Radial release control method for tail end load in space tether dragging process

Abstract

The invention discloses a radial release control method for a tail end load in a space tether dragging process, which aims at a tether dragging system in a track plane, simplifies the system into a two-degree-of-freedom in-plane system, exports a set of analytic rope length change control rate according to the definition of singular points, and ensures smooth release of a tow rope by setting constraint conditions. Solving a single-valued matrix of the system by using a variation equation, and judging the convergence condition of the in-plane swing angle in the release process according to the characteristic root of the single-valued matrix; and meanwhile, analyzing and giving a condition for keeping the tow rope tight so as to provide a parameter domain to guide the tow rope to be stably released. The invention can asymptotically stabilize and keep the positive tension of the tow rope at any time, and effectively avoids the adverse effects of in-plane large-amplitude swing, negative tension and the like on release in the release process.

Description

Technical Field

[0001] The present invention relates to the technical field of spacecraft flight technology, and particularly to a method for controlling the radial release of an end load during a space tether towing process. Background Art

[0002] An on-orbit towing system is a strongly non-linear structure, and there must be a large number of complex dynamics and control problems during its operation, which has currently attracted great attention from scientific researchers. For example, Liu et al. studied the deorbiting control problem of a class of towing systems, and they successfully realized towing and transferring a decommissioned geostationary satellite to a graveyard orbit while avoiding risks such as tether entanglement, breakage, and impact. Based on the particle-spring model of the tether, Jasper et al. discussed the dynamics of a debris active removal system, and the spectral energy of the tether was effectively dissipated at the natural frequency of the tether. Mantellato et al. explored the stability of the towing system during the maneuvering process, and problems such as the orbital decay of the system, the coupling of lateral and radial dynamics, and the internal resonance of the debris attitude were analyzed in detail. Yang et al. studied the dynamics of a class of orbital towing systems with a hierarchical tether structure, the equilibrium configurations of the systems were derived, and their stability was evaluated through linearization theory. Through numerical calculations, Faizullin et al. obtained the dynamic response results of the debris attitude in the towing system. Cui et al. explored the release control of a three-dimensional electrodynamic tether towing system and gave a numerical safety zone during the rope release process. Shahbazzadeh et al. simulated the dynamic response of a towing system with two solar panels, and they conducted a parametric study on the satellite platform mass and the length of the solar sail.

[0003] Previous studies have shown that there is no research on the release control method of the on-orbit tether towing system, and there is currently no literature on the stable release strategy for such systems. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for controlling the radial release of an end load during a space tether towing process in view of the defects involved in the background art.

[0005] The present invention adopts the following technical solutions to solve the above technical problems:

[0006] A method for controlling the radial release of an end load during a space tether towing process, the center of mass o of the space tether towing system always orbits the earth along an undisturbed Kepler orbit, including a tugboat with a low traction force T towing, a captured end load, and a towing rope for connection. The end load will be stably released through the towing rope and finally reach the specified orbit. The method for controlling the radial release of the end load during the space tether towing process includes the following steps:

[0007] Step 1), Let point P and point E be the perigee of the orbit and the Earth's centroid respectively, and the angle ∠PEo be the true anomaly ν; establish a dimensionless coordinate system o t -χζ, whose origin o t is fixed to the tugboat, the χ-axis points in the opposite direction of the operation of the space tethered towing system, and the ζ-axis points to the Earth's centroid E;

[0008] Step 2), Conduct dynamic modeling on the space tethered towing system to obtain the dimensionless dynamic equation of the space tethered towing system;

[0009] Step 2.1), Establish the total kinetic energy model of the space tethered towing system, and the total kinetic energy K = K t +K r +K d , where K t is the kinetic energy generated by the translation of the centroid o of the space tethered towing system along the orbit, K r is the kinetic energy generated by the rotation of the space tethered towing system around its own centroid o, and K d is the kinetic energy generated by the retraction and deployment of the tow rope;

[0010]

[0011] In the formula, "′" represents the derivative with respect to time t; m1 and m2 are the masses of the tugboat and the end load respectively; r c represents the orbital radius of the current position of the centroid o of the space tethered towing system, δ represents the orbital inclination of the space tethered towing system, l represents the current tow rope length, and α represents the in-plane swing angle of the space tethered towing system;

