A method for identifying chaos of a non-orbital-plane perturbed tether debris drag system

Abstract

The application discloses a chaos identification method for a non-orbital surface disturbed tethered debris drag system, and comprises the following steps: S1, constructing a debris drag system, wherein the debris drag system is composed of a drag device and debris through a drag rope, and a centroid o of the debris drag system is always operated on a Kepler circular orbit around a centroid E of the earth; S2, constructing a mathematical equation of the debris drag system according to a total kinetic energy expression of the debris drag system; S3, solving a balance point and a homoclinic orbit of the debris drag system according to the mathematical equation of the debris drag system; and S4, combining the homoclinic orbit with a Mayne-cov method to construct a chaos discriminant; if the chaos discriminant has a simple zero point, then chaotic motion will occur to a non-orbital surface roll angle of the debris drag system. The application can effectively predict chaos and avoid adverse effects of a disturbed traction force on the non-orbital surface debris drag system.

Description

TECHNICAL FIELD

[0001] The present application relates to non-orbital surface disturbed debris drag system, especially to a non-orbital surface disturbed tethered debris drag system chaos identification method. BACKGROUND

[0002] In-orbit tethered systems are strongly nonlinear structures, and there are a large number of complex dynamic phenomena in the process of working operation, which have attracted much attention of researchers. For example, Liu et al. (Liu HT, Yang LP, Zhang QB, Zhu YW. An investigation on tether-tugging de-orbit of defunct geostationary satellites. Science China Technological Sciences 2012; 55(7): 2019-2027) studied the de-orbit dynamics of a class of tethered systems, and successfully achieved the de-orbit of a defunct geostationary satellite to a graveyard orbit, while avoiding the risks of tether entanglement, breakage, and impact. Based on the mass-spring model of the tether, Jasper et al. (Jasper L, Schaub H. Input shaped large thrust maneuver with a tethered debris object. Acta Astronautica 2014; 96: 128-137) discussed the dynamics of a debris removal system, and the spectral energy of the tether was effectively dissipated at the natural frequency of the tether. Mantellato et al. (Mantellato R, Olivieri L, Lorenzini EC. Input shaped large thrust maneuver with a tethered debris object. Acta Astronautica 2017; 138: 559-569) discussed the stability of the tethered system during the maneuver, and the problems of orbital decay, lateral-radial dynamic coupling, and debris attitude internal resonance were analyzed in detail. Yang et al. (Yang KY, Misra AK, Zhang JR, Qi R, Lu S, Liu Y. Dynamics of a debris towing system with hierarchical tether architecture. Acta Astronautica 2020; 177: 891-905) studied the dynamics of a class of orbital tethered systems with hierarchical tether architecture, and the equilibrium configurations of the system were derived, and their stability was evaluated by linearization theory.Faizullin et al. (Faizullin LF, Pikalov RS. Tether system motion analysis of the various space debris’ anchor point positions. Journal of Physics: Conference Series 2021; 1745:012060) obtained the dynamic response of the debris attitude in the tether system through numerical calculation. Cui et al. (Cui J, Shen T, Wei T, Chu ZY. Tangling and instability effect analysis of initial in-plane / out-of-plane angles onelectrodynamic tether deployment under gravity gradient. Chinese Journal of Aeronautics 2021; 34(1): 1-9) explored the release dynamics of a three-dimensional electrodynamic tether system, and gave a numerical safety zone in the process of releasing the tether. Shahbazzadeh et al. (Shahbazzadeh ZJ, Vatankhah R, Eghtesad M, Assadian N. Development and verification of a flexible tetheredsatellite system model considering the fuel slosh. Multibody System Dynamics 2022; 56(3): 289-312) simulated the dynamic response of a tether system with two solar panels, and they made a parameter study on the mass of the satellite platform and the length of the solar panel. Whether the space tether system has chaotic motion has not been revealed, and there is no literature on identifying the chaos of such systems. SUMMARY

[0003] The purpose of the present application is to provide a non-orbital plane disturbed tethered debris tether system chaos identification method that can identify whether there is non-orbital plane chaotic behavior and effectively predict chaos.

