CN110162069B - Sunlight reflection staring expected attitude analysis solving method for near-earth orbit spacecraft - Google Patents

Sunlight reflection staring expected attitude analysis solving method for near-earth orbit spacecraft Download PDF

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CN110162069B
CN110162069B CN201910391506.2A CN201910391506A CN110162069B CN 110162069 B CN110162069 B CN 110162069B CN 201910391506 A CN201910391506 A CN 201910391506A CN 110162069 B CN110162069 B CN 110162069B
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张福桢
金磊
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Beihang University
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Abstract

The invention discloses a sunlight reflection staring expected attitude analysis solving method for a near-earth orbit spacecraft, which comprises the steps of establishing a target coordinate system according to orbit information of the spacecraft and longitude and latitude information of a target, establishing three-axis components of the target coordinate system under an inertial coordinate system, so as to establish an expected quaternion of the target coordinate system relative to the inertial coordinate system, further combining a rigid body dynamics basic theory, and finally determining angular velocity and angular acceleration of the target coordinate system by solving primary guidance and secondary guidance, so as to determine an analysis form of complete target attitude information. According to the method, when the spacecraft executes the sunlight reflection staring task, the target angular velocity and angular acceleration do not need to be solved through methods such as a difference method, and the defects of large errors and the like generated by a traditional difference method are overcome; when the target attitude is solved, the invention considers the shape factors such as the oblateness of the earth and the like, is not based on the assumption of the spherical shape of the earth, and is more in line with the engineering practice.

Description

Sunlight reflection staring expected attitude analysis solving method for near-earth orbit spacecraft
[ technical field ] A method for producing a semiconductor device
The invention relates to a method for solving sunlight reflection staring expected attitude analysis of a near-earth orbit spacecraft, in particular to a novel method for solving an expected attitude analysis solution when the near-earth orbit spacecraft executes a sunlight reflection staring task, and belongs to the field of spacecraft attitude dynamics.
[ background of the invention ]
With the development of aerospace technology, the tasks faced by spacecrafts are becoming more and more complex. In recent years, video satellites have been widely regarded and rapidly developed, mainly because of the capability of outputting continuous video signals, realizing real-time monitoring of ground target points, assisting in handling natural disasters, and other functions. The gaze task is the core mode of operation of the video satellite. The posture of the spacecraft is adjusted in real time, so that the components (cameras, video cameras and the like) carried by the spacecraft and used for staring realize continuous observation on ground target points in a period of time, and the task is completed. In 1999, the German DLR-Tubsat satellite realizes the acquisition of video signals by assembling three cameras, and the successful transmission of the first video satellite 'Tiantu No. two' in China realizes the breakthrough of the video satellite in China from scratch. Thereafter, the "Jilin No. one" developed by the Chinese academy of sciences and Long-time satellite technology Co., Ltd also realizes the video acquisition function.
On the basis of a video satellite, Russia provides a novel satellite (MayakCubeSat) provided with a sunlight reflector, and sunlight is reflected by the reflector to form a marker in night and can be used for space garbage removal and other work. The above task is mainly achieved by reflecting sunlight to achieve the "staring" task. The staring task is mainly realized by adjusting the attitude of the spacecraft, so how to obtain the target attitude of the spacecraft during attitude adjustment is particularly important. At present, similar research is mainly carried out on video minisatellites, different algorithms are provided for calculating the target attitude, and the target angular velocity and the target angular acceleration are numerically solved in a differential mode. However, the difference also has the effect of amplifying the error, thereby reducing the control accuracy.
[ summary of the invention ]
Aiming at a low earth orbit satellite, the invention provides a method for resolving the sunlight reflection staring expected attitude of the low earth orbit satellite, and the method provided by the invention can determine the analytic forms of the quaternion, the angular velocity and the angular acceleration of the expected attitude of a target coordinate system by constructing a reasonable target coordinate system according to the orbit information of the spacecraft and the longitude and latitude of a target point, so that solar rays can point to the earth to fix the target point within a period of time after passing through the satellite to finish staring.
Aiming at the problems, the technical scheme of the invention is as follows:
establishing a target coordinate system according to the orbit information of the spacecraft and the target longitude and latitude information, establishing the three-axis component of the target coordinate system under the inertial coordinate system, thereby establishing the expected quaternion of the target coordinate system relative to the inertial coordinate system, further combining with the rigid body dynamics basic theory, and finally determining the angular velocity and the angular acceleration of the target coordinate system by solving the primary guidance and the secondary guidance, thereby determining the complete analytic form of the target attitude information.