[0012] Step 2.2), Let the gravitational potential energy of the space tethered towing system be zero at infinity, then In the formula, V is the gravitational potential energy of the space tethered towing system, and μ E = 3.9885×10 14 m 3 / s 2 represents the Earth's gravitational constant;

[0013] Step 2.3), Select the in-plane swing angle α and the current tow rope length l as the generalized coordinates, introduce the Lagrangian function L = K - V, and substitute it into the second Lagrangian equation to obtain the dynamic equation of the space tethered towing system as follows:

[0014]

[0015] In the formula, Q α and Q l are the generalized forces corresponding to α and l respectively;

[0016] Step 2.4), Conduct the following dimensionless transformation: The dimensionless dynamic equations of the space tethered drag system are as follows:

[0017]

[0018] where the true anomaly ν is taken as the dimensionless time, ξ is the dimensionless tether length, l r is the reference tether length, “·” represents the derivative with respect to the dimensionless time ν, κ = 1 + ecosv, e is the orbital eccentricity, ω c is the orbital angular velocity of the center of mass o of the space tethered drag system around the Earth, and u is the dimensionless control force;

[0019] Step 3), establish the release control law of the space tethered drag system and carry out the release or recovery of the tether;

[0020] Step 3.1), substitute α = α a , into the first expression of the dimensionless dynamic equations of the space tethered drag system and analytically solve the following release control law:

[0021]

[0022] where,

[0023] Step 3.2), establish the stability constraint of the release control law;

[0024] Step 3.2.1), substitute the release control law into the first expression of the dimensionless dynamic equations, let α 1* = α and obtain the following state - space equations:

[0025]

[0026] Introduce the transformation α1 = α 1* - α a and α2 = α 2* , and rewrite the state - space equations as:

[0027]

[0028] Step 3.2.2), list the variational equations

[0029] where the Jacobian matrix

[0030] α p (t)=(α p1 ,α p2) represents the periodic solution of the space tethered drag system, where f1 and f2 are two vector components of the rewritten state space equation;

[0031] Step 3.2.3), integrate the variational vector Φ for one period of 2π to obtain the monodromy matrix Β = Φ| v＝2π ; Then the stability constraint of the release control law is that the modulus of all eigenvalues of the monodromy matrix Β is less than 1;

[0032] Step 3.3), establish the feasibility constraint of the release control law;

[0033] Step 3.3.1), based on the second expression of the dimensionless dynamic equation of the space tethered drag system, derive the dimensionless control force

[0034] Step 3.3.2), substitute and its derivative into the equation of the dimensionless control force, and reorganize the dimensionless control force as:

[0035]

[0036] where,

[0037] Step 3.3.3), let u > 0, and the feasibility constraint of the release control law is as follows:

[0038] α a ranges from (0, π / 2), and for parameter α a and e satisfy the following conditions:

[0039]

[0040] Step 3.4), based on the stability constraint and feasibility constraint of the release control law, carry out the release or recovery of the tow rope according to the release control law.

[0041] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0042] The present invention discloses a method for radial release control of the end load during the space tethered drag process, which can be asymptotically stable and maintain a positive tension of the tow rope at all times, effectively avoiding the adverse effects on the release such as large in-plane swing and negative tension during the release process. Description of the Drawings

[0043] Figure 1 is a schematic diagram of the in-plane tethered drag system;

[0044] Figure 2 is a schematic diagram of the parameter domain;

[0045] Figure 3 (a), Figure 3 (b), Figure 3 (c), Figure 3 Figures (a), (b), (c), and (d) are respectively the schematic diagrams of the end load release trajectory, the in-plane swing angle time history, the in-plane angular velocity time history, and the dimensionless control force time history in the present invention. Detailed implementation manners

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0047] The present invention can be implemented in many different forms and should not be considered limited to the embodiments described herein. On the contrary, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present invention to those skilled in the art. In the drawings, components are enlarged for clarity.

[0048] Study an in-plane tethered towing system with a variable rope length. Assume that the center of mass o of the system always orbits the Earth along an undisturbed Kepler orbit, as Figure 1 shown. The system consists of a tugboat towed by a low traction force T, a captured end load, and a connecting tow rope. The end load will be stably released through the tow rope and finally reach a specified orbit. Since the volumes of the two ends of the load are much smaller than the length of the tow rope, they are both regarded as point masses. In addition, the lightweight tow rope is regarded as a rigid rod with a variable length, which is wound on a release mechanism installed in the tugboat, and the release of the tow rope is controlled by this mechanism. The angle ∠PEo is defined as the true anomaly v, where points P and E are the perigee of the orbit and the Earth's center of mass. At the same time, a dimensionless coordinate system o t -χζ is defined, with its origin o t fixed to the tugboat, the χ-axis pointing in the opposite direction of the system's operation, and the ζ-axis pointing to the Earth's center of mass E. This coordinate system will be used to display the end load release process.

