Relative motion control and angular momentum management method for electromagnetic formation

Abstract

The application relates to a relative motion control and angular momentum management method of an electromagnetic formation, which comprises the following steps: establishing a 6-degree-of-freedom relative dynamics model of an electromagnetic satellite formation, determining a leader satellite and a follower satellite; selecting a leader satellite to be installed with a reaction wheel, designing a magnetic moment for the satellite based on a frequency division multiplexing method, generating effective electromagnetic force and torque, and designing a direct current component for angular momentum unloading; adopting an active disturbance rejection control method to design a 6-degree-of-freedom relative motion controller of the electromagnetic formation system under the condition of external disturbance and model uncertainty; 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 application, the relative position control and the angular momentum management are decoupled by using the idea of frequency division multiplexing, and the angular momentum unloading and the formation control are carried out in parallel.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of electromagnetic formation flying, and in particular to a relative motion control and angular momentum management method for electromagnetic formation flying. BACKGROUND

[0002] Electromagnetic formation flying (EMFF) is a formation flying mode without consuming propellant. Each satellite in the electromagnetic formation flying is equipped with an electromagnetic coil to generate a magnetic field. The magnetic field interacts with the magnetic field of other satellites to generate electromagnetic force and electromagnetic torque between satellites. Since the electromagnetic coupling between satellites is an internal force of the system, the relative position and attitude control of satellites needs to be combined with reaction wheels.

[0003] However, the relative motion dynamics equation of the electromagnetic formation flying satellite and the inter-satellite electromagnetic force and electromagnetic torque model have strong nonlinearity and coupling. The electromagnetic force and electromagnetic torque are closely related to the relative position / attitude of the satellite. The electromagnetic force is inversely proportional to the fourth power of the distance between satellites, and the electromagnetic torque is inversely proportional to the third power of the distance between satellites. There is a clear nonlinear relationship between the electromagnetic force and electromagnetic torque and the magnetic moment vector, and the attitude and orbit of the formation satellite 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 disturbances such as gravity gradient torque and aerodynamic torque during on-orbit flight. In addition, in the on-orbit flight mission of the EMFF in the low earth orbit, the strong magnetic field of the earth will also generate a large disturbance torque on the electromagnetic satellite. In the process of satellite attitude control, the angular momentum of the attitude actuator such as the control moment gyro and the reaction wheel increases with time, which leads to the saturation of the angular momentum of the attitude actuator. It is very important to take some measures to manage the angular momentum of the reaction wheel. For EMFF, the magnetic moment of the electromagnetic satellite can be used to realize the function of the magnetic torque device, that is, the electromagnetic torque generated by the geomagnetic field on the satellite magnetic moment is used to unload the angular momentum, and the frequency division multiplexing method provides a good research idea for the decoupling control of the dynamics of the electromagnetic formation flying satellite and the management of the angular momentum of the satellite. SUMMARY

[0004] To solve the above technical problems in the prior art, the purpose of the present application is to provide a relative motion control and angular momentum management method for electromagnetic formation flying, which decouples the relative position control and angular momentum management by using the idea of frequency division multiplexing, and realizes the parallel operation of angular momentum unloading and formation control.

[0005] To achieve the above application purpose, the present application provides a relative motion control and angular momentum management method for electromagnetic formation flying, comprising the following steps:

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

[0007] Step S2, select the leader satellite to install the reaction wheel, design the magnetic moment for the satellite based on the frequency division multiplexing method, generate effective electromagnetic force and torque, and design the direct current component for angular momentum unloading;

[0008] Step S3, design the 6-DOF relative motion controller of the electromagnetic formation system under external disturbance and model uncertainty by using the active disturbance rejection control method;

[0009] Step S4, design the unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solve the direct current component of the magnetic moment of the leader satellite in step S2.

