CN101891018A - Single frame control moment gyro control method based on moment output capability optimization - Google Patents

Abstract

Based on the single-gimbal control momentum gyro method of operating that torque output capability is optimal, it is related to quick satellite gravity anomaly field, it solves existing single-gimbal control momentum gyro manipulation rule cannot effective singularity avoidance, and the problem of fugitive odd different time can bring biggish torque error, the technology specifically comprises the steps of: satellite attitude control system obtains CMG system according to current posture information and exports current expectation moment Tc; Calculate the optimal frame angular speed of CMG system output expectation moment Tc ability

; According to desired moment Tc and optimal frame angular speed

, obtain the frame rotary speed instruction of CMG system

; Realize the gesture stability to satellite. The method of the invention efficiently avoids singular problem, while reducing output torque error, to improve the attitude control accuracy of satellite.

Description

Single frame control moment gyroscope method of operating based on moment fan-out capability optimum

Technical field

The present invention relates to quick satellite attitude control field, be specifically related to single frame control moment gyroscope group's the high precision moment output and the method for operating of effective unusual avoidance.

Background technology

Along with the development of aerospace industry, people are more and more higher to the accuracy requirement of taking pictures over the ground of following earth observation satellite.The high-precision resolution of taking pictures over the ground means less relatively field aperture angle, just require satellite to have the fast speed maneuverability and want to realize the sensitizing range taken pictures on a large scale, also just require satellite attitude control system that enough big control torque can be provided.And the attitude control actuator that big moment can be provided up to now mainly contains two kinds of control moment gyroscope (Control Moment Gyros is hereinafter to be referred as CMG) and jet thrust devices.Compare with the jet thrust device, CMG had both had the moment amplifying power, and bigger control torque can be provided, again can be accurately output torque continuously, and consume fuel is not polluted optical device and flight environment of vehicle.In addition, control moment gyroscope is compared with the speed change reaction wheel because the rotor constant speed is rotated, and will suppress the small chatter of celestial body, and this will help to improve the pointing accuracy and the lasting accuracy of celestial body.Therefore, for the quick satellite of fast reserve, CMG is first-selected attitude control actuator.The WorldView-1 satellite of U.S.'s emission in 2007, the WorldView-2 satellite of emission in 2009 and France are just planning all to adopt CMG as the attitude actuating unit in the Pleiades-HR satellite of emission in the end of the year 2010.Wherein, the Pleiades satellite can be in 25s along the axis of rolling motor-driven 60 °, and pointing accuracy can reach ± 0.03 °, and this is inconceivable for traditional employing counteraction flyback as the satellite of attitude actuating unit.

Though CMG has above plurality of advantages, but the CMG system exists serious singular problem, make it can't produce expectation moment on some frame corners configuration, this has brought very big difficulty to research and the design that CMG handles rule, has also further influenced CMG in spaceborne application.So-called unusual being meant: on some frame corners configuration, no matter how whole C MG system moves, and all can't produce along the moment on a certain direction, and this direction is called unusual direction, and this moment, whole attitude control system was lost the three-axis attitude control ability.The task that control moment gyroscope is handled rule is according to current frame corners of CMG system and the expectation control torque that is provided by control system, the frame corners velocity magnitude of each control moment gyroscope of reasonable distribution, make that the moment of total system output is consistent with the expectation moment of control system, avoid unusual state simultaneously effectively.Therefore, adopt the design of the satellite attitude control system of CMG at first must design the manipulation rule that to avoid unusual state, accurate output expectation moment effectively.

