Relative motion control and angular momentum management method for electromagnetic formation

Abstract

The invention relates to a relative motion control and angular momentum management method for an electromagnetic formation, and the method comprises the steps: building a six-degree-of-freedom relative dynamics model of the electromagnetic satellite formation, and determining a leader satellite and a follower satellite; selecting a leader satellite to install a reaction wheel, designing a magnetic moment for the satellite based on a frequency division multiplexing method, generating effective electromagnetic force and moment, and designing a direct current component for angular momentum unloading; designing a six-degree-of-freedom relative motion controller under the conditions that the electromagnetic formation system has external interference and the model is uncertain by adopting an active disturbance rejection control method; and designing an unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solving the direct current component of the magnetic moment of the leader satellite. According to the invention, relative position control and angular momentum management are decoupled by using the idea of frequency division multiplexing, so that angular momentum unloading and formation control are carried out in parallel.

Description

Technical Field

[0001] The present invention relates to the technical field of electromagnetic formations, and in particular to a relative motion control and angular momentum management method of an electromagnetic formation. Background Art

[0002] Electromagnetic formation flying (EMFF) is a formation method that does not require the consumption of propellant. Each satellite in the electromagnetic formation is equipped with an electromagnetic coil to generate a magnetic field. The magnetic field interacts with the magnetic fields of other satellites to generate electromagnetic forces and electromagnetic torques between satellites. Since the electromagnetic coupling between satellites is an internal force of the system, it is necessary to combine reaction wheels to achieve the relative position and attitude control of satellites.

[0003] However, the relative motion dynamics equations of electromagnetic formation satellites and the intersatellite electromagnetic force and electromagnetic torque models are highly nonlinear and coupled. The electromagnetic force and electromagnetic torque are closely related to the relative position / attitude of the satellites. The electromagnetic force is inversely proportional to the fourth power of the distance between satellites, and the electromagnetic torque is inversely proportional to the cube of the distance between satellites. There is an obvious nonlinear relationship between the electromagnetic force and electromagnetic torque and the magnetic moment vector, and the attitude and orbit of the formation satellites are coupled. The complex coupling relationship brings great challenges to the magnetic moment solution and controller design. In addition, the formation system is also affected by external interference such as gravity gradient torque and aerodynamic torque during on-orbit flight. In the EMFF on-orbit mission in low-Earth orbit, the strong magnetic field of the earth will also produce a large disturbance torque on the electromagnetic satellite. In the process of satellite attitude control, the angular momentum of attitude actuators such as control torque gyroscopes and reaction wheels continues to increase with the accumulation of time, resulting in saturation of the angular momentum of the attitude actuators. It is crucial to take some measures to manage the angular momentum of the reaction wheels. For EMFF, the magnetic moment of the electromagnetic satellite can be used to realize the function of the magnetic torquer, that is, the electromagnetic torque generated by the magnetic moment of the satellite using the geomagnetic field is used to unload the angular momentum. The frequency division multiplexing method provides a good research idea for the dynamic decoupling control of electromagnetic formation satellites and the management of satellite angular momentum. Summary of the invention

[0004] In order to solve the technical problems existing in the above-mentioned prior art, the purpose of the present invention is to provide a method for relative motion control and angular momentum management of an electromagnetic formation, which utilizes the idea of ​​frequency division multiplexing to decouple relative position control and angular momentum management, thereby realizing parallel angular momentum unloading and formation control.

[0005] To achieve the above-mentioned object of the invention, the present invention provides a method for relative motion control and angular momentum management of an electromagnetic formation, comprising the following steps:

[0006] Step S1, establishing a 6-DOF relative dynamics model of the electromagnetic satellite formation, and determining the leader satellite and the follower satellite;

[0007] Step S2, selecting a leader satellite to install a reaction wheel, designing a magnetic moment for the satellite based on a frequency division multiplexing method, generating an effective electromagnetic force and torque, and designing a DC component for angular momentum unloading;

[0008] Step S3, using an anti-disturbance control method to design a controller for 6-DOF relative motion of an electromagnetic formation system in the presence of external interference and model uncertainty;

[0009] Step S4, designing an unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solving the DC component of the magnetic moment of the leader satellite in step S2.

