CN104697510B - High-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate - Google Patents

High-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate Download PDF

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CN104697510B
CN104697510B CN201510006191.7A CN201510006191A CN104697510B CN 104697510 B CN104697510 B CN 104697510B CN 201510006191 A CN201510006191 A CN 201510006191A CN 104697510 B CN104697510 B CN 104697510B
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rotor
satellite
magnetic suspension
moment
magnetic
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CN104697510A (en
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任元
王平
陈晓岑
王英广
田希晖
蔡远文
王�华
李新洪
汪洲
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Space Engineering University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C19/00—Gyroscopes; Turn-sensitive devices using vibrating masses; Turn-sensitive devices without moving masses; Measuring angular rate using gyroscopic effects
    • G01C19/02—Rotary gyroscopes
    • G01C19/42—Rotary gyroscopes for indicating rate of turn; for integrating rate of turn
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C19/00—Gyroscopes; Turn-sensitive devices using vibrating masses; Turn-sensitive devices without moving masses; Measuring angular rate using gyroscopic effects
    • G01C19/02—Rotary gyroscopes
    • G01C19/04—Details
    • G01C19/16—Suspensions; Bearings
    • G01C19/24—Suspensions; Bearings using magnetic or electrostatic fields

Abstract

The invention relates to a high-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate. Establishing a single-axis satellite magnetic suspension control sensitive gyroscope dynamic model according to rigid body dynamics and a coordinate transformation principle; the radial resultant external moment of the magnetic suspension rotor and the deflection moment of the magnetic suspension rotor, which are convenient for direct high-precision measurement and calculation, are utilized to indirectly obtain the acting moment of the satellite single-axis motion on the magnetic suspension rotor; and according to the relationship between the satellite uniaxial attitude angular rate and the moment of action of the satellite uniaxial motion on the magnetic suspension rotor, providing a high-precision high-bandwidth analytical expression of the satellite uniaxial attitude angular rate. Establishing a single-shaft magnetic suspension rotor dynamic model according to rigid body dynamics and a coordinate transformation principle; the invention belongs to the technical field of aerospace control, and can be applied to high-precision satellite attitude control, measurement and ground gyroscope calibration.

Description

High-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate
Technical Field
The invention relates to a high-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate, which is suitable for high-precision high-bandwidth attitude angular rate measurement of a satellite.
Technical Field
The stability and the hyperstatic performance of the satellite attitude control system are two important indexes of satellite attitude control and are also important factors influencing the satellite performance. The detection and control of the traditional attitude control system are separated, the whole attitude control system is of a single closed loop structure, and the bandwidth of the attitude control system is limited. Therefore, the conventional attitude control system is difficult to suppress high-frequency small-amplitude disturbance of the satellite. In addition, the detection and control of the existing attitude control system are separated, and the flexible structure of the satellite is added, so that the problem of out-of-position control is inevitably caused, and the stability and robustness of the whole attitude control system are inevitably influenced.
In order to solve the problems, Zhengshiqiang combines moment execution and attitude measurement through a double-frame magnetic suspension control moment gyroscope, but the research reuses measurement and control in a time-sharing way, the magnetic suspension control moment gyroscope can only work in one state at a certain moment, and the measurement and the control cannot be carried out at the same time; liu bin proposes a design scheme of a magnetic suspension gyro flywheel, although the magnetic suspension gyro flywheel can be controlled and measured at the same time, the method does not obtain an analytic expression of the three-axis attitude angular velocity, not only the practicability is not strong, but also the analysis of the relationship between the attitude angular velocity and the system parameters is inconvenient in mechanism.