[0004] Technical scheme: A non-orbital plane disturbed tethered debris tether system chaos identification method, comprising the following steps:

[0005] S1, constructing a debris towing system composed of a towing device and a debris through a towing rope, and setting the center of mass o of the debris towing system to run on a Keplerian circular orbit around the center of mass E of the Earth at all times;

[0006] S2, constructing a mathematical equation of the debris towing system according to a total kinetic energy expression of the debris towing system;

[0007] S3, solving a balance point and a homoclinic orbit of the debris towing system according to the mathematical equation of the debris towing system;

[0008] S4, constructing a chaos discriminant by combining the homoclinic orbit with the method of Meniscus, and if the chaos discriminant has a simple zero point, chaotic motion of a non-orbital plane roll angle of the debris towing system occurs.

[0009] Further, in step S1, the towing rope is regarded as a light rigid rod, and a traction force applied to the debris towing system is along a tangential direction of the orbit.

[0010] Further, in step S2, the total kinetic energy expression of the debris towing system is:

[0011] K = K t + K r + K d

[0012] K t represents kinetic energy generated by translation of the center of mass O of the debris towing system along the orbit; K r represents kinetic energy generated by rotation of the center of mass o of the debris towing system around itself; and K d represents kinetic energy generated by winding and unwinding of the towing rope; and specific expressions of the kinetic energies are:

[0013]

[0014] In the formula, "'" represents derivation with respect to time t; m t is the mass of the towing device, and m d is the mass of the debris; l represents the length of the towing rope; r c and δ respectively represent the orbital radius and the orbital inclination angle; ν represents the orbital true anomaly; α is the included angle from the extension line of the ray EO to the projection of the towing rope on the orbital plane; β is the included angle from the orbital plane to the towing rope; and γ is defined as the non-orbital plane roll angle;

[0015] The gravitational potential energy expression of the system is where μ E represents the Earth's gravitational constant;

[0016] The independent variables α, β and l are selected as generalized coordinates, and the Lagrange function L = K - J is substituted into the second type of Lagrange equation, so that the expression of the dynamic differential equation of the debris drag system is as follows:

[0017]

[0018] Q α , Q β and Q l are generalized forces corresponding to the three independent variables α, β and l.

[0019] Further, in step S2, the dynamic differential equation of the debris drag system is dimensionless transformed, so that:

[0020]

[0021] In the formula, “·” represents derivation with respect to dimensionless time; ξ represents dimensionless length corresponding to the length l of the drag rope, L r represents reference length, and are dimensionless generalized forces corresponding to Q α , Q β and Q l .

[0022] When performing the on-orbit drag task, the debris drag system keeps the in-plane pitch angle at a fixed angle α a during the drag process by an orthogonal force acting on the debris; when the system is in the state-keeping stage, the dimensionless change rate of the length of the drag rope The dimensionless expression of the dynamic differential equation of the debris drag system is a two-dimensional non-autonomous system dynamic differential equation:

[0023]

[0024] In the formula, σ 2 = 1 + 3cos 2 α a , b = -(T0 sinα a ) / (m t lω c 2 ).

[0025] Further, in step S3, the detailed steps for solving the equilibrium point and homoclinic orbit of the debris drag system are as follows:

[0026] The variable vector is introduced, so that the two-dimensional non-autonomous system dynamic differential equation is rewritten in the state space form:

[0027]

[0028]

[0029]

[0030] wherein f(beta), g(beta, nu) represent two-dimensional vector field and periodic perturbation term respectively;

[0031] Let Then the stable non-orbital plane roll angle beta 1s and unstable non-orbital plane roll angle beta 1u are calculated.

[0032]

[0033] The stable non-orbital plane roll angle beta 1s and unstable non-orbital plane roll angle beta 1u correspond to stable equilibrium point (beta 1s , 0) and unstable equilibrium point (beta 1u , 0) respectively.

