CN112068419A - Flexible satellite pointing tracking control method containing six-degree-of-freedom vibration isolation platform - Google Patents

Flexible satellite pointing tracking control method containing six-degree-of-freedom vibration isolation platform Download PDF

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CN112068419A
CN112068419A CN202010736828.9A CN202010736828A CN112068419A CN 112068419 A CN112068419 A CN 112068419A CN 202010736828 A CN202010736828 A CN 202010736828A CN 112068419 A CN112068419 A CN 112068419A
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吴晗
金磊
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Beihang University
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Abstract

The invention discloses a flexible satellite pointing tracking control method containing a six-degree-of-freedom vibration isolation platform, and designs a pointing tracking control method easy for engineering realization on the basis of time domain and frequency domain indexes aiming at a flexible satellite containing a six-degree-of-freedom vibration isolation platform. Firstly, considering the vibration characteristics of a control moment gyro, respectively establishing dynamic models of an upper platform and a lower platform of a six-degree-of-freedom vibration isolation system; then, the vibration isolation support rod is equivalent to a three-parameter model, a transfer function of the flexible satellite vibration isolation platform is obtained through simplification, and parameters of the vibration isolation platform are designed according to vibration isolation indexes; and finally, aiming at the problem that the target posture and the angular speed change rapidly in a certain period of time in the pointing control process, designing the form and the parameters of the controller based on a frequency domain theory according to the performance index of the controller. The invention establishes the transfer function of the flexible satellite three-parameter vibration isolation platform, the designed vibration isolation parameters can realize effective attenuation of high-frequency vibration of the actuating mechanism, and the designed pointing tracking controller can meet frequency domain indexes and time domain precision indexes, and can be widely applied to actual engineering.

Description

Flexible satellite pointing tracking control method containing six-degree-of-freedom vibration isolation platform
Technical Field
The invention relates to a flexible satellite pointing tracking control method with a six-degree-of-freedom vibration isolation platform. The integrated control of the upper platform and the lower platform for pointing tracking is realized for the flexible satellite which takes the control moment gyroscope as an actuating mechanism to carry out integral vibration isolation. The method can be applied to attitude tracking control with fast target attitude and angular speed change in a certain period of time, inhibits flexible vibration of the sailboard in the attitude maneuver process, realizes high-frequency vibration isolation of the actuating mechanism, and has strong robustness on inertia parameter change and actuating mechanism disturbance. The invention belongs to the field of spacecraft attitude control.
Background
With the continuous deepening of space applications and space detection activities such as laser communication, space remote sensing, deep space telescopes and the like, the functions and the structures of satellites become more complex, the carried scientific detection instruments become more and more precise, and the requirements on the attitude pointing accuracy and the attitude pointing stability become higher. The installation and manufacturing errors of the actuating mechanism and the elastic deformation of the solar sailboard are main factors causing the micro-vibration of the satellite attitude, and it is a hot point of current research to adopt corresponding measures to improve the control precision. The Stewart platform is a parallel mechanism with six-degree-of-freedom motion capability, has the advantages of high precision, high rigidity, stable structure, strong bearing capability, small motion inertia, good dynamic characteristic and the like, and is widely applied to the problem of vibration isolation of satellites.
For a flexible satellite comprising a Stewart vibration isolation platform, parameter design and design of a pointing tracking controller of the vibration isolation platform are two key problems. In a document (Yao, research on high-frequency vibration isolation of attitude of a high-precision and high-stability satellite: Beijing university of aerospace, 2012), aiming at a single rigid body satellite containing a Stewart platform, considering the vibration characteristics of a control moment gyroscope, establishing a vibration isolation platform transfer function of a two-parameter, three-parameter and two-parameter tuning mass damper, giving a parameter design rule, and realizing attitude stability control by using a PID (proportion integration differentiation) controller; for a flexible satellite containing a Stewart platform, a vibration isolation platform transfer function based on a two-parameter model is established, and attitude stability control is realized. For the directional tracking control of the master satellite and the slave satellite, the existing control methods include sliding mode control, adaptive control, active disturbance rejection control, backstepping control, combined control of various methods and the like, and although many advanced theories and methods are developed, the design method of the controller based on the frequency domain method is still dominant in engineering application.
Therefore, for a flexible satellite using a control moment gyro as an actuating mechanism, a Stewart platform is adopted for overall vibration isolation, and performance indexes of a frequency domain and a time domain are comprehensively considered.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at high-frequency vibration generated by a control moment gyro due to installation and manufacturing errors, a Stewart platform based on a three-parameter model is utilized for overall vibration isolation, a transfer function of the control moment gyro under a flexible satellite is established, and a vibration isolation parameter design rule is discussed; then, aiming at the problems that a flexible satellite containing a Stewart platform is high in target attitude and angular speed change in the pointing tracking control process, flexible vibration of a sailboard in the maneuvering process and the like, the periodicity of controller output and the disturbance of an actuating mechanism to inertia are considered, and the pointing tracking control method integrating the upper platform and the lower platform is provided based on frequency domain theoretical design and can be used for high-precision and high-stability pointing tracking control of a flexible spacecraft.
The technical scheme adopted by the invention for solving the technical problems is as follows: aiming at a flexible satellite comprising a six-degree-of-freedom vibration isolation platform, firstly, considering the vibration characteristics of an actuating mechanism, and establishing a dynamic model of an upper platform and a lower platform of the flexible satellite comprising the six-degree-of-freedom vibration isolation platform; then, the vibration isolation supporting rod is equivalent to a three-parameter model, the transfer function of the flexible satellite vibration isolation platform is simplified and obtained, and the parameters of the vibration isolation platform are designed; and finally, designing an integrated pointing tracking controller of an upper platform and a lower platform of the flexible satellite containing the six-degree-of-freedom vibration isolation platform according to performance indexes of a frequency domain and a time domain of a system, and realizing high-precision high-stability pointing tracking control of the flexible satellite. The specific implementation steps are as follows:
step 1: and establishing a dynamic model of the upper platform and the lower platform of the flexible satellite comprising the six-degree-of-freedom vibration isolation platform.
