CN110990943B - Singular point judgment method based on singular geometric meaning of control moment gyro group - Google Patents
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Abstract
The invention discloses a singular point judging method based on singular geometric meaning of a control moment gyro group, which comprises the following steps: step 1: respectively calculating a moment vector and an angular momentum vector according to the frame angle, the initial moment vector and the initial angular momentum vector; step 2: calculating a quasi singular vector according to the moment vector; step 3: calculating singular coefficients according to the quasi singular vectors and the moment vectors, and judging whether the frame angle is a singular point or not; step 4: if the frame angle is not the singular point, ending the judgment; if the frame angle is a singular point, calculating a judging coefficient, judging the type of the singular point, and ending the judgment. The method solves the problems of complex singular point judgment operation and low operation rate of the different control force rejection gyro groups, selects the coplanar state of any two moment gyroscopes in the control moment gyro groups to judge whether any frame angle of the different control moment gyro groups is a singular point, determines the types of the singular points, simplifies the judgment operation amount, and improves the accuracy and the reliability of a control system.
Description
Technical Field
The invention relates to the technical field of satellite control force rejection, in particular to a singular point judging method based on singular geometric meaning of a control moment gyro group.
Background
With the increasing complexity of satellite tasks, the requirements for satellite maneuver performance are also increasing. The control moment gyro group (SGCMG) can provide large and continuous control moment and has larger angular momentum space, so that the control moment gyro group is an ideal actuator of a large-angle fast maneuvering satellite.
However, when the control moment gyro group calculates the frame angle of the moment, a singular problem exists, so that the condition that the output moment is inconsistent with the command moment appears, and how to judge the existence of singular points in the control moment gyro group is very important. The singular points of the gyro groups with different moments are judged at present, so that the operation amount is large, the operation is complex, and the operation rate is low.
Disclosure of Invention
The invention aims to provide a singular point judging method based on singular geometric meaning of a control moment gyro group. The method aims to solve the problems of complex singular point judgment operation and low operation rate of the different control force rejection gyro groups, selects the coplanar state of any two moment gyroscopes in the control moment gyro groups to judge whether any frame angle of the different control moment gyro groups is a singular point, determines the types of the singular points, simplifies the judgment operation amount, and improves the accuracy and the reliability of a control system.
In order to achieve the above purpose, the invention provides a singular point judging method based on singular geometric meaning of a control moment gyro group, which comprises the following steps:
step 1: according to the frame angle, the initial moment vector and the initial angular momentum vector of each moment gyro in the SGCMG, respectively calculating the moment vector and the angular momentum vector of each moment gyro in the SGCMG;
step 2: calculating quasi singular vectors of the SGCMG according to moment vectors of the moment gyroscopes;
step 3: calculating singular coefficients of the SGCMG according to the quasi-singular vectors and the moment vectors of the moment gyroscopes, and judging whether the frame angles of the moment gyroscopes are singular points of the SGCMG according to the singular coefficients;
step 4: if the frame angle of each moment gyro is not the singular point of the SGCMG, ending the judgment; if the frame angle of each moment gyro is the singular point of the SGCMG, calculating a judging coefficient according to the angular momentum vector and the quasi singular vector of each moment gyro, judging the type of the singular point of the SGCMG according to the judging coefficient, and ending the judgment.
Most preferably, the angular momentum vector of each moment gyro is h, and the angular momentum vector of the ith moment gyro is h i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
h i ＝h i 0 cosδ i +c i 0 sinδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i Is the frame angle of the ith moment gyro.
Most preferably, the moment vector of each moment gyro is c, and the ith moment gyroThe moment vector of the screw is c i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
c i ＝-h i 0 sinδ i +c i 0 cosδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i Is the frame angle of the ith moment gyro.
