Modeling method of space tether system in restricted three-body problem for deep space exploration

Abstract

The application relates to a space tether system modeling method in a restricted three-body problem for deep space exploration. Firstly, the motion equation of a space three-body problem is studied, and then the three-body system is simplified into an elliptical restricted three-body problem. Then, the space tether system is affected by orbit perturbation, and the influence of J2 perturbation on the system is mainly studied. Finally, the Lagrange libration point of the system is obtained by solving the special solution of the system, and the space tether system is modeled based on the libration point. The core of the modeling method is to comprehensively consider the restricted three-body problem and propose a modeling method for the space tether system near the Lagrange libration point. On the basis of the previous application, the influence of the orbit perturbation J2 is considered, and a three-dimensional dynamics model of the space tether system in the restricted three-body problem is established.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology and relates to a modeling method for space tethered systems in the restricted three-body problem for deep space exploration. Background Technology

[0002] Deep space exploration is one of the most rapidly challenging fields in the world's high-tech field today, highly integrating many advanced technologies. With the rapid development of aerospace technology, it has attracted the attention of major spacefaring nations worldwide. According to the guidelines for the development of my country's aerospace industry, deep space exploration is one of the important directions for future aerospace development. Therefore, basic and applied research closely related to deep space exploration has become a hot research area in my country's aerospace industry.

[0003] For example, Chinese patent application CN201710355157.X proposes a Lagrangian dynamic model and controller for a space tethered system. Addressing the modeling problem of space tethered systems, it considers six parameters: three target attitude angles, in-plane and out-of-plane angles of the tether, and tether length. It derives the generalized force model of the tether in detail using the Lagrangian method and designs a generalized state-stabilized controller. This model can be used to solve the target attitude modeling problem of tethered systems when the platform and target masses are equal, and its controller can control the generalized variables of the tethered system after target capture. However, it is only a two-body model and is not applicable to restricted three-body problems in space.

[0004] Unlike the classic two-body problem in near-Earth spaceflight, the motion models involved in deep space exploration are more often three-body problems. Since restricted three-body problems are prevalent in deep space exploration, and given the irregular and complex gravitational fields, we utilize a space tethered detection system near the translational point in the restricted three-body problem to conduct deep space exploration, studying a class of space tethered detection systems in restricted three-body scenarios.

[0005] Currently, the dynamic model of a spatial rope system based on the L1 translation point involves two dimensions. Since it only considers the rope length and in-plane angles and does not account for the effects of orbital perturbations, it affects the control of the spatial rope system. Therefore, this invention, building upon previous inventions, considers the influence of orbital perturbation J2 and establishes a three-dimensional dynamic model of a spatial rope system in the restricted three-body problem. Summary of the Invention

[0006] Technical problems to be solved

[0007] To avoid the shortcomings of existing technologies, this invention proposes a modeling method for space tethered systems in the restricted three-body problem for deep space exploration.

[0008] Technical solution

[0009] A method for modeling space tethered systems in the restricted three-body problem for deep space exploration, characterized by the following steps:

[0010] Step 1: Establish an inertial coordinate system OXYZ with the center of mass O of the three-body system as the origin. Define r1, r2, and r3 as the position vectors of the three particles in the inertial coordinate system. Then, the equations of motion of the three-body system can be rewritten in Euler form as the three-body motion equations:

[0011]

[0012] Where: r ij =r j -r i Let i ≠ j, i = 1, 2, 3, representing the i-th particle P. i Pointing to the j-th mass P j The position vector, and r ij =||r ij ||;m1, m2, and m3 are the masses of the three point masses; G is the gravitational constant;

[0013] Step 2: Redefine the inertial coordinate system OXYZ: The origin is located at the common centroid of the two mass points P1 and P2, the xy plane is the plane of motion of P1 and P2, and the scalar form of the elliptical restricted three-body system is:

[0014]

[0015] Where Ω is the potential energy function, e is the eccentricity of the two-body orbit, and f is the true perihelion angle;

[0016] Step 3: The dynamic equations of the restricted three-body system considering the influence of orbital perturbation J2 are as follows:

[0017]

[0018] in and These represent the components of orbital perturbation in different directions.

[0019] Step 4: Establish a dynamic model of the spatial rope system based on the L1 translational point:

[0020] Substituting x = σ + lk cosβcosα, y = lk cosβsinα, z = lk sinβ into the restricted three-body system dynamics equations of step 3, and applying control actions for the interior angles, exterior angles, and rope length respectively, we obtain the dynamics equations as follows:

[0021] α″-F α =u α

[0022] β″-Fβ =u β

[0023] l″-F l =u l

[0024] Where σ is the x-coordinate of the translation point L1, and k = 1 + e cos f, It is a dimensionless distance, L is the length of the tether, α is the interior angle of the orbital plane, and β is the exterior angle of the orbital plane;

[0025] The The The

[0026]

[0027] Where μ = m2 / (m1+m2), and m1 and m2 are the masses of the two main celestial bodies, respectively.

[0028] The equations of motion for the three-body system consisting of the three particles in space are:

[0029]

[0030] Where q i and m i Let $\mathbf$ be the position vector and mass of the $i$-th particle (i = 1, 2, 3).

