Electric propulsion GEO satellite thrust descent fixed-point transfer reconstruction guidance method

Abstract

The invention discloses an electric thrust GEO satellite thrust decline fixed-point transfer reconstruction guidance method, which comprises the following steps: S1, updating an orbital transfer kinetic model of an electric thrust GEO satellite by adopting a volume Kalman filter (CKF) method to obtain the real-time estimated thrust of the electric thrust GEO satellite; s2, according to the real-time estimated thrust, a fault state function is adopted to judge whether an electric thrust descent fault occurs in the electric thrust GEO satellite, if yes, the step S3 is executed, and if not, the step S5 is executed; s3, determining whether reconstruction guidance is needed or not by adopting a reconstruction guidance judgment function constructed on the basis of the thrust reduction degree, the fault occurrence stage and the thrust abnormal and stable duration time, if so, entering step S4, and otherwise, entering step S5; s4, a fixed-point transfer reconstruction guidance method based on the Lyapunov stability theory is adopted, adaptive adjustment is conducted on an electric propulsion GEO satellite orbit under the thrust fault, and then the step S5 is executed; and S5, judging whether the electric propulsion GEO satellite reaches a target point or not, if so, terminating the algorithm, and otherwise, returning to the step S1.

Description

Technical Field

[0001] The present invention belongs to the technical field of satellite orbit dynamics, and in particular relates to a thrust descent fixed-point transfer and reconstruction guidance method for an electric propulsion GEO satellite. Background Art

[0002] Reconfiguration guidance for thrust descent fixed-point transfer of an electrically propelled GEO satellite primarily involves identifying the occurrence of a thrust descent failure caused by an abnormality in the electric thruster during the transfer from a low orbit to a geostationary orbit (GEO) and arrival at a designated orbital position. This involves using a specific state estimation and judgment algorithm to identify the fault and, based on the estimation results, replan and adjust the original orbit transfer guidance strategy to ensure the successful completion of the intended GEO fixed-point transfer mission. Thrust descent failure detection is a prerequisite for reconfiguration guidance. Only by accurately identifying the extent and moment of thrust descent can effective guidance strategy reconstruction be performed. Guidance reconfiguration is key to ensuring that an electrically propelled satellite can still complete its intended fixed-point transfer mission despite the fault.

[0003] In the field of thrust-down fault detection for electric propulsion satellites, existing research primarily estimates thrust magnitude based on orbital parameters and sets fault determination based on the thrust magnitude characteristics of the electric thruster system to achieve thrust-down fault detection. However, this research focuses on estimating the nominal thrust of satellites maneuvering in the orbital plane and is difficult to directly apply to the dynamic thrust estimation during the fixed-point transfer process of electric propulsion satellites. In the field of reconfiguration guidance for electric propulsion satellites, existing research focuses on optimizing the position retention of electric propulsion GEO satellites in the event of thrust failures after they enter orbit. However, for thrust-down faults during the orbit transfer phase, optimizing the electric thrust direction and the electric thruster switching sequence for orbital reconfiguration guidance is computationally intensive and has low robustness.

[0004] In order to ensure the smooth completion of the autonomous fixed-point transfer mission of the electric propulsion GEO satellite and avoid excessive extension of the fixed-point transfer time or additional fuel consumption due to thrust drop failure, it is urgent to propose a thrust drop failure fault detection and reconstruction guidance method during the electric propulsion GEO satellite fixed-point transfer process. Summary of the Invention

[0005] In view of the above-mentioned deficiencies in the prior art, the electric propulsion GEO satellite thrust descent fixed-point transfer reconstruction guidance method provided by the present invention solves the problems of excessive extension of the fixed-point transfer time or additional fuel consumption under the existing thrust descent failure.

[0006] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: A thrust descent fixed-point transfer and reconstruction guidance method for an electric propulsion GEO satellite is provided, which comprises the following steps: S1. Using the cubature Kalman filter method (CKF) to update the orbit transfer dynamics model of the electric thrust GEO satellite, the real-time estimated thrust of the electric thrust GEO satellite is obtained. S2. Based on the real-time estimated thrust, a fault status function is used to determine whether the electric thrust GEO satellite has an electric thrust descent fault. If so, proceed to step S3; otherwise, proceed to step S5. S3. Determine whether guidance reconstruction is required using a guidance reconstruction decision function based on the thrust drop degree, the fault occurrence stage, and the duration of abnormal and stable thrust. If so, proceed to step S4; otherwise, proceed to step S5. S4. Adopting a fixed-point transfer reconstruction guidance method based on Lyapunov stability theory, the orbit of the electric propulsion GEO satellite under thrust failure is adaptively adjusted, and then proceeding to step S5.

[0007] S5. Determine whether the electric propulsion GEO satellite has reached the target point. If so, terminate the algorithm; otherwise, return to step S1.

[0008] Furthermore, the expression of the fault status function is: , , , in, is the fault status function, a value of 1 indicates the presence of a fault, and a value of 0 indicates the absence of a fault; is the deviation of the electric thrust at time t; is the thrust magnitude threshold; is the duration of thrust anomaly; is the abnormal duration threshold; is the real-time estimated thrust at time t; is the nominal thrust; The electric thrust magnitude deviation triggers the thrust magnitude threshold for the first time time point; To continuously exceed the thrust threshold The ending moment; and are all constants greater than 0; is the sampling period.

[0009] Furthermore, the expression of the reconstruction guidance decision function is: , , , in, is the reconstruction guidance decision function, a value of 1 indicates that the guidance needs to be reconstructed, and a value of 0 indicates that the guidance does not need to be reconstructed; is the fault status function; is the degree of thrust reduction; is the thrust reduction threshold; is the relative progress at the time of failure; is the task progress threshold; is the duration of relative stability after the thrust decreases; The time threshold for starting reconstruction guidance; is the deviation of the electric thrust at time t; is the nominal thrust; The time when the electric thruster begins to experience thrust reduction failure; The total estimated duration of the electric propulsion satellite's fixed-point transfer mission; is a constant greater than 0; is the sampling period; is the moment when the stability condition is first met; The moment of determining thrust reduction failure; is the estimated mean thrust within the sliding window; for The estimated mean thrust magnitude in the previous window; for and The absolute deviation of is the estimated thrust standard deviation; is the standard deviation threshold; is the duration; is a constant and is taken as 50 in this simulation.