[0010] According to a technical solution of the present invention, the step S1 specifically includes:

[0011] If the electromagnetic formation system is regarded as a mass center moving in a near-earth circular motion, the relative dynamic equation of the electromagnetic formation system can be expressed as:

[0012]

[0013] in The angular velocity of the system's center of mass moving around the center of the earth is μ g =3.986×10 14 m 3 / s 2 is the gravitational constant, r CM is the distance between the center of the earth and the center of mass of the binary star system, X1=[xyz] T represents the position vector of the satellite relative to the system center of mass, represents the relative velocity vector of satellite i, f d =[f dx f dy f dz ] T is the disturbance acceleration received by the satellite;

[0014] X 1d =[x d y d z d ] T represents the satellite's desired relative position vector, represents the desired relative velocity vector of the satellite, then the control target of the satellite's relative translational motion is expressed as:

[0015]

[0016] The error dynamics equation of the satellite is:

[0017]

[0018] in, represents the control torque, Δ represents the total uncertainty of the system,

[0019] The satellite's attitude control target can be expressed as Because ln q e =[0,θ / 2] T , the attitude control target of the satellite can be expressed as

[0020]

[0021] According to a technical solution of the present invention, the leader satellite is satellite A, the follower satellite is satellite B, and in step S2, the magnetic moment of the satellite is designed as:

[0022]

[0023] Among them, μ1 and μ2 are the magnetic moments of satellite A and satellite B respectively, ω1 and ω2 are two different AC frequencies, and μ lg is the DC component of the magnetic moment used to unload the angular momentum of the reaction wheel of satellite A, μ 11 , μ 12 are the two AC magnetic moment amplitude components of satellite A, μ 21 , μ 22 are the two AC magnetic moment amplitude components of satellite B;

[0024] Based on the electromagnetic model, the electromagnetic force and torque are calculated, and the electromagnetic force of the geomagnetic field on the electromagnetic satellite is ignored. The electromagnetic force on satellite A and satellite B is expressed as:

[0025]

[0026] Among them, F1 and F2 are the electromagnetic forces on satellite A and satellite B respectively;

[0027] The electromagnetic torque on satellite A and satellite B is expressed as:

[0028]

[0029] Among them, R1 and R2 are the position vectors from the center of the earth to satellite A and satellite B respectively. are the electromagnetic torques on satellite A and satellite B respectively;

[0030] The cosine terms in equations (7) and (8) are treated as external interference of the system, and the effective control force and control torque provided by the electromagnetic formation system to satellite B are expressed as:

[0031]

[0032] Among them, F ec2 , are the effective control force and control torque acting on satellite B respectively.

[0033] According to a technical solution of the present invention, in step S2, the electromagnetic force, electromagnetic moment and magnetic moment are expressed in the electromagnetic coordinate system for analysis, and the two magnetic moments are expressed as μ in the electromagnetic coordinate system. i =[μ ix μ iy μ iz ] T , μ j =[μ jx μ jy μ jz ] T According to the electromagnetic model, the electromagnetic force and electromagnetic torque are expressed as:

[0034]

[0035] Let μ i =[m1 m2 m3] T μ j =[m4 m5 m6] T , Based on formula (10), we can get:

[0036]

[0037] Equation group (11) is a multivariate quadratic equation group, whose independent variables are {m1, m2, m3, m4, m5, m6} and dependent variables are {c1, c2, c3, c4, c5, c6}. Arranging and simplifying equation group (11) yields:

[0038]

[0039] Equation group (12) represents the coupling relationship between magnetic moment, electromagnetic force and electromagnetic moment components, then:

[0040] When c5-c3=0, c2+c6=0, and c2 and c3 are not 0 at the same time, m4 is selected as the free variable, and the solution of equation group (12) is m1=0.

[0041] When 2c2+c6=0, 2c3-c5=0, and c2 and c3 are not 0 at the same time, m1 is selected as the free variable, and the solution of equation group (12) is m4=0.

[0042] When c2=c3=c5=c6=0, m2 and m3 can be selected as free variables, and the solution of equation group (8) is m1=0, m4=0,

[0043] For formula (9), let μ 11 =[a1 a2 a3] T , μ 21 =[a4 a5 a6] T , μ 12 =[b1 b2 b3] T , μ 22 =[b4 b5b6] T , According to equation (12), we can get:

[0044]

[0045] Let n k =d k +l k , k=1, 2, …6, in order to make the equation system have a solution under the condition that {n1, n2, n3, n4, n5, n6} takes any value, according to the solution analysis of equation system (12), we can set l5-l3=0, -l2-l6=0, 2d2+d6=0, 2d3-d5=0, and we can get:

[0046]

[0047] Among them, k1 and k2 are arbitrary constants, and for {n1, n2, n3, n4, n5, n6}, they take arbitrary values. According to equation (14), {d2, d3, d1, d4, l2, l3, l1, l4} can be set, and then the magnetic moment is solved according to the analysis of equation group (12), that is, the magnetic moment solution of equation group (13) is obtained.