[0034] Ignoring the periodic perturbation term, let a T = 0, gamma β = 0, then the corresponding Hamiltonian system is obtained:

[0035]

[0036] The initial integral of the Hamiltonian system is as follows:

[0037]

[0038] Then the homoclinic orbit of the Hamiltonian system is as follows:

[0039]

[0040] In the formula,

[0041] Further, in step S4, the homoclinic orbit equation is substituted into the Mneimikov function to obtain a chaos discriminant:

[0042]

[0043] When M(nu0) has a simple zero point, the non-orbital plane roll angle of the debris drag system will have chaotic motion.

[0044] Compared with the prior art, the present application has the following remarkable effects:

[0045] 1. Based on the Melnikov function, it is analyzed whether the homoclinic orbit breaks down near the unstable equilibrium point of the debris-tether system under the perturbation of the traction force, so as to identify whether the non-planar chaotic behavior exists in the system.

[0046] 2. By solving the equilibrium point and homoclinic orbit of the debris-tether system, the chaos can be effectively predicted, and the adverse effects of the perturbation of the traction force on the non-planar debris-tether system can be avoided. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 Fig. 1 is a schematic diagram of a debris-tether system in orbit around the earth;

[0048] Figure 2 Fig. 2 is a schematic diagram of static bifurcation of the debris-tether system;

[0049] Figure 3 Fig. 3 is a schematic diagram of the parameter chaotic domain;

[0050] Fig. 4(a) is a schematic diagram of the variation of the non-planar roll angle when α a = π / 6, a T = 0.01 and ω T = 15;

[0051] Fig. 4(b) is a schematic diagram of the Poincare section of the dynamic response of the debris-tether system when α a = π / 6, a T = 0.01 and ω T = 15;

[0052] Fig. 4(c) is a schematic diagram of the power spectral density of the debris-tether system when α a = π / 6, a T = 0.01 and ω T = 15;

[0053] Fig. 4(d) is a schematic diagram of the maximum Lyapunov exponent of the debris-tether system when α a = π / 6, a T = 0.01 and ω T = 15;

[0054] Fig. 5(a) is a schematic diagram of the variation of the non-planar roll angle when α a = π / 6, a T = 0.01 and ω T = 30;

[0055] Fig. 5(b) is a schematic diagram of the Poincare section of the dynamic response of the debris-tether system when α a = π / 6, a T = 0.01 and ω T = 30;

[0056] Figure 5(c) shows the a =π / 6, a T =0.01 and ω T Schematic diagram of the power spectrum density of the debris dragging system when =30;

[0057] Figure 5(d) shows the a =π / 6, a T =0.01 and ω T Schematic diagram of the maximum Lyapunov exponent of the debris-dragging system when =30. DETAILED DESCRIPTION

[0058] The present invention will be described in further detail below with reference to the accompanying drawings and specific implementations.

[0059] like Figure 1 The following figure shows a debris drag system consisting of a mass m t The towing device and mass m d The debris is towed by a tow rope. In this embodiment, the tow rope is considered to be a lightweight rigid rod. Since the volume of the towing device and the debris is much smaller than the length of the tow rope, the attitude motion of the rigid bodies at both ends is ignored. The traction force applied to the debris towing system pulls the debris towing system tangentially along the system orbit. At the same time, it is assumed that the center of mass O of the debris towing system always moves around the center of mass E of the Earth in a Keplerian orbit. In addition, Figure 1 The angle from the extension line of ray EO to the projection of the tow rope on the orbital plane represents the pitch angle α in the orbital plane, and the angle from the orbital plane to the tow rope is defined as the non-orbital roll angle β.

[0060] In order to establish the mathematical equation of the system, we need to first give the kinetic energy and potential energy expressions of the debris dragging system. The total kinetic energy expression of the system is:

[0061] K=K t +K r +K d (1)

[0062] Among them, K t represents the kinetic energy generated by the translation of the system's center of mass O along the orbit; K r K represents the kinetic energy generated by the system rotating around its own center of mass O; d It represents the kinetic energy generated by the retraction and extension of the towing rope; the specific expressions of each kinetic energy are as follows:

[0063]

[0064] In the formula, “'” represents the derivative with respect to time t; and represents the system export mass; l represents the length of the towing rope; r c, δ and δ denote the orbital radius and the orbital inclination of the system, respectively; ν denotes the true anomaly of the orbit.