Neglecting external disturbance to the upper platform system, considering rotor static and dynamic unbalance of the control moment gyroscope, and non-perpendicularity and non-intersection of the frame shaft and the rotor shaft, the dynamic equation of the vibration isolation upper platform system is as follows:
Figure BDA0002605373550000021
Figure BDA0002605373550000022
Figure BDA0002605373550000023
wherein v isuThe component array representing the velocity of the upper stage in the upper stage system,
Figure BDA0002605373550000024
denotes vuA first derivative with respect to time; omegauThe component array of the upper platform system is shown in the attitude angular velocity of the upper platform,
Figure BDA0002605373550000025
represents omegauA first derivative with respect to time; muFor the quality of the upper platform system, SuIs a representation of the static moment of the upper platform system in the upper platform system, IuFor a representation of the system in the upper platform without the moment of inertia of the control moment gyro, JuRepresenting the total rotational inertia of the upper platform in a system on the upper platform in order to consider the vibration characteristic of the control moment gyroscope; p is a radical ofiRepresenting a position vector array from the center of mass of the upper platform to the connecting point of the upper platform;
Figure BDA0002605373550000026
is the viscous damping moment of the upper platform ball pivot, csiIs the coefficient of the viscous damping of the spherical hinge,
Figure BDA0002605373550000027
angular velocity of the strut relative to the inertial system, AueA coordinate transformation matrix representing the inertial system to the upper platform body system;
Figure BDA0002605373550000028
is shown as a wholeMomentum of the platform system removes the remaining part, m, with the interfering source termgiRepresents the frame mass, mwiRepresenting the rotor mass, rugiThe position vector array from the center of mass of the upper platform to the center of mass of the frame is represented, and m represents the number of control moment gyroscopes; fsiIs the restraining force of the ball joint on the upper support rod, FdRepresenting disturbance force, T, produced by a control moment gyrodAnd the coupling moment and the control moment generated by the control moment gyroscope are shown, and N is the number of the supporting rods of the vibration isolation platform. In the formula, superscript "to" represents a cross-product antisymmetric diagonal matrix of the array, i.e., x ═ x for any three-dimensional array1 x2 x3]TIs provided with
Figure BDA0002605373550000031
The lower platform system is a flexible spacecraft which is substantially a central rigid body and a sailboard, and assuming that the sailboard is in a locked state, a dynamic model of translation, rotation and sailboard vibration of the lower platform system can be expressed as follows:
Figure BDA0002605373550000032
Figure BDA0002605373550000033
wherein v isbThe component array representing the velocity of the lower platform system in the satellite system,
Figure BDA0002605373550000034
denotes vbA first derivative with respect to time; omegabThe component array of the attitude angular velocity of the lower platform system in the satellite system is shown,
Figure BDA0002605373550000035
represents omegabA first derivative with respect to time; etaakThe modal coordinates of the windsurfing board k are represented,
Figure BDA0002605373550000036
respectively represent ηakFirst and second derivatives with respect to time; msRepresents the total mass of the lower platform system, SsAnd IsRespectively representing the total static moment and the total rotational inertia of the lower platform system in a satellite system; lambdaakIs a diagonal array of modal frequencies, Λ, of sailboard kak＝diag([Λk1；Λk2；…Λkj；…Λkn])，
Figure BDA0002605373550000037
The j-th order flexible modal frequency is represented, and n represents the modal order; xiakThe flexible modal damping coefficient of the sailboard k; frbakA flexible coupling coefficient matrix of the kth flexible sailboard vibration to the central rigid body rotation; ftbakA flexible coupling coefficient matrix of the kth flexible sailboard vibration to the central rigid body translation; r isbdA position vector array, q, representing the origin of a fixed coordinate system of the lower platform system from the center of mass of the lower platform system to the lower platformiRepresenting a position vector array from the original point of the lower platform fixed connection coordinate system to the connecting point of the lower platform;
Figure BDA0002605373550000038
is the viscous damping moment of the lower platform universal hinge, cuiIs the damping coefficient of universal hinge viscosity, AbeA coordinate transformation matrix representing the inertial system to the satellite body system; a. thebdA coordinate transformation matrix representing the fixed connection coordinate system of the lower platform to the satellite body system; fuiIs the restraining force of the universal hinge on the lower support rod, FextIs the interference force of the external environment, does not include the universal gravitation item, MuihiIs the restraining moment of the universal hinge on the lower supporting rod, TextDisturbing the moment for the external environment; w represents the number of windsurfing boards.
Step 2: the vibration isolation supporting rod is equivalent to a three-parameter model, the transfer function of the flexible satellite vibration isolation platform is simplified and obtained, and the parameters of the vibration isolation platform are designed.
The vibration characteristic of the actuating mechanism is not considered, the first five-order modal vibration is taken, and F is maderbak＝[k1 k2 k3 k4 k5]The value of k is l and r, which are respectively referred to as left and right sailboards, wherein
Figure BDA0002605373550000041
Representing a 3 x 1 column vector, the flexible satellite attitude dynamics transfer function can be expressed as
Figure BDA0002605373550000042
Where s represents a complex variable in the laplace transform. In practical engineering, the left and right sailboards are usually symmetrically installed and have consistent parameters, so that the flexible modal damping coefficients of the left and right sailboards can be represented by xi, namely xiar＝ξalXi, the flexible modal frequency of j th order of the left sailboard and the right sailboard can be used by ΛjIs represented byj＝Λrj＝Λlj，j＝1,...,5。
For the transfer function of the flexible satellite vibration isolation platform, the vibration isolation supporting rod is equivalent to a three-parameter model, and the following assumptions are made: (1) the mass and inertia of the supporting rod of the vibration isolation platform have little influence on the transfer function of the vibration isolation platform, and can be ignored in the approximation process; (2) the attitude angles of the upper platform and the lower platform meet the small angle assumption, and a coordinate transformation matrix from a system fixedly connected with the upper platform to a system fixedly connected with the star body to an inertial coordinate system can be regarded as a unit matrix; (3) the amplitude of the disturbance torque is small, and the configuration of the vibration isolation platform is assumed to be unchanged. The dynamics of the upper and lower platform systems can be simplified as follows:
Figure BDA0002605373550000043
Figure BDA0002605373550000044
wherein, t, θuRepresenting the position and attitude angle of the upper platform system in the inertial system, b, thetabRepresenting the position and attitude angle of the lower platform system in the inertial system,
Figure BDA0002605373550000045
respectively represent t and thetau、b、θbA second derivative with respect to time; restraining force F acting on the upper platformsiIt can be abbreviated as follows:
Figure BDA0002605373550000046
wherein k isAIs the main spring rate, k, of the strutBIs the additional spring rate of the strut, cAIs the damping coefficient of the strut, suiThe component array in the inertial system of the unit vector along the axial direction of the strut i is shown.
Order:
Figure BDA0002605373550000047
then FsiCan be written as follows
Figure BDA0002605373550000051
Wherein JiRepresenting a 3 x 22 matrix, which is an intermediate variable,
Figure BDA0002605373550000052
the dynamics model of the simplified context platform system can be written in the form of
Figure BDA0002605373550000053
Wherein
Figure BDA0002605373550000054
Respectively representing the third, second and first time derivatives of x with respect to time,
Figure BDA0002605373550000055
respectively representing a mass matrix, a damping matrix and a rigidity matrix of the system,
Figure BDA0002605373550000056
occurring during the three-parameter model approximation process, represent
Figure BDA0002605373550000057
The coefficient of (a) is determined,
Figure BDA0002605373550000058
input forces and moments representing equations of translation, rotation and vibration of the upper and lower platforms, and having
Figure BDA0002605373550000059
Figure BDA00026053735500000510
Figure BDA00026053735500000511
Figure BDA00026053735500000512
Wherein EiUnit array of i × i, 0iZero matrix, 0, representing i x ii×jA zero matrix representing i multiplied by j, wherein i and j represent positive integers; mak＝2ξakΛakIs a damping matrix for the windsurfing board k,
Figure BDA00026053735500000513
is the stiffness matrix of the windsurfing board k.
