A multi-constraint coupled orbit design method for a return spacecraft

Abstract

The application discloses a return type spacecraft multi-constraint coupling orbit design method, and comprises the following steps: determining orbit design input parameters, analytically determining an orbit inclination and an ascending node right ascension, analytically determining a perigee amplitude, iteratively solving an eccentricity and determining a semi-major axis, aligning a phase of the earth rotation, and numerically verifying and outputting orbit roots. The application reduces five-dimensional iterative optimization to three root analytical determination plus one-dimensional root seeking, has small calculation amount, high convergence determination and high constraint satisfaction precision.

Description

Technical Field

[0001] This invention relates to the field of spacecraft orbit design technology, and more specifically to a multi-constraint coupled orbit design method for recoverable spacecraft. Background Technology

[0002] After completing their on-orbit missions, recoverable spacecraft (such as recoverable satellites, manned spacecraft, and reusable experimental spacecraft) need to undergo processes such as braking deorbiting, transition flight, and atmospheric reentry to return to a predetermined landing site on Earth. The design of the return trajectory is one of the key aspects of the overall design of recoverable spacecraft, and its design results directly affect the success or failure of the return mission and the landing accuracy.

[0003] From an engineering application perspective, many constraints on the reentry trajectory have clear physical origins. The constraint that the nadir trajectory passes through the launch point stems from the mission prerequisite that the spacecraft must be launched into orbit from a predetermined launch site. The constraint that the nadir trajectory passes through the landing point stems from the mission requirement that the spacecraft must be recovered to a predetermined landing site. For manned spacecraft, the landing site is typically equipped with telemetry, tracking, and command (TT&C), search and rescue, and recovery facilities; deviation from the landing site will severely impact recovery efficiency and astronaut safety. The constraint that the braking point is located at the apogee stems from the energy characteristics of orbital mechanics. Braking at the apogee allows for the maximum change in perigee altitude with the minimum velocity increment, thereby reducing propellant consumption of the braking engine. The constraint that the reentry angle equals a preset value stems from the limitations of the reentry corridor. An excessively large reentry angle will cause the spacecraft to experience overload and aerodynamic heating exceeding the structural limits, while an excessively small reentry angle may cause the spacecraft to eject from the atmosphere and fail to land on the predetermined trajectory. Because these constraints involve different geometric and energy characteristics such as orbital plane orientation, in-plane azimuth, and orbital shape, their influence on orbital elements is intertwined, making the reentry trajectory design problem a long-standing reliance on empirical iteration or high-dimensional numerical optimization.

[0004] The design of a reentry orbit typically requires the simultaneous fulfillment of multiple constraints. Taking a typical reentry mission as an example, the constraints to be met include at least the following: (1) The nadir trajectory of the reentry orbit must pass through both the launch point and the landing point. The nadir trajectory passing through the launch point is a prerequisite for ensuring that the spacecraft can enter the designed orbit from the launch point, and the nadir trajectory passing through the landing point is a prerequisite for ensuring that the spacecraft can land at the predetermined landing site after reentry; (2) The braking point needs to be located at the apogee of the reentry orbit in order to take advantage of the high braking efficiency of the apogee to reduce the braking speed increment and to ensure that the trajectory after braking meets the reentry corridor requirements; (3) The reentry angle when the spacecraft reaches the reentry point needs to be equal to the pre-designed value to ensure that the overload, heat flux and landing accuracy during the reentry process are all within the allowable range. The above multiple constraints simultaneously have a coupled effect on multiple of the six orbital elements (semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean perigee), so the reentry orbit design is a typical multi-constraint coupled design problem.

[0005] In existing technologies, the design of reentry orbits mainly adopts two technical approaches. The first approach treats the spacecraft's orbit as a known input, using guidance and control variables such as braking point location, braking time, braking speed increment, or aiming parameters as design variables. Iterative adjustments are made to ensure the reentry angle and landing point meet the constraints. This approach cannot answer the forward orbit design question during the mission validation phase: what kind of orbit should be designed so that its sub-satellite point trajectory simultaneously passes through the launch and landing points, the braking point is precisely at the apogee, and the reentry angle equals the preset value? The second approach directly performs fully free iterative optimization on the first five of the six orbital elements—semi-major axis, eccentricity, orbital inclination, right ascension of the ascending node, and argument of perigee—to ensure the design meets the aforementioned multiple constraints. While this technical approach is intuitive, the five design variables constitute a five-dimensional search space, and each constraint is highly coupled with the five variables. Blindly iterating to find the best results in slow convergence, easy getting stuck in local optima, or even failing to converge. Furthermore, it is difficult to strictly satisfy all constraints in an analytical sense, which cannot meet the needs of rapid solution demonstration and rapid replanning before returning to the previous state in engineering.