[0010] According to one of the technical solutions of the application, the step S1 specifically comprises:

[0011] The electromagnetic formation system is regarded as a center of mass doing a near-earth circular motion, and the relative dynamics equation of the electromagnetic formation system is represented as:

[0012]

[0013] Wherein The angular velocity of the system center of mass around the center of the earth is μ g = 3.986 x 10 14 m 3 / s 2 is the gravitational constant of the center of the earth, r CM is the distance between the center of the earth and the center of mass of the double-star system, X1 = [x y z] T represents the position vector of the satellite relative to the center of mass of the system, represents the relative velocity vector of satellite i, f d = [f dx f dy f dz ] T is the disturbance acceleration suffered by the satellite;

[0014] X 1d = [x d y d z d ] T represents the relative position vector of the satellite, represents the relative velocity vector of the satellite, and the control target of the satellite relative to the translational motion is represented as:

[0015]

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

[0017]

[0018] wherein, denotes the control torque, and Δ denotes the total uncertainty of the system,

[0019] The attitude control target of the satellite can be represented as Since ln q e = [0, θ / 2] T The attitude control target of the satellite can be represented as

[0020]

[0021] According to one technical solution of the present application, 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:

[0022]

[0023] wherein μ1 and μ2 are the magnetic moments of satellite A and satellite B respectively, ω1 and ω2 are two different alternating frequencies, μ lg is the magnetic moment DC component for reaction wheel angular momentum unloading of satellite A, μ 11 and μ 12 are two alternating magnetic moment amplitude components of satellite A, μ 21 and μ 22 are two alternating magnetic moment amplitude components of satellite B.

[0024] The electromagnetic force and torque are calculated based on the electromagnetic model, and the effect of the geomagnetic field on the electromagnetic satellite is ignored. The electromagnetic force received by satellite A and satellite B is represented as:

[0025]

[0026] wherein F1 and F2 are the electromagnetic forces received by satellite A and satellite B respectively.

[0027] The electromagnetic torque received by satellite A and satellite B is represented as:

[0028]

[0029] wherein R1 and R2 are the position vectors from the center of the earth to satellite A and satellite B respectively, are the electromagnetic torques received by satellite A and satellite B respectively.

[0030] The cosine terms in equations (7) and (8) are treated as external disturbances of the system. The effective control force and control torque provided by the electromagnetic formation system for satellite B are represented as:

[0031]

[0032] wherein F ec2 , These represent the effective control force and control torque experienced by satellite B, respectively.

[0033] According to one technical solution of the present invention, in step S2, the electromagnetic force, electromagnetic torque, and magnetic moment are all expressed in an electromagnetic coordinate system for analysis, and the two magnetic moments are respectively 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 Based on the electromagnetic model, the electromagnetic force and electromagnetic torque are expressed as follows:

[0034]

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

[0036]

[0037] The system of equations (11) is a system of quadratic equations with independent variables {m1, m2, m3, m4, m5, m6} and dependent variables {c1, c2, c3, c4, c5, c6}. Simplifying the system of equations (11) yields:

[0038]

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

[0040] When c5-c3=0, c2+c6=0, and c2 and c3 are not both 0, m4 is chosen as the free variable, and the solution to the system of equations (12) is m1=0.

[0041] When 2c2+c6=0, 2c3-c5=0, and c2 and c3 are not both 0, if m1 is chosen as the free variable, the solution to the system of equations (12) is m4=0.

[0042] When c2 = c3 = c5 = c6 = 0, m2 and m3 can be chosen as free variables, and the solution to the system of equations (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 group (12), the following can be obtained:

[0044]

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

[0046]

[0047] Wherein, k1 and k2 are arbitrary constants, for {n1, n2, n3, n4, n5, n6} taking any value, {d2, d3, d1, d4, l2, l3, l1, l4} can be set according to formula (14), 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 one of the technical solutions of the application, in step S3, the self-disturbance control method is used to design the controller of 6 degrees of freedom relative motion of the electromagnetic formation system under the condition of external disturbance and model uncertainty, wherein the required control force and torque of the follower satellite are provided by electromagnetic force and electromagnetic torque, the control torque of the leader satellite is provided by electromagnetic torque and reaction wheel, and the specific steps include:

[0049] A linear extended state observer LESO is designed for the electromagnetic formation system, and the following is obtained:

[0050]

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

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

[0053] According to the online estimation of total uncertainty, the active disturbance rejection control is designed as:

[0054]

[0055] where the control coefficient can be simply set as k3= k4= ω k2= 2ω tc , ω tc = diag(ω tc1 , ω tc2 , ω tc3 ), where ω tcj > 0, j = 1, 2, 3 are controller bandwidths;

