Lanbert transfer singularity avoidance method based on track splicing

Abstract

The invention provides a Lanbert transfer singularity avoidance method based on orbit splicing, and belongs to the technical field of spacecraft orbit design and orbit control, and the method comprises the steps: obtaining an initial moment position and velocity vector, a terminal moment position vector, and flight time between two points; calculating an initial center transfer angle; solving a Lanbert problem of an initial state; calculating the flight time and the position velocity vector of each center transfer angle interval node; initializing a terminal moment and a terminal position; starting from the initial moment, for each control moment, orbit control calculation is carried out according to the following steps; solving an interval Lanbert problem; when the center transfer angle of flight in the current interval reaches half of the interval transfer angle, the terminal switching moment is the moment of the next interval node, and the terminal position is the position of the next interval node; and the two steps are repeated until the whole-course track transfer is completed. According to the invention, whole-course non-singular control can be realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of spacecraft orbit design and orbit control, and particularly relates to a Lambert transfer singularity avoidance method based on orbit splicing. BACKGROUND

[0002] The Lambert problem originates from a famous theorem proposed by a Swiss astronomer Lambert in 1761: in a central gravitational field, the flight time required for a transfer orbit connecting two points in space only depends on the distance between the two points, the sum of the distances from the two points to the gravitational center, and the semi-major axis of the orbit connecting the two points, and is irrelevant to the eccentricity of the orbit. The depth of this theorem lies in that it directly relates the flight time to the geometric parameters of the orbit, and decouples the flight time from the specific shape of the orbit, greatly simplifying the problem. The Lambert problem, also known as the Gauss problem or the orbit boundary value problem, is a classic and crucial basic problem in celestial mechanics and space dynamics, and its core can be summarized as: in a central gravitational field, given the initial position vector, the terminal position vector, and the flight time between the two points, the Kepler orbit connecting the two points can be determined. The Lambert problem belongs to the two-point boundary value problem, and it has wide applications in orbit determination and orbit control, etc. It is the theoretical cornerstone of modern space mission design and orbit analysis, and its solution algorithm development and optimization has always been one of the key technologies to promote the advancement of the space industry, from the Apollo moon landing to deep space exploration.

[0003] However, in the Lambert problem, when the initial position vector and the terminal position vector are collinear, i.e., the transfer angle of the starting point and the terminal point about the gravitational center is 180°, there are infinitely many Kepler orbits connecting the two points, and at this time, two types of singular problems will occur.

[0004] The first type is the calculation singularity problem of the shape of the orbit. The Lambert equation is a nonlinear transcendental equation, which is generally solved by numerical iteration method. In the general Lambert solution with a central transfer angle of 180°, the semi-major axis is usually selected as the iteration variable, the flight time between the two points is the dependent variable, but when the central transfer angle is 180°, the rate of change of the dependent variable with respect to the independent variable becomes infinite, causing the iteration algorithm to fail. The classical solution method is the independent variable transformation method, which successfully changes the rate of change of the dependent variable with respect to the new independent variable to be finite by transforming the independent variable from the semi-major axis to the auxiliary independent variable (where is half the perimeter of the triangle formed by the two points and the gravitational center), and then the iteration calculation can be completed.

[0005] The second type is the orientation singularity problem of the orbit in space. After solving the calculation singularity problem of the first type, the semi-major axis and the eccentricity While the orbital shape is determined, the inclination parameters that determine the orbit's orientation in space still have infinitely many solutions. Choosing the reasonable solution is a crucial problem in orbital control engineering. Furthermore, in the engineering implementation of orbital transfers, a common control method involves cyclically calculating the Lambert problem according to a certain control cycle. When the initial center transfer angle is greater than 180°, as time progresses, it will inevitably reach a point where the transfer angle approaches or equals 180°, thus encountering this singular problem. Therefore, researching this indeterminate singular problem is of significant engineering importance for large-scale Lambert transfers with center transfer angles exceeding 180°. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a Lambert transfer singularity avoidance method based on track splicing, which can achieve singularity-free control throughout the entire track transfer process for a wide range of track transfers with a center transfer angle exceeding 180°.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A Lambert transfer singularity avoidance method based on orbit splicing includes the following steps:

[0009] Step 1: Obtain the initial position and velocity vectors, the final position vector, and the flight time between the two points;

[0010] Step 2: Calculate the initial center transfer angle and divide the center transfer angle interval;

[0011] Step 3: Solve the Lambert problem for the initial state to obtain the theoretical transfer trajectory;

[0012] Step 4: Calculate the flight time and position-velocity vector of each node in the central transfer angle interval;

[0013] Step 5: Initialize the terminal time and terminal position, using the time of the nearest node in the rightmost interval as the initial terminal time and the position of the nearest node in the rightmost interval as the initial terminal position;

[0014] Step 6: Starting from the initial moment, for each control moment, perform track control calculations according to steps 7 to 9 below;

[0015] Step 7: Using the current position as the starting position, the set terminal position as the terminal position, and the time between the current time and the set terminal time as the flight time, solve the interval Lambert problem;

[0016] Step 8: When the center transfer angle of the flight in the current interval reaches half of the interval transfer angle, switch the terminal time to the time of the next interval node and the terminal position to the position of the next interval node;

[0017] Step 9: Repeat steps 7 and 8 until the entire orbital transfer is completed.

[0018] Beneficial effects:

[0019] This invention addresses the challenges of large-scale orbital transfers with a center transfer angle exceeding 180°. Traditional methods struggle to calculate orbital shape parameters and uniquely determine orbital orientation parameters when the angle between the current and terminal positions approaches 180°. This invention plans the entire trajectory, then divides and splices intervals, ensuring that new calculations and control always occur within sub-intervals less than 180° where singularities are unlikely. This achieves singularity-free calculations and continuous orbital transfer control throughout the entire process, possessing significant practical implications for the design of Earth orbit spacecraft and deep space probes, and providing strong technical support for the successful implementation of engineering missions. Attached Figure Description

[0020] Figure 1 This is a flowchart of the Lambert transfer singularity avoidance method based on track splicing, according to an embodiment of the present invention.

[0021] Figure 2 A schematic diagram of the center transition angle curve for the entire Lambert solution. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0023] like Figure 1 As shown, the present invention provides a Lambert transfer singularity avoidance method based on orbit splicing, comprising the following steps:

[0024] Step 1: Obtain the initial position and velocity vectors, the final position vector, and the flight time between the two points;

[0025] Step 2: Calculate the initial center transfer angle and divide the center transfer angle interval;

[0026] Step 3: Solve the Lambert problem for the initial state to obtain the theoretical transfer trajectory;

[0027] Step 4: Calculate the flight time and position-velocity vector of each node in the central transfer angle interval;

[0028] Step 5: Initialize terminal time and terminal position: Use the time of the nearest node in the rightmost interval as the initial terminal time, and use the position of the nearest node in the rightmost interval as the initial terminal position;

[0029] Step 6: Starting from the initial moment, for each control moment, perform track control calculations according to steps 7 to 9 below;

[0030] Step 7: Using the current position as the starting position, the set terminal position as the terminal position, and the time between the current time and the set terminal time as the flight time, solve the interval Lambert problem;

[0031] Step 8: When the center transfer angle of the flight in the current interval reaches half of the interval transfer angle, switch the terminal time to the time of the next interval node and the terminal position to the position of the next interval node;

[0032] Step 9: Repeat steps 6 and 7 until the entire orbital transfer is completed.

[0033] Specifically, step 1 includes:

[0034] Obtain the initial position vector Initial velocity vector Terminal position vector and flight time between the two points .

[0035] Specifically, step 2 includes:

[0036] Step 2.1: Determine the center transfer angle Is it greater than 180°?

[0037] ;

[0038] in, This represents the magnitude of a calculated vector. Represents the vector cross product. This represents the vector dot product.