[0049] The following conducts dynamic modeling on the space tethered towing system. First, write out the kinetic energy and potential energy of the system. The total kinetic energy of the system can be written as K = K t + K r + K d , which consists of K t , K r , and K d in three parts. Here, K t is the kinetic energy generated by the translation of the center of mass o of the system along the orbit, K r is the kinetic energy generated by the rotation of the system around its own center of mass o, and K d is the kinetic energy generated by the retraction and release of the tow rope. Their specific expressions are as follows:

[0050]

[0051] In the formula, "′" represents the derivative with respect to time t. and represents the derived mass of the system, and m1 and m2 are the tugboat mass and the end load mass. r c represents the orbital radius of the current position of the system's center of mass o, δ represents the orbital inclination of the system, l represents the current towrope length, and α represents the in-plane swing angle of the system.

[0052] Suppose the gravitational potential energy of the system is zero at infinity, and it can be specifically written as:

[0053]

[0054] In the formula, μ E = 3.9885×10 14 m 3 / s 2 represents the Earth's gravitational constant.

[0055] Select the in-plane swing angle α and the current towrope length l as the generalized coordinates, introduce the Lagrangian function L = K - V, and substitute it into the second Lagrangian equation, then the dynamic equation of the tethered towing system can be written:

[0056]

[0057] Among them, Q α and Q l are the generalized forces corresponding to the two independent variables α and l respectively. For the convenience of research, the following dimensionless transformation is carried out:

[0058]

[0059] In the formula, the true anomaly v is used as the dimensionless time, ξ represents the dimensionless towrope length, and l r represents the reference towrope length. In this way, the following dimensionless dynamic equation of the system can be obtained:

[0060]

[0061] Among them, "·" represents the derivative with respect to the dimensionless time v, κ = 1 + ecosv is the defined variable, e is the orbital eccentricity, ω c represents the orbital angular velocity of the system's center of mass o around the Earth, and u is the dimensionless control force.

[0062] The system simplified model constructed by equation (5) can effectively characterize the release dynamic behavior of the two-degree-of-freedom tethered towing system under the action of the control input.

[0063] The following designs a release control law for the tethered towing system and deeply discusses its stability and feasibility. The radial release of the towrope means that the end load will move along a specified in-plane angle αa Away from the tugboat. At the same time, it is required that the oscillation of the towing rope gradually disappears, that is, the in-plane swing angular velocity and angular acceleration both tend to 0 eventually.

[0064] According to the definition of the singularity, substitute the above-mentioned α = α a , and all into the first expression of equation (5). In this way, we can analytically solve the following release control law:

[0065]

[0066] The above release control law can be used to carry out the release or recovery of the towing rope. Here, if the variable satisfies it indicates that the external towing rope is continuously lengthening, that is, the towing rope remains released. Therefore, substitute the inequality into expression (6) to obtain the following condition:

[0067]

[0068] Obviously, only when the system parameters satisfy inequality (7) can the release of the towing rope be guaranteed.

[0069] Immediately afterwards, analyze the stability of the system under the action of this control law (6). Substitute expression (6) into the first expression of equation (5), and assume α 1* = α and It can be reconstructed into the following state-space form:

[0070]

[0071] Then introduce the transformation α1 = α 1* - α a and α2 = α 2* . Then the singularity (α a , 0) on the phase plane is translated to the origin (0, 0). Therefore, the state-space equation (8) is rewritten as:

[0072]

[0073] So far, the following variational equation can be listed:

[0074]

[0075] In the formula, the Jacobian matrix is specifically written as:

[0076]

[0077] Here, α p (t) = (α p1 , α p2) represents the periodic solution of the system, and f1 and f2 are two vector components of Equation (9). It is not difficult to see that this Jacobian matrix is periodic, that is, Df(v + 2π) = Df(v). Integrating the variational vector Φ over a period of 2π, the monodromy matrix can be defined as:

[0078] Β = Φ| v＝2π (12)

[0079] According to the Frechet theory, the stability of the above periodic solution depends on the eigenvalues of the monodromy matrix. That is, only when the modulus of all eigenvalues is less than 1, the in-plane oscillation of the system is stable. Obviously, the stability depends on the system parameters in Expression (11).