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

[0057]

[0058] where, 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 parameter satisfies L aij = s 2 + γ 1j s + γ 2j , j = 1, 2, 3 are Hurwitz polynomials, and let γ 1j = 2ω oaj , ω oaj > 0, j = 1, 2, 3, then L aj = (s + ω oaj ) 2 ;

[0059] Based on the online estimation of "total disturbance" or extended state, the controller is designed as:

[0060]

[0061] where is the virtual control, and the control coefficient can be simply set as 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 formula (3) and (18), the reference error system is obtained as:

[0063]

[0064] wherein, The reference error system is a hierarchical system, according to the contraction theorem, formula (19) is exponentially convergent to a small ball around the desired trajectory.

[0065] According to one of the technical solutions of the present application, the step S4 specifically comprises:

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

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

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

[0069] The torque that is expected to be unloaded by the geomagnetic field is denoted as τ EA , and according to formula (20), it can be obtained that

[0070]

[0071] When it is set that μ lg is perpendicular to the geomagnetic field B E , according to formula (21), the magnetic moment required for angular momentum unloading can be calculated as:

[0072]

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

[0074] The present application proposes a relative motion control and angular momentum management method for electromagnetic formation, which decouples the relative position control and angular momentum management by using the idea of frequency division multiplexing, and realizes parallel angular momentum unloading and formation control.

[0075] The application designs the magnetic moment of the satellite based on the frequency division multiplexing method, and gives the magnetic moment solution corresponding to the arbitrary effective electromagnetic force and moment between satellites, so that the electromagnetic formation can control the relative position and attitude of the electromagnetic satellite by using the electromagnetic force and electromagnetic moment.

[0076] The application only needs to install a set of reaction wheels on the leader satellite to realize the attitude control of the whole formation, and the required control force and moment of the follower satellite are provided by the electromagnetic force and electromagnetic moment.

[0077] The application designs the unloading law to unload the angular momentum of the reaction wheels of the leader satellite, and the direct current component of the magnetic moment of the leader satellite interacts with the geomagnetic field, thereby realizing the angular momentum unloading in the process of 6-degree-of-freedom relative motion control of the satellite, and the angular momentum of the reaction wheels can be unloaded, so that the angular momentum of the reaction wheels can be kept in a small range for a long time. BRIEF DESCRIPTION OF DRAWINGS

[0078] In order to more clearly illustrate the technical solutions in the embodiments of the application or in the prior art, the following will briefly introduce the drawings needed in the embodiments. Obviously, the drawings in the following description only constitute some embodiments of the application, and for those skilled in the art, other drawings can also be obtained from these drawings without creative labor.

[0079] Figure 1 A flow chart schematically showing a relative motion control and angular momentum management method of an electromagnetic formation provided in an embodiment of the application;

[0080] Figure 2 A coordinate system diagram of the application;

[0081] Figure 3a A relative position error change curve diagram of a double-satellite in an embodiment of the application;

[0082] Figure 3b A relative velocity error change curve diagram of a double-satellite in an embodiment of the application;

[0083] Figure 3c A relative translation control acceleration change curve diagram in an embodiment of the application;

[0084] Figure 3d An attitude control moment change curve diagram in an embodiment of the application;

[0085] Figure 3e A magnetic moment change curve diagram of satellite A in an embodiment of the application;

[0086] Figure 3f A magnetic moment change curve diagram of satellite B in an embodiment of the application;

[0087] Figure 4a Figure 4 is a plot of the attitude error variation of satellite A in an embodiment of the present application;

[0088] Figure 4b Figure 5 is a plot of the attitude error variation of satellite B in an embodiment of the present application;

[0089] Figure 4c Figure 6 is a plot of the angular velocity error variation of satellite A in an embodiment of the present application;

[0090] Figure 4d Figure 7 is a plot of the angular velocity error variation of satellite B in an embodiment of the present application;

[0091] Figure 4e Figure 8 is a plot of the direct current component magnetic moment variation for angular momentum unloading of satellite A in an embodiment of the present application;

[0092] Figure 4f Figure 9 is a plot of the angular momentum variation of satellite in an embodiment of the present application. DETAILED DESCRIPTION

[0093] In order to make the objects, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