Most preferably, the calculation of the quasi singular vector further comprises the steps of:
step 2.1: moment vectors of any two moment gyroscopes are selected from SGCMG, namely an e-th moment gyro and an f-th moment gyro c respectively e And c f ，e,f∈i，i＝1,2,3...n；
Step 2.2: moment gyro c according to e and f e And c f Calculating a quasi singular vector of the SGCMG; the quasi singular vector is u, and satisfies:
most preferably, the calculation of the singular coefficients further comprises the steps of:
step 3.1: selective removal of the e and f moment gyroscopes c from SGCMG e And c f Moment vectors of the rest moment gyroscopes except the moment vectors; the moment vector of the residual moment gyro is c i-e-f ,e,f∈i,i＝1,2,3...n；
Step 3.2: moment vector c based on quasi singular vector u and residual moment gyro i-e-f Calculating singular coefficients of the SGCMG; singular coefficient k i And satisfies:
k i ＝u·c i-e-f ，i＝1,2,3...n。
most preferably, the singular point is determined by determining the singular coefficient k i Is a numerical value of (2); when the singular coefficient k i When the moment gyroscopes are zero, the frame angles of the moment gyroscopes are singular points of the SGCMG; conversely, the frame angle of each moment gyro is not the singular point of the SGCMG。
Most preferably, the determination coefficient is P -1 And satisfies:
wherein h is i An angular momentum vector of the ith moment gyro; u is a quasi singular vector.
Most preferably, the determination of the singular point type further comprises the steps of:
step 4.1: from i rows and i columns -1 Extracting all diagonal elements from the diagonal array; the ith diagonal element is S i And satisfies:
S i ＝h i ·u，i＝1,2,3...n；
step 4.2: for i diagonal elements S i For the first judgment, if i diagonal elements S i Are all positive, i.e. S i The singular point type is saturated singular, and the judgment is ended; if i diagonal elements S i If there is a negative value in (i) diagonal elements S i Judging the number of negative values in the number of the negative values for the second time;
step 4.3: in the second judgment, if i diagonal elements S i There are three or more diagonal elements S i Negative, the singular point type is hyperbolic singular, and the judgment is ended; if i diagonal elements S i There are one or two diagonal elements S i Is negative and needs to be based on the angular momentum vector h of the ith moment gyro i Performing Gaussian calculation on the quasi-singular vector u to calculate Gaussian curvature lambda;
step 4.4: at i diagonal elements S i There is a diagonal element S i Under the condition of negative values, the type of the singular point is judged for the third time according to the positive and negative values of the Gaussian curvature lambda; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type is elliptical singular, and judging is finished; if the Gaussian curvature lambda is positive, namely lambda is more than 0, the singular point type is hyperbolic singular, and the judgment is ended;
step 4.5: among iDiagonal element S i There are two diagonal elements S i Under the condition of negative values, the type of the singular point is judged for the fourth time according to the positive and negative values of the Gaussian curvature lambda; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type is hyperbolic singular, and the judgment is ended; if the Gaussian curvature lambda is positive, namely lambda > 0, the method is also based on the Gaussian curvature lambda and the angular momentum vector h i Calculating the average curvature with the quasi singular vector u to calculate the average curvature v;
step 4.6: the type of the singular point is judged for the fifth time according to the positive and negative values of the average curvature v; if the average curvature v is a negative value, namely v is less than 0, the singular point type is hyperbolic singular, and the judgment is ended; if the average curvature v is positive, namely v is more than 0, the singular point type is elliptical singular, and the judgment is finished.
Most preferably, the gaussian calculation further comprises the steps of:
step 4.3.1: according to the angular momentum vector h of the ith moment gyro i And quasi singular vector u to calculate Gaussian coefficient; the Gaussian coefficient is q i And satisfies:
step 4.3.2: according to Gaussian coefficient q i Calculating Gaussian curvature lambda; the gaussian curvature λ satisfies:
wherein e, f are any two moment gyroscopes in the SGCMG, e, f e i, i=1, 2, 3..n respectively; g e And g f The frame axis vectors of any two moment gyroscopes in the SGCMG are respectively; and is also provided with
[g e g f u]＝g e ·(g f ×u)。
Most preferably, the average curvature v satisfies:
wherein q i Is a gaussian coefficient.