[0031] The elliptical restricted three-body system is based on two assumptions: ① Among the three point masses, the mass of point mass P3 is much smaller than that of P1 and P2, and P3 moves under the gravitational pull of the two larger point masses, while the gravitational pull of P3 on P1 and P2 is negligible; ② Point masses P1 and P2 move in a circular motion around their common center of mass.

[0032] The potential energy function Where μ = m2 / (m1+m2), and m1 and m2 are the masses of the two main celestial bodies, respectively.

[0033] Beneficial effects

[0034] This invention proposes a modeling method for a space tethered system in a restricted three-body problem for deep space exploration. First, the equations of motion for the space three-body problem are studied. Second, the three-body system is simplified into an elliptical restricted three-body problem. Then, the space tethered system is affected by orbital perturbations, with the J2 perturbation primarily affecting the system. Finally, by finding a particular solution to the system, the Lagrange translation point of the system is obtained, and the space tethered system is modeled based on this translation point. The core of this modeling method is to propose a modeling approach for the space tethered system near the Lagrange translation point, based on a comprehensive consideration of the restricted three-body problem.

[0035] Based on previous inventions, this invention considers the influence of orbital perturbation J2 and establishes a three-dimensional dynamic model of a space rope system in a restricted three-body problem. Attached Figure Description

[0036] Figure 1 Step 1: Schematic diagram of the three-body system

[0037] Figure 2 In step 2, the inertial coordinate system and the rendezvous coordinate system...

[0038] Figure 3 In step 3, the five Lagrange translation points in the three-body problem

[0039] Figure 4 The position of particle P relative to the translational point L1 in polar coordinates. Detailed Implementation

[0040] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0041] To achieve the above objectives, the technical solution adopted by the present invention includes the following steps:

[0042] Step 1: Study of the motion of the three-body problem in space

[0043] Step Two: Study of the Elliptical Restricted Three-Body Problem

[0044] Step 3: Study of the restricted three-body problem considering the influence of orbital perturbation J2

[0045] Step 4: Establish a dynamic model of the spatial rope system based on the L1 translation point.

[0046] In step one, the schematic diagram of the three-body system is as follows: Figure 1 As shown:

[0047] The classic two-body problem in near-Earth spaceflight discusses the motion of a system consisting of two point masses, specifically the motion of a probe orbiting the Earth. When the probe is relatively close to Earth, Earth's gravity plays a dominant role in its motion (without considering active control by the probe), and the gravitational forces of other celestial bodies can be considered as minor perturbations. However, for deep-space probes, such as those exploring one of Mars' moons, the gravitational interactions between Mars and Phobos will affect the probe's motion, thus involving a three-body system consisting of Mars, Phobos, and the probe.

[0048] Consider N point masses moving in space, each subject only to the gravitational pull of the other point masses. In a certain inertial coordinate system, define q... i and m i Let be the position vector and mass of the i-th particle, respectively. Then, according to Newton's second law, the equation of motion of the N-body system composed of N points can be written as:

[0049]

[0050] Where G is the gravitational constant, and U is the gravitational potential energy, expressed as:

[0051]

[0052] When N=3, the equations of motion for the three-body system consisting of three point masses can be written as follows:

[0053]

[0054] Where m1, m2 and m3 are the masses of the three particles.

[0055] Establish an inertial coordinate system OXYZ with the center of mass O of the three-body system as the origin. Define r1, r2, and r3 as the position vectors of the three particles in the inertial coordinate system, respectively. Then, the equations of motion can be rewritten as the Euler form of the three-body motion equations.

[0056]

[0057] Where, r ij =r j -r i Let i ≠ j, i = 1, 2, 3, representing the i-th particle P. i Pointing to the j-th mass P j The position vector, and r ij =||r ij ||.

[0058] In step two, the inertial coordinate system and the rendezvous coordinate system are established as follows: Figure 2 As shown:

[0059] The general three-body problem is complex, making it difficult to find general rules. Poincaré, Hill, and others' research on the three-body problem mainly focused on a simplified case: the circularly restricted three-body problem. However, most satellites orbit elliptical planes, meaning the Mars-Phobos distance periodically expands and contracts with Phobos's revolution around the sun. The elliptical restricted three-body problem is based on the following two assumptions:

[0060] ① Of the three point masses, the mass of point mass P3 is much smaller than that of P1 and P2. P3 is among the two largest point masses.

[0061] The mass point mass moves under the influence of gravity, while the gravitational force of P3 on P1 and P2 can be ignored.

[0062] ②Particles P1 and P2 move in a circular motion around their common center of mass.

[0063] Based on the above assumptions, the inertial coordinate system OXYZ is redefined as follows: the origin is located at the common centroid of the two mass particles P1 and P2, and the xy plane is the motion plane of P1 and P2.

[0064] According to the Euler form of the equations of motion, the equations of motion of particle P in the inertial coordinate system are as follows:

[0065]

[0066] Where r1 and r2 are the position vectors pointing from particles P1 and P2 to particle P, respectively.

