CN105388902A - Control moment gyro singularity avoidance method based on instruction moment vector adjustment - Google Patents

Abstract

A control moment gyro singularity avoidance method based on instruction moment vector adjustment comprises the steps of first acquiring a frame angular vector of control moment gyros, then calculating a Jacobian matrix, a singularity metric value and an instruction moment vector adjustment coefficient of a control moment gyro motion equation, next obtaining a control moment instruction adjustment vector and a zero motion singularity avoidance strength coefficient based on a control moment instruction given by an attitude controller, and finally obtaining a control moment gyro frame angular velocity instruction vector based on the control moment instruction adjustment vector and the zero motion singularity avoidance strength coefficient so as to control the moment gyro frame angular velocity. The invention overcomes the situation in which a moment instruction coincides with a particular direction thereof when a frame is "locked", thereby making separation impossible; solves the problem of a "locked" frame angle during singularity avoidance in the prior art; and achieves effective avoidance of the control moment gyro singularity.

Description

The unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control

Technical field

The present invention relates to spacecraft attitude deterministic finite automata field, particularly the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control.

Background technology

The quick spacecraft with attitude fast reserve requirement generally adopts control-moment gyro group as topworks.Because attitude of satellite maneuverability is strong, when control-moment gyro group frame movement to during certain configuration can close to and be in the unusual state of framework, thus cause topworks cannot by expection output spacecraft three axle control moment.For avoiding the singular problem of control-moment gyro frame movement, existing technical method is the zero Kinematics singularity bypassing method and the unusual bypassing method of robust researched and proposed for singular problem, wherein, zero Kinematics singularity bypassing method only can realize effectively evading hidden unusual the evading in unusual type, the unusual bypassing method of robust is mainly for the insurmountable aobvious unusual evasion of zero Kinematics singularity bypassing method, but the latter relatively produce certain disturbance to spacecraft attitude evading in process the former.Although although therefore adopt zero conventional Kinematics singularity to evade the unusual algorithm of evading with robust can avoid this problem to a certain extent, spacecraft can be caused temporarily to lose gesture stability ability because the method exists control-moment gyro framework configuration " locked " phenomenon and affect attitude maneuver performance.

When take conventional unusual evade algorithm after occur that framework " locked " phenomenon is found and (WieBong after obtaining reasonable mathematic(al) treatment, et.al., SingularityRobustSteeringLogicforRedundantSingle-GimbalC ontrolMomentGyros, AIAAGuidance, Navigation, andControlConferenceandExhibit, Denver, 2000), relevant scholar has carried out the research avoiding this phenomenon.Follow-up major part research mainly improves the unusual algorithm of evading of robust, wherein, the most typically American scholar WieBong, its technological approaches taked is that the off diagonal element being zero is transformed into multi-form nonzero element (WieBong in singular divisor matrix anti-in unusual for traditional robust bypassing method, Newsingularityescape/avoidancesteeringlogicforcontrolmom entgyrosystems, JournalofGuidanceControlandDynamics, 28 (5), 2005), particularly off diagonal element is that time to time change is with the problem avoiding framework locked, and applied for that multinomial patent is (as WieB., et.al., RobustSingularityAvoidanceinSatelliteAttitudeControl, U.S.Patent6, 039, 290, 2000, WieB., SingularityEscape/AvoidanceSteeringLogicforControlMoment GyroSystems, U.S.PatentNo.6,917,862,2005. etc.).Existingly to improve one's methods above-mentioned, unusual evading and simultaneously as far as possible little to celestial body attitude disturbance requirement is realized in order to reach, need to select the multiparameter emulation examination mode of gathering in algorithm, therefore evade the locked and follow-up study of framework and improve one's methods generally to be difficult to take into account for conventional frame is unusual simultaneously and unusually evade the problem little with spacecraft attitude disturbance.

Summary of the invention

The technical matters that the present invention solves is: overcome the deficiencies in the prior art, provide and a kind ofly evade the unusual bypassing method with robust, unusual basis of evading algorithm introduced and independently regulate with unusual tolerance based on control moment instruction vector at zero Kinematics singularity, can enter unusual or hightail from singular point by anti-locking system, avoiding the phenomenon occurring control-moment gyro framework configuration " locked ".

