CN111942619B - Angular momentum distribution method of redundant flywheel combination based on modified pseudo-inverse matrix - Google Patents

Angular momentum distribution method of redundant flywheel combination based on modified pseudo-inverse matrix Download PDF

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CN111942619B
CN111942619B CN202010928003.7A CN202010928003A CN111942619B CN 111942619 B CN111942619 B CN 111942619B CN 202010928003 A CN202010928003 A CN 202010928003A CN 111942619 B CN111942619 B CN 111942619B
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刘刚
张文政
张家巍
陈殿印
张泽涛
尹海宁
叶立军
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Shanghai Aerospace Control Technology Institute
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    • B—PERFORMING OPERATIONS; TRANSPORTING
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    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
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    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
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Abstract

The invention relates to a redundant flywheel combination angular momentum distribution method based on a modified pseudo-inverse matrix, which comprises the following steps: s1, calculating a reference driving torque of each flywheel in the redundant flywheel combination according to the triaxial instruction torque based on a pseudo-inverse matrix and a zero-motion flywheel angular momentum distribution algorithm; s2, constructing a saturation variable according to the reference driving moment of each flywheel, the current angular momentum of each flywheel and the upper limit of the angular momentum of each flywheel, and evaluating the degree of each flywheel approaching the saturation of the angular momentum; calculating a performance index correction coefficient by using the saturation variable, and obtaining a corrected pseudo-inverse matrix by using the saturation variable and the performance index correction coefficient; and S3, calculating the driving torque of each flywheel in the redundant flywheel combination according to the corrected pseudo-inverse matrix. The invention can effectively solve the problem that the utilization rate of the angular momentum of a single flywheel is obviously reduced after the redundant flywheel combination introduces zero motion, and can obviously prolong the output time of given torque.

Description

Angular momentum distribution method of redundant flywheel combination based on modified pseudo-inverse matrix
Technical Field
The invention relates to an angular momentum distribution method of a redundant flywheel combination, in particular to an angular momentum distribution method of a redundant flywheel combination based on a modified pseudo-inverse matrix.
Background
At present, a reaction flywheel outputting torque based on the principle of angular momentum exchange is the most used attitude control actuating mechanism on a three-axis attitude stabilization satellite. However, modern space missions put higher demands on the service life of satellites, so in order to improve the reliability of the system, more than 3 flywheels are usually installed on the satellites to form a redundant flywheel combination as an actuating mechanism of the attitude control system. Because the number of the flywheels is larger than the dimension of the output torque, the redundant flywheel combination can bias the flywheel rotating speed to be close to the optimal working rotating speed according to instructions while outputting the given torque. At present, the redundant flywheel combination adopts a mode of adding zero motion in a pseudo-inverse distribution algorithm to change the central working rotating speed of the flywheel, for example, domestic patent 'an autonomous management method of redundant flywheel angular momentum' introduces how to calculate the zero motion of the redundant flywheel combination to realize rotating speed offset of the flywheel.
The introduction of zero motion can change the central operating speed of the flywheel assembly, but because there is an upper limit to the angular momentum of each flywheel assembly, the result of this will tend to reduce the angular momentum envelope of the flywheel assembly. Taking the simplest two flywheels outputting one-dimensional torque as an example, suppose that the output torque of a single flywheel is 0.2Nm, and the upper limit of angular momentum is 25 Nms. When the flywheel combination is required to output a constant torque of 0.2Nm in the positive direction, if the angular momentum of each flywheel is 0Nms in the initial state, 250s can be continuously output under the allocation algorithm of pseudo-inverse plus zero motion. However, if the flywheel is initially biased at plus or minus 15Nms angular momentum due to zero motion, only 100s can be continuously output. In both cases, the initial synthetic angular momentum of the flywheel combination is 0Nms, but the time for outputting equivalent moment is obviously reduced under the condition of zero motion offset, which indicates that when zero motion is matched with a pseudo-inverse matrix to carry out redundant flywheel angular momentum distribution, if the initial angular momentum of a single flywheel is not zero, the utilization rate of the flywheel angular momentum is reduced, and the flywheel combination cannot output given moment under the condition that the angular momentum in the direction of outputting the moment is still remained.