[0065] Assuming that the gravitational potential of the debris-tether system is zero at infinity, the gravitational potential can be written as

[0066]

[0067] where μ E denotes the gravitational constant of the Earth.

[0068] Selecting the independent variables α, β and l as the generalized coordinates, and substituting the Lagrangian L = K - J into the second type of Lagrange equation, the expression of the kinetic differential equation of the debris-tether system is as follows:

[0069]

[0070] where Q α , Q β and Q l are the generalized forces corresponding to the three independent variables α, β and l. The following dimensionless transformation is performed:

[0071]

[0072] Equation (4) can be converted into the following dimensionless form:

[0073]

[0074] where “·” denotes the derivative with respect to the dimensionless time, i.e., the derivative with respect to the true anomaly ν of the orbit; ξ denotes the dimensionless length corresponding to the length l of the tether, L r denotes the reference length,

[0075] The dimensionless generalized force expressions corresponding to Q α , Q β and Q l are as follows:

[0076]

[0077] where denotes the angular velocity of the center of mass O of the debris-tether system around the Earth; T = T0(1 + a T cosω T ν) is the disturbed time-varying traction force acting on the tether device, T0 denotes the desired constant traction force, a T cosω T ν is the disturbance term due to undesirable internal factors, a T denotes the micro-amplitude and |a T | << 1, ω T denotes the disturbance frequency.

[0078] Based on the principle of virtual work, the atmospheric damping perturbation generated by the debris drag system and the atmospheric friction around the earth can be introduced, and the related parameters in formula (7) are specifically represented as

[0079]

[0080] and

[0081]

[0082] wherein, is the mass ratio; C t , C d respectively represent the damping coefficients of the drag device and the debris; ρ t , ρ d respectively represent the atmospheric density corresponding to the altitude of the drag device and the debris; A t , A d respectively represent the windward area of the drag device and the debris; ω E is the spin angular velocity of the earth around the rotation axis, and in the present embodiment, it is assumed that the angular velocity of the atmosphere is consistent with the spin angular velocity of the earth.

[0083] When performing the on-orbit dragging task, the debris drag system generates an orthogonal force acting on the debris during the dragging process, so that the in-plane pitch angle is kept at a fixed angle α a When the system is in the state-keeping phase, i.e., the dimensionless change rate of the drag rope length formula (6) can be written as the following two-dimensional non-autonomous system dynamics differential equation, and the expression is as follows:

[0084]

[0085] In the formula, σ 2 = 1 + 3cos 2 α a , Formula (10) can depict the non-orbital plane rolling dynamics characteristics of the debris drag system.

[0086] Introducing the variable vector formula (10) can be rewritten as the state space form expression:

[0087]

[0088] In the formula,

[0089]

[0090] and

[0091]

[0092] denote the two-dimensional vector field and the periodic perturbation term, respectively. Obviously, the period of the periodic perturbation term g(β,ν) is π.

[0093] First, we need to solve the equilibrium points and the homoclinic orbits of the debris-towing system. Ignoring the perturbation term, i.e., letting a T = 0, γ β = 0, we can obtain the corresponding Hamiltonian system:

[0094]

[0095] The initial integral is

[0096]

[0097] Letting the two-dimensional vector field expression (12) equal to 0, we can calculate the stable non-orbital plane roll angle β 1s and the unstable non-orbital plane roll angle β 1u :

[0098]

[0099] The stability of the non-orbital plane roll angle can be evaluated by the potential energy function of the following Hamiltonian system:

[0100]

[0101] When the potential energy function is a minimum value, the non-orbital plane roll angle is stable; when the potential energy function is a maximum value, the non-orbital plane roll angle is unstable. The stable non-orbital plane roll angle β 1s , the unstable non-orbital plane roll angle β 1u correspond to the stable equilibrium point (β 1s , 0) and the unstable equilibrium point (β 1u , 0), respectively.