Since three more than ten years, people have carried out big quantity research at singular problem how to avoid the CMG system.1978, the geometrical property and zero problems such as structure of moving of CMG system singular surface that Margulies and Aubrun have adopted differential geometry and topological method labor.Yet the SGCMG internal system exists the non-existent oval singular surface of zero motion, makes zero motion algorithm can't effectively avoid unusual.Nineteen ninety, Bedrossian has pointed out that the disengaging singular surface must satisfy two conditions: at first, zero motion exists at this singular surface place; Secondly,, can change the Jacobian rank of matrix, and utilize the Binet-Cauchy theorem to provide to judge the sufficient condition of separating existence of these two problems by zero motion.The unusual robust manipulation rule that Bedrossian also is applied in Nakamura in the mechanical arm manipulation is applied in the SGCMG manipulation.This manipulation rule can be fled from some singular surface with the cost of bringing the moment error, but when expectation moment is parallel with unusual direction, the frame corners speed that unusual robust is handled rule output is 0, total system rests on the singular surface place all the time, can't flee from singular surface, this is referred to as " framework locking " phenomenon." framework locking " phenomenon is having a strong impact on the application that unusual robust is handled rule.1991, Paradiso proposed and can the overall situation avoid the unusual overall unusual manipulation rule of avoiding based on the heuristic intelligent search algorithm of A star.This handles rule according to expectation moment of momentum track, realizes and can the overall situation avoid unusual optimum frame corners track in each node choose reasonable zero motion size.Yet, the overall situation handle rule all to carry out at each node place zero motion select with trajectory predictions relatively, calculate extremely complicated, only suitable calculated off-line, controller performance can't be guaranteed in real time.2000, Ford proposed unusual direction according to the svd theory and has avoided handling rule.This manipulation rule is handled rule with the robust pseudoinverse and is compared, and only having introduced the moment error in one direction escapes singular surface, has improved control accuracy, and still " framework locking " problem is not effectively solved yet.Calendar year 2001, Wie handles at unusual robust and has proposed broad sense robust manipulation rule on the basis of restraining.The broad sense robust is handled rule and is passed through the internal system singular surface smoothly by introducing non-diagonal angle weighting matrix item, effectively avoids " framework locking " problem simultaneously, but to saturated unusual can't the passing through in outside.2005, Wie proposed the unusual robust manipulation in non-diagonal angle rule based on least square fong theory and perturbation matrix theory.This handles rule near singular surface, can produce the ever-increasing periodic disturbance signal of amplitude, is cost to introduce the moment error, and it is saturated unusual not only can to flee from the outside, also can flee from smoothly the oval singular surface in inside.Yet the unusual robust in non-diagonal angle is handled rule can introduce bigger moment error when the escape singular surface, and attitude control accuracy can't be guaranteed, and requires very high to the robust performance of control system.

In a word, above mentioned manipulation rule always exists the some shortcomings part, or can not effectively avoid unusual, or the fugitive odd different time can bring very large moment error.Therefore, how designing and a kind ofly can effectively avoid unusually, can more accurately export the manipulation rule of expectation moment again, is the theme of current control moment gyroscope application facet.

Flywheel that single frame CMG unit is rotated by constant speed and support flying wheel and rotary framework are formed; When framework drives the constant speed flywheel with cireular frequency

During rotation, CMG moment of momentum direction changes, and will produce moment of precession τ, can be expressed as:

$\mathrm{\τ}=-\stackrel{\·}{h}=-\stackrel{\·}{\mathrm{\δ}}\×h=(h\×g)\stackrel{\·}{\mathrm{\δ}}---\left(1\right)$

Wherein, g is the gimbal axis direction vector,

Be the frame corners velocity vector, h is a CMG moment of momentum vector.As can be seen, along with the rotation of framework, output torque τ is vertical with h with g all the time, and forms a plane in the space, as shown in Figure 1.

Because the output torque of a CMG is positioned at a plane, and, need three CMG unit at least in order to realize the control of satellite three-axis attitude.For the spacecraft that adopts CMG as actuating unit, generally adopt the individual CMG of n (n＞3) to be combined into redundant CMG system by certain configuration.CMG commonly used installs configuration and mainly contains pyramid configuration, two parallel configuration, pentagonal pyramid configuration and hexagonal pyramid configuration etc.The pyramid configuration has minimum redundancy, and inner singular surface is very complicated, often is used to check and handles the unusual performance of rule.Pyramid configuration CMG system adopts four CMG unit symmetries to install, and configuration is installed as shown in Figure 2.