[0010] Furthermore, the fault model of the electric thrust drop fault is: in, is the working state of the electric thruster; The time when the electric thruster begins to experience thrust reduction failure; It is the minimum working state of the electric thruster during thrust failure; The time it takes for the thrust of the electric thruster to drop to the minimum value; is the end time of work; t is the current working time; Furthermore, the orbital transfer dynamics model The expression is: Where r is the position of the electric propulsion GEO satellite; v is the velocity vector; m is the weight of the electric propulsion GEO satellite; T is the thrust of the electric propulsion GEO satellite; is the transpose of the matrix; Orbital transfer dynamics model Equation of state Expressed as: in, 、 、 and are the derivatives of satellite position, velocity, mass and thrust respectively; is the gravity acceleration vector; is the Earth’s J2 perturbation acceleration vector; is the solar gravitational acceleration vector; is the lunar gravitational acceleration vector; is the atmospheric drag acceleration vector; is the solar pressure perturbation acceleration vector; is the acceleration vector generated by the estimated electric thrust in the satellite orbit coordinate system; is the average gravitational acceleration at sea level; is the specific impulse of the electric thruster of the electric propulsion satellite; is the transformation matrix from the orbital coordinate system to the J2000 ECI coordinate system.

[0011] The beneficial effects of the above technical solution are as follows: This solution adopts the state equation of the orbit transfer dynamics model that includes thrust as a state quantity, which can more comprehensively and accurately describe the dynamic characteristics of the electric propulsion GEO satellite during the orbit transfer process. This state equation incorporates the satellite thrust into the state quantity for estimation, and can capture the dynamic changes of the thrust in real time. Especially in the electric thrust drop failure scenario, it can accurately reflect the impact of thrust anomalies on the satellite orbit. By introducing the thrust state quantity, the model not only takes into account external factors such as the earth's gravity and perturbation acceleration, but also realizes real-time tracking of the key control variable thrust, providing a precise dynamic basis for subsequent fault state function judgment, reconstruction guidance judgment and orbit adaptive adjustment based on Lyapunov stability theory.

[0012] Furthermore, the transformation matrix from the orbital coordinate system to the J2000 ECI coordinate system is The expression is: in, is the instantaneous angular momentum vector of the electrically propelled GEO satellite; 、 and are all unit vectors; is the mold length; Gravitational acceleration vector The expression is: in, is the Earth's gravitational constant; Solar gravitational acceleration The expression is: in, is the solar gravitational constant; is the position vector of the Sun in the J2000 ECI coordinate system; lunar gravitational acceleration The expression is: in, is the lunar gravitational constant; is the position vector of the moon in the J2000 ECI coordinate system; The expression of J2 perturbation acceleration is: in, 、 and is the vector component of the J2 perturbation acceleration; is the harmonic coefficient, ; x, y and z are the components of the position vector; is the mean radius of the Earth; Atmospheric drag acceleration vector The expression is: in, is the ratio of the effective area S to the mass m of the electric propulsion GEO satellite; is the atmospheric drag coefficient; is the functional relationship between atmospheric density and the altitude of the electric propulsion GEO satellite, and its expression is , , To electrically propel GEO satellite altitude, is the reference height; is the atmospheric elevation; is the velocity vector of the electrically propelled GEO satellite relative to the atmosphere, and its expression is: , is the Earth's rotation angular velocity vector; is the angular velocity of the Earth; Solar pressure perturbation acceleration vector The expression is: in, is the shadow coefficient. When the satellite is in the umbra, , when the satellite is in the penumbra, , when the satellite is outside the umbra and penumbra, ; is the SRP reference value when starting from the sun; is the reflection coefficient; The expression for estimating the acceleration vector generated by the electric thrust is: in, 、 and are radial, along-track, and normal acceleration components respectively; 、 is the optimal control thrust angle of electric thrust in orbital coordinate system.

[0013] Furthermore, step S1 further includes: S11. The position and velocity estimates of the electric thruster GEO satellite in the J2000 ECI coordinate system are used as the measurement values ​​of the CKF-based electric thruster real-time on-orbit thrust estimation method. The measurement equation is: Expressed as: in, are the estimated values ​​of the satellite's position vector and velocity vector in the J2000 ECI coordinate system; are the true position vector and velocity vector of the satellite in the J2000 ECI coordinate system; is the estimation accuracy; S12. Discretize the state space model of the orbital transfer dynamics model: in, and are the discretized state quantities at time k and time k+1 respectively; is the estimated value of the satellite's position vector and velocity vector in the J2000 ECI coordinate system at time k+1; is the state transfer function, which means predicting the satellite state at time k+1 from the state of the electric propulsion GEO satellite at time k; is the observation function; are the system noise at time k and the measurement noise at time k+1 respectively; S13, given initial state variables of the filter , the initial covariance matrix , process noise covariance matrix , the measurement noise covariance matrix ; S14. Calculate the predicted value of the state variable based on the initial state variable and the initial covariance matrix: , in, is the predicted value of the state variable at time k+1; and They are respectively volume points; n is the dimension of state variables; is the estimated value of the state variable at time k; is the Cholesky decomposition of the state variable covariance matrix at time k; is a matrix composed of unit vectors; S15. Predicting values ​​based on state variables and volume points , calculate the state prediction covariance matrix at time k+1 : S16, according to the volume point , calculate the measured predicted value at time k+1 : , in, The observation function is weighted and summed after propagation of volume points to obtain the measurement prediction value; S17. Predicting values ​​based on state variables , volume point and measured predicted values , update the measurement covariance matrix and cross covariance matrix: , in, and are the measurement covariance matrix and cross covariance matrix at time k+1 respectively; S18, according to the measurement covariance matrix and the cross-covariance matrix , calculate the Kalman gain: in, is the Kalman gain at time k+1; S19, according to 、 、 and , the estimated value of the state variable at time k+1 and the state covariance matrix To update: , ; Estimated value of the state variable at time k+1 The thrust estimate value of the electric propulsion GEO satellite in is the real-time estimated thrust of the electric propulsion GEO satellite.