[0048] According to a technical solution of the present invention, in step S3, an anti-disturbance control method is used to design a controller for 6-DOF relative motion of an electromagnetic formation system in the presence of external interference and model uncertainty, wherein the control force and torque required by the follower satellite are provided by electromagnetic force and electromagnetic torque, and the magnetic torque is solved according to step S2, and the control torque of the leader satellite is provided by the electromagnetic torque and the reaction wheel, specifically including:

[0049] Designing a linear extended state observer LESO for the electromagnetic formation system, we have:

[0050]

[0051] in, are the estimated values ​​of X1, X2, f, and β k =diag(β k1 , β k2 , βk3 ), k = 1, 2, 3 are observer gains;

[0052] Design Beta 1j =3ω otj ω otj >0,j=1,2,3,,ω otj is the observer bandwidth;

[0053] Based on the online estimation of the total uncertainty, the ADRC is designed as:

[0054]

[0055] Among them, k2=2ω tc ,ω tc =diag(ω tc1 ,ω tc2 ,ω tc3 ), where ω tcj >0, j=1, 2, 3 is the controller bandwidth;

[0056] Designing the linear extended state observer LESO for the error dynamics system of formula (3), we have:

[0057]

[0058] in, and are the estimated values ​​of ω and Δ, and the observer gain is γ k =diag(γ k1 , γ k2 , γ k3 ), k = 1, 2; the design gain parameters satisfy L aij =s 2 +γ 1j s+γ 2j , j = 1, 2, 3 are the conditions of Hurwitz polynomial, let γ 1j =2ω oaj , ω oaj >0, j=1,2,3, then L aj =(s+ω oaj ) 2 ;

[0059] Based on an online estimate of the “total disturbance” or extended state, the controller is designed to:

[0060]

[0061] in For virtual control, the control coefficient can be simply set to k3 = k4 = ωca ,ω ca =diag(ω ca1 ,ω ca2 ,ω ca3 ),ω caj >0, j=1, 2, 3 is the controller bandwidth;

[0062] Based on equations (3) and (18), the reference error system is obtained as:

[0063]

[0064] in, The reference error system is a hierarchical system. According to the contraction theorem, Equation (19) converges exponentially to a small sphere around the desired trajectory.

[0065] According to a technical solution of the present invention, the step S4 specifically includes:

[0066] The angular momentum that satellite A needs to unload is:

[0067] τ un =-k h (hh n )=-k h Δh (20)

[0068] Among them, k h is the unloading gain to be determined, h is the projection of the angular momentum vector of the reaction flywheel on the three axes of the satellite body coordinate system, and h n is the preset nominal angular momentum vector, Δh is the excess angular momentum to be unloaded;

[0069] The torque expected to be unloaded by the Earth's magnetic field is denoted by τ EA According to formula (20), we can get

[0070]

[0071] When set to μ lg With the Earth's magnetic field B E When vertical, the magnetic moment required for angular momentum unloading can be calculated according to formula (21):

[0072]

[0073] Compared with the prior art, the present invention has the following beneficial effects:

[0074] The present invention proposes a relative motion control and angular momentum management method for an electromagnetic formation, which decouples relative position control and angular momentum management by utilizing the idea of ​​frequency division multiplexing, thereby achieving parallel angular momentum unloading and formation control.

[0075] The present invention designs magnetic moments for satellites based on a frequency division multiplexing method and provides magnetic moment solutions corresponding to arbitrary effective electromagnetic forces and moments generated between satellite pairs, so that the electromagnetic formation can use electromagnetic forces and electromagnetic moments to control the relative positions and attitudes of electromagnetic satellites.

[0076] The present invention only requires a set of reaction wheels to be installed on the leader satellite to realize the attitude control of the entire formation, and the control force and torque required by the follower satellite are provided by the electromagnetic force and electromagnetic torque.

[0077] The present invention designs an unloading law to unload the angular momentum of the reaction wheel of the leader satellite. The DC component of the magnetic moment of the leader satellite interacts with the geomagnetic field, thereby realizing the angular momentum unloading during the satellite's 6-DOF relative motion control process. It can be used to unload the angular momentum of its reaction wheel, so that the angular momentum of the reaction wheel can be maintained in a smaller range for a longer period of time. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0079] Figure 1 A flowchart schematically showing a relative motion control and angular momentum management method of an electromagnetic formation provided in accordance with one embodiment of the present invention;

[0080] Figure 2 It is a schematic diagram of the coordinate system of the present invention;

[0081] Figure 3a A curve diagram showing a change in the relative position error of two satellites in one embodiment of the present invention;