In the satellite body system of axes, gimbal axis is installed vector and can be expressed as:

g ₁＝isinβ+kcosβ，

g ₂＝jsinβ+kcosβ，

g ₃＝-isinβ+kcosβ， (2)

g ₄＝-jsinβ+kcosβ，

Whole pyramid configuration CMG system can be expressed as with the total angular momentum h that frame corners changes:

$h=h_1\left({\mathrm{\δ}}_1\right)+h_2\left({\mathrm{\δ}}_2\right)+h_3\left({\mathrm{\δ}}_3\right)+h_4\left({\mathrm{\δ}}_4\right)$

$=\left[\begin{array}{c}-\cos \mathrm{\β}\sin {\mathrm{\δ}}_1\\ \cos {\mathrm{\δ}}_1\\ \sin \mathrm{\β}{\sin \mathrm{\δ}}_1\end{array}\right]+\left[\begin{array}{c}-{\cos \mathrm{\δ}}_2\\ -\cos \mathrm{\β}{\sin \mathrm{\δ}}_2\\ \sin \mathrm{\β}\sin {\mathrm{\δ}}_2\end{array}\right]+\begin{array}{}\end{array}\left[\begin{array}{c}\cos \mathrm{\β}{\sin \mathrm{\δ}}_3\\ -\cos {\mathrm{\δ}}_3\\ \sin \mathrm{\β}{\sin \mathrm{\δ}}_3\end{array}\right]+\left[\begin{array}{c}\cos {\mathrm{\δ}}_4\\ \cos \mathrm{\β}\sin {\mathrm{\δ}}_4\\ \sin \mathrm{\β}\sin {\mathrm{\δ}}_4\end{array}\right]---\left(3\right)$

H asks local derviation to have to the CMGs total angular momentum,

$\frac{\mathrm{dh}}{\mathrm{dt}}=\frac{\∂h}{\∂\mathrm{\δ}}\frac{\mathrm{d\δ}}{\mathrm{dt}}=\underset{i=1}{\overset{4}{\mathrm{\Σ}}}\frac{\∂h_i}{\∂{\mathrm{\δ}}_i}{\stackrel{\·}{\mathrm{\δ}}}_i=\underset{i=1}{\overset{4}{\mathrm{\Σ}}}{J}_i{\stackrel{\·}{\mathrm{\δ}}}_i=J\stackrel{\·}{\mathrm{\δ}}---\left(4\right)$

Then the control torque that acts on the celestial body of whole C MG system generation is:

$T_c=-\stackrel{\·}{h}=-J\stackrel{\·}{\mathrm{\δ}}---\left(5\right)$

Wherein, J is a Jacobian matrix, is defined as:

$J=\frac{\∂h}{\∂\mathrm{\δ}}=\left[\begin{array}{cccc}{J}_1& J_2& J_3& J_4\end{array}\right]$

$=\left[\begin{array}{cccc}\begin{array}{c}-\cos \mathrm{\β}{\cos \mathrm{\δ}}_1\\ -\sin {\mathrm{\δ}}_1\\ \sin \mathrm{\β}{\cos \mathrm{\δ}}_1\end{array}& \begin{array}{c}\sin {\mathrm{\δ}}_2\\ -\cos \mathrm{\β}{\cos \mathrm{\δ}}_2\\ \sin \mathrm{\β}\cos {\mathrm{\δ}}_2\end{array}& \begin{array}{c}\cos \mathrm{\β}\cos {\mathrm{\δ}}_3\\ \sin {\mathrm{\δ}}_3\\ \sin \mathrm{\β}{\cos \mathrm{\δ}}_3\end{array}& \begin{array}{c}-\sin {\mathrm{\δ}}_4\\ \cos \mathrm{\β}{\cos \mathrm{\δ}}_4\\ \sin \mathrm{\β}{\cos \mathrm{\δ}}_4\end{array}\end{array}\right]---\left(6\right)$

The control torque τ of i CMG unit output is shown in the i tabulation of Jacobian matrix _iReciprocal unit vector, described i=1,2,3,4.