The invention solves the problems of complex singular point judgment operation and low operation rate of the different control force rejection gyro groups, selects the coplanar state of any two moment gyroscopes in the control moment gyro groups to judge whether any frame angle of the different control moment gyro groups is a singular point, determines the types of the singular points, simplifies the judgment operation amount and improves the precision and the reliability of a control system.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention designs the coplanar state of the moment of any two gyroscopes in the control moment gyro group to judge whether any frame angle of different control moment gyro groups is a singular point or not, determines the types of the singular points, provides a basis for the design of the control moment gyro group operation rate, ensures that all the singular points can be avoided as much as possible, and improves the accuracy and the reliability of a control system.
2. The method for quickly judging whether any frame angle of the different control moment gyro groups is the singular point of the control moment gyro groups is simple and reliable, has small operation amount and is easy to realize engineering.
Drawings
FIG. 1 is a flow chart of the singular point determination method provided by the invention;
fig. 2 is a schematic flow chart of singular point type judgment provided by the invention.
Detailed Description
The invention is further described by the following examples, which are given by way of illustration only and are not limiting of the scope of the invention.
The invention relates to a singular point judging method based on singular geometric meaning of a control moment gyro group (SGCMG), which is shown in figure 1 and comprises the following steps:
step 1: and respectively calculating the moment vector and the angular momentum vector of each moment gyro in the SGCMG according to the frame angle, the initial moment vector and the initial angular momentum vector of each moment gyro in the SGCMG.
The moment vector of each moment gyro is c, and the moment vector of the ith moment gyro is c i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
c i ＝-h i 0 sinδ i +c i 0 cosδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i Is the frame angle of the ith moment gyro.
The angular momentum vector of each moment gyro is h, and the angular momentum vector of the ith moment gyro is h i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
h i ＝h i 0 cosδ i +c i 0 sinδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i Is the frame angle of the ith moment gyro.
Step 2: according to the moment vector of the i moment gyroscopes, c i Calculating a quasi singular vector of the SGCMG; the calculation of the quasi-singular vectors further comprises the steps of:
step 2.1: moment vectors of any two moment gyroscopes are selected from SGCMG, namely an e-th moment gyro and an f-th moment gyro c respectively e And c f ，e,f∈i，i＝1,2,3...n；
Step 2.2: moment gyro c according to e and f e And c f Calculating a quasi singular vector of the SGCMG; the quasi singular vector is u, and satisfies:
step 3: according to the quasi singular vector u and the moment vectors of the i moment gyroscopes, the moment vector is c i Calculating singular coefficients of SGCMGk i The method comprises the steps of carrying out a first treatment on the surface of the Singular coefficient k i The calculation of (2) further comprises the steps of:
step 3.1: selective removal of the e and f moment gyroscopes c from SGCMG e And c f Moment vectors of the rest moment gyroscopes except the moment vectors; the moment vector of the residual moment gyro is c i-e-f Wherein e, f e i, i=1, 2, 3..n;
step 3.2: moment vector c based on quasi singular vector u and residual moment gyro i-e-f Calculating singular coefficients of the SGCMG; singular coefficient k i And satisfies:
k i ＝u·c i-e-f ，i＝1,2,3...n。
and according to singular coefficient k i Judging whether the frame angle of the ith moment gyro is a singular point of SGCMG or not by judging a singular coefficient k i Is a numerical value of (2); when the singular coefficient k i When the moment gyroscopes are zero, the frame angle of the ith moment gyroscope is the singular point of the SGCMG; conversely, the frame angle of the ith moment gyro is not the singular point of the SGCMG.