[0006] Therefore, how to overcome the shortcomings of existing five-parameter fully free iterative optimization orbit design methods, such as large search space, strong constraint coupling, slow convergence, easy getting trapped in local optima, and difficulty in strictly satisfying all constraints, and provide a returnable spacecraft orbit design method with small computational cost, strong convergence determinism, and design results that can strictly satisfy multiple constraints such as the nadir trajectory passing through the launch point and landing point, the braking point being located at the apogee, and the reentry angle being equal to the preset value, is an urgent technical problem to be solved in this field. Summary of the Invention

[0007] To address the aforementioned shortcomings of existing technologies, this invention provides a multi-constraint coupled orbit design method for recoverable spacecraft. This method overcomes the problems of existing five-parameter fully free iterative optimization methods, such as large search space, strong constraint coupling, slow convergence, susceptibility to local optima, and difficulty in strictly satisfying all constraints. It achieves forward orbit design for recoverable spacecraft with low computational cost, strong convergence determinism, and high constraint satisfaction accuracy. This invention proposes a multi-constraint coupled orbit design method for recoverable spacecraft, comprising the following steps:

[0008] Step S1, determine the trajectory design input parameters: determine the longitude and latitude of the launch point, the longitude and latitude of the landing point, the preset re-entry angle, the re-entry point altitude, and the braking point altitude; Step S2, Analyze and determine the orbital inclination and right ascension of the ascending node: Based on the constraint that the nadir point trajectory passes through the launch point and the landing point, and based on the spherical geometry of the great circle defined by two points, analyze and determine the orbital inclination and right ascension of the ascending node of the return orbit; Step S3, Analyze and determine the perigee argument: Based on the constraint that the braking point is located at the apogee of the return orbit, and based on the geometric relationship that the true perigee angle of the spacecraft at the apogee is always 180°, the perigee argument of the return orbit is analytically determined from the latitude argument of the braking point; Step S4, iteratively solve for the eccentricity and determine the semi-major axis: Based on the constraint that the reentry angle is equal to the preset reentry angle, and based on the conservation of angular momentum and energy, establish an analytical relationship between the reentry angle, the semi-major axis of the return trajectory, and the eccentricity; determine the relationship between the semi-major axis and the eccentricity based on the constraint that the braking point is located at the apogee of the return trajectory, substitute the semi-major axis into the analytical relationship to eliminate the semi-major axis, and obtain a one-dimensional equation with the eccentricity as the unknown; iteratively solve the one-dimensional equation within the eccentricity value range determined by the perigee safety height constraint to obtain the eccentricity, and then determine the semi-major axis from the eccentricity; Step S5, Register Earth's rotation phase: Register the Earth's rotation phase by selecting the braking moment, so that the nadir point trajectory of the return orbit passes through the landing point at the return moment; Step S6, numerical verification and output of orbital elements: Substitute the orbital inclination, right ascension of the ascending node, argument of perigee, eccentricity and semi-major axis determined in steps S2 to S5 into the perturbation orbital dynamics model for numerical verification. When the nadir trajectory passes through the launch point and the landing point, the braking point is located at the apogee and the reentry angle is equal to the preset reentry angle, output the determined orbital elements of the return orbit.

[0009] The core concept of this invention lies in the fact that instead of using the first five of the six orbital roots as five free variables in the joint optimization, it utilizes the inherent analytical correspondence between each constraint and a specific orbital root to perform layered decoupling of the multi-constraint coupled orbital design problem. Specifically, the orbital plane parameters (orbital inclination and right ascension of the ascending node) are analytically determined by the constraint that the sub-satellite trajectory passes through the launch and landing points; the in-plane azimuth parameters (argument of perigee) are analytically determined by the constraint that the braking point is located at the apogee; and the orbital shape parameters (semi-major axis and eccentricity) are reduced to iterative root finding of a one-dimensional equation by the constraint that the re-entry angle equals a preset value. Thus, the five-dimensional iterative optimization of the prior art is reduced to three analytically determined roots plus one-dimensional root finding.

[0010] Further, in step S2, the specific method for analytically determining the orbital inclination and the right ascension of the ascending node is as follows: the longitude of the landing point is corrected according to the phase angle caused by the Earth's rotation from the launch time to the return time to obtain the registration longitude; the geocentric unit vector r of the launch point is determined from the longitude and latitude of the launch point. L The geocentric unit vector r of the landing point is determined by the latitude of the landing point and the registration longitude. T The unit vector h normal to the orbital plane is determined by the following formula:

[0011] The track inclination angle i is determined by the following formula:

[0012] In the formula, k is the unit vector in the direction of the Earth's axis, and the unit vector normal to the orbital plane h is the solution that causes the spacecraft to fly from the launch point to the landing point, and the orbital inclination i satisfies:

[0013] In the formula φ L φ is the latitude of the launch point. T The latitude of the landing point.

[0014] Further, in step S2, the specific method for determining the right ascension Ω of the ascending node is as follows: It is determined based on the longitude difference between the launch point and the landing point, the phase angle caused by the Earth's rotation between the launch time and the braking time, and the precession of the right ascension of the ascending node caused by the Earth's non-spherical J2 term perturbation, so that the nadir trajectory passes through the launch point at the launch time and through the landing point at the return time. The precession of the right ascension of the ascending node is calculated using the following formula:

[0015] In the formula, J2 is the coefficient of the second-order zone harmonic term of the Earth's gravitational field, and R... ELet Ω be the Earth's radius, p be the semi-circle of the return orbit, i be the orbital inclination, n be the average angular velocity of the return orbit, and Δt be the time interval between launch and return. The unit of ΔΩ is radians, the unit of n is rad / s, and the unit of Δt is seconds.