[0039] Step 2.2: Calculate the center transfer angle :

[0040] ;

[0041] Step 2.3: Divide the center transfer angle interval:

[0042] According to equal intervals not greater than 90° Divide the center transfer angle interval and set... Divide the interval ( The following is an example, in which: The interval number, To obtain complete division Total number of intervals:

[0043] ;

[0044] Corresponding interval node transition angle ( The set of ) is .

[0045] Specifically, step 3 includes:

[0046] Solving the Lambert problem for the initial state yields the theoretical transfer trajectory. This invention is applicable to cases where the initial center transfer angle is not 180°. For the Lambert problem of the initial state, any conventional method can be used for solving it. The initial velocity vector of the obtained transfer trajectory is denoted as... .

[0047] Specifically, step 4 includes:

[0048] Let the initial time be... =0, terminal time is , It is the flight time between the initial moment and the terminal moment, for any moment on the transfer orbit. According to the initial state Using the two-body orbital dynamics equations, it can be extrapolated to obtain Vector of position and velocity at time .

[0049] Specifically, setting the control cycle The initial values ​​of the entire transfer trajectory are extrapolated according to the control cycle, and the center transfer angle of the current position relative to the initial position at each control moment is calculated. ,when equal to the node transition angle of a certain interval ( Record the flight time at that moment. ( ) and position vector ( ), velocity vector ( ). Calculate the position at each control moment. Relative initial position center transfer angle The method is as follows:

[0050] ;

[0051] Specifically, step 5 includes:

[0052] Initialize terminal time and terminal position: Use the time of the rightmost nearest interval node as the initial terminal time, and use the position of the rightmost nearest interval node as the initial terminal position. Specifically: equal (Right now The time of the interval node) and location The initial terminal time and terminal position.

[0053] Specifically, step 7 includes:

[0054] Using the current position as the starting position and a set terminal position as the ending position, and the time between the current moment and the set terminal moment as the flight time, this invention solves the interval Lambert problem. The center transition angle of this interval Lambert problem is always less than 180°, and no singular cases will occur; therefore, any conventional method can be used to solve it. Furthermore, it should be noted that this invention primarily addresses the Lambert problem involving position intersections, and only calculates the initial velocity vector of the transfer trajectory; the calculation of the terminal velocity vector is not included in this invention.

[0055] Specifically, step 8 includes:

[0056] Let the center transition angle of the current position relative to the initial position be... Located in the interval When the first satisfaction At that time, the terminal state is switched to... equal ( When not a final node, it equals If it is the final node, then it equals The time of the interval node) and location For the new terminal time and terminal location.

[0057] Step 9: Repeat steps 6 and 7 until the entire orbital transfer is completed.

[0058] Example:

[0059] This embodiment of the Lambert transfer singularity avoidance method based on orbit splicing includes the following steps:

[0060] Step 1: In this embodiment, we consider a deep space exploration scenario within the solar system, with the Sun as the central gravitational body. The projected coordinate system for the following vectors is taken as the J2000 inertial coordinate system with the Sun as the origin. The specific selection of each parameter is as follows:

[0061] The initial time is 00:00 UTC on August 1, 2028, the terminal time is 00:00 UTC on March 9, 2029, and the flight time is... =220 days.

[0062] The initial position and velocity vectors are taken as the Earth's position at that moment. and speed The parameters are:

[0063] The unit is km;

[0064] The unit is km / s.

[0065] The terminal position vector is taken as the position of asteroid 2015XF261 at the terminal time. The parameters are:

[0066] The unit is km.

[0067] Step 2: In this embodiment, because Therefore, the center transfer angle Based on this, further calculations The result is:

[0068] ;

[0069] In this embodiment, the interval is taken. Dividing the center transfer angle into equal intervals of 90°, the results are as follows:

[0070] ;

[0071] Corresponding interval node transition angle ( The set of ) is:

[0072] ;

[0073] Step 3: Solve the Lambert problem for the initial state to obtain the theoretical transition trajectory. Solve the Lambert problem for the initial state using a conventional iterative method, with the transition time as the dependent variable in the iteration. The independent variable is taken as the semi-major axis of the transfer orbit. Calculations yielded km, and obtain the initial velocity vector of the transfer orbit as The unit is km / s.

