CN109032161B - Small-inertia spacecraft attitude jitter determination method based on fourth-order Runge Kutta method - Google Patents

Small-inertia spacecraft attitude jitter determination method based on fourth-order Runge Kutta method Download PDF

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CN109032161B
CN109032161B CN201810870904.8A CN201810870904A CN109032161B CN 109032161 B CN109032161 B CN 109032161B CN 201810870904 A CN201810870904 A CN 201810870904A CN 109032161 B CN109032161 B CN 109032161B
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吴爱国
白子扬
张颖
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Shenzhen Graduate School Harbin Institute of Technology
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Abstract

The invention provides a small inertia spacecraft attitude jitter determination method based on a fourth-order Runge Kutta method, which comprises the following steps of: s1, establishing an attitude kinematics equation of the spacecraft based on the direction cosine matrix; s2, dispersing the attitude kinematics equation by using a fourth-order Runge Kutta method to obtain a recursion equation of a direction cosine matrix; and S3, correcting the recursion equation by using two different types of sensors, wherein the measurement data of one type of sensor is used as a dominant term, and the sensor of the other type is used as a correction term through a data fusion algorithm. The invention has the beneficial effects that: the measurement bandwidth of the attitude jitter is improved, and the attitude jitter of the small-inertia spacecraft under a small angle can be accurately measured in a relatively wide bandwidth range.

Description

Small-inertia spacecraft attitude jitter determination method based on fourth-order Runge Kutta method
Technical Field
The invention relates to a spacecraft, in particular to a small-inertia spacecraft attitude jitter determination method based on a four-order Runge Kutta method.
Background
Small spacecraft are beginning to become more and more powerful for a variety of reasons. Such spacecraft possess less inertia than conventional spacecraft, which makes them highly susceptible to disturbances that produce attitude jitter that is more severe than large spacecraft. This level of jitter is unacceptable to scientific instruments and must be compensated for. The attitude jitter must be measured over a wide bandwidth before compensation.
The existing attitude sensors have the problem of bandwidth limitation, and can only measure within respective bandwidth ranges.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a small inertia spacecraft attitude jitter determination method based on a four-order Runge Kutta method, which is used for solving the problem of expanding the measurement bandwidth of an attitude sensor.
The invention provides a small inertia spacecraft attitude jitter determination method based on a fourth-order Runge Kutta method, which comprises the following steps of:
s1, establishing an attitude kinematics equation of the spacecraft based on the direction cosine matrix;
s2, dispersing the attitude kinematics equation by using a fourth-order Runge Kutta method to obtain a recursion equation of a direction cosine matrix;
and S3, correcting the recursion equation by using two different types of sensors, wherein the measurement data of one type of sensor is used as a dominant term, and the sensor of the other type is used as a correction term through a data fusion algorithm.
As a further improvement of the present invention, step S1 includes:
establishing an attitude kinematics equation of the spacecraft based on the direction cosine matrix as follows:
Figure BDA0001752121380000021
wherein C is a direction cosine matrix of the spacecraft,
Figure BDA0001752121380000023
is the cosine of the directionFirst derivative of the matrix with respect to time, ω×Is the angular velocity vector ω ═ ωx ωy ωz]TCross-product matrix of (a).
As a further improvement of the present invention, step S2 includes:
a four-step Runge Kutta method is used to perform discrete approximation on the formula (1):
Figure BDA0001752121380000022
wherein, K1Is the slope at time n; k2Introducing K by an Euler method for the slope of the midpoint between the n moment and the n +1 moment1To obtain the value of (1); k3Also is the slope of the midpoint between the n time and the n +1 time, and introduces K by an Euler method2To obtain the value of (1); k4The slope at the time n + 1; and T is the sampling time of the sensor, and because two different types of sensors are involved in the data fusion algorithm, if the sampling time of the two different types of sensors is different, the T is taken as the least common multiple of the sampling time of the two different types of sensors.
