A satellite initial orbit determination method embedded with a physical knowledge neural network

Abstract

The application relates to a satellite initial orbit determination method embedded with a physical knowledge neural network, which is used for short-arc initial orbit determination under optical observation conditions and provides support for subsequent precise orbit determination. The network comprises an encoder-decoder structure, the encoder predicts the motion state of a satellite in an orbit according to observation information, the decoder converts the predicted satellite motion state into observation values, evaluates the accuracy of the encoder prediction, and introduces physical information of satellite orbit dynamics in a loss function, so that satellite orbit determination based on a neural network is realized. The input of the encoder is satellite observation angle (i.e. azimuth and elevation), observation time and position information (i.e. longitude, latitude and altitude) of an observer, the output of the encoder, i.e. the input of the decoder, is the absolute motion state quantity (i.e. the position and speed of the satellite in an earth-centered inertial coordinate system) of the satellite in the orbit and the position information of the observer, and the output of the decoder is the observation quantity of the satellite relative to the observer. During network training, firstly, the mean square error between the predicted observation value and the true observation value is taken as a loss function to train the decoder, the transformation relationship from the satellite motion state to the observation value is learned, then the structure and parameters of the decoder are fixed, the whole encoder-decoder network is trained, the derivative of the predicted satellite absolute state quantity relative to the observation time is calculated, the physical information loss of the velocity term and the acceleration term is calculated, the physical information loss is added to the mean square error of the observation quantity to construct a total loss function, and the network parameters are optimized through error back propagation, in the inference stage, the encoder network is used for inference, and the absolute motion state quantity of the satellite predicted by the encoder is converted into an orbit element form, i.e. initial orbit determination is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of satellite initial orbit determination, and in particular to a satellite initial orbit determination method based on an embedded physical knowledge neural network. BACKGROUND

[0002] With the continuous progress of aerospace technology, especially the significant improvement in carrying capacity, the number of satellites entering space has shown explosive growth. This provides more opportunities for human exploration and utilization of space, but also brings an increasingly complex operating environment. In this context, accurately determining the orbit parameters of satellites becomes particularly important. This is not only a key means to ensure the normal operation of satellites, but also plays an important role in preventing potential spacecraft collisions and avoiding space debris. By optimizing the monitoring and management of orbit parameters, the order of space operations can be more effectively maintained to ensure the safety of future space activities.

[0003] Traditional orbit determination methods are based on modeling the state of the satellite, including the Gibbs method, Lambert method and Newton iteration method. The Gibbs method uses a cubic position vector to determine the orbit, directly calculates the orbit parameters through vector operations, and is suitable for two-body problems. However, when the measurement time interval is large, the error will increase significantly, and it is not suitable for cases with large perturbations. The Lambert method solves the orbit by giving two positions and a time interval, and is commonly used for orbit transfer calculations. It can be applied to complex orbit design, but the equation solving process is complex and sensitive to initial guess values, which may cause convergence problems. The Newton iteration method iteratively approximates the exact orbit solution, which is suitable for high-precision orbit determination problems. However, due to the need for multiple iterations, the computational load is large, and if the initial conditions are not chosen properly, the method may not converge or converge too slowly.

[0004] With the rapid application of deep learning technology in multiple fields, in recent years, data-driven neural network methods have been used to estimate the motion state of satellites. This type of method no longer relies on accurate modeling of perturbation forces, but instead extracts the relationship between observations and satellite motion states from a large amount of data. However, neural network methods are sensitive to the quality of observation data and lack generalization ability, making it difficult to be widely applied in various complex environments. Therefore, further improvement and optimization of satellite orbit determination technology still has important practical significance.

[0005] Existing initial orbit determination techniques have some shortcomings in practical applications, mainly in the following aspects: (1) traditional algorithms have large computational load and slow convergence speed; (2) the selection of initial values is sensitive, which can easily affect the accuracy and convergence of the algorithm; (3) deep learning algorithms require high-quality data and are difficult to generalize in various scenarios. SUMMARY

[0006] The technical problem solved by the present application is to overcome the shortcomings of the prior art and provide an initial orbit determination method for satellites, which combines physical knowledge and a deep learning model to improve the efficiency and accuracy of orbit parameter estimation and achieve high-precision orbit determination of satellites.

