CN102788597B - Error suppressing method of rotary strap-down inertial navigation system based on space stabilization - Google Patents

Abstract

The invention provides an error suppressing method of a rotary strap-down inertial navigation system based on space stabilization. The error suppressing method comprises the following steps: determining an initial position parameter of a carrier through a GPS (global positioning system) and binding the initial position parameter of the carrier in a navigation computer; preparing to preheat a strap-down inertial navigation system, collecting data output by a fiber-optic gyroscope and a quartz accelerometer and processing the data; rotating an IMU (inertial measurement unit) and then converting the data generated by the fiber-optic gyroscope and the quartz accelerometer to be in a navigation coordinate system to obtain modulation format of constant deviation of an inertial unit; analyzing calibration factor error and installment error of the gyroscope in a modulation type inertial navigation system with steady space; and calculating attitude error caused by the calibration factor error and the installment error of the gyroscope during a conversion progress between a coordinate system of the IMU and the inertial system. With the adoption of the error suppressing method disclosed by the invention, constant deviation of inertial units in three-axis directions is modulated, and al the calibration factor error and the mounting error are also prevented from being coupled with the rotational angular velocity of the earth, consequently, the system has better stabilization, and navigation positioning precision is improved.

Description

Rotation strapdown inertial navitation system (SINS) error inhibition method based on spatial stability

(1) technical field

What the present invention relates to is a kind of measuring method, in particular a kind of rotation strapdown inertial navitation system (SINS) error inhibition method based on spatial stability.

(2) background technology

Spatial stability type inertial navigation system claims again analytic expression inertial navigation system.It has a gyrostabilized platform, this platform relative inertness spatial stability, and it is to utilize gyroscope to keep the constant gyroscopic inertia of direction, the spatial stability inertial platform of realizing by three cover servomechanisms at inertial space.Three orthogonal accelerometers are housed on stable platform.Because inertial platform does not have rotational angular velocity with respect to inertial space, therefore accelerometer output signal needn't be eliminated the impact of harmful acceleration.Because platform stable is at inertial space, under diverse location, terrestrial gravitation field vector changes at the component of inertial system, in the output signal of accelerometer, will there is gravitational acceleration component like this, so obtain speed and the positional information of carrier after gravitational acceleration component is compensated through integral action.

High-precision strapdown inertial navitation system (SINS) need to adopt high performance inertial sensor and advanced systems technology.Due to the restriction of China's processing technology and manufacture level, manufacture high performance inertia device difficulty large, high performance inertia device can cause the cost of whole strapdown inertial navitation system (SINS) to improve simultaneously, and therefore advanced systems technology always is the study hotspot of strapdown inertial navitation system (SINS).Because the relevant photoelectric device of optical fibre gyro is in technology with quantitatively can not meet the general requirement of Gyroscope Design, the development of optical fibre gyro is restricted, existing optical fibre gyro precision cannot meet again while growing boat, high-precision requirement, is very important so find a kind of method that improves navigation accuracy under existing Gyro Precision condition.

Error modulation technique is a kind of technology of a kind of inertial device error auto-compensation based on rotatory inertia measuring unit (IMU).The error of inertia measurement device is the main determining factor of INS errors.Be subject to the restriction of technological and manufacturing level, manufacture high performance inertia device difficulty very large, develop high performance inertia device simultaneously and can make the cost of whole strapdown inertial navitation system (SINS) improve, therefore advanced systems technology is all the study hotspot of strapdown inertial navitation system (SINS) all the time.Rotation error modulation technique is exactly a kind of advanced person's systems technology, and it,, by adding rotation and control gear at inertance element or IMU outside, then utilizes rotation on average to fall the drift of inertance element to the impact of navigation performance.

(3) summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, a kind of rotation strapdown inertial navitation system (SINS) error inhibition method based on spatial stability is provided.