According to formula (1), the moment of i CMG unit output can be expressed as again:

${\mathrm{\τ}}_i=(h_i\×g_i){\stackrel{\·}{\mathrm{\δ}}}_i=-J_i{\stackrel{\·}{\mathrm{\δ}}}_i---\left(7\right)$

The generalized inverse of satisfying the frame corners speed of formula (5) separate into:

$\stackrel{\·}{\mathrm{\δ}}=-J^T{\left({\mathrm{JJ}}^T\right)}^{-1}{T}_c=J^T{\left({\mathrm{JJ}}^T\right)}^{-1}\stackrel{\·}{h}---\left(8\right)$

This pseudoinverse is separated the minimum quadratic form problem below satisfying simultaneously:

$\underset{\stackrel{\·}{\mathrm{\δ}}}{\min }{\left|\right|\stackrel{\·}{\mathrm{\δ}}\left|\right|}^2,s.t.\stackrel{\·}{h}=J\stackrel{\·}{\mathrm{\δ}}---\left(9\right)$

Yet, as rank (J)＜3, this moment det (JJ ^T)=0, the generalized inverse of Jacobian matrix does not exist, and pseudoinverse is handled and restrained the frame corners speed of finding the solution is infinitely great, has surpassed the fan-out capability of hardware, and system is absorbed in unusual state.

In order to weigh the distance of odd metachromatic state, people have defined the unusual D of measuring:

D＝det(JJ ^T) (10)

D is big more, and odd antarafacial is far away more, and D levels off to zero more, shows that odd antarafacial is near more.The physical significance of D=0 is that the output torque vector of four CMG is positioned on the plane, can't produce expectation moment on the normal direction (unusual direction) on this plane, has lost the three-axis attitude control ability.Yet when expectation moment was positioned at this plane, the CMG system still can export expectation moment.Therefore, the unusual reasonably ability of evaluation system output expectation moment of D of measuring.

Summary of the invention

The present invention can not effectively avoid unusual for solving existing single frame control moment gyroscope manipulation rule, and the fugitive odd different time can bring the problem of bigger moment error, and a kind of single frame control moment gyroscope method of operating based on moment fan-out capability optimum is provided.

Based on the single frame control moment gyroscope method of operating of moment fan-out capability optimum, this method is realized by following steps:

Step 1, satellite attitude control system obtain the current expectation moment T of CMG system outlet according to current attitude information _c

Step 2, the current expectation moment T that obtains according to step 1 _c, calculate CMG system outlet expectation moment T _cThe frame corners speed of ability optimum

The frame corners speed of the described optimum of step 2

Computation process be:

According to expectation moment T _cWith i the angular momentum h that the CMG unit is current _i, obtain i CMG unit output expectation moment T _cThe angular momentum h of ability optimum _Di, that is:

$h_{\mathrm{di}}=\frac{g_i\×T_c}{\left|\right|g_i\×T_c\left|\right|}$

Current angular momentum h _iAngular momentum h with the optimum of expecting _DiBetween angular distance δ _DiFor:

${\mathrm{\&delta;}}_{\mathrm{di}}=\arccos \left(\frac{<h_i,h_{\mathrm{di}}>}{\left|\right|h_i\left|\right|\&CenterDot;\left|\right|h_{\mathrm{di}}\left|\right|}\right)\&CenterDot;\mathrm{sgn}(<h_i\&times;h_{\mathrm{di}},g_i>)$

After a control cycle Δ t, described current frame corners δ _iOverlap the frame corners speed of expectation with the frame corners of expectation For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{di}}=\frac{{\mathrm{\&delta;}}_{\mathrm{di}}}{\mathrm{\&Delta;t}}$

Satisfy The maximum frame corners speed that can provide for hardware.Then moment T is expected in output _cThe frame corners speed of ability optimum

For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d={\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="1"\; label="d"\; filenames="HSA00000179644300021.png,HSA00000179644300042.png"\; state="\{\{state\}\}">d</figure-callout>}1}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="HSA00000179644300042.png,HSA00000179644300052.png"\; state="\{\{state\}\}">d</figure-callout>}2}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="3"\; label="d"\; filenames="HSA00000179644300062.png"\; state="\{\{state\}\}">d</figure-callout>}3}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d4}\end{array}\right]}^T$

Step 3, the current expectation moment T that obtains according to step 1 and step 2 _cWith optimum frame corners speed

Obtain the framework rotary speed instruction of CMG system

The described framework rotary speed instruction that obtains the CMG system

Computation process be:

Propose to mix quadratic form and optimize index L:

$L=\underset{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}{\min }\frac{1}{2}\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e^T& A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e+& T_e^T& {\mathrm{BT}}_e\end{array}\right]$

