CN104006787A - High-precision attitude determination method for spacecraft attitude motion simulation platform - Google Patents

Abstract

The objective of the invention is to provide a high-precision attitude determination method for a spacecraft attitude motion simulation platform. Equipment used in the method comprises a laser tracker, an intelligent probe, a gyroscope and two industrial control computers, so a ground fully-physical simulation experimental facility is constructed. The method is an improved method for attitude determination based on extended Kalman filtering, acquires attitude information of the spacecraft attitude motion simulation platform by carrying out data-fusion filtering on output information of the laser tracker and the gyroscope, better accords with actual working conditions and has high instantaneity. Measurement equipment used in the invention is easy to install, has high measurement precision and can realize dynamic contactless measurement of the attitude angle and angular rate information of the spacecraft attitude motion simulation platform.

Description

Spacecraft Attitude motion simulation platform high-precision attitude is determined method

Technical field

The present invention relates to measuring technique, is exactly that a kind of spacecraft Attitude motion simulation platform high-precision attitude is determined method specifically.

Background technology

The air film that spacecraft Attitude motion simulation platform (also referred to as three-axis air-bearing table) relies on pressurized air to form between air-bearing and bearing seat, making to simulate stage body floats, thereby realize approximate friction free relative motion condition, with the spacecrafts such as analog satellite in outer space the very little mechanical environment of suffered disturbance torque.As the universal simulator of spacecraft Attitude motion, three-axis air-bearing table can carry out the physical simulation experiment check analysis of attitude of flight vehicle control system performance, is important means and method in spacecraft development process.

Spacecraft Attitude motion simulation platform needs dynamically to provide by attitude measurement system the attitude informations such as attitude angle, angular speed in process of the test, to complete Control loop, special construction due to spacecraft Attitude motion simulation platform, the device (as rotary transformer, inductosyn, photoelectric code disk, grating etc.) of in the past measuring for turntable is not suitable for the measurement of spacecraft Attitude motion simulation platform, need to consider new measuring method and device.And in practical application at present, the height of attitude measurement system precision is directly connected to the effect of l-G simulation test.

Through searching document, find, Chinese invention patent application number: 201110249979.2, patent name is three-axis air-bearing table high-precision attitude angle measuring devices and methods therefor, this patent is at three-axis air-bearing table table top bottom surrounding mounting guide rail, window-blind and marker, and artificial light system and digital CCD video camera are arranged on lifting platform.The image of digital CCD camera acquisition marker is also transferred on the computing machine of being responsible for image processing, by marker is carried out to sub-pixel positioning, obtains the accurate coordinates of marker, thereby calculates the relative attitude angle between marker and digital CCD video camera.But due to the defect on system constructing, measuring accuracy is restricted, thereby affect its actual usable range.

Chinese invention patent application number: 201310134631.8, patent name is: three-axis air-bearing table high-precision attitude angle and angular velocity measurement device, this patent is installed intelligent side head, gyroscope and kaleidoscope prism on the instrument platform of three-axis air-bearing table, laser tracker and two photoelectric auto-collimators are installed under platform, according to laser tracker, two photoelectric auto-collimators and gyrostatic data, are directly obtained attitude information.But this patent, in data filtering processing section, is not considered the constraint condition that deviation quaternary digital-to-analogue is 1, directly adopts Kalman filtering, easily cause error covariance matrix to occur unusual, cause data to be dispersed, attitude is determined unsuccessfully.This patent does not consider that attitude parameter conversion, the gyroscope of laser tracker and photoelectric auto-collimator output information measure the processes such as coordinate system demarcation.In addition, data processing and communications portion are arranged under platform and are carried out, do not meet actual conditions.

At document " the servo angle measurement systematic research of three-axis air-bearing table single frame ", (be published in aerospace journal, 1996, the 17th volume, the 4th phase, the page number: 71-74), the Li Jisu of the Zhang Xiaoyou of Harbin Institute of Technology, Liu Dun and Beijing Control Engineering Inst. etc. has proposed the servo measurement scheme of a kind of single frame, this system is installed an arc arms that can rotate around air floating table center pedal line on air floating table base, and transportable balladeur train is installed thereon, by the attitude information of the rotation of responsive arc arms and the traverse measurement air floating table of balladeur train.When this system need to increase complicated mechanical system and sensor system, mechanism is complicated, and engineering application is more difficult, and its precision is subject to the restriction of mechanical hook-up and sensor, is difficult to reach high precision.

Summary of the invention

Attitude problem identificatioin for spacecraft Attitude motion simulation platform, the object of the present invention is to provide a kind of spacecraft Attitude motion simulation platform high-precision attitude to determine method, the method is definite the improving one's methods of a kind of attitude based on EKF, by the output information of laser tracker and gyro is carried out to Data Fusion Filtering processing, obtain the attitude information of spacecraft Attitude motion simulation platform.

The present invention is achieved in that a kind of spacecraft Attitude motion simulation platform high-precision attitude determines method, the equipment adopting comprises laser tracker, intelligent gauge head, gyroscope and two industrial control computers, and laser tracker is installed under the platform of spacecraft Attitude motion simulation platform; Before attitude is determined, intelligence gauge head is arranged under platform, the measurement coordinate system that coordinates laser tracker Calibration of Laser tracker, in attitude deterministic process, intelligence gauge head is fixed on the instrument platform of spacecraft Attitude motion simulation platform, on intelligence gauge head, there is target positioning system, three-dimensional perspective and mobile three-dimensional distance that laser tracker coordinates intelligent gauge head can follow the tracks of and determine that intelligent gauge head rotates; Gyroscope is fixed on the instrument platform of spacecraft Attitude motion simulation platform, the angular speed information of output region attitude of flight vehicle motion simulation platform; The first industrial control computer is arranged on the instrument platform of spacecraft Attitude motion simulation platform, carries out data processing work; The second industrial control computer is arranged under platform, gathers the output data of laser tracker, and by wireless network, sends to the first industrial control computer; The first industrial control computer receives laser tracker and gyrostatic attitude data, and carry out Data Fusion Filtering processing, the attitude angle information of output region attitude of flight vehicle motion simulation platform is demarcated gyrostatic drift simultaneously, revises in real time gyrostatic output angle speed.