Technical solution of the present invention is: a kind of rotation strapdown inertial navitation system (SINS) error inhibition method based on spatial stability, it is characterized in that Inertial Measurement Unit to be stabilized in equatorial plane, adopt Fourth Ring structure isolation carrier angular motion and the impact of rotational-angular velocity of the earth on rotation modulation type strapdown inertial navitation system (SINS) error modulation effect, avoid the coupling of gyrostatic scale factor error and alignment error and rotational-angular velocity of the earth, make system there is better stability, be conducive to alliance error and go to zero gradually.Its concrete steps are as follows:

(1) determine the initial position parameters of carrier by GPS, they are bound to navigational computer;

(2) strapdown inertial navitation system (SINS) is carried out preheating preparation, gathers the data of fibre optic gyroscope and quartz accelerometer output and data are processed;

(3) data that after IMU rotation, fibre optic gyroscope and quartz accelerometer generate are transformed under navigation coordinate system, obtain the modulation format that inertia device is often worth deviation;

The ox of Inertial Measurement Unit coordinate system _sy _splane is parallel with the equatorial plane of the earth, oz _saxle is parallel to earth's axis, and sensing consistent with earth rotation angular velocity direction (as accompanying drawing 3), determines the transformational relation of IMU coordinate system and navigation coordinate system:

$C_s^n=C_e^n{C}_i^e{C}_s^i$

The posture renewal process of the modulation type strapdown system based on spatial stability can be summed up as matrix with ask for.Wherein, for the transformation matrix between navigation coordinate system and terrestrial coordinate system; for the transition matrix between terrestrial coordinate system and inertial system, can be determined by the longitude λ of carrier position, latitude L and time interval t.

$C_e^n=\left[\begin{array}{ccc}-\sin \mathrm{\λ}& \cos \mathrm{\λ}& 0\\ -\sin L\cos \mathrm{\λ}& -\sin L\sin \mathrm{\λ}& \cos L\\ \cos L\cos \mathrm{\λ}& \cos L\sin \mathrm{\λ}& \sin L\end{array}\right]$

$C_i^e=\left[\begin{array}{ccc}\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ -\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ 0& 0& 1\end{array}\right]$

Set initial time IMU coordinate system and overlap with inertial coordinates system, subsequently IMU with Constant Angular Velocity ω the oz around inertial coordinates system _iaxle continues to rotate, and the relative position relation of two coordinate systems is:

$C_s^i=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]$

When Inertial Measurement Unit is during around inertial system continuous rotation process, can obtain the projection form that gyro drift is fastened in navigation:

${\mathrm{\ϵ}}^n=C_s^n{\mathrm{\ϵ}}^s=C_e^n{C}_i^e{C}_s^i{\mathrm{\ϵ}}^s=\left[\begin{array}{c}{\mathrm{\ϵ}}_x^n\\ {\mathrm{\ϵ}}_y^n\\ {\mathrm{\ϵ}}_z^n\end{array}\right]$

Wherein,

${\mathrm{\ϵ}}_x^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(-\sin \mathrm{\λ}\cos \mathrm{\ωt}-\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}\sin \mathrm{\ω}t{\mathrm{\ϵ}}_x^s-$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s-$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\sin \mathrm{\λ}\cos \mathrm{\ωt}+\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

${\mathrm{\ϵ}}_y^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s-$

$\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

${\mathrm{\ϵ}}_z^n=\cos L\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}+\cos L){\mathrm{\ϵ}}_x^s+\sin L{\mathrm{\ϵ}}_z^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s-\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\sin \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+$

$\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\sin \mathrm{\ωt}-\sin L\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\cos \mathrm{\ωt}+\sin L\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

Horizontal gyro is often worth the component that deviation fastens at navigation coordinate after the rotation in Inertial Measurement Unit relative inertness space to be modulated completely, is zero through the action effect after complete cycle integration; Gyroscope on azimuth axis is on azimuth axis, to have produced normal value deviation at navigation coordinate after being often worth the latitude coupling of deviation and carrier position.

(4) gyroscope scale factor error and alignment error in the modulation type inertial navigation system of spatial stability are analyzed to the attitude error that in calculating IMU coordinate system and inertial system transfer process, gyroscope scale factor error and alignment error cause.

1) in Inertial Measurement Unit forward continuous rotation process, the attitude error causing due to the existence of scale factor error is transformed into inertial coordinates system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\mathrm{\δ}{K}_{\mathrm{gx}}& 0& 0\\ 0& \mathrm{\δ}{K}_{\mathrm{gy}}& 0\\ 0& 0& \mathrm{\δ}{K}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ \mathrm{\δ}{K}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]$

In like manner can obtain during Inertial Measurement Unit rotates backward, the attitude error that gyroscope scale factor error causes is at the component of inertial system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\mathrm{\δ}{K}_{\mathrm{gx}}& 0& 0\\ 0& \mathrm{\δ}{K}_{\mathrm{gy}}& 0\\ 0& 0& \mathrm{\δ}{K}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ -\mathrm{\δ}{K}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]$

Suppose that a rotation period is T '=2T in lasting rotating scheme, attitude error angle being projected as in inertial system that the output error causing due to gyroscope scale factor error so produces through integration in a complete positive and negative continuous rotation period:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Adopt the continuous positive and negative rotation of Inertial Measurement Unit, disappeared by positive negative with the scale factor error of angular velocity of rotation coupling, owing to adopting the spatial stability method of relative equatorial plane, the namely spatial stability type inertial navigation system of four framed structures, do not have the coupling of rotational-angular velocity of the earth and gyroscope scale factor error, the attitude error that the attitude error under inertial system obtains the lower carrier of navigation system through transfer process is zero.