[0074] Step 4: In this embodiment, the control cycle is set. =1s, for the initial state , Perform two-body orbit extrapolation to calculate the current position. relative to initial position center transfer angle equal to the interval node transition angle Flight time (relative to the initial time) (Time) and position vector The results are shown in Table 1:

[0075] Table 1. Motion parameters where the center transition angle relative to the initial position is equal to the interval node transition angle.

[0076]

[0077] Step 5: In this embodiment, using The moment equal to 90° =80 days 8 hours 28 minutes 23 seconds and location The initial terminal time and terminal position.

[0078] Step 6: Starting from the initial moment, for each control moment, perform track control calculations according to the following steps.

[0079] Step 7: In this embodiment, the first typical current time is taken as... =40 days, at which point the center transfer angle that has been flown relative to the initial state is 42.384°, located in the interval The first half of the interval, its terminal time and terminal position are respectively... =80 days 8 hours 28 minutes 23 seconds and The interval Lambert problem is solved using the conventional iterative method, with the interval transition time as the dependent variable. =40 days 8 hours 28 minutes 23 seconds, with the independent variable being the semi-major axis of the transfer orbit. Calculations yielded The unit of 'a' is km, and the initial velocity vector of the transfer orbit is obtained as follows: Its unit is km / s; the spacecraft's current actual speed is Its unit is km / s, comparing the current speed with the commanded speed. The required speed increment is obtained as The unit is km / s. Transfer control can be continued in the direction of this speed increment. When the start-up threshold is set to be no less than 0.01 km / s, the track control does not need to be started.

[0080] In this embodiment, the second typical current time is taken as =60 days, at which point the center transfer angle that has been flown relative to the initial state is 65.708°, located in the interval In the first half of the interval, when passing through the midpoint of the interval, the terminal time and terminal position are determined by the node in the subsequent step 8. The state at 90° switches to a node. Given a state of 180°, the new terminal time and terminal position are respectively... =165 days 14 hours 13 minutes 42 seconds and The interval Lambert problem is solved using the conventional iterative method, with the interval transition time as the dependent variable. =105 days 14 hours 13 minutes 42 seconds, with the independent variable being the semi-major axis of the transfer orbit. Calculations yielded Its unit is km, and the initial velocity vector of the transfer orbit is obtained as follows: Its unit is km / s; the spacecraft's current actual speed is Its unit is km / s, comparing the current speed with the commanded speed. The required speed increment is obtained as The unit is km / s. Transfer control can be continued in the direction of this speed increment. When the start-up threshold is set to be no less than 0.01 km / s, the track control does not need to be started.

[0081] Step 8: In this embodiment, taking the first terminal state switch as an example, the switch time is the center transfer angle relative to the initial position. In the interval The moment when the center shift angle relative to the initial position first exceeds 45° beyond the midpoint of the interval is 42 days, 6 hours, 57 minutes, and 46 seconds. =45.00001°, the terminal time and terminal state changed from the original node The state at =90° switches to node. =180° state, see Table 1 for details.

[0082] Step 9: Repeat steps 6 and 7 until the entire orbital transfer is completed.

[0083] Based on the preceding analysis, the initial center transfer angle of the spacecraft relative to the target is 223.759°. This embodiment presents the Lambert-calculated center transfer angle curves for all orbital transfers with a center transfer angle greater than 25° relative to the target, as shown below. Figure 2 As shown in the figure, the maximum center transfer angle Y = 135°, corresponding to a flight time of X = 1015 hours, and the minimum center transfer angle Y = 25.02°, corresponding to a flight time of X = 4504 hours, are marked in the figure. Figure 2 As can be seen, after track partitioning and splicing, the center transfer angle is never greater than 135°, thus always staying away from the singular state of 180°, verifying the effectiveness of the method provided by the present invention.