[0016] Furthermore, in step S3, the perigee argument ω is determined by the following formula:

[0017] In the formula, u P The latitude argument of the braking point is determined by the sub-satellite position of the braking point, which lies on the sub-satellite trajectory of the return orbit. The geocentric angle between the sub-satellite position of the braking point and the landing point is equal to the preset braking range angle. Preferably, the braking range angle is predetermined based on the geometry of the return orbit according to the braking point altitude, reentry point altitude, and preset reentry angle, and its value ranges from 40° to 110°.

[0018] Furthermore, in step S4, the analytical relation is:

[0019]

[0020]

[0021] In the formula, γ E Let be the angle of reentry, e be the eccentricity, a be the semi-major axis, and V be the angle of reentry. E Let r be the true perimeter angle of the spacecraft at the reentry point. E r is the geocentric distance of the reentry point determined by the reentry point altitude. a The distance from the center of the braking point to the ground is determined by the height of the braking point.

[0022] Furthermore, in step S4, the specific methods for expressing the semi-major axis a as a function of the eccentricity e, constructing the one-dimensional equation, and iteratively finding the root are as follows: The semi-major axis a can be expressed as a function of the eccentricity e by the following formula:

[0023] Substitute the remaining analytical relations and eliminate the true anterior angle V E This yields a one-dimensional equation with only the eccentricity e as the unknown:

[0024] The range of values ​​for the eccentricity e is determined by the perigee safety height constraint:

[0025] In the formula, r pTo determine the geocentric distance from the perigee of the return orbit, R E h is the Earth's radius. min As the lower limit of the safe height for near-geometry, γ h is the preset reentry angle. min It can take negative values, ranging from -300 km to 0 km.

[0026] Furthermore, the one-dimensional equation is iteratively solved using the secant method. The two initial values ​​for the iteration are respectively the eccentricity value e1 that makes the perigee height equal to the lower limit of the perigee safe height and the eccentricity value e2 that makes the perigee height equal to the reentry point height.

[0027]

[0028] The convergence criterion for the iterative root finding is that the absolute value of the difference between the reentry angles obtained from two adjacent iterations is not greater than a preset threshold, and the preset threshold ranges from 10. -40 ~10 -30 The number of iterations for finding the root should not exceed 10.

[0029] Further, in step S5, the specific method for registering the Earth's rotation phase is as follows: calculate the longitude drift caused by the Earth's rotation from the launch time to the return time, select the number of waiting orbits the spacecraft takes on the return orbit, and ensure that the absolute value of the difference between the longitude drift and the longitude registration residual of the nadir point trajectory is not greater than a preset registration threshold; when there is no waiting orbit that meets the condition, fine-tune the launch time until the registration residual meets the preset registration threshold.

[0030] Furthermore, the reentry point altitude ranges from 100 km to 120 km, and the braking point altitude ranges from 200 km to 500 km.

[0031] Furthermore, in step S6, the perturbation orbital dynamics model is an orbital dynamics model that takes into account the perturbation of the Earth's non-spherical J2 term.

[0032] Compared with the prior art, the beneficial effects of adopting the solution of the present invention are as follows: (1) The search dimension is significantly reduced. The present invention reduces the five-dimensional joint iterative optimization of the existing technology to three analytical determinations of the roots plus one-dimensional equation root finding. The iteration variables are reduced from five to one, and the amount of computation is reduced by about one order of magnitude. This can meet the real-time requirements of rapid scheme design in the mission demonstration stage and rapid replanning on the ground or on the satellite before returning to Earth.

[0033] (2) Strong convergence determinism, no local optimum problem. The three roots of orbital inclination, right ascension of ascending node and argument of perigee are analytically determined by the constraint conditions. The initial value of the remaining one-dimensional root-finding problem can be directly given by the perigee safety height constraint. The iteration is stable and converges, which fundamentally avoids the defect of five-dimensional blind search that is prone to getting trapped in local optima or even infeasibility.

[0034] (3) High accuracy in satisfying constraints. The three constraints of the ground point trajectory passing through the launch point and landing point, the braking point being located at the apogee, and the reentry angle being equal to the preset value are all embedded in the solution process with analytical relationships. The design results strictly satisfy all constraints within the accuracy of the dynamic model.

[0035] (4) The scheme is simple and easy to implement in engineering. All calculations in this invention only involve closed analytical formulas such as spherical trigonometric relations, Keplerian orbital geometry relations, and conservation of angular momentum and energy. There is no need for complex optimization tools such as nonlinear programming and pseudospectral methods. It can be quickly implemented on ordinary spaceborne computers or ground workstations, which facilitates its application in missions such as recoverable satellites, manned spacecraft, and reusable experimental spacecraft. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of the overall process of the multi-constraint coupled orbit design method for recoverable spacecraft provided in an embodiment of the present invention.