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

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

[0096] Taking a two-satellite formation as an example, for a close-range electromagnetic formation in which the system centroid performs a near-circular orbit, the relative dynamics equation of the satellites is expressed as:

[0097]

[0098] wherein, The angular velocity of the system centroid around the Earth center is μ g = 3.986 x 10 14 m 3 / s 2 is the Earth's gravitational constant, rCM Let X1 be the distance between the Earth's center and the center of mass of the binary system, where X1 = [xyz]. T The position vector representing the satellite relative to the system's center of mass. f represents the relative velocity vector of satellite i. d =[f dx f dy f dz ] T The acceleration caused by interference experienced by the satellite;

[0099] With X 1d =[x d y d z d ] T This represents the satellite's expected relative position vector. Let the desired relative velocity vector of the satellite be represented, then the control objective for the satellite's relative translational motion is expressed as:

[0100]

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

[0102]

[0103] in, The control torque is represented by Δ, and the total uncertainty of the system is represented by Δ.

[0104] The satellite's attitude control objective can be represented as: Because ln q e = [0, θ / 2] T The satellite's attitude control objective can be represented as

[0105]

[0106] Step S2: Select the leader satellite to install the reaction wheel, design the magnetic moment for the satellite based on the frequency division multiplexing method, generate effective electromagnetic force and torque, and design the 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 Earth's magnetic field is used to unload the angular momentum of the reaction wheel. The satellite's magnetic moment is designed as follows:

[0108]

[0109] Where μ1 and μ2 are the magnetic moments of satellite A and satellite B, respectively, and ω1 and ω2 are two different AC frequencies, μ lg For the DC component of the magnetic moment used for unloading the angular momentum of the reaction wheel of satellite A, μ11 , μ 12 are two components of the amplitude of the alternating magnetic moment of satellite A, μ 21 , μ 22 are two components of the amplitude of the alternating magnetic moment of satellite B;

[0110] According to the far-field electromagnetic model, the magnetic moment μ i The electromagnetic force and torque action on the magnetic moment μ j is represented as

[0111]

[0112] where r ij is the relative position between the magnetic moment μ i and the magnetic moment μ j , d ij =||r ij ||;

[0113] Ignoring the geomagnetic field's electromagnetic force action on the electromagnetic satellite, according to equation (6), the electromagnetic forces received by satellite A and satellite B are represented as:

[0114]

[0115] where F1, F2 are the electromagnetic forces received by satellite A and satellite B, respectively;

[0116] The electromagnetic torque received by satellite A and satellite B is represented as:

[0117]

[0118] where R1, R2 are the position vectors from the center of the Earth to satellite A and satellite B, respectively, are the electromagnetic torques received by satellite A and satellite B, respectively;

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

[0120]

[0121] where F ec2 , are the effective control force and control torque received by satellite B, respectively;

[0122] The electromagnetic force, electromagnetic torque, and magnetic moment are all represented in the electromagnetic coordinate system for analysis. The two magnetic moments in the electromagnetic coordinate system are represented as μ i = [μ ix μ iy μ iz ] T , μj = [m1 m2 m3] jx μ jy = [m4 m5 m6] 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 , Then based on equation (10), we have:

[0125]

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

[0127]

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

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

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

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

[0132] For equation (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 the equation group (12), we have:

[0133]

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

[0135]

[0136] Where k1 and k2 are arbitrary constants, for {n1, n2, n3, n4, n5, n6} taking any value, we can set {d2, d3, d1, d4, l2, l3, l1, l4} according to formula (14), and then solve the magnetic moment according to the analysis of equation group (12), that is, the magnetic moment solution of equation group (13) is obtained.

[0137] Step S3, the self-disturbance control method is used to design the controller of 6 degrees of freedom relative motion of the electromagnetic formation system under the condition of external disturbance and model uncertainty.