[0084] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A Lambert transfer singularity-avoiding method based on track splicing, characterized in that, The method comprises the following steps: Step 1: obtaining initial time position and velocity vector, terminal time position vector and flight time between two points; Step 2: calculating initial central transfer angle and dividing central transfer angle interval; Step 3: solving Lambert problem of the initial state to obtain a theoretical transfer orbit; Step 4: calculating flight time and position and velocity vector of each central transfer angle interval node; Step 5: initializing terminal time and terminal position, taking the time of the nearest interval node at the right end as the initialized terminal time, and taking the position of the nearest interval node at the right end as the initialized terminal position; Step 6: starting from the initial time, for each control time, the following steps 7-9 are performed for orbit control calculation; Step 7: solving interval Lambert problem by taking the current position as the starting position, taking the set terminal position as the terminal position, and taking the time between the current time and the set terminal time as the flight time; Step 8: when the central transfer angle that has been flown in the current interval reaches half of the interval transfer angle, the terminal time is switched to the time of the next interval node, and the terminal position is switched to the position of the next interval node; Step 9: repeating steps 7-8 until the whole orbit transfer is completed.

2. The Lambert Conformal map transition method based on track splicing according to claim 1, wherein, The step 2 comprises: Step 2.1 : Determine the center transfer angle whether it is greater than 180°; Step 2.2: Calculate the center shift angle based on the judgement of step 2.1 ; Step 2.3: Determining the center shift angle based on the center shift angle interval Divide the center shift angle interval.

3. The Lambert Conformal map transition method based on track splicing according to claim 2, wherein, The judgment formula of the step 2.1 is: ； is the initial time position vector, is the initial time velocity vector, is the terminal time position vector, denotes the modulus of a vector, denotes the vector cross product, denotes the vector dot product.

4. The Lambert Conformal map transition singularity-avoiding method based on track splicing according to claim 3, characterized in that, In step 2.2, the center transfer angle is calculated as follows: 。 5. The Lambert Conformal map transition method based on track stitching according to claim 4, wherein, The step 2.3 comprises: According to equal intervals not greater than 90° Divide the center transfer angle interval and set... Divide the interval , As follows, among which, The interval number, To obtain complete division Total number of intervals: 。 6. The Lambert Conformal map transition method based on track stitching according to claim 5, wherein, corresponding interval node transfer angle , the set of .

7. The Lambert Conformal map transition method based on track stitching according to claim 1, wherein, The step 4 comprises: according to the theoretical transfer orbit obtained in step 3, taking the initial time as the zero point and the terminal time as the total flight time endpoint, and gradually extrapolating the whole orbit according to the set control period; at each extrapolation, the central transfer angle of the current position relative to the initial position is calculated in real time, and once the angle is equal to any interval node angle divided in step 2.3, the corresponding flight time, position vector and velocity vector are recorded.

8. The Lambert Conformal map transition method based on track stitching according to claim 7, wherein, In the step 5, after completing the extrapolation and recording of all interval nodes, the first interval node closest to the end point in the flight direction is selected from the recorded node set, the flight time corresponding to the node is set as the initial terminal time of the current orbit control, and the position vector of the node is set as the initial terminal position of the current orbit control, thereby completing the initialization of the terminal state.

9. The Lambert Conformal map transition singularity-avoiding method based on track splicing according to claim 1, wherein, In the loop advancing process of steps 6-7, the flown central transfer angle of the spacecraft is continuously monitored; once the angle first crosses the half-region threshold of the current interval, the terminal time and terminal position used in step 7 are replaced by the recorded values corresponding to the next interval node, so that the subsequent orbit control target is automatically updated to the new node, thereby completing a terminal state switching, and the loop continues until the end of the whole process.

10. The Lambert Conformal map transition singularity-avoiding method based on track splicing according to claim 1, wherein, The steps 6-9 constitute a closed loop: after completing the solving of step 7 once, it returns to step 6 to enter the next control period, until the current time reaches the terminal time and the whole orbit transfer is completed.

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