A stable control method for the libration of an electrodynamically tethered satellite

Abstract

The present invention provides a stable control method for the libration of an electrodynamic tethered satellite. Compared with existing control methods based on optimization theory, this control method has an analytical form, a small computational effort, and can handle the effects of control current saturation nonlinearity and unknown system disturbances. The method comprises: establishing orbital dynamics equations based on the six elements of the vernal equinox; establishing attitude dynamics equations based on in-plane and out-of-plane angles; obtaining control equations describing the libration of the tethered satellite through dimensionless transformation; designing an auxiliary dynamics system to compensate for the saturation nonlinearity of the control current; using a radial basis function neural network to approximate multiple disturbances, such as model perturbations and external interferences, and estimating the weight matrix and approximation error to compensate for the disturbances; designing an analytical control law for the current in the electrodynamic tether; correcting singular terms in the analytical control law using a dynamic scaling generalized inverse method; and controlling the libration process of the electrodynamic tethered satellite system using the analytical control law.

Description

Technical Field

[0001] The invention relates to a stable control method for the libration of an electrodynamic tethered satellite. Background Art

[0002] As we all know, the development of space technology faces a significant threat from the increasing number of space debris. To mitigate the threat of space debris, the engineering and academic communities have proposed various deorbiting strategies. Among these solutions, the electrodynamic tethered propulsion method based on tethered satellites is widely considered a promising space debris deorbiting technology due to its low cost, pollution-free nature, and ease of use. When an electrodynamic tether orbits in low Earth orbit, the charge exchange between the conductor and the ionosphere generates a current in the conductor. The interaction between this current and the Earth's magnetic field generates the Lorentz force. Clearly, the Lorentz force can deorbit space debris by regulating the direction of the current. However, due to the complex librational motion (i.e., constant-amplitude oscillations near the equilibrium point) induced by the Lorentz force, electrodynamic tethered satellite systems are subject to instability, which becomes increasingly pronounced as the current increases. It is important to note that the numerous control laws proposed for the complex librational motion of electrodynamic tethered satellite systems fail to simultaneously compensate for current saturation nonlinearities and suppress unknown system disturbances. Summary of the Invention

[0003] Purpose of the Invention: The technical problem to be solved by the present invention is to address the shortcomings of the existing technology and provide a method for stabilizing the libration of an electro-dynamic tethered satellite. The electro-dynamic tethered satellite structure includes a primary satellite, a satellite, and an electro-dynamic tether wrapped with an insulating material. The primary satellite and the satellite are connected by the electro-dynamic tether and operate together in Earth orbit. The two satellites and the tether are respectively regarded as two concentrated masses and a rigid rod. The method includes the following steps:

[0004] Step 1: For the electrodynamic tethered satellite, define the coordinate system and parameters;

[0005] Step 2, define the six numbers of the vernal equinox, e, a, i, w, f and u;

[0006] Step 3, using the six numbers of the vernal equinox defined in step 2, to express the orbital dynamics equation of the electrodynamic tethered satellite;

[0007] Step 4: Design the acceleration vector μ = [μ x μ y μ z ] T and the Lorentz force Θ, then the Lorentz force Θ = mμ, where m = m1 + m2 + m3;

[0008] Step 5, representing the geomagnetic intensity vector Ω;

[0009] Step 6, re-express the vector Ω;

[0010] Step 7, express the Lorentz force Θ;

[0011] Step 8: Based on the Euler-Lagrange equation, establish the attitude dynamics equation of the electrodynamic tethered satellite;

[0012] Step 9, design a new set of dimensionless parameters;

[0013] Step 10: Substitute the parameters obtained in step 9 into the attitude dynamics equation in step 8 to obtain the dimensionless attitude dynamics equation, that is, the control equation that can describe the system's libration motion;

[0014] Step 11: Obtain an analytical control law for the tether current that can effectively control the libration of the electrodynamic tethered satellite.