[0007] The technical solution of the present application is an initial orbit determination method for satellites based on a neural network embedded with physical knowledge, comprising the following steps:

[0008] S1: Construct a decoder with a structure of a multi-layer perceptron neural network, the input of which is the absolute motion state quantity of the satellite (i.e. its position and velocity in the geocentric inertial coordinate system) and the latitude, longitude and altitude of the observer, and the output of which is the observation quantity of the satellite relative to the observer, the mean square error between the predicted observation value and the true observation value being used as the loss function to train the neural network. After the training process is completed, the structure and parameters of the decoder network are fixed;

[0009] S2: Construct an encoder with a structure of a neural network of transformer structure, the input of which is the observation quantity of the satellite relative to the observer, the observation time and the latitude, longitude and altitude of the observer, and the output of which is the absolute motion state quantity of the satellite (i.e. its position and velocity in the geocentric inertial coordinate system);

[0010] S3: Connect the encoder in S2 and the decoder with fixed structure and parameters in sequence, i.e. the output of the encoder is used as the input of the decoder, to construct an encoder-decoder network for orbit determination;

[0011] S4: Calculate the first derivative of the absolute state quantity of the satellite relative to the observation time based on the absolute state quantity of the satellite predicted by the encoder-decoder network constructed in S3, and calculate the velocity term and acceleration term physical information loss according to the velocity equation and acceleration equation;

[0012] S5: Add the physical information term loss calculated in S4 to the mean square error of the observation quantity to construct a total loss function. In the training phase, the network performs error backpropagation to optimize the network parameters of the encoder, and in the inference phase, the absolute motion state quantity of the satellite predicted by the encoder is converted into the form of orbit elements.

[0013] Further, the specific implementation method of S1 is as follows:

[0014] The data is arranged into the format of the input and output of the decoder, the input including the position parameters of the observer: longitude lat, latitude lon and altitude alt, and the position and velocity of the satellite in the geocentric inertial coordinate system:

[0015] r = [x, y, z]

[0016] v = [v x ,v y ,vz ]

[0017] The output is the observation of the satellite relative to the observer, which can be expressed by the observation angle, i.e., the elevation angle e and the azimuth angle a, under the optical observation condition.

[0018] A multi-layer perception neural network is constructed as a decoder. Specifically, the decoder includes a plurality of parameter-shared and structure-identical sub-networks, the number of the sub-networks being the same as the number of the measurement data points in the observation arc segment, each sub-network decoding a single measurement data point, and the parameter sharing reducing the size of the parameters.

[0019] The loss function of the decoder is the mean square error between the predicted observation angle and the true observation angle:

[0020]

[0021] wherein e and e i are the network-predicted elevation angle value and the true observed elevation angle value of the i-th observation data point, respectively, and a i are the network-predicted azimuth angle value and the true observed azimuth angle value of the i-th observation data point, respectively, and n is the number of the measurement data points in the arc segment.

[0022] The error is back-propagated to optimize the network parameters of the decoder.

[0023] Further, the specific implementation method of S2 is as follows:

[0024] A transformer structure neural network is constructed as an encoder to capture the relationship between the sequence measurement data and predict the motion state. The input is the elevation angle e and the azimuth angle a of the satellite relative to the observer, the observation time t, and the longitude lat, the latitude lon, and the altitude alt of the observer, and the output is the position and velocity of the satellite in the Earth-Centered Inertial coordinate system, and both the input and the output are the sequence data in the observation arc segment:

[0025] r=[x,y,z]

[0026] v=[v x ,v y ,v z ]

[0027] Further, the specific implementation method of S3 is as follows:

[0028] The encoder in S2 and the decoder with fixed structure and parameters in S1 are connected in turn, that is, the output of the encoder is used as the input of the decoder. A path is added between the input and output of the encoder, and the longitude lat, the latitude Ion and the altitude alt of the observer are used as part of the output of the encoder. A path is added between the input and output of the decoder, and the position r and the speed v of the satellite in the Earth-Centered Inertial coordinate system are used as part of the output of the decoder, to construct an encoder-decoder network for orbit determination. The input of the encoder-decoder network is the pitch angle e and the direction angle a of the satellite relative to the observer, the observation time t, the longitude lat, the latitude Ion and the altitude alt of the observer, and the output is the position r and the speed v of the satellite in the Earth-Centered Inertial coordinate system, and the pitch angle e and the direction angle a of the satellite relative to the observer.