Based on the above, it is desirable to provide an angular momentum allocation method for a redundant flywheel assembly based on a modified pseudo-inverse matrix, which can modify the pseudo-inverse matrix in real time according to the current state of the redundant flywheel assembly, so as to suppress the influence of zero motion on the utilization rate of angular momentum of a single flywheel, thereby effectively solving the disadvantages and limitations existing in the prior art.
Disclosure of Invention
The invention aims to provide a method for distributing angular momentum of a redundant flywheel combination based on a modified pseudo-inverse matrix, which can effectively solve the problem that the utilization rate of the angular momentum of a single flywheel is obviously reduced after zero motion is introduced into the redundant flywheel combination, and can obviously prolong the output time of given torque.
In order to achieve the above object, the present invention provides a method for distributing angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix, comprising the following steps:
s1, calculating a reference driving torque of each flywheel in the redundant flywheel combination according to the triaxial instruction torque based on a pseudo-inverse matrix and a zero-motion flywheel angular momentum distribution algorithm;
s2, correcting the pseudo-inverse matrix according to the current angular momentum of each flywheel in the redundant flywheel combination and the reference driving torque of each flywheel in the redundant flywheel combination; comprises the following steps:
s21, constructing a saturation variable according to the reference driving moment of each flywheel, the current angular momentum of each flywheel and the upper limit of the angular momentum of each flywheel, and evaluating the degree of each flywheel approaching the saturation of the angular momentum;
s22, calculating a performance index correction coefficient by using the saturation variable, and obtaining a corrected pseudo-inverse matrix by using the saturation variable and the performance index correction coefficient;
and S3, calculating the driving torque of each flywheel in the redundant flywheel combination according to the corrected pseudo-inverse matrix.
The step S1 specifically includes the following steps:
s11, in a control period TsAnd decomposing the angular momentum increment delta h of each flywheel in the redundant flywheel combination to obtain the angular momentum increment delta h of the output torque of each flywheelTAnd angular momentum increment of zero motion of each flywheel Δ hz；
S12 passing through pseudo inverse matrix CT(CCT)-1Calculating the angular momentum gain of the output torque of each flywheelQuantity Δ hT：
ΔhT＝CT(CCT)-1MTs；
Wherein C is a flywheel mounting matrix; m is a control moment which needs to be applied in the current control period and is given by the attitude controller; t issIs the control period of the attitude controller;
s13, calculating the angular momentum increment delta h of the zero motion of each flywheelz；
S14, calculating reference driving moment m of each flywheel*：
Figure BDA0002669141040000031
Wherein the content of the first and second substances,
Figure BDA0002669141040000032
respectively, the reference driving torque of each flywheel.
In the step S21, a saturation variable alpha is constructediThe method comprises the following steps:
Figure BDA0002669141040000033
wherein alpha isiThe saturation variable of the ith flywheel represents the degree of the ith flywheel approaching the angular momentum saturation; exp (…) is an exponential function; min (…) is a minimum function; sign (…) is a sign function; h isimaxRepresenting the maximum angular momentum of the ith flywheel;
Figure BDA0002669141040000034
representing a reference driving torque of the ith flywheel; h ismiRepresenting the current angular momentum of the ith flywheel; n is the number of flywheels in the redundant flywheel combination.
The step S22 specifically includes the following steps:
s221, according to a saturation variable alphaiCalculating a performance index correction coefficient eta:
Figure BDA0002669141040000035
s222, according to the saturation variable alphaiAnd a performance index correction coefficient eta, constructing a performance index weighting coefficient matrix R:
Figure BDA0002669141040000036
Figure BDA0002669141040000037
s223, according to the performance index weighting coefficient matrix R, carrying out comparison on the pseudo-inverse matrix CT(CCT)-1Correcting to obtain a corrected pseudo-inverse matrix RCT(CRCT)-1。
The step S3 specifically includes the following steps:
s31, correcting the pseudo-inverse matrix RC according to the correctionT(CRCT)-1Calculating the angular momentum increment of the corrected output torque of each flywheel
Figure BDA0002669141040000038
Figure BDA0002669141040000039
S32, increasing angular momentum according to the corrected output torque of each flywheel
Figure BDA0002669141040000041
Calculating the driving moment m of each flywheel based on a flywheel angular momentum distribution algorithm combining a pseudo-inverse matrix and zero motion:
Figure BDA0002669141040000042
wherein m ═[m1 m2 ... mn]TAnd m is1，m2，...mnThe driving torque of each flywheel is respectively.