Wherein,

A, B are respectively weighting matrix,

Satisfy the framework rotating speed of this optimization index For:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(A+J^T\mathrm{BJ})}^{-1}(A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^{T}B\stackrel{\&CenterDot;}{h})$

Get B=E ₃, A=λ E ₄, have:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(\mathrm{\&lambda;}{E}_4+J^{T}J)}^{-1}(A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^T\stackrel{\&CenterDot;}{h})$

Following formula is transformed to:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}=[E_4-\frac{1}{\mathrm{\&lambda;}}{J}^T{(E_3+\frac{1}{\mathrm{\&lambda;}}{\mathrm{JJ}}^T)}^{-1}J]({\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+\frac{1}{\mathrm{\&lambda;}}{J}^T\stackrel{\&CenterDot;}{h})$

λ is taken as in the formula:

λ＝λ ₀exp(0.5det(JJ ^T))，

Wherein, λ ₀Be constant;

Step 4, the framework rotary speed instruction that obtains according to step 3

, handle CMG system outlet expectation moment T _c, realize attitude control to satellite.

Principle of the present invention: in conjunction with Fig. 4, the attitude information that attitude controller obtains according to attitude sensor in the spacecraft is obtained the control torque of expectation, then, handles rule and obtains the frame corners speed command that can produce this control torque; This frame corners speed command in the CMG frame system, makes CMG system outlet control torque act on the celestial body through the framework kinetics function, makes star rotation, and repeats this process, reaches the expectation attitude up to celestial body.The present invention is based on the output torque principle of optimization, at first provided the frame corners speed that can make the CMG system outlet expect the moment capacity optimum

Propose then to optimize index, make the CMG system when output expectation moment, frame corners speed

As far as possible with optimum frame corners speed

Unanimity also will guarantee moment output error minimum simultaneously.It is unusual to avoid the CMG system to be absorbed in effectively like this, and accurate expectation control torque can be provided, and does not have simultaneously " framework locking " phenomenon, can bring less moment error when unusual.

Beneficial effect of the present invention:

One, because this is invented when guaranteeing the output control torque, and system can suitably adjust the framework configuration, makes the ability optimum of output expectation moment, guaranteed that output torque performance figure S gets maxim, thereby avoided unusual effectively.

Two, therefore this invention can avoid unusual robust to handle " framework locking " phenomenon that rule exists owing to added optimum frame corners speed term effectively.Be absorbed in when unusual in system, though expectation moment along unusual direction, the frame corners speed that this invention is exported is not 0 yet.Therefore, system can flee from unusual rapidly.

Three, the output torque error is relevant with parameter lambda, suitably select parameter lambda can greatly reduce the output torque error to and external interference moment on a magnitude, simultaneously can also guarantee to avoid effectively unusual, improved the precision that attitude is controlled.

The present invention can be according to current expected force moment vector, distribution frame cireular frequency reasonably, guarantee making output expect to make output torque error minimum in the moment capacity optimum, it is unusual to avoid the CMG system to be absorbed in effectively, accurate expectation control torque can be provided, and there is not " framework locking " phenomenon, can bringing less moment error when unusual.

Description of drawings

Fig. 1 is a single frame CMG fundamental diagram.

Fig. 2 is a pyramid configuration CMG system scheme of installation.

Fig. 3 is a CMG moment output geometric relationship scheme drawing of the present invention.

Fig. 4 is a program flow diagram of the present invention.

Fig. 5 is a satellite attitude control schematic diagram of the present invention.

Fig. 6 is the output control torque change curve scheme drawing of CMG of the present invention system.

Fig. 7 is the frame corners speed change curves scheme drawing of CMG of the present invention system.

Fig. 8 is the output torque ability performance figure change curve scheme drawing of CMG of the present invention system.

Fig. 9 is the unusual change curve scheme drawing of measuring.

Figure 10 is the frame corners change curve scheme drawing of CMG of the present invention system.

Figure 11 is a CMG system angle momentum change curve synoptic diagram of the present invention.