[0082] Figure 3b A curve diagram showing a change in the relative velocity error of two stars in one embodiment of the present invention;

[0083] Figure 3c A curve diagram of relative translation control acceleration variation in one embodiment of the present invention;

[0084] Figure 3d A graph showing a change in attitude control torque in one embodiment of the present invention;

[0085] Figure 3e is a graph showing a change in magnetic moment of satellite A in one embodiment of the present invention;

[0086] Figure 3f is a graph showing a change in magnetic moment of satellite B in one embodiment of the present invention;

[0087] Figure 4a is a graph showing a change in attitude error of a satellite A in one embodiment of the present invention;

[0088] Figure 4b is a graph showing a change in attitude error of satellite B in one embodiment of the present invention;

[0089] Figure 4c is a curve diagram of angular velocity error variation of satellite A in one embodiment of the present invention;

[0090] Figure 4d is a curve diagram of the angular velocity error variation of satellite B in one embodiment of the present invention;

[0091] Figure 4e A curve diagram of the change of the DC component magnetic moment of satellite A for angular momentum unloading in one embodiment of the present invention;

[0092] Figure 4f Graph showing the change in angular momentum of a satellite in one embodiment of the present invention. DETAILED DESCRIPTION

[0093] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0094] like Figure 1 and Figure 2 As shown, a relative motion control and angular momentum management method of an electromagnetic formation of the present invention comprises the following steps:

[0095] Step S1, establishing a 6-DOF relative dynamics model of the electromagnetic satellite formation and determining the leader satellite and the follower satellite.

[0096] Taking the binary satellite formation as an example, for a close-range electromagnetic formation where the center of mass of the system makes a near-earth circular motion, the relative dynamic equation of the satellite is expressed as:

[0097]

[0098] in, The angular velocity of the system's center of mass moving around the center of the earth is μ g =3.986×10 14 m 3 / s 2 is the gravitational constant, rCM is the distance between the center of the earth and the center of mass of the binary star system, X1=[xyz] T represents the position vector of the satellite relative to the system center of mass, represents the relative velocity vector of satellite i, f d =[f dx f dy f dz ] T is the disturbance acceleration received by the satellite;

[0099] X 1d =[x d y d z d ] T represents the satellite's desired relative position vector, represents the desired relative velocity vector of the satellite, then the control target of the satellite's relative translational motion is expressed as:

[0100]

[0101] The error dynamics equation of the satellite is:

[0102]

[0103] in, represents the control torque, Δ represents the total uncertainty of the system,

[0104] The satellite's attitude control target can be expressed as Because ln q e =[0,θ / 2] T , the attitude control target of the satellite can be expressed as

[0105]

[0106] Step S2: Select a leader satellite to install a reaction wheel, design a magnetic moment for the satellite based on a frequency division multiplexing method, generate an effective electromagnetic force and torque, and design a DC component for angular momentum unloading.

[0107] The leader satellite is satellite A, and the follower satellite is satellite B. Only satellite A is equipped with a reaction wheel, and the geomagnetic field is used to unload the angular momentum of the reaction wheel. The magnetic moment of the satellite is designed as:

[0108]

[0109] Among them, μ1 and μ2 are the magnetic moments of satellite A and satellite B respectively, ω1 and ω2 are two different AC frequencies, and μ lg is the DC component of the magnetic moment used to unload the angular momentum of the reaction wheel of satellite A, μ11 , μ 12 are the two AC magnetic moment amplitude components of satellite A, μ 21 , μ 22 are the two AC magnetic moment amplitude components of satellite B;

[0110] According to the far-field electromagnetic model, the magnetic moment μ i Magnetic moment μ j The electromagnetic force and torque are expressed as

[0111]

[0112] Among them, r ij is the magnetic moment μ i With magnetic moment μ j The relative position between ij =||r ij ||;

[0113] Ignoring the electromagnetic force of the geomagnetic field on the electromagnetic satellite, according to formula (6), the electromagnetic force on satellite A and satellite B is expressed as:

[0114]

[0115] Among them, F1 and F2 are the electromagnetic forces on satellite A and satellite B respectively;

[0116] The electromagnetic torque on satellite A and satellite B is expressed as:

[0117]

[0118] Among them, R1 and R2 are the position vectors from the center of the earth to satellite A and satellite B respectively. are the electromagnetic torques on satellite A and satellite B respectively;

[0119] The cosine terms in equations (7) and (8) are treated as external interference of the system, and the effective control force and control torque provided by the electromagnetic formation system to satellite B are expressed as:

[0120]