[0014] The beneficial effects of the above technical solution are as follows: combining the state equation of the orbit transfer dynamics model with the cubature Kalman filter (CKF) method can fully leverage the CKF's advantages in accurately processing state estimation in nonlinear systems. Through discretization of the state equation and recursive iterative calculation, it achieves real-time, high-precision thrust estimation for electric propulsion GEO satellites. This method constructs volume points through Cholesky decomposition and performs state propagation, effectively capturing the nonlinear characteristics of satellite orbital dynamics while being highly robust to system noise and measurement noise. Incorporating thrust as a state variable in the state equation, combined with the CKF's recursive update mechanism, it can track the dynamic changes of thrust in real time under complex perturbation environments, providing a reliable data foundation for the precise detection of electric thrust descent faults. This ensures the accuracy and timeliness of subsequent fault state judgment, reconstruction guidance determination, and orbit adaptive adjustment, significantly improving the satellite's orbit estimation accuracy and mission execution reliability under abnormal thrust conditions.

[0015] Furthermore, the matrix composed of unit vectors The expression is: .

[0016] Furthermore, the step S4 further includes: S41: Determine the guidance mode as the shortest time guidance law or the limited time fuel consumption least guidance law according to the current mission requirements, and proceed to step S42 and step S43 respectively; S42. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: in, Maximum flight time for electric propulsion GEO satellite fixed-point transfer; The flight time for the orbital phase; The maximum allowable deviation of the terminal of the electric propulsion satellite fixed-point transfer; and are all constants, Take 0.97, Take 0.01; S43. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: ; S44: Output the optimal control thrust angle under thrust drop fault according to the current state variables of the electric propulsion satellite, the current orbit information of the target point, and the optimized weight coefficient. : , Where D1, D2 and D3 are auxiliary variables respectively; Q is the RQ-Law function; is the orbital element number of the electric propulsion GEO satellite; is the maximum rate of change of the eccentricity component f with respect to the thrust direction; is the maximum rate of change of the eccentricity component g with respect to the thrust direction; is the maximum rate of change of the five orbital elements of an electric propulsion satellite with respect to the thrust direction and true longitude; is the acceleration vector in the orbital coordinate system; is the true longitude of the GEO satellite; is an auxiliary variable; a is the right ascension of the ascending node; f and g are the eccentricity vector components; 、 is the orbital inclination vector component; L is the true longitude; is the Earth's gravitational constant; S45, optimal control of thrust angle under thrust drop fault , adaptively adjust the orbit of electric propulsion GEO satellite.

[0017] The beneficial effects of the above technical solution are as follows: by constructing a flexible guidance mode selection mechanism and optimization strategy, efficient adaptive guidance of the electric propulsion GEO satellite orbit under thrust-down failure is achieved. This solution switches between the "shortest time" or "limited time and fuel-saving" guidance law according to mission requirements. By constructing an optimal performance indicator that includes constraints such as orbit insertion deviation, the RQ-Law function weight parameters are optimized, enabling the system to balance time efficiency and fuel consumption under different operating conditions. The optimal control thrust angle is calculated in real time based on the satellite's current state and target orbit information, fully considering dynamic parameters such as the orbital root change rate and acceleration vector to ensure the accuracy of thrust direction control. Finally, through an adaptive adjustment mechanism, the orbit is dynamically corrected in the event of a thrust-down failure, effectively avoiding the problems of excessively extended fixed-point transfer time or additional fuel consumption in traditional methods. This significantly improves the satellite's mission adaptability and guidance accuracy under failure conditions, and achieves multi-objective optimization of time, fuel, and orbit control accuracy.

[0018] Furthermore, the initial state constraints include: Initial state constraints: Among them, t0 is the initial time, a C (t0) is the semi-major axis of the orbit at the initial moment, f C (t0) is the vector of orbital eccentricity at the initial moment, g C (t0) is the component of orbital eccentricity at the initial moment, h C (t0) is the vector of the orbital inclination at the initial moment, k C (t0) is the component of the orbital inclination at the initial moment, L C (t0) is the true longitude at the initial moment, a0, f0, g0, h0, k0, L0 are the orbital elements of the initial orbit; End constraints: Among them, t f is the transfer time; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the GEO satellite respectively; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the target point respectively; Equations of motion constraints: in, 、 、 、 、 、 、 are the semi-major axis, eccentricity components f and g, orbital inclination components h and k, true longitude, and mass change rate; is the right ascension of the ascending node; p is the semi-diameter; is the specific impulse of the electric thruster of the electric propulsion satellite.

[0019] Furthermore, the expression for optimizing the weight coefficient is: in, and are all relevant weights of the penalty function; are the weights of the semi-major axis, eccentricity components f, g, and orbital inclination components h, k, respectively; and is the phase parameter; , and are the associated scaling weights; The expression of the Lypunov function Q is: in, are penalty functions, The smaller it is, the shorter the time it takes to complete orbit transfer and positioning; is the scaling function; is the weight corresponding to the improved vernal equinox orbital elements; is the orbital element number of the electric propulsion GEO satellite; is the augmented set of target orbital elements; It is the maximum rate of change of each orbital element of an electrically propulsed GEO satellite relative to the thrust direction and true longitude.

[0020] The beneficial effects of the present invention are: in order to avoid excessive extension of the fixed-point transfer time and additional fuel consumption caused by the thrust descent failure of the electric propulsion GEO satellite, the fixed-point transfer reconstruction guidance method designed in this scheme covers three links: thrust size estimation, fault descent fault judgment and guidance reconstruction. This method uses the CKF filtering algorithm to estimate the thrust size online and formulates a two-factor judgment strategy, that is, by analyzing the dynamic deviation characteristics of the thrust estimation value and the nominal thrust and the continuous deviation duration, the thrust descent fault is judged, so as to accurately judge the thrust descent fault and improve the reliability of fault identification; at the same time, unnecessary calculations and guidance law adjustments are avoided by reconstructing the guidance judgment.