The present invention also has following feature:

1, a kind of spacecraft Attitude motion simulation platform high-precision attitude as above is determined method, and the method that wherein Data Fusion Filtering is processed is as follows:

Step 1: according to attitude parameter conversion formula, adopt 312 to turn order, utilize laser tracker output air floating table attitude angle θ, ψ calculates the coordinate conversion matrix A of air supporting stage body coordinate system relative inertness system _br:

Wherein, θ, ψ is the air floating table attitude angle of laser tracker output;

Further calculate attitude quaternion:

$\begin{array}{c}{q}_0=\sqrt{1+A_{11}+A_{22}+A_{33}}/2\\ q_1=(A_{23}-A_{32})/4q_0\\ q_2=(A_{31}-A_{13})/4q_0\\ q_1=(A_{12}-A_{21})/4q_0\end{array}---\left(2\right)$

Step 2: the state equation of tectonic system

Select the vector part Q of deviation hypercomplex number _eform sextuple state variable X (t), the state equation of tectonic system with gyroscopic drift deviation delta b:

$\left[\begin{array}{c}{\stackrel{.}{Q}}_e\\ \mathrm{\Δ}\stackrel{.}{b}\end{array}\right]=\left[\begin{array}{cc}-[\mathrm{\ω}\×]& -\frac{1}{2}{C}_{\mathrm{bs}}^g\\ 0_{3\×3}& 0_{3\×3}\end{array}\right]\left[\begin{array}{c}{Q}_e\\ \mathrm{\Δb}\end{array}\right]+\left[\begin{array}{cc}-\frac{1}{2}{C}_{\mathrm{bs}}^g& 0_{3\×3}\\ 0_{3\×3}& C_{\mathrm{bs}}^g\end{array}\right]\left[\begin{array}{c}{n}_g\\ n_s\end{array}\right]---\left(3\right)$

$\stackrel{.}{X}\left(t\right)=\mathrm{FX}\left(t\right)+\mathrm{GW}\left(t\right)---\left(4\right)$

Wherein

$[\mathrm{\ω}\×]=\left[\begin{array}{ccc}0& -{\mathrm{\ω}}_z& {\mathrm{\ω}}_y\\ {\mathrm{\ω}}_z& 0& -{\mathrm{\ω}}_x\\ -{\mathrm{\ω}}_y& {\mathrm{\ω}}_x& 0\end{array}\right]---\left(5\right)$

F is the state matrix of system, G be systematic procedure noise drive matrix, ω be the angle of rotation speed of body coordinate system relative inertness system of spacecraft Attitude motion simulation platform in the projection of body coordinate system, the coordinate conversion matrix of body coordinate system of the measurement coordinate system space attitude of flight vehicle motion simulation platform of gyro, n _gthe measurement noise of gyro, n _sit is the driving noise of the associated drift of gyro;

Discretize is processed:

X _k＝Φ _k，k-1X _k-1+Γ _k-1W _k-1 (6)

Wherein

$\begin{array}{c}{\mathrm{\Φ}}_{k,k-1}=I_{6\×6}+F\·T+\frac{T^2}{2}{F}^2\\ {\mathrm{\Γ}}_{k,k-1}=(I_{6\×6}\·T+\frac{T^2}{2}F+\frac{T^3}{6}{F}^2)G\end{array}---\left(7\right)$

T is sampling time interval, W _k-1it is the process noise of system;

Step 3: the measurement equation of tectonic system

Choose the vector part Q of deviation hypercomplex number _efor measuring value Z _k, the measurement equation of tectonic system:

$Z_k=H_k{X}_k+C_{\mathrm{is}}^s{V}_k---\left(8\right)$

Wherein

H _k＝[I _3×3 0 _3×3] (9)

H _kmeasurement matrix, V _kthe measurement noise of laser tracker, it is the coordinate conversion matrix of the measurement coordinate system relative inertness coordinate system of laser tracker;

Step 4: a step stepwise predict

At k constantly, known k-1 attitude quaternion estimated value constantly the estimated value of gyroscopic drift the measurement angular speed u of gyro _k-1, measure and obtain the current k output hypercomplex number q of laser tracker constantly _k, gyrostatic output angle speed u _k;

Because hypercomplex number exists mould, be 1 constraint condition, and the multiplication cross operation relation of deviation hypercomplex number, for a step stepwise predict process, need to separately discuss to the gyroscopic drift in state variable and attitude quaternion:

(4.1) a step stepwise predict value of gyroscopic drift

Because gyroscopic drift changes slowly, k gyroscopic drift is constantly exactly k-1 estimated value constantly that is:

${\hat{b}}_{k|k-1}={\hat{b}}_{k-1}---\left(10\right)$

(4.2) a step stepwise predict value of attitude quaternion

According to the kinematical equation of attitude quaternion, obtain the k-1 rate of change of attitude quaternion constantly

${\stackrel{.}{\hat{q}}}_{k-1}=\frac{1}{2}{\hat{q}}_{k-1}\⊗\left\{\begin{array}{cc}0& {\widehat{\mathrm{\ω}}}_{k-1}\end{array}\right\}---\left(11\right)$

Wherein, be the constantly modified value of gyro output angle speed of k-1, i.e. the estimated value of spacecraft Attitude motion simulation platform angular speed, by utilizing gyroscopic drift to estimate gyro is exported to u _k-1in drift partly reject and obtain:

${\widehat{\mathrm{\ω}}}_{k-1}=u_{k-1}-{\hat{b}}_{k-1}---\left(12\right)$

Utilize adopt the method for correlation values integration, can obtain a k step stepwise predict value of attitude quaternion constantly

Further, according to the definition of deviation hypercomplex number, utilize a step stepwise predict value of attitude quaternion output q with laser tracker _k, obtain a step stepwise predict value of deviation hypercomplex number

${\hat{q}}_{e,k|k-1}={\hat{q}}_{k|k-1}^{-1}\⊗q_k---\left(13\right)$

Step 5: state upgrades

According to following formula, carry out state renewal, obtain the filtering estimated value of deviation hypercomplex number vector part and the filtering estimated value of gyroscope (3) drift bias