The magnetic suspension control sensitive gyroscope is a multifunctional new concept inertial mechanism integrating attitude sensing and control, vibration detection and inhibition, and has the potential advantages of high precision, long service life and extremely micro vibration. Theoretically, a single magnetic suspension control sensitive gyroscope has the measurement capability of the attitude angular rate of two degrees of freedom, but like the traditional gyroscope, the second-order small term and the coupling term are ignored during the calculation of the attitude angular rate, so that the measurement accuracy and the bandwidth of the attitude angular rate are influenced inevitably.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: in order to avoid the problem that the measurement precision and bandwidth are influenced by neglecting high-order quadratic terms and partial coupling terms when the attitude angular rate of the spacecraft is measured by the conventional gyroscope, a high-precision high-bandwidth measurement method for the satellite single-axis attitude angular rate based on a magnetic suspension control sensitive gyroscope is provided. The method can realize high-precision and high-bandwidth measurement of the single-axis attitude angular rate, can obtain an analytical expression of the attitude angular rate, and provides a brand-new technical approach for high-precision and high-bandwidth measurement and control of the satellite attitude.
The technical solution of the invention is as follows: establishing a single-shaft magnetic suspension control sensitive gyroscope kinetic model according to rigid body dynamics and a coordinate transformation principle; the radial resultant external moment borne by the magnetic suspension rotor and the deflection moment of the magnetic suspension rotor, which are convenient for direct high-precision measurement and calculation, are utilized to indirectly obtain the acting moment on the magnetic suspension rotor during the single-axis motion of the satellite; and according to the relationship between the satellite uniaxial attitude angular rate and the rotor action moment caused by the satellite uniaxial motion, giving an analytical expression of the satellite uniaxial attitude angular rate. The method specifically comprises the following steps:
(1) the magnetic suspension rotor dynamic equation according to the rigid body dynamics and the coordinate transformation principle is as follows:
Figure BSA0000112806400000021
wherein:
Hr＝IΩi
Figure BSA0000112806400000022
Figure BSA0000112806400000023
Figure BSA0000112806400000024
Figure BSA0000112806400000025
Figure BSA0000112806400000026
in the formula, MrIndicating the combined external torque of the magnetic levitation rotor, HrRepresenting the angular momentum of the rotor under the rotor train,
Figure BSA0000112806400000027
expressing the angular momentum of the rotor under the rotor system, I expressing the moment of inertia of the rotor rotating around the reference coordinate system of the magnetically suspended control sensitive gyroscope, IrExpressing the radial moment of inertia of the rotor, IZ the axial moment of inertia of the rotor, omega the axial speed of rotation of the rotor, omegarExpressing the rotor speed, ΩiRepresenting the absolute angular velocity of the rotor,
Figure BSA0000112806400000028
representing the absolute angular speed rate of change of the rotor,
Figure BSA0000112806400000029
representing the absolute angular velocity of the rotor coordinate system, i.e. the rotational speed with respect to the inertia space,
Figure BSA0000112806400000031
indicating the deflection speed of the rotor relative to the magnetic bearings,
Figure BSA0000112806400000032
for magnetically suspending the speed of the reference system of the control sensitive gyro relative to the inertial space,
Figure BSA0000112806400000033
is the angular velocity, ω, of the rotor relative to the inertial spacecmgIn order to be the angular velocity of the frame,
Figure BSA0000112806400000034
is a transformation matrix from the magnetic bearing coordinate system to the rotor coordinate system,
Figure BSA0000112806400000035
is a transformation matrix from the frame coordinate system to the magnetic bearing coordinate system,
Figure BSA0000112806400000036
is a transformation matrix from a reference system of the magnetically suspended control sensitive gyroscope to a frame coordinate system,
Figure BSA0000112806400000037
a transformation matrix from a star system to a magnetic suspension control sensitive gyroscope reference system;