[0102] When the number of equilibrium points changes, it means that the system undergoes a static bifurcation, and the bifurcation diagram is shown in Figure 2 . It can be seen from Figure 2 that this is a subcritical pitchfork bifurcation. Obviously, chaos can only occur near the unstable equilibrium point.

[0103] In addition, the homoclinic orbit of the Hamiltonian system can also be calculated as follows:

[0104]

[0105] In the formula, there are

[0106] Based on the Melnikov method, the homoclinic orbit equation (i.e. equation (18)) can be substituted into the Melnikov function, and then the chaos criterion is obtained as

[0107]

[0108] As long as equation (19) has simple zero points, the non-orbital plane roll angle of the debris-tether system will have chaotic motion. This is because the appearance of simple zero points means that the homoclinic orbit of the system will break under the excitation of a small perturbation, so that the stable manifold and the unstable manifold will intersect near the unstable equilibrium point (β 1u ,0), and a large number of intersecting homoclinic points will be generated, thus causing chaos.

[0109] Based on equation (19), the present application can efficiently identify the chaotic motion of the non-orbital plane roll, and then avoid the occurrence of chaos by means of avoiding parameters.

[0110] To test the correctness of the above chaos criterion-equation (19), the following examples are used to study it. Let the mass of the tether device and the debris be m t =100 kg and m d =1500 kg, the length of the tether be l=100 m, the desired constant traction force be T0=1 N, the damping coefficients of the tether device and the debris be C t =2.2 and C d =2.2, the windward areas of the tether device and the debris be A t =0.25 m 2 and A d =1.5 m 2 , the orbital radius of the system mass center be r c =7071 km, and the orbital inclination be δ=π / 3.

[0111] For equation (19), a numerical parameter chaos domain is calculated, as shown in Figure 3 It can be seen that the volume of the parameter chaos domain decreases with the increase of parameters α a and ω T . Therefore, appropriately increasing parameters α a and ω T can reduce the possibility of chaos.

[0112] Let the specified orbital plane pitch angle, micro-amplitude and perturbation frequency be α a =π / 6, a T =0.01 and ω T =15, and at this time, the system parameters are located in the parameter chaos domain of Figure 3 , and the corresponding chaotic algorithm is shown in Figures 4(a)-4(d) It can be calculated that the above parameters satisfy |b / σ 2|>1, so according to formula (17), the equilibrium point (0,0) at this time is an unstable equilibrium point. Figure 4(a) shows the variation of the non-orbital roll angle over time, which shows that this is an irregular motion; Figure 4(b) shows the Poincare section of the system's dynamic response. The local magnification shows that there are a large number of transverse homoclinic points near the unstable equilibrium point (0,0); Figure 4(c) shows the power spectrum density. The dense power spectrum in the figure indicates that the system has a large number of unstable periodic orbits; Figure 4(d) shows the variation of the system's maximum Lyapunov exponent, which can be seen to be always greater than 0. The above results show that this is a typical chaotic motion.

[0113] Just change the perturbation frequency to ω T =30, other parameters remain unchanged, then the debris drag system parameters will no longer be Figure 3 In the parameter chaos domain, the corresponding system motion is as follows Figures 5(a)-5(b) As shown in Figure 5(a), the roll angle variation of the non-orbital surface appears to be a regular motion; the Poincare section in Figure 5(b) does not show a transverse homoclinic point, but rather a closed periodic orbit; the power spectrum density in Figure 5(c) has only three peaks; and the maximum Lyapunov exponent of the system in Figure 5(d) eventually approaches 0. This result indicates that the system is no longer in a chaotic motion but in a quasi-periodic motion.

[0114] The above two sets of example results are consistent with the parametric chaos domain analysis results, which demonstrates the correctness of the method of the present invention.

