CN103954351A - Measuring method of micro angle vibration of spacecraft - Google Patents

Abstract

The invention discloses a measuring system and method of micro angle vibration of a spacecraft. An acceleration measurement combination is arranged in a spacecraft carrier coordinate system, low-frequency minimal-level line vibration, directly related to micro angle vibration, of a carrier is measured, data processing is performed by combining the relative position relationship of the acceleration measurement combination and a spacecraft carrier barycenter, and micro angle vibration of the spacecraft is calculated. The measuring system and method achieves measurement of micro angle vibration of the spacecraft under the space environment, and have the advantages of being high in reliability and accuracy, small in size and low in cost.

Description

The measuring method of the micro-angular oscillation of a kind of spacecraft

Technical field

The present invention relates to the measuring method of the micro-angular oscillation of a kind of spacecraft, belong to inertial technology field.

Background technology

May there is flutter in-orbit time in spacecraft, main manifestations is the angle jitter of steady-state sine response, random fluctuation or damped oscillation under movable part or outside mechanics environment force.The value (generally more than 0.1 rad) of flutter and frequency spectrum (0.1～500Hz) because of interference source and Flight Vehicle Structure different.Micro-angular oscillation disturbance meeting of spacecraft affects imaging precision or the pointing accuracy of camera or sighting system, has also reduced the accuracy of space science experiment simultaneously.Therefore, the small magnitude angular oscillation of accurate, real-time measurement space aircraft, and as analyzing and compensate with reference to data diagonal angle Vibration Condition, significant in aerospace application.

At present, the satellites such as China ZY, GF use vibration transducer to measure aircraft microgravity acceleration environment, by measuring the trace level line vibration in certain frequency band at orbit aerocraft, for every scientific research provides foundation, but cannot directly measure the angular oscillation of aircraft, and then aircraft angle jitter conditions is analyzed and compensated.Laser gyro can meet the requirement of micro-angular oscillation measurement index on measuring accuracy and Measurement bandwidth, but due to the restriction of its volume, power consumption and environmental suitability, yet there are no both at home and abroad at present and be applied to the report that the micro-angular oscillation of spacecraft is measured.China has developed the micro-angular oscillation survey sensor for spacecraft based on magnetohydrodynamics principle, and due to the restriction of technology and technological level, the index short-terms such as its precision, reliability, volume and life-span are also difficult to meet the requirement of practical engineering application.Therefore, along with Chinese Space technology is in the continuous expansion of military and civilian field application, in the urgent need to exploring the micro-angular oscillation measuring method of effective spacecraft, the requirement of measuring for micro-angular oscillation with the aircraft under meeting spatial environment.

Summary of the invention

Technology of the present invention is dealt with problems and is: for the deficiencies in the prior art, a kind of highly reliable, high precision, small size and the micro-angular oscillation measuring method of spacecraft are cheaply provided, have solved the domestic difficult problem that cannot carry out exactly micro-angular oscillation measurement at spacecraft.

Technical solution of the present invention is:

A measuring system for the micro-angular oscillation of spacecraft, comprises acceleration analysis combination, resistance sampling circuit, every straight amplifying circuit;

Acceleration analysis combination is exported the aircraft carrier line vibration measuring with current forms, current signal is converted to voltage signal by resistance sampling circuit, the DC component of every straight amplifying circuit, this voltage signal being carried out in secondary amplification filtered signal obtains vibrating corresponding voltage signal with line, then utilize low frequency wire vibration calibrating device to carry out calibration to accelerometer measures combined, voltage signal can be converted to low frequency, the vibration of trace level line.

Described acceleration analysis combination adopts three axle high-precision quartz flexure accelerometerses to realize, and accelerometer measures combined is positioned over respectively on spacecraft carrier coordinate system X-axis, Y-axis and Z axis.

Described resistance sampling circuit comprises operational amplifier A 1, resistance R 1, R2, R3, capacitor C 1; The positive input ground connection of operational amplifier A 1, reverse input end connects one end of R2 and C1, and the other end of R2 connects R1 and is connected with one end of R3, and the other end of R3 and the other end of C1 are all connected to the output terminal of operational amplifier A 1, the other end ground connection of R1.