Step 4: if the frame angle of the ith moment gyro is not the singular point of the SGCMG, ending the judgment; if the frame angle of the ith moment gyro is the singular point of the SGCMG, according to the angular momentum vector h of the ith moment gyro i And quasi singular vector u to calculate determination coefficient P -1 The method comprises the steps of carrying out a first treatment on the surface of the The judgment coefficient is P -1 The method meets the following conditions:
as shown in FIG. 2, and according to the determination coefficient P -1 Judging the singular point type of the SGCMG, and ending the judgment; the judging of the singular point type further includes the steps of:
step 4.1: determining the coefficient P from i rows and i columns -1 Extracting all diagonal elements from the diagonal array; the ith diagonal element is S i And satisfies:
S i ＝h i ·u，i＝1,2,3...n。
step 4.2: for i diagonal elements S i Is of (3)Negative values are determined for the first time, if i diagonal elements S i Are all positive, i.e. S i The singular point type of the SGCMG is saturated singular, and the judgment is finished; if i diagonal elements S i If there is a negative value in (i) diagonal elements S i The number of negative values in the set is determined for the second time.
Step 4.3: in the second judgment, if i diagonal elements S i There are three or more diagonal elements S i The singular point type of the SGCMG is hyperbolic singular and the judgment is ended; if i diagonal elements S i There are one or two diagonal elements S i Is negative and needs to be based on the angular momentum vector h of the ith moment gyro i And performing Gaussian calculation on the quasi-singular vector u to calculate the Gaussian curvature lambda of the SGCMG.
Wherein the gaussian calculation further comprises the steps of:
step 4.3.1: according to the angular momentum vector h of the ith moment gyro i And quasi singular vector u to calculate Gaussian coefficient; the Gaussian coefficient is q i And satisfies:
step 4.3.2: according to Gaussian coefficient q i Calculating Gaussian curvature lambda; the gaussian curvature λ satisfies:
wherein e, f are any two moment gyroscopes in the SGCMG, respectively, and e, f e i, i=1, 2, 3..n; g e And g f The frame axis vectors of any two moment gyroscopes in the SGCMG are respectively; and is also provided with
[g e g f u]＝g e ·(g f ×u)。
Step 4.4: at i diagonal elements S i There is a diagonal element S i In the case of negative values, the singular point type of the SGCMG is entered according to the positive and negative values of the Gaussian curvature lambdaJudging for the third time; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type of the SGCMG is elliptical singular, and judging is finished; if the Gaussian curvature lambda is positive, namely lambda is more than 0, the singular point type of the SGCMG is hyperbolic singular, and the judgment is finished.
Step 4.5: at i diagonal elements S i There are two diagonal elements S i Under the condition of negative values, the singular point type of the SGCMG is judged for the fourth time according to the positive and negative values of the Gaussian curvature lambda; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type of the SGCMG is hyperbolic singular, and judging is finished; if the Gaussian curvature lambda is positive, i.e. lambda > 0, it is also necessary to calculate the Gaussian curvature lambda according to the Gaussian coefficient q i Calculating the average curvature, and calculating the average curvature v of the SGCMG, wherein the average curvature v meets the following conditions:
step 4.6: carrying out fifth judgment on the singular point type of the SGCMG according to the positive and negative values of the average curvature v; if the average curvature v is a negative value, namely v is less than 0, the singular point type of the SGCMG is hyperbolic singular, and the judgment is ended; if the average curvature v is positive, namely v is more than 0, the singular point type of the SGCMG is elliptical singular, and the judgment is finished.