Output is defined as the generalized force experienced by the star and is written as follows:
Figure BDA00026053735500000514
wherein
Figure BDA0002605373550000061
A damping matrix, a stiffness matrix representing a generalized force expression,
Figure BDA0002605373550000062
taking a state quantity
Figure BDA0002605373550000063
The input u is the disturbance force and moment borne by the upper platform system, the output Y represents the disturbance force and moment transmitted to the lower platform system, and the dynamic equations of the upper and lower platform systems can be written in the form of state equations:
Figure BDA0002605373550000064
wherein A represents a system matrix, B represents an input matrix, CtRepresenting the output matrix, D representing the direct transfer matrix
Figure BDA0002605373550000065
Figure BDA0002605373550000066
By GVIP(s) represents the transfer function matrix of the flexible satellite vibration isolation platform, namely the transfer function matrix of the upper platform system disturbed to the star body, GVIP(s)＝Ct(sE-A)-1B + D, which includes the effect of the flexible windsurfing board; e represents an identity matrix of the same order as the matrix A;
after a transfer function of the flexible satellite vibration isolation platform is obtained, k is obtained by using a control variable methodA、kB、cAThree parameters and main channelDetermining parameter values according to the variation rule between bode graphs of the vibration isolation transfer function and the performance index of vibration isolation;
and step 3: aiming at the problem that the target posture and the angular speed change rapidly in a certain period of time in the pointing control process, according to the performance indexes of a control system, including pointing accuracy, flexible vibration suppression requirements and the like, a pointing tracking controller integrating an upper platform and a lower platform of a flexible satellite with a six-degree-of-freedom vibration isolation platform is designed.
Step 3.1: defining error attitude quaternion and angular velocity
The quaternion of the body coordinate system relative to the inertial coordinate system is recorded as QbIThe angular velocity of the body coordinate system relative to the inertial coordinate system is denoted as ωb(ii) a Quaternion of the desired coordinate system relative to the inertial coordinate system is denoted as QTIThe angular velocity of the desired coordinate system relative to the inertial coordinate system is denoted as ωTAngular acceleration is noted
Figure BDA0002605373550000067
Let QeAnd ωeRepresenting the attitude quaternion and the triaxial angular velocity of the spacecraft body system relative to a desired coordinate system, and defining the following error attitude quaternion and error angular velocity
Figure BDA0002605373550000068
In the formula qe0、qeRespectively representing error quaternion QeThe target and vector of (a) are,
Figure BDA0002605373550000069
is QTIA conjugated quaternion of (3), C (Q)e) A transformation matrix for the desired coordinate system to the spacecraft body coordinate system can be written as
Figure BDA0002605373550000071
Wherein
Figure BDA0002605373550000072
Is qeCross-multiplication antisymmetric oblique square matrix of (E)3Is a 3 × 3 unit array. The corresponding error attitude equation can be written as
Figure BDA0002605373550000073
Step 3.2: PD + feedforward controller design
Writing the attitude dynamics equation of the satellite into the form of error quaternion and error angular velocity, and recording C (q)e) Is C. The upper platform and the lower platform are regarded as a whole to carry out pointing control, and the following attitude tracking controller is designed:
Figure BDA0002605373550000074
wherein k isp、kdSgn (·) represents a sign function for the control parameter; from J'uRepresenting the total moment of inertia, r, of the upper platform irrespective of the vibrational characteristics of the actuatorbuPosition vector array representing the center of mass of the star body to the center of mass of the upper platform, IsumThe total moment of inertia of the upper and lower platforms can be obtained according to the parallel axis theorem
Figure BDA0002605373550000075
Let Tc＝-2kpIsum sgn(qe0)qe-kdIsumωeRecord Kp＝kpIsum,Kd＝kdIsumThen the transfer function of the PD controller can be written as
Figure BDA0002605373550000076
Wherein T isc(s)、Θe(s) each represents Tc、Θe(ii) a laplace transform of; k is a radical ofp、kdThe selection rule of the parameters is as follows: the lower platform can be regarded as a rigid body and is static relative to an inertial system, and then the value ranges of the three main channel control parameters are determined according to a root track theory. Here, k is determined according to the second order system theoryp、kdThe parameter (c) of (c).
Step 3.3: design of hysteresis correction link
In order to further improve the speed and accuracy of the pointing tracking, the bandwidth of the control system needs to be increased, but the stability margin of the control system should be avoided from being greatly influenced. Therefore, a hysteresis correction link is adopted, on one hand, the gain of a low frequency band can be improved, and the precision of the pointing control is effectively improved; on the other hand, the bandwidth of the control system can be increased, and the pointing tracking speed is improved. The transfer function of the lag correction element can be expressed as:
Figure BDA0002605373550000077
t and alpha are adjustable parameters of a hysteresis correction link, and the performance of the control system can be improved by adjusting T and alpha.
Step 3.4: structural filter design
The flexible vibration of the sailboard can cause fluttering of the satellite attitude angular velocity and the control moment, and in order to reduce the influence of the vibration on the satellite attitude, a structural filter is usually introduced to suppress the flexible vibration. The classical structural filter form is as follows
Figure BDA0002605373550000081
Wherein ζz、ζp、ωz、ωpDifferent design rules can obtain different forms of filters for filter parameters. The invention adopts the minimum phase notch filter to restrain the flexible vibration of the sailboard and improves the stability of attitude control.
And 3.2-3.4, forming a complete controller, considering the periodicity of control instruction output, and performing combined design of control parameters on the flexible satellite containing the six-degree-of-freedom vibration isolation platform according to performance indexes.
Compared with the prior art, the invention has the advantages that:
1. aiming at the flexible satellite, the vibration isolation platform transfer function based on the three-parameter model is established, the parameter design rule is analyzed, and the high-frequency vibration of the actuating mechanism can be effectively attenuated;
2. aiming at the problem of large dynamic pointing tracking of a flexible satellite with a vibration isolation platform, the invention designs a controller combination of PD + feedforward + hysteresis correction + structural filter, can effectively inhibit the flexible vibration of a sailboard, and realizes high-precision and high-stability pointing tracking control.
3. The invention considers the periodicity output by the controller and the inertia disturbance caused by the vibration of the actuating mechanism, can meet various performance indexes of a system frequency domain and a system time domain, and has great engineering application value.
Drawings
FIG. 1 is a block flow diagram of the present invention.
Fig. 2 is a structural diagram of a flexible satellite with a six-degree-of-freedom vibration isolation platform.
Fig. 3 is a three-parameter vibration isolation model.
FIG. 4 is a block diagram of a control system for the z-channel.
Fig. 5 is a description diagram of a point tracking task.
FIG. 6a shows a fixed parameter kA、cABode plot of x-channel isolation transfer function of (a).
FIG. 6b shows a fixed parameter kB、cABode plot of x-channel isolation transfer function of (a).
FIG. 6c shows a fixed parameter kA、kBBode plot of x-channel isolation transfer function of (a).
Fig. 7a shows the pointing accuracy of the lower platform system before the vibration isolation platform is added.
Fig. 7b shows the pointing accuracy of the lower platform system after the vibration isolation platform is added.
Fig. 8a shows the pointing stability (error angular velocity) of the lower stage system before vibration isolation is added.
Fig. 8b shows the pointing stability (error angular velocity) of the lower stage system after vibration isolation.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
As shown in fig. 1, the invention relates to a flexible satellite pointing tracking control method with a six-degree-of-freedom vibration isolation platform, which comprises the following specific implementation steps:
step 1: according to the system shown in fig. 2, a dynamic model of the upper platform and the lower platform of the flexible satellite with the six-degree-of-freedom vibration isolation platform is established.