$P_{k/k-1}={\mathrm{\Φ}}_{k,k-1}{P}_{k-1}{\mathrm{\Φ}}_{k,k-1}^{|T}+{\mathrm{\Γ}}_{k-1}{Q}_{k-1}{\mathrm{\Γ}}_{k-1}^T---\left(14\right)$

$K_k=P_{k/k-1}{H}_k^T{(H_k{P}_{k/k-1}{H}_k^T+R_k)}^{-1}---\left(15\right)$

P _k＝(1-K _kHk ₎P _k/k-1 (16)

$\left[\begin{array}{c}{\hat{Q}}_{e,k}\\ \mathrm{\Δ}{\hat{b}}_k\end{array}\right]=\left[\begin{array}{c}{\hat{Q}}_{e,k|k-1}\\ \mathrm{\Δ}{\hat{b}}_{k|k-1}\end{array}\right]+K_k\·{\hat{Q}}_{e,k|k-1}---\left(17\right)$

Q wherein _kprocess noise statistical matrix, r _kmeasurement noise statistical matrix, p _k/k-1it is a step stepwise predict matrix of error statistics characteristic; K _kit is filter gain matrix; P _kit is error covariance matrix;

Step 6: attitude filtering information merges

Because hypercomplex number exists mould, be 1 constraint condition, and the multiplication cross operation relation of deviation hypercomplex number, attitude filtering information fusion process also needs gyroscopic drift and attitude quaternion separately to discuss:

(6.1) the filtering estimated value of gyroscopic drift

Gyroscopic drift deviation filtering estimated value with gyroscopic drift one step stepwise predict value be added, obtain the filtering estimated value of gyroscopic drift

${\hat{b}}_k={\hat{b}}_{k|k-1}+\mathrm{\Δ}{\hat{b}}_k={\hat{b}}_{k-1}+\mathrm{\Δ}{\hat{b}}_k---\left(18\right)$

(6.2) the filtering estimated value of attitude quaternion

According to the definition of deviation hypercomplex number, obtain the filtering estimated value of attitude quaternion

${\hat{q}}_k={\hat{q}}_{k|k-1}\⊗{\hat{q}}_{e,k}---\left(19\right)$

${\hat{q}}_{e,k}={\left[\begin{array}{cc}1& {\hat{Q}}_{e,k}\end{array}\right]}^T---\left(20\right)$

Step 7: output attitude parameter

(7.1) spacecraft Attitude motion simulation platform angular speed

Utilize the filtering estimated value of gyroscopic drift by gyro output angle speed u _kin drift value exclude, obtain spacecraft Attitude motion simulation platform angular speed

${\widehat{\mathrm{\ω}}}_k=u_k-{\hat{b}}_k---\left(21\right)$

(7.2) spacecraft Attitude motion simulation platform attitude angle θ, ψ:

According to the conversion formula of attitude parameter, utilize the filtering estimated value of attitude quaternion can obtain the attitude angle of spacecraft Attitude motion simulation platform current time:

2, a kind of spacecraft Attitude motion simulation platform high-precision attitude as above is determined method, wherein the coordinate conversion matrix of the measurement coordinate system relative inertness coordinate system of laser tracker scaling method be:

When using the attitude angle of laser tracker measurement space attitude of flight vehicle motion simulation platform, need the measurement coordinate system of known laser tracker with respect to the position relation between inertial coordinates system, i.e. the measurement coordinate system O of Calibration of Laser tracker _s-X _sy _sz _s, concrete grammar is as follows:

On initial point in inertial coordinates system and east northeast both direction axle, survey and on measured point of fixity, place respectively three target balls, measure three target balls and measure the coordinate in coordinate system at laser tracker, thereby calibrate laser tracker, measure the position relation between coordinate system relative inertness coordinate system

First adjust laser tracker and make its level, then measure O by laser tracker _starget ball [0 y on point and north orientation axle ₁0] ^tcoordinate in laser tracker coordinate system is respectively [a b c] ^t[x y z] ^t, by homogeneous coordinate transformation relation, have:

$\left[\begin{array}{c}0\\ y1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{cccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& -\sin {\mathrm{\θ}}_{\mathrm{is}}& 0& a\\ \sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0& b\\ 0& 0& 1& c\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]---\left(23\right)$

Solve thus θ _is, due to laser tracker level, so laser tracker measures coordinate system O _s-X _sy _sz _srelative inertness coordinate system O _i-X _iy _iz _ibetween coordinate conversion matrix for:

$C_{\mathrm{is}}^s=\left[\begin{array}{ccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& \sin {\mathrm{\θ}}_{\mathrm{is}}& 0\\ -\sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0\\ 0& 0& 1\end{array}\right]---\left(24\right)$

3, a kind of spacecraft Attitude motion simulation platform high-precision attitude as above is determined method, and gyro measures the coordinate conversion matrix of coordinate system space attitude of flight vehicle motion simulation platform body coordinate system scaling method, as follows:

According to gyrostatic Installation and Measurement, show that gyroscope measures coordinate system O _gx _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle α between axle forward ₁, α ₁, α ₃, O _gy _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle β between axle forward ₁, β ₂, β ₃, O _gz _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle γ between axle forward ₁, γ ₂, γ ₃, gyroscope measures coordinate system O _g-X _gy _gz _gspace attitude of flight vehicle motion simulation platform body coordinate system O _b-X _by _bz _brotational transform matrix for:

$C_{\mathrm{bs}}^g=\left[\begin{array}{ccc}\cos {\mathrm{\α}}_1& \cos {\mathrm{\β}}_1& \cos {\mathrm{\γ}}_1\\ \cos {\mathrm{\α}}_2& \cos {\mathrm{\β}}_2& \cos {\mathrm{\γ}}_2\\ \cos {\mathrm{\α}}_3& \cos {\mathrm{\β}}_3& \cos {\mathrm{\γ}}_3\end{array}\right]---\left(25\right).$

The invention provides a kind of spacecraft Attitude motion simulation platform high-precision attitude and determine method, measuring equipment is installed simple, measuring accuracy is high, the dynamically attitude angle of noncontacting measurement spacecraft Attitude motion simulation platform and angular speed information, and demarcate gyrostatic drift simultaneously, revise in real time gyrostatic output angle speed.The present invention builds ground physical simulation experiment device, on platform, the output information of laser tracker and gyro is carried out to Data Fusion Filtering processing, and more realistic working condition has higher real-time.The present invention is also for the accurate measurement of single axle table attitude angle, angular speed.

