A method for identifying the motion form of a two-body tethered system in an atypical plane on orbit

Abstract

The present invention discloses a method for identifying the motion form of a two-body tethered system in an atypical plane on orbit. First, a billiard ball model that can describe the dynamic behavior of a tethered system in any atypical plane is constructed in a local orbit coordinate system; second, the dynamic system is converted into a two-dimensional motion plane coordinate system based on Euler's theorem; finally, its motion form is identified based on the motion stability theory and the motion characteristics of the system. The present invention can identify the motion form of a system in any plane, with any initial in-plane pitch angle, and with any initial angular velocity.

Description

Technical Field

[0001] The invention belongs to the technical field of spacecraft flight, and in particular relates to a method for identifying the motion form of a two-body tethered system in an atypical plane on orbit. Background Art

[0002] The on-orbit tether system has the characteristics of high reliability, low cost and reusability. It has great application prospects in ground reconnaissance, three-dimensional imaging, space debris cleaning and other aspects. More and more scientific and technological personnel have shown great interest in it. For example, Kojima et al. studied the chaotic libration of the electrodynamic tether system in an elliptical orbit and pointed out that unless an out-of-plane control orthogonal to the orbital plane is applied, in-plane coupling motion will inevitably occur. Nakanishi et al. analyzed the bifurcation behavior of the pitch motion of the tether system in the orbital plane and found that as the orbital eccentricity increases, a path leading to chaos will appear. For the open / closed axis-spoke tether formation system with the normal of the rotation plane pointing to the earth (that is, the rotation plane is perpendicular to the orbital plane), Avanzini et al. numerically discussed the influence of the tether mass, damping and stiffness on the system motion stability. Zukovic et al. explored the nonlinear characteristics of the linear tether formation in the orbital plane under parameter excitation, and globally demonstrated the bounded motion, tumbling and monotonic spin through a parameter domain. For an in-plane tethered return capsule mission, Aslanov derived a tether release control law, in which the required tether length is significantly less than that of the on-orbit YES2 mission. Based on the feedback linearization method, Paul et al. designed a nonlinear controller to suppress the in-plane coupled oscillations of the tethered system, but pointed out that this method is only effective for in-plane oscillations. Lim et al. analyzed the recovery dynamics of orbital debris captured by an in-plane space tether, and numerically evaluated the orbital radius, capture speed, and recovery speed of the debris. Qi et al. deeply considered the strong coupling and nonlinear characteristics of the in-plane tether, discussed the stability of the tethered towing system, and built a ground experimental platform to verify the numerical results.

[0003] Existing academic achievements show that the dynamic research of on-orbit tethered systems is mostly limited to the orbital plane or a typical plane perpendicular to the orbital plane, while there is insufficient research on the system motion in other atypical planes. Currently, there is no effective method to discuss the motion form of the system in an arbitrary plane. Summary of the invention

[0004] Purpose of the invention: The present invention provides a method for identifying the motion form of a two-body tethered system in an atypical plane on orbit. For a system in any plane, with any initial in-plane pitch angle, and with any initial angular velocity, this method can identify the motion form of the system.

[0005] Technical solution: The present invention provides a method for identifying the motion form of a two-body tethered system in an atypical plane on orbit, which specifically includes the following steps:

[0006] (1) A table tennis model is constructed in the local orbital coordinate system to describe the dynamic behavior of any atypical in-plane tethered system.

[0007] (2) Based on Euler's theorem, the dynamic system is transformed into a two-dimensional motion plane coordinate system;

[0008] (3) Identify the motion form based on the theory of motion stability and the system motion characteristics.

[0009] Furthermore, the implementation process of step (1) is as follows:

[0010] Construct an in-plane tether system orbiting the Earth. The system consists of a mother star M, a satellite S, and an elastic tether. The system moves in the plane Π. Its out-of-plane motion is suppressed by the control force acting on the satellite orthogonal to the plane Π. The normal vector n between the orbital plane Λ and the motion plane Π is o and n m It is possible not to satisfy n o ||n m Not satisfied with n o ⊥ m ; The in-plane pitch angle θ is defined in the motion plane Π; The parent star and the daughter star are both regarded as point masses, and their masses are expressed as m M and m S Indicates that m S <<m M ; Assume that the parent star always flies in a circular Keplerian orbit, and the energy dissipation caused by the stretching and impact of the massless tether is negligible;