[0029] Further, the specific implementation method of S4 is as follows:

[0030] The first-order derivative of the position r and the speed v of the satellite relative to the observation time t is calculated according to the absolute state quantity of the satellite predicted by the encoder-decoder network constructed according to S3, and according to the dynamics equation of the satellite under the influence of the Earth center gravity and the J2 perturbation force:

[0031]

[0032]

[0033] Where r is the size of the position r, μ is the gravitational constant, J2 is the J2 perturbation coefficient, R e is the radius of the Earth.

[0034] Further, the specific implementation method of S5 is as follows:

[0035] According to the first-order derivative equation of the position r and the speed v of the satellite relative to the observation time t in S4, the satellite position r and speed v predicted by the neural network should satisfy the above speed equation and acceleration equation, so the physical information item loss is constructed according to the equation to constrain the network output. The form of the speed item loss is:

[0036]

[0037] Where and are the position components of the satellite in the Earth-Centered Inertial coordinate system predicted by the network, and are the speed components of the satellite in the Earth-Centered Inertial coordinate system predicted by the network. The loss form of the acceleration item is:

[0038]

[0039] The total loss of the network contains the mean square error loss between the predicted observation and the true observation, the velocity term loss and the acceleration term loss:

[0040] L = λ d L d + λ v L v + λ a L a

[0041] where λ d , λ v and λ a are the weights of the three loss terms, respectively.

[0042] In the training process, the network structure and weights of the decoder are kept unchanged, and the network parameters of the encoder are updated.

[0043] In the inference stage, the absolute motion state quantity of the satellite predicted by the encoder is converted into the form of orbital elements:

[0044] The size of the geocentric distance r is calculated:

[0045] The size of the velocity v is calculated:

[0046] The total energy is calculated:

[0047] The semi-major axis a is calculated:

[0048] The component matrix of the momentum moment H = r x v is calculated:

[0049] The modulus H of the momentum moment is calculated:

[0050] The semi-latus rectum p is calculated:

[0051] The eccentricity e is calculated:

[0052] The component matrix of the ascending node vector b = k i x H is calculated:

[0053]

[0054] where k i is the unit vector of the z i axis, and

[0055] (k i ) i = (0, 0, 1) T

[0056] eccentricity vector column array of components

[0057]

[0058] calculate the orbital inclination i:

[0059] calculate the ascending node right ascension Omega:

[0060] calculate the argument of perigee omega:

[0061] calculate the argument of latitude u:

[0062] calculate the true anomaly theta: theta = u - omega

[0063] calculate the eccentric anomaly E:

[0064] calculate the mean anomaly M: M = E - esinE

[0065] The advantages of the present application compared with the prior art are:

[0066] (1) The present application uses the learning and reasoning ability of neural network, reduces the cumbersome calculation amount in traditional algorithm, and can estimate the orbit parameters more quickly.

[0067] (2) The present application enhances the physical consistency of the model by introducing the physical information loss term in satellite orbit dynamics. This method ensures the close connection between the network output and the physical equation, thereby significantly improving the prediction accuracy of the satellite motion state.

[0068] (3) The present application adopts an encoder-decoder architecture and optimizes the loss by combining physical information and observation errors, so that the model not only has strong learning ability, but also has good generalization ability, and is suitable for different types of satellites and observation environments.

[0069] (4) Compared with the traditional orbit determination method, the neural network method of the present application does not depend on high-precision initial state modeling, can flexibly handle complex orbit environment and nonlinear perturbation effects, and makes its application scenarios more extensive.

[0070] In summary, the method adopted by the present application is simple in principle and can achieve the purpose of quickly and accurately determining the satellite motion state and orbit parameters. BRIEF DESCRIPTION OF DRAWINGS

[0071] Figure 1 is a flow chart of the satellite initial orbit determination method of the present application embedded with physical knowledge neural network.

[0072] Figure 2 The flow chart of the training phase of the satellite initial orbit determination method embedded with a physical knowledge neural network.

[0073] Figure 3 The flow chart of the inference phase of the satellite initial orbit determination method embedded with a physical knowledge neural network.