Neglecting external disturbance to the upper platform system, considering rotor static and dynamic unbalance of the control moment gyroscope, and non-perpendicularity and non-intersection of the frame shaft and the rotor shaft, the dynamic equation of the vibration isolation upper platform system is as follows:
Figure BDA0002605373550000091
Figure BDA0002605373550000092
Figure BDA0002605373550000093
wherein v isuThe velocity of the upper stage is expressed in the component array of the upper stage system, and the initial value is vu0＝[0；0；0]，
Figure BDA0002605373550000094
Denotes vuA first derivative with respect to time; omegauRepresenting the component array of the upper platform system by the attitude angular velocity of the upper platform, and taking omega as the initial valueu0＝[0.4；-0.5；0.4]°/s，
Figure BDA0002605373550000095
Represents omegauA first derivative with respect to time; muIs the quality of the upper platform system, take Mu＝150kg，IuFor the representation of the system rotating on the upper platform without control moment gyro, take
Iu＝diag([9.47；9.47；18.72])kg·m2；
Figure BDA0002605373550000096
Is the viscous damping moment of the upper platform ball pivot, csiIs the viscous damping coefficient of the spherical hinge, and c is takensi＝0.0001，AueThe coordinate transformation matrix representing the inertial system to the upper platform body system can be obtained by the attitude quaternion of the upper platform relative to the inertial system, and the initial value of the attitude quaternion of the upper platform is Qu0＝[0.663；-0.6；0.2；0.4]；
Figure BDA0002605373550000097
Representing the remaining part of the momentum of the entire upper platform system after removal of the terms with interfering sources, rugiThe position vector array from the upper platform center of mass to the frame center of mass can be obtained according to the control moment gyroscope configuration.
Controlling moment gyro parameters: m represents the number of control moment gyros, and for a pentagonal pyramid configuration, m is 6, the pentagonal pyramid cone angle beta is 63.4 degrees, and the sagittal diameter from the mass center of the upper platform body system to the mass center of the frame is rug＝0.8m，mgiRepresenting the mass of the frame, take mgi＝6.7kg，mwiRepresenting rotor mass, taking mwiThe moment of inertia of the frame relative to its centre of mass is denoted I in the frame system at 9kgg＝diag([0.0217；0.0197；0.0177])kg·m2The moment of inertia of the rotor relative to the center of mass is expressed in a rotor inertia principal axis coordinate system
Figure BDA0002605373550000101
The rotating speed of the rotor is equal to 9000rpm, and the maximum rotating speed of the frame max60 °/s; vibration parameters: static unbalance amount ρw＝[0.8；1.0；0.8]×10-6m, dynamic unbalance amount
η＝5.8824×10-6rad、μ＝8.8235×10-6rad, non-intersecting extent r of rotor and frame axesfg＝[6；8；8]×10-6m, non-perpendicularity α is 0.0001rad、β＝0.0001rad、ψ＝0.0001rad。SuRepresentation of the upper platform body system for the static moment of the upper platform system, JuRepresentation of the total moment of inertia of the upper platform in the system of the upper platform in order to take account of the vibration characteristics of the control moment gyro, FdRepresenting disturbance force, T, produced by a control moment gyrodThe coupling moment and the control moment generated by the control moment gyro can be obtained according to parameters of the upper platform and the control moment gyro, and are not given in detail here.
The lower platform system is essentially a flexible spacecraft with a central rigid body and a sailboard, and assuming that the sailboard is in a locked state, a dynamic model of translation, rotation and sailboard vibration of the lower platform system can be expressed as follows:
Figure BDA0002605373550000102
Figure BDA0002605373550000103
wherein v isbRepresenting the component array of the speed of the lower platform system in the satellite system, and taking v as the initial valueb0＝[0；0；0]，
Figure BDA0002605373550000104
Denotes vbA first derivative with respect to time; omegabThe component array of the attitude angular velocity of the lower platform system in the satellite system is expressed, and the initial value is omegab0＝[0.4；-0.5；0.4]°/s，
Figure BDA0002605373550000105
Represents omegabA first derivative with respect to time; etaakRepresenting modal coordinates of sailboards, with initial value ηak＝[0；0；0；0；0]，
Figure BDA0002605373550000106
Respectively represent ηakFirst and second derivatives with respect to time; msThe total mass of the lower platform system is expressed as
Figure BDA0002605373550000107
SsAnd IsRespectively representing the total static moment and the total rotational inertia of the lower platform system in a satellite system, and the expressions are
Figure BDA0002605373550000108
mbRepresenting the mass of the central rigid body, SbAnd IbRespectively representing the representation of the static moment and the inertia matrix of the central rigid body in the satellite system, and the value is mb＝2500kg、Sb＝[0；0；0]kg·m、Ib＝diag([3900；7000；5000])kg·m2，makRepresenting the mass of the sailboard k, SbakThe expression of the static moment of the sailboard k relative to the star body is
Figure BDA0002605373550000109
Is the viscous damping moment of the lower platform universal hinge, cuiIs the universal hinge viscous damping coefficient, take cui＝0.0001，AbeThe coordinate transformation matrix representing the inertial system to the satellite body system can be obtained by the attitude quaternion of the upper platform relative to the inertial system, and the initial value of the attitude quaternion of the upper platform is Qb0＝[0.663；-0.6；0.2；0.4]；FextIs the interference force of the external environment, does not include the universal gravitation item, TextDisturbing the moment for the external environment.
Parameters of the sailboard: w represents the number of sailboards, where W is 2; the mass of the left sailboard and the right sailboard is mak＝27.2kg，ΛakThe modal frequency diagonal matrix of the sailboards is obtained, and the left sailboard and the right sailboard are both taken as lambdaak＝diag([0.34；1.65；1.90；5.34；9.35])Hz，ξakThe modal damping ratio of the sailboards is set as xi (xi) 0.005 for both the left sailboard and the right sailboard; translation coupling coefficient B expressed under sailboard fixed connection coordinate systemtranTaking the following steps:
Figure BDA0002605373550000111
rotary coupling expressed under sailboard fixed coordinate systemCoefficient BrotGet
Figure BDA0002605373550000112
FtbakThe flexible coupling coefficient matrix of the flexible sailboard k vibration to the central rigid body translation is expressed in the system, Ftbak＝AbakBtran；FrbakThe flexible coupling coefficient matrix for the vibration of the flexible sailboard k to the rotation of the central rigid body is expressed in the system,
Figure BDA0002605373550000113
wherein A isbakShowing the mounting matrix of the sailboards k, two each, relative to the lower platform body system
Figure BDA0002605373550000114
rbakShowing the mounting position of the sailboard k on the lower platform body system, two sailboards are respectively taken
rbal＝[0.65；1.2615；0]m rbar＝[0.65；-1.2615；0]m
IbakThe inertia matrix representing the inertia of the windsurfing board relative to the star body can be approximated as
Figure BDA0002605373550000115
Vibration isolation platform parameters: n is the number of the supporting rods of the vibration isolation platform, N is 6, and the height of the platform
Figure BDA0002605373550000116
Nominal length of strut
Figure BDA0002605373550000117
Upper strut central mass mui1kg, lower support concentrating mass mdi＝2kg，piRepresenting the position vector array from the upper platform center of mass to the upper platform connecting point, and taking
Figure BDA0002605373550000118
qiExpressing the position vector array from the original point of the fixed coordinate system of the lower platform to the connecting point of the lower platform, and taking
Figure BDA0002605373550000121
The unit vector of the fixed rotating shaft of the universal hinge of each strut is expressed as
Figure BDA0002605373550000122
rbdExpressing the position vector array from the center of mass of the lower platform system to the origin of the fixed coordinate system of the lower platform, and taking rbd＝[0 0 0.5]Tm, the moment of inertia of the upper supporting rod under the upper supporting rod fixedly connected coordinate system and the moment of inertia of the lower supporting rod under the lower supporting rod fixedly connected coordinate system are respectively as follows
Iu0i＝diag([1.7×10-4；4.885×10-3；4.885×10-3])kg·m2
Id0i＝diag([9.87×10-4；1.7449×10-2；1.7449×10-2])kg·m2
The ith strut is fixedly connected with the vector array r from the coordinate system center to the mass concentration position of the upper strutu0iA vector array r fixedly connected with the center of the coordinate system of the lower support rod to the mass concentration position of the lower support rodd0iAre respectively as follows
ru0i＝[-0.06 0 0]Tm，rd0i＝[0.08 0 0]Tm
Coordinate conversion matrix A for fixedly connecting coordinate system to satellite body system by lower platformbdGet
Figure BDA0002605373550000123
FsiIs the restraining force of the ball joint on the upper support rod, FuiIs the restraining force of the universal hinge on the lower supporting rod, MuihiIs the restraining moment of the universal hinge to the lower supporting rod,
Figure BDA0002605373550000124
the angular velocity of the strut relative to the inertial system can be obtained according to the dynamics and kinematics of the Stewart platform, which is not shown in detail here.