[0121] Among them, F ec2 , are the effective control force and control torque acting on satellite B respectively;

[0122] The electromagnetic force, electromagnetic torque and magnetic moment are all expressed in the electromagnetic coordinate system for analysis. The two magnetic moments are expressed as μ in the electromagnetic coordinate system. i =[μ ix μ iy μ iz ] T , μj =[μ jx μ jy μ jz ] T According to the electromagnetic model, the electromagnetic force and electromagnetic torque are expressed as:

[0123]

[0124] Let μ i =[m1 m2 m3] T μ j =[m4 m5 m6] T , Based on formula (10), we can get:

[0125]

[0126] Equation group (11) is a multivariate quadratic equation group, whose independent variables are {m1, m2, m3, m4, m5, m6} and dependent variables are {c1, c2, c3, c4, c5, c6}. Arranging and simplifying equation group (11) yields:

[0127]

[0128] Equation group (12) represents the coupling relationship between magnetic moment, electromagnetic force and electromagnetic moment components, then:

[0129] When c5-c3=0, c2+c6=0, and c2 and c3 are not 0 at the same time, m4 is selected as the free variable, and the solution of equation group (12) is m1=0.

[0130] When 2c2+c6=0, 2c3-c5=0, and c2 and c3 are not 0 at the same time, m1 is selected as the free variable, and the solution of equation group (12) is m4=0.

[0131] When c2=c3=c5=c6=0, m2 and m3 can be selected as free variables, and the solution of equation group (8) is m1=0, m4=0,

[0132] For formula (9), let μ 11 =[a1 a2 a3] T , μ 21 =[a4 a5 a6] T , μ 12 =[b1 b2 b3] T , μ 22 =[b4 b5b6] T , According to equation (12), we can get:

[0133]

[0134] Let n k =d k +l k , k=1, 2, …6, in order to make the equation system have a solution under the condition that {n1, n2, n3, n4, n5, n6} takes any value, according to the solution analysis of equation system (12), we can set l5-l3=0, -l2-l6=0, 2d2+d6=0, 2d3-d5=0, and we can get:

[0135]

[0136] Among them, k1 and k2 are arbitrary constants, and for {n1, n2, n3, n4, n5, n6}, they take arbitrary values. According to equation (14), {d2, d3, d1, d4, l2, l3, l1, l4} can be set, and then the magnetic moment is solved according to the analysis of equation group (12), that is, the magnetic moment solution of equation group (13) is obtained.

[0137] Step S3: adopt an anti-disturbance control method to design a controller for the 6-DOF relative motion of the electromagnetic formation system in the presence of external interference and model uncertainty.

[0138] The anti-disturbance control method is used to solve the tracking control problem of 6-DOF relative motion in the electromagnetic formation system with external interference and model uncertainty. The control force and torque required by the follower satellite are provided by the electromagnetic force and electromagnetic torque. The magnetic torque is solved according to step 2. The control torque of the leader satellite is provided by the electromagnetic torque and the reaction wheel, which includes:

[0139] Designing a linear extended state observer LESO for the electromagnetic formation system, we have:

[0140]

[0141] in, are the estimated values ​​of X1, X2, f, and β k =diag(β k1 , β k2 , β k3 ), k = 1, 2, 3 are observer gains;

[0142] Design Beta 1j =3ω otj ω otj >0,j=1,2,3,ω otj is the observer bandwidth;

[0143] Based on the online estimation of the total uncertainty, the ADRC is designed as:

[0144]

[0145] Among them, k2=2ω tc ,ω tc =diag(ω tc1 ,ω tc2 ,ω tc3 ), where ω tcj >0, j=1, 2, 3 is the controller bandwidth;

[0146] Designing the linear extended state observer LESO for the error dynamics system of formula (3), we have:

[0147]

[0148] in, and are the estimated values ​​of ω and Δ, and the observer gain is γ k =diag(γ k1 , γ k2 , γ k3 ), k = 1, 2; the design gain parameters satisfy L aj =s 2 +γ 1j s+γ 2j , j = 1, 2, 3 are the conditions of Hurwitz polynomial, let γ 1j =2ω oaj , ω oaj >0, j=1,2,3, then L aj =(s+ω oaj ) 2 ;

[0149] Based on an online estimate of the “total disturbance” or extended state, the controller is designed to:

[0150]

[0151] in For virtual control, the control coefficient can be simply set to k3 = k4 = ω ca ,ω ca =diag(ω ca1 ,ω ca1 ,ω ca3 ),ω oaj >0, j=1, 2, 3 is the controller bandwidth;

[0152] Based on equations (3) and (18), the reference error system is obtained as:

[0153]

[0154] in, The reference error system is a hierarchical system. According to the contraction theorem, Equation (19) converges exponentially to a small sphere around the desired trajectory.