The specific embodiment

The specific embodiment one, in conjunction with Fig. 1 to Fig. 5 present embodiment is described, based on the single frame control moment gyroscope method of operating of moment fan-out capability optimum, this method is realized by following steps:

Step 1, satellite attitude control system obtain the current expectation moment T of CMG system outlet according to current attitude information _c

Step 2, the current expectation moment T that obtains according to step 1 _c, calculate and make CMG system outlet expectation moment T _cThe frame corners speed of ability optimum

The frame corners speed of the described optimum of step 2

Computation process be:

According to expectation moment T _cWith i the angular momentum h that the CMG unit is current _i, obtain i CMG unit output expectation moment T _cThe angular momentum h of ability optimum _Di, that is:

$h_{\mathrm{di}}=\frac{g_i\&times;T_c}{\left|\right|g_i\&times;T_c\left|\right|}$

Current angular momentum h _iAngular momentum h with the optimum of expecting _DiBetween angular distance δ _DiFor:

${\mathrm{\&delta;}}_{\mathrm{di}}=\arccos \left(\frac{<h_i,h_{\mathrm{di}}>}{\left|\right|h_i\left|\right|\&CenterDot;\left|\right|h_{\mathrm{di}}\left|\right|}\right)\&CenterDot;\mathrm{sgn}(<h_i\&times;h_{\mathrm{di}},g_i>)$

Therefore, in order to make the fan-out capability optimum, after a control cycle Δ t, described current frame corners δ _iOverlap the frame corners speed of expectation with the frame corners of expectation

For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{di}}=\frac{{\mathrm{\&delta;}}_{\mathrm{di}}}{\mathrm{\&Delta;t}}$

Satisfy

The maximum frame corners speed that can provide for hardware; Then moment T is expected in output _cThe frame corners speed of ability optimum

For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d={\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="1"\; label="d"\; filenames="HSA00000179644300021.png,HSA00000179644300042.png"\; state="\{\{state\}\}">d</figure-callout>}1}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="HSA00000179644300042.png,HSA00000179644300052.png"\; state="\{\{state\}\}">d</figure-callout>}2}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{<figure-callout\; id="3"\; label="d"\; filenames="HSA00000179644300062.png"\; state="\{\{state\}\}">d</figure-callout>}3}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d4}\end{array}\right]}^T$

Step 3, the current expectation moment T that obtains according to step 1 and step 2 _cWith optimum frame corners speed

Obtain the framework rotary speed instruction of CMG system

In order to make moment fan-out capability performance figure S get maxim, the angular momentum h of each CMG _iShould be as far as possible and the optimum angular momentum h of expectation _DiUnanimity, i.e. frame corners speed

Should be as far as possible and the frame corners speed of expectation Unanimity also will guarantee the moment error minimum of CMG system outlet simultaneously, so proposes to mix quadratic form optimization index L:

$L=\underset{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}{\min }\frac{1}{2}\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e^T& A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e+& T_e^T& {\mathrm{BT}}_e\end{array}\right]$

Wherein,

A, B are respectively weighting matrix,

Satisfy the framework rotating speed of this optimization index

For:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(A+J^T\mathrm{BJ})}^{-1}(A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^{T}B\stackrel{\&CenterDot;}{h})$

Get B=E ₃, A=λ E ₄, have:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(\mathrm{\&lambda;}{E}_4+J^{T}J)}^{-1}(\mathrm{\&lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^T\stackrel{\&CenterDot;}{h})$

Parameter lambda is more little, and the output torque error is more little, however the frame corners speed that λ reduces to make this manipulation rule to try to achieve increase, surpassed the fan-out capability of hardware; Otherwise λ increases will bring bigger moment error, influences the controller performance of attitude control system.Especially, will bring bigger moment error in the time of near singular surface.In order to be reduced in the moment error that the singular surface vicinity produces, λ is taken as:

λ＝λ ₀exp(0.5det(JJ ^T))

Wherein, λ ₀Be constant;

Can draw according to above-mentioned formula, near singular surface, the unusual D=det of measuring (JJ be arranged ^T) → 0, λ reduces rapidly, and the moment error will reduce.Simultaneously, λ ≠ 0 will make the united control rule always have certain moment output error, reduce the value of λ, can correspondingly reduce the moment output error.