Step 2: as shown in fig. 3, the vibration isolation support rod is equivalent to a three-parameter model, so that the transfer function of the vibration isolation platform can be simply obtained, the parameters of the vibration isolation platform are designed, and the performance indexes of the vibration isolation platform are as follows
Figure BDA0002605373550000125
The vibration characteristic of the actuating mechanism is not considered, the first five-order modal vibration is taken, and F is maderbak＝[k1 k2 k3 k4 k5]The value of k is l and r, which are respectively referred to as left and right sailboards, wherein
Figure BDA0002605373550000131
Representing a 3 x 1 column vector, the flexible satellite attitude dynamics transfer function can be expressed as
Figure BDA0002605373550000132
Where s represents a complex variable in the laplace transform. In practical engineering, the left and right sailboards are usually symmetrically installed and have consistent parameters, so that the flexible modal damping coefficient of the left and right sailboards can be expressed by xi, namely xiar＝ξalXi, the flexible mode frequency of j th order of left and right sailboards can be used as ΛjIs represented byj＝Λrj＝Λlj，j＝1,...,5。
For the transfer function of the flexible satellite vibration isolation platform, the vibration isolation supporting rod is equivalent to a three-parameter model, and the following assumptions are made: (1) the mass and inertia of the vibration isolation supporting rod have little influence on the transfer function of the vibration isolation platform, and can be ignored in the approximation process; (2) the attitude angles of the upper platform and the lower platform meet the small angle assumption, and a coordinate transformation matrix from a system fixedly connected with the upper platform to a system fixedly connected with the star body to an inertial coordinate system can be regarded as a unit matrix; (3) the amplitude of the disturbance torque is small, and the configuration of the vibration isolation platform is assumed to be unchanged. The dynamics of the upper and lower platforms can be simplified as follows:
Figure BDA0002605373550000133
Figure BDA0002605373550000134
wherein, t, θuRepresenting the position and attitude angle of the upper platform system in the inertial system, b, thetabRepresenting the position and attitude angle of the lower platform system in the inertial system,
Figure BDA0002605373550000135
respectively represent t and thetau、b、θbA second derivative with respect to time; restraining force F acting on the upper platformsiIt can be abbreviated as follows:
Figure BDA0002605373550000136
wherein k isAIs the main spring rate, k, of the strutBIs the additional spring rate of the strut, cAIs the damping coefficient of the strut, suiThe component array in the inertial system of the unit vector along the axial direction of the strut i is shown.
Order:
Figure BDA0002605373550000137
then FsiCan be written as follows
Figure BDA0002605373550000141
Wherein JiA 3 x 22 matrix is represented by,
Figure BDA0002605373550000142
the dynamics model of the simplified context platform system can be written in the form of
Figure BDA0002605373550000143
Wherein
Figure BDA0002605373550000144
Respectively representing the third, second and first time derivatives of x with respect to time,
Figure BDA0002605373550000145
respectively representing a mass matrix, a damping matrix and a rigidity matrix of the system,
Figure BDA0002605373550000146
occurring during the three-parameter model approximation process, represent
Figure BDA0002605373550000147
The coefficient of (a) is determined,
Figure BDA0002605373550000148
input forces and moments representing equations of translation, rotation and vibration of the upper and lower platforms, and having
Figure BDA0002605373550000149
Figure BDA00026053735500001410
Figure BDA00026053735500001411
Figure BDA00026053735500001412
Wherein EiUnit array of i × i, 0iZero matrix, 0, representing i x ii×jA zero matrix representing i multiplied by j, wherein i and j represent positive integers; mak＝2ξakΛakIs a damping matrix for the windsurfing board k,
Figure BDA00026053735500001413
is the stiffness matrix of the windsurfing board k.
Output is defined as the generalized force experienced by the star and is written as follows:
Figure BDA00026053735500001414
wherein
Figure BDA00026053735500001415
A damping matrix, a stiffness matrix representing a generalized force expression,
Figure BDA0002605373550000151
taking a state quantity
Figure BDA0002605373550000152
The input u is the disturbance force and moment borne by the upper platform system, the output Y represents the disturbance force and moment transmitted to the lower platform system, and the dynamic equations of the upper and lower platform systems can be written in the form of state equations:
Figure BDA0002605373550000153
wherein A represents a system matrix, B represents an input matrix, CtRepresenting output matrices, D representing direct transfer momentsMatrix of
Figure BDA0002605373550000154
Figure BDA0002605373550000155
By GVIP(s) represents the transfer function matrix of the flexible satellite vibration isolation platform, namely the transfer function matrix of the upper platform system disturbed to the star body, GVIP(s)＝Ct(sE-A)-1B + D, which includes the effect of the flexible windsurfing board; e represents an identity matrix of the same order as the matrix A;
after the transfer function of the flexible satellite vibration isolation platform is obtained, taking an x-rotation channel as an example, k is obtained by using a control variable methodA、kB、cAThe law of variation between the three parameters and the bode diagram of the main channel vibration isolation transfer function, as shown in fig. 6a, 6b and 6c, can be concluded as follows: (1) at main spring rate kAAnd damping coefficient cAConstant additional spring rate kBThe larger the frequency is, the lower the high-frequency attenuation rate is, and the influence on the resonance peak value and the vibration isolation frequency is small; (2) at additional spring rate kBAnd damping coefficient cAConstant main spring rate kAThe larger the vibration isolation frequency is, the larger the high frequency attenuation rate is; (3) at main spring rate kAAnd an additional spring rate kBConstant damping parameter cAThe larger the resonance peak, the smaller the resonance peak, but the high frequency attenuation ratio is unchanged. According to the rule and the performance index of vibration isolation, k is takenA＝25000，kB＝30000，cA＝600。
And step 3: aiming at the problem that the target attitude and the angular speed change rapidly in a certain period of time in the pointing control process, according to the performance indexes of a control system, including pointing accuracy, flexible vibration suppression requirements and the like, a pointing tracking controller integrating an upper platform and a lower platform of a flexible satellite with a six-degree-of-freedom vibration isolation platform is designed, the block diagram of the control system is shown in figure 4, and the performance indexes are as follows
Figure BDA0002605373550000156
Figure BDA0002605373550000161
Wherein omegaiRepresenting the spectral range of the input signal.