[0021] This approach, through real-time thrust estimation, fault diagnosis, and guidance reconfiguration, avoids excessive extension of fixed-point transfer time or fuel consumption, ensuring the successful completion of autonomous fixed-point transfer missions for electrically propulsed GEO satellites. This method accurately estimates the actual electric thrust, providing a basis for determining thrust-down faults. By comprehensively considering engineering constraints such as orbit determination accuracy, attitude control precision, and thrust fluctuations, this method significantly reduces the increase in flight time and fuel consumption caused by thrust-down faults and enhances the robustness of the fixed-point transfer process. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 Flowchart of the reconstructed guidance method for thrust descent fixed-point transfer of an electrically propulsed GEO satellite.

[0023] Figure 2 Schematic diagram of the thrust reduction fault model of the electric thruster; among them, (a) is a schematic diagram of thrust reduction fault mode one, (b) is a schematic diagram of thrust reduction fault mode two, (c) is a schematic diagram of thrust reduction fault mode three, and (d) is a schematic diagram of thrust reduction fault mode four.

[0024] Figure 3 Schematic diagram of the error in estimating the magnitude of the electric thrust.

[0025] Figure 4 Schematic diagram of thrust reduction fault judgment results.

[0026] Figure 5 Schematic diagram of the degree of electric thrust reduction.

[0027] Figure 6 Schematic diagram of reconstructed guidance judgment results.

[0028] Figure 7 Schematic diagram of a portion of the results of the fixed-point transfer of an electric propulsion satellite under thrust reduction failure; including: (a) 3D trajectory without reconstruction guidance, (b) 3D trajectory with reconstruction guidance, (c) schematic diagram of the change of the semi-major axis, (d) schematic diagram of the change of the eccentricity, (e) schematic diagram of the change of the orbital inclination, and (f) schematic diagram of the change of the total mass of the satellite; Figure 8 Another partial schematic diagram of the results of the electric propulsion satellite's fixed-point transfer under a thrust drop failure; among them, (g) a schematic diagram of the in-plane control thrust angle α when no reconstruction is performed; (h) a schematic diagram of the out-of-plane control thrust angle β when no reconstruction is performed; (i) a schematic diagram of the in-plane control thrust angle α after reconstruction; (j) a schematic diagram of the out-plane control thrust angle β after reconstruction. DETAILED DESCRIPTION

[0029] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0030] refer to Figure 1 , Figure 1 FIG. 1 shows a flow chart of a thrust descent fixed-point transfer and reconstruction guidance method for an electric propulsion GEO satellite; FIG. Figure 1 As shown, the method S includes steps S1 to S4.

[0031] In step S1, the orbit transfer dynamics model of the electric thrust GEO satellite is updated using the cubature Kalman filter method CKF to obtain the real-time estimated thrust of the electric thrust GEO satellite; In one embodiment of the present invention, the thrust generated by the electric thruster is As a state variable modeled in the orbital dynamics model, the state variable is the orbital transfer dynamics model The expression is: Where r is the position of the electric propulsion GEO satellite; v is the velocity vector; m is the weight of the electric propulsion GEO satellite; T is the thrust of the electric propulsion GEO satellite; is the transpose of the matrix; Orbital transfer dynamics model Equation of state Expressed as: in, 、 、 and are the derivatives of satellite position, velocity, mass and thrust respectively; is the gravity acceleration vector; is the Earth’s J2 perturbation acceleration vector; is the solar gravitational acceleration vector; is the lunar gravitational acceleration vector; is the atmospheric drag acceleration vector; is the solar pressure perturbation acceleration vector; is the acceleration vector generated by the estimated electric thrust in the satellite orbit coordinate system; is the average gravitational acceleration at sea level; is the specific impulse of the electric thruster of the electric propulsion satellite; is the transformation matrix from the orbital coordinate system to the J2000 ECI coordinate system.

[0032] The state equation of this scheme The expressions of the parameters are as follows: Transformation matrix from orbital coordinate system to J2000 ECI coordinate system The expression is: in, is the instantaneous angular momentum vector of the electrically propelled GEO satellite; 、 and are all unit vectors; is the mold length; Gravitational acceleration vector The expression is: in, is the Earth's gravitational constant; Solar gravitational acceleration The expression is: in, is the solar gravitational constant; is the position vector of the Sun in the J2000 ECI coordinate system; lunar gravitational acceleration The expression is: in, is the lunar gravitational constant; is the position vector of the moon in the J2000 ECI coordinate system; The expression of J2 perturbation acceleration is: in, 、 and is the vector component of the J2 perturbation acceleration; is the harmonic coefficient, ; x, y and z are the components of the position vector; is the mean radius of the Earth; Atmospheric drag acceleration vector The expression is: in, is the ratio of the effective area S to the mass m of the electric propulsion GEO satellite; is the atmospheric drag coefficient; is the functional relationship between atmospheric density and the altitude of the electric propulsion GEO satellite, and its expression is , , To electrically propel GEO satellite altitude, is the reference height; is the atmospheric elevation; is the velocity vector of the electrically propelled GEO satellite relative to the atmosphere, and its expression is: , is the Earth's rotation angular velocity vector; is the angular velocity of the Earth; Solar pressure perturbation acceleration vector The expression is: in, is the shadow coefficient. When the satellite is in the umbra, , when the satellite is in the penumbra, , when the satellite is outside the umbra and penumbra, ; is the SRP reference value when starting from the sun; is the reflection coefficient; The expression for estimating the acceleration vector generated by the electric thrust is: in, 、 and are radial, along-track, and normal acceleration components respectively; 、 is the optimal control thrust angle of electric thrust in orbital coordinate system.