The formula of considering is calculated comparatively complexity to 4 * 4 matrix inversions, and it is transformed to:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}=[E_4-\frac{1}{\mathrm{\&lambda;}}{J}^T{({\mathrm{<figure-callout\; id="3"\; label="E"\; filenames="HSA00000179644300062.png"\; state="\{\{state\}\}">E</figure-callout>}}_3+\frac{1}{\mathrm{\&lambda;}}{\mathrm{JJ}}^T)}^{-1}J]({\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+\frac{1}{\mathrm{\&lambda;}}{J}^T\stackrel{\&CenterDot;}{h})$

Step 4, the framework rotary speed instruction that obtains according to step 3 , handle CMG system outlet expectation moment T _c, realize attitude control to satellite.

In order more reasonably to weigh the ability of CMG system outlet expectation moment, defined output torque ability performance figure S in the present embodiment:

$S=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{<\frac{{\mathrm{\&tau;}}_i}{\left|\right|{\mathrm{\&tau;}}_i\left|\right|},\frac{T_c}{\left|\right|T_c\left|\right|}>}^2=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{<-J_i,\frac{T_c}{\left|\right|T_c\left|\right|}>}^2=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HSA00000179644300042.png,HSA00000179644300052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_i$

Wherein, θ _iExpression τ _iWith expectation moment T _cBetween angle.S is big more, shows CMG system outlet expectation this moment moment T _cAbility big more.S=0 represents that this CMG system can't provide expectation moment T _c, expect moment T this moment _cAlong unusual direction, the unusual D=0 of measuring must be arranged, but when D=0, output torque ability performance figure S needs not be equal to 0.Measure D more with respect to unusual, output torque performance figure S has more reasonably reflected the ability size of system outlet expectation moment.In order to make the moment fan-out capability maximum of whole C MG system, we should make output torque ability performance figure S get maxim as far as possible, just will make the output torque τ of each CMG unit _iWith expectation moment T _cBetween angle as far as possible little.At this moment, the output torque τ of each CMG unit _iAt expectation moment T _cComponent maximum on the direction, the ability optimum of system outlet expectation moment.

The framework rotary speed instruction that present embodiment obtains according to step 3

The choose reasonable parameter lambda can guarantee to reduce the output torque error under the limited situation of framework rotating speed, more accurately output expectation moment T _cThe designed manipulation of the present invention is restrained when guaranteeing moment output error minimum, will make the control moment gyroscope group towards the strongest configuration motion of output torque ability, thereby avoid unusual effectively.

Concrete reality executes mode two, in conjunction with Fig. 6 to Figure 11 present embodiment is described, present embodiment is the embodiment of the described single frame control moment gyroscope method of operating based on moment fan-out capability optimum of the specific embodiment one:

In order to verify the unusual avoidance and the moment output performance of designed manipulation rule, present embodiment adopts pyramid configuration CMG system; The described CMG mounted angle β of system=53.13 °, initial frame corners is δ ₀=[0 °, 0 °, 0 °, 0 °], the moment of momentum of each CMG unit is 1Nms, maximum frame corners speed

Parameter carried out mathematical simulation; Expectation moment in the described emulation is taken as T _c=[0.200] Nm, parameter lambda is taken as λ=0.0001exp (0.5det (JJ ^T)).

By analysis, draw to draw a conclusion to simulation result in the foregoing description:

One, designed manipulation rule does not run into that pseudoinverse handles that rule and unusual robust handle that rule can run in the present embodiment is positioned at δ=[90 °, 0 °, 90 °, 0 °] unusual robust, do not take place yet and handle rule " framework locking " phenomenon can take place in the oval singular surface located.

Two, under the situation of frame corners limited speed, though the unusual D of measuring of system levels off to 0, system is near singular surface, but output torque performance figure S is about 0.5, and system still can export expectation moment, afterwards, the unusual D of measuring increases sharply, and system hightails singular surface.

Three, in whole simulation process, the CMG system can accurately export expectation moment, and the output torque error is very little, that is: system when approximately 6s is near singular surface, X-axis maximum torque output error Δ T=0.002Nm, Y-axis, Z axle maximum torque output error are about 10 ^-4The Nm magnitude.