2) Inertial Measurement Unit relative inertness space is rotated in the forward in process continuously, and the component that the gyroscope output error that gyroscope alignment error causes is fastened at inertial coordinate is:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}{K}_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\sin \mathrm{\ωt}+K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]$

In like manner can obtain in the continuous reverse rotation process of Inertial Measurement Unit, because the gyroscope that alignment error causes is exported:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}-K_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\mathrm{\ω}\sin \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]$

Adopt forward and reverse rotation angle to be the lasting rotating scheme of 360 °, a complete rotation period elapsed time is T '=2T, and wherein T represents the cycle of unidirectional complete rotation.The attitude error causing due to gyroscope alignment error is:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Adopt in the continuous positive and negative scheme of rotation in Inertial Measurement Unit relative inertness space, gyroscope alignment error can not cause attitude of carrier error.

The present invention's advantage is compared with prior art: the present invention has broken traditional rotation modulation method and can not effectively isolate carrier movement and earth rotation angular motion and can not avoid system to occur locking phenomenon, propose a kind of Inertial Measurement Unit to be stabilized in to the error rotation modulation scheme in equatorial plane, now in the angular velocity of gyroscope sensitivity, only there is the rotational angular velocity of Inertial Measurement Unit, and do not have rotational-angular velocity of the earth information.The method can often be worth deviation by the inertia device on three direction of principal axis modulates, and effectively improves navigation and positioning accuracy.

The effect useful to the present invention is described as follows:

Under VC++ simulated conditions, the method is carried out to emulation experiment:

Carrier remains static, the error model parameters of the scheme of the continuous rotating of IMU based on spatial stability:

The time consuming when unidirectional positive and negative rotation one week is: T=12 second;

In each positive and negative rotation transfer process, Acceleration and deceleration time is respectively 4 seconds;

Carrier initial position: 42.0124 ° of north latitude, 121.6481 ° of east longitudes;

Initial attitude error angle: three initial attitude error angles are zero;

Equatorial radius: R _e=6378393.0 meters;

Ellipsoid degree: e=3.367e-3;

By the available earth surface acceleration of gravity of universal gravitation: g ₀=9.78049;

Rotational-angular velocity of the earth (radian per second): 7.2921158e-5;

Gyro drift: 0.01 degree/hour;

Accelerometer bias: 10 ^-4g ₀;

Constant: π=3.1415926;

Utilize the described method of invention to obtain carrier positions graph of errors as shown in Figure 4.Result shows under the positive and negative continuous rotation condition of the IMU based on spatial stability, adopts the inventive method can obtain higher positioning precision.

(4) brief description of the drawings

Fig. 1 is the rotation strapdown inertial navitation system (SINS) error inhibition method process flow diagram based on spatial stability of the present invention;

Fig. 2 is the rotation strapdown inertial navitation system (SINS) schematic diagram based on spatial stability of the present invention;

Fig. 3 is the rotation strapdown inertial navitation system (SINS) coordinate system relative position figure based on spatial stability of the present invention;

Contrast experiment's curve of carrier positioning error when Fig. 4 is the carrier positions error of the positive and negative scheme of rotation of IMU based on spatial stability of the present invention and IMU stationary state.

(5) embodiment

Below in conjunction with accompanying drawing, the specific embodiment of the present invention is described in detail:

(1) determine the initial position parameters of carrier by GPS, they are bound to navigational computer;

(2) strapdown inertial navitation system (SINS) is carried out preheating preparation, gathers the data of fibre optic gyroscope and quartz accelerometer output and data are processed;

(3) data that after IMU rotation, fibre optic gyroscope and quartz accelerometer generate are transformed under navigation coordinate system, obtain the modulation format that inertia device is often worth deviation;

The ox of Inertial Measurement Unit coordinate system _sy _splane is parallel with the equatorial plane of the earth, oz _saxle is parallel to earth's axis, and sensing consistent with earth rotation angular velocity direction (as accompanying drawing 3), determines the transformational relation of IMU coordinate system and navigation coordinate system:

$C_s^n=C_e^n{C}_i^e{C}_s^i---\left(1\right)$

The posture renewal process of the modulation type strapdown system based on spatial stability can be summed up as matrix with ask for.Wherein, for the transformation matrix between navigation coordinate system and terrestrial coordinate system; for the transition matrix between terrestrial coordinate system and inertial system, can be determined by the longitude λ of carrier position, latitude L and time interval t.