A method for resolving sunlight reflection staring expected attitude of a near-earth orbit spacecraft comprises the following specific operation steps:
step 1: and establishing a target coordinate system according to the orbit information of the spacecraft and the target longitude and latitude information. The method specifically comprises the following steps:
step 1.1 definition of coordinate System
a. Earth's center inertial coordinate system (F)i(OiXiYiZi))
Origin O of the earth's center inertial coordinate systemiIs fixedly connected to the center of the earth OiXiThe axis being in the equatorial plane and pointing towards the vernal equinox, OiZiThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth, OiYiThe axis being in the equatorial plane and being co-incident withiXiShaft, OiZiThe axes form a rectangular coordinate system.
b. Coordinate system of rotation of the earth (F)E(OEXEYEZE))
Origin O of earth's rotating coordinate systemEIs fixedly connected with the earth core (and O)iCoincidence), OEZEThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth (with O)iZiCoincident axes), OEXEAxis directed at the Greenwich meridian, OEYEThe axis being in the equatorial plane and being co-incident withEXEShaft, OEZEThe axes form a rectangular coordinate system.
c. Orbital coordinate system (F)O(OOXOYOZO))
Origin O of the orbital coordinate systemOLocated in the centre of mass of the spacecraft, OOZOThe axis lying in the plane of the track and pointing towards the centre of the earth, OoXoThe axis lying in the plane of the track, perpendicular to OOZOAxial and in the direction of spacecraft speed, OoYoThe axis being perpendicular to the orbital plane and to the axis OoXoShaft, OOZOThe axes form a rectangular coordinate system.
Step 1.2 establishing a target coordinate system
Set with spacecraft OBZBWith axis directed towards the centre of the earth by ORT(ΛT,ΦT) Indicating the target point to which pointing is required, ΛT,ΦTRepresenting the longitude and latitude of the target point, respectively, and defining a vector p pointing from the satellite to the target pointTDefining the vector of the earth center pointing to the center of mass of the spacecraft as rsDefining the center of the earth pointing to the target point ORTIs a vector of rTThen ρT＝rT-rs. Defining the sun direction vector as SsunThen the target coordinate system FT(OTXTYTZT) Origin O ofTThree axes x of a target coordinate system at the position of the center of mass of the spacecraftT,yT,zTAre respectively defined as follows:
Figure BDA0002055990070000031
step 1.3 calculation of rotation matrix between coordinate systems
Defining a geocentric inertial frame FiTo the earth rotation coordinate system FEA rotation matrix ofEIEarth rotation coordinate system FETo-center inertial frame FiA rotation matrix ofIEDefining a geocentric inertial frame FiTo the orbital coordinate system FOA rotation matrix ofOIOrbital coordinate system FOTo-center inertial frame FiA rotation matrix ofIODefining a geocentric inertial frame FiTo the target coordinate system FTA rotation matrix ofTIObject coordinate system FTTo-center inertial frame FiA rotation matrix ofIT. If the orbit information of the spacecraft is given in the form of six elements of the orbit, the orbit information is respectively recorded as an orbit inclination angle phiiSemi-major axis a of the track, eccentricity e of the track, right ascension omega of the intersection point, argument phi of the perigeeωLatitude argument phiuDefining the spring division point Greenwich mean Red channel as αG. Thereby to obtain
Figure BDA0002055990070000041
Figure BDA0002055990070000042
Step 2: and calculating the attitude quaternion of the target coordinate system relative to the inertial coordinate system. The method comprises the following specific steps:
step 2.1 representation of the target coordinate system in the centroid inertial coordinate system
In order to calculate the attitude quaternion of the target coordinate system relative to the inertial coordinate system, three axes of the target coordinate system need to be expressed in the geocentric inertial coordinate system, and x needs to be expressedT,yT,zTThe expressions in the centroid inertial coordinate system are respectively recorded asIxT,IyT,IzTThus having
Figure BDA0002055990070000043
Wherein:ISsunrepresents the sun direction vector SsunThe representation under the geocentric inertial coordinate system can be obtained by calculating the orbit parameters and the ephemeris;Ixoo representing an orbital coordinate systemoXoThe representation of the axis in the geocentric inertial coordinate system can be determined according to the orbit parameters; i | · | represents taking the 2-norm of the vector,IρTvector p representing the pointing of a satellite at a target pointTThe representation under the earth's center inertial coordinate system,IρT＝IrT-Irs，IrT,Irsrespectively represent centroid vector r of earth center pointing spacecraftsAnd the center of the earth points to the target point ORTIs a vector of rTRepresentation in an inertial coordinate system.IrsCan be obtained by actual measurement of a GPS,IrTthe method can be used for calculating by combining earth oblateness and other earth spherical information according to the longitude and latitude of a given target point, and comprises the following specific processes:
vector r of center of earth pointing to target pointTIn the earth's rotating coordinate system FEBelow can be expressed asErT：
Figure BDA0002055990070000051
Wherein,
Figure BDA0002055990070000052
RE6378137m represents the earth's equivalent radius,
Figure BDA0002055990070000053
fE1/298.2572 denotes the earth oblateness, so the earth center points to the target point vector rTIn the centroid inertial frame FIBelow can be expressed asIrT：
IrT＝AIE ErT(6)
Wherein A isIERepresenting a rotation matrix from the earth's rotating coordinate system to the earth's centered inertial coordinate system. Thus, the rotation matrix A from the target coordinate system to the centroid inertia coordinate systemITCan be expressed as:
AIT＝[IxT IyT IzT](7)
and the desired pose quaternion for the target coordinate system may be determined from the correspondence between the coordinate system and the quaternion.
Step 2.2 calculation of target attitude quaternion
According to equation (7), the spacecraft target attitude quaternion can be calculated according to the following scheme, assuming that the attitude quaternion of the target coordinate system relative to the inertial system is represented as QTI＝[qT0qT1qT2qT3]T. Suppose a rotation matrix A in equation (7)ITEach component is expressed as follows:
Figure BDA0002055990070000054
at this time, it is calculated by the following form:
Figure BDA0002055990070000061
and comparing q in the above formulaT0,qT1,qT2,qT3The size of (2).
a) If q isT0At maximum, then
Figure BDA0002055990070000062
b) If q isT1At maximum, then
Figure BDA0002055990070000063
c) If q isT2At maximum, then
Figure BDA0002055990070000064
d) If q isT3At maximum, then
Figure BDA0002055990070000065
Therefore, the attitude quaternion of the target coordinate system relative to the geocentric inertial coordinate system can be obtained through calculation.
And step 3: determining a target angular velocity of a target coordinate system
Definition ofIρ1＝||IρT||ISsun+IρT，Iρ2＝IzT×IxoThus having
IzT＝Iρ1/||Iρ1||,IyT＝Iρ2/||Iρ2|| (14)
Since the motion of the sun direction vector is much slower than the orbital motion and attitude motion of the spacecraft in the inertial system, the sun direction vector can be approximately considered to remain unchanged in the inertial system in a short time, i.e. the time derivative of the sun direction vector is 0 and is expressed as
Figure BDA0002055990070000071
Since the target point is located on the earth's surface, there are
Figure BDA0002055990070000072
Wherein
Figure BDA0002055990070000073
Representing the time derivative of the geocentric-to-target point vector represented under the geocentric inertial system,IωEis the representation of the rotational angular velocity of the earth under the geocentric inertial system, and the value isIωE＝[0 0 7.292×10-5]Trad/s. Thus, pairIρ1The formula is used for derivation, which is
Figure BDA0002055990070000074
Wherein,
Figure BDA0002055990070000075
IVsthe representation of the velocity of the spacecraft in an inertial frame of coordinates can be obtained by GPS measurements. Further, it is possible to calculateIzT,IyTNamely:
Figure BDA0002055990070000076
wherein
Figure BDA0002055990070000077
Can be given by the formula (16), E3Representing a third order identity matrix, d (-) dt represents a differential operator. Similarly, there are
Figure BDA0002055990070000078
Wherein,
Figure BDA0002055990070000079
in the formulaIzT,
Figure BDA00020559900700000710
Can be obtained from the formulae (14) and (17),Iωorepresenting the representation of the angular velocity of the spacecraft orbit under the inertial system,Ixoshowing a track system FOO of (A) to (B)oXoThe representation of the axis in the inertial system can be calculated from the orbit parameters. Thus, the angular velocity of the target coordinate system relative to the inertial system can be calculated by:
a) the angular velocity of the target coordinate system relative to the earth center inertial coordinate system is assumed to be expressed in the inertial system asIωTIThen, the following relationship holds:
Figure BDA0002055990070000081
b) according to the formula (19), a
Figure BDA0002055990070000082
Where a · represents a vector dot product operation. Thus, the absolute angular velocity of the target coordinate system is expressed in the inertial system as
Figure BDA0002055990070000083
Reuse of coordinate transformation matrix relationships
Figure BDA0002055990070000084
Is provided with
TωTI＝ATI IωTI(22)
Therefore, the representation of the angular speed of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be calculated.