Describedly comprise operational amplifier A 2 every straight amplifying circuit, resistance R 4, R5, R6, capacitor C 2; The output terminal of one end concatenation operation amplifier A1 of capacitor C 2, positive input and the resistance R 4 of the other end concatenation operation amplifier of capacitor C 2, other one end ground connection of R4, the reverse input end of operational amplifier A 2 connects R5 and R6, the other end ground connection of R5, the other end of R6 is connected with the output terminal of operational amplifier A 2.

Based on a measuring method for micro-angular oscillation measuring system, comprise that step is as follows:

(1) degree of will speed up is measured combination and is arranged in spacecraft carrier coordinate system, measures low frequency, the vibration of trace level line of carrier;

The line vibration that described acceleration analysis multiple measurement obtains can be expressed as:

$a=\mathrm{\ω}\×\mathrm{\ω}\×r+\stackrel{\·}{\mathrm{\ω}}\×r$

Wherein, ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration, r is the position vector at the relative carrier barycenter O of particle place.

(2) utilize the relative position relation of acceleration analysis combination and spacecraft carrier barycenter to carry out data processing, and then calculate micro-angular oscillation of spacecraft the low frequency measuring in step (1), the vibration of trace level line.

Accelerometer measures combined is positioned over respectively the B on spacecraft carrier coordinate system X-axis, Y-axis and Z axis, C, and D 3 points, the position vector of space aircraft carrier barycenter O is R _bO, R _cO, R _dO, the line vibration that each measurement combination records is as follows:

$a_B=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{BO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{BO}}$

$a_C=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{CO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{CO}}$

$a_D=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{DO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{DO}}$

A _b, a _cand a _dfor the line vibration vector that accelerometer measures combined measures, direction is respectively spacecraft carrier coordinate system X, Y and Z axis, and ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration.

In described step (2) to low frequency, trace level line vibration to carry out the method for data processing as follows: set up X, Y and Z axis angular acceleration with line vibration a _b, a _c, a _dbetween relation:

${\stackrel{\·}{\mathrm{\ω}}}_X=\frac{a_C^Z}{{2R}_{\mathrm{CO}}}-\frac{a_D^Y}{{2R}_{\mathrm{DO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Y=\frac{a_D^X}{{2R}_{\mathrm{DO}}}-\frac{a_B^Z}{{2R}_{\mathrm{BO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Z=\frac{a_B^Y}{{2R}_{\mathrm{BO}}}-\frac{a_C^X}{{2R}_{\mathrm{CO}}}$

Wherein, the vibration of the line along aircraft carrier coordinate system Y-axis, Z axis that the accelerometer measures combined of placing for B point in aircraft carrier coordinate system X-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Z axis that the accelerometer measures combined of placing for C point in aircraft carrier coordinate system Y-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Y-axis that the accelerometer measures combined of placing for D point on aircraft carrier coordinate system Z axis records;

Utilize quadravalence Runge-Kutta method to carry out integral operation, can obtain micro-angular oscillation of spacecraft:

${\mathrm{\ω}}_X\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_X(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_X(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_X\left(t\right)$

${\mathrm{\ω}}_Y\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y\left(t\right)$

${\mathrm{\ω}}_Z\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z\left(t\right)$

Wherein, t represents current sampling instant, and T is the sampling period.

The present invention compared with prior art tool has the following advantages:

(1) the present invention has set up the corresponding relation between the vibration of angular acceleration and line under space microgravity environment, for the micro-angular oscillation measurement of spacecraft provides theoretical foundation, solve existing equipment and cannot carry out the difficult problem that the micro-angular oscillation of high precision spacecraft is measured.

(2) the acceleration analysis combination the present invention relates to selects high-precision quartz flexure accelerometers as sensitive element, has the advantages that precision is high, volume is little and environmental suitability is strong.

(3) resistance sampling circuit adopts T-shaped resistor network to reduce circuit aliasing noise and external environment condition temperature, humidity changes the impact on signal.What adopt that first isolated DC component amplifies again has avoided the amplification to direct current signal every straight amplifying circuit form, has improved the signal to noise ratio (S/N ratio) of signal.

(4) the present invention measures directly related with micro-angular oscillation low frequency, the vibration of trace level line by accelerometer measures combined, utilize mathematical operation to obtain the micro-angular oscillation of spacecraft, simple in structure, method economy, limited at carrier bulk, working environment is special and measuring accuracy requires the micro-angular oscillation fields of measurement of high spacecraft to have broad application prospects.