Claims (2)

1. based on the single frame control moment gyroscope method of operating of moment fan-out capability optimum, it is characterized in that this method may further comprise the steps:

Step 1, satellite attitude control system obtain the current expectation moment T of CMG system outlet according to current attitude information _c

Step 2, the current expectation moment T that obtains according to step 1 _c, calculate CMG system outlet expectation moment T _cThe frame corners speed of ability optimum

The frame corners speed of the described optimum of step 2

Computation process be:

According to expectation moment T _cWith i the angular momentum h that the CMG unit is current _i, obtain i CMG unit output expectation moment T _cThe angular momentum h of ability optimum _Di, that is:

$h_{\mathrm{di}}=\frac{g_i\&times;T_c}{\left|\right|g_i\&times;T_c\left|\right|}$

In the formula, g _iBe the gimbal axis direction vector of i CMG unit, current angular momentum h _iAngular momentum h with the optimum of expecting _DiBetween angular distance δ _DiFor:

${\mathrm{\&delta;}}_{\mathrm{di}}=\arccos \left(\frac{<h_i,h_{\mathrm{di}}>}{\left|\right|h_i\left|\right|\&CenterDot;\left|\right|h_{\mathrm{di}}\left|\right|}\right)\&CenterDot;\mathrm{sgn}(<h_i\&times;h_{\mathrm{di}},g_i>)$

After a control cycle Δ t, described i the current frame corners δ in CMG unit _iOverlap with the frame corners of expectation, then Qi Wang frame corners speed

For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{\mathrm{di}}=\frac{{\mathrm{\&delta;}}_{\mathrm{di}}}{\mathrm{\&Delta;t}}$

Satisfy

The maximum frame corners speed that can provide for hardware; Then moment T is expected in output _cThe frame corners speed of ability optimum For:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d={\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d1}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d2}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d3}& {\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_{d4}\end{array}\right]}^T$

Step 3, the current expectation moment T that obtains according to step 1 and step 2 _cWith optimum frame corners speed

Obtain the framework rotary speed instruction of CMG system

The described framework rotary speed instruction that obtains the CMG system

Computation process be:

Propose to mix quadratic form and optimize index L:

$L=\underset{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}{\min }\frac{1}{2}\left[\begin{array}{cccc}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e^T& A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_e+& T_e^T& {\mathrm{BT}}_e\end{array}\right]$

Wherein,

Described J is a Jacobian matrix, and A, B are respectively weighting matrix, satisfies the framework rotating speed of this optimization index

For:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(A+J^T\mathrm{BJ})}^{-1}(A{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^{T}B\stackrel{\&CenterDot;}{h})$

Get B=E ₃, A=λ E ₄, have:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}={(\mathrm{\&lambda;}{E}_4+J^{T}J)}^{-1}(\mathrm{\&lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+J^T\stackrel{\&CenterDot;}{h})$

Following formula is transformed to:

$\stackrel{\&CenterDot;}{\mathrm{\&delta;}}=[E_4-\frac{1}{\mathrm{\&lambda;}}{J}^T{(E_3+\frac{1}{\mathrm{\&lambda;}}{\mathrm{JJ}}^T)}^{-1}J]({\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d+\frac{1}{\mathrm{\&lambda;}}{J}^T\stackrel{\&CenterDot;}{h})$

λ is taken as in the formula:

λ＝λ ₀exp(0.5det(JJ ^T))，

Wherein, λ ₀Be constant;

Step 4, the framework rotary speed instruction that obtains according to step 3

Handle CMG system outlet expectation moment T _c, realize attitude control to satellite.

2. the single frame control moment gyroscope method of operating based on moment fan-out capability optimum according to claim 1 is characterized in that, has defined the state of the system critical for the evaluation: output torque performance figure S, and S is defined as:

$S=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{<\frac{{\mathrm{\&tau;}}_i}{\left|\right|{\mathrm{\&tau;}}_i\left|\right|},\frac{T_c}{\left|\right|T_c\left|\right|}>}^2=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{<-J_i,\frac{T_c}{\left|\right|T_c\left|\right|}>}^2=\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{\cos }^2{\mathrm{\&theta;}}_i$

In the formula, θ _iThe output torque τ that represents i CMG unit _iWith expectation moment T _cBetween angle; S increases, current C MG system outlet expectation moment T _cAbility increase; Output torque τ when each CMG unit _iAt expectation moment T _cWhen the component on the direction is maximum, the ability optimum of CMG system outlet expectation moment.

Images