[0015] Step 1 includes: the masses of the primary and satellite are denoted as m1 and m2 respectively, the length and mass of the electrodynamic tether are denoted as L and m3 respectively; O-xyz represents the orbital coordinate system, where point O is located at the center of mass of the system, Ox, Oy and Oz are the three axes of the orbital coordinate system, the in-plane angle φ and the out-plane angle represents the instantaneous direction of the system, d0 represents the eccentricity, k0 represents the semi-major axis, g0 represents the inclination, Λ0 represents the longitude of the ascending node, ν0 represents the pericentric angle, and n0 represents the true anomaly angle. Represents a real matrix space of m×n dimensions, where m and n are both positive integers, and m and n represent the number of rows and columns of the matrix space respectively. represents an m×1 real vector space, and t represents time.

[0016] Step 2 includes: Based on the classical orbital elements, the six numbers of the vernal equinox, e, a, i, w, f, and u are defined as follows:

[0017]

[0018] Step 3 includes: ignoring the influence of attitude motion, using the six numbers of the vernal equinox defined in step 2 to express the orbital dynamics equation of the electrodynamic tethered satellite as follows:

[0019]

[0020] Among them, ψ=icosu+asinu+1, ν 2 =w 2 +f 2+1, (·)′ is the first derivative of the polynomial (·) with respect to time t, (·)” is the second derivative of (·) with respect to t, so e′, a′, i′, w′, f′, u′ represent the first derivatives of the six roots of the vernal equinox with respect to t, Γ1 is the gravitational constant, and μ x ,μ y ,μ z are the perturbation accelerations along the three axes of the orbital coordinate system.

[0021] Step 5 includes: expressing the geomagnetic intensity vector Ω as follows based on the non-tilted dipole model of the Earth's geomagnetic field:

[0022]

[0023] Among them, Ω x ,Ω y ,Ω z are the geomagnetic intensity vectors along the three axes of the orbital coordinate system, Γ2 represents the magnetic moment of the Earth dipole, and r represents the orbital radius.

[0024] Step 6 involves rewriting the vector Ω using the six numbers of the vernal equinox:

[0025]

[0026]

[0027]

[0028] Step 7 includes: considering that the current inside the insulating tether is uniform, the Lorentz force Θ is expressed as:

[0029]

[0030] Where γ is the unit length vector pointing from the primary star to the secondary star, dl is the infinitesimal length of the insulating tether L, and Ω × is the antisymmetric matrix of Ω, expressed as:

[0031]

[0032] Q represents the current value in the tether, and δ(Q) represents the current saturation nonlinear function, which is expressed as:

[0033]

[0034] Among them, Q max Indicates the maximum value that the current can actually reach.

[0035] Step 8 includes: the attitude dynamics equation of the electrodynamic tethered satellite is expressed as:

[0036]

[0037] in,

[0038]

[0039]

[0040]

[0041] Where M, η are intermediate variables, and χ is the orbital perturbation; They represent the second-order derivatives of the in-plane angle and out-of-plane angle with respect to time t, They represent the first-order derivatives of the in-plane angle and out-of-plane angle with respect to time t, G φ and represents the generalized moment due to the Lorentz force, κ φ and represent the unknown terms caused by modeling uncertainty and external disturbances, respectively.

[0042] Step 9 includes: the new set of dimensionless parameters is:

[0043]

[0044]

[0045]

[0046]

[0047] J i =(m1-m2)(2mMΓ1) -1 Γ2,

[0048]

[0049] in, is the dimensionless parameter, and its physical meaning is the same as before dimensionless; is the polynomial (·) with respect to The first derivative of is (·) relative to The second derivative of is the intermediate parameter.