[0080] Next, we further evaluate the feasibility of the proposed control law. During the tethered towing process, the tensile force control law can be effectively implemented only when the tow rope is always in a taut state (i.e., maintaining the control force u > 0). Therefore, based on the second expression of Equation (5), the dimensionless control force can be derived as:

[0081]

[0082] Substitute and its derivative into Equation (13), and it can be rearranged as:

[0083]

[0084] where the coefficient

[0085]

[0086] To confirm the sign of Expression (14), the following conditions can be obtained first:

[0087]

[0088] For If the parameters α a and e satisfy the above conditions, the first term of Expression (14) is positive. In addition, as long as α a is in the range (0, π / 2), when there is only a sufficiently small perturbation near the in-plane swing angle α a , the second term of Expression (14) is also greater than 0.

[0089] The present invention can stably release the tow rope along the specified direction, and at the same time can ensure that the tow rope tension is always positive during the whole process, thus ensuring the feasibility of the release process.

[0090] Perform numerical simulation on the tow rope release control law proposed by the present invention. First, give some important parameters of the on-orbit towing system. Let the average radius of the earth be R E= 6371 km, the perigee altitude of the system is H p = 500 km, the mass of the tugboat is m1 = 100 kg, and the length of the towrope is l = 200 m.

[0091] For this controlled system, a parameter domain (α a , T, e) is calculated and presented in Figure 2 . Here, the value ranges of Ψ1 and Ψ2 are (0, π / 2) and (-0.2, 0.2) respectively. When the system parameters α a , T, e are within it, it indicates that the radial release can be carried out stably, and the towrope remains taut throughout the process. As can be seen from Figure 2 , for different specified in-plane angles, this parameter domain is divided into two parts, and the parameter domain changing with the specified in-plane angle changes violently; as the orbital eccentricity increases, this parameter domain will gradually disappear; while the influence of the traction force on this parameter domain is not very significant.

[0092] The following gives an example of the release process of an on-orbit controlled towing system. Let the orbital eccentricity of the system be e = 0.05, the traction force acting on the tugboat be T = 0.02 N, and the initial dimensionless rope length and its change rate be ξ0 = 0.1 and It is expected that the end load is released along the specified in-plane angle α a = π / 10 in a straight line. Here, it can be first calculated that the system parameters are located in the Figure 2 parameter domain.

[0093] Let the initial perturbation of the in-plane swing angle be π / 20, Figure 3 (a) shows the release trajectory of the end load in the dimensionless coordinate system o t -χζ. It can be seen that the end load is released in a straight line along the specified in-plane angle α a . Figure 3 (b) is the time history of the in-plane swing angle, and the towrope completes the release within 1.68 orbital periods. After experiencing certain oscillations, the swing angle finally converges to the specified angle α a = π / 10. Figure 3 (c) shows the change of the in-plane angular velocity of the towrope during the release process, which also finally tends to 0. Moreover, their amplitudes do not exceed the ranges of Ψ1 and Ψ2 given before. Figure 3 (d) plots the time history of the dimensionless control force. The numerical results show that this control force is always greater than 0 throughout the release process. This means that the towrope remains taut during the release process. Therefore, we can conclude that the release control law proposed in the present invention is stable and feasible.

[0094] The above numerical examples show that when the system parameters are within the parameter domain, the end load can be asymptotically stably released along the specified in-plane swing angle, and the tow rope always remains taut, thus verifying the feasibility and correctness of the method of the present invention.

[0095] Those skilled in the art of the present technology can understand that unless otherwise defined, all terms (including technical terms and scientific terms) used herein have the same meaning as the general understanding of those of ordinary skill in the art to which the present invention pertains. It should also be understood that terms such as those defined in a general dictionary should be understood to have a meaning consistent with the meaning in the context of the prior art, and will not be interpreted with an idealized or overly formal meaning unless defined as such herein.