[0074] Figure 4 The encoder and decoder network structure diagram of the satellite initial orbit determination method embedded with a physical knowledge neural network. DETAILED DESCRIPTION

[0075] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative effort belong to the protection scope of the present application.

[0076] As shown in the satellite initial orbit determination method embedded with a physical knowledge neural network, the specific implementation steps are as follows: Figures 1-4

[0077] S1: The data is arranged into the format of the decoder input and output, the input contains the observer position parameters: longitude lat, latitude lon, and altitude alt, and the position and velocity of the satellite in the Earth-Centered Inertial coordinate system:

[0078] r = [x, y, z]

[0079] v = [v x x, v y y, v z z]

[0080] The output is the observation of the satellite relative to the observer, which can be expressed by the observation angle, i.e., the pitch angle e and the direction angle a under the optical observation condition.

[0081] A multi-layer perceptron neural network is constructed as the decoder. Specifically, the decoder contains multiple parameter-shared and structure-identical sub-networks, the number of sub-networks is the same as the number of measurement data points in the observation arc segment, each sub-network decodes a single measurement data point, and the parameter sharing can reduce the scale of parameters.

[0082] The loss function of the decoder is the mean square error between the predicted observation angle and the true observation angle:

[0083]

[0084] ​wherein, and e i are the network predicted and the real observed pitch angle value of the i th observation data point, respectively, and a i are the network predicted and the real observed azimuth angle value of the i th observation point, respectively, n is the number of measurement data points in the arc segment;

[0085] The error is back-propagated to optimize the network parameters of the decoder.

[0086] S2: A transformer structure neural network is constructed as an encoder to capture the relationship between the sequence measurement data and predict the motion state. The input is the satellite relative to the observer's pitch angle e and azimuth angle a, observation time t, and the observer's longitude lat, latitude lon and altitude alt, and the output is the satellite position and velocity in the Earth-Centered Inertial coordinate system, and both the input and the output are the sequence data in the observation arc segment:

[0087] r = [x, y, z]

[0088] v = [v x , v y , v z ]

[0089] S3: The encoder in S2 and the decoder with fixed structure and parameters trained in S1 are connected in turn, that is, the output of the encoder is used as the input of the decoder. A path is added between the input and output of the encoder, and the observer's longitude lat, latitude lon and altitude alt are used as part of the encoder output. A path is added between the input and output of the decoder, and the satellite position r and velocity v in the Earth-Centered Inertial coordinate system are used as part of the decoder output, to construct an encoder-decoder network for orbit determination. The input of the encoder-decoder network is the satellite relative to the observer's pitch angle e and azimuth angle a, observation time t, and the observer's longitude lat, latitude lon and altitude alt, and the output is the satellite position r and velocity v in the Earth-Centered Inertial coordinate system and the satellite relative to the observer's pitch angle e and azimuth angle a.

[0090] S4: According to the satellite absolute state quantity predicted by the encoder-decoder network constructed in S3, the first derivative of the satellite position r and velocity v with respect to the observation time t is calculated, and according to the dynamics equation of the satellite under the influence of the Earth's center gravity and J2 perturbation force:

[0091]

[0092] wherein r is the size of the position r, μ is the gravitational constant, J2 is the J2 perturbation coefficient, R e is the radius of the Earth.

[0093] S5: According to the position r and the first derivative of velocity v with respect to observation time t of the satellite in S4, the satellite position r and velocity v predicted by the neural network should satisfy the above velocity equation and acceleration equation, so the physical information item loss is constructed according to the equation to constrain the network output, and the form of the velocity item loss is:

[0094]

[0095] where and are the position components of the satellite predicted by the network in the geocentric inertial coordinate system, and are the velocity components of the satellite predicted by the network in the geocentric inertial coordinate system, and the loss form of the acceleration item is:

[0096]

[0097] The total loss of the network contains the mean square error loss between the predicted observation value and the true observation value, the velocity item loss and the acceleration item loss:

[0098] L = λ d L d + λ v L v + λ a L a

[0099] where λ d , λ v and λ a are the weights of the three loss items,

[0100] In the training process, the network structure and weights of the decoder are kept unchanged, and the network parameters of the encoder are updated.