[0050] Step 10 includes: the dimensionless posture dynamics equation is:

[0051]

[0052] Step 11 includes: designing the state vector and T represents the matrix transpose, which expresses the equation in step 10 in the form of state space:

[0053]

[0054] in, For state vectors q1,q2 The first-order derivative of , D is the state space matrix, κ is the unknown term of the state equation, s, c are intermediate variables; the expression of the parameters is as follows:

[0055]

[0056]

[0057]

[0058]

[0059] Rewrite the current saturation nonlinear function δ(Q) in step 7 as δ(Q) = -Q + σ Q ,in,

[0060]

[0061] Design the following auxiliary dynamics system:

[0062]

[0063] in, is the state vector of the auxiliary system, It's about The first derivative of , ρ>0 is the design parameter;

[0064] Design a new state variable h, the expression is as follows:

[0065]

[0066] Among them, the design intermediate parameters K1 is the gain matrix, k1>0 and k2>0 are design parameters;

[0067] Substituting the auxiliary system and state variable h into the state space equation, the closed-loop system state equation is obtained, which is expressed as:

[0068]

[0069] in, For the state variable h The first derivative of for about The first derivative of ;

[0070] For the unknown term κ of the state equation, the following radial basis function neural network is used for approximation:

[0071]

[0072] in, is the expected weight matrix, α is the number of neuron nodes; b=[b1 b2…b θ ] T is the input vector, b θ represents the θth element of vector b, and Π b is the domain of b, θ is the dimension of b vector; Υ(b)=[υ1(b) υ2(b)…υ α (b)] T represents the basis function vector, υ α (b) represents the αth basis function of the vector Υ(b), where the jth basis function υ j (b) is defined as a Gaussian function, j = 1, 2, ... α, and is expressed as:

[0073]

[0074] Among them, a j and ξ j >0 represents the center vector and function width respectively, and is the approximate error vector, satisfying is an unknown constant, and exp is an exponential function with the natural constant e as the base;

[0075] P * The estimated values ​​of and κ are expressed as and Then the expected weight matrix P * Determined by the following formula:

[0076]

[0077] The required electrodynamic tether current control law is designed as:

[0078]

[0079] in, τ>0, k3>0 and k4>0 are design parameters, c + is the Moore-Penrose inverse matrix of c, yes estimates;

[0080] parameter and Updated according to the following adaptation rules:

[0081]

[0082] in, Representation parameters Relative to The first derivative of θ1>0, θ2>0, θ3>0 and θ4>0 are design parameters. express The initial value of

[0083] Design the following dynamic scale generalized inverse matrix

[0084]

[0085] in, I is the intermediate variable, ||q1|| λ is the λ norm of q1, and the power λ is called the dynamic scale index. hour,

[0086] Will Instead of c + , the analytical control law of the tether current that can effectively control the libration of the electrodynamic tethered satellite is obtained, and the expression is:

[0087]

[0088] Beneficial Effects: This invention discloses a neural network-based adaptive feedback control law for electrodynamic tethered satellites. This control method suppresses the system's libration by adjusting the current in the electrodynamic tether. This control method has an analytical form and minimal computational effort, making it ideal for satellite applications with limited computing resources. The control current curve is smooth, free of high-frequency chattering, and robust to model perturbations and unknown interference. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, and the above and / or other advantages of the present invention will become more apparent.

[0090] Figure 1 This is the schematic diagram of the electrodynamic tethered satellite.

[0091] Figure 2 It is a schematic diagram of the response curve of the in-plane angle φ.

[0092] Figure 3 is the in-plane angular velocity Schematic diagram of the response curve.

[0093] Figure 4is the external angle Schematic diagram of the response curve.

[0094] Figure 5 is the out-of-plane angular velocity Schematic diagram of the response curve.

[0095] Figure 6 It is a schematic diagram of the changing curve of the current in the tether. DETAILED DESCRIPTION

[0096] The present invention is directed to an electro-dynamic tethered satellite, which is constructed as follows: Figure 1 As shown, the present invention includes a main satellite, a satellite, and an electrodynamic tether wrapped with insulating material. The main satellite and the satellite are connected by the electrodynamic tether and operate together in Earth orbit. The two satellites and the tether are respectively regarded as two concentrated masses and a rigid rod. The present invention specifically includes the following steps:

[0097] Step 1, for Figure 1 For the electrodynamic tethered satellite shown in the figure, the following coordinate system and parameters are defined first:

[0098] The masses of the primary and secondary stars are denoted as m1 and m2, respectively, and the length and mass of the electrodynamic tether are denoted as L and m3, respectively. O-xyz represents the orbital coordinate system, where point O is located at the center of mass of the system, Ox, Oy, and Oz are the three axes of the orbital coordinate system, and the in-plane angle φ and out-plane angle represents the instantaneous direction of the system, d0 represents the eccentricity, k0 represents the semi-major axis, g0 represents the inclination, Λ0 represents the longitude of the ascending node, ν0 represents the pericentric angle, and n0 represents the true anomaly angle. represents the m×n dimensional real matrix space, and Represents an m×1 real vector space.

[0099] Step 2: The six classical orbital elements (i.e., d0, k0, g0, Λ0, ν0, and n0) are commonly used to describe the orbital motion of aerospace systems. However, this orbital description method has singularities when k0 = 0 or g0 = 0. The six equinox elements (i.e., e, a, i, w, f, and u) provide an effective solution, ensuring that all orbital motions with g0 less than 180° are free of singularities. Based on the classical orbital elements, the six equinox elements are defined as follows:

[0100]

[0101] Step 3: Ignoring the influence of attitude motion, the orbital dynamics equation of the electrodynamic tethered satellite is expressed as follows using the six numbers of the vernal equinox defined in step 2:

[0102]

[0103] Among them, ψ=icosu+asinu+1, ν 2 =w 2 +f 2 +1, (·)′ is the first-order time derivative, (·)” is the second-order time derivative, Γ1 is the gravitational constant, and μ x ,μ y ,μ z are the perturbation accelerations along the three axes of the orbital coordinate system.

[0104] Step 4: Design the acceleration vector μ = [μ x μ y μ z ] T and the Lorentz force Θ, then the Lorentz force Θ = mμ, where m = m1 + m2 + m3.

[0105] Step 5: Based on the non-tilted dipole model of the Earth's magnetic field, the geomagnetic intensity vector Ω is expressed as follows:

[0106]

[0107] Where Γ2 represents the magnetic moment of the Earth's dipole and r represents the orbital radius.

[0108] Step 6. Using the six numbers of the vernal equinox from step 5, rewrite the vector Ω as:

[0109]

[0110]

[0111]

[0112] Step 7: Considering the uniform current inside the insulating tether, the Lorentz force Θ is expressed as:

[0113]

[0114] Where γ is the unit length vector pointing from the primary star to the secondary star, dl is the infinitesimal length of the insulating tether L, and Ω × is the antisymmetric matrix of Ω, expressed as:

[0115]

[0116] Q represents the current value in the tether, and δ(Q) represents the current saturation nonlinear function, which is expressed as:

[0117]

[0118] Among them, Q max Indicates the maximum value that the current can actually reach.

[0119] Step 8: Based on the Euler-Lagrange equation, the attitude dynamics equation of the electrodynamic tethered satellite is established as:

[0120]

[0121] in,

[0122]

[0123]

[0124]

[0125] G φ and represents the generalized moment due to the Lorentz force, κ φ and represents the unknown terms caused by modeling uncertainties and external disturbances.

[0126] Step 9, design a new set of dimensionless parameters:

[0127]

[0128]

[0129]

[0130]

[0131] J i =(m1-m2)(2mMΓ1) -1 Γ2,

[0132]

[0133] Step 10: Substitute the parameters obtained in step 9 into the attitude dynamics equation in step 8 to obtain the dimensionless attitude dynamics equation, which is also the control equation that can describe the system's libration:

[0134]

[0135] Step 11, design state vector and Express the equations in step 10 in state space form:

[0136]

[0137] in,

[0138]

[0139]

[0140]

[0141]

[0142] Step 12: rewrite the current saturation nonlinear function δ(Q) in step 7 as δ(Q)=-Q+σ Q ,in,

[0143]

[0144] Step 13: To compensate for the nonlinear effect of current saturation, design the following auxiliary dynamics system:

[0145]

[0146] in, is the state vector of the auxiliary system, and ρ>0 is the design parameter.