[0011] Construct a local orbital coordinate system ox o y o z o , the origin of the coordinate system is fixed at the parent star o, and the orbital angular velocity Ω o Rotate around the center of the earth; x o Axis and z o The axes are all located in the orbital plane Λ, pointing from the origin o to the direction of motion of the parent star and the center of mass of the Earth respectively; o The axis can be determined by the right-hand rule, and the normal vector n o Overlapping at y o Axis; define two rotation angles, where the rotation angle ψ represents the rotation from n o to n m Angle, and the other rotation angle Indicates that o Axis to normal vector n m The angle of projection on plane Λ, the two rotation angles can uniquely determine the normal vector nm ;

[0012] Applying Newton's second law to the satellite S, its local orbital coordinate system ox o y o z o The dynamic equation is:

[0013]

[0014] In the formula, “··” represents the second derivative with respect to time t, represents the position vector of the sub-star; the gravity pointing to the center of mass of the Earth is:

[0015]

[0016] In the formula, μ E represents the Earth's gravity parameter, represents the position vector of the Earth's center of mass in the local orbital coordinate system, R E and H M represent the average radius of the Earth and the orbital height of the parent star respectively; the tension generated by the stretching and deformation of the tether is:

[0017]

[0018] In the formula, E t and A t are Young's modulus and cross-sectional area of ​​the tether, L u represents the stress-free length of the tether; when the tether is relaxed, When the tether tension is equal to 0, the local orbital coordinate system ox o y o z o The inertial force caused by rotating around the earth is:

[0019]

[0020] In the formula, is the orbital angular velocity; since the local orbital coordinate system where the satellite is located rotates around the earth, there is a Coriolis force:

[0021]

[0022] In the formula, Relative velocity of the satellite; out-of-plane control force satisfy:

[0023]

[0024] In the formula, represents the component of gravity perpendicular to the plane π.

[0025] Furthermore, the implementation process of step (2) is as follows:

[0026] Construct a transformation coordinate system ox t y t z t , x t Axis and z t The axes are located in the motion plane Π and are orthogonal to each other, y t The axis satisfies the right-hand rule when the rotation angle satisfies ψ=0 and When x t Axis, y t Axis, z t The axis coincides with x o Axis, y o Axis, z o Axis; Based on Euler's theorem, the transformation relationship between the sub-star position vector and two different coordinate systems is derived:

[0027]

[0028] Finite rotation tensor and rotation angle ψ and Related to:

[0029]

[0030] In the formula, For the satellite S in the transformed coordinate system ox t y t z t The position vector under the transformation; after the transformation, the second component of the sub-star position vector is always equal to 0; therefore, choose and their derivatives define a vector variable:

[0031]

[0032] Then, construct a Poincaré map P that satisfies:

[0033] P(χ (w) ) = x (w+1) ∈Γ (10)

[0034] In the formula, Γ={(χ1,χ2,χ3,χ4)|χ3=χ 3,p ,χ1<0} represents the Poincaré section, χ 3,p is a chosen constant, χ (w) represents the w-th Poincaré point on the Poincaré section.

[0035] Furthermore, the implementation process of step (3) is as follows:

[0036] To evaluate stability, the Poincaré map is linearized:

[0037] χ (w+1) -x p =DP(χ p )(x (w) -x p ) (11)

[0038] In the formula, χ p represents a fixed point on the Poincaré section, that is, DP(χ p ) means that the Poincaré map P at point χ p The Jacobian matrix at the position; calculate the characteristic root λ of the Jacobian matrix r (r=1,2,3,4), we get the following system in-plane motion stability criterion:

[0039]

[0040] According to the motion trajectory of the sub-star, if the trajectory of the sub-star is closed, the system is in spin motion; if the trajectory of the sub-star is not closed, the system exhibits pendulum-like motion.

[0041] Furthermore, the It can also be the component of tension, inertia force or Coriolis force perpendicular to plane π.