[0138] The self-disturbance control method is used to solve the tracking control problem of 6 degrees of freedom relative motion of the electromagnetic formation system under the condition of external disturbance and model uncertainty, wherein the required control force and torque of the follower satellite are provided by electromagnetic force and electromagnetic torque. According to step 2, the magnetic moment is solved, and the control torque of the leader satellite is provided by electromagnetic torque and reaction wheel, which specifically includes:

[0139] A linear extended state observer LESO is designed for the electromagnetic formation system, so that:

[0140]

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

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

[0143] According to the online estimation of total uncertainty, the active disturbance rejection control is designed as:

[0144]

[0145] where the control coefficient can be set as k2= 2ω 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 a linear extended state observer LESO for the error dynamics system of formula (3), we have:

[0147]

[0148] where, 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 parameter satisfies the condition L aj = s 2 + γ 1j s + γ 2j , j = 1, 2, 3 is a Hurwitz polynomial, let γ 1j = 2ω oaj , ω oaj > 0, j = 1, 2, 3, then L aj = (s + ω oaj ) 2 ;

[0149] Based on the online estimation of "total disturbance" or extended state, the controller is designed as:

[0150]

[0151] where is the virtual control, and the control coefficient can be simply set as 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 formula (3) and (18), the reference error system is obtained as:

[0153]

[0154] wherein, The reference error system is a hierarchical system, according to the contraction theorem, formula (19) is exponentially convergent to a small ball around the desired trajectory.

[0155] In step S4, the angular momentum of the reaction wheel of the leader satellite is unloaded, and the direct current component of the magnetic moment of the leader satellite in step S2 is solved, specifically comprising:

[0156] The angular momentum that needs to be unloaded by satellite A is:

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

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

[0159] The torque expected to be unloaded by the geomagnetic field is denoted as τ EA , and according to formula (20), the following formula can be obtained:

[0160]

[0161] When it is set that μ lg is perpendicular to the geomagnetic field B E , according to formula (21), the magnetic moment required for angular momentum unloading is:

[0162]

[0163] Based on formula (22), it can be seen that only one parameter k h needs to be set in the known geomagnetic field during unloading, and the to-be-determined unloading gain k h is a preset related parameter, that is, a determined value.

[0164] The above calculation formula takes the double-satellite electromagnetic formation as an example, and for a multi-satellite electromagnetic formation, a person skilled in the art can design based on the design process of the double-satellite electromagnetic formation.

[0165] Simulation experiment

[0166] For the convenience of illustrating the application effect, the control of the circular formation of the dual-star low earth orbit is taken as an example for simulation. In the simulation, it is assumed that the mass center of the satellite formation runs on a 700 km circular orbit, and the satellite attitude keeps consistent with the mass center orbit coordinate system. The relative position vector of the dual-star is designed as wherein d=15 m, T=5 h, and ω=2π / T. The mass of each satellite is 150 kg, the inertia matrix is J=diag(20, 17, 22) kg·m 2 , the inertia 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 disturbance force and disturbance moment caused by the earth magnetic field, it is assumed that the satellite also receives other external periodic external disturbance force and moment as

[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 dual-star is r e (0)=[1 2 -2] T m, the initial error of the relative velocity is The initial attitude error of satellite A is q e1 (0)=[0.843 -0.2 0.3 -0.4] T , and 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 , and 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, and has a significant impact on the control accuracy of the system. In order to solve this problem, a larger alternating frequency value is considered to be designed, and a second-order low-pass filter is added to the control input before entering the magnetic dipole solving stage. The solved magnetic dipole will enter the object, significantly reducing the high-frequency disturbance directly entering the controller. The control parameters of the simulation are designed as ω otj =0.2, ω ctj =0.03, ω oaj =1, ω caj =0.1, j=1, 2, 3, and the carrier frequency of the satellite magnetic dipole is designed as ω1=100Hz, ω2=200Hz. The transfer function of the second-order low-pass filter is Wherein k=1, ξ=1, ω=5Hz. Figures 3(a-f) and 4(a-f) show the simulation results of relative position, attitude and reaction wheel angular momentum. The formation satellite can achieve good control accuracy and keep a low angular momentum range within one orbit period.