Brief description of the drawings

Fig. 1 is systematic schematic diagram of the present invention;

Fig. 2 is resistance sampling circuit diagram of the present invention;

Fig. 3 is of the present invention every straight amplification circuit diagram.

Embodiment

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

As shown in Figure 1, the measuring system of the micro-angular oscillation of a kind of spacecraft of the present invention, comprises acceleration analysis combination, resistance sampling circuit, every straight amplifying circuit; Acceleration analysis combination is exported the aircraft carrier line vibration measuring with current forms, current signal is converted to voltage signal by resistance sampling circuit, the DC component of every straight amplifying circuit, this voltage signal being carried out in secondary amplification filtered signal obtains vibrating corresponding voltage signal with line, then utilize low frequency wire vibration calibrating device to carry out calibration to accelerometer measures combined, voltage signal can be converted to low frequency, the vibration of trace level line.

Acceleration analysis combination adopts three axle high-precision quartz flexure accelerometerses to realize, and accelerometer measures combined is positioned over respectively on spacecraft carrier coordinate system X-axis, Y-axis and Z axis.

As shown in Figure 2, resistance sampling circuit comprises operational amplifier A 1, resistance R 1, R2, R3, capacitor C 1; The positive input ground connection of operational amplifier A 1, reverse input end connects one end of R2 and C1, and the other end of R2 connects R1 and is connected with one end of R3, and the other end of R3 and the other end of C1 are all connected to the output terminal of operational amplifier A 1, the other end ground connection of R1.

As shown in Figure 3, comprise operational amplifier A 2 every straight amplifying circuit, resistance R 4, R5, R6, capacitor C 2; The output terminal of one end concatenation operation amplifier A1 of capacitor C 2, positive input and the resistance R 4 of the other end concatenation operation amplifier of capacitor C 2, other one end ground connection of R4, the reverse input end of operational amplifier A 2 connects R5 and R6, the other end ground connection of R5, the other end of R6 is connected with the output terminal of operational amplifier A 2.

As shown in Figure 1, a kind of measuring method based on micro-angular oscillation measuring system, comprises that step is as follows:

(1) degree of will speed up is measured combination and is arranged in spacecraft carrier coordinate system, measures low frequency, the vibration of trace level line of carrier;

The line vibration that acceleration analysis multiple measurement obtains can be expressed as:

$a=\mathrm{\ω}\×\mathrm{\ω}\×r+\stackrel{\·}{\mathrm{\ω}}\×r$

Wherein, ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration, r is the position vector at the relative carrier barycenter O of particle place.

(2) utilize the relative position relation of acceleration analysis combination and spacecraft carrier barycenter to carry out data processing, and then calculate micro-angular oscillation of spacecraft the low frequency measuring in step (1), the vibration of trace level line.

Accelerometer measures combined is positioned over respectively the B on spacecraft carrier coordinate system X-axis, Y-axis and Z axis, C, and D 3 points, the position vector of space aircraft carrier barycenter O is R _bO, R _cO, R _dO, the line vibration that each measurement combination records is as follows:

$a_B=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{BO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{BO}}$

$a_C=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{CO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{CO}}$

$a_D=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{DO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{DO}}$

A _b, a _cand a _dfor the line vibration vector that accelerometer measures combined measures, direction is respectively spacecraft carrier coordinate system X, Y and Z axis, and ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration.

In step (2) to low frequency, trace level line vibration to carry out the method for data processing as follows: set up X, Y and Z axis angular acceleration with line vibration a _b, a _c, a _dbetween relation:

${\stackrel{\·}{\mathrm{\ω}}}_X=\frac{a_C^Z}{{2R}_{\mathrm{CO}}}-\frac{a_D^Y}{{2R}_{\mathrm{DO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Y=\frac{a_D^X}{{2R}_{\mathrm{DO}}}-\frac{a_B^Z}{{2R}_{\mathrm{BO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Z=\frac{a_B^Y}{{2R}_{\mathrm{BO}}}-\frac{a_C^X}{{2R}_{\mathrm{CO}}}$

Wherein, the vibration of the line along aircraft carrier coordinate system Y-axis, Z axis that the accelerometer measures combined of placing for B point in aircraft carrier coordinate system X-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Z axis that the accelerometer measures combined of placing for C point in aircraft carrier coordinate system Y-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Y-axis that the accelerometer measures combined of placing for D point on aircraft carrier coordinate system Z axis records;

Utilize quadravalence Runge-Kutta method to carry out integral operation, can obtain micro-angular oscillation of spacecraft:

${\mathrm{\ω}}_X\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_X(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_X(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_X\left(t\right)$

${\mathrm{\ω}}_Y\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y\left(t\right)$

${\mathrm{\ω}}_Z\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z\left(t\right)$

Wherein, t represents current sampling instant, and T is the sampling period.