[0042] Beneficial effects: Compared with the prior art, the present invention has the following beneficial effects: For a system in any plane, with any initial in-plane pitch angle, and with any initial angular velocity, the present invention can identify the motion form of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a schematic diagram of the billiard ball model of a two-body rope system;

[0044] Figure 2 is the system motion in the local orbital coordinate system in the form of spin motion;

[0045] Figure 3 The motion trajectory of the sub-star in the transformed coordinate system under the form of spin motion;

[0046] Figure 4 The system motion in the local orbital coordinate system under the irregular motion form

[0047] Figure 5 The motion trajectory of the sub-satellite in the transformed coordinate system under the irregular motion form;

[0048] Figure 6 is the system motion in the local orbital coordinate system in the form of simple pendulum motion;

[0049] Figure 7 It is the motion trajectory of the sub-star in the transformed coordinate system under the form of simple pendulum motion. DETAILED DESCRIPTION

[0050] The present invention is further described in detail below with reference to the accompanying drawings.

[0051] The present invention provides a method for identifying the motion form of a two-body tethered system in an atypical plane on orbit. First, a billiard ball model that can describe the dynamic behavior of a tethered system in any atypical plane is constructed in a local orbit coordinate system. Then, based on Euler's theorem, the dynamic system is converted into a two-dimensional motion plane coordinate system. Then, its motion form is identified based on the motion stability theory and the system motion characteristics, instead of using ground equivalent experiments to observe the system motion form. The specific implementation process is as follows:

[0052] like Figure 1 As shown in Figure 1, an in-plane tether system orbiting the Earth is studied. The system consists of a parent star M, a satellite S, and an elastic tether, and moves in a plane Π. Its out-of-plane motion is suppressed by the control force acting on the satellite orthogonal to the plane Π. It should be emphasized that the normal vector n between the orbital plane Λ and the motion plane Π is o and n m It is possible not to satisfy n o ||n m Not satisfied with n o ⊥ m In addition, the in-plane pitch angle θ is defined in the motion plane Π. The parent star and the daughter star are both regarded as point masses, and their masses are expressed as m M and m S Indicates that m S <<m M Therefore, it can be assumed that the parent star always flies in a circular Keplerian orbit. At the same time, the energy dissipation caused by the stretching and impact of the massless tether is negligible.

[0053] In order to describe the motion behavior of the system more clearly, a local orbital coordinate system ox is constructed. o y o z o ,like Figure 1 The origin of the coordinate system is fixed at the parent star o, and the orbital angular velocity Ω o Rotate around the center of the earth; x o Axis and z o The axes are all located in the orbital plane Λ, and they point from the origin o to the direction of motion of the parent star and the center of mass of the Earth respectively; o The axis can be determined by the right-hand rule. Note that the normal vector n o Overlapping at y o At the same time, two rotation angles are defined, where the rotation angle ψ represents the rotation from n oto n m Angle, and the other rotation angle Indicates that o Axis to normal vector n m The angle of projection on plane Λ. Obviously, the normal vector n can be uniquely determined by these two rotation angles. m .

[0054] Applying Newton's second law to the satellite S, its local orbital coordinate system ox o y o z o The dynamic equation can be written as:

[0055]

[0056] In the formula, “··” represents the second derivative with respect to time t, represents the position vector of the sub-star. The gravity pointing to the center of mass of the Earth can be written as:

[0057]

[0058] In the formula, μ E represents the Earth's gravity parameter, represents the position vector of the Earth's center of mass in the local orbital coordinate system, where R E and H M They represent the average radius of the Earth and the orbital height of the parent star respectively. The tension generated by the stretching and deformation of the tether is expressed as:

[0059]

[0060] In the formula, E t and A t are Young's modulus and cross-sectional area of ​​the tether, L u represents the stress-free length of the tether. Obviously, when the tether is relaxed, When , the tether tension is equal to 0. In addition, from the local orbital coordinate system ox o y o z o The inertial force caused by rotating around the earth is expressed as:

[0061]

[0062] In the formula, is the orbital angular velocity. Since the local orbital coordinate system where the satellite is located rotates around the earth, there is a Coriolis force:

[0063]

[0064] in, Represents the relative speed of the satellite. satisfy:

[0065]

[0066] here, It represents the component of gravity (tension, inertia or Coriolis force) perpendicular to plane π.

[0067] The table tennis model constructed above can be used to discuss the motion form of a tethered system flying in an atypical plane.