[0172] The present application adopts the above-mentioned electromagnetic formation relative motion control and angular momentum management method, designs a magnetic moment for the satellite based on frequency division multiplexing method, and gives a magnetic moment solution corresponding to the electromagnetic force and torque generated between the satellites, so that the electromagnetic formation can control the relative position and attitude of the electromagnetic satellite by using electromagnetic force and electromagnetic torque, and only needs to install a set of reaction wheels on the leader satellite to realize the attitude control of the entire formation, and the required control force and torque of the follower satellite are provided by the electromagnetic force and electromagnetic torque; the unloading law is designed to unload the angular momentum of the reaction wheels of the leader satellite, so as to realize the angular momentum unloading in the process of 6-degree-of-freedom relative motion control of the satellite. The direct current component of the magnetic moment of the leader satellite interacts with the geomagnetic field, which can be used to unload the angular momentum of the reaction wheels, so that the angular momentum of the reaction wheels can be kept in a small range for a long period of time.

[0173] Further, the present application uses the idea of frequency division multiplexing to decouple the relative position control and angular momentum management, and realizes parallel angular momentum unloading and formation control.

[0174] It should be noted that although the above embodiments of the present application are illustrative, this is not a limitation of the present application, and therefore the present application is not limited to the above specific embodiments. Any other embodiments obtained by those skilled in the art under the inspiration of the present application without departing from the principles of the present application are considered to be within the protection scope of the present application.

Claims

1. A method for relative motion control and angular momentum management of electromagnetic formations, characterized in that, Includes the following steps: Step S1: Establish a 6-DOF relative dynamic model of the electromagnetic satellite formation and determine the leader satellite and follower satellites; Step S2: Select the leader satellite to install the reaction wheel, design the magnetic moment for the satellite based on the frequency division multiplexing method, generate effective electromagnetic force and torque, and design the DC component for angular momentum unloading; Step S3: Design a controller for the 6-DOF relative motion of the electromagnetic formation system under conditions of external interference and model uncertainty using the active disturbance rejection control method; Step S4: Design an unloading law to unload the angular momentum of the reaction wheel of the leader satellite, and solve for the DC component of the magnetic moment of the leader satellite in step S2. Step S1 specifically includes: If we consider the electromagnetic formation system as having its center of mass undergoing near-Earth circular motion, then the relative dynamic equations of the electromagnetic formation system can be expressed as follows: in The angular velocity of the system's center of mass orbiting the Earth's center is μ g =3.986×10 14 m 3 / s 2 r is the gravitational constant of the Earth's core. CM Let X1 be the distance between the Earth's center and the center of mass of the binary system, where X1 = [xyz]. T The position vector representing the satellite relative to the system's center of mass. f represents the relative velocity vector of satellite i. d =[f dx f dy f dz ] T The acceleration caused by interference experienced by the satellite; With X 1d =[x d y d z d ] T This represents the satellite's expected relative position vector. Let the desired relative velocity vector of the satellite be represented, then the control objective for the satellite's relative translational motion is expressed as: The error dynamics equation of the satellite is: in, The control torque is represented by Δ, and the total uncertainty of the system is represented by Δ. The satellite's attitude control objective can be represented as: Because ln q e = [0, θ / 2] T The satellite's attitude control objective can be represented as 2. The method for relative motion control and angular momentum management of electromagnetic formations according to claim 1, 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 follows: Where μ1 and μ2 are the magnetic moments of satellite A and satellite B, respectively, and ω1 and ω2 are two different AC frequencies, μ lg For the DC component of the magnetic moment used for unloading the angular momentum of the reaction wheel of satellite A, μ 11 μ 12 For satellite A, μ represents the two AC magnetic moment amplitude components. 21 μ 22 These are the two AC magnetic moment amplitude components of satellite B; Based on the electromagnetic model, the electromagnetic force and torque are calculated. Ignoring the effect of the Earth's magnetic field on the electromagnetic satellites, the electromagnetic forces acting on satellites A and B are expressed as follows: Where F1 and F2 are the electromagnetic forces acting on satellite A and satellite B, respectively; The electromagnetic torques acting on satellites A and B are expressed as follows: Where R1 and R2 are the position vectors from the Earth's center to satellite A and satellite B, respectively. These are the electromagnetic torques acting on satellite A and satellite B, respectively. Treating the cosine terms in equations (7) and (8) as external disturbances to the system, the effective control force and control torque provided by the electromagnetic formation system for satellite B are expressed as follows: Among them, F ec2 , These represent the effective control force and control torque experienced by satellite B, respectively.