In summary, compared with the prior art, the angular momentum allocation method of the redundant flywheel combination based on the modified pseudo-inverse matrix provided by the invention has the following advantages and beneficial effects:
1. the invention can effectively solve the problem that the utilization rate of the angular momentum of a single flywheel is obviously reduced after the redundant flywheel combination introduces zero motion, and can obviously prolong the output time of given torque;
2. the method does not need repeated iteration optimization, has small calculated amount, occupies less satellite resources and is easy to be applied practically;
3. the method is provided in an analytic form, can perform calculation in real time, and does not have the problem of unstable numerical values.
Drawings
FIG. 1 is a schematic diagram of the output torque of a redundant flywheel assembly without the use of a modified pseudo-inverse matrix according to the present invention;
FIG. 2 is a schematic diagram of the output torque of the redundant flywheel assembly after the pseudo-inverse matrix correction is used in the present invention;
FIG. 3 is a schematic diagram of the angular momentum of each flywheel of the redundant flywheel assembly of the present invention without the use of a modified pseudo-inverse matrix;
FIG. 4 is a schematic diagram of the angular momentum of each flywheel of the redundant flywheel assembly after the pseudo-inverse matrix correction is used in the present invention;
FIG. 5 is a flow chart of the angular momentum distribution method of the redundant flywheel assembly based on the modified pseudo-inverse matrix according to the present invention.
Detailed Description
The technical contents, construction features, achieved objects and effects of the present invention will be described in detail by preferred embodiments with reference to fig. 1 to 5.
First, in the present invention, the capital-plus-bold vector is used as the combined angular momentum or output/drive torque of the redundant flywheel assembly, and the small-plus-bold vector is used as the angular momentum or output/drive torque of each individual flywheel in the redundant flywheel assembly.
As shown in fig. 5, the method for distributing angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix provided by the present invention comprises the following steps:
s1, calculating a reference driving torque of each flywheel in the redundant flywheel combination according to a three-axis command torque based on a pseudo-inverse matrix and zero-motion flywheel angular momentum distribution algorithm in the prior art;
s2, correcting the pseudo-inverse matrix according to the current angular momentum of each flywheel in the redundant flywheel combination and the reference driving torque of each flywheel in the redundant flywheel combination calculated in S1, so that the angular momentum utilization rate of each flywheel is improved;
and S3, calculating the driving torque of each flywheel in the redundant flywheel combination according to the corrected pseudo-inverse matrix.