[0068] The motion stability of this on-orbit tether system is discussed further. For the convenience of analysis, a transformation coordinate system ox is constructed. t y t z t ,like Figure 1 Here, x t Axis and z t The axes are located in the motion plane Π and are orthogonal to each other, y t The axis satisfies the right-hand rule. It is easy to see that when the rotation angle satisfies ψ=0 and When x t Axis, y t Axis, z t The axis coincides with x o Axis, y o Axis, z o axis.

[0069] Based on Euler's theorem, the coordinate system transformation problem can be deduced from the transformation relationship between the sub-satellite position vector and two different coordinate systems:

[0070]

[0071] Here, the finite rotation tensor is related to the rotation angle ψ and Related, written as:

[0072]

[0073] In the formula, For the satellite S in the transformed coordinate system ox t y t z t It can be seen that after the transformation, the second component of the sub-star position vector is always equal to 0. Therefore, choose and their derivatives define a vector variable:

[0074]

[0075] Then, construct a Poincaré map P that satisfies:

[0076] P(χ (w)) = x (w+1) ∈Γ (10)

[0077] Here, Γ={(χ1,χ2,χ3,χ4)|χ3=χ 3,p ,χ1<0} represents the Poincaré section, χ 3,p is a chosen constant, χ (w) represents the w-th Poincaré point on the Poincaré section.

[0078] To evaluate stability, this Poincaré map is linearized:

[0079] χ (w+1) -x p =DP(χ p )(x (w) -x p ) (11)

[0080] In the formula, χ p represents a fixed point on the Poincaré section, that is, DP(χ p ) means that the Poincaré map P at point χ p The Jacobian matrix at . We only need to calculate the characteristic root λ of the Jacobian matrix r (r=1,2,3,4), we can get the following system in-plane motion stability criterion:

[0081]

[0082] Based on the motion trajectory of the sub-star, the following identification can be further made: if the trajectory of the sub-star is closed, the system is in spin motion; if the trajectory of the sub-star is not closed, the system exhibits pendulum-like motion.

[0083] In this embodiment, it is assumed that the parent star M is at a distance H from the earth. M =500km in a circular Keplerian orbit, and the mass of the satellite is m S =0.1×10 3 kg, the Young's modulus and cross-sectional area of ​​the space tether are E t =200Gpa and A t =1×10 -6 m 2 The length of the stress-free tether between the two stars is L u = 5km. At the initial moment, the tether outside the star is neither loose nor tight. Now take the rotation angle ψ = π / 6 and The initial in-plane pitch angle θ0 = π / 8 and the initial angular velocity ratio is ω0 / Ω o =2.0 to analyze the motion form. The four characteristic roots of the system can be first calculated as Obviously, there is |λ r | max ≤1, therefore, it is necessary to further discuss the numerical results of the system. The corresponding motion form of the system is as follows Figure 2 , Figure 3 As shown, it can be seen that in Figure 2 The middle tether is always in a taut state. Figure 3 The neutron star trajectory is a closed orbit, which means that the system's motion form under the action of this parameter is spin motion.

[0084] Similarly, the rotation angle ψ = π / 6 and The initial in-plane pitch angle θ0 = π / 8, and only the initial angular velocity ratio is changed to ω0 / Ω o =-2.0. The four characteristic roots of the system can be obtained as Obviously, there is |λ r v max >1. The corresponding motion form of the system is as follows Figure 4 , Figure 5 As shown in Figure 2, this is an irregular motion. It is not difficult to see that the numerical results are consistent with the theoretical identification results.

[0085] Then take the rotation angle ψ = π / 6 and The initial in-plane pitch angle θ0 = π / 8, and the initial angular velocity ratio is changed to ω0 / Ω o =1.0. The four characteristic roots of the system can be obtained as Obviously, there is |λ r v max <1, the system movement form is as follows Figure 6 , Figure 7 As shown. Figure 6 The middle tether is tight and Figure 7 The neutron star trajectory is not closed, which indicates that under this parameter, the system's motion is pendulum-like.