As a further improvement of the present invention, step S2 includes:
the recursive equation of the direction cosine matrix obtained by the fourth-order Runge Kutta method is as follows:
Figure BDA0001752121380000031
wherein, Cn+1A direction cosine matrix of the spacecraft at time n +1, CnIs a spacecraft direction cosine matrix at the time of n,
Figure BDA0001752121380000032
is a cross-product matrix of the attitude angular velocity vector at the moment n,
Figure BDA0001752121380000033
is a cross-product matrix of the attitude angular velocity vector at the midpoint of the time n and the time n +1,
Figure BDA0001752121380000034
a cross-product matrix of the attitude angular velocity vector at the moment n + 1;
Figure BDA0001752121380000035
and
Figure BDA0001752121380000036
all the attitude angle displacement measured by the sensor is obtained by difference; since the sampling time of the sensor is T, the attitude angular velocity at the midpoint between n and n +1 times
Figure BDA0001752121380000037
The sensor can not be obtained and can only be used
Figure BDA0001752121380000038
Approximately, so there are:
Figure BDA0001752121380000039
wherein,
Figure BDA0001752121380000041
is a cross-multiplication matrix of the spacecraft attitude angular displacement vector measured by the sensor at the moment of n +1,
Figure BDA0001752121380000042
is a cross-product matrix of the attitude angular displacement vector of the spacecraft measured by the sensor at the moment n,
Figure BDA0001752121380000043
a cross-multiplication matrix of the spacecraft attitude angular displacement vector measured by the sensor at the n-1 moment;
the formula (4) can be substituted for the formula (3):
Figure BDA0001752121380000044
as a further improvement of the present invention, step S3 includes:
when the angular displacement sensor and the data measured by the inertial reference unit are used for data fusion, the data measured by the angular displacement sensor is used as a leading term, the data measured by the inertial reference unit is used as a low-frequency correction term, and the following steps are performed:
Figure BDA0001752121380000045
wherein,
Figure BDA0001752121380000046
are cross multiplication matrixes of angular displacement vectors of the spacecraft measured by the angular displacement sensor at the moments of n-1, n and n +1 respectively,
Figure BDA0001752121380000047
the correction terms of the angular displacement are respectively obtained after data measured by the inertia reference unit at the moments of n-1, n and n +1 pass through the closed-loop controller;
the specific expression is as follows:
Figure BDA0001752121380000048
Figure BDA0001752121380000051
whereinADSθx、ADSθy、ADSθzThe angular displacement of a rolling shaft, a pitching shaft and a yawing shaft of the spacecraft is measured by an angular displacement sensor; whilecorθx，corθy，corθzIt is the low frequency correction term produced by the closed loop controller.
By substituting equation (6) for equation (5), a recurrence equation after data fusion can be obtained:
Figure BDA0001752121380000052
the invention has the beneficial effects that: the measurement bandwidth of the attitude jitter is improved, and the attitude jitter of the small-inertia spacecraft under a small angle can be accurately measured in a relatively wide bandwidth range.
Drawings
Fig. 1 is a schematic block diagram of a data fusion algorithm of a small inertia spacecraft attitude jitter determination method based on a fourth-order longstota method according to the present invention.
FIG. 2 is a simulink simulation block diagram of a data fusion algorithm of a small inertia spacecraft attitude jitter determination method based on a fourth-order Runge Kutta method.
Detailed Description
The invention is further described with reference to the following description and embodiments in conjunction with the accompanying drawings.
As shown in fig. 1 to 2, a method for determining small inertia spacecraft attitude jitter based on a fourth-order longge stoke method includes the following steps:
firstly, establishing an attitude kinematics equation of the spacecraft based on a direction cosine matrix as follows:
Figure BDA0001752121380000061
wherein C is a direction cosine matrix of the spacecraft,
Figure BDA0001752121380000063
is the first derivative of the directional cosine matrix with respect to time, ω×Is the angular velocity vector ω ═ ωx ωy ωz]TCross-product matrix of (a). And because the attitude sensors have sampling time, carrying out discretization processing on the continuous equation by using a four-order Runge Kutta method. According to the fourth-order longgedusta method, there are:
Figure BDA0001752121380000062
wherein, K1Is the slope at time n; k2Introducing K by an Euler method for the slope of the midpoint between the n moment and the n +1 moment1To obtain the value of (1); k3Also is the slope of the midpoint between the n time and the n +1 time, and introduces K by an Euler method2To obtain the value of (1); k4The slope at time n + 1. T is the sampling time of the sensor, and because two different types of sensors are involved in the data fusion algorithm provided by the patent, the sampling time of the two sensors may be different, and the T can be taken as the least common multiple of the sampling time of the two sensors. The discrete form of the spacecraft attitude kinematics equation obtained by the fourth-order longge stoke method is:
Figure BDA0001752121380000071
wherein, Cn+1A direction cosine matrix of the spacecraft at time n +1, CnIs a spacecraft direction cosine matrix at the time of n,
Figure BDA0001752121380000072
is a cross-product matrix of the attitude angular velocity vector at the moment n,
Figure BDA0001752121380000073
is a cross-product matrix of the attitude angular velocity vector at the midpoint time of n time and n +1 time,
Figure BDA0001752121380000074
is a cross-product matrix of the attitude angular velocity vector at the moment n + 1.