[0037] Explanation of reference numerals in the attached figures The specific steps of the multi-constraint coupled orbit design method for S1-S6 recoverable spacecraft. Detailed Implementation

[0038] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0039] Example 1 This embodiment provides a method for designing multi-constraint coupled orbits for recoverable spacecraft, such as... Figure 1 As shown, the method includes steps S1 to S6, and the specific implementation process of each step is as follows.

[0040] Step S1: Determine the track design input parameters.

[0041] The design input parameters include: the longitude λ of launch point A. L and latitude φ L Longitude λ of landing point B T and latitude φ T Preset reentry angle γ Reentry point height h E and braking point height h P The latitude and longitude of launch point A and landing point B are determined by the geographical locations of the launch site and landing site; the preset reentry angle γ... Determined by the reentry corridor design, the reentry angle is typically -2° to -8° for ballistic reentry; the reentry point height h E The atmospheric boundary altitude is taken, with a value ranging from 100 km to 120 km; the braking point altitude h P The value ranges from 200 km to 500 km, determined by the overall mission requirements for the braking point location. Then, the reentry point altitude h is used as the reference. E and braking point height h P The reentry point distance r is determined by the following formulas respectively. E and the distance r between the center and the braking point a :

[0042]

[0043] In the formula, R E The radius is the Earth's radius.

[0044] Step S2: Analyze and determine the orbital inclination and right ascension of the ascending node.

[0045] Based on the constraint that the sub-satellite trajectory passes through the launch point and the landing point, and using the spherical geometry of a great circle defined by two points, the orbital inclination and right ascension of the ascending node of the return orbit are analytically determined.

[0046] Specifically, considering that the spacecraft passes through launch point A at launch and landing point B at return, and that the Earth rotates a certain phase angle between these two moments, the longitude λ of landing point B is first determined. T The registration longitude is obtained by correcting for the phase angle caused by Earth's rotation from launch to return time using the following formula:

[0047] In the formula, ω E Let Δt be the Earth's rotational angular velocity, and Δt be the time interval between launch and return. When the calculated result is negative, it is modulo 360°. Then, the geocentric unit vector r of launch point A is determined using its longitude and latitude. L The geocentric unit vector r of the landing point is determined by the latitude of landing point B and the registration longitude. T The geocentric unit vector r is calculated using the following formula:

[0048] In the formula, φ and λ represent the latitude and longitude of the point, respectively. The unit vector h normal to the orbital plane is determined by the following formula:

[0049] The track inclination angle i is determined by the following formula:

[0050] In the formula, k is the unit vector in the direction of the Earth's axis. Since there are two solutions for the orbital plane normal determined by the two points, which are opposite to each other, this embodiment takes the solution corresponding to the direction that makes the spacecraft fly from launch point A to landing point B (i.e., the forward motion solution). To ensure that the great circle of the orbital plane can indeed pass through launch point A and landing point B, the orbital inclination angle i must also satisfy the following condition:

[0051] If the conditions are not met, it indicates that there is no solution for the given registration longitude, and it is necessary to adjust the return time in step S5 or wait for the number of orbits before recalculating.

[0052] The right ascension Ω of the ascending node is directly determined by the orientation of the orbital plane normal unit vector h in the geocentric equatorial inertial coordinate system, that is:

[0053] In the formula, h x h y Let h be the components of the unit vector normal to the orbital plane along the X and Y axes of the inertial coordinate system. Furthermore, to improve design accuracy, the determination of the right ascension Ω of the ascending node also needs to take into account the precession of the right ascension of the ascending node caused by the J2 term perturbation of the Earth's non-sphericity. This precession is calculated using the following formula:

[0054] In the formula, J2 is the coefficient of the second-order zone harmonic term of the Earth's gravitational field, and R... E Let Ω be the Earth's radius, p be the semi-major diameter of the return orbit, i be the orbital inclination, n be the average angular velocity of the return orbit, and Δt be the time interval between launch and return. ΔΩ is in radians, n is in rad / s, and Δt is in seconds. By superimposing these precessions onto the right ascension of the ascending node, determined by the two-point great circle rule, the nadir trajectory will pass through launch point A at launch and landing point B at return.

[0055] After step S2, the two roots, orbital inclination angle i and right ascension Ω of the ascending node, are analytically determined and no longer participate in subsequent iterative solutions.

[0056] Step S3: Analyze and determine the argument of perigee.

[0057] Based on the constraint that the braking point is located at the apogee of the return orbit, and based on the geometric relationship that the true perigee angle of the spacecraft at the apogee is always 180°, the perigee argument of the return orbit is analytically determined by the latitude argument of the braking point; and the perigee argument ω is analytically determined by the constraint that the braking point is located at the apogee.