In the present invention, accelerometer measures combined range is that measurement range is ± 0.1g, and resolution is 5ug, and the position vector of space aircraft carrier barycenter O is R _bO, R _cO, R _dObe 1m, be output as ± 0.125mA of accelerometer measures combined full scale ± 0.1g (selecting sensitive element constant multiplier is 1.25mA/g), resistance sampling circuit is as Fig. 2, resistance sampling circuit adopts T-shaped resistor network to realize, wherein capacitor C 1 size is 47pF, be used for preventing circuit self-excitation, R1=1K Ω, R2=500 Ω, R3=10K Ω, circuit gain is: G=R2 × (1+R3/R1+R3/R2), as calculated, resistance sampling circuit gain is 15500V/A, and the voltage signal scope of output is-1.9375～+ 1.9375V.

Because the micro-angular oscillation Measurement bandwidth of spacecraft requires as 0.1-150Hz, need the DC component of the voltage signal that filtering resistance sampling obtains.In order to improve Signal-to-Noise, the signal after filtering DC component is carried out to secondary amplification, enlargement factor is 2 times, and the voltage signal scope of output is-3.875～+ 3.875V, and every straight amplifying circuit, as Fig. 3, block isolating circuit signal cutoff frequency is 0.0096Hz.Wherein capacitor C 2 sizes are 3.3uF, R4=5M Ω, R5=5K Ω, R5=6K Ω.

The present embodiment has been applied to certain type high-resolution satellite, flying quality shows, micro-angular oscillation Measurement bandwidth of the present embodiment spacecraft reaches 0.1-150Hz, and angular resolution is 0.01 rad, measuring accuracy is better than 10%, can measure requirement by the micro-angular oscillation of meeting spatial aircraft.

The undocumented technology of the present invention belongs to techniques well known.

Claims (9)

1. a measuring system for the micro-angular oscillation of spacecraft, is characterized in that comprising: acceleration analysis combination, resistance sampling circuit, every straight amplifying circuit;

Acceleration analysis combination is exported the aircraft carrier line vibration measuring with current forms, current signal is converted to voltage signal by resistance sampling circuit, the DC component of every straight amplifying circuit, this voltage signal being carried out in secondary amplification filtered signal obtains vibrating corresponding voltage signal with line, then utilize low frequency wire vibration calibrating device to carry out calibration to accelerometer measures combined, voltage signal can be converted to low frequency, the vibration of trace level line.

2. the measuring system of the micro-angular oscillation of a kind of spacecraft according to claim 1, it is characterized in that: described acceleration analysis combination adopts three axle high-precision quartz flexure accelerometerses to realize, and accelerometer measures combined is positioned over respectively on spacecraft carrier coordinate system X-axis, Y-axis and Z axis.

3. the measuring system of the micro-angular oscillation of a kind of spacecraft according to claim 1, is characterized in that: described resistance sampling circuit comprises operational amplifier A 1, resistance R 1, R2, R3, capacitor C 1; The positive input ground connection of operational amplifier A 1, reverse input end connects one end of R2 and C1, and the other end of R2 connects R1 and is connected with one end of R3, and the other end of R3 and the other end of C1 are all connected to the output terminal of operational amplifier A 1, the other end ground connection of R1.

4. the measuring system of the micro-angular oscillation of a kind of spacecraft according to claim 3, is characterized in that: describedly comprise operational amplifier A 2 every straight amplifying circuit, resistance R 4, R5, R6, capacitor C 2; The output terminal of one end concatenation operation amplifier A1 of capacitor C 2, positive input and the resistance R 4 of the other end concatenation operation amplifier of capacitor C 2, other one end ground connection of R4, the reverse input end of operational amplifier A 2 connects R5 and R6, the other end ground connection of R5, the other end of R6 is connected with the output terminal of operational amplifier A 2.

