initial value, adopting a Newton iterative targeting algorithm based on differential approximation to carry out targeting correction on the initial guess value of the initial speed until the terminal precision meets the requirement. A schematic diagram of the step 4 newton iterative targeting process is shown in fig. 2.
The invention uses a differential approximation method to calculate the Jacobian matrix, and the specific process is as follows: first, a small disturbance δv=10 is applied to each component of the initial velocity vector -6 km/s; then, integrating to the end state and recording the deviation δr of the end position; finally, the obtainedThe jacobian matrix is:
then, the start-end speed correction amount is:
Δv i ＝J -1 Δr fi
wherein Deltar fi Representing the end position error at the ith iteration.
Then, the i-th corrected start speed vector v di ＝v di-1 +Δv i 。
Examples of the method of the invention: example verification of the invention is described in connection with fig. 3a, 3b, 4 and 5, setting the following calculation conditions and technical parameters:
(1) Taking the wooden star as a central celestial body, wherein the average equatorial radius of the wooden star is R J 71492km, the star attraction constant is μ J ＝126686543.922km 3 /s 2 The J2 perturbation term coefficient is j2= 0.01475.
(2) The parameter value range of the random sample is set as follows:
where r is the initial orbit radius, e is the initial orbit eccentricity, i is the initial orbit inclination angle, Ω is the initial ascent intersection point right ascent, ω is the initial near-spot argument, u is the initial true near-spot argument, tof is the time of flight, T is the orbit period of the initial orbit,(a is the initial orbit semi-major axis).
(3) The track end position targeting accuracy was set to 0.001km.
(4) The deep neural network comprises four hidden layers, each hidden layer comprises 50 neurons, the input layer comprises 10 neurons, the output layer comprises 3 neurons, the activation function of the hidden layers adopts a hyperbolic sine function (tanh), and the activation function of the output layer adopts a linear rectification function (Relu).
(5) The number of training samples of the neural network is 200000, the training optimizer selects an adaptive moment estimation algorithm (Adaptive moment estimation, adam), the maximum training times is 50000, and the initial learning rate is set to 0.001.
(6) The Monte Carlo simulation number was set to 1000.
Based on the J2 perturbation Lambert solving method and the set calculation conditions and technical parameters, except that the neural network training is carried out by adopting a Tensorflow module of Python, other parts adopt Matlab software for simulation verification. Fig. 3a and 3b show the error distribution of the components of the initial velocity vector and the final position vector before and after correction of the deep neural network, respectively. After correction of the neural network, the standard deviation of the initial speed error is not more than 0.01km/s, the component error of the tail end position is not more than 100km, and the initial value precision is obviously improved. Fig. 4 and fig. 5 respectively compare the convergence effect and the calculation time of the existing newton iterative targeting algorithm, the homolunar iterative algorithm and the J2 perturbation Lambert problem solving algorithm based on the deep neural network and the targeting algorithm adopted in the invention when the J2 perturbation Lambert problem with different turns is solved. From fig. 4, it can be seen that the present invention can still stably converge with the increase of the number of turns thanks to the high-precision initial value provided by the neural network. As can be seen from fig. 5, the calculation time of the present invention increases slightly with the increase of the number of turns, and the increase trend is far smaller than that of the homolunar iterative algorithm, so that the present invention has higher calculation efficiency when solving the problem of multi-turn J2 perturbation Lambert.
The present invention has been described in terms of the preferred embodiments thereof, and it should be understood by those skilled in the art that various modifications can be made without departing from the principles of the invention, and such modifications should also be considered as being within the scope of the invention.