[0101] In the inference stage, the absolute motion state quantity of the satellite predicted by the encoder is converted into the form of orbital elements:

[0102] The geocentric distance r is calculated:

[0103] The velocity v is calculated:

[0104] The total energy is calculated:

[0105] The semi-major axis a is calculated:

[0106] The component array of the momentum moment H = r x v is calculated:

[0107] The modulus H of the momentum moment is calculated:

[0108] Calculate semi-major axis p:

[0109] Calculate eccentricity e:

[0110] Calculate the components of the ascending node vector b = k i × H:

[0111]

[0112] where k i is the unit vector along the z i axis, and

[0113] (k i ) i = (0, 0, 1) T

[0114] Calculate the components of the eccentricity vector e

[0115]

[0116] Calculate the inclination i:

[0117] Calculate the longitude of the ascending node Ω:

[0118] Calculate the argument of perigee ω:

[0119] Calculate the argument of latitude u:

[0120] Calculate the true anomaly θ: θ = u - ω

[0121] Calculate the eccentric anomaly E:

[0122] Calculate the mean anomaly M: M = E - esin E

[0123] The details of the application not described in the specification are within the knowledge of the person skilled in the art. Although the above describes the specific embodiments of the application in order to facilitate the understanding of the application by the person skilled in the art, it should be clear that the application is not limited to the scope of the specific embodiments, and for the person skilled in the art, any changes that are obvious within the spirit and scope of the application as defined and determined by the claims attached are obvious, and all the applications that utilize the concept of the application are within the scope of protection.​

Claims

1. A satellite initial orbit determination method embedded with a physical knowledge neural network, characterized in that, The method comprises the following steps: S1: Construct a decoder with a structure of a multi-layer perception neural network, input of absolute motion state quantity of a satellite (i.e. position and velocity of the satellite in an earth-centered inertial coordinate system) and longitude, latitude and altitude of an observer, and output of observation quantity of the satellite relative to the observer, take the mean square error between the predicted observation value and the true observation value as a loss function, and train the neural network; After the training process is completed, the structure and parameters of the decoder network are fixed; S2: Construct an encoder with a structure of a neural network of a transformer, input of observation quantity of the satellite relative to the observer, observation time and longitude, latitude and altitude of the observer, and output of absolute motion state quantity of the satellite (i.e. position and velocity of the satellite in the earth-centered inertial coordinate system); S3: sequentially connect the encoder in S2 and the decoder with the fixed structure and parameters, i.e. the output of the encoder is taken as the input of the decoder, to construct an encoder-decoder network for orbit determination; S4: calculate the first derivative of the absolute state quantity of the satellite relative to the observation time according to the absolute state quantity of the satellite predicted by the encoder-decoder network constructed in S3, and calculate the velocity term and the acceleration term physical information loss according to the velocity equation and the acceleration equation; S5: add the mean square error of the observation quantity to the calculation result of the physical information term loss in S4 to construct a total loss function, perform error back propagation in the training stage, optimize the network parameters of the encoder, and convert the absolute motion state quantity of the satellite predicted by the encoder into an orbit element form in the inference stage.

2. The satellite initial orbit determination method with the embedded physical knowledge neural network according to claim 1, wherein: The specific implementation method of S1 is as follows: The data is arranged into the format of the input and output of the decoder, the input includes the position parameters of the observer: longitude lat, latitude lon and altitude alt, and the position and velocity of the satellite in the earth-centered inertial coordinate system: r=[x, y, z] v = [v x ,v y ,v z ] The output is the observation quantity of the satellite relative to the observer, which can be represented by the observation angle quantity, i.e. the elevation angle ele and the direction angle azi under the condition of optical observation; A multi-layer perception neural network is constructed as the decoder, specifically, the decoder comprises a plurality of parameter-shared and structure-identical sub-networks, the number of the sub-networks is the same as the number of the measurement data points in the observation arc segment, each sub-network decodes a single measurement data point, and the scale of the parameters can be reduced through the parameter sharing mode; The loss function of the decoder is in the form of the mean square error between the predicted observation angle quantity and the true observation angle quantity: wherein, and e i are the network predicted and the true observed pitch angle values, respectively, for the i-th observation data point, and a i are the network predicted and the true observed azimuth angle values, respectively, for the i-th observation point, and n is the number of measurement data points in the arc segment. Error back propagation is performed to optimize the network parameters of the decoder.