[0058] According to the geometric properties of Keplerian orbits, the true anomaly angle f of the spacecraft at its apogee is always equal to 180°. Therefore, the constraint that the braking point P is located at the apogee of the return orbit is equivalent to the latitude argument at the braking point P satisfying the following:

[0059] On the other hand, the nadir position of braking point P is located on the nadir trajectory determined in step S2, and the geocentric angle between the nadir of braking point P and landing point B is equal to the preset braking range angle η. The braking range angle η is the geocentric angle corresponding to the nadir trajectory from braking point P to landing point B, composed of the geocentric angle of the free flight segment from braking point P to reentry point E and the geocentric angle of the reentry segment from reentry point E to landing point B. Its value is determined by the braking point height h. P Reentry point height h E Preset reentry angle γ The reentry trajectory characteristics are predetermined based on the geometry of the return trajectory, typically ranging from 40° to 110°. Once the sub-satellite position of braking point P is determined, its latitude argument u... P Therefore, the perigee argument ω is determined analytically using the following formula:

[0060] After step S3, the perigee argument ω is analytically determined and no longer participates in subsequent iterative solutions.

[0061] Step S4: Iteratively solve for the eccentricity and determine the semi-major axis.

[0062] Based on the constraint that the reentry angle equals the preset reentry angle, an analytical relation is established and the eccentricity e and semi-major axis a are solved iteratively by reducing the dimension.

[0063] After braking at braking point P (i.e., the apogee of the return orbit), the spacecraft flies freely along the return orbit to reentry point E. According to the conservation of angular momentum and energy, the reentry angle γ at reentry point E is... E The shape of the return trajectory completely determines the following analytical relationship: (Equation 1) (Equation 2) (Equation 3) In the formula, e is the eccentricity, a is the semi-major axis, and V E Let r be the true perimeter angle of the spacecraft at reentry point E.E r is the distance from the reentry point to the Earth's center. a denoted as r, the distance from the center of the braking point to the earth. In Equation 3, r... a The braking point height h determined by step S1 P Since it is a known quantity, the semi-major axis a can be expressed as a function of the eccentricity e using this formula:

[0064] That is, there is only one independent variable between the semi-major axis *a* and the eccentricity *e*. Substituting the above equation into equation 2 and combining it with equation 1, we can eliminate the true anterior angle *V*. E Then, the reentry angle γ is obtained. E The univariate functional relationship regarding the eccentricity e:

[0065] This will make the reentry angle equal to the preset reentry angle γ. The constraints are transformed into a root-finding problem of a one-dimensional equation:

[0066] The range of values ​​for the eccentricity e is determined by the perigee safety height constraint:

[0067] In the formula, h min This is the lower limit of the safe altitude for perigee, ranging from -300 km to 0 km. Since the spacecraft enters the atmosphere at the reentry point before reaching perigee, the perigee of the return orbit is usually a virtual geometric point located below the ground. This constraint is used to limit the maximum eccentricity of the return orbit, and thus limit the feasible range of braking speed increment and reentry angle.

[0068] The secant method is used to iteratively find the root of the above one-dimensional equation. The two initial values ​​for the iteration are taken as the eccentricity value e1 that makes the perigee altitude equal to the lower limit of the perigee safe altitude, and the eccentricity value e2 that makes the perigee altitude equal to the reentry point altitude, respectively:

[0069]

[0070] γ is calculated in each iteration E (e) and γ Compare and update e using the secant method until the absolute value of the difference between the reentry angles obtained from two adjacent iterations is not greater than a preset threshold ε (ε ranges from 10). -40 ~10 -30 Stop iteration when γ is reached, output the eccentricity e, and determine the semi-major axis a; the number of iterations should not exceed 10. Because γ E(e) If the function is monotonically continuous within the range of values, the above one-dimensional root finding will necessarily converge and the convergence speed is faster than the five-dimensional iteration in the prior art.

[0071] Once the eccentricity e and semi-major axis a are determined, the orbital velocity of the spacecraft at the braking point P can be calculated using the following formula:

[0072] In the formula, μ is the Earth's gravitational constant. The required braking speed increment can be determined by combining the velocity of the orbit before braking, which is then used for the design of the braking engine.

[0073] Step S5: Register the Earth's rotation phase.

[0074] Due to the Earth's rotation, the nadir trajectory continuously shifts westward relative to the ground. After solving for all five roots in steps S2 to S4, the longitude drift caused by the Earth's rotation from launch to return is calculated. The number of orbits the spacecraft waits in its return orbit is selected such that the absolute value of the difference between the longitude drift and the longitude registration residual of the nadir trajectory does not exceed a preset registration threshold. If no number of orbits meets the condition, the launch time is fine-tuned (equivalent to shifting the entire mission sequence) until the registration residual meets the preset registration threshold, ensuring that the nadir trajectory strictly passes through landing point B at the return time.

[0075] Step S6: Numerical verification and output of orbital elements.