[0155] Step S4, designing an unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solving the DC component of the magnetic moment of the leader satellite in step S2, specifically includes:

[0156] The angular momentum that satellite A needs to unload is:

[0157] τ un =-k h (hh n )=-k h Δh (20)

[0158] Among them, k h is the unloading gain to be determined, h is the projection of the angular momentum vector of the reaction flywheel on the three axes of the satellite body coordinate system, and h n is the preset nominal angular momentum vector, Δh is the excess angular momentum to be unloaded;

[0159] The torque expected to be unloaded by the Earth's magnetic field is denoted by τ EA According to formula (20), we can get

[0160]

[0161] When set to μ lg With the Earth's magnetic field B E When vertical, the magnetic moment required for angular momentum unloading can be calculated according to formula (21):

[0162]

[0163] Based on formula (22), it can be seen that when unloading, only one parameter k needs to be set under the condition of known geomagnetic field: h can be achieved, and the unloading gain k to be determined h is the preset relevant parameter, that is, the determined value.

[0164] The above calculation formula uses a dual-star electromagnetic formation as an example. For a multi-star electromagnetic formation, those skilled in the art can design it based on the design process of the dual-star electromagnetic formation.

[0165] Simulation experiment

[0166] In order to illustrate the application effect of the present invention, the control of a circular formation of two satellites in low-Earth orbit is used as an example for simulation. In the simulation, it is assumed that the center of mass of the satellite formation is running on a circular orbit of 700 km, and the satellite attitude is kept consistent with the center of mass orbit coordinate system. The relative position vector of the two satellites is designed as Where, d = 15m, T = 5h, ω = 2π / T. The mass of each satellite is 150kg, and the inertia matrix is ​​J = diag(20, 17, 22)kg·m 2 , the inertial uncertainty of the satellite is ΔJ=diag(1,-2,3), and the uncertainty of the electromagnetic far-field model is 8%.

[0167] In addition to the external interference force and torque caused by the earth's magnetic field, it is assumed that the satellite is also subject to other external periodic external interference forces and torques:

[0168] f d =10 -5 ×[6sin(ω0t+π / 6)-5×sin(ω0t+π / 3))-4sinω0t] T m / s 2

[0169] τ d =0.01×[sin(ω0t+π / 4)-sin(ω0t-π / 3)2cos(ω0t+π / 6)] T N·m

[0170] The initial error of the relative position of the binary star is r e (0) = [1 2 -2] T m, the initial error of relative velocity is The initial attitude error of satellite A is q e1 (0) = [0.843 -0.2 0.3 -0.4] T , the initial angular velocity error is ω e1 (0) = [0.3 0.2 0.4] T rad / s. The initial attitude error of satellite B is q e2 (0) = [0.911 0.2 -0.2 -0.3] T , the initial angular velocity error is ω e2 (0) = [-0.3 -0.1 0.2] T rad / s.

[0171] The high-frequency disturbance term contained in the magnetic dipole will enter the controller through the object state, which will have a significant impact on the control accuracy of the system. To solve this problem, consider designing a larger AC frequency value and adding a second-order low-pass filter to the control input before entering the magnetic dipole solution stage. The solved magnetic dipole will enter the object, significantly reducing the high-frequency disturbance that directly enters the controller. The control parameter of the simulation is designed to be ω otj =0.2,ω ctj =0.03,ω oaj =1,ω caj =0.1, j=1, 2, 3, the carrier frequency of the satellite magnetic dipole is designed to be ω1=100Hz, ω2=200Hz. The transfer function of the second-order low-pass filter is Where k = 1, ξ = 1, ω = 5 Hz. Figures 3(af) and 4(af) show the simulation results of relative position, attitude, and reaction wheel angular momentum. The formation satellite can achieve good control accuracy and maintain a low angular momentum range within one orbital period.

[0172] The present invention adopts the above-mentioned relative motion control and angular momentum management method of an electromagnetic formation, designs magnetic moments for satellites based on the frequency division multiplexing method, and provides magnetic moment solutions corresponding to arbitrary effective electromagnetic forces and torques generated between satellite pairs, so that the electromagnetic formation can use electromagnetic forces and electromagnetic torques to control the relative position and attitude of electromagnetic satellites. Only a set of reaction wheels needs to be installed on the leader satellite to achieve attitude control of the entire formation, and the control force and torque required for the follower satellite are provided by electromagnetic forces and electromagnetic torques. An unloading law is designed to unload the angular momentum of the reaction wheels of the leader satellite, thereby realizing angular momentum unloading during the satellite's 6-degree-of-freedom relative motion control process. The DC component of the leader satellite's magnetic moment interacts with the geomagnetic field, which can be used to unload the angular momentum of its reaction wheels, so that the angular momentum of the reaction wheels can be maintained in a smaller range for a longer period of time.