And 4, step 4: determining a target angular acceleration of a target coordinate system
In order to calculate the angular acceleration of the target coordinate system relative to the centroid inertial coordinate system, the angular acceleration needs to be calculated firstIyTAndIzTsecond derivative with respect to time. According to the formulae (17), (18), there are
Figure BDA0002055990070000091
Figure BDA0002055990070000092
Wherein,
Figure BDA0002055990070000093
Figure BDA0002055990070000094
Figure BDA0002055990070000095
from equations (25) to (27), the final calculation can be made
Figure BDA0002055990070000096
And
Figure BDA0002055990070000097
on the basis, further, the formula is subjected to derivation, including
Figure BDA0002055990070000098
Reuse of coordinate transformation matrix relationships
Figure BDA0002055990070000099
Is provided with
Figure BDA00020559900700000910
Therefore, the representation of the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be calculated.
Thereby finally obtaining the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system.
The invention provides an analysis method for resolving a target attitude aiming at a near-earth orbit spacecraft executing a sunlight reflection staring task, which has the main advantages that:
1) according to the target attitude analysis method provided by the invention, when the spacecraft executes a sunlight reflection staring task, the solution of the angular velocity and the angular acceleration of the target is not required to be carried out by methods such as a difference method, and the defects of large error and the like generated by a traditional difference method are overcome;
2) according to the method, when the target attitude is solved, the shape factors such as the oblateness of the earth are considered, the assumption of the spherical shape of the earth is not made, and the method is more suitable for engineering practice.
[ description of the drawings ]
Fig. 1 is a sunlight reflection gaze fixation diagram.
Fig. 2 is a sunlight reflection gaze task coordinate system definition.
FIG. 3 is a sunlight reflecting target coordinate system definition.
FIG. 4 is a diagram illustrating a transformation relationship between coordinate systems
FIG. 5 is a schematic diagram of a target pose calculation process.
[ detailed description ] embodiments
The following will specifically describe the implementation process of the present invention by taking a certain type of near-earth orbit spacecraft as an example, as shown in fig. 1 to 5. Assuming the initial time spacecraft orbit parameters are as follows:
Figure BDA0002055990070000101
let Λ be the longitude and latitude of the gaze target point where reflected rays are expectedT＝108deg，ΦTWhen the time is 0deg, the initial time is 2018, 10, 4, 2: 30. Determining the representation of the sun direction vector in the geocentric inertial coordinate system according to the orbit information and the ephemerisISsun(unit vector, backlight direction), the track parameters evolve over time. The initial time target attitude calculation process is specifically described below according to the description.
1. And establishing a target coordinate system according to the orbit information of the spacecraft and the target longitude and latitude information. The method specifically comprises the following steps:
step 1.1 defining a coordinate system:
a. earth's center inertial coordinate system (F)i(OiXiYiZi))
Origin O of the earth's center inertial coordinate systemiIs fixedly connected to the center of the earth OiXiThe axis being in the equatorial plane and pointing towards the vernal equinox, OiZiThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth, OiYiThe axis being in the equatorial plane and being co-incident withiXiShaft, OiZiThe axes form a rectangular coordinate system.
b. Coordinate system of rotation of the earth (F)E(OEXEYEZE))
Origin O of earth's rotating coordinate systemEIs fixedly connected with the earth core (and O)iCoincidence), OEZEThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth (with O)iZiCoincident axes), OEXEAxis directed at the Greenwich meridian, OEYEThe axis being in the equatorial plane and being co-incident withEXEShaft, OEZEThe axes form a rectangular coordinate system.
c. Orbital coordinate system (F)O(OOXOYOZO))
Origin O of the orbital coordinate systemOLocated in the centre of mass of the spacecraft, OOZOThe axis lying in the plane of the track and pointing towards the centre of the earth, OoXoThe axis lying in the plane of the track, perpendicular to OOZOAxial and in the direction of spacecraft speed, OoYoThe axis being perpendicular to the orbital plane and to the axis OoXoShaft, OOZOThe axes form a rectangular coordinate system.