3. The satellite initial orbit determination method with the embedded physical knowledge neural network according to claim 1, wherein: The specific implementation method of S2 is as follows: A transformer structure neural network is constructed as the encoder, which is used to capture the relationship between the sequence measurement data and predict the motion state; the input of the encoder is the elevation angle e and the direction angle a of the satellite relative to the observer, the observation time t and the longitude lat, the latitude lon and the altitude alt of the observer, and the output of the encoder is the position and velocity of the satellite in the earth-centered inertial coordinate system, and the input and the output are both the serialized data in the observation arc segment: r=[x, y, z] v = [v x ,v y ,v z ].

4. The satellite initial orbit determination method with the embedded physical knowledge neural network according to claim 1, wherein: The specific implementation method of S3 is as follows: The encoder in S2 and the decoder in S1 with fixed structure and parameters after training are connected in turn, that is, the output of the encoder is used as the input of the decoder, a path is added between the input and output of the encoder, the longitude lat, the latitude lon and the altitude alt of the observer are used as part of the output of the encoder, a path is added between the input and output of the decoder, the position r and the velocity v of the satellite in the Earth-Centered Inertial coordinate system are used as part of the output of the decoder, and an encoder-decoder network for orbit determination is constructed, wherein the input of the encoder-decoder network is the elevation angle e and the azimuth angle a of the satellite relative to the observer, the observation time t, the longitude lat, the latitude lon and the altitude alt of the observer, and the output of the encoder-decoder network is the position r and the velocity v of the satellite in the Earth-Centered Inertial coordinate system and the elevation angle e and the azimuth angle a of the satellite relative to the observer.

5. The satellite initial orbit determination method with the inlaid physical knowledge neural network according to claim 1, wherein: The specific implementation method of S4 is as follows: The first-order derivative of the position r and the velocity v of the satellite relative to the observation time t is calculated according to the absolute state quantity of the satellite predicted by the encoder-decoder network constructed according to S3, and according to the dynamic equation of the satellite under the influence of the gravity of the Earth center and the J2 perturbation force, it can be known that: where r is the magnitude of position r, μ is the gravitational constant, J2 is the J2 perturbation coefficient, R e is the radius of the Earth.

6. The satellite initial orbit determination method with an in-line physical knowledge neural network according to claim 1, wherein: The specific implementation method of S5 is as follows: According to the first-order derivative equation of the position r and the velocity v of the satellite relative to the observation time t calculated by S4, it can be known that the position r and the velocity v of the satellite predicted by the neural network should satisfy the above velocity equation and acceleration equation, and therefore a physical information item loss is constructed according to the equation to constrain the output of the network, wherein the form of the velocity item loss is: where and are the network-predicted position components of the satellite in the geocentric inertial coordinate system, and are the network-predicted velocity components of the satellite in the geocentric inertial coordinate system, the loss form of the acceleration term is: The total loss of the network includes the mean square error loss between the predicted observation value and the true observation value, the velocity item loss and the acceleration item loss: L = λ d L d + λ v L v + λ a L a where λ d , λ v , and λ a are the weights of the three losses, respectively, In the training process, the network structure and the weight of the decoder are kept unchanged, and the network parameters of the encoder are updated; In the reasoning stage, the absolute motion state quantity of the satellite predicted by the encoder is converted into the form of orbit elements; Computing the geocentric distance r size: The magnitude of the calculation speed v is: Total energy is calculated: Calculate half-length axis a: The component array of the angular momentum H = r x v is calculated: The module H of the angular momentum is calculated: Compute half radius p: Calculating the eccentricity e: Compute the ascending node vector b = k i The column matrix of components of H: where k i is the unit vector of the z i axis, and (k i ) i = (0,0,1) T eccentricity vector component column vector The orbital inclination i is calculated as: 0 ≤ i ≤ 180° Calculate the ascending node right ascension Ω: 0 ≤ Ω ≤ 360° Calculate the argument of perigee ω: Calculate the latitude amplitude u: The true anomaly angle θ is calculated: θ = u-ω Calculate the eccentricity E: The mean anomaly angle M is calculated: M = E-esinE.

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