5. the measuring system of the micro-angular oscillation of a kind of spacecraft according to claim 3, is characterized in that: described capacitor C 1 size is used for preventing circuit self-excitation, R1=1K Ω, R2=500 Ω, R3=10K Ω for 47pF.

6. the measuring system of the micro-angular oscillation of a kind of spacecraft according to claim 4, is characterized in that: described capacitor C 2 sizes are 3.3uF, R4=5M Ω, R5=5K Ω, R6=5K Ω.

7. the measuring method based on micro-angular oscillation measuring system claimed in claim 1, is characterized in that step is as follows:

(1) degree of will speed up is measured combination and is arranged in spacecraft carrier coordinate system, measures low frequency, the vibration of trace level line of carrier;

The line vibration that described acceleration analysis multiple measurement obtains can be expressed as:

$a=\mathrm{\ω}\×\mathrm{\ω}\×r+\stackrel{\·}{\mathrm{\ω}}\×r$

Wherein, ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration, r is the position vector at the relative carrier barycenter O of particle place.

(2) utilize the relative position relation of acceleration analysis combination and spacecraft carrier barycenter to carry out data processing, and then calculate micro-angular oscillation of spacecraft the low frequency measuring in step (1), the vibration of trace level line.

8. the measuring method of the micro-angular oscillation of a kind of spacecraft according to claim 7, it is characterized in that: accelerometer measures combined is positioned over respectively the B on spacecraft carrier coordinate system X-axis, Y-axis and Z axis, C, D 3 points, the position vector of space aircraft carrier barycenter O is RBO, RCO, RDO, and the line vibration that each measurement combination records is as follows:

$a_B=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{BO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{BO}}$

$a_C=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{CO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{CO}}$

$a_D=\stackrel{\·}{\mathrm{\ω}}\×R_{\mathrm{DO}}+\mathrm{\ω}\×\mathrm{\ω}\×R_{\mathrm{DO}}$

A _b, a _cand a _dfor the line vibration vector that accelerometer measures combined measures, direction is respectively spacecraft carrier coordinate system X, Y and Z axis, and ω is spacecraft carrier rotational angular velocity, for spacecraft carrier angle of rotation acceleration.

9. the measuring method of the micro-angular oscillation of a kind of spacecraft according to claim 8, is characterized in that: in described step (2) to low frequency, trace level line vibration to carry out the method for data processing as follows: set up X, Y and Z axis angular acceleration with line vibration a _b, a _c, a _dbetween relation:

${\stackrel{\·}{\mathrm{\ω}}}_X=\frac{a_C^Z}{{2R}_{\mathrm{CO}}}-\frac{a_D^Y}{{2R}_{\mathrm{DO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Y=\frac{a_D^X}{{2R}_{\mathrm{DO}}}-\frac{a_B^Z}{{2R}_{\mathrm{BO}}}$

${\stackrel{\·}{\mathrm{\ω}}}_Z=\frac{a_B^Y}{{2R}_{\mathrm{BO}}}-\frac{a_C^X}{{2R}_{\mathrm{CO}}}$

Wherein, the vibration of the line along aircraft carrier coordinate system Y-axis, Z axis that the accelerometer measures combined of placing for B point in aircraft carrier coordinate system X-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Z axis that the accelerometer measures combined of placing for C point in aircraft carrier coordinate system Y-axis records; the vibration of the line along aircraft carrier coordinate system X-axis, Y-axis that the accelerometer measures combined of placing for D point on aircraft carrier coordinate system Z axis records;

Utilize quadravalence Runge-Kutta method to carry out integral operation, can obtain micro-angular oscillation of spacecraft:

${\mathrm{\ω}}_X\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_X(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_X(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_X\left(t\right)$

${\mathrm{\ω}}_Y\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Y(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Y\left(t\right)$

${\mathrm{\ω}}_Z\left(t\right)=\mathrm{\ω}(t-T)+\frac{T}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-T)+\frac{2T}{3}{\stackrel{\·}{\mathrm{\ω}}}_Z(t-\frac{1}{2}T)+\frac{1}{6}{\stackrel{\·}{\mathrm{\ω}}}_Z\left(t\right)$

Wherein, t represents current sampling instant, and T is the sampling period.