the magnetic bearing installation system, the frame coordinate system, the magnetic suspension control sensitive gyro reference system are coincided and only the single-axis angular rate of the satellite is available
Figure BSA0000112806400000038
Under the conditions of (a):
Figure BSA0000112806400000039
although α, β are very small under the rotor train, to improve measurement accuracy, the magnetic bearings mount a transformation matrix tied to the rotor train
Figure BSA00001128064000000318
No approximation is made:
Figure BSA00001128064000000310
and also
Figure BSA00001128064000000311
Figure BSA00001128064000000312
Figure BSA00001128064000000313
Then:
Figure BSA00001128064000000314
Figure BSA00001128064000000315
Figure BSA00001128064000000316
Figure BSA00001128064000000317
rotor radial resultant external moment
Figure BSA0000112806400000041
The expression is as follows:
Figure BSA0000112806400000042
Figure BSA0000112806400000043
(2) measurement and calculation of rotor radial resultant external moment
The magnetic suspension rotor is subjected to the following combined external moment:
Figure BSA0000112806400000044
Figure BSA0000112806400000045
the magnetic force applied to the magnetic suspension rotor can be expressed in the following linear form:
fλ＝kiλiλ+khλhλ(λ＝ax，ay，bx，by)
in the formula, kiλAnd khλ(λ ═ ax, ay, bx, by) are shown in tablesThe current rigidity and the displacement rigidity of radial ax, ay, bx and by channels of the magnetic suspension rotor can be calibrated through experiments; i.e. iax、ibx、iayAnd ibyIs the winding current of four radial channels, hax、hbx、hayAnd hbyIs the linear displacement of the magnetic levitation rotor in the ax, bx, ay and by directions, respectively, lmRepresenting the distance from the center of the magnetically levitated rotor to the center of the radial magnetic bearing; h isax、hbx、hay、hbyCan be measured by an eddy current displacement sensor, iax、ibx、iay、ibyThe external torque applied to the rotor can be calculated by measuring through the current sensor;
(3) measurement and calculation of rotor deflection information
The expression for the rotor deflection angle is:
α＝(hay-hby)/(2lm)
β＝(hax-hbx)/(2lm)
hax、hbx、hayand hbyIs the linear displacement of the magnetic levitation rotor in the ax, bx, ay and by directions, respectively, lmDenotes the distance, h, from the center of the magnetically levitated rotor to the center of the radial magnetic bearingax、hbx、hay、hbyCan be measured by an eddy current displacement sensor, so that the deflection information α, β of the rotor can be calculated,
Figure BSA0000112806400000046
Figure BSA0000112806400000051
sinα、sinβ、cosα、cosβ；
(4) Single axis satellite attitude angular rate analytic solution
Will be provided with
Figure BSA0000112806400000052
Is converted into
Figure BSA0000112806400000053
Then:
a＝(Ir-Iz)sinβcosβ
Figure BSA0000112806400000054
Figure BSA0000112806400000055
in the expression of a, IrIs less than IzTherefore:
Figure BSA0000112806400000056
the principle of the invention is as follows: according to the theorem of moment of inertia, the angular momentum of a high-speed rotor changes direction in the inertial space only depending on the external moment to which it is subjected. The moment borne by the magnetic suspension rotor is caused by the rotation of a satellite and the relative deflection of the rotor, the magnitude of the moment borne by the magnetic suspension rotor is uniquely determined by the force of a magnetic bearing, and the attitude angular rate of the satellite can be obtained by detecting the current of the magnetic bearing and the displacement of the rotor in real time and with high precision and resolving. Establishing a single-shaft magnetic suspension rotor dynamic model according to rigid body dynamics and a coordinate transformation principle; the radial resultant external moment borne by the magnetic suspension rotor and the deflection moment of the magnetic suspension rotor, which are convenient for direct high-precision measurement and calculation, are utilized to indirectly obtain the acting moment on the magnetic suspension rotor during the single-axis motion of the satellite; according to the relationship between the satellite uniaxial attitude angular rate and the rotor acting torque caused by the satellite uniaxial motion, an analytical expression of the satellite uniaxial attitude angular rate is given, so that the high-precision high-bandwidth detection of the satellite uniaxial attitude angular rate is realized.