3. The method for relative motion control and angular momentum management of electromagnetic formations according to claim 2, characterized in that, In step S2, the electromagnetic force, electromagnetic torque, and magnetic moment are all represented in the electromagnetic coordinate system for analysis. The two magnetic moments are represented 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 follows: Let μ i =[m1 m2 m3] T μ j =[m4 m5 m6] T , Based on equation (10), we can obtain: The system of equations (11) is a system of quadratic equations with independent variables {m1, m2, m3, m4, m5, m6} and dependent variables {c1, c2, c3, c4, c5, c6}. Simplifying the system of equations (11) yields: Equation set (12) represents the coupling relationship between the magnetic moment, electromagnetic force, and electromagnetic torque components, then: When c5-c3=0, c2+c6=0, and c2 and c3 are not both 0, m4 is chosen as the free variable, and the solution to the system of equations (12) is m1=0. When 2c2+c6=0, 2c3-c5=0, and c2 and c3 are not both 0, if m1 is chosen as the free variable, the solution to the system of equations (12) is m4=0. When c2 = c3 = c5 = c6 = 0, m2 and m3 can be chosen as free variables, and the solution to the system of equations (8) is m1 = 0, m4 = 0. For equation (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 the system of equations (12), we can obtain: Let n k =d k +lk, k=1,2,…6, In order to ensure that the system of equations has a solution under the condition that {n1,n2,n3,n4,n5,n6} takes any value, according to the analysis of solving the system of equations (12), we can let l5-l3=0, -l2-l6=0, 2d2+d6=0, 2d3-d5=0, and we can get: Where k1 and k2 are arbitrary constants, and {n1, n2, n3, n4, n5, n6} can take any value. {d2, d3, d1, d4, l2, l3, l1, l4} can be set according to equation (14). Then, the magnetic moment is solved according to the analysis of equation set (12), that is, the magnetic moment solution of equation set (13) is obtained.

4. The method for relative motion control and angular momentum management of electromagnetic formations according to claim 3, characterized in that, In step S3, a controller for the 6-DOF relative motion of the electromagnetic formation system under conditions of external interference and model uncertainty is designed using an active disturbance rejection control method. The control force and torque required by the follower satellite are provided by electromagnetic force and electromagnetic torque. Based on the solution for the magnetic moment in step S2, the control torque of the leader satellite is provided by both electromagnetic torque and the reaction wheel, specifically including: To design a linear extended state observer (LESO) for an electromagnetic formation system, we have: in, The estimated values ​​of X1, X2, and f are respectively, and β is the sum of the values ​​of f and f. k =diag(β) k1 ,β k2 ,β k3 ), k = 1, 2, 3 are the observer gains; design ω otj For observer bandwidth; Based on the online estimation of the total uncertainty, the active disturbance rejection control is designed as follows: Among them, the countable Where ω tcj >0, j=1, 2, 3 represents the controller bandwidth; To design a linear extended state observer (LESO) for the error dynamics system of equation (3), we have: in, and Here 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; design gain parameters satisfy L aj =s 2 +γ 1j s+γ 2j Let j = 1, 2, 3 be the conditions of the Hurwitz polynomial. Then L aj =(s+ω oaj ) 2 ; Based on online estimation of the "total disturbance" or extended state, the controller is designed as follows: in For virtual control, the control coefficients can be simply set as k3 = k4 = ω ca ω ca =diag(ω ca1 ω ca2 ω ca3 ), ω caj >0, j=1, 2, 3 represents the controller bandwidth; Based on equations (3) and (18), the reference error system can be obtained as follows: in, The reference error system is a hierarchical system. According to the contraction theorem, equation (19) converges exponentially to the ball surrounding the desired trajectory.

5. The method for relative motion control and angular momentum management of electromagnetic formations according to claim 4, characterized in that, 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) Where, k h Let h be the unloading gain to be determined, and h be the projection of the angular momentum vector of the reaction flywheel onto the three axes of the satellite body coordinate system. n The nominal angular momentum vector is a preset value, and Δh is the excess angular momentum that needs to be unloaded. The torque expected to be unloaded using the Earth's magnetic field is denoted as τ. EA According to formula (20), we can obtain When set to μ lg With the geomagnetic field B E When perpendicular, the magnetic moment required for angular momentum unloading can be calculated according to equation (21):

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