[0076] Substitute the orbital inclination i, right ascension Ω of the ascending node, argument ω of perigee, eccentricity e, and semi-major axis a determined in steps S2 to S5, along with the sixth root number (mean perigee angle M or perigee time) determined by the mission timing, into the perturbation orbital dynamics model (including the J2 perturbation term) for numerical integration verification: check whether the nadir trajectory passes through launch point A and landing point B, whether the braking point P is located at apogee, and the reentry angle γ. E Is it equal to the preset reentry angle γ? When all three constraints are satisfied, the number of orbital elements of the determined return trajectory is output, and the design is completed. If individual constraints are slightly deviated due to unmodeled factors such as higher-order perturbations, and the constraints are not satisfied, a small-scale local correction is performed with the root result of the one-dimensional equation in step S4 as the center, and the result is re-verified until all constraints are satisfied, and then the final design result is output.

[0077] Example 2 This embodiment uses a specific return trajectory design example to illustrate the implementation process of the method of the present invention.

[0078] Step S1: Determine the trajectory design input parameters. Assume the launch point A's latitude and longitude are (40.9°N, 100.3°E), the landing point B's latitude and longitude are (41.8°N, 112.4°E), and the preset re-entry angle γ... = -3.0°, reentry point height h E = 100 km, braking point height h P = 400 km, and the time interval Δt between launch and return is approximately 1 day. Take the Earth's radius R. E = 6371 km, Earth's gravitational constant μ = 398600.4 km³ / s², and the daily phase angle corresponding to Earth's rotation angular velocity is 360.9856°. Based on this, the geocentric distance r from the reentry point is determined. E = R E + h E = 6471 km, distance from the center of the braking point to the ground r a = R E + h P = 6771 km.

[0079] Step S2: Analyze and determine the orbital inclination i and the right ascension Ω of the ascending node. First, correct the longitude of landing point B according to the Earth's rotation phase angle, and calculate the registration longitude using the following formula: .

[0080] Then, the geocentric unit vectors of launch point A (40.9°N, 100.3°E) and landing point B (41.8°N, 111.41°E) are calculated according to... The unit normal vector to the orbital plane is obtained, and the orbital inclination angle i = 41.86° is thus calculated, satisfying the geometrically necessary condition i ≥ max(40.9°, 41.8°) = 41.8°; the right ascension of the ascending node Ω0 = 25.10° is not considered during perturbations. Further considering the precession of the right ascension of the ascending node caused by the J2 term perturbation, the calculation is as follows:

[0081] Let J2 = 1.0826 × 10 -3 The semi-major diameter p ≈ 6414 km, and the average angular velocity n ≈ 1.224 × 10⁻⁶ km. -3 The precession is approximately -7.2° / day. After compensating for this precession, the right ascension of the ascending node is set to Ω = 32.33°, which ensures that the nadir trajectory passes through launch point A at launch time and landing point B at return time.

[0082] Step S3: Analyze and determine the perigee argument ω. This is based on the braking point height h. P = 400 km, reentry point altitude h E = 100km and preset reentry angle γ = -3.0°. Based on the geometry of the return trajectory, the geocentric angle of the free flight segment from braking point P to reentry point E is pre-determined to be approximately 80.4°, and the geocentric angle of the reentry segment from reentry point E to landing point B is approximately 12.0°. Therefore, the braking distance angle η ≈ 92.4°. On the sub-satellite trajectory determined in step S2, the latitude argument of landing point B is approximately 90° (the latitude of landing point B is close to the highest latitude of the great circle of the orbital plane). Therefore, the latitude argument u of braking point P is determined. P = 90° - 92.4° + 360° = 357.6°, and then analytically determine the perigee argument ω = u P -180° = 177.6°.

[0083] Step S4: Establish analytical relationships and iteratively solve for the eccentricity e and semi-major axis a using dimension reduction. (The last part, "r", appears to be a typo and can be omitted.) a = 6771 km, r E =6471 km Substitute into equations 1 to 3, and take the lower limit of the safe altitude at near-ground h min = -300 km, determine the feasible interval for eccentricity e. The initial values ​​for the two iterations of the secant method are respectively:

[0084]

[0085] Iterative solution of the one-dimensional equation f(e) = γ E (e) - γ The process of getting 0: After the first iteration, the residual is approximately 0.86°; after the second iteration, the residual is approximately 0.46°; after the third iteration, the residual is approximately 0.062°; and after the fourth iteration, the residual is approximately 4.3 × 10⁻⁶. -30 The residual after the 5th iteration is approximately 4.0 × 10⁻⁶. -50 The preset threshold ε = 10 has been met. -40 The convergence criterion is used to stop the iteration. The final eccentricity is calculated as e = 0.0527, and the semi-major axis a is determined by the following formula:

[0086] The corresponding perigee altitude is approximately -278 km (a virtual perigee below the ground, satisfying perigee safety altitude constraints), and the spacecraft's true perigee angle V at reentry point E is... E = 260.42°, reentry velocity approximately 7.83 km / s, reentry angle γ E = -3.00°, consistent with the preset reentry angle.

[0087] Furthermore, assuming the spacecraft is in a circular orbit at an altitude of 400 km before braking, with an orbital velocity of 7.673 km / s; and its velocity at the apogee of the return orbit after braking is 7.468 km / s, the required braking velocity increment Δv ≈ 205 m / s. This value is within the capability range of conventional braking engines, verifying the engineering feasibility of the design. The return orbit period is approximately 85.6 minutes, and the free flight time from braking point P to reentry point E is approximately 20.5 minutes.