The schematic diagram of the magnetically levitated control sensitive gyroscope is shown in fig. 1, the installation positions of the radial magnetic bearings are symmetrical relative to the center of mass of the rotor, the rotor realizes the suspension control through the 5-freedom-degree magnetic bearing, the radial 4 magnetic bearings (respectively represented by ax, ay, bx and by) control two radial translational degrees of freedom and two rotational degrees of freedom of the magnetically levitated rotor, the axial bearing (represented by z) controls one translational degree of freedom, and the rotational degree of freedom is driven by the motor to provide the angular momentum of the rotor. The magnetic suspension control sensitive gyro rotor resultant moment is caused by satellite rotation and rotor deflection, and by applying an Euler kinetic equation, a magnetic suspension rotor kinetic equation under a rotor system can be obtained as follows:
Figure BSA0000112806400000061
wherein:
Hr＝IΩi
Figure BSA0000112806400000062
Figure BSA0000112806400000063
Figure BSA0000112806400000064
Figure BSA0000112806400000065
Figure BSA0000112806400000066
the magnetic bearing installation system, the frame coordinate system, the magnetic suspension control sensitive gyro reference system are coincided and only the single-axis angular rate of the satellite is available
Figure BSA0000112806400000067
Under the conditions of (a):
Figure BSA0000112806400000068
although α, β are very small under the rotor train, to improve measurement accuracy, the magnetic bearings mount a transformation matrix tied to the rotor train
Figure BSA0000112806400000069
No approximation is made:
Figure BSA00001128064000000610
and also
Figure BSA00001128064000000611
Figure BSA00001128064000000612
Figure BSA00001128064000000613
Then:
Figure BSA00001128064000000614
Figure BSA00001128064000000615
Figure BSA0000112806400000071
Figure BSA0000112806400000072
rotor radial resultant external moment
Figure BSA0000112806400000073
The expression is as follows:
Figure BSA0000112806400000074
Figure BSA0000112806400000075
the magnetic suspension rotor is subjected to the following combined external moment:
Figure BSA0000112806400000076
Figure BSA0000112806400000077
the magnetic suspension rotor is subjected to the following electromagnetic force:
fλ＝kiλiλ+khλhλ(λ＝ax，ay，bx，by)
the expression for the rotor deflection angle is:
α＝(hay-hby)/(2lm)
β＝(hax-hbx)/(2lm)
other deflection information of the rotor can be calculated according to α and β
Figure BSA0000112806400000078
sinα、sinβ、cosα、cosβ。
Will be provided with
Figure BSA0000112806400000079
Is converted into
Figure BSA00001128064000000710
Then:
a＝(Ir-Iz)sinβcosβ
Figure BSA0000112806400000081
Figure BSA0000112806400000082
α expression of (I)rIs less than IzTherefore:
Figure BSA0000112806400000083
therefore, the attitude angular rate of the satellite is obtained, and high-precision and high-bandwidth detection of the single-axis attitude angular rate of the satellite is realized.
Compared with the prior art, the scheme of the invention has the main advantages that: aiming at the problems of ex-position control and the like caused by non-co-location detection and control of the existing satellite attitude control system, a high-precision high-bandwidth measurement method for satellite single-axis attitude angular rate based on a magnetic suspension control sensitive gyroscope is provided, the problem that the conventional gyroscope ignores high-order quadratic terms and partial coupling terms to influence the detection precision and bandwidth of the system is solved, an analytical expression of the attitude angular rate is provided, and a new technical approach is provided for the high-precision high-bandwidth measurement and control of the satellite.