Technical solution of the present invention is: the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control, comprises the steps:

(1) control-moment gyro group frame corners vector δ is gathered, then according to the Jacobian matrix J of control-moment gyro equation of motion Computational frame angular motion;

(2) calculating Singularity Degree value Sv is

Sv＝det(J·J ^T)

And then obtain command force moment vector adjustment factor υ and be

$\mathrm{\υ}=\left\{\begin{array}{cc}0& ifSv>D_{s3}\\ -k_{tmp\mathrm{\υ}}\·(Sv-D_{s3})& ifSv\≤D_{s3}\end{array}\right.$

When | υ | > υ _limttime, υ=sgn (υ) υ _limt, wherein, k _{tmp υ}for regulating gain, D _s3for vector regulates threshold value, υ _limtfor vector adjustment factor amplitude limit value, sgn (υ) is for getting the symbolic operation of υ;

(3) according to the control moment instruction τ that attitude controller provides _cobtain control moment instruction and regulate vector τ _cAdfor

${\mathrm{\τ}}_{cAd}=\left[\begin{array}{ccc}1& -\mathrm{\υ}& \mathrm{\υ}\\ \mathrm{\υ}& 1& -\mathrm{\υ}\\ -\mathrm{\υ}& \mathrm{\υ}& 1\end{array}\right]\·{\mathrm{\τ}}_c$

(4) as Sv < D _s1time, zero Kinematics singularity evades strength factor

Work as D _s1≤ Sv < D _s2time, zero Kinematics singularity evades strength factor

As Sv>=D _s2time, zero Kinematics singularity evades strength factor α _s1=0;

Wherein, D _s1startup threshold value is evaded, D for robust is unusual _s2be that zero Kinematics singularity evades startup threshold value, C ₁be that zero Kinematics singularity evades intensity gain coefficient, α _s10for amount of bias;

(5) as Sv < D _s1time, robust is unusual evades strength factor α _s2=

As Sv>=D _s1time, robust is unusual evades strength factor α _s2=0, wherein, C ₂intensity gain coefficient is evaded for robust is unusual;

(6) vector τ is regulated according to control moment instruction _cAd, unusually evade strength factor α _s1and α _s2obtain control-moment gyro frame corners speed command vector for

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d=-\frac{1}{H_{cmg0}}{J}^T{\left(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3\right)}^{-1}{\mathrm{\&tau;}}_{cAd}+{\mathrm{\&alpha;}}_{s1}\left(I_3-J^T{(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3)}^{-1}J\right)\frac{\&part;Sv\left(\mathrm{\&delta;}\right)}{\&part;\mathrm{\&delta;}}.$

Use control-moment gyro frame corners speed command vector control-moment gyro frame corners speed in adjustment control-moment gyro group, wherein, H _cmg0for control-moment gyro angular momentum, I ₃be 3 rank unit matrixs, for Sv is to the partial derivative of frame corners vector δ ,-1 is matrix inversion operation symbol.

Described adjustment gain k _{tmp υ}span be-100≤k _{tmp υ}≤ 100, vector regulates threshold value D _s3> 0, vector adjustment factor amplitude limit value υ _limtspan be 0≤υ _limt< 1, robust unusual evading starts threshold value D _s1> 0, zero Kinematics singularity evades startup threshold value D _s2>=D _s1, zero Kinematics singularity evades intensity gain coefficient C ₁>=0, amount of bias α _s10>=0, robust is unusual evades intensity gain coefficient C ₂>=0.

Described step (5) also comprises amplitude limit value α _s20, work as α _s2> α _s20time, α _s2=α _s20, wherein, α _s20> 0.

Described k _{tmp υ}=2, D _s3=0.7, υ _limt=0.7, D _s1=0.5, D _s2=2.0, C ₁=0.18, C ₂=0.1, α _s10=0, α _s20=0.5.