Accompanying drawing explanation

Fig. 1 is the composition schematic diagram of spacecraft Attitude motion simulation platform high-precision attitude determining device;

Fig. 2 is spacecraft Attitude motion simulation platform attitude angle precision curve map;

Fig. 3 is spacecraft Attitude motion simulation platform angle speed precision curve map;

Fig. 4 is the schematic diagram that Calibration of Laser tracker measures coordinate system;

Fig. 5 is each coordinate system transformational relation schematic diagram;

Fig. 6 is the communication pattern schematic diagram of spacecraft Attitude motion simulation platform high-precision attitude determining device.

Embodiment

Below in conjunction with accompanying drawing, the invention will be further described for example.

Embodiment 1:

In conjunction with Fig. 1, a kind of spacecraft Attitude motion simulation platform of the present invention high-precision attitude is determined method, the equipment adopting comprises laser tracker 1, intelligent gauge head 2, gyroscope 3 and two industrial control computers 4.5, and laser tracker 1 is installed under the platform of spacecraft Attitude motion simulation platform; Before attitude is determined, intelligence gauge head 2 is arranged under platform, the measurement coordinate system that coordinates laser tracker 1 Calibration of Laser tracker 1, in attitude deterministic process, intelligence gauge head 2 is fixed on the instrument platform of spacecraft Attitude motion simulation platform, on intelligence gauge head 2, have target positioning system, laser tracker 1 coordinates intelligent gauge head 2 can follow the tracks of and determine three-dimensional perspective and mobile three-dimensional distance that intelligent gauge head 2 rotates; Gyroscope 3 is fixed on the instrument platform of spacecraft Attitude motion simulation platform, the angular speed information of output region attitude of flight vehicle motion simulation platform; The first industrial control computer 4 is arranged on the instrument platform of spacecraft Attitude motion simulation platform, carries out data processing work; The second industrial control computer 5 is arranged under platform, gathers the output data of laser tracker 1, and by wireless network, sends to the first industrial control computer 4; The first industrial control computer 4 receives the attitude data of laser tracker 1 and gyroscope 3, and carry out Data Fusion Filtering processing, the attitude angle information of output region attitude of flight vehicle motion simulation platform, the drift of simultaneously demarcating gyroscope 3, revises the output angle speed of gyroscope 3 in real time.

Performing step is as follows:

Step 1: open device power supply (DPS), carry out next step after each equipment preheating completes;

Step 2: the measurement coordinate system of Calibration of Laser tracker 1;

Step 3: the measurement coordinate system of demarcating gyroscope 3;

Step 4: build wireless network, realize the network service of two industrial control computers 4.5 above and below the stage;

Step 5: utilize the output data of laser tracker 1 and gyroscope 3, obtain attitude angle and the angular speed information of air floating table by Data Fusion Filtering software 6.

The present invention utilizes the high-precision attitude of laser tracker 1 and gyroscope 3 associating implementation space attitude of flight vehicle motion simulation platforms to determine, wherein laser tracker 1 can be selected the product of the Switzerland's AT901 of Leica company product or U.S. API company, this sentences the AT901 of Leica company product is example, the measuring accuracy of its 3 d pose angle is 3 " (3 σ), measurement range reaches ± and 60 °.During use, laser tracker 1 is arranged on the support under air floating table, intelligence gauge head 2 is arranged on air floating table instrument platform and moves together with stage body, during installation, to guarantee the visual field demand between laser tracker 1 optical axis and intelligent gauge head 2 target balls, with assurance test macro, do not lose in the course of the work measurement degree of freedom, laser tracker 1 calculates the relative attitude information of intelligent gauge head 2 in real time according to measurement data.

The measurement of spacecraft Attitude motion simulation platform angular speed is realized by high precision hemispherical resonant gyro.Hemispherical resonant gyro is vertically mounted on to the spacecraft Attitude motion simulation platform center after leveling, and its sensitive axes is vertical with the rotating shaft of spacecraft Attitude motion simulation platform.The output packet of hemispherical resonant gyro has contained the constant value drift of spacecraft Attitude motion simulation platform, associated drift, the drift noises such as measurement noise, when reality is used, adopt the attitude information of laser tracker through the output angle speed of Data Fusion Filtering software correction gyro.

Embodiment 2:

The present invention proposes a kind of spacecraft Attitude motion simulation platform attitude based on EKF and determine improvement project, build ground physical simulation experiment device, by laser tracker and gyrostatic output information are carried out to Data Fusion Filtering processing, the attitude information that obtains spacecraft Attitude motion simulation platform, concrete steps are as follows:

Step 1: according to attitude parameter conversion formula, adopt 312 to turn order, utilize laser tracker output air floating table attitude angle θ, ψ calculates the coordinate conversion matrix A of air supporting stage body coordinate system relative inertness system _br:

Wherein, θ, ψ is the air floating table attitude angle of laser tracker output;

Further calculate attitude quaternion:

$\begin{array}{c}{q}_0=\sqrt{1+A_{11}+A_{22}+A_{33}}/2\\ q_1=(A_{23}-A_{32})/4q_0\\ q_2=(A_{31}-A_{13})/4q_0\\ q_1=(A_{12}-A_{21})/4q_0\end{array}---\left(27\right)$

Step 2: the state equation of tectonic system

Select the vector part Q of deviation hypercomplex number _eform sextuple state variable X (t), the state equation of tectonic system with gyroscopic drift deviation delta b:

$\left[\begin{array}{c}{\stackrel{.}{Q}}_e\\ \mathrm{\Δ}\stackrel{.}{b}\end{array}\right]=\left[\begin{array}{cc}-[\mathrm{\ω}\×]& -\frac{1}{2}{C}_{\mathrm{bs}}^g\\ 0_{3\×3}& 0_{3\×3}\end{array}\right]\left[\begin{array}{c}{Q}_e\\ \mathrm{\Δb}\end{array}\right]+\left[\begin{array}{cc}-\frac{1}{2}{C}_{\mathrm{bs}}^g& 0_{3\×3}\\ 0_{3\×3}& C_{\mathrm{bs}}^g\end{array}\right]\left[\begin{array}{c}{n}_g\\ n_s\end{array}\right]---\left(28\right)$

$\stackrel{.}{X}\left(t\right)=\mathrm{FX}\left(t\right)+\mathrm{GW}\left(t\right)---\left(29\right)$

Wherein

$[\mathrm{\ω}\×]=\left[\begin{array}{ccc}0& -{\mathrm{\ω}}_z& {\mathrm{\ω}}_y\\ {\mathrm{\ω}}_z& 0& -{\mathrm{\ω}}_x\\ -{\mathrm{\ω}}_y& {\mathrm{\ω}}_x& 0\end{array}\right]---\left(30\right)$

F is the state matrix of system, G be systematic procedure noise drive matrix, ω be the angle of rotation speed of body coordinate system relative inertness system of spacecraft Attitude motion simulation platform in the projection of body coordinate system, the coordinate conversion matrix of body coordinate system of the measurement coordinate system space attitude of flight vehicle motion simulation platform of gyro, n _gthe measurement noise of gyro, n _sit is the driving noise of the associated drift of gyro;

Discretize is processed:

X _k＝Φ _k，k-1X _k-1+Γ _k-1W _k-1 (31)

Wherein

$\begin{array}{c}{\mathrm{\Φ}}_{k,k-1}=I_{6\×6}+F\·T+\frac{T^2}{2}{F}^2\\ {\mathrm{\Γ}}_{k,k-1}=(I_{6\×6}\·T+\frac{T^2}{2}F+\frac{T^3}{6}{F}^2)G\end{array}---\left(32\right)$

T is sampling time interval, W _k-1it is the process noise of system;

Step 3: the measurement equation of tectonic system

Choose the vector part Q of deviation hypercomplex number _efor measuring value Z _k, the measurement equation of tectonic system:

$Z_k=H_k{X}_k+C_{\mathrm{is}}^s{V}_k---\left(33\right)$

Wherein

H _k＝[I _3×3 0 _3×3] (34)

H _kmeasurement matrix, V _kthe measurement noise of laser tracker, it is the coordinate conversion matrix of the measurement coordinate system relative inertness coordinate system of laser tracker;

Step 4: a step stepwise predict

At k constantly, known k-1 attitude quaternion estimated value constantly the estimated value of gyroscopic drift the measurement angular speed u of gyro _k-1, measure and obtain the current k output hypercomplex number q of laser tracker constantly _k, gyrostatic output angle speed u _k;

Because hypercomplex number exists mould, be 1 constraint condition, and the multiplication cross operation relation of deviation hypercomplex number, for a step stepwise predict process, need to separately discuss to the gyroscopic drift in state variable and attitude quaternion:

(4.1) a step stepwise predict value of gyroscopic drift

Because gyroscopic drift changes slowly, k gyroscopic drift is constantly exactly k-1 estimated value constantly that is:

${\hat{b}}_{k|k-1}={\hat{b}}_{k-1}---\left(35\right)$

(4.2) a step stepwise predict value of attitude quaternion

According to the kinematical equation of attitude quaternion, obtain the k-1 rate of change of attitude quaternion constantly

${\stackrel{.}{\hat{q}}}_{k-1}=\frac{1}{2}{\hat{q}}_{k-1}\⊗\left\{\begin{array}{cc}0& {\widehat{\mathrm{\ω}}}_{k-1}\end{array}\right\}---\left(36\right)$

Wherein, be the constantly modified value of gyro output angle speed of k-1, i.e. the estimated value of spacecraft Attitude motion simulation platform angular speed, by utilizing gyroscopic drift to estimate gyro is exported to u _k-1in drift partly reject and obtain:

${\widehat{\mathrm{\ω}}}_{k-1}=u_{k-1}-{\hat{b}}_{k-1}---\left(37\right)$

Utilize adopt the method for correlation values integration, can obtain a k step stepwise predict value of attitude quaternion constantly

Further, according to the definition of deviation hypercomplex number, utilize a step stepwise predict value of attitude quaternion output q with laser tracker _k, obtain a step stepwise predict value of deviation hypercomplex number

${\hat{q}}_{e,k|k-1}={\hat{q}}_{k|k-1}^{-1}\⊗q_k---\left(38\right)$

Step 5: state upgrades

According to following formula, carry out state renewal, obtain the filtering estimated value of deviation hypercomplex number vector part and the filtering estimated value of gyroscope (3) drift bias

$P_{k/k-1}={\mathrm{\Φ}}_{k,k-1}{P}_{k-1}{\mathrm{\Φ}}_{k,k-1}^{|T}+{\mathrm{\Γ}}_{k-1}{Q}_{k-1}{\mathrm{\Γ}}_{k-1}^T---\left(39\right)$

$K_k=P_{k/k-1}{H}_k^T{(H_k{P}_{k/k-1}{H}_k^T+R_k)}^{-1}---\left(40\right)$

P _k＝(1-K _kH _k)P _k/k-1 (41)

$\left[\begin{array}{c}{\hat{Q}}_{e,k}\\ \mathrm{\Δ}{\hat{b}}_k\end{array}\right]=\left[\begin{array}{c}{\hat{Q}}_{e,k|k-1}\\ \mathrm{\Δ}{\hat{b}}_{k|k-1}\end{array}\right]+K_k\·{\hat{Q}}_{e,k|k-1}---\left(42\right)$

Q wherein _kprocess noise statistical matrix, r _kmeasurement noise statistical matrix, p _k/k-1it is a step stepwise predict matrix of error statistics characteristic; K _kit is filter gain matrix; P _kit is error covariance matrix;

Step 6: attitude filtering information merges

Because hypercomplex number exists mould, be 1 constraint condition, and the multiplication cross operation relation of deviation hypercomplex number, attitude filtering information fusion process also needs gyroscopic drift and attitude quaternion separately to discuss:

(6.1) the filtering estimated value of gyroscopic drift

Gyroscopic drift deviation filtering estimated value with gyroscopic drift one step stepwise predict value be added, obtain the filtering estimated value of gyroscopic drift

${\hat{b}}_k={\hat{b}}_{k|k-1}+\mathrm{\Δ}{\hat{b}}_k={\hat{b}}_{k-1}+\mathrm{\Δ}{\hat{b}}_k---\left(43\right)$

(6.2) the filtering estimated value of attitude quaternion

According to the definition of deviation hypercomplex number, obtain the filtering estimated value of attitude quaternion

${\hat{q}}_k={\hat{q}}_{k|k-1}\⊗{\hat{q}}_{e,k}---\left(44\right)$

${\hat{q}}_{e,k}={\left[\begin{array}{cc}1& {\hat{Q}}_{e,k}\end{array}\right]}^T---\left(45\right)$

Step 7: output attitude parameter

(7.1) spacecraft Attitude motion simulation platform angular speed

Utilize the filtering estimated value of gyroscopic drift by gyro output angle speed u _kin drift value exclude, obtain spacecraft Attitude motion simulation platform angular speed

${\widehat{\mathrm{\ω}}}_k=u_k-{\hat{b}}_k---\left(46\right)$

(7.2) spacecraft Attitude motion simulation platform attitude angle θ, ψ:

According to the conversion formula of attitude parameter, utilize the filtering estimated value of attitude quaternion can obtain the attitude angle of spacecraft Attitude motion simulation platform current time:

In conjunction with Fig. 2-3, the measuring accuracy of laser tracker is 3 " (3 σ); gyrostatic constant value drift is 1.5 °/h (3 σ); random drift is 0.003 °/h (3 σ); measurement noise is 10 " (3 σ), through mathematical simulation, spacecraft Attitude motion simulation platform attitude angle precision is 0.05 ", angle speed precision is 0.7 ".

Embodiment 3:

In conjunction with Fig. 4, introduce the measurement coordinate system O of laser tracker _s-X _sy _sz _srelative inertness coordinate system O _i-X _iy _iz _icoordinate conversion matrix scaling method.

First adjust laser tracker and make its level, then measure O by laser tracker _starget ball [0 y on point and north orientation axle ₁0] ^tcoordinate in laser tracker coordinate system is respectively [a b c] ^t[x y z] ^t, by homogeneous coordinate transformation relation, have:

$\left[\begin{array}{c}0\\ y1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{cccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& -\sin {\mathrm{\θ}}_{\mathrm{is}}& 0& a\\ \sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0& b\\ 0& 0& 1& c\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]---\left(23\right)$

Can solve θ thus _is, due to laser tracker level, therefore by laser tracker, measure coordinate system O _s-X _sy _sz _srelative inertness coordinate system O _i-X _iy _iz _ibetween coordinate conversion matrix for:

$C_{\mathrm{is}}^s=\left[\begin{array}{ccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& \sin {\mathrm{\θ}}_{\mathrm{is}}& 0\\ -\sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0\\ 0& 0& 1\end{array}\right]---\left(24\right)$

Embodiment 4:

In conjunction with Fig. 5, introduce gyro and measure coordinate system O _g-X _gy _gz _gspace attitude of flight vehicle motion simulation platform body coordinate system O _b-X _by _bz _bcoordinate conversion matrix scaling method, as follows

According to gyrostatic Installation and Measurement, show that gyroscope measures coordinate system O _gx _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle α between axle forward ₁, α ₂, α ₃, O _gy _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle β between axle forward ₁, β ₂, β ₃, O _gz _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle γ between axle forward ₁, γ ₂, γ ₃, gyroscope measures coordinate system O _g-X _gy _gz _gspace attitude of flight vehicle motion simulation platform body coordinate system O _b-X _by _bz _brotational transform matrix for:

$C_{\mathrm{bs}}^g=\left[\begin{array}{ccc}\cos {\mathrm{\α}}_1& \cos {\mathrm{\β}}_1& \cos {\mathrm{\γ}}_1\\ \cos {\mathrm{\α}}_2& \cos {\mathrm{\β}}_2& \cos {\mathrm{\γ}}_2\\ \cos {\mathrm{\α}}_3& \cos {\mathrm{\β}}_3& \cos {\mathrm{\γ}}_3\end{array}\right]---\left(25\right).$

Embodiment 5:

In conjunction with Fig. 6, introduce the communication pattern of spacecraft Attitude motion simulation platform high-precision attitude determining device.

Industrial control computer on platform gathers the angular rate data from gyroscope output by data acquisition scale.Industrial control computer under platform gathers the attitude angle data from laser tracker output by data acquisition scale, and sends to the industrial control computer on platform by wireless network.Industrial control computer on platform utilizes wireless network to receive the data from the industrial control computer under platform, and carry out Data Fusion Filtering processing, the attitude angle information of output region attitude of flight vehicle motion simulation platform, demarcate gyrostatic drift simultaneously, revise in real time gyrostatic output angle speed.

Claims (4)

1. a spacecraft Attitude motion simulation platform high-precision attitude is determined method, the equipment adopting comprises laser tracker, intelligent gauge head, gyroscope and two industrial control computers, it is characterized in that: laser tracker is installed under the platform of spacecraft Attitude motion simulation platform; Before attitude is determined, intelligence gauge head is arranged under platform, the measurement coordinate system that coordinates laser tracker Calibration of Laser tracker, in attitude deterministic process, intelligence gauge head is fixed on the instrument platform of spacecraft Attitude motion simulation platform, on intelligence gauge head, there is target positioning system, three-dimensional perspective and mobile three-dimensional distance that laser tracker coordinates intelligent gauge head can follow the tracks of and determine that intelligent gauge head rotates; Gyroscope is fixed on the instrument platform of spacecraft Attitude motion simulation platform, the angular speed information of output region attitude of flight vehicle motion simulation platform; The first industrial control computer is arranged on the instrument platform of spacecraft Attitude motion simulation platform, carries out data processing work; The second industrial control computer is arranged under platform, gathers the output data of laser tracker, and by wireless network, sends to the first industrial control computer; The first industrial control computer receives laser tracker and gyrostatic attitude data, and carry out Data Fusion Filtering processing, the attitude angle information of output region attitude of flight vehicle motion simulation platform is demarcated gyrostatic drift simultaneously, revises in real time gyrostatic output angle speed.