[0173] Furthermore, the present invention utilizes the concept of frequency division multiplexing to decouple relative position control and angular momentum management, thereby achieving parallel execution of angular momentum unloading and formation control.

[0174] It should be noted that although the embodiments of the present invention described above are illustrative, they are not intended to limit the present invention, and therefore the present invention is not limited to the above specific embodiments. Without departing from the principles of the present invention, any other embodiments obtained by those skilled in the art under the guidance of the present invention are deemed to be within the protection of the present invention.

Claims

1. A relative motion control and angular momentum management method for an electromagnetic formation, characterized in that: The following steps are involved: Step S1, establishing a 6-DOF relative dynamics model of the electromagnetic satellite formation, and determining the leader satellite and the follower satellite; Step S2, selecting a leader satellite to install a reaction wheel, designing a magnetic moment for the satellite based on a frequency division multiplexing method, generating an effective electromagnetic force and torque, and designing a DC component for angular momentum unloading; Step S3, using an anti-disturbance control method to design a controller for 6-DOF relative motion of an electromagnetic formation system in the presence of external interference and model uncertainty; Step S4, designing an unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solving the DC component of the magnetic moment of the leader satellite in step S2.

2. The relative motion control and angular momentum management method of the electromagnetic formation according to claim 1, characterized in that: The step S1 specifically includes: If the electromagnetic formation system is regarded as a mass center moving in a near-earth circular motion, the relative dynamic equation of the electromagnetic formation system can be expressed as: in The angular velocity of the system's center of mass moving around the center of the earth is μ g =3.986×10 14 m 3 / s 2 is the gravitational constant, r CM is the distance between the center of the earth and the center of mass of the binary star system, X1=[xyz] T represents the position vector of the satellite relative to the system center of mass, represents the relative velocity vector of satellite i, f d =[f dx f dy f dz ] T is the disturbance acceleration received by the satellite; X 1d =[x d y d z d ] T represents the satellite's desired relative position vector, represents the desired relative velocity vector of the satellite, then the control target of the satellite's relative translational motion is expressed as: The satellite error dynamics equation is: in, represents the control torque, Δ represents the total uncertainty of the system, The satellite's attitude control target can be expressed as Because ln q e =[0,θ / 2] T , the satellite's attitude control target can be expressed as 3. The relative motion control and angular momentum management method of the electromagnetic formation according to claim 2, characterized in that: The leader satellite is satellite A, and the follower satellite is satellite B. In step S2, the magnetic moment of the satellite is designed as: Among them, μ1 and μ2 are the magnetic moments of satellite A and satellite B respectively, ω1 and ω2 are two different AC frequencies, and μ lg is the DC component of the magnetic moment used to unload the angular momentum of the reaction wheel of satellite A, μ 11 , μ 12 are the two AC magnetic moment amplitude components of satellite A, μ 21 , μ 22 are the two AC magnetic moment amplitude components of satellite B; Based on the electromagnetic model, the electromagnetic force and torque are calculated, and the electromagnetic force of the geomagnetic field on the electromagnetic satellite is ignored. The electromagnetic force on satellite A and satellite B is expressed as: Among them, F1 and F2 are the electromagnetic forces on satellite A and satellite B respectively; The electromagnetic torque on satellite A and satellite B is expressed as: Among them, R1 and R2 are the position vectors from the center of the earth to satellite A and satellite B respectively. are the electromagnetic torques on satellite A and satellite B respectively; The cosine terms in equations (7) and (8) are treated as external interference of the system, and the effective control force and control torque provided by the electromagnetic formation system to satellite B are expressed as: Among them, F ec2 , are the effective control force and control torque acting on satellite B respectively.