The present invention's advantage is compared with prior art:

(1) the present invention unusual to control-moment gyro framework by instruction Torque vector control under control moment instruction direction vector and amplitude carry out with unusual tolerance size from main deflection regulate means, overcome the torque command when framework " locked " to overlap with its specific direction and the situation that cannot depart from, solve and evade frame corners " locked " problem existing in process in prior art unusual, achieve unusual effectively the evading of control-moment gyro, there is form simple, explicit physical meaning and make parameter choose easy, the advantages such as engineer applied is strong,

(2) the present invention selects off diagonal element to be the technological means of antisymmetric matrix operator by the algorithm that regulates at vector, solve vector regulate after cause contrary with former instruction polarity of control moment instruction round pass through singular surface and the possibility problem making Spacecraft During Attitude Maneuver failure cannot be departed from, achieve control-moment gyro framework in subsequent control and can to depart from completely and away from corresponding unusual state;

(3) the present invention by comprising, zero Kinematics singularity is evaded, robust unusual evade call with the reasonable logic of instruction Torque vector control algorithm, vector adjustment factor and evade strength factor continuously adjustable design means, avoiding the unusual initial stage directly calls unusual algorithm and the vector evaded of robust and regulates algorithm or celestial body attitude is brought to the problem of disturbance because the adjustment factor that respectively calls algorithm and action intensity suddenly change, and achieves unusually to evade target little to celestial body attitude disturbance in overall process.

Accompanying drawing explanation

Fig. 1 is the unusual bypassing method process flow diagram of a kind of control-moment gyro based on instruction Torque vector control of the present invention;

Fig. 2 is the control-moment gyro frame corners curve that the present invention gathers;

Fig. 3 is control-moment gyro group configuration singularity value curve of the present invention;

Fig. 4 is that controller of the present invention exports comparison diagram before and after control moment instruction adjustment;

Fig. 5 is the control-moment gyro frame corners speed command curve that the present invention calculates.

Embodiment

The present invention is directed to the deficiencies in the prior art, propose the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control, to comprise concrete implementing procedure as follows for the inventive method as shown in Figure 1:

1) (control-moment gyro is by the momentum flywheel of permanent rotating speed to gather control-moment gyro group frame corners vector δ, the framework of support flying wheel and frame member servo-drive system composition, the vector be made up of control-moment gyro frame corner each in control-moment gyro group is referred to as control-moment gyro group frame corners vector), according to the Jacobian matrix J of control-moment gyro equation of motion Computational frame angular motion (equation of motion describe and Jacobian matrix ask for can be see: ParadisoJ.A., GlobalSteeringofSingleGimballedControlMomemtGyroscopeUsi ngaDirectedSearch, J.ofGuidance, Control, andDynamics, 15 (5), 1992:1236-1244), Singularity Degree value Sv and Sv is to the partial derivative of frame corners vector δ value (local derviation specific algorithm can see document: Zhang Renwei, satellite orbit and attitude dynamics and control, publishing house of BJ University of Aeronautics & Astronautics, 1998:293), and by Singularity Degree value Sv computations Torque vector control coefficient υ, and control moment instruction τ is provided to Spacecraft Control device _ccarry out vector and regulate the adjusted vector τ of computing _cAd.Be specially:

(1) Sv is matrix J J ^tdeterminant, namely

Sv＝det(J·J ^T)

Wherein, J ^tfor the transposed matrix of matrix J.

(2) by Singularity Degree value Sv computations Torque vector control coefficient υ be

$\mathrm{\&upsi;}=\left\{\begin{array}{cc}0& ifSv>D_{s3}\\ -k_{tmp\mathrm{\&upsi;}}\&CenterDot;(Sv-D_{s3})& ifSv\&le;D_{s3}\end{array}\right.$

If | υ | > υ _limtshi Ze carries out amplitude limiting processing: υ=sgn (υ) υ _limt, wherein, k _{tmp υ}(-100≤k _{tmp υ}≤ 100) for regulating gain, D _s3(D _s3> 0) be vector adjustment threshold value, υ _limt(0≤υ _limt< 1) be vector adjustment factor amplitude limit value, sgn (υ), for getting the symbolic operation of υ, is namely taken as 1 when υ is greater than zero, otherwise is taken as 0.