The step S1 specifically includes the following steps:
s11, decomposing the composite angular momentum increment of the redundant flywheel combination;
in a control period TsComposite angular momentum increment of internal and redundant flywheel combination
Figure BDA0002669141040000051
Is composed of two parts, namely:
ΔH＝ΔHT+ΔHz
wherein the content of the first and second substances,
Figure BDA0002669141040000052
an incremental angular momentum representing the output torque of the redundant flywheel combination,
Figure BDA0002669141040000053
Figure BDA0002669141040000054
an angular momentum increment representing zero motion of the redundant flywheel assembly;
Figure BDA0002669141040000055
express a plurality ofA real number matrix space is defined, wherein the upper right-hand number 3 x 1 indicates the number of rows and columns of the matrix (hereinafter referred to as "column" or "column")
Figure BDA0002669141040000056
All representing a multidimensional real matrix space);
in a control period TsIn the embodiment, the angular momentum increment Δ h of each flywheel in the redundant flywheel combination is composed of two parts, namely:
CΔh＝CΔhT+CΔhz
wherein C represents a flywheel mounting matrix; Δ hTAn angular momentum increment representing an output torque of each flywheel; Δ hzAn angular momentum increment representing zero motion of each flywheel;
it should be noted that Δ h is an n-dimensional vector composed of elements of angular momentum increment of each flywheel; Δ hTThe angular momentum increment of the output torque of each flywheel is used as an n-dimensional vector consisting of elements; Δ hzAn n-dimensional vector composed of elements of the angular momentum increment of zero motion of each flywheel; n represents the number of flywheels in the redundant flywheel combination;
s12, calculating the angular momentum increment delta h of the output torque of each flywheelT；
Angular momentum increment Delta H of output torque according to redundant flywheel combinationTBy means of a pseudo-inverse matrix CT(CCT)-1And calculating to obtain:
ΔhT＝CT(CCT)-1ΔHT＝CT(CCT)-1MTs
wherein M is a control moment which needs to be applied in the current control period and is given by the attitude controller; t issIs the control period of the attitude controller;
s13, calculating the angular momentum increment delta h of the zero motion of each flywheelz；
Let the central angular momentum of each flywheel be hzSame, hzAn n-dimensional vector composed of elements of the central angular momentum of each flywheel; order:
hz＝be
Ce＝0
b is an artificially set scalar, engineering experience is selected to be relied on, and the design principle is that each flywheel works near the ideal rotating speed to the greatest extent according to the change and accumulation conditions of the interference torque of the satellite in one orbit period; e is a zero-space base solution of the flywheel mounting matrix C, and the design principle is that the sizes of all elements in e are as close as possible;
angular momentum increment of zero motion Δ h of each flywheelzThe method can be obtained by solving the following quadratic optimization problem with equality constraint by adopting a QR decomposition-based method:
Figure BDA0002669141040000061
S.t.CΔhz＝0
wherein h ism＝[hm1 hm2 ... hmn]TFor the current angular momentum of the respective flywheel, hm1，hm2，...hmnRespectively representing the current angular momentum of each flywheel;
solving to obtain:
Figure BDA0002669141040000062
where K is a control gain matrix; q2Can be obtained by the following method;
carrying out QR decomposition on the flywheel installation matrix C:
Figure BDA0002669141040000063
q is a unitary matrix obtained by decomposition, and R is an upper triangular matrix; while the unitary matrix Q can be written in the form of a block matrix as follows:
Figure BDA0002669141040000071
wherein Q is1And Q2Are all matrixes with the number of columns being n/2;
suppose that
Figure BDA0002669141040000072
The control gain matrix K is expressed in the form of an equal-proportion amplitude limit, i.e.
Figure BDA0002669141040000073
Wherein,. DELTA.hmaxIs the upper limit of the angular momentum zero motion of a single flywheel in a control period; k1,...,KnIs zero motion control gain; diag (…) represents a diagonal matrix made up of bracketed elements; max (…) is the maximum function;
the amplitude limiting processing is carried out in order to prevent the single flywheel from outputting torque normally due to too large zero motion of the single flywheel in consideration of the upper limit of the output torque of the single flywheel.
S14 angular momentum increment delta h according to output torque of each flywheelTAnd the angular momentum increment Δ h of zero motion of each flywheelzCalculating a reference driving torque of each flywheel; the method specifically comprises the following steps:
Figure BDA0002669141040000074
m*＝ΔhT/Ts+Δhz/Ts
wherein m is*Is a reference drive torque of each flywheel, and m*Is a vector composed of elements of the reference driving moment of each flywheel;
Figure BDA0002669141040000075
respectively, the reference driving torque of each flywheel.