Claims

1. A method for chaotic identification of a non-planar perturbed tether debris drag system, characterized in that, The steps comprise the following: S1, constructing a debris towing system composed of a towing device and a debris through a towing rope, and setting the center of mass o of the debris towing system to run on a Keplerian circular orbit around the center of mass E of the earth; S2, constructing a mathematical equation of the debris towing system according to a total kinetic energy expression of the debris towing system; S3, solving the equilibrium point and the homoclinic orbit of the debris towing system according to the mathematical equation of the debris towing system; S4, substituting the homoclinic orbit into a Mneimkov function to obtain a chaos discriminant; if the chaos discriminant has a simple zero point, chaotic motion of a non-orbital surface roll angle of the debris towing system occurs; In step S2, the total kinetic energy expression of the debris towing system is: K = K t + K r + K d where K t represents the kinetic energy generated by the translation of the center of mass O of the debris-towing system along the orbit; K r represents the kinetic energy generated by the rotation of the center of mass O of the debris-towing system around itself; K d represents the kinetic energy generated by the extension and retraction of the tow rope; the specific expression of each kinetic energy is: where "'" denotes the derivative with respect to time t; m t m is the mass of the debris and m d is the mass of the debris; l denotes the length of the tether; r c and δ denote the orbital radius and the orbital inclination of the system, respectively; v denotes the true anomaly of the orbit; α is the angle from the extension of the ray EO to the projection of the tether on the orbital plane; β is the angle from the orbital plane to the tether; and γ is defined as the non-orbital plane roll angle; The gravitational potential expression of the system is where μ E denotes the Earth's gravitational constant; Selecting independent variables alpha, beta and l as generalized coordinates, and substituting a Lagrange function L = K-J into a second type of Lagrange equation, an expression of a dynamic differential equation of the debris towing system is as follows: Q α , Q β , and Q l are generalized forces corresponding to the three independent variables a, b, and l.

2. The method of claim 1, wherein, In step S1, the towing rope is regarded as a light rigid rod, and a traction force applied to the debris towing system is in a tangential direction of the orbit.

3. The method of claim 1, wherein, In step S2, a dimensionless transformation is performed on the dynamic differential equation of the debris towing system, and then there is: where "•" denotes differentiation with respect to dimensionless time; ξ denotes a dimensionless length corresponding to the length of the towrope, L r denotes a reference length, and are dimensionless generalized forces corresponding to Q α , Q β , and Q l , respectively. During the on-orbit drag mission, the debris drag system keeps the in-plane pitch angle at a fixed angle α by a normal force acting on the debris during the drag process a ; and when the system is in the state-maintaining phase, the dimensionless drag rope length change rate The dimensionless expression of the dynamic differential equation of the debris drag system is a two-dimensional non-autonomous system dynamic differential equation: where σ 2 = 1 + 3 cos 2 α a , 4. The method of claim 3, wherein, In step S3, detailed steps for solving the equilibrium point and the homoclinic orbit of the debris towing system are as follows: Introducing the vector of variables The two-dimensional non-autonomous system dynamics differential equation is rewritten in state space form: Wherein, f(beta), g(beta, nu) represent a two-dimensional vector field and a periodic perturbation term, respectively; Let then the stable non-plane roll angle β 1s and the unstable non-plane roll angle β 1u is calculated then the stable non-plane-of-orbit roll angle β 1s , the unstable non-plane-of-orbit roll angle β 1u correspond to the stable equilibrium point (β 1s , 0) and the unstable equilibrium point (β 1u , 0), respectively. Neglecting the periodic perturbation term, let a T = 0, γ β = 0, we get the corresponding Hamiltonian system: The initial integral is as follows: Then the homoclinic orbit of the Hamiltonian system is as follows: In the formulae, 5. The method of claim 4, wherein the method further comprises: In step S4, substituting the homoclinic orbit equation into the Mneimkov function to obtain a chaos discriminant: When M(nu0) has a simple zero point, chaotic motion of a non-orbital surface roll angle of the debris towing system occurs.

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