4. The relative motion control and angular momentum management method of the electromagnetic formation according to claim 3 is characterized in that: In step S2, the electromagnetic force, electromagnetic moment and magnetic moment are expressed in the electromagnetic coordinate system for analysis. The two magnetic moments are expressed as μ and μ respectively in the electromagnetic coordinate system. i =[μ ix μ iy μ iz ] T , μ j =[μ jx μ jy μ jz ] T According to the electromagnetic model, the electromagnetic force and electromagnetic torque are expressed as: Let μ i =[m1 m2 m3] T μ j =[m4 m5 m6] T , Based on formula (10), we can get: Equation group (11) is a multivariate quadratic equation group, whose independent variables are {m1, m2, m3, m4, m5, m6} and dependent variables are {c1, c2, c3, c4, c5, c6}. Arranging and simplifying equation group (11) yields: Equation group (12) represents the coupling relationship between magnetic moment, electromagnetic force and electromagnetic moment components, then: When c5-c3=0, c2+c6=0, and c2 and c3 are not 0 at the same time, m4 is selected as the free variable, and the solution of equation group (12) is m1=0. When 2c2+c6=0, 2c3-c5=0, and c2 and c3 are not 0 at the same time, m1 is selected as the free variable, and the solution of equation group (12) is m4=0. When c2=c3=c5=c6=0, m2 and m3 can be selected as free variables, and the solution of equation group (8) is m1=0, m4=0, For formula (9), let μ 11 =[a1 a2 a3] T , μ 21 =[a4 a5 a6] T , μ 12 =[b1 b2 b3] T , μ 22 =[b4 b5 b6] T , According to equation (12), we can get: Let n k =d k +l k , k=1, 2, …6, in order to make the equation system have a solution under the condition that {n1, n2, n3, n4, n5, n6} takes any value, according to the solution analysis of equation system (12), we can set l5-l3=0, -l2-l6=0, 2d2+d6=0, 2d3-d5=0, and we can get: Among them, k1 and k2 are arbitrary constants, and for {n1, n2, n3, n4, n5, n6}, they take arbitrary values. According to equation (14), {d2, d3, d1, d4, l2, l3, l1, l4} can be set, and then the magnetic moment is solved according to the analysis of equation group (12), that is, the magnetic moment solution of equation group (13) is obtained.

5. The relative motion control and angular momentum management method of the electromagnetic formation according to claim 4, characterized in that: In the step S3, an anti-disturbance control method is used to design a controller for the 6-DOF relative motion of the electromagnetic formation system in the presence of external interference and model uncertainty, wherein the control force and torque required by the follower satellite are provided by the electromagnetic force and the electromagnetic torque, and the magnetic torque is solved according to step S2, and the control torque of the leader satellite is provided by the electromagnetic torque and the reaction wheel, specifically including: Designing a linear extended state observer LESO for the electromagnetic formation system, we have: in, X1, X 2，f The estimated value of β k =diag(β k1 , β k2 , β k3 ), k = 1, 2, 3 are observer gains; Design Beta 1j =3ω otj ω otj >0,j=1,2,3,ω otj is the observer bandwidth; Based on the online estimation of the total uncertainty, the ADRC is designed as: Among them, k2=2ω tc ,ω tc =diag(ω tc1 ,ω tc2 ,ω tc3 ), where ω tcj >0, j=1, 2, 3 is the controller bandwidth; Designing the linear extended state observer LESO for the error dynamics system of formula (3), we have: in, and are the estimated values ​​of ω and Δ, and the observer gain is γ k =diag(γ k1 , γ k2 , γ k3 ), k = 1, 2; the design gain parameters satisfy L aj =s 2 +γ 1j s+γ 2j , j = 1, 2, 3 are the conditions of Hurwitz polynomial, let γ 1j =2ω oαj , ω oaj >0,j=1,2,3, then L aj =(s+ω oaj ) 2 ; Based on an online estimate of the "total disturbance" or extended state, the controller is designed to: in For virtual control, the control coefficient can be simply set to k3 = k4 = ω ca ,ω ca =diag(ω ca1 ,ω ca2 ,ω ca3 ),ω caj >0, j=1, 2, 3 is the controller bandwidth; Based on equations (3) and (18), the reference error system is obtained as: in, The reference error system is a hierarchical system. According to the contraction theorem, Equation (19) converges exponentially to a small sphere around the desired trajectory.

6. The relative motion control and angular momentum management method of the electromagnetic formation according to claim 5, characterized in that: The step S4 specifically includes: The angular momentum that satellite A needs to unload is: τ un =-k h (h-h n )=-k h Δh (20) Among them, k h is the unloading gain to be determined, h is the projection of the angular momentum vector of the reaction flywheel on the three axes of the satellite body coordinate system, and h n is the preset nominal angular momentum vector, Δh is the excess angular momentum to be unloaded; The torque expected to be unloaded by the Earth's magnetic field is denoted by τ EA According to formula (20), we can get When set to μ lg With the Earth's magnetic field B E When vertical, the magnetic moment required for angular momentum unloading can be calculated according to formula (21):

Images