Step 3.1: defining error attitude quaternion and angular velocity
The quaternion of the body coordinate system relative to the inertial coordinate system is recorded as QbIThe angular velocity of the body coordinate system relative to the inertial coordinate system is denoted as ωb(ii) a Quaternion of the desired coordinate system relative to the inertial coordinate system is denoted as QTIThe angular velocity of the desired coordinate system relative to the inertial coordinate system is denoted as ωTAngular acceleration is noted
Figure BDA0002605373550000162
Let QeAnd ωeRepresenting the attitude quaternion and the triaxial angular velocity of the spacecraft body system relative to a desired coordinate system, and defining the following error attitude quaternion and error angular velocity
Figure BDA0002605373550000163
In the formula qe0、qeRespectively representing error quaternion QeThe target and vector of (a) are,
Figure BDA0002605373550000164
is QTIA conjugated quaternion of (3), C (Q)e) A transformation matrix for the desired coordinate system to the spacecraft body coordinate system can be written as
Figure BDA0002605373550000169
Wherein
Figure BDA0002605373550000165
Is qeCross-multiplication antisymmetric oblique square matrix of (E)3Is a 3 × 3 unit array. The corresponding error attitude equation can be written as
Figure BDA0002605373550000166
Step 3.2: PD + feedforward controller design
Writing the attitude dynamics equation of the satellite into the form of error quaternion and error angular velocity, and recording C (q)e) Is C. The upper platform and the lower platform are regarded as a whole to carry out pointing control, and the following attitude tracking controller is designed:
Figure BDA0002605373550000167
wherein k isp、kdSgn (·) represents a sign function for the control parameter; by Ju' means the total moment of inertia of the upper platform, r, irrespective of the vibrational characteristics of the actuatorbuPosition vector array representing the center of mass of the star body to the center of mass of the upper platform, IsumThe total moment of inertia of the upper and lower platforms can be obtained according to the parallel axis theorem
Figure BDA0002605373550000168
Let Tc＝-2kpIsumsgn(qe0)qe-kdIsumωeRecord Kp＝kpIsum,Kd＝kdIsumThen the transfer function of the PD controller can be written as
Figure BDA0002605373550000171
Wherein T isc(s)、Θe(s) is dividedRespectively represents Tc、Θe(ii) a laplace transform of; k is a radical ofp、kdThe selection rule of the parameters is as follows: the lower platform can be regarded as a rigid body and is static relative to an inertial system, and then the value ranges of the three main channel control parameters are determined according to a root track theory. Here, according to the second order system theory, there are
Figure BDA0002605373550000172
kd＝2ζωnZeta is a second-order system damping ratio, and usually takes a value near 0.707, omeganThe natural oscillation frequency is usually less than 1/3 times of the first-order modal frequency of the windsurfing board. Where ζ is 0.8, ωn＝0.1。
Step 3.3: design of hysteresis correction link
To further improve the speed and accuracy of the pointing tracking, the bandwidth of the control system needs to be increased, but at the same time, the stability margin should be avoided from being greatly affected. Therefore, a hysteresis correction link is adopted, on one hand, the gain of a low frequency band can be improved, and the precision of the pointing control is effectively improved; on the other hand, the bandwidth of the control system can be increased, and the pointing tracking speed is improved. The transfer function of the lag correction element can be expressed as:
Figure BDA0002605373550000174
wherein T and alpha are both adjustable parameters of a hysteresis correction link, and the principle of selecting T and alpha is as follows: (1) the influence of phase angle lag on the stability margin should be avoided as much as possible; (2) the system cut-off frequency is improved as much as possible, and the gain at the first-order modal frequency of the sailboard is prevented from being influenced. For the above system, the hysteresis correction parameter is T ═ 0.9, and 1/α ═ 4.4.
Step 3.4: structural filter design
The flexible vibration of the sailboard can cause fluttering of the satellite attitude angular velocity and the control moment, and in order to reduce the influence of the vibration on the satellite attitude, a structural filter is usually introduced to suppress the flexible vibration. The classical structural filter form is as follows
Figure BDA0002605373550000173
Wherein ζz、ζp、ωz、ωpDifferent design rules can obtain different forms of filters for filter parameters. The invention adopts a minimum phase notch filter to restrain the flexible vibration of a sailboard, and the parameter selection principle is as follows: (1) for minimum phase notch filters, parameter ωz、ωpEqual and equal to the frequency corresponding to the peak at resonance in the open loop system; (2) zetaz、ζpThe amplitude of the trap is influenced and should be adjusted according to the peak at resonance. For the above system, the x-channel filter parameter takes ζ because the x and z channels exhibit significant resonancez＝0.01，ζp＝0.4，ωz＝ωpZeta is taken as the parameter of the z-channel filter as 13.1rad/sz＝0.005，ζp＝0.13，ωz＝ωp＝2.28rad/s。
The chosen pointing tracking control task is shown in fig. 5. Desired coordinate system oTxTyTzTIs defined as:
Figure BDA0002605373550000181
wherein the content of the first and second substances,
Figure BDA0002605373550000182
a representation of the vector from the primary star centroid to the secondary star centroid in the inertial frame,
Figure BDA0002605373550000183
y representing the orbital system of the main staroThe component array of the unit vector of the axis under the inertial system. The process of obtaining the target angular velocity and angular acceleration will not be described in detail here.
The orbit parameters of the master and slave stars are taken as:
Figure BDA0002605373550000184
and (4) bringing the controller combination designed in the step (3.2) to the step (3.4) into the transfer function of the vibration isolation platform and the flexible satellite, and verifying the frequency domain index of vibration isolation and control by using the bode diagram.
Finally, according to the established expected attitude and motion rule, time domain simulation is carried out on the flexible satellites before and after the six-degree-of-freedom vibration isolation platform is added, the periodicity of a control system is considered, the simulation step length is 0.001s, and T is taken as the control step lengths0.1s, the simulation time is 2000 s.
Fig. 7a and 7b, and fig. 8a and 8b show simulation results of the pointing control process before and after vibration isolation, wherein a posture maneuvering stage is before 160s, and 900-1100 s are moments when the main star passes through the top from right above the star, so that the target posture and angular speed change is large in amplitude, the change speed is fastest, and the pointing tracking accuracy is worst. As can be seen from fig. 7a and 7b and fig. 8a and 8b, compared with a flexible satellite without a six-degree-of-freedom vibration isolation platform, the system with the vibration isolation platform has significantly improved pointing accuracy and error angular velocity jitter in the attitude maneuver stage; in the time period of the main satellite passing the top, after the vibration isolation platform is added, the pointing accuracy and the pointing stability of the lower platform system can be improved by 50%, and the jitter of the error angular velocity can be attenuated by 3 orders of magnitude.