[0147] Step 14: Design a new state variable h, which is expressed as follows:

[0148]

[0149] in, k1>0 and k2>0 are design parameters.

[0150] In step 15, the auxiliary system in step 13 and the state variable h in step (14) are substituted into the state space equation in step 11 to obtain the closed-loop system state equation, which is expressed as follows:

[0151]

[0152] Step 16: For the unknown term κ of the state equation in step 15, the following radial basis function (RBF) neural network is used for approximation:

[0153]

[0154] in, is the expected weight matrix, α is the number of neuron nodes; b=[b1 b2…b θ ] T is the input vector, and Π b is the domain of b, θ is the dimension of b vector; Υ(b)=[υ1(b) υ2(b)…υ α (b)] T represents the basis function vector, where the jth (j=1,2,…α) basis function υj (b) is defined as a Gaussian function, expressed as follows:

[0155]

[0156] Among them, a j and ξ j >0 indicates the center vector and function width, and is the approximate error vector, satisfying is an unknown constant.

[0157] Step 17, P * The estimated values ​​of and κ are expressed as and Then the expected weight matrix P * It can be determined by the following formula:

[0158]

[0159] Step 18: Based on the equations and parameters established in steps 11 to 17, the required electrodynamic tether current control law is designed as follows:

[0160]

[0161] in, τ>0, k3>0 and k4>0 are design parameters, c + is the Moore-Penrose inverse (pseudo-inverse) matrix of c, yes Estimates.

[0162] Step 19, the parameters in the current control law shown in step 18 and Updated according to the following adaptation rules:

[0163]

[0164] in, θ1>0, θ2>0, θ3>0 and θ4>0 are design parameters, express The initial value of .

[0165] Step 20, the matrix cc in the current control law shown in step 18 T The rank of is less than 2, and there is a singularity problem. In order to solve this problem, a dynamic scale generalized inverse matrix is ​​designed. as follows:

[0166]

[0167] in, ||q1|| λ is the λ norm of q1, and the power λ is called the dynamic scale index. hour,

[0168] Step 21, the Replace c in step (18) + , the analytical control law of the tether current that can effectively control the libration of the electrodynamic tethered satellite is obtained, and the expression is as follows:

[0169]

[0170] Simulation verification:

[0171] In one embodiment of the present invention, the effectiveness of the proposed current analytical control law is verified by the following example. Assuming that the tether current is used only to control the system's libration, the physical parameters of the electrodynamic tethered satellite are defined as follows: m1 = 50 kg, m2 = 5 kg, m3 = 2 kg, L = 2 km. The initial orbital parameters are set as follows: k0 = 7001 km, d0 = 0.01, Λ0=0,ν0=0,n0=0,Γ1=398600.4km 3 / s 2 ,r=6878km,Γ2=8×10 6 Tesla km 3 . Set the initial posture parameters as follows: For the RBF radial basis function (RBF) neural network, the design parameters are selected as follows:

[0172] α=7,

[0173] ξ j ={0.4 0.4 0.5 0.5 30},

[0174]

[0175]

[0176] The control parameters of the designed controller are as follows: k1=1, k2=5, k3=1, k4=3, Q max =0.5A,ρ=2,τ=0.01,λ=2,θ1=0.025,θ2=0.01,θ3=0.03,θ4=0.01. In addition, in order to verify the robustness of the control law, the disturbance term is set to and Based on the above design parameters, the design method proposed in this invention can be used to obtain the analytical control law of the current in the electrodynamic tether. Applying this control law, the orbital dynamics and attitude dynamics of the electrodynamic tethered satellite system are simulated using MATLAB (version 2018b), and the following can be obtained: Figures 2 to 6 The system state response curve is shown. Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 、 Figure 6 It can be seen that under the action of the designed current control law, the inboard / outboard angle of the electrodynamic tether is and its angular velocity They can be driven to the equilibrium point quickly, and the motion amplitude can be quickly and effectively suppressed within the bounded range, that is, the balance kinetic energy of the system is quickly and stably controlled, and the control current in the tether has a relatively smooth response curve, satisfying the saturation constraint Q max =0.5A.