Claims

1. A method for identifying the motion form of a two-body tethered system in an atypical plane on orbit, characterized in that: The following steps are involved: (1) A table tennis model is constructed in the local orbital coordinate system to describe the dynamic behavior of any atypical in-plane tethered system. (2) Based on Euler's theorem, the dynamic system is transformed into a two-dimensional motion plane coordinate system; (3) Identify the motion form based on the theory of motion stability and the system motion characteristics; The implementation process of step (1) is as follows: Construct an in-plane tether system orbiting the Earth. The system consists of a mother star M, a satellite S, and an elastic tether. The system moves in the plane Π. Its out-of-plane motion is suppressed by the control force acting on the satellite orthogonal to the plane Π. The normal vector n between the orbital plane Λ and the motion plane Π is o and n m It is possible not to satisfy n o ||n m Not satisfied with n o ⊥ m ; The in-plane pitch angle θ is defined in the motion plane Π; The parent star and the daughter star are both regarded as point masses, and their masses are expressed as m M and m S Indicates that m S <<m M ; Assume that the parent star always flies in a circular Keplerian orbit, and the energy dissipation caused by the stretching and impact of the massless tether is negligible; Construct a local orbital coordinate system ox o y o z o , the origin of the coordinate system is fixed at the parent star o, and the orbital angular velocity Ω o Rotate around the center of the earth; x o Axis and z o The axes are all located in the orbital plane Λ, pointing from the origin o to the direction of motion of the parent star and the center of mass of the Earth respectively; o The axis can be determined by the right-hand rule, and the normal vector n o Overlapping at y o Axis; define two rotation angles, where the rotation angle ψ represents the rotation from n o to n m Angle, and the other rotation angle Indicates that o Axis to normal vector n m The angle of projection on plane Λ, the two rotation angles can uniquely determine the normal vector n m ; Applying Newton's second law to the satellite S, its local orbital coordinate system ox o y o z o The dynamic equation is: In the formula, "." represents the second derivative with respect to time t, represents the position vector of the sub-star; the gravity pointing to the center of mass of the Earth is: In the formula, μ E represents the Earth's gravity parameter, represents the position vector of the Earth's center of mass in the local orbital coordinate system, R E and H M represent the average radius of the Earth and the orbital height of the parent star respectively; the tension generated by the stretching and deformation of the tether is: In the formula, E t and A t are Young's modulus and cross-sectional area of ​​the tether, L u represents the stress-free length of the tether; when the tether is relaxed, When the tether tension is equal to 0, the local orbital coordinate system ox o y o z o The inertial force generated by rotating around the earth is: In the formula, is the orbital angular velocity; since the local orbital coordinate system where the satellite is located rotates around the earth, the Coriolis force exists: In the formula, Relative velocity of the satellite; out-of-plane control force satisfy: In the formula, represents the component of gravity perpendicular to the plane Π; The implementation process of step (2) is as follows: Construct a transformation coordinate system ox t y t z t , x t Axis and z t The axes are located in the motion plane Π and are orthogonal to each other, y t The axis satisfies the right-hand rule when the rotation angle satisfies ψ=0 and When x t Axis, y t Axis, z t The axis coincides with x o Axis, y o Axis, z o Axis; Based on Euler's theorem, the transformation relationship between the sub-star position vector and two different coordinate systems is derived: Finite rotation tensor and rotation angle ψ and Related to: In the formula, For the satellite S in the transformed coordinate system ox t y t z t The position vector under the transformation; after the transformation, the second component of the sub-star position vector is always equal to 0; therefore, choose and their derivatives define a vector variable: Then, construct a Poincaré map P that satisfies: In the formula, Γ={(χ1,χ2,χ3,χ4)|χ3=χ 3,p ,χ1<0} represents the Poincaré section, χ 3,p is a chosen constant, χ (w) represents the w-th Poincaré point on the Poincaré section; The implementation process of step (3) is as follows: To evaluate stability, the Poincaré map is linearized: x (w+1) -x p =DP(x p )(x (w) -x p ) (11) In the formula, χ p represents a fixed point on the Poincaré section, that is, DP(χ p ) means that the Poincaré map P at point χ p The Jacobian matrix at the position; calculate the characteristic root λ of the Jacobian matrix r (r=1,2,3,4), we get the following system in-plane motion stability criterion: According to the motion trajectory of the sub-star, if the trajectory of the sub-star is closed, the system is in spin motion; if the trajectory of the sub-star is not closed, the system exhibits pendulum-like motion.

2. The method for identifying the motion form of a two-body tethered system in an atypical plane on-orbit according to claim 1, characterized in that: Said It can also be the component of tension, inertia force or Coriolis force perpendicular to the plane ∏.

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