The simulation result proves the effectiveness of the vibration isolation parameter and the pointing tracking controller designed for the flexible satellite with the six-degree-of-freedom vibration isolation platform. The system with high-frequency vibration of the control moment gyro, dynamic change of the moment inertia and rapid change of the target attitude and the angular speed can realize high-precision and high-stability pointing tracking control, meet given performance indexes and have extremely high engineering application value.
Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (1)

1. A flexible satellite pointing tracking control method containing a six-degree-of-freedom vibration isolation platform is characterized by comprising the following steps:
step 1: establishing a dynamic equation of an upper platform system and a lower platform system of the flexible satellite containing the six-degree-of-freedom vibration isolation platform;
neglecting external disturbance to the upper platform system, considering rotor static and dynamic unbalance of the control moment gyroscope, and non-perpendicularity and non-intersection of the frame shaft and the rotor shaft, the dynamic equation of the vibration isolation upper platform system is as follows:
Figure FDA0002605373540000011
Figure FDA0002605373540000012
Figure FDA0002605373540000013
wherein v isuThe component array representing the velocity of the upper stage in the upper stage system,
Figure FDA0002605373540000014
denotes vuA first derivative with respect to time; omegauThe component array of the upper platform system is shown in the attitude angular velocity of the upper platform,
Figure FDA0002605373540000015
represents omegauA first derivative with respect to time; muFor the quality of the upper platform system, SuIs a representation of the static moment of the upper platform system in the upper platform system, IuFor a representation of the system in the upper platform without the moment of inertia of the control moment gyro, JuRepresenting the total rotational inertia of the upper platform in a system on the upper platform in order to consider the vibration characteristic of the control moment gyroscope; p is a radical ofiRepresenting a position vector array from the center of mass of the upper platform to the connecting point of the upper platform;
Figure FDA0002605373540000016
is the viscous damping moment of the upper platform ball pivot, csiIs the coefficient of the viscous damping of the spherical hinge,
Figure FDA0002605373540000017
angular velocity of the strut relative to the inertial system, AueA coordinate transformation matrix representing the inertial system to the upper platform body system;
Figure FDA0002605373540000018
representing the remaining part of the momentum of the entire upper platform system after removal of the terms with interfering sources, mgiRepresents the frame mass, mwiRepresenting the rotor mass, rugiThe position vector array from the center of mass of the upper platform to the center of mass of the frame is represented, and m represents the number of control moment gyroscopes; fsiIs the restraining force of the ball joint on the upper support rod, FdRepresenting disturbance force, T, produced by a control moment gyrodThe coupling moment and the control moment generated by the control moment gyroscope are represented, and N is the number of the supporting rods of the vibration isolation platform; in the formula, superscript "to" represents a cross-product antisymmetric diagonal matrix of the array, i.e., x ═ x for any three-dimensional array1 x2 x3]TIs provided with
Figure FDA0002605373540000019
The lower platform system is a flexible spacecraft which is substantially a central rigid body plus a sailboard, and if the sailboard is in a locked state, the dynamic equations of the translation, rotation and sailboard vibration of the lower platform system are expressed as follows:
Figure FDA0002605373540000021
Figure FDA0002605373540000022
Figure FDA0002605373540000023
wherein v isbThe component array representing the velocity of the lower platform system in the satellite system,
Figure FDA0002605373540000024
denotes vbA first derivative with respect to time; omegabThe component array of the attitude angular velocity of the lower platform system in the satellite system is shown,
Figure FDA0002605373540000025
represents omegabA first derivative with respect to time; etaakThe modal coordinates of the windsurfing board k are represented,
Figure FDA0002605373540000026
respectively represent ηakFirst and second derivatives with respect to time; msRepresents the total mass of the lower platform system, SsAnd IsRespectively representing the total static moment and the total rotational inertia of the lower platform system in a satellite system; lambdaakIs a diagonal array of modal frequencies, Λ, of sailboard kak＝diag([Λk1；Λk2；…Λkj；…Λkn])，ΛkjThe j-th order flexible modal frequency is represented, and n represents the modal order; xiakThe flexible modal damping coefficient of the sailboard k; frbakA flexible coupling coefficient matrix of the kth flexible sailboard vibration to the central rigid body rotation; ftbakA flexible coupling coefficient matrix of the kth flexible sailboard vibration to the central rigid body translation; r isbdA position vector array, q, representing the origin of a fixed coordinate system of the lower platform system from the center of mass of the lower platform system to the lower platformiRepresenting a position vector array from the original point of the lower platform fixed connection coordinate system to the connecting point of the lower platform;
Figure FDA0002605373540000027
is the viscosity of the universal hinge of the lower platformDamping moment, cuiIs the damping coefficient of universal hinge viscosity, AbeA coordinate transformation matrix representing the inertial system to the satellite body system; a. thebdA coordinate transformation matrix representing the fixed connection coordinate system of the lower platform to the satellite body system; fuiIs the restraining force of the universal hinge on the lower support rod, FextIs the interference force of the external environment, does not include the universal gravitation item, MuihiIs the restraining moment of the universal hinge on the lower supporting rod, TextDisturbing the moment for the external environment; w represents the number of sailboards;
step 2: the vibration isolation support rod is equivalent to a three-parameter model, a transfer function of the flexible satellite vibration isolation platform is simplified and obtained, and parameters of the vibration isolation platform are designed;
the vibration characteristic of the actuating mechanism is not considered, the first five-order modal vibration is taken, and
Frbak＝[k1 k2 k3 k4 k5]the value of k is l and r, which are respectively referred to as left and right sailboards, whereinkj(j ═ 1,2, … 5) represents a column vector of 3 × 1, and the transfer function of the attitude dynamics of the flexural satellites is expressed as
Figure FDA0002605373540000028
Wherein s represents a complex variable in the laplace transform; in practical engineering, the left and right sailboards are symmetrically installed and have consistent parameters, so that the flexible modal damping coefficients of the left and right sailboards are both expressed by xi, namely xiar＝ξalXi, the j-th order flexible modal frequency of the left sailboard and the right sailboard is respectively expressed by ΛjIs represented byj＝Λrj＝Λlj，j＝1,...,5；
For the transfer function of the flexible satellite vibration isolation platform, the vibration isolation supporting rod is equivalent to a three-parameter model, and the following parameters are set: (1) the mass and inertia of the supporting rod of the vibration isolation platform have small influence on the transfer function of the vibration isolation platform and are ignored in the approximation process; (2) setting the attitude angle of the upper platform and the lower platform to meet a small angle, and considering a coordinate transformation matrix from a system fixed connection coordinate system of the upper platform and a satellite fixed connection coordinate system to an inertial coordinate system as a unit matrix; (3) the amplitude of the disturbance torque is small, and the configuration of the vibration isolation platform is assumed to be unchanged; then the kinetic equation of the upper and lower platform systems is simplified as:
Figure FDA0002605373540000031
Figure FDA0002605373540000032