[0096] The specific embodiments described above have further elaborated on the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only for the specific embodiments of the present invention and is not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A radial release control method for the end load during the space tether drag process. The center of mass O of the space tether drag system always orbits the Earth along an undisturbed Kepler orbit, including a tugboat with a low traction force T for towing, a captured end load, and a towing cable for connection. The end load will be stably released through the towing cable and finally reach the designated orbit. It is characterized in that, The radial release control method for the end load during the space tether drag process includes the following steps: Step 1), let point P and point E be the perigee of the orbit and the Earth's centroid respectively, and the angle ∠PEo be the true anomaly v; establish a dimensionless coordinate system o t -χζ, whose origin o t is fixed to the tugboat, the χ-axis points in the opposite direction of the operation of the space tethered towing system, and the ζ-axis points to the Earth's centroid E; Step 2), perform dynamic modeling on the space tether drag system to obtain the dimensionless dynamic equation of the space tether drag system; Step 2.1), establish the total kinetic energy model of the space tethered drag system. The total kinetic energy \(K = K\) t + \(K\) r + \(K\) d , where \(K\) t is the kinetic energy generated by the translational motion of the center of mass \(o\) of the space tethered drag system along the orbit, \(K\) r is the kinetic energy generated by the rotation of the space tethered drag system about its own center of mass \(o\), and \(K\) d is the kinetic energy generated by the pay-in and pay-out of the tether; where "′" represents the derivative with respect to time t; m1 and m2 are the mass of the tugboat and the mass of the end load respectively; r c represents the orbital radius of the current position of the centroid o of the space tethered drag system, δ represents the orbital inclination angle of the space tethered drag system, l represents the current length of the drag tether, and α represents the in-plane swing angle of the space tethered drag system; Step 2.2), let the gravitational potential energy of the space tethered drag system be zero at infinity, then where V is the gravitational potential energy of the space tethered drag system, and μ E = 3.9885×10 14 m 3 / s 2 represents the Earth's gravitational constant; Step 2.3), select the in-plane swing angle α and the current tether length l as the generalized coordinates, introduce the Lagrangian function L = K - V, substitute it into the second Lagrangian equation, and obtain the dynamic equation of the space tether drag system as follows: where Q α and Q l are the generalized forces corresponding to α and l, respectively; Step 2.4), perform the following dimensionless transformation: The dimensionless dynamic equation of the space tethered drag system is obtained as follows: where the true anomaly \(v\) is taken as the dimensionless time, \(\xi\) is the dimensionless rope length, \(l\) r is the reference towrope length, a dot “·” represents the derivative with respect to the dimensionless time \(v\), \(\kappa = 1 + e\cos\nu\), \(e\) is the orbital eccentricity, \(\omega\) c is the orbital angular velocity of the center of mass \(o\) of the space tethered towing system around the Earth, and \(u\) is the dimensionless control force; Step 3), establish the release control law of the space tether drag system and carry out the release or recovery of the tether; Step 3.1), substitute α = α a and into the first expression of the dimensionless dynamic equation of the space tethered drag system, and analytically solve the following release control law: Wherein, Step 3.2), establish the stability constraint of the release control law; Step 3.2.1), substitute the release control law into the first expression of the dimensionless dynamics equation, and let α 1* = α and obtain the following state-space equation: Introduce the transformation α1 = α 1* -α a and α2 = α 2* , and rewrite the state - space equation as: Step 3.2.2), list the variational equation In the formula, the Jacobian matrix α p \( \tau(t) = (\alpha p1 , \alpha p2 )\) represents the periodic solution of the space tethered towing system, and \(f_1\), \(f_2\) are the two vector components of the rewritten state space equation; Step 3.2.3), integrate the variational vector Φ over one period of 2π to obtain the monodromy matrix Β = Φ| v＝2π ; then the stability constraint of the release control law is that the modulus of all the eigenvalues of the monodromy matrix Β is less than 1; Step 3.3), establish the feasibility constraint of the release control law; Step 3.3.1), based on the second expression of the dimensionless dynamic equation of the space tethered drag system, derive the dimensionless control force Step 3.3.2), substitute and its derivative into the equation of the dimensionless control force, and rearrange the dimensionless control force as: In the formula, Step 3.3.3), let u > 0, and obtain the feasibility constraint of the release control law as follows: α a The value range of is (0, π / 2), and for parameter α a and e satisfy the following conditions: Step 3.4), based on the stability constraint and feasibility constraint of the release control law, carry out the release or recovery of the tether according to the release control law.

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