(3) to the control moment instruction τ that attitude controller provides _ccarry out vector and regulate computing:

${\mathrm{\&tau;}}_{cAd}=\left[\begin{array}{ccc}1& -\mathrm{\&upsi;}& \mathrm{\&upsi;}\\ \mathrm{\&upsi;}& 1& -\mathrm{\&upsi;}\\ -\mathrm{\&upsi;}& \mathrm{\&upsi;}& 1\end{array}\right]\&CenterDot;{\mathrm{\&tau;}}_c$

2) calculate zero Kinematics singularity and evade strength factor α _s1strength factor α is evaded with robust is unusual _s2.Be specially:

(1) zero Kinematics singularity evades strength factor α _s1computing formula is

If Sv < is D _s1time, ${\mathrm{\&alpha;}}_{s1}=C_1\left(\frac{1}{D_{s1}}-\frac{1}{D_{s2}}\right);$

If D _s1≤ Sv < D _s2time, ${\mathrm{\&alpha;}}_{s1}=C_1\left(\frac{1}{Sv}-\frac{1}{D_{s2}}\right)+{\mathrm{\&alpha;}}_{s10};$

If Sv>=D _s2time, α _s1=0;

Wherein, D _s1(D _s1> 0) evade startup threshold value for robust is unusual, D _s2(D _s2> 0, and meet D _s2>=D _s1) be that zero Kinematics singularity evades startup threshold value, C ₁(C ₁>=0) be that zero Kinematics singularity evades intensity gain coefficient, amount of bias α _s10>=0.

(2) robust is unusual evades strength factor α _s2computing formula is

If Sv < is D _s1time, ${\mathrm{\&alpha;}}_{s2}=C_2\&CenterDot;\left(\frac{1}{Sv+{10}^{-6}}-\frac{1}{D_{s1}+{10}^{-6}}\right)$

If Sv>=D _s1time, α _s2=0

If above-mentioned result of calculation meets α _s2> α _s20time, then put α _s2=α _s20, wherein, C ₂(C ₂>=0) intensity gain coefficient is evaded for robust is unusual, α _s20(α _s20> 0) be α _s2amplitude limit value.

3) the control moment instruction provided according to attitude controller regulates vector τ _cAd, unusually evade strength factor α _s1, α _s2, take zero Kinematics singularity to evade frame corners speed command that the unusual bypassing method with robust carries out control-moment gyro group calculate, be specially:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d=-\frac{1}{H_{cmg0}}{J}^T{\left(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3\right)}^{-1}{\mathrm{\&tau;}}_{cAd}+{\mathrm{\&alpha;}}_{s1}\left(I_3-J^T{(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3)}^{-1}J\right)\frac{\&part;Sv\left(\mathrm{\&delta;}\right)}{\&part;\mathrm{\&delta;}}.$

Wherein, for control-moment gyro frame corners speed command to be asked vector, H _cmg0for control-moment gyro angular momentum, I ₃be 3 rank unit matrixs, for Sv is to the partial derivative (specific formula for calculation can see document: Zhang Renwei, satellite orbit and attitude dynamics and control, publishing house of BJ University of Aeronautics & Astronautics, 1998:P293) of frame corners vector δ, upper right footmark "-1 " is matrix inversion operation symbol.Below in conjunction with embodiment, the inventive method is described in detail.

Embodiment 1: adopt angular momentum H _cmg0the system of 6 control-moment gyro composition pentagonal pyramid configuration topworkies configuration of=25Nms is carried out+45 °/-45 ° (rolling/pitching) twin shafts by zero attitude and is combined motor-driven.In order to make control-moment gyro group configuration more easily close to unusual state, only adopt 5 control-moment gyro ginseng Spacecraft During Attitude Maneuvers to control at this, wherein control-moment gyro 1 frame corners is locked in 0 ° and does not participate in control.The unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control is specifically implemented as follows:

Setup parameter: k _{tmp υ}=2, D _s3=0.7, υ _limt=0.7, D _s1=0.5, D _s2=2.0, C ₁=0.18, C ₂=0.1, α _s10=0, α _s20=0.5.