wherein, t, θuRepresenting the position and attitude angle of the upper platform system in the inertial system, b, thetabRepresenting the position and attitude angle of the lower platform system in the inertial system,
Figure FDA0002605373540000033
respectively represent t and thetau、b、θbA second derivative with respect to time; restraining force F acting on the upper platformsiIt is abbreviated as follows:
Figure FDA0002605373540000034
wherein k isAIs the main spring rate, k, of the strutBIs the additional spring rate of the strut, cAIs the damping coefficient of the strut, suiRepresenting the component array of the unit vector along the axial direction of the strut i in the inertial system;
order:
Figure FDA0002605373540000035
then FsiWritten in the form of
Figure FDA0002605373540000036
Wherein JiRepresenting a 3 x 22 matrix, which is an intermediate variable,
Figure FDA0002605373540000037
the simplified kinetic equations of the upper and lower platform systems are written in the form
Figure FDA0002605373540000041
Wherein
Figure FDA0002605373540000042
Respectively representing the third, second and first time derivatives of x with respect to time,
Figure FDA0002605373540000043
respectively representing a mass matrix, a damping matrix and a rigidity matrix of the system,
Figure FDA0002605373540000044
occurring during the three-parameter model approximation process, represent
Figure FDA0002605373540000045
The coefficient of (a) is determined,
Figure FDA0002605373540000046
input forces and moments representing equations of translation, rotation and vibration of the upper and lower platforms, and having
Figure FDA0002605373540000047
Figure FDA0002605373540000048
Figure FDA0002605373540000049
Figure FDA00026053735400000410
Wherein EiUnit array of i × i, 0iZero matrix, 0, representing i x ii×jA zero matrix representing i multiplied by j, wherein i and j represent positive integers; mak＝2ξakΛakIs a damping matrix for the windsurfing board k,
Figure FDA00026053735400000411
a stiffness matrix of the sailboard k;
output is defined as the generalized force experienced by the star and is written as follows:
Figure FDA00026053735400000412
wherein
Figure FDA00026053735400000413
A damping matrix, a stiffness matrix representing a generalized force expression,
Figure FDA00026053735400000414
taking a state quantity
Figure FDA00026053735400000415
The input u is the disturbance force and moment borne by the upper platform system, the output Y represents the disturbance force and moment transmitted to the lower platform system, and the dynamic equations of the upper and lower platform systems are written into the form of a state equation:
Figure FDA0002605373540000051
wherein A represents a system matrix, B represents an input matrix, CtRepresenting an output matrix, and D representing a direct transfer function matrix;
Figure FDA0002605373540000052
Figure FDA0002605373540000053
by GVIP(s) represents the transfer function matrix of the flexible satellite vibration isolation platform, namely the transfer function matrix of the upper platform system disturbed to the star body, GVIP(s)＝Ct(sE-A)-1B + D, which includes the effect of the flexible windsurfing board; e represents an identity matrix of the same order as the matrix A;
after a transfer function of the flexible satellite vibration isolation platform is obtained, k is obtained by using a control variable methodA、kB、cADetermining parameter values according to the performance indexes of vibration isolation according to the change rule between the three parameters and the bode graph of the vibration isolation transfer function of the main channel;
and step 3: aiming at the problem that the target attitude and the angular speed change rapidly in a certain period of time in the pointing control process, according to the performance indexes of a control system, including pointing accuracy and flexible vibration suppression requirements, a pointing tracking controller integrating an upper platform and a lower platform of a flexible satellite with a six-degree-of-freedom vibration isolation platform is designed;
step 3.1: defining error attitude quaternion and angular velocity
The quaternion of the body coordinate system relative to the inertial coordinate system is recorded as QbIThe angular velocity of the body coordinate system relative to the inertial coordinate system is denoted as ωb(ii) a Quaternion of the desired coordinate system relative to the inertial coordinate system is denoted as QTIThe angular velocity of the desired coordinate system relative to the inertial coordinate system is denoted as ωTAngular acceleration is noted
Figure FDA0002605373540000054
Let QeAnd ωeRepresenting the attitude quaternion and the triaxial angular velocity of the spacecraft body system relative to a desired coordinate system, and defining the following error attitude quaternion and error angular velocity
Figure FDA0002605373540000059
ωe＝ωb-C(Qe)ωT
In the formula qe0、qeRespectively representing error quaternion QeThe target and vector of (a) are,
Figure FDA0002605373540000055
is QTIA conjugated quaternion of (3), C (Q)e) For the transformation matrix of the desired coordinate system into the spacecraft body coordinate system, write
Figure FDA0002605373540000056
Wherein
Figure FDA0002605373540000057
Is qeCross-multiplication antisymmetric oblique square matrix of (E)33 × 3 unit array; the corresponding error attitude equation is written as
Figure FDA0002605373540000058
Step 3.2: PD + feedforward controller design
Writing the attitude dynamics equation of the satellite into the form of error quaternion and error angular velocity, and recording C (q)e) Is C; the upper platform and the lower platform are regarded as a whole to carry out pointing control, and the following attitude tracking controller is designed:
Figure FDA0002605373540000061
wherein k isp、kdSgn (·) represents a sign function for the control parameter; from J'uRepresenting the total moment of inertia, r, of the upper platform irrespective of the vibrational characteristics of the actuatorbuPosition vector array representing the center of mass of the star body to the center of mass of the upper platform, IsumThe total moment of inertia of the upper and lower platforms is obtained according to the parallel axis theorem
Figure FDA0002605373540000062
Let Tc＝-2kpIsumsgn(qe0)qe-kdIsumωeRecord Kp＝kpIsum,Kd＝kdIsumThen the transfer function of the PD controller is written as
Figure FDA0002605373540000063
Wherein T isc(s)、Θe(s) each represents Tc、Θe(ii) a laplace transform of; k is a radical ofp、kdThe selection rule of the parameters is as follows: the lower platform is regarded as a rigid body and is static relative to an inertial system, and then the value ranges of the three main channel control parameters are determined according to a root track theory; here, k is determined according to the second order system theoryp、kdThe parameters of (1);
step 3.3: design of hysteresis correction link
In order to further improve the speed and the precision of the pointing tracking, the bandwidth of the control system needs to be increased, but the stability margin of the control system is prevented from being greatly influenced; therefore, a hysteresis correction link is adopted, on one hand, the gain of a low frequency band can be improved, and the precision of the pointing control is effectively improved; on the other hand, the bandwidth of a control system can be increased, and the pointing tracking speed is improved; the transfer function of the lag correction element is expressed as:
Figure FDA0002605373540000064
t and alpha are adjustable parameters of a hysteresis correction link, and the performance of a control system is improved by adjusting T and alpha;
step 3.4: structural filter design
The flexible vibration of the sailboard can cause the flutter of the satellite attitude angular velocity and the control moment, and a structural filter is introduced to inhibit the flexible vibration in order to reduce the influence of the vibration on the satellite attitude; the classical structural filter form is as follows
Figure FDA0002605373540000065
Wherein ζz、ζp、ωz、ωpObtaining filters in different forms for filter parameters and different design rules; a minimum phase notch filter is adopted to inhibit the flexible vibration of the sailboard, and the stability of attitude control is improved;
and 3.2-3.4, forming a complete controller, and simultaneously considering the periodicity of control instruction output and performing combined design of control parameters on the flexible satellite containing the six-degree-of-freedom vibration isolation platform according to performance indexes.
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