Figure BDA0001752121380000075
And
Figure BDA0001752121380000076
can be obtained by differentiating the angular displacement of the attitude measured by the sensor. Since attitude sensor sampling time T is constant, the time is the midpoint between n and n +1Attitude angular velocity
Figure BDA0001752121380000077
The sensors are not directly available, as used herein
Figure BDA0001752121380000078
Approximately, so there are:
Figure BDA0001752121380000079
wherein,
Figure BDA00017521213800000710
and the cross multiplication matrixes are respectively expressed by the three-axis attitude angular displacement vectors of the spacecraft measured by the attitude sensor at the moments of n-1, n and n + 1.
By substituting equation (13) into equation (12), a recursive form of the spacecraft attitude kinematics equation can be obtained:
Figure BDA00017521213800000711
in order to expand the measurement bandwidth of the attitude sensor, two sensors with different bandwidths can be used for data fusion, data of one sensor is used as a leading term, data obtained by the other sensor through a closed-loop controller is used as a correction term, a schematic block diagram of the attitude sensor is shown in fig. 1, a sensor 1 is used as the leading term, and a sensor 2 is used as the correction term.
For example, when the Angular Displacement Sensor (ADS) is selected to be fused with the Inertial Reference Unit (IRU) data, the ADS data is used as a dominant term, and the IRU data is used as a correction term. In the course of updating directional cosine matrix, order
Figure BDA0001752121380000081
Wherein,
Figure BDA0001752121380000082
are cross multiplication matrixes of spacecraft angular displacement vectors measured by the ADS sensor at the moments of n-1, n and n +1 respectively,
Figure BDA0001752121380000083
the correction terms of the angular displacement are respectively obtained after data measured by the IRU sensor at the time of n-1, n and n +1 passes through the closed-loop controller. The specific expression is as follows:
Figure BDA0001752121380000084
Figure BDA0001752121380000085
by substituting equation (15) into equation (14), a recursive algorithm of the attitude kinematics equation after data fusion can be obtained:
Figure BDA0001752121380000091
when the attitude angle of the spacecraft is calculated by the direction cosine matrix obtained after data fusion, because the problem of attitude jitter is concerned, the attitude jitter is very small, and the method can be suitable for small-angle approximation of the attitude angle. The attitude angle of the spacecraft can be directly extracted from the directional cosine matrix:
θ＝f(C)＝[θx θy θz]T (19)
wherein,
Figure BDA0001752121380000092
Figure BDA0001752121380000093
Figure BDA0001752121380000094
fig. 2 shows a block diagram of a computer simulation program constructed by simulink, in which the M code of the fcn module:
function[x,y,z,c]＝fcn(ads0,ads1,ads2,cor0,cor1,cor2,c0)
adsx0＝[0 -ads0(3) ads0(2)；ads0(3) 0 -ads0(1)；-ads0(2) ads0(1) 0]；
adsx1＝[0 -ads1(3) ads1(2)；ads1(3) 0 -ads1(1)；-ads1(2) ads1(1) 0]；
adsx2＝[0 -ads2(3) ads2(2)；ads2(3) 0 -ads2(1)；-ads2(2) ads2(1) 0]；
corx0＝[0 -cor0(3) cor0(2)；cor0(3) 0 -cor0(1)；-cor0(2) cor0(1) 0]；
corx1＝[0 -cor1(3) cor1(2)；cor1(3) 0 -cor1(1)；-cor1(2) cor1(1) 0]；
corx2＝[0 -cor2(3) cor2(2)；cor2(3) 0 -cor2(1)；-cor2(2) cor2(1) 0]；
cx0＝[c0(1) c0(2) c0(3)；c0(4) c0(5) c0(6)；c0(7) c0(8) c0(9)]；
fused0＝adsx0+corx0；
fused1＝adsx1+corx1；
fused2＝adsx2+corx2；
err1＝fused1-fused0；
err2＝fused2-fused1；
cx1＝(eye(3)-err1+((err1^2)/3)-((err1^3)/12)+(err2*err1/6)-(err2*(err1^2)/12)+(err2*(err1^3)/24))*cx0；
x＝0.5*(cx1(2,3)-cx1(3,2))；
y＝0.5*(cx1(3,1)-cx1(1,3))；
z＝0.5*(cx1(1,2)-cx1(2,1))；
c＝[cx1(1,1)；cx1(1,2)；cx1(1,3)；cx1(2,1)；cx1(2,2)；cx1(2,3)；cx1(3,1)；cx1(3,2)；cx1(3,3)]。
the invention provides a small inertia spacecraft attitude jitter determination method based on a fourth-order Runge Kutta method, aiming at the problem that a single sensor has measurement bandwidth limitation, a data fusion algorithm of two sensors is designed based on the fourth-order Runge Kutta method. The invention aims to expand the measurement bandwidth of the attitude jitter, thereby laying a foundation for inhibiting the attitude jitter. The method adopts a spacecraft attitude kinematics equation expressed by a directional cosine matrix, uses a four-order Runge Kutta method to carry out discretization, uses the measurement data of one sensor as a leading item and the data of the other sensor as a correction item, processes the discretized attitude kinematics equation, and obtains an attitude kinematics recurrence equation of a data fusion algorithm. The ADS sensor and the IRU sensor are taken as examples, and specific expression forms of the recurrence equation are given. And finally, verifying the effectiveness of the data fusion algorithm through simulink computer simulation design.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (1)