Implementing procedure when following steps are the inventive method application in a control cycle:

1) control-moment gyro group frame corners vector is gathered:

δ＝[0.00.5450689-0.468502.573360.577711.30145] ^T(rad)；

The instruction moment exported by Spacecraft Control device is:

τ _c＝[0.041250.03748-0.65202] ^T；

Can calculate according to the equation of motion:

$J=\left[\begin{array}{cccccc}0& 0.12433106& 0.512692519& -0.93114051& -0.66814621& -0.2661014\\ 0& -0.93949406& -0.81231845& -0.15234310& 0.66372979& -0.96394501\\ 0& -0.31920635& 0.278001998& -0.3313139& -0.3362192& 0\end{array}\right];$

Sv＝0.69644；

$\frac{\&part;Sv}{\&part;\mathrm{\&delta;}}={\left[\begin{array}{cccccc}0& 2.7919939& -1.0193131& 0.8514346& 0.3620309& -0.7564965\end{array}\right]}^T;$

υ＝0.007116；

2) zero motion is evaded strength factor calculate with the unusual motion of robust:

α _s1＝0.16845669，α _s2＝0。

3) according to τ _cAd, α _s1, α _s2, take zero Kinematics singularity to evade frame corners speed command that the unusual bypassing method with robust carries out control-moment gyro group for

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d={\left[\begin{array}{cccccc}0& -3.0220& -0.61098& 4.3513& -6.36854& -1.51161\end{array}\right]}^T(\mathrm{deg}/s)$

Whole attitude maneuver application overall process the results are shown in Figure shown in 2 ~ Fig. 5.Wherein, Fig. 2 (a), Fig. 2 (b), Fig. 2 (c), Fig. 2 (d), Fig. 2 (e), Fig. 2 (f) sets forth the frame corners (unit: degree) of the control-moment gyro 1 ~ 6 collected; Regulating and unusually evading algorithm appears reducing and triggers vector in framework Singularity Degree value in Spacecraft During Attitude Maneuver process as shown in Figure 3, will raise to some extent evading unusual tolerance under algorithm effect; As shown in Figure 4, under regulating action, the deflection of control moment vector travel direction and amplitude are to a certain degree regulated, wherein, Fig. 4 (a), Fig. 4 (b), Fig. 4 (c) sets forth the component (unit: Nm) of control moment vector on celestial body rolling, pitching and driftage three axle after former control moment vector and vector adjustment, solid line is the component of control moment vector on celestial body rolling, pitching and driftage three axle of former control moment vector, and dotted line is the component of control moment vector on celestial body rolling, pitching and driftage three axle after vector regulates; As shown in Figure 5, the severe degree of the frame corners speed command change of calculating when Singularity Degree value is less compare singular value greatly time change little, thus describe the inventive method and have and good unusually evade characteristic, wherein, Fig. 5 (a), Fig. 5 (b), Fig. 5 (c), Fig. 5 (d), Fig. 5 (e), 5 (f) sets forth the frame corners speed command (unit: degree/second) of control-moment gyro 1 ~ 6; From experimental verification, the inventive method achieves close to effectively evading during unusual state, and overcomes frame corners " locked " situation that conventional unusual bypassing method exists, and ensures the motor-driven superperformance of system.

The content be not described in detail in instructions of the present invention belongs to the known technology of those skilled in the art.

Claims (4)

1., based on the unusual bypassing method of control-moment gyro of instruction Torque vector control, it is characterized in that comprising the steps:

(1) control-moment gyro group frame corners vector δ is gathered, then according to the Jacobian matrix J of control-moment gyro equation of motion Computational frame angular motion;

(2) calculating Singularity Degree value Sv is

Sv＝det(J·J ^T)

And then obtain command force moment vector adjustment factor υ and be

$\mathrm{\&upsi;}=\left\{\begin{array}{cc}0& ifSv>D_{s3}\\ -k_{tmp\mathrm{\&upsi;}}\&CenterDot;(Sv-D_{s3})& ifSv\&le;D_{s3}\end{array}\right.$