[0177] The present invention provides a method for stabilizing the libration of an electrodynamically tethered satellite. While numerous methods and approaches exist for implementing this technical solution, the foregoing merely represents a preferred embodiment of the present invention. It should be noted that improvements and modifications could be made by those skilled in the art without departing from the principles of the present invention, and such improvements and modifications are also within the scope of protection of the present invention. Components not specified in this embodiment may be implemented using existing technologies.

Claims

1. A method for stabilizing the libration of an electrodynamic tethered satellite, wherein the electrodynamic tethered satellite comprises a primary satellite, a secondary satellite, and an electrodynamic tether wrapped with an insulating material, wherein the primary satellite and the secondary satellite are connected by the electrodynamic tether and operate together in Earth orbit, wherein the two satellites and the tether are respectively regarded as two concentrated masses and a rigid rod, and wherein: The following steps are involved: Step 1: For the electrodynamic tethered satellite, define the coordinate system and parameters; Step 2, define the six numbers of the vernal equinox, e, a, i, w, f and u; Step 3, using the six numbers of the vernal equinox defined in step 2, to express the orbital dynamics equation of the electrodynamic tethered satellite; Step 4: Design the acceleration vector μ = [μ x μ y μ z ] T and the Lorentz force Θ, then the Lorentz force Θ = mμ, where m = m1 + m2 + m3; Step 5, representing the geomagnetic intensity vector Ω; Step 6, re-express the vector Ω; Step 7, express the Lorentz force Θ; Step 8: Based on the Euler-Lagrange equation, establish the attitude dynamics equation of the electrodynamic tethered satellite; Step 9, design a new set of dimensionless parameters; Step 10: Substitute the parameters obtained in step 9 into the attitude dynamics equation in step 8 to obtain the dimensionless attitude dynamics equation, that is, the control equation that can describe the system's libration motion; Step 11, obtaining an analytical control law for the tether current that can effectively control the libration of the electrodynamic tethered satellite; Step 1 includes: the masses of the primary and satellite are denoted as m1 and m2 respectively, the length and mass of the electrodynamic tether are denoted as L and m3 respectively; O-xyz represents the orbital coordinate system, where point O is located at the center of mass of the system, Ox, Oy and Oz are the three axes of the orbital coordinate system, the in-plane angle φ and the out-plane angle represents the instantaneous direction of the system, d0 represents the eccentricity, k0 represents the semi-major axis, g0 represents the inclination, Λ0 represents the longitude of the ascending node, ν0 represents the pericentric angle, and n0 represents the true anomaly angle. Represents a real matrix space of m×n dimensions, where m and n are both positive integers, and m and n represent the number of rows and columns of the matrix space respectively. represents the m×1 real vector space, and t represents time; Step 2 includes: Based on the classical orbital elements, the six numbers of the vernal equinox, e, a, i, w, f, and u are defined as follows: Step 3 includes: ignoring the influence of attitude motion, using the six numbers of the vernal equinox defined in step 2 to express the orbital dynamics equation of the electrodynamic tethered satellite as follows: Among them, ψ=icosu+asinu+1, ν 2 =w 2 +f 2 +1, is a polynomial The first derivative with respect to time t, for The second-order derivative with respect to t, so e′, a′, i′, w′, f′, u′ represent the first-order derivatives of the six numbers of the vernal equinox with respect to t, Γ1 is the gravitational constant, and μ x ,μ y ,μ z are the perturbation accelerations along the three axes of the orbital coordinate system; Step 8 includes: the attitude dynamics equation of the electrodynamic tethered satellite is expressed as: in, Where r is the orbit radius; M, η are intermediate variables, χ is the orbit perturbation; φ”, They represent the second-order derivatives of the in-plane angle and out-of-plane angle with respect to time t, φ', They represent the first-order derivatives of the in-plane angle and out-of-plane angle with respect to time t, G φ and represents the generalized moment due to the Lorentz force, κ φ and represent the unknown terms caused by modeling uncertainty and external disturbances, respectively.