$C_e^n=\left[\begin{array}{ccc}-\sin \mathrm{\λ}& \cos \mathrm{\λ}& 0\\ -\sin L\cos \mathrm{\λ}& -\sin L\sin \mathrm{\λ}& \cos L\\ \cos L\cos \mathrm{\λ}& \cos L\sin \mathrm{\λ}& \sin L\end{array}\right]---\left(2\right)$

$C_i^e=\left[\begin{array}{ccc}\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ -\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ 0& 0& 1\end{array}\right]---\left(3\right)$

Set initial time IMU coordinate system and overlap with inertial coordinates system, subsequently IMU with Constant Angular Velocity ω the oz around inertial coordinates system _iaxle continues to rotate, and the relative position relation of two coordinate systems is:

$C_s^i=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]---\left(4\right)$

When Inertial Measurement Unit is during around inertial system continuous rotation process, can obtain the projection form that gyro drift is fastened in navigation:

${\mathrm{\ϵ}}^n=C_s^n{\mathrm{\ϵ}}^s=C_e^n{C}_i^e{C}_s^i{\mathrm{\ϵ}}^s=\left[\begin{array}{c}{\mathrm{\ϵ}}_x^n\\ {\mathrm{\ϵ}}_y^n\\ {\mathrm{\ϵ}}_z^n\end{array}\right]---\left(5\right)$

Wherein,

${\mathrm{\ϵ}}_x^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(-\sin \mathrm{\λ}\cos \mathrm{\ωt}-\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ\ωt}\sin {\mathrm{\ϵ}}_x^s-$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s----\left(6\right)$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\sin \mathrm{\λ}\cos \mathrm{\ωt}+\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

${\mathrm{\ϵ}}_y^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s-$

$\left(7\right)$

$\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

${\mathrm{\ϵ}}_z^n=\cos L\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}+\cos L){\mathrm{\ϵ}}_x^s+\sin L{\mathrm{\ϵ}}_z^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s-\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\sin \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+$

$\left(8\right)$

$\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\sin \mathrm{\ωt}-\sin L\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+$

$\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\cos \mathrm{\ωt}+\sin L\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s$

Horizontal gyro is often worth the component that deviation fastens at navigation coordinate after the rotation in Inertial Measurement Unit relative inertness space to be modulated completely, is zero through the action effect after complete cycle integration; Gyroscope on azimuth axis is on azimuth axis, to have produced normal value deviation at navigation coordinate after being often worth the latitude coupling of deviation and carrier position.

(4) gyroscope scale factor error and alignment error in the modulation type inertial navigation system of spatial stability are analyzed to the attitude error that in calculating IMU coordinate system and inertial system transfer process, gyroscope scale factor error and alignment error cause.

1) in Inertial Measurement Unit forward continuous rotation process, the attitude error causing due to the existence of scale factor error is transformed into inertial coordinates system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\mathrm{\δ}{K}_{\mathrm{gx}}& 0& 0\\ 0& \mathrm{\δ}{K}_{\mathrm{gy}}& 0\\ 0& 0& \mathrm{\δ}{K}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ \mathrm{\δ}{K}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]---\left(9\right)$

In like manner can obtain during Inertial Measurement Unit rotates backward, the attitude error that gyroscope scale factor error causes is at the component of inertial system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\mathrm{\δ}{K}_{\mathrm{gx}}& 0& 0\\ 0& \mathrm{\δ}{K}_{\mathrm{gy}}& 0\\ 0& 0& \mathrm{\δ}{K}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ -\mathrm{\δ}{K}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]---\left(10\right)$

Suppose that a rotation period is T '=2T in lasting rotating scheme, attitude error angle being projected as in inertial system that the output error causing due to gyroscope scale factor error so produces through integration in a complete positive and negative continuous rotation period:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]---\left(11\right)$

Adopt the continuous positive and negative rotation of Inertial Measurement Unit, disappeared by positive negative with the scale factor error of angular velocity of rotation coupling, owing to adopting the spatial stability method of relative equatorial plane, the namely spatial stability type inertial navigation system of four framed structures, do not have the coupling of rotational-angular velocity of the earth and gyroscope scale factor error, the attitude error that the attitude error under inertial system obtains the lower carrier of navigation system through transfer process is zero.