S2 is the core step of the present invention, and S2 specifically includes the following steps:
s21, on the basis of the reference driving torque of each flywheel, combining the current angular momentum of each flywheel and the upper limit of the angular momentum of each flywheel, and constructing a saturation variable alpha to evaluate the degree of the each flywheel approaching the angular momentum saturation;
let the maximum angular momentum of each flywheel be hmaxAnd h ismaxThe vector composed of the maximum angular momentum of each flywheel as an element includes:
hmax＝[h1max h2max ... hnmax]T
wherein h is1max，h2max，...hnmaxRespectively the maximum angular momentum of each flywheel;
reference drive torque based on ith flywheel
Figure BDA0002669141040000081
And the current angular momentum hmiTaking a saturation variable alphaiRepresenting the proximity of the ith flywheel to the saturation of angular momentum, alphaiThe larger the flywheel is, the closer the ith flywheel is to the angular momentum saturation; wherein, i is 1.. and n is the number of flywheels in the redundant flywheel combination;
the saturation variable α is calculated using the following formulai：
Figure BDA0002669141040000082
Wherein exp (…) is an exponential function; min (…) is a minimum function; sign (…) is a sign function; h isimaxRepresenting the maximum angular momentum of the ith flywheel;
s22, calculating a performance index correction coefficient eta by using the saturation variable alpha, and obtaining a corrected pseudo-inverse matrix by using the saturation variable alpha and the performance index correction coefficient eta;
calculating angular momentum increment delta h of output torque of each flywheelTThe pseudo-inverse matrix of (a) is obtained by solving the following optimal problem:
Figure BDA0002669141040000083
S.t.CΔhT＝MTs
according to the saturation variable alpha and the performance index correction coefficient eta, correcting the optimal problem into the following steps:
Figure BDA0002669141040000084
S.t.CΔhT＝MTs
wherein, R is a performance index weighting coefficient matrix; and has the following components:
Figure BDA0002669141040000085
Figure BDA0002669141040000086
Figure BDA0002669141040000087
according to the performance index weighting coefficient matrix R, the pseudo inverse matrix C is subjected toT(CCT)-1Correcting to obtain a corrected pseudo-inverse matrix RCT(CRCT)-1。
The step S3 specifically includes the following steps:
s31, calculating the angular momentum increment of the output torque of each flywheel after correction according to the pseudo-inverse matrix after correction
Figure BDA0002669141040000091
Figure BDA0002669141040000092
Wherein M is the current given by the attitude controllerControl torque to be applied in a control cycle; t issIs the control period of the attitude controller;
s32, increasing angular momentum according to the corrected output torque of each flywheel
Figure BDA0002669141040000093
Calculating the driving moment m of each flywheel based on a flywheel angular momentum distribution algorithm combining a pseudo-inverse matrix and zero motion:
Figure BDA0002669141040000094
wherein m ═ m1 m2 ... mn]TAnd m is1，m2，...mnThe driving torque of each flywheel is respectively.
The effectiveness of the angular momentum distribution method of the redundant flywheel combination based on the modified pseudo-inverse matrix can be verified through simulation. As shown in fig. 1 to 4, 4 flywheels are selected to form a redundant flywheel combination, and the upper limit of angular momentum of a single flywheel is 25 Nms. The angular momentum of each flywheel is first biased around 7.5Nms by zero motion and starts to output 0.1Nm of torque in the X direction continuously at 500 s. It can be seen that for the angular momentum distribution method without using the modified pseudo-inverse matrix, there is a flywheel with angular momentum saturation at 690 s. In the angular momentum distribution method using the modified pseudo-inverse matrix, the angular momentum saturation of a single flywheel does not occur until 760 s. It is therefore apparent that the method of the present invention can significantly extend the output time for a given torque.
In conclusion, the angular momentum distribution method of the redundant flywheel combination based on the modified pseudo-inverse matrix, provided by the invention, can effectively solve the problem that the utilization rate of the angular momentum of a single flywheel is obviously reduced after zero motion is introduced into the redundant flywheel combination, and can increase the range of the angular momentum envelope of the redundant flywheel combination after the zero motion is used.
While the present invention has been described in detail with reference to the preferred embodiments, it should be understood that the above description should not be taken as limiting the invention. Various modifications and alterations to this invention will become apparent to those skilled in the art upon reading the foregoing description. Accordingly, the scope of the invention should be determined from the following claims.