[0033] In one embodiment of the present invention, step S1 further comprises: S11. The position and velocity estimates of the electric thruster GEO satellite in the J2000 ECI coordinate system are used as the measurement values ​​of the CKF-based electric thruster real-time on-orbit thrust estimation method. The measurement equation is: Expressed as: in, are the estimated values ​​of the satellite's position vector and velocity vector in the J2000 ECI coordinate system; are the true position vector and velocity vector of the satellite in the J2000 ECI coordinate system; is the estimation accuracy; S12. Discretize the state space model of the orbital transfer dynamics model: in, and are the discretized state quantities at time k and time k+1 respectively; is the estimated value of the satellite's position vector and velocity vector in the J2000 ECI coordinate system at time k+1; is the state transfer function, which means predicting the satellite state at time k+1 from the state of the electric propulsion GEO satellite at time k; is the observation function; are the system noise at time k and the measurement noise at time k+1 respectively; S13, given initial state variables of the filter , the initial covariance matrix , process noise covariance matrix , the measurement noise covariance matrix ; S14. Calculate the predicted value of the state variable based on the initial state variable and the initial covariance matrix: , in, is the predicted value of the state variable at time k+1; and They are respectively volume points; n is the dimension of state variables; is the estimated value of the state variable at time k; is the Cholesky decomposition of the state variable covariance matrix at time k; a matrix composed of unit vectors The expression is: .

[0034] S15. Predicting values ​​based on state variables and volume points , calculate the state prediction covariance matrix at time k+1 : S16, according to the volume point , calculate the measured predicted value at time k+1 : , in, The observation function is weighted and summed after propagation of volume points to obtain the measurement prediction value; S17. Predicting values ​​based on state variables , volume point and measured predicted values , update the measurement covariance matrix and cross covariance matrix: , in, and are the measurement covariance matrix and cross covariance matrix at time k+1 respectively; S18, according to the measurement covariance matrix and the cross-covariance matrix , calculate the Kalman gain: in, is the Kalman gain at time k+1; S19, according to 、 、 and , the estimated value of the state variable at time k+1 and the state covariance matrix To update: , ; Estimated value of the state variable at time k+1 The thrust estimate value of the electric propulsion GEO satellite in is the real-time estimated thrust of the electric propulsion GEO satellite.

[0035] According to existing research, there are four main failure modes of thrust reduction failure caused by electric thrusters. The thrust reduction failure diagram is as follows: Figure 2 As shown in Figure 2, the four failure modes are: Thrust reduction failure mode 1 Figure 2 Middle (a): The working state of the electric thruster suddenly changes from 100% to 0. Its fault model can be expressed as follows: .

[0036] Thrust reduction failure mode 2 Figure 2 Middle (b): The working state of the electric thruster gradually decreases from 100% to 0. Its fault model can be expressed as follows: .

[0037] Thrust reduction failure mode three Figure 2 Middle (c): In this mode, the thruster working state drops from full state to a certain value, and then recovers from this value to 100%. Its fault model can be expressed as follows: in, It is the minimum working state of the electric thruster during thrust failure; The time it takes for the thrust of the electric thruster to drop to the minimum value; The thrust recovery time of the electric thruster.

[0038] Four thrust reduction failure modes Figure 2 Middle (d): In this mode, the thruster working state drops from full state to a certain value and remains unchanged. Its fault model can be expressed as follows: .

[0039] In this scheme, the electric propulsion GEO satellite thrust descent fixed-point transfer reconstruction guidance method of this scheme is mainly applicable to thrust descent failure mode four.

[0040] In step S2, based on the real-time estimated thrust, a fault status function is used to determine whether the electric thrust GEO satellite has an electric thrust descent fault. If so, the process proceeds to step S3; otherwise, the process proceeds to step S5. During implementation, the preferred expression of the fault status function in this solution is: , , , in, is the fault status function, a value of 1 indicates the presence of a fault, and a value of 0 indicates the absence of a fault; is the deviation of the electric thrust at time t; is the thrust magnitude threshold; is the duration of thrust anomaly; is the abnormal duration threshold; is the real-time estimated thrust at time t; is the nominal thrust; The electric thrust magnitude deviation triggers the thrust magnitude threshold for the first time time point; To continuously exceed the thrust threshold The ending moment; and are all constants greater than 0; is the sampling period.

[0041] In step S3, a reconstruction guidance decision function constructed based on the thrust drop degree, the fault occurrence stage, and the duration of abnormal and stable thrust is used to determine whether reconstruction guidance is needed. If so, proceed to step S4; otherwise, proceed to step S5.

[0042] In one embodiment of the present invention, the expression of the reconstructed guidance decision function is: , , , in, is the reconstruction guidance decision function, a value of 1 indicates that the guidance needs to be reconstructed, and a value of 0 indicates that the guidance does not need to be reconstructed; is the fault status function; is the degree of thrust reduction; is the thrust reduction threshold; The closer it is to 1, the failure occurred in the early stage of the task, and the closer it is to 0, the failure occurred in the later stage of the task; is the task progress threshold; is the duration of relative stability after the thrust decreases; The time threshold for starting reconstruction guidance; is the deviation of the electric thrust at time t; is the nominal thrust; The time when the electric thruster begins to experience thrust reduction failure; The total estimated duration of the electric propulsion satellite's fixed-point transfer mission; is a constant greater than 0; is the sampling period; is the moment when the stability condition is first met; The moment of determining thrust reduction failure; is the estimated mean thrust within the sliding window; for The estimated mean thrust magnitude in the previous window; for and The absolute deviation of is the estimated thrust standard deviation; is the standard deviation threshold; is the duration; is a constant and is taken as 50 in this simulation.

[0043] By accurately identifying thrust reduction failures through real-time estimation of electric thrust and the subsequent identification of thrust reduction failures, reconfiguration guidance becomes crucial for ensuring that electric propulsion satellites successfully complete their designated orbit transfer and positioning missions. A thrust reduction failure in an electric thruster derails the orbit transfer and positioning phase, and different degrees of thrust reduction have varying impacts on the positioning transfer mission. When the impact of a thrust reduction failure on the positioning transfer mission is minimal, reconfiguration guidance can be omitted to avoid unnecessary calculations and guidance law adjustments, thereby reducing system burden. Conversely, when the impact of a thrust reduction failure is significant, reconfiguration guidance must be initiated to adjust the orbit control strategy and ensure the smooth completion of the positioning transfer mission. Therefore, it is necessary to establish a reasonable judgment criterion based on the degree of thrust reduction and the stage of the thrust reduction failure to determine whether to initiate reconfiguration guidance.