2. The method according to claim 1, characterized in that Step 5 includes: expressing the geomagnetic intensity vector Ω as follows based on the non-tilted dipole model of the Earth's geomagnetic field: Among them, Ω x ,Ω y ,Ω z are the geomagnetic intensity vectors along the three axes of the orbital coordinate system, and Γ2 represents the magnetic moment of the Earth dipole.

3. The method according to claim 2, characterized in that Step 6 involves rewriting the vector Ω using the six numbers of the vernal equinox:

4. The method according to claim 3, characterized in that Step 7 includes: considering that the current inside the insulating tether is uniform, the Lorentz force Θ is expressed as: Where γ is the unit length vector pointing from the primary star to the secondary star, dl is the infinitesimal length of the insulating tether L, and Ω × is the antisymmetric matrix of Ω, expressed as: Q represents the current value in the tether, and δ(Q) represents the current saturation nonlinear function, which is expressed as: Among them, Q max Indicates the maximum value that the current can actually reach.

5. The method according to claim 4, characterized in that Step 9 includes: the new set of dimensionless parameters is: J i =(m1-m2)(2mMΓ1) -1 Γ2, in, is the dimensionless parameter; is the polynomial (·) with respect to The first derivative of is (·) relative to The second derivative of J i , is the intermediate parameter.

6. The method according to claim 5, characterized in that Step 10 includes: the dimensionless posture dynamics equation is: Step 11 includes: designing the state vector and T represents the matrix transpose, which expresses the equation in step 10 in the form of state space: in, For state vectors q1,q2 The first-order derivative of , D is the state space matrix, κ is the unknown term of the state equation, s, c are intermediate variables; the expression of the parameters is as follows: Rewrite the current saturation nonlinear function δ(Q) in step 7 as δ(Q) = -Q + σ Q ,in, Design the following auxiliary dynamics system: in, is the state vector of the auxiliary system, It's about The first derivative of , ρ>0 is the design parameter; Design a new state variable h, the expression is as follows: Among them, the design intermediate parameters K1 is the gain matrix, k1>0 and k2>0 are design parameters; Substituting the auxiliary system and state variable h into the state space equation, the closed-loop system state equation is obtained, which is expressed as: in, For the state variable h The first derivative of for about The first derivative of ; For the unknown term κ of the state equation, the following radial basis function neural network is used for approximation: in, is the expected weight matrix, α is the number of neuron nodes; b=[b1b2…b θ ] T is the input vector, b θ represents the θth element of vector b, and Π b is the domain of b, θ is the dimension of b vector; γ(b)=[υ1(b) υ2(b) … υ α (b)] T represents the basis function vector, υ α (b) represents the αth basis function of the vector Υ(b), where the jth basis function υ j (b) is defined as a Gaussian function, j = 1, 2, ... α, and is expressed as: Among them, a j and ξ j >0 represents the center vector and function width respectively, and is the approximate error vector, satisfying is an unknown constant, and exp is an exponential function with the natural constant e as the base; P * The estimated values ​​of and κ are respectively The required electrodynamic tether current control law is designed as: in, τ>0, k3>0 and k4>0 are design parameters, c + is the Moore-Penrose inverse matrix of c, yes estimates; parameter and Updated according to the following adaptation rules: in, Representation parameters Relative to The first derivative of θ1>0, θ2>0, θ3>0 and θ4>0 are design parameters, express The initial value of Design the following dynamic scale generalized inverse matrix in, I is the intermediate variable, ||q1|| λ is the λ norm of q1, and the power λ is called the dynamic scale index. hour, Will Instead of c + , the analytical control law of the tether current that can effectively control the libration of the electrodynamic tethered satellite is obtained, and the expression is:

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