[0088] Step S5: Register the Earth's rotation phase. Calculated based on the return orbit period, the longitude drift caused by the Earth's rotation between launch and return times is approximately 360.99° (including J2 precession correction). After selecting a waiting period of N = 16 orbits, the longitude registration residual of the nadir trajectory is approximately 0.3°, which does not meet the preset registration threshold of 0.1°. After fine-tuning the launch time for approximately 2 minutes, the registration residual decreases to 0.05°, meeting the preset registration threshold, and the nadir trajectory strictly passes through landing point B at the return time.

[0089] Step S6: Numerical verification and output of orbital elements. Substituting i = 41.86°, Ω = 32.33°, ω = 177.6°, e = 0.0527, a = 6432.1 km, and the mean perihelion angle M determined by the mission timing into the orbital dynamics model incorporating the J2 term perturbation, numerical integration is performed. The results show that the nadir trajectory passes through launch point A at launch time (deviation less than 0.01°) and land point B at return time (deviation less than 0.05°). The braking point P is located at the apogee of the return trajectory, and the reentry angle γ... E = -3.00°, all three constraints are satisfied, output the above orbital elements, and the design is complete.

[0090] Using the method of this invention, convergence to 10 is achieved in only about 5 one-dimensional iterations. -40 The method of this invention is of metric level, and the three roots are directly determined by analytical relations, with no convergence problem. The convergence speed and convergence determinism of the method of this invention are significantly better than the existing five-dimensional iterative optimization methods.

[0091] Example 3 This embodiment illustrates the application of the method of the present invention in a scenario of rapid replanning before returning.

[0092] When a recoverable spacecraft encounters an emergency requiring early return during its orbital flight (such as spacecraft malfunction, space debris collision warning, or mission change), a return trajectory that satisfies all constraints needs to be redesigned within a short period. Using existing five-dimensional iterative optimization methods, a single reprogramming calculation typically takes tens of seconds to several minutes and carries the risk of non-convergence, making it difficult to meet the real-time requirements of emergency scenarios.

[0093] The process of rapid replanning using the method of this invention is as follows: Starting from the current moment, the design input parameters are reacquired, the latitude and longitude of launch point A and landing point B remain unchanged, and the re-entry angle γ is preset. Reentry point height h E The braking point height h remains unchanged. P The orbital altitude can be selected within the range of 200 km to 500 km based on the spacecraft's current orbital altitude. Then, steps S2 to S4 are executed sequentially: the orbital inclination i and the right ascension Ω of the ascending node are analytically determined using the principle of two points defining a great circle (these two calculations only require a few trigonometric function operations), and the perigee argument ω is analytically determined using the latitude argument of the braking point (only one subtraction operation is required). Then, the one-dimensional equation is solved using the secant method. Subsequently, step S5 is followed to select the braking time and complete the Earth's rotation phase registration. This generates a new return orbit scheme that satisfies all constraints, which can be executed autonomously by the spacecraft or uploaded after confirmation by the ground.

[0094] The solution of this invention reduces the orbit design problem to three analytical determinations of the roots plus one-dimensional equation root finding. The calculation process does not require complex optimization tools such as nonlinear programming. It has low computational load, short time consumption, and convergence determination. It is particularly suitable for emergency rapid replanning scenarios before return, which significantly improves the mission adaptability and emergency return capability of recoverable spacecraft.

[0095] It should be noted that the logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be specifically implemented in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0096] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0097] In the description of this specification, references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0098] It should also be further understood that the term "and / or" as used in this specification refers to any combination of one or more of the associated listed items, as well as all possible combinations, and includes such combinations.

[0099] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A method for designing multi-constraint coupled orbits for recoverable spacecraft, characterized in that, Includes the following steps: Step S1, determine the trajectory design input parameters: determine the longitude and latitude of the launch point, the longitude and latitude of the landing point, the preset re-entry angle, the re-entry point altitude, and the braking point altitude; Step S2, Analyze and determine the orbital inclination and right ascension of the ascending node: Based on the constraint that the nadir point trajectory passes through the launch point and the landing point, and based on the spherical geometry of the great circle defined by two points, analyze and determine the orbital inclination and right ascension of the ascending node of the return orbit; Step S3, Analyze and determine the perigee argument: Based on the constraint that the braking point is located at the apogee of the return orbit, and based on the geometric relationship that the true perigee angle of the spacecraft at the apogee is always 180°, the perigee argument of the return orbit is analytically determined from the latitude argument of the braking point; Step S4, iteratively solve for the eccentricity and determine the semi-major axis: Based on the constraint that the reentry angle is equal to the preset reentry angle, and based on the conservation of angular momentum and energy, establish an analytical relationship between the reentry angle, the semi-major axis of the return trajectory, and the eccentricity; determine the relationship between the semi-major axis and the eccentricity based on the constraint that the braking point is located at the apogee of the return trajectory, substitute the semi-major axis into the analytical relationship to eliminate the semi-major axis, and obtain a one-dimensional equation with the eccentricity as the unknown; iteratively solve the one-dimensional equation within the eccentricity value range determined by the perigee safety height constraint to obtain the eccentricity, and then determine the semi-major axis from the eccentricity; Step S5, Register Earth's rotation phase: Register the Earth's rotation phase by selecting the braking moment, so that the nadir point trajectory of the return orbit passes through the landing point at the return moment; Step S6, numerical verification and output of orbital elements: Substitute the orbital inclination, right ascension of the ascending node, argument of perigee, eccentricity and semi-major axis determined in steps S2 to S5 into the perturbation orbital dynamics model for numerical verification. When the nadir trajectory passes through the launch point and the landing point, the braking point is located at the apogee and the reentry angle is equal to the preset reentry angle, output the determined orbital elements of the return orbit.

2. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S2, the specific method for analytically determining the orbital inclination and the right ascension of the ascending node is as follows: The registration longitude is obtained by correcting the longitude of the landing point for the phase angle caused by the Earth's rotation from the launch time to the return time. The geocentric unit vector r of the launch point is then determined from the longitude and latitude of the launch point. L The geocentric unit vector r of the landing point is determined by the latitude of the landing point and the registration longitude. T The unit vector h normal to the orbital plane is determined by the following formula: The track inclination angle i is determined by the following formula: In the formula, k is the unit vector in the direction of the Earth's axis, the unit vector h normal to the orbital plane is the solution that causes the spacecraft to fly from the launch point towards the landing point, and the orbital inclination i satisfies: In the formula, φ L φ is the latitude of the launch point. T The latitude of the landing point.

3. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S2, the specific method for determining the right ascension Ω of the ascending node is as follows: The trajectory of the sub-satellite point is determined based on the longitude difference between the launch point and the landing point, the phase angle caused by the Earth's rotation between the launch time and the braking time, and the precession of the ascending node's right ascension caused by the Earth's non-spherical J2 term perturbation, so that the sub-satellite point trajectory passes through the launch point at the launch time and the landing point at the return time; the precession of the ascending node's right ascension is calculated by the following formula: In the formula, J2 is the coefficient of the second-order zone harmonic term of the Earth's gravitational field, and R... E Let p be the Earth's radius, i be the semi-circular diameter of the return orbit, n be the orbital inclination, n be the average angular velocity of the return orbit, and Δt be the time interval between launch and return.

4. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S3, the perigee argument ω is determined by the following formula: In the formula, u P The latitude argument of the braking point is determined by the sub-satellite position of the braking point, which is located on the sub-satellite trajectory of the return orbit, and the geocentric angle between the sub-satellite of the braking point and the landing point is equal to the preset braking range angle.

5. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 4, characterized in that, The braking range angle is predetermined based on the braking point altitude, the reentry point altitude, and the preset reentry angle, using the geometric relationship of the return trajectory, and its value ranges from 40° to 110°.

6. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S4, the analytical relation is: In the formula, γ E Let be the angle of reentry, e be the eccentricity, a be the semi-major axis, and V be the angle of reentry. E Let r be the true perimeter angle of the spacecraft at the reentry point. E r is the geocentric distance of the reentry point determined by the reentry point altitude. a The distance from the center of the braking point to the ground is determined by the height of the braking point.

7. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 6, characterized in that, In step S4, the specific methods for expressing the semi-major axis a as a function of eccentricity e, constructing the one-dimensional equation, and iteratively finding the root are as follows: The semi-major axis a can be expressed as a function of the eccentricity e by the following formula: Substitute the remaining analytical relations and eliminate the true anterior angle V E This yields a one-dimensional equation with only the eccentricity e as the unknown: The range of values ​​for the eccentricity e is determined by the perigee safety height constraint: In the formula, r p To determine the geocentric distance from the perigee of the return orbit, R E h is the Earth's radius. min As the lower limit of the safe height for near-geometry, γ The preset reentry angle is given.

8. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 7, characterized in that, The one-dimensional equation is solved iteratively using the secant method. The two initial values ​​for the iteration are the eccentricity value e1 that makes the perigee height equal to the lower limit of the perigee safe height and the eccentricity value e2 that makes the perigee height equal to the reentry point height. The convergence criterion for the iterative root finding is that the absolute value of the difference between the reentry angles obtained from two adjacent iterations is not greater than a preset threshold, and the preset threshold ranges from 10. -40 ~10 -30 .

9. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S5, the specific method for registering the Earth's rotation phase is as follows: Calculate the longitude drift caused by Earth's rotation from launch time to return time, select the number of orbits the spacecraft needs to complete in its return orbit, and ensure that the absolute value of the difference between the longitude drift and the longitude registration residual of the nadir point trajectory is not greater than a preset registration threshold; if there is no number of orbits that meet the condition, fine-tune the launch time until the registration residual meets the preset registration threshold.

10. The multi-constraint coupled orbit design method for recoverable spacecraft according to claim 1, characterized in that, In step S1, the reentry point altitude ranges from 100 km to 120 km, and the braking point altitude ranges from 200 km to 500 km.

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