Step 1.2 establishing a target coordinate system
Target point latitude and longitude Λ selected by way of exampleT＝108deg，ΦTThe target coordinate system is established according to the above step 1.2, assuming 0deg and the sun direction vector.
Step 1.3 calculation of rotation matrix between coordinate systems
According to the six-element parameter of the spacecraft orbit, the orbit inclination angle phiiSemi-major axis a of the track, eccentricity e of the track, right ascension omega of the intersection point, argument phi of the perigeeωLatitude argument phiuAnd spring division point Greenwich mean Chi channel αGDetermining the geocentric inertial coordinate system F according to the step 1.3iEarth rotation coordinate system FEAnd an orbital coordinate system FOBy a rotation matrix therebetween, i.e. determining the earth's center inertial frame FiTo the earth rotation coordinate system FEA rotation matrix ofEIEarth rotation coordinate system FETo-center inertial frame FiA rotation matrix ofIEEarth center inertial coordinate system FiTo the orbital coordinate system FOA rotation matrix ofOIOrbital coordinate system FOTo-center inertial frame FiA rotation matrix ofIOThe Greenwich mean Chin at spring break point is αG50.24(deg), so that the initial time coordinate system transformation matrix is:
Figure BDA0002055990070000121
Figure BDA0002055990070000122
step 2: and calculating the attitude quaternion of the target coordinate system relative to the inertial coordinate system. The method comprises the following specific steps:
step 2.1 representation of the target coordinate system in the centroid inertial coordinate system
In order to calculate the attitude quaternion of the target coordinate system relative to the inertial coordinate system, three axes of the target coordinate system need to be expressed in the geocentric inertial coordinate system. The calculation is as follows:
ISsunrepresents the sun direction vector SsunThe representation in the geocentric inertial coordinate system can be given by a table lookup asISsun＝[-0.9822 -0.1724 -0.0748]T。IxoShow the track seatO of the systemoXoRepresentation of axes in the Earth's center inertial frame due to O of the orbital frameoXoThe axes can be expressed in an orbital coordinate system asOxo＝[1 0 0]TThus havingIxo＝AIO Oxo＝[0.0497 0.1231 -0.9911]T. Centroid vector r of earth-centered pointing spacecraftsCan be actually measured according to GPS, and can be calculated asIrs＝106[3.4026 -5.3947 0]T(m) of the reaction mixture. According to the longitude and latitude information and the earth spherical shape information of the target point, the earth center pointing target point vector r can be calculated according to the formula (5)TIn the earth's rotating coordinate system FEBelow can be expressed asErT＝106[-1.9710 -6.0660 0]T(m), which can be expressed as in the inertial coordinate system
IrT＝AIE ErT＝106[3.4026 -5.3947 0]T(m) (32)
Thus, the vector ρ of the satellite pointing to the target pointTRepresentation in the geocentric inertial frameIρT＝IrT-Irs＝106[9.8186 7.9820 0]T. Thus, according to formula (4), xT,yT,zTExpressed as in the geocentric inertial coordinate system
Figure BDA0002055990070000131
Rotation matrix A from target coordinate system to earth center inertial coordinate systemITCan be expressed as:
Figure BDA0002055990070000132
step 2.2 calculation of target attitude quaternion
Assuming that the attitude quaternion of the target coordinate system relative to the inertial system is denoted as QTI＝[qT0qT1qT2qT3]T. Suppose rotation matrix A in equation (34)ITEach component is expressed as follows:
Figure BDA0002055990070000133
at this time, it is calculated by the following form:
Figure BDA0002055990070000134
obviously, the target quaternion should be calculated according to equation (11) with the result that
Figure BDA0002055990070000141
Thus the attitude quaternion of the target coordinate system relative to the centroid inertial coordinate system
QTI＝[0.4181 0.5872 0.4473 -0.5294]T。
And step 3: determining a target angular velocity of a target coordinate system
According to step 3, there areIρ1＝||IρT||ISsun+IρT＝107[-0.2610 1.0164 -0.0946]T，Iρ2＝IzT×Ixo＝[0.9672 -0.2499 0.0174]T，
Since the motion of the sun direction vector is much slower than the orbital motion and attitude motion of the spacecraft in the inertial system, the sun direction vector can be approximately considered to remain unchanged in the inertial system in a short time, i.e. the time derivative of the sun direction vector is 0 and is expressed as
Figure BDA0002055990070000142