2. a kind of spacecraft Attitude motion simulation platform high-precision attitude according to claim 1 is determined method, it is characterized in that, the method that described Data Fusion Filtering is processed is as follows:

Step 1: according to attitude parameter conversion formula, adopt 312 to turn order, utilize laser tracker output air floating table attitude angle θ, ψ calculates the coordinate conversion matrix A of air supporting stage body coordinate system relative inertness system _br:

Wherein, θ, ψ is the air floating table attitude angle of laser tracker output;

Further calculate attitude quaternion:

$\begin{array}{c}{q}_0=\sqrt{1+A_{11}+A_{22}+A_{33}}/2\\ q_1=(A_{23}-A_{32})/4q_0\\ q_2=(A_{31}-A_{13})/4q_0\\ q_1=(A_{12}-A_{21})/4q_0\end{array}---\left(2\right)$

Step 2: the state equation of tectonic system

Select the vector part Q of deviation hypercomplex number _eform sextuple state variable X (t), the state equation of tectonic system with gyroscopic drift deviation delta b:

$\left[\begin{array}{c}{\stackrel{.}{Q}}_e\\ \mathrm{\Δ}\stackrel{.}{b}\end{array}\right]=\left[\begin{array}{cc}-[\mathrm{\ω}\×]& -\frac{1}{2}{C}_{\mathrm{bs}}^g\\ 0_{3\×3}& 0_{3\×3}\end{array}\right]\left[\begin{array}{c}{Q}_e\\ \mathrm{\Δb}\end{array}\right]+\left[\begin{array}{cc}-\frac{1}{2}{C}_{\mathrm{bs}}^g& 0_{3\×3}\\ 0_{3\×3}& C_{\mathrm{bs}}^g\end{array}\right]\left[\begin{array}{c}{n}_g\\ n_s\end{array}\right]---\left(3\right)$

$\stackrel{.}{X}\left(t\right)=\mathrm{FX}\left(t\right)+\mathrm{GW}\left(t\right)---\left(4\right)$

Wherein

$[\mathrm{\ω}\×]=\left[\begin{array}{ccc}0& -{\mathrm{\ω}}_z& {\mathrm{\ω}}_y\\ {\mathrm{\ω}}_z& 0& -{\mathrm{\ω}}_x\\ -{\mathrm{\ω}}_y& {\mathrm{\ω}}_x& 0\end{array}\right]---\left(5\right)$

F is the state matrix of system, G be systematic procedure noise drive matrix, ω be the angle of rotation speed of body coordinate system relative inertness system of spacecraft Attitude motion simulation platform in the projection of body coordinate system, the coordinate conversion matrix of body coordinate system of the measurement coordinate system space attitude of flight vehicle motion simulation platform of gyro, n _gthe measurement noise of gyro, n _sit is the driving noise of the associated drift of gyro;

Discretize is processed:

X _k＝Φ _k，k-1X _k-1+Γ _k-1W _k-1 (6)

Wherein

$\begin{array}{c}{\mathrm{\Φ}}_{k,k-1}=I_{6\×6}+F\·T+\frac{T^2}{2}{F}^2\\ {\mathrm{\Γ}}_{k,k-1}=(I_{6\×6}\·T+\frac{T^2}{2}F+\frac{T^3}{6}{F}^2)G\end{array}---\left(7\right)$

T is sampling time interval, W _k-1it is the process noise of system;

Step 3: the measurement equation of tectonic system

Choose the vector part Q of deviation hypercomplex number _efor measuring value Z _k, the measurement equation of tectonic system:

$Z_k=H_k{X}_k+C_{\mathrm{is}}^s{V}_k---\left(8\right)$

Wherein

H _k＝[I _3×3 0 _3×3] (9)

H _kmeasurement matrix, V _kthe measurement noise of laser tracker, it is the coordinate conversion matrix of the measurement coordinate system relative inertness coordinate system of laser tracker;

Step 4: a step stepwise predict

At k constantly, known k-1 attitude quaternion estimated value constantly the estimated value of gyroscopic drift the measurement angular speed u of gyro _k-1, measure and obtain the current k output hypercomplex number q of laser tracker constantly _k, gyrostatic output angle speed u _k;

For a step stepwise predict process, need to separately discuss to the gyroscopic drift in state variable and attitude quaternion:

(4.1) a step stepwise predict value of gyroscopic drift

Because gyroscopic drift changes slowly, k gyroscopic drift is constantly exactly k-1 estimated value constantly that is:

${\hat{b}}_{k|k-1}={\hat{b}}_{k-1}---\left(10\right)$

(4.2) a step stepwise predict value of attitude quaternion

According to the kinematical equation of attitude quaternion, obtain the k-1 rate of change of attitude quaternion constantly

${\stackrel{.}{\hat{q}}}_{k-1}=\frac{1}{2}{\hat{q}}_{k-1}\⊗\left\{\begin{array}{cc}0& {\widehat{\mathrm{\ω}}}_{k-1}\end{array}\right\}---\left(11\right)$

Wherein, be the constantly modified value of gyro output angle speed of k-1, i.e. the estimated value of spacecraft Attitude motion simulation platform angular speed, by utilizing gyroscopic drift to estimate gyro is exported to u _k-1in drift partly reject and obtain:

${\widehat{\mathrm{\ω}}}_{k-1}=u_{k-1}-{\hat{b}}_{k-1}---\left(12\right)$

Utilize adopt the method for correlation values integration, can obtain a k step stepwise predict value of attitude quaternion constantly

Further, according to the definition of deviation hypercomplex number, utilize a step stepwise predict value of attitude quaternion output q with laser tracker _k, obtain a step stepwise predict value of deviation hypercomplex number