Drawings
FIG. 1 is a schematic structural diagram of a magnetically suspended control sensitive gyroscope;
FIG. 2 is a functional block diagram of the present invention;
FIG. 3 is a comparison curve of actual and measured attitude angular rates at 1Hz bandwidth;
FIG. 4 is a plot of actual attitude angular rate values and measured value error for a bandwidth of 1 Hz;
FIG. 5 is a comparison curve of actual and measured attitude angular rates at a bandwidth of 10 Hz;
FIG. 6 is a plot of actual attitude angular rate values and measured value error for a bandwidth of 10 Hz;
FIG. 7 is a comparison graph of actual and measured attitude angular rates at 50Hz bandwidth;
FIG. 8 is a plot of actual and measured attitude angular rate error at 50Hz bandwidth;
detailed description of the preferred embodiments
The implementation object of the invention is shown in fig. 1, the radial magnetic bearings are installed symmetrically relative to the center of mass of the rotor, the radial 4 magnetic bearings (respectively represented by ax, ay, bx and by) control two radial translational degrees of freedom and two rotational degrees of freedom of the magnetic suspension rotor, and the magnetic suspension control sensitive gyro rotor resultant moment in the actuating mechanism is caused by satellite rotation, gyro frame rotation and rotor deflection. The specific implementation scheme of the invention is shown in figure 2, and the specific implementation steps are as follows:
(1) the magnetic suspension rotor dynamic equation according to the rigid body dynamics and the coordinate transformation principle is as follows:
Figure BSA0000112806400000091
wherein:
Hr＝IΩi
Figure BSA0000112806400000092
Figure BSA0000112806400000093
Figure BSA0000112806400000094
Figure BSA0000112806400000096
Figure BSA0000112806400000097
in the formula, MrIndicating the combined external torque of the magnetic levitation rotor, HrRepresenting the angular momentum of the rotor under the rotor train,
Figure BSA0000112806400000098
expressing the angular momentum of the rotor under the rotor system, I expressing the moment of inertia of the rotor rotating around the reference coordinate system of the magnetically suspended control sensitive gyroscope, IrRepresenting the radial moment of inertia of the rotor, IzRepresenting the axial moment of inertia of the rotor, omega representing the axial speed of rotation of the rotor, omegarExpressing the rotor speed, ΩiRepresenting the absolute angular velocity of the rotor,
Figure BSA0000112806400000099
representing the absolute angular speed rate of change of the rotor,
Figure BSA00001128064000000910
representing the absolute angular velocity of the rotor coordinate system, i.e. the rotational speed with respect to the inertia space,
Figure BSA00001128064000000911
indicating the deflection speed of the rotor relative to the magnetic bearings,
Figure BSA00001128064000000912
for magnetically suspending the speed of the reference system of the control sensitive gyro relative to the inertial space,
Figure BSA00001128064000000913
is the angular velocity, ω, of the rotor relative to the inertial spacecmgIn order to be the angular velocity of the frame,
Figure BSA00001128064000000914
is a transformation matrix from the magnetic bearing coordinate system to the rotor coordinate system,
Figure BSA00001128064000000915
is a transformation matrix from the frame coordinate system to the magnetic bearing coordinate system,
Figure BSA00001128064000000916
is a transformation matrix from a reference system of the magnetically suspended control sensitive gyroscope to a frame coordinate system,
Figure BSA00001128064000000917
a transformation matrix from a star system to a magnetic suspension control sensitive gyroscope reference system;
the magnetic bearing installation system, the frame coordinate system, the magnetic suspension control sensitive gyro reference system are coincided and only the single-axis angular rate of the satellite is available
Figure BSA0000112806400000101
Under the conditions of (a):
Figure BSA0000112806400000102
although α, β are very small in the rotor system, the magnetic axis is used to improve the measurement accuracyBearing mounting tie to rotor tie transition matrix
Figure BSA0000112806400000103
No approximation is made:
Figure BSA0000112806400000104
and also
Figure BSA0000112806400000105
Figure BSA0000112806400000106
Figure BSA0000112806400000107
Then:
Figure BSA0000112806400000108
Figure BSA0000112806400000109
Figure BSA00001128064000001010
rotor radial resultant external moment
Figure BSA00001128064000001011
The expression is as follows:
Figure BSA00001128064000001012
Figure BSA0000112806400000111