Since the target point is located on the earth surface, the representation of the rotational angular velocity of the earth under the geocentric inertial systemIωE＝[0 0 7.292×10-5]Trad/s, thus representation of the geocentric-to-target point vector under the geocentric inertial systemIrTTime derivative of (1)
Figure BDA0002055990070000143
Figure BDA0002055990070000144
In addition to this, the present invention is,
Figure BDA0002055990070000145
thus, pairIρ1The formula is used for derivation, which is
Figure BDA0002055990070000146
Further, it is possible to calculateIzT,IyTNamely:
Figure BDA0002055990070000147
similarly, pair
Figure BDA0002055990070000148
Is provided with
Figure BDA0002055990070000149
Thus, it is possible to provide
Figure BDA0002055990070000151
Therefore, according to equations (19) - (21), the absolute angular velocity of the target coordinate system in the inertial system is expressed as
Figure BDA0002055990070000152
Reuse of coordinate transformation matrix relationships
Figure BDA0002055990070000153
Is provided with
TωTI＝ATI IωTI＝[0 -0.0007 0.0011]T(43)
Therefore, the representation of the angular speed of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be calculated.
And 4, step 4: determining a target angular acceleration of a target coordinate system
In order to calculate the angular acceleration of the target coordinate system relative to the centroid inertial coordinate system, the angular acceleration needs to be calculated firstIyTAndIzTsecond derivative with respect to time. According to (25) to (27), there are
Figure BDA0002055990070000154
Figure BDA0002055990070000155
Figure BDA0002055990070000156
According to the formulae (17), (18), there are
Figure BDA0002055990070000157
Figure BDA0002055990070000158
Can finally find
Figure BDA0002055990070000159
And
Figure BDA00020559900700001510
on the basis of this, further derivation is carried out on the formula (42) in
Figure BDA00020559900700001511
Reuse of coordinatesTransforming matrix relations
Figure BDA00020559900700001512
Is provided with
Figure BDA00020559900700001513
Therefore, the representation of the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be calculated. Therefore, analytical expressions of the attitude quaternion, the angular velocity and the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system can be given, and the error amplification effect caused by the fact that the angular velocity and the angular acceleration are solved through a difference method in the prior art is overcome.

Claims (1)

1. A method for resolving sunlight reflection staring expected attitude of a near-earth orbit spacecraft comprises the steps of establishing a target coordinate system according to orbit information of the spacecraft and longitude and latitude information of a target, establishing three-axis components of the target coordinate system under an inertial coordinate system, and determining an expected quaternion of the target coordinate system relative to the inertial coordinate system;
the method is characterized in that: the method comprises the following specific operation steps:
step 1: establishing a target coordinate system according to the orbit information of the spacecraft and the target longitude and latitude information:
step 1.1 definition of coordinate System
a. Earth's center inertial coordinate system (F)i(OiXiYiZi))
Origin O of the earth's center inertial coordinate systemiIs fixedly connected to the center of the earth OiXiThe axis being in the equatorial plane and pointing towards the vernal equinox, OiZiThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth, OiYiThe axis being in the equatorial plane and being co-incident withiXiShaft, OiZiThe shaft is formed straightAn angular coordinate system;
b. coordinate system of rotation of the earth (F)E(OEXEYEZE))
Origin O of earth's rotating coordinate systemEIs fixedly connected with the earth core (and O)iCoincidence), OEZEThe axis being perpendicular to the equatorial plane and aligned with the direction of the rotational angular velocity of the earth (with O)iZiCoincident axes), OEXEAxis directed at the Greenwich meridian, OEYEThe axis being in the equatorial plane and being co-incident withEXEShaft, OEZEThe axes form a rectangular coordinate system;
c. orbital coordinate system (F)O(OOXOYOZO))
Origin O of the orbital coordinate systemOLocated in the centre of mass of the spacecraft, OOZOThe axis lying in the plane of the track and pointing towards the centre of the earth, OoXoThe axis lying in the plane of the track, perpendicular to OOZOAxial and in the direction of spacecraft speed, OoYoThe axis being perpendicular to the orbital plane and to the axis OoXoShaft, OOZOThe axes form a rectangular coordinate system;
step 1.2 establishing a target coordinate system