[0044] Since the electric thrust suddenly drops to 0 in only fault mode 1 among the electric thruster failure modes, the electric thrust in other failure modes decreases gradually. At the same time, the reconstruction guidance needs to input the electric thrust, which requires the stable electric thrust data. Therefore, it is necessary to introduce the relatively stable duration after the thrust decreases. As the basis for judgment.

[0045] In step S4, a fixed-point transfer reconstruction guidance method based on Lyapunov stability theory is used to adaptively adjust the orbit of the electric propulsion GEO satellite under thrust failure, and then the process proceeds to step S5.

[0046] In one embodiment of the present invention, step S4 further includes: S41: Determine the guidance mode as the shortest time guidance law or the limited time fuel consumption least guidance law according to the current mission requirements, and proceed to step S42 and step S43 respectively; S42. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: in, Maximum flight time for electric propulsion GEO satellite fixed-point transfer; The flight time for the orbital phase; The maximum allowable deviation of the terminal of the electric propulsion satellite fixed-point transfer; and are all constants, Take 0.97, Take 0.01; S43. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: ; S44: Output the optimal control thrust angle under thrust drop fault according to the current state variables of the electric propulsion satellite, the current orbit information of the target point, and the optimized weight coefficient. : , Where D1, D2 and D3 are auxiliary variables respectively; Q is the RQ-Law function; is the orbital element number of the electric propulsion GEO satellite; is the maximum rate of change of the eccentricity component f with respect to the thrust direction; is the maximum rate of change of the eccentricity component g with respect to the thrust direction; is the maximum rate of change of the five orbital elements of an electric propulsion satellite with respect to the thrust direction and true longitude; is the acceleration vector in the orbital coordinate system; is the true longitude of the GEO satellite; is an auxiliary variable; a is the right ascension of the ascending node; f and g are the eccentricity vector components; 、 is the orbital inclination vector component; L is the true longitude; is the Earth's gravitational constant; S45, optimal control of thrust angle under thrust drop fault , adaptively adjust the orbit of electric propulsion GEO satellite.

[0047] In this solution, the initial state constraints include three constraints: Initial state constraints: Among them, t0 is the initial time, a C (t0) is the semi-major axis of the orbit at the initial moment, f C (t0) is the vector of orbital eccentricity at the initial moment, g C (t0) is the component of orbital eccentricity at the initial moment, h C (t0) is the vector of the orbital inclination at the initial moment, k C (t0) is the component of the orbital inclination at the initial moment, LC (t0) is the true longitude at the initial moment, a0, f0, g0, h0, k0, L0 are the orbital elements of the initial orbit; End constraints: Among them, t f is the transfer time; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the GEO satellite respectively; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the target point respectively; Equations of motion constraints: in, 、 、 、 、 、 、 are the semi-major axis, eccentricity components f and g, orbital inclination components h and k, true longitude, and mass change rate; is the right ascension of the ascending node; p is the semi-diameter; is the specific impulse of the electric thruster of the electric propulsion satellite.

[0048] In step S42 and step S43, the expression of the optimization weight coefficient of this solution is: in, and are all relevant weights of the penalty function; are the weights of the semi-major axis, eccentricity components f, g, and orbital inclination components h, k, respectively; and is the phase parameter; , and are the associated scaling weights; The expression of the Lypunov function Q is: in, are penalty functions, The smaller it is, the shorter the time it takes to complete orbit transfer and positioning; is the scaling function; is the weight corresponding to the improved vernal equinox orbital elements; is the orbital element number of the electric propulsion GEO satellite; is the augmented set of target orbital elements; It is the maximum rate of change of each orbital element of an electrically propulsed GEO satellite relative to the thrust direction and true longitude.

[0049] In step S5, it is determined whether the electrically propulsed GEO satellite has reached the destination. If so, the algorithm is terminated, otherwise it returns to step S1.

[0050] To verify the effectiveness of the proposed fixed-point transfer and reconfiguration guidance method for an electrically propelled GEO satellite under thrust-drop failure conditions, numerical simulations were conducted using Python 3.8 on a desktop computer equipped with an Intel(R) Core(TM) i7-10700F CPU at 2.90 GHz and 16 GB of memory. The physical parameters of the electrically propelled satellite are shown in Tables 1 and 2.

[0051] Table 1 Initial state variables and initial errors of electric propulsion satellite Table 2 Physical parameters of electric propulsion satellite Aiming at the thrust drop failure situation of the electric thruster, combined with the analysis of different failure modes, a simulation analysis is carried out, taking failure mode 4 as the representative, which has the most significant impact on the fixed-point transfer mission and the cumulative effect cannot be ignored.

[0052] The simulation results are shown in Figure Figures 3 to 8 , Figure 3 This is the error curve of the estimated magnitude of the electric thrust; Figure 4 This is the result diagram of thrust reduction fault identification; Figure 5 and Figure 6 This is the result diagram of the GEO satellite electric thrust reduction degree and reconstruction guidance. When the reconstruction guidance judgment function is 1, guidance reconstruction is required; Figure 7 and Figure 8 This paper presents an integrated approach and segmented considerations for the change in satellite eccentricity, orbital inclination, and thrust angle for reconfiguration guidance of GEO satellites. Simulation results show that when an electric thruster thrust reduction failure occurs on an electrically propelled GEO satellite, the available thrust acceleration decreases, thereby increasing the time required to complete a designated fixed-point transfer mission. Without reconfiguration guidance, the time required for an electrically propelled GEO satellite to complete a fixed-point transfer mission is 270.06 days, an increase of 81.69 days compared to the case without the thrust reduction failure.