${\hat{q}}_{e,k|k-1}={\hat{q}}_{k|k-1}^{-1}\⊗q_k---\left(13\right)$

Step 5: state upgrades

According to following formula, carry out state renewal, obtain the filtering estimated value of deviation hypercomplex number vector part filtering estimated value with gyroscopic drift deviation

$P_{k/k-1}={\mathrm{\Φ}}_{k,k-1}{P}_{k-1}{\mathrm{\Φ}}_{k,k-1}^{|T}+{\mathrm{\Γ}}_{k-1}{Q}_{k-1}{\mathrm{\Γ}}_{k-1}^T---\left(14\right)$

$K_k=P_{k/k-1}{H}_k^T{(H_k{P}_{k/k-1}{H}_k^T+R_k)}^{-1}---\left(15\right)$

P _k＝(1-K _kH _k)P _k/k-1 (16)

$\left[\begin{array}{c}{\hat{Q}}_{e,k}\\ \mathrm{\Δ}{\hat{b}}_k\end{array}\right]=\left[\begin{array}{c}{\hat{Q}}_{e,k|k-1}\\ \mathrm{\Δ}{\hat{b}}_{k|k-1}\end{array}\right]+K_k\·{\hat{Q}}_{e,k|k-1}---\left(17\right)$

Q wherein _kprocess noise statistical matrix, r _kmeasurement noise statistical matrix, p _k/k-1it is a step stepwise predict matrix of error statistics characteristic; K _kit is filter gain matrix; P _kit is error covariance matrix;

Step 6: attitude filtering information merges

Attitude filtering information fusion process also needs gyroscopic drift and attitude quaternion separately to discuss:

(6.1) the filtering estimated value of gyroscopic drift

Gyroscopic drift deviation filtering estimated value with gyroscopic drift one step stepwise predict value be added, obtain the filtering estimated value of gyroscopic drift

${\hat{b}}_k={\hat{b}}_{k|k-1}+\mathrm{\Δ}{\hat{b}}_k={\hat{b}}_{k-1}+\mathrm{\Δ}{\hat{b}}_k---\left(18\right)$

(6.2) the filtering estimated value of attitude quaternion

According to the definition of deviation hypercomplex number, obtain the filtering estimated value of attitude quaternion

${\hat{q}}_k={\hat{q}}_{k|k-1}\⊗{\hat{q}}_{e,k}---\left(19\right)$

${\hat{q}}_{e,k}={\left[\begin{array}{cc}1& {\hat{Q}}_{e,k}\end{array}\right]}^T---\left(20\right)$

Step 7: output attitude parameter

(7.1) spacecraft Attitude motion simulation platform angular speed

Utilize the filtering estimated value of gyroscopic drift by gyro output angle speed u _kin drift value exclude, obtain spacecraft Attitude motion simulation platform angular speed

${\widehat{\mathrm{\ω}}}_k=u_k-{\hat{b}}_k---\left(21\right)$

(7.2) spacecraft Attitude motion simulation platform attitude angle θ, ψ:

According to the conversion formula of attitude parameter, utilize the filtering estimated value of attitude quaternion obtain the attitude angle of spacecraft Attitude motion simulation platform current time:

3. a kind of spacecraft Attitude motion simulation platform high-precision attitude according to claim 1 is determined method, it is characterized in that the coordinate conversion matrix of the measurement coordinate system relative inertness coordinate system of laser tracker scaling method, i.e. the measurement coordinate system O of Calibration of Laser tracker _s-X _sy _sz _s, concrete grammar is as follows:

On initial point in inertial coordinates system and east northeast both direction axle, survey and on measured point of fixity, place respectively three target balls, measure three target balls and measure the coordinate in coordinate system at laser tracker, thereby calibrate laser tracker, measure the position relation between coordinate system relative inertness coordinate system

First adjust laser tracker and make its level, then measure O by laser tracker _starget ball [0 y on point and north orientation axle ₁0] ^tcoordinate in laser tracker coordinate system is respectively [a b c] ^t[x y z] ^t, by homogeneous coordinate transformation relation, have:

$\left[\begin{array}{c}0\\ y1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{cccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& -\sin {\mathrm{\θ}}_{\mathrm{is}}& 0& a\\ \sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0& b\\ 0& 0& 1& c\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]---\left(23\right)$

Solve thus θ _is, due to laser tracker level, so laser tracker measures coordinate system O _s-X _sy _sz _srelative inertness coordinate system O _i-X _iy _iz _ibetween coordinate conversion matrix for:

$C_{\mathrm{is}}^s=\left[\begin{array}{ccc}\cos {\mathrm{\θ}}_{\mathrm{is}}& \sin {\mathrm{\θ}}_{\mathrm{is}}& 0\\ -\sin {\mathrm{\θ}}_{\mathrm{is}}& \cos {\mathrm{\θ}}_{\mathrm{is}}& 0\\ 0& 0& 1\end{array}\right]---\left(24\right).$

4. a kind of spacecraft Attitude motion simulation platform high-precision attitude according to claim 1 is determined method, it is characterized in that, gyro measures the coordinate conversion matrix of coordinate system space attitude of flight vehicle motion simulation platform body coordinate system scaling method, as follows:

According to gyrostatic Installation and Measurement, show that gyroscope measures coordinate system O _gx _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle α between axle forward ₁, α ₂, α ₃, O _gy _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle β between axle forward ₁, β ₂, β ₃, O _gz _gaxle forward and instrument platform body coordinate system O _bx _baxle, O _by _baxle, O _bz _bangle γ between axle forward ₁, γ ₂, γ ₃, gyroscope measures coordinate system O _g-X _gy _gz _gspace attitude of flight vehicle motion simulation platform body coordinate system O _b-X _by _bz _brotational transform matrix for:

$C_{\mathrm{bs}}^g=\left[\begin{array}{ccc}\cos {\mathrm{\α}}_1& \cos {\mathrm{\β}}_1& \cos {\mathrm{\γ}}_1\\ \cos {\mathrm{\α}}_2& \cos {\mathrm{\β}}_2& \cos {\mathrm{\γ}}_2\\ \cos {\mathrm{\α}}_3& \cos {\mathrm{\β}}_3& \cos {\mathrm{\γ}}_3\end{array}\right]---\left(25\right).$