(2) measurement and calculation of rotor radial resultant external moment
The magnetic suspension rotor is subjected to the following combined external moment:
Figure BSA0000112806400000112
Figure BSA0000112806400000113
the magnetic force applied to the magnetic suspension rotor can be expressed in the following linear form:
fλ＝kiλiλ+khλhλ(λ＝ax，ay，bx，by)
in the formula, kiλAnd khλThe (λ ═ ax, ay, bx, by) respectively represents the current stiffness and the displacement stiffness of radial ax, ay, bx and by channels of the magnetic suspension rotor, and can be calibrated through experiments; i.e. iax、ibx、iayAnd ibyIs the winding current of four radial channels, hax、hbx、hayAnd hbyIs the linear displacement of the magnetic levitation rotor in the ax, bx, ay and by directions, respectively, lmRepresenting the distance from the center of the magnetically levitated rotor to the center of the radial magnetic bearing; h isax、hbx、hay、hbyCan be measured by an eddy current displacement sensor, iax、ibx、iay、ibyThe external torque applied to the rotor can be calculated by measuring through the current sensor;
(3) measurement and calculation of rotor deflection information
The expression for the rotor deflection angle is:
α＝(hay-hby)/(2lm)
β＝(hax-hbx)/(2lm)
hax、hbx、hayand hbyIs the linear displacement of the magnetic suspension rotor in the Ax, Bx, Ay and By directions, respectively, lmDenotes the distance, h, from the center of the magnetically levitated rotor to the center of the radial magnetic bearingax、hbx、hay、hbyCan be measured by an eddy current displacement sensor, so that the rotor can be obtained by calculationDeflection information α, β,
Figure BSA0000112806400000114
sinα、sinβ、cosα、cosβ；
(4) Single axis satellite attitude angular rate analytic solution
Will be provided with
Figure BSA0000112806400000115
Is converted into
Figure BSA0000112806400000116
Then:
a＝(Ir-Iz)sinβcosβ
Figure BSA0000112806400000121
Figure BSA0000112806400000122
in the expression of a, IrIs less than IzTherefore:
Figure BSA0000112806400000123
in order to verify the effect of the measurement method, the satellite uniaxial attitude angular rate bandwidths are respectively 1Hz, 10Hz and 50Hz, the actual angular rate curve of the satellite and the angular rate curve calculated by the method are compared, and the test results are respectively shown in fig. 3, fig. 4, fig. 5, fig. 6, fig. 7 and fig. 8.
In fig. 3, 4, 5, 6, 7, and 8, the abscissa indicates time in units of s, and the ordinate indicates roll angle rate in units of °/s. Comparing the actual satellite single-axis attitude angular rate with the angular rate measured by the method of the invention, it can be seen that the precision of the measured value obtained by the method of the invention is very high, and the precision of the measurement is not affected under the condition of improving the bandwidth, which shows that the method of the invention can well realize the high-precision high-bandwidth measurement of the satellite single-axis attitude angular rate, and the calculation is simpler and the engineering is strong.
Those skilled in the art will appreciate that the details of the present invention not described in detail herein are well within the skill of those in the art.

Claims (1)

1. A high-precision high-bandwidth measurement method for satellite uniaxial attitude angular rate is characterized by comprising the following steps: establishing a single-shaft magnetic suspension control sensitive gyroscope kinetic model according to rigid body dynamics and a coordinate transformation principle; the radial resultant external moment of the magnetic suspension rotor and the deflection moment of the magnetic suspension rotor, which are convenient for direct high-precision measurement and calculation, are utilized to indirectly obtain the acting moment on the magnetic suspension rotor during the single-axis motion of the satellite; according to the relationship between the satellite uniaxial attitude angular rate and the rotor action moment caused by the satellite uniaxial motion, an analytic expression of the satellite uniaxial attitude angular rate is given, and the method specifically comprises the following steps:
(1) according to the rigid body dynamics and the coordinate transformation principle, the dynamic equation of the magnetic suspension control sensitive gyroscope is as follows:
Figure FDA0002365073890000011
wherein:
Hr＝IΩi
Figure FDA0002365073890000012
Figure FDA0002365073890000013
Figure FDA0002365073890000014
Figure FDA0002365073890000015
Figure FDA0002365073890000016
in the formula, MrIndicating the magnetic levitation control sensitive gyro external moment, HrRepresenting the angular momentum of the rotor under the rotor train,