Claims (3)
1. A J2 perturbation Lambert problem solving method based on a deep neural network and a targeting algorithm is characterized by comprising the following steps:
1) According to the initial and final position vectors and the flight time, solving a two-body Lambert problem by using a universal variable method to obtain an initial value of initial speed, and performing orbit recursion based on the initial value of the initial speed to obtain a final position error under J2 perturbation interference;
2) According to the end position error obtained in the step 1) and the start and end positions and the flight time in the initial conditions, the error of the initial value of the start speed is estimated by using the trained deep neural network, and the initial value of the start speed obtained in the step 1) is corrected by taking the error as a correction amount, so that a corrected initial guess value of the start speed is obtained;
3) Performing target shooting correction on the initial guess value of the initial speed by utilizing a Newton iterative target shooting algorithm based on differential approximation until the precision of the position of the tail end meets the requirement;
the training sample of the deep neural network in the step 2) is represented by a start position vector r 0 And a starting end speed initial value v obtained based on two-body Lambert problem solving 0 End position error vector Deltar of Kepler solution under time of flight tof and simple two-body dynamics f For input, the initial velocity error vector Deltav of Kepler solution under simple two-body dynamics 0 For output, all the position and speed vectors are described by spherical coordinates; initial guess value of initial end speed v d ＝v 0 +Δv 0 ；
The training sample obtaining step of the deep neural network in the step 2) specifically comprises the following steps:
21 Randomly generating an initial orbit state and a time of flight r 0 ；v s0 ；tof]；
22 To the initial state [ r ] 0 ；v s0 ]Recursion under J2 perturbation kinetic model to obtain end state [ r ] f ；v f ]；
23 Based on the start-end state and the flight time parameters, calculating the initial value v of the start-end speed by solving the two-body Lambert problem 0 ；
24 Initial state [ r ] with initial end position and initial end speed 0 ；v 0 ]Recursion under J2 perturbation dynamics model to obtain the actual end position r a ；
25 Calculating initial value v of initial end speed 0 And end position r a Error of Deltav 0 ＝v s0 -v 0 ，Δr f ＝r f -r a ；
26 With a start position vector r 0 And initial value v of initial end speed 0 Time of flight tof and end position error vector Δr f For input, an error vector Deltav of initial value of initial end speed 0 For output, forming training samples, and all vectors are described in a spherical coordinate system;
the specific form of the training sample S in the step 2) is as follows:
S＝{[r 00 ，α r0 ，β r0 ，v 00 ，α v0 ，β v0 ，Δr ff ，Δα f ，Δβ f ，tof]，[Δv 00 ，Δα 0 ，Δβ 0 ]}
wherein alpha and beta respectively represent azimuth angle and polar angle of the vector, subscripts 0 and f are respectively start and end marks, and Deltav 00 ＝||Δv 0 The modulus length of the initial velocity correction amount, v 00 For the initial velocity of the mould length, r 00 ＝||r 0 The || represents the module length of the start end position, Δr ff ＝||Δr f And I is the modular length of the end position error, then:
the step 3) specifically comprises the following steps: calculating a jacobian matrix using a differential approximation method, and applying a small disturbance δv=10 to each component of the initial velocity vector -6 km/s; integrating to the end state and recording the deviation δr of the end position; the resulting jacobian matrix is:
then, the start-end speed correction amount is:
Δv i ＝J -1 Δr fi
wherein Deltar fi Representing the end position error at the ith iteration.
2. The method for solving the J2 perturbation Lambert problem based on the deep neural network and the targeting algorithm according to claim 1, wherein the step 1) specifically comprises: according to the start-end position vector (r) 0 ，r f ) And flight time tof, solving a two-body Lambert problem by using a universal variable method to obtain initial value v of initial end speed 0 ；
v 0 ＝L(r 0 ，r f ，tof)
Where L represents the Lambert solution process.
3. The method for solving the J2 perturbation Lambert problem based on the deep neural network and the targeting algorithm according to claim 1, wherein the step 1) specifically further comprises: according to the initial value v of the obtained initial end speed 0 And a known start position r 0 Performing orbit recursion, wherein the recursion time is equal to the flight time, and the recursion dynamics model adopts a J2 perturbation dynamics model to obtain the actual arrival end position and velocity vector (r) a ，v a ) In combination with known ideal end position r f Obtaining the end position error Deltar f ＝r f -r a 。
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