Set with spacecraft OOZOWith axis directed towards the centre of the earth by ORT(ΛT,ΦT) Indicating the target point to which pointing is required, ΛT,ΦTRepresenting the longitude and latitude of the target point, respectively, and defining a vector p pointing from the satellite to the target pointTDefining the vector of the earth center pointing to the center of mass of the spacecraft as rsDefining the center of the earth pointing to the target point ORTIs a vector of rTThen, then
Figure FDA0002489903610000024
Defining the sun direction vector as SsunThen the target coordinate system FT(OTXTYTZT) Origin O ofTThree axes x of a target coordinate system at the position of the center of mass of the spacecraftT,yT,zTAre respectively defined as follows:
Figure FDA0002489903610000021
step 1.3 calculation of rotation matrix between coordinate systems
Defining a geocentric inertial frame FiTo the earth rotation coordinate system FEA rotation matrix ofEIEarth rotation coordinate system FETo-center inertial frame FiA rotation matrix ofIEDefining a geocentric inertial frame FiTo the orbital coordinate system FOA rotation matrix ofOIOrbital coordinate system FOTo-center inertial frame FiA rotation matrix ofIODefining a geocentric inertial frame FiTo the target coordinate system FTA rotation matrix ofTIObject coordinate system FTTo-center inertial frame FiA rotation matrix ofIT(ii) a If the orbit information of the spacecraft is given in the form of six elements of the orbit, the orbit information is respectively recorded as an orbit inclination angle phiiSemi-major axis a of the track, eccentricity e of the track, right ascension omega of the intersection point, argument phi of the perigeeωLatitude argument phiuDefining the spring division point Greenwich mean Red channel as αG(ii) a Thereby to obtain
Figure FDA0002489903610000022
Figure FDA0002489903610000023
Step 2: calculating the attitude quaternion of the target coordinate system relative to the inertial coordinate system;
step 2.1 representation of the target coordinate system in the centroid inertial coordinate system
In order to calculate the attitude quaternion of the target coordinate system relative to the inertial coordinate system, three axes of the target coordinate system need to be expressed in the geocentric inertial coordinate system, and x needs to be expressedT,yT,zTExpressed points under the geocentric inertial coordinate systemIs otherwise noted asIxT,IyT,IzTThus having
Figure FDA0002489903610000031
Wherein:ISsunrepresents the sun direction vector SsunThe representation under the geocentric inertial coordinate system can be obtained by calculating the orbit parameters and the ephemeris;Ixoo representing an orbital coordinate systemoXoThe representation of the axis in the geocentric inertial coordinate system can be determined according to the orbit parameters; i | · | represents taking the 2-norm of the vector,IρTvector p representing the pointing of a satellite at a target pointTThe representation under the earth's center inertial coordinate system,IρT＝IrT-Irs，IrT,Irsrespectively represent centroid vector r of earth center pointing spacecraftsAnd the center of the earth points to the target point ORTIs a vector of rTRepresentation in an inertial coordinate system;Irscan be obtained by actual measurement of a GPS,IrTthe method can be used for calculating by combining earth oblateness and other earth spherical information according to the longitude and latitude of a given target point, and comprises the following specific processes:
vector r of center of earth pointing to target pointTIn the earth's rotating coordinate system FEBelow can be expressed asErT：
Figure FDA0002489903610000032
Wherein,
Figure FDA0002489903610000033
RE6378137m represents the earth's equivalent radius,
Figure FDA0002489903610000034
fE1/298.2572, therefore,vector r of center of earth pointing to target pointTIn the centroid inertial frame FIBelow can be expressed asIrT：
IrT＝AIE ErT(6)
Wherein A isIERepresenting a rotation matrix from the earth rotation coordinate system to the earth center inertia coordinate system; thus, the rotation matrix A from the target coordinate system to the centroid inertia coordinate systemITCan be expressed as:
AIT＝[IxT IyT IzT](7)
and determining the quaternion of the expected attitude of the target coordinate system according to the corresponding relation between the coordinate system and the quaternion;
step 2.2 calculation of target attitude quaternion
According to equation (7), the spacecraft target attitude quaternion can be calculated according to the following scheme, assuming that the attitude quaternion of the target coordinate system relative to the inertial system is represented as QTI＝[qT0qT1qT2qT3]T(ii) a Suppose a rotation matrix A in equation (7)ITEach component is expressed as follows:
Figure FDA0002489903610000041
at this time, it is calculated by the following form:
Figure FDA0002489903610000042
and comparing q in the above formulaT0,qT1,qT2,qT3The size of (d);
a) if q isT0At maximum, then
Figure FDA0002489903610000043
b) If q isT1At maximum, then
Figure FDA0002489903610000051
c) If q isT2At maximum, then
Figure FDA0002489903610000052
d) If q isT3At maximum, then
Figure FDA0002489903610000053
Therefore, the attitude quaternion of the target coordinate system relative to the geocentric inertial coordinate system can be obtained through calculation;
and step 3: determining a target angular velocity of a target coordinate system
Definition ofIρ1＝||IρT||ISsun+IρT，Iρ2＝IzT×IxoThus having
IzT＝Iρ1/||Iρ1||,IyT＝Iρ2/||Iρ2|| (14)
Since the motion of the sun direction vector is much slower than the orbital motion and attitude motion of the spacecraft in the inertial system, the sun direction vector can be approximately considered to remain unchanged in the inertial system in a short time, i.e. the time derivative of the sun direction vector is 0 and is expressed as
Figure FDA0002489903610000054
Since the target point is located on the earth's surface, there are
Figure FDA0002489903610000055
Wherein
Figure FDA0002489903610000056
Representing the time derivative of the geocentric-to-target point vector represented under the geocentric inertial system,IωEis the representation of the rotational angular velocity of the earth under the geocentric inertial system, and the value isIωE＝[0 0 7.292×10-5]Trad/s; thus, pairIρ1The formula is used for derivation, which is
Figure FDA0002489903610000061
Wherein,
Figure FDA0002489903610000062
IVsthe representation of the speed of the spacecraft in an inertial coordinate system can be obtained through GPS measurement; further, it is possible to calculateIzT,IyTNamely:
Figure FDA0002489903610000063
wherein
Figure FDA0002489903610000064
Can be given by the formula (16), E3Representing a third order identity matrix, d (-) dt representing a differential operator; similarly, there are
Figure FDA0002489903610000065
Wherein,
Figure FDA0002489903610000066
in the formulaIzT,
Figure FDA0002489903610000067
Can be obtained from the formulae (14) and (17),Iωoindicating the angular velocity of the spacecraft orbit atThe representation under the inertial system is that,Ixoshowing a track system FOO of (A) to (B)oXoThe representation of the shaft under the inertial system can be obtained by calculating the orbit parameters; thus, the angular velocity of the target coordinate system relative to the inertial system can be calculated by:
a) the angular velocity of the target coordinate system relative to the earth center inertial coordinate system is assumed to be expressed in the inertial system asIωTIThen, the following relationship holds:
Figure FDA0002489903610000068
b) according to the formula (19), a
Figure FDA0002489903610000071
Where · represents a vector dot product operation; thus, the absolute angular velocity of the target coordinate system is expressed in the inertial system as
Figure FDA0002489903610000072
Reuse of coordinate transformation matrix relationships
Figure FDA0002489903610000073
Is provided with
TωTI＝ATI IωTI(22)
Therefore, the representation of the angular speed of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be obtained through calculation;
and 4, step 4: determining a target angular acceleration of a target coordinate system
In order to calculate the angular acceleration of the target coordinate system relative to the centroid inertial coordinate system, the angular acceleration needs to be calculated firstIyTAndIzTa second derivative with respect to time; according to the formulae (17), (18), there are
Figure FDA0002489903610000074
Figure FDA0002489903610000075
Wherein,
Figure FDA0002489903610000081
Figure FDA0002489903610000082
Figure FDA0002489903610000083
from equations (25) to (27), the final calculation can be made
Figure FDA0002489903610000084
And
Figure FDA0002489903610000085
on the basis, further, the formula is subjected to derivation, including
Figure FDA0002489903610000086
Reuse of coordinate transformation matrix relationships
Figure FDA0002489903610000087
Is provided with
Figure FDA0002489903610000088
Therefore, the representation of the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system in the target coordinate system can be calculated; and finally obtaining the angular acceleration of the target coordinate system relative to the geocentric inertial coordinate system.
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