Figure FDA0002365073890000017
representing the change rate of the angular momentum of the rotor under the rotor system, I representing the rotational inertia of the rotor rotating around the reference coordinate system of the magnetic suspension control sensitive gyroscope, IrRepresenting the radial moment of inertia of the rotor, IzRepresenting the axial moment of inertia of the rotor, omega representing the axial speed of rotation of the rotor, omegarExpressing the rotor speed, ΩiRepresenting the absolute angular velocity of the rotor,
Figure FDA0002365073890000018
representing the absolute angular speed rate of change of the rotor,
Figure FDA0002365073890000019
representing the absolute angular velocity of the rotor system, i.e. the rotational speed relative to the inertia space,
Figure FDA00023650738900000110
for magnetically suspending the speed of the reference system of the control sensitive gyro relative to the inertial space,
Figure FDA00023650738900000111
is the angular velocity, ω, of the rotor relative to the inertial spacecmgIn order to be the angular velocity of the frame,
Figure FDA00023650738900000112
is a transformation matrix from a magnetic bearing coordinate system to a rotor-system,
Figure FDA0002365073890000021
is a transformation matrix from the frame coordinate system to the magnetic bearing coordinate system,
Figure FDA0002365073890000022
is a transformation matrix from a reference system of the magnetically suspended control sensitive gyroscope to a frame coordinate system,
Figure FDA0002365073890000023
a transformation matrix from a star system to a magnetic suspension control sensitive gyroscope reference system;
coincidence between a magnetic bearing coordinate system, a frame coordinate system, a magnetic suspension control sensitive gyro reference system and a satellite with only a single-axis angular rate
Figure FDA0002365073890000024
Under the conditions of (a):
Figure FDA0002365073890000025
although α, β are very small under the rotor system, in order to improve the measurement accuracy, the conversion matrix of the magnetic bearing coordinate system to the rotor system
Figure FDA0002365073890000026
No approximation is made:
Figure FDA0002365073890000027
and also
Figure FDA0002365073890000028
Then:
Figure FDA0002365073890000029
Figure FDA00023650738900000210
Figure FDA00023650738900000211
Figure FDA00023650738900000212
rotor radial resultant external moment
Figure FDA00023650738900000213
The expression is as follows:
Figure FDA0002365073890000031
Figure FDA0002365073890000032
(2) measurement and calculation of rotor radial resultant external moment
The magnetic suspension rotor is subjected to the following combined external moment:
Figure FDA0002365073890000033
Figure FDA0002365073890000034
the magnetic force applied to the magnetic suspension rotor can be expressed in the following linear form:
fλ＝kiλiλ+khλhλ
wherein λ ═ ax, ay, bx, by, kiλAnd khλRespectively representing the current rigidity and the displacement rigidity of radial ax, ay, bx and by channels of the magnetic suspension rotor, and calibrating through experiments; i.e. iax、ibx、iayAnd ibyIs the winding current of four radial channels, hax、hbx、hayAnd hbyIs the linear displacement of the magnetic levitation rotor in the ax, bx, ay and by directions, respectively, lmRepresenting the distance from the center of the magnetically levitated rotor to the center of the radial magnetic bearing; h isax、hbx、hay、hbyBy passingEddy current displacement sensor measurement, iax、ibx、iay、ibyThe external torque applied to the rotor is calculated by measuring through a current sensor;
(3) measurement and calculation of rotor deflection information
The expression for the rotor deflection angle is:
α＝(hay-hby)/(2lm)
β＝(hax-hbx)/(2lm)
hax、hbx、hayand hbyIs the linear displacement of the magnetic levitation rotor in the ax, bx, ay and by directions, respectively, lmDenotes the distance, h, from the center of the magnetically levitated rotor to the center of the radial magnetic bearingax、hbx、hay、hbyMeasured by an eddy current displacement sensor, thereby obtaining rotor deflection information α, β,
Figure FDA0002365073890000035
Figure FDA0002365073890000041
sinα、sinβ、cosα、cosβ；
(4) Single axis satellite attitude angular rate analytic solution
In the step (1)
Figure FDA0002365073890000042
Is converted into
Figure FDA0002365073890000043
Then a, b, c are defined as follows:
a＝(Ir-Iz)sinβcosβ
Figure FDA0002365073890000044
Figure FDA0002365073890000045
since in the expression of a, IrIs less than IzTherefore:
Figure FDA0002365073890000046
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