The working principle of the invention is as follows:
according to the frame angle, the initial moment vector and the initial angular momentum vector of each moment gyro in the SGCMG, respectively calculating the moment vector and the angular momentum vector of each moment gyro in the SGCMG; calculating quasi singular vectors of the SGCMG according to moment vectors of the moment gyroscopes; calculating singular coefficients of the SGCMG according to the quasi-singular vectors and the moment vectors of the moment gyroscopes, and judging whether the frame angles of the moment gyroscopes are singular points of the SGCMG according to the singular coefficients; if the frame angle of each moment gyro is not the singular point of the SGCMG, ending the judgment; if the frame angle of each moment gyro is the singular point of the SGCMG, calculating a judging coefficient according to the angular momentum vector and the quasi singular vector of each moment gyro, judging the type of the singular point of the SGCMG according to the judging coefficient, and ending the judgment.
In summary, the singular point judging method based on the singular geometric meaning of the control moment gyro group solves the problems of complex singular point judging operation and low operation rate of the control moment gyro groups, selects any two moment gyroscopes in the control moment gyro group to judge whether any frame angle of the control moment gyro groups is singular point or not in a coplanar state, determines the types of the singular points, simplifies judging operation quantity, and improves the accuracy and reliability of a control system.
While the present invention has been described in detail through the foregoing description of the preferred embodiment, it should be understood that the foregoing description is not to be considered as limiting the invention. Many modifications and substitutions of the present invention will become apparent to those of ordinary skill in the art upon reading the foregoing. Accordingly, the scope of the invention should be limited only by the attached claims.
Claims (7)
1. The singular point judging method based on the singular geometric meaning of the control moment gyro group is characterized by comprising the following steps of:
step 1: according to the frame angle, the initial moment vector and the initial angular momentum vector of each moment gyro in the SGCMG, respectively calculating the moment vector and the angular momentum vector of each moment gyro in the SGCMG; the angular momentum vector of each moment gyro is h, and the angular momentum vector of the ith moment gyro is h i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
h i ＝h i 0 cosδ i +c i 0 sinδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i The frame angle of the ith moment gyro;
the moment vector of each moment gyro is c, and the moment vector of the ith moment gyro is c i I=1, 2, 3..n, n is the number of moment gyroscopes, and satisfies:
c i ＝-h i 0 sinδ i +c i 0 cosδ i
wherein c i 0 An initial moment vector h of the ith moment gyro i 0 Initial angular momentum vector, delta, for the ith moment gyro i The frame angle of the ith moment gyro;
step 2: calculating the quasi singular vector of the SGCMG according to the moment vector of each moment gyro;
the quasi singular vector calculation further comprises the following steps:
step 2.1: moment vectors of any two moment gyroscopes are selected from SGCMG, namely an e-th moment gyro and an f-th moment gyro c respectively e And c f ，e,f∈i，i＝1,2,3...n；
Step 2.2: according to the e and f moment gyro c e And c f Calculating a quasi singular vector of the SGCMG; the quasi singular vector is u, and satisfies:
step 3: calculating singular coefficients of the SGCMG according to the quasi-singular vectors and the moment vectors of the moment gyroscopes, and judging whether the frame angles of the moment gyroscopes are singular points of the SGCMG according to the singular coefficients;
step 4: if the frame angle of each moment gyro is not the singular point of the SGCMG, ending the judgment; if the frame angle of each moment gyro is the singular point of the SGCMG, calculating a judging coefficient according to the angular momentum vector of each moment gyro and the quasi singular vector, judging the type of the singular point of the SGCMG according to the judging coefficient, and ending the judgment.
2. The singular point determination method based on the singular geometric meaning of a control moment gyro group according to claim 1, wherein the calculation of the singular coefficients further comprises the steps of:
step 3.1: selective removal of e and e from SGCMGf moment gyroscopes c e And c f Moment vectors of the rest moment gyroscopes except the moment vectors; the moment vector of the residual moment gyro is c i-e-f ,e,f∈i,i＝1,2,3...n；
Step 3.2: moment vector c of the residual moment gyro according to the quasi singular vector u and the residual moment gyro i-e-f Calculating singular coefficients of the SGCMG; the singular coefficient is k i And satisfies:
k i ＝u·c i-e-f ，i＝1,2,3...n。
3. the singular point determination method based on the singular geometric meaning of a control moment gyro group according to claim 2, wherein the singular point is determined by determining a singular coefficient k i Is a numerical value of (2); when the singular coefficient k i When the moment gyroscopes are zero, the frame angles of the moment gyroscopes are singular points of the SGCMG; and on the contrary, the frame angle of each moment gyro is not the singular point of the SGCMG.