[0053] After implementing reconfigured guidance, the time required for an electrically propelled GEO satellite to complete a fixed-point transfer was reduced to 254.01 days, a reduction of 16.05 days compared to the unreconfigured case. Considering time as the primary performance metric, the electrically propelled satellite operated continuously throughout the entire transfer process, excluding restrictions imposed by the Earth's shadow, thereby minimizing fuel consumption while minimizing time.

[0054] pass Figures 3 to 8 The simulation results show that after the electric propulsion satellite is reconstructed and guided for fixed-point transfer, not only is the fixed-point transfer time shortened, but 22.99 kg of fuel is also saved, which can be further used for subsequent tasks such as position maintenance, thereby improving the overall execution efficiency and reliability of the mission.

Claims

1. A thrust descent fixed-point transfer and reconstruction guidance method for an electric propulsion GEO satellite, characterized in that: Including steps: S1. Using the cubature Kalman filter method (CKF) to update the orbit transfer dynamics model of the electric thrust GEO satellite, the real-time estimated thrust of the electric thrust GEO satellite is obtained. S2. Based on the real-time estimated thrust, a fault status function is used to determine whether the electric thrust GEO satellite has an electric thrust descent fault. If so, proceed to step S3; otherwise, proceed to step S5. S3. Determine whether guidance reconstruction is required using a guidance reconstruction decision function based on the degree of thrust reduction, the stage of fault occurrence, and the duration of abnormal and stable thrust. If so, proceed to step S4; otherwise, proceed to step S5. S4, using a fixed-point transfer reconstruction guidance method based on Lyapunov stability theory to adaptively adjust the orbit of the electric propulsion GEO satellite under thrust failure, and then entering step S5; S5. Determine whether the electric propulsion GEO satellite has reached the target point. If so, terminate the algorithm; otherwise, return to step S1.

2. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 1 is characterized in that: The expression of the fault status function is: ， ， ， in, is the fault status function, a value of 1 indicates the presence of a fault, and a value of 0 indicates the absence of a fault; is the deviation of the electric thrust at time t; is the thrust magnitude threshold; is the duration of thrust anomaly; is the abnormal duration threshold; is the real-time estimated thrust at time t; is the nominal thrust; The electric thrust magnitude deviation triggers the thrust magnitude threshold for the first time time point; To continuously exceed the thrust threshold The ending moment; and are all constants greater than 0; is the sampling period.

3. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 1 is characterized in that: The expression of the reconstructed guidance decision function is: ， ， ， in, is the reconstruction guidance decision function, a value of 1 indicates that the guidance needs to be reconstructed, and a value of 0 indicates that the guidance does not need to be reconstructed; is the fault status function; is the degree of thrust reduction; is the thrust reduction threshold; is the relative progress at the time of failure; is the task progress threshold; is the duration of relative stability after the thrust decreases; The time threshold for starting reconstruction guidance; is the deviation of the electric thrust at time t; is the nominal thrust; The time when the electric thruster begins to experience thrust reduction failure; The total estimated duration of the electric propulsion satellite's fixed-point transfer mission; is a constant greater than 0; is the sampling period; is the moment when the stability condition is first met; The moment of determining thrust reduction failure; is the estimated mean thrust within the sliding window; for The estimated mean thrust magnitude in the previous window; for and The absolute deviation of is the estimated thrust standard deviation; is the standard deviation threshold; is the duration; It is a constant and takes the value of 50.

4. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to any one of claims 1 to 3, characterized in that: The fault model of the electric thrust drop fault is: in, is the working state of the electric thruster; The time when the electric thruster begins to experience thrust reduction failure; It is the minimum working state of the electric thruster during thrust failure; The time it takes for the thrust of the electric thruster to drop to the minimum value; is the end time of work; t is the current working time.

5. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to any one of claims 1 to 3, characterized in that: The orbital transfer dynamics model The expression is: Where r is the position of the electric propulsion GEO satellite; v is the velocity vector; m is the weight of the electric propulsion GEO satellite; T is the thrust of the electric propulsion GEO satellite; is the transpose of the matrix; Orbital transfer dynamics model Equation of state Expressed as: in, 、 、 and are the derivatives of satellite position, velocity, mass and thrust respectively; is the gravity acceleration vector; is the Earth’s J2 perturbation acceleration vector; is the solar gravitational acceleration vector; is the lunar gravitational acceleration vector; is the atmospheric drag acceleration vector; is the solar pressure perturbation acceleration vector; is the acceleration vector generated by the estimated electric thrust in the satellite orbit coordinate system; is the average gravitational acceleration at sea level; is the specific impulse of the electric thruster of the electric propulsion satellite; is the transformation matrix from the orbital coordinate system to the J2000 ECI coordinate system.

6. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 5 is characterized in that: Transformation matrix from orbital coordinate system to J2000 ECI coordinate system The expression is: in, is the instantaneous angular momentum vector of the electrically propelled GEO satellite; 、 and are all unit vectors; is the mold length; Gravitational acceleration vector The expression is: in, is the Earth's gravitational constant; Solar gravitational acceleration The expression is: in, is the solar gravitational constant; is the position vector of the Sun in the J2000 ECI coordinate system; lunar gravitational acceleration The expression is: in, is the lunar gravitational constant; is the position vector of the moon in the J2000 ECI coordinate system; The expression of J2 perturbation acceleration is: in, 、 and is the vector component of the J2 perturbation acceleration; is the harmonic coefficient, ; x, y and z are the components of the position vector; is the mean radius of the Earth; Atmospheric drag acceleration vector The expression is: in, is the ratio of the effective area S to the mass m of the electric propulsion GEO satellite; is the atmospheric drag coefficient; is the functional relationship between atmospheric density and the altitude of the electric propulsion GEO satellite, and its expression is , , To electrically propel GEO satellite altitude, is the reference height; is the atmospheric elevation; is the velocity vector of the electrically propelled GEO satellite relative to the atmosphere, and its expression is: , is the Earth's rotation angular velocity vector; is the angular velocity of the Earth; Solar pressure perturbation acceleration vector The expression is: in, is the shadow coefficient. When the satellite is in the umbra, , when the satellite is in the penumbra, , when the satellite is outside the umbra and penumbra, ; is the SRP reference value when starting from the sun; is the reflection coefficient; The expression for estimating the acceleration vector generated by the electric thrust is: in, 、 and are radial, along-track, and normal acceleration components respectively; 、 is the optimal control thrust angle of electric thrust in orbital coordinate system.

7. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 5, characterized in that: Step S1 further comprises: S11. The position and velocity estimates of the electric thruster GEO satellite in the J2000 ECI coordinate system are used as the measurement values ​​of the CKF-based electric thruster real-time on-orbit thrust estimation method. The measurement equation is: Expressed as: in, are the estimated values ​​of the satellite's position vector and velocity vector in the J2000 ECI coordinate system; are the true position vector and velocity vector of the satellite in the J2000 ECI coordinate system; is the estimation accuracy; S12. Discretize the state space model of the orbital transfer dynamics model: in, and are the discretized state quantities at time k and time k+1 respectively; is the estimated value of the satellite's position vector and velocity vector in the J2000ECI coordinate system at time k+1; is the state transfer function, which means predicting the satellite state at time k+1 from the state of the electric propulsion GEO satellite at time k; is the observation function; are the system noise at time k and the measurement noise at time k+1 respectively; S13, given initial state variables of the filter , the initial covariance matrix , process noise covariance matrix , the measurement noise covariance matrix ; S14. Calculate the predicted value of the state variable based on the initial state variable and the initial covariance matrix: ， in, is the predicted value of the state variable at time k+1; and They are respectively the k+1th volume points; n is the dimension of state variables; is the estimated value of the state variable at time k; is the Cholesky decomposition of the state variable covariance matrix at time k; is a matrix composed of unit vectors; S15. Predicting values ​​based on state variables and volume points , calculate the state prediction covariance matrix at time k+1 : S16, according to the volume point , calculate the measured predicted value at time k+1 : ， in, The observation function is weighted and summed after propagation of volume points to obtain the measurement prediction value; S17. Predicting values ​​based on state variables , volume point and measured predicted values , update the measurement covariance matrix and cross covariance matrix: ， in, and are the measurement covariance matrix and cross covariance matrix at time k+1 respectively; S18, according to the measurement covariance matrix and the cross-covariance matrix , calculate the Kalman gain: in, is the Kalman gain at time k+1; S19, according to 、 、 and , the estimated value of the state variable at time k+1 and the state covariance matrix To update: ， ； Estimated value of the state variable at time k+1 The thrust estimate of the electric propulsion GEO satellite in is the real-time estimated thrust of the electric propulsion GEO satellite.

8. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 5 is characterized in that: The step S4 further comprises: S41: Determine the guidance mode as the shortest time guidance law or the limited time fuel consumption least guidance law according to the current mission requirements, and proceed to step S42 and step S43 respectively; S42. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: in, Maximum flight time for electric propulsion GEO satellite fixed-point transfer; The flight time for the orbital phase; The maximum allowable deviation of the terminal of the electric propulsion satellite fixed-point transfer; and are all constants, Take 0.97, Take 0.01; S43. Constructing initial state constraints and optimal performance indicators , and use the initial state constraints and the optimal performance index to optimize the weight parameters of the RQ-Law function to obtain the optimized weight coefficient, and then enter step S44, the optimal performance index The expression is: ； S44: Output the optimal control thrust angle under thrust drop fault according to the current state variables of the electric propulsion satellite, the current orbit information of the target point, and the optimized weight coefficient. : ， Where D1, D2 and D3 are auxiliary variables respectively; Q is the RQ-Law function; is the orbital element number of the electric propulsion GEO satellite; is the maximum rate of change of the eccentricity component f with respect to the thrust direction; is the maximum rate of change of the eccentricity component g with respect to the thrust direction; is the maximum rate of change of the five orbital elements of an electric propulsion satellite with respect to the thrust direction and true longitude; is the acceleration vector in the orbital coordinate system; is the true longitude of the GEO satellite; is an auxiliary variable; a is the right ascension of the ascending node; f and g are the eccentricity vector components; 、 is the orbital inclination vector component; L is the true longitude; is the Earth's gravitational constant; S45, optimal control of thrust angle under thrust drop fault , adaptively adjust the orbit of electric propulsion GEO satellite.

9. The electric propulsion GEO satellite thrust descent fixed-point transfer and reconstruction guidance method according to claim 8, characterized in that: Initial state constraints include: Initial state constraints: Among them, t0 is the initial time, a C (t0) is the semi-major axis of the orbit at the initial moment, f C (t0) is the vector of orbital eccentricity at the initial moment, g C (t0) is the component of orbital eccentricity at the initial moment, h C (t0) is the vector of the orbital inclination at the initial moment, k C (t0) is the component of the orbital inclination at the initial moment, L C (t0) is the true longitude at the initial moment, a0, f0, g0, h0, k0, L0 are the orbital elements of the initial orbit; End constraints: Among them, t f is the transfer time; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the GEO satellite respectively; 、 、 and are the semi-major axis, eccentricity, orbit inclination and true longitude of the target point respectively; Equations of motion constraints: in, 、 、 、 、 、 、 are the semi-major axis, eccentricity components f and g, orbital inclination components h and k, true longitude, and mass change rate; is the right ascension of the ascending node; p is the semi-diameter; is the specific impulse of the electric thruster of the electric propulsion satellite.

10. The electric propulsion GEO satellite thrust descent fixed-point transfer reconstruction guidance method according to claim 8, characterized in that: The expression for optimizing the weight coefficient is: in, and are all relevant weights of the penalty function; are the weights of the semi-major axis, eccentricity components f, g, and orbital inclination components h, k, respectively; and is the phase parameter; , and are the associated scaling weights; The expression of the Lypunov function Q is: in, are penalty functions, The smaller it is, the shorter the time it takes to complete orbit transfer and positioning; is the scaling function; is the weight corresponding to the improved vernal equinox orbital elements; is the orbital element number of the electric propulsion GEO satellite; is the augmented set of target orbital elements; It is the maximum rate of change of each orbital element of an electrically propulsed GEO satellite relative to the thrust direction and true longitude.

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