When | υ | > υ _limttime, υ=sgn (υ) υ _limt, wherein, k _{tmp υ}for regulating gain, D _s3for vector regulates threshold value, υ _limtfor vector adjustment factor amplitude limit value, sgn (υ) is for getting the symbolic operation of υ;

(3) according to the control moment instruction τ that attitude controller provides _cobtain control moment instruction and regulate vector τ _cAdfor

${\mathrm{\&tau;}}_{cAd}=\left[\begin{array}{ccc}1& -\mathrm{\&upsi;}& \mathrm{\&upsi;}\\ \mathrm{\&upsi;}& 1& -\mathrm{\&upsi;}\\ -\mathrm{\&upsi;}& \mathrm{\&upsi;}& 1\end{array}\right]\&CenterDot;{\mathrm{\&tau;}}_c$

(4) determine that zero Kinematics singularity evades strength factor α according to Singularity Degree value Sv _s1, robust is unusual evades strength factor α _s2, strength factor α is evaded for zero Kinematics singularity _s1,

Work as Sv<D _s1time, zero Kinematics singularity evades strength factor

Work as D _s1≤ Sv<D _s2time, zero Kinematics singularity evades strength factor

As Sv>=D _s2time, zero Kinematics singularity evades strength factor α _s1=0;

Wherein, D _s1startup threshold value is evaded, D for robust is unusual _s2be that zero Kinematics singularity evades startup threshold value, C ₁be that zero Kinematics singularity evades intensity gain coefficient, α _s10for amount of bias;

(5) strength factor α is evaded for robust is unusual _s2, work as Sv<D _s1time, robust is unusual evades strength factor ${\mathrm{\&alpha;}}_{s2}=C_2\&CenterDot;(\frac{1}{Sv+{10}^{-6}}-\frac{1}{D_{s1}+{10}^{-6}});$

As Sv>=D _s1time, robust is unusual evades strength factor α _s2=0, wherein, C ₂intensity gain coefficient is evaded for robust is unusual;

(6) vector τ is regulated according to control moment instruction _cAd, unusually evade strength factor α _s1and α _s2obtain control-moment gyro frame corners speed command vector for ${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_d=-\frac{1}{H_{cmg0}}{J}^T{(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3)}^{-1}{\mathrm{\&tau;}}_{cAd}+{\mathrm{\&alpha;}}_{s1}(I_3-J^T{\left(J\&CenterDot;J^T+{\mathrm{\&alpha;}}_{s2}{I}_3\right)}^{-1}J)\frac{\&part;Sv\left(\mathrm{\&delta;}\right)}{\&part;\mathrm{\&delta;}}.$

Use control-moment gyro frame corners speed command vector control-moment gyro frame corners speed in adjustment control-moment gyro group, wherein, H _cmg0for control-moment gyro angular momentum, I ₃be 3 rank unit matrixs, for Sv is to the partial derivative of frame corners vector δ ,-1 is matrix inversion operation symbol.

2. the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control according to claim 1 and 2, is characterized in that: described adjustment gain k _{tmp υ}span be-100≤k _{tmp υ}≤ 100, vector regulates threshold value D _s3>0, vector adjustment factor amplitude limit value υ _limtspan be 0≤υ _limt<1, robust unusual evading starts threshold value D _s1>0, zero Kinematics singularity evades startup threshold value D _s2>=D _s1, zero Kinematics singularity evades intensity gain coefficient C ₁>=0, amount of bias α _s10>=0, robust is unusual evades intensity gain coefficient C ₂>=0.

3. the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control according to claim 1 and 2, is characterized in that: described step (5) also evades strength factor α to robust is unusual _s2limit, work as α _s2> α _s20time, α _s2=α _s20, wherein, amplitude limit value α _s20>0.

4. the unusual bypassing method of a kind of control-moment gyro based on instruction Torque vector control according to claim 1 and 2, is characterized in that: described k _{tmp υ}=2, D _s3=0.7, υ _limt=0.7, D _s1=0.5, D _s2=2.0 _,c ₁=0.18, C ₂=0.1, α _s10=0, α _s20=0.5.