Claims (5)

1. A redundant flywheel combination angular momentum distribution method based on a modified pseudo-inverse matrix is characterized by comprising the following steps:
s1, calculating a reference driving torque of each flywheel in the redundant flywheel combination according to the triaxial instruction torque based on a pseudo-inverse matrix and a zero-motion flywheel angular momentum distribution algorithm;
s2, correcting the pseudo-inverse matrix according to the current angular momentum of each flywheel in the redundant flywheel combination and the reference driving torque of each flywheel in the redundant flywheel combination; comprises the following steps:
s21, constructing a saturation variable according to the reference driving moment of each flywheel, the current angular momentum of each flywheel and the upper limit of the angular momentum of each flywheel, and evaluating the degree of each flywheel approaching the saturation of the angular momentum;
s22, calculating a performance index correction coefficient by using the saturation variable, and obtaining a corrected pseudo-inverse matrix by using the saturation variable and the performance index correction coefficient;
and S3, calculating the driving torque of each flywheel in the redundant flywheel combination according to the corrected pseudo-inverse matrix.
2. The method for allocating angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix as claimed in claim 1, wherein said step S1 comprises the following steps:
s11, in a control period TsAnd decomposing the angular momentum increment delta h of each flywheel in the redundant flywheel combination to obtain the angular momentum increment delta h of the output torque of each flywheelTAnd angular momentum increment of zero motion of each flywheel Δ hz；
S12 passing through pseudo inverse matrix CT(CCT)-1Calculating each flywheelAngular momentum increment of output torque Δ hT：
ΔhT＝CT(CCT)-1MTs；
Wherein C is a flywheel mounting matrix; m is a control moment which needs to be applied in the current control period and is given by the attitude controller; t issIs the control period of the attitude controller;
s13, calculating the angular momentum increment delta h of the zero motion of each flywheelz；
S14, calculating reference driving moment m of each flywheel*：
Figure FDA0002669141030000011
Wherein the content of the first and second substances,
Figure FDA0002669141030000012
respectively, the reference driving torque of each flywheel.
3. The method for allocating angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix as claimed in claim 2, wherein in S21, a saturation variable α is constructediThe method comprises the following steps:
Figure FDA0002669141030000021
wherein alpha isiThe saturation variable of the ith flywheel represents the degree of the ith flywheel approaching the angular momentum saturation; exp (…) is an exponential function; min (…) is a minimum function; sign (…) is a sign function; h isimaxRepresenting the maximum angular momentum of the ith flywheel;
Figure FDA0002669141030000022
representing a reference driving torque of the ith flywheel; h ismiRepresenting the current angular momentum of the ith flywheel; n is the number of flywheels in the redundant flywheel combination.
4. The method for allocating angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix as claimed in claim 3, wherein the step of S22 comprises the following steps:
s221, according to a saturation variable alphaiCalculating a performance index correction coefficient eta:
Figure FDA0002669141030000023
s222, according to the saturation variable alphaiAnd a performance index correction coefficient eta, constructing a performance index weighting coefficient matrix R:
Figure FDA0002669141030000024
Figure FDA0002669141030000025
s223, according to the performance index weighting coefficient matrix R, carrying out comparison on the pseudo-inverse matrix CT(CCT)-1Correcting to obtain a corrected pseudo-inverse matrix RCT(CRCT)-1。
5. The method for allocating angular momentum of a redundant flywheel assembly based on a modified pseudo-inverse matrix as claimed in claim 4, wherein said step of S3 comprises the following steps:
s31, correcting the pseudo-inverse matrix RC according to the correctionT(CRCT)-1Calculating the angular momentum increment of the corrected output torque of each flywheel
Figure FDA0002669141030000026
Figure FDA0002669141030000027
S32, increasing angular momentum according to the corrected output torque of each flywheel
Figure FDA0002669141030000028
Calculating the driving moment m of each flywheel based on a flywheel angular momentum distribution algorithm combining a pseudo-inverse matrix and zero motion:
Figure FDA0002669141030000031
wherein m ═ m1 m2...mn]TAnd m is1，m2，...mnThe driving torque of each flywheel is respectively.
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