[0067] Since P1 and P2 undergo circular motion around the Z-axis, and the angular velocity of this circular motion is defined as ω, the derivatives of the position vectors of particle P in the two coordinate systems satisfy the following relationship:

[0068]

[0069] By differentiation, we can obtain the equation of motion of particle P in the synoptic coordinate system as follows:

[0070]

[0071] Normalizing the parameters yields

[0072]

[0073] According to the basic theory of the two-body problem, the distance between particles P1 and P2 is...

[0074]

[0075] Where p is the eccentricity of the two-body orbit, e is the eccentricity of the orbit, and f is the true anterior angle.

[0076] Expanding the normalized equations of motion into scalar form, we get

[0077]

[0078] Where Ω is the potential energy function.

[0079]

[0080]

[0081] In step three, due to the aspherical nature of the planet, the gravitational force on the object no longer points towards the planet's center of mass. Integrating using the method of infinitesimal elements, the gravitational potential function can be obtained as follows:

[0082]

[0083] Where μ = GM, P k It is the kth Legendre polynomial.

[0084] The J2 perturbation dominates the orbital perturbation. Restricting the Legendre polynomial series to k = 2, then P2(ν) = (3ν) 2 -1) / 2, we can obtain the gravitational potential generated by the J2 perturbation as

[0085]

[0086] From the J2 perturbation gravitational potential, the acceleration can be obtained as follows:

[0087]

[0088] The system dynamic equation is

[0089]

[0090] In step four, the five Lagrange translation points in the three-body problem are as follows: Figure 3 As shown:

[0091] The translation point is the position where gravity and inertial forces are in equilibrium during the three-body motion. If a point mass P is placed at one of these positions with an initial velocity of zero, it will remain stationary relative to the two main celestial bodies in the coordinate system.

[0092] The coordinates of the collinear translational points L1, L2, and L3 can be obtained from the following formula.

[0093]

[0094] We place the rope system at the translation point L1. To determine the coordinates of the translation point L1, we can solve the above equation to obtain the coordinates as (σ, 0).

[0095] The position of particle P relative to the translational point L1 in polar coordinates is as follows: Figure 4 The following variables can be used to replace the definition.

[0096] x=σ+lk cosβcosα, y=lk cosβsinα, z=lk sinβ

[0097] Where k = 1 + e cos f, It is a dimensionless distance, and L is the length of the tether.

[0098] By applying control actions to the interior angle, exterior angle, and rope length, the dynamic equation can be obtained as follows:

[0099] α″-F α =u α

[0100] β″-F β =u β

[0101] l″-F l =u l

[0102] in

[0103]

[0104]

[0105]

[0106] Where σ is the x-coordinate of the translation point L1, and k = 1 + e cos f, It is a dimensionless distance, L is the length of the tether, α is the interior angle of the orbital plane, and β is the exterior angle of the orbital plane.

[0107] Based on previous inventions, this invention considers the influence of orbital perturbation J2 and establishes a three-dimensional dynamic model of a space rope system in a restricted three-body problem.

Claims

1. A modeling method for space tethered systems in the restricted three-body problem for deep space exploration, characterized in that... The steps are as follows: Step 1: Using the center of mass of the three-body system Establish an inertial coordinate system with the origin. Define respectively , and Let be the position vectors of the three point masses in the inertial coordinate system. Then, the equations of motion of the three-body system can be rewritten in Euler form as the three-body equations of motion: Among them: Among them, , , , indicating the first A point mass Pointing to the A point mass The position vector, and has ; , and The masses of the three particles are respectively. It is the gravitational constant; Step 2: Set the inertial coordinate system Redefinition: The origin lies at two large mass points. and The common centroid, plane is and The scalar form of the elliptical restricted three-body system, given the plane of motion, is: in, Let be the potential energy function. Let be the eccentricity of the two-body orbit. It is a true near angle; Step 3: Consider orbital perturbations The dynamic equations of the restricted three-body system under the influence are: in , and These represent the components of orbital perturbation in different directions; Step 4, based on Establish a dynamic model of the spatial rope system using a translational point: Will Substituting the restricted three-body system dynamics equations from step 3, and applying control actions at the in-plane angles, out-of-plane angles, and rope length respectively, we obtain the following dynamics equations: in for x-coordinate of the translation point , It is a dimensionless distance. This is the length of the rope. Angle in the orbital plane It is the exterior angle of the orbital plane; The The The in , and These represent the masses of the two main celestial bodies.

2. The method for modeling space tethered systems in the restricted three-body problem for deep space exploration according to claim 1, characterized in that: The equations of motion for the three-body system consisting of the three particles in space are: in and The first The position vector and mass of a point mass .

3. The method for modeling space tethered systems in the restricted three-body problem for deep space exploration according to claim 1, characterized in that: The elliptical restricted three-body system is based on two assumptions: ① Of the three point masses, point masses... Compared to quality and Very small It moves under the gravitational pull of two large point masses, and right and Gravitational force is negligible; ② Point mass and They revolve in a circle around their common center of mass.

4. The method for modeling space tethered systems in the restricted three-body problem for deep space exploration according to claim 1, characterized in that: The potential energy function ,in , and These represent the masses of the two main celestial bodies.

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