4. The singular point determination method based on the singular geometric meaning of the control moment gyro group as claimed in claim 2, wherein the determination coefficient is P -1 And satisfies:
wherein h is i An angular momentum vector for the ith moment gyro; u is the quasi singular vector.
5. The singular point determination method based on the singular geometric meaning of a control moment gyro group according to claim 4, wherein the determination of the type of the singular point further comprises the steps of:
step 4.1: from i rows and i columns -1 Extracting all diagonal elements from the diagonal array;
the ith diagonal element is S i And satisfies:
S i ＝h i ·u，i＝1,2,3...n；
step 4.2: for the i diagonal elements S i First judging the positive and negative values of the i diagonal elements S i Are all positive, i.e. S i > 0, wherein the singular point type is saturated singular, and the judgment is ended; if the i diagonal elements S i Negative values exist in (a), then for the i diagonal elements S i Judging the number of negative values in the number of the negative values for the second time;
step 4.3: in the second judgment, if i diagonal elements S i There are three or more diagonal elements S i Is a negative value, the type of the singular point is hyperbolic singular, and the judgment is ended; if i diagonal elements S i There are one or two diagonal elements S i Is negative and needs to be based on the angular momentum vector h of the ith moment gyro i Carrying out Gaussian calculation on the quasi singular vector u to calculate Gaussian curvature lambda;
step 4.4: at the i diagonal elements S i There is a diagonal element S i Under the condition of negative values, the singular point type is judged for the third time according to the positive and negative values of the Gaussian curvature lambda; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type is elliptical singular, and judgment is finished; if the Gaussian curvature lambda is positive, namely lambda is more than 0, the singular point type is hyperbolic singular, and judgment is finished;
step 4.5: at the i diagonal elements S i There are two diagonal elements S i In the case of a negative value, judging the type of the singular point for the fourth time according to the positive and negative values of the Gaussian curvature lambda; if the Gaussian curvature lambda is a negative value, namely lambda is less than 0, the singular point type is hyperbolic singular, and the judgment is ended; if the Gaussian curvature lambda is positive, namely lambda > 0, the angular momentum vector h is also required according to the Gaussian curvature lambda i Calculating the average curvature of the quasi singular vector u, and calculating the average curvature v;
step 4.6: carrying out fifth judgment on the singular point type according to the positive and negative values of the average curvature v; if the average curvature v is a negative value, namely v is less than 0, the singular point type is hyperbolic singular, and the judgment is ended; and if the average curvature v is a positive value, namely v is more than 0, the singular point type is elliptical singular, and the judgment is finished.
6. The singular point determination method based on the singular geometric meaning of a control moment gyro group according to claim 5, wherein the gaussian calculation further comprises the steps of:
step 4.3.1: according to the angular momentum vector h of the ith moment gyro i And the quasi singular vector u calculates a Gaussian coefficient; the Gaussian coefficient is q i And satisfies:
step 4.3.2: according to the Gaussian coefficient q i Calculating Gaussian curvature lambda; the gaussian curvature λ satisfies:
wherein e, f are any two moment gyroscopes in the SGCMG, e, f e i, i=1, 2, 3..n respectively; g e And g f The frame axis vectors of any two moment gyroscopes in the SGCMG are respectively; and is also provided with
[g e g f u]＝g e ·(g f ×u)。
7. The singular point determination method based on the singular geometric meaning of a control moment gyro group according to claim 6, wherein the average curvature v satisfies:
wherein q i Is the gaussian coefficient.
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