1. A small inertia spacecraft attitude jitter determination method based on a fourth-order Runge Kutta method is characterized by comprising the following steps:
s1, establishing an attitude kinematics equation of the spacecraft based on the direction cosine matrix;
s2, dispersing the attitude kinematics equation by using a fourth-order Runge Kutta method to obtain a recursion equation of a direction cosine matrix;
s3, correcting the recursion equation by using two sensors of different types, wherein the measurement data of one sensor is used as a leading term, and the sensor of the other sensor is used as a correction term;
wherein,
step S1 includes:
establishing an attitude kinematics equation of the spacecraft based on the direction cosine matrix as follows:
Figure FDA0002968168000000011
wherein C is a direction cosine matrix of the spacecraft,
Figure FDA0002968168000000012
is the first derivative of the directional cosine matrix with respect to time, ω×Is the angular velocity vector ω ═ ωx ωy ωz]TA cross-product matrix of;
step S2 includes:
a four-step Runge Kutta method is used to perform discrete approximation on the formula (1):
Figure FDA0002968168000000021
wherein, K1Is the slope at time n; k2Introducing K by an Euler method for the slope of the midpoint between the n moment and the n +1 moment1To obtain the value of (1); k3Also is the slope of the midpoint between the n time and the n +1 time, and introduces K by an Euler method2To obtain the value of (1); k4The slope at the time n + 1; t is sampling time of the sensors, and because two different types of sensors are involved in the data fusion algorithm, if the sampling time of the two different types of sensors is different, the T is taken as the least common multiple of the sampling time of the two different types of sensors;
step S2 includes:
the recursive equation of the direction cosine matrix obtained by the fourth-order Runge Kutta method is as follows:
Figure FDA0002968168000000022
wherein, Cn+1A direction cosine matrix of the spacecraft at time n +1, CnIs a spacecraft direction cosine matrix at the time of n,
Figure FDA0002968168000000023
is a cross-product matrix of the attitude angular velocity vector at the moment n,
Figure FDA0002968168000000024
is a cross-product matrix of the attitude angular velocity vector at the midpoint of the time n and the time n +1,
Figure FDA0002968168000000025
a cross-product matrix of the attitude angular velocity vector at the moment n + 1;
Figure FDA0002968168000000031
and
Figure FDA0002968168000000032
all the attitude angle displacement measured by the sensor is obtained by difference; since the sampling time of the sensor is T, the attitude angular velocity at the midpoint between n and n +1 times
Figure FDA0002968168000000033
The sensor can not be obtained and can only be used
Figure FDA0002968168000000034
Approximately, so there are:
Figure FDA0002968168000000035
wherein,
Figure FDA0002968168000000036
is a cross-multiplication matrix of the spacecraft attitude angular displacement vector measured by the sensor at the moment of n +1,
Figure FDA0002968168000000037
fork for spacecraft attitude angular displacement vector measured by sensor at n momentThe multiplication matrix is used to multiply the data,
Figure FDA0002968168000000038
a cross-multiplication matrix of the spacecraft attitude angular displacement vector measured by the sensor at the n-1 moment;
the formula (4) can be substituted for the formula (3):
Figure FDA0002968168000000039
step S3 includes:
when the angular displacement sensor and the data measured by the inertial reference unit are used for data fusion, the data measured by the angular displacement sensor is used as a leading term, the data measured by the inertial reference unit is used as a low-frequency correction term, and the following steps are performed:
Figure FDA00029681680000000310
wherein,
Figure FDA0002968168000000041
are cross multiplication matrixes of angular displacement vectors of the spacecraft measured by the angular displacement sensor at the moments of n-1, n and n +1 respectively,
Figure FDA0002968168000000042
the correction terms of the angular displacement are respectively obtained after data measured by the inertia reference unit at the moments of n-1, n and n +1 pass through the closed-loop controller;
the specific expression is as follows:
Figure FDA0002968168000000043
Figure FDA0002968168000000044
whereinADSθx、ADSθy、ADSθzThe angular displacement of a rolling shaft, a pitching shaft and a yawing shaft of the spacecraft is measured by an angular displacement sensor; whilecorθx，corθy，corθzThen it is the low frequency correction term produced by the closed loop controller;
by substituting equation (6) for equation (5), a recurrence equation after data fusion can be obtained:
Figure FDA0002968168000000045
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