2) Inertial Measurement Unit relative inertness space is rotated in the forward in process continuously, and the component that the gyroscope output error that gyroscope alignment error causes is fastened at inertial coordinate is:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}{K}_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\sin \mathrm{\ωt}+K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]---\left(12\right)$

In like manner can obtain in the continuous reverse rotation process of Inertial Measurement Unit, because the gyroscope that alignment error causes is exported:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}-K_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\mathrm{\ω}\sin \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]---\left(13\right)$

Adopt forward and reverse rotation angle to be the lasting rotating scheme of 360 °, a complete rotation period elapsed time is T '=2T, and wherein T represents the cycle of unidirectional complete rotation.The attitude error causing due to gyroscope alignment error is:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]---\left(14\right)$

Adopt in the continuous positive and negative scheme of rotation in Inertial Measurement Unit relative inertness space, gyroscope alignment error can not cause attitude of carrier error.

Claims (1)

1. the rotation strapdown inertial navitation system (SINS) error inhibition method based on spatial stability, is characterized in that comprising the following steps:

(1) determine the initial position parameters of carrier by GPS, they are bound to navigational computer;

(2) strapdown inertial navitation system (SINS) is carried out preheating preparation, gathers the data of fibre optic gyroscope and quartz accelerometer output and data are processed;

(3) data that after IMU rotation, fibre optic gyroscope and quartz accelerometer generate are transformed under navigation coordinate system, obtain the modulation format that inertia device is often worth deviation;

The ox of Inertial Measurement Unit coordinate system _sy _splane is parallel with the equatorial plane of the earth, oz _saxle is parallel to earth's axis, and sensing is consistent with earth rotation angular velocity direction, determines the transformational relation of IMU coordinate system and navigation coordinate system:

$C_s^n=C_e^n{C}_i^e{C}_s^i$

The posture renewal process of the modulation type strapdown system based on spatial stability can be summed up as matrix with ask for, wherein, for the transformation matrix between navigation coordinate system and terrestrial coordinate system; for the transition matrix between terrestrial coordinate system and inertial system, can be determined by the longitude λ of carrier position, latitude L and time interval t;

$C_e^n=\left[\begin{array}{ccc}-\sin \mathrm{\λ}& \cos \mathrm{\λ}& 0\\ -\sin L\cos \mathrm{\λ}& -\sin L\sin \mathrm{\λ}& \cos L\\ \cos L\cos \mathrm{\λ}& \cos L\sin \mathrm{\λ}& \sin L\end{array}\right]$

$C_i^e=\left[\begin{array}{ccc}\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ -\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& \cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)& 0\\ 0& 0& 1\end{array}\right]$

Set initial time IMU coordinate system and overlap with inertial coordinates system, subsequently IMU with Constant Angular Velocity ω the oz around inertial coordinates system _iaxle continues to rotate, and the relative position relation of two coordinate systems is:

$C_s^i=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]$

When Inertial Measurement Unit is during around inertial system continuous rotation process, can obtain the projection form that gyro drift is fastened in navigation:

${\mathrm{\ϵ}}^n=C_s^n{\mathrm{\ϵ}}^s=C_e^n{C}_i^e{C}_s^i{\mathrm{\ϵ}}^s=\left[\begin{array}{c}{\mathrm{\ϵ}}_x^n\\ {\mathrm{\ϵ}}_y^n\\ {\mathrm{\ϵ}}_z^n\end{array}\right]$

Wherein,

$\begin{array}{c}{\mathrm{\ϵ}}_x^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(-\sin \mathrm{\λ}\cos \mathrm{\ωt}-\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}\sin \mathrm{\ωt}{\mathrm{\ϵ}}_x^s-\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s-\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\sin \mathrm{\λ}\cos \mathrm{\ωt}+\cos \mathrm{\λ}\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s\end{array}$

$\begin{array}{c}{\mathrm{\ϵ}}_y^n=\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s+\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_x^s-\\ \cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\cos \mathrm{\λ}\sin \mathrm{\ωt}+\sin \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin L(\sin \mathrm{\λ}\sin \mathrm{\ωt}-\cos \mathrm{\λ}\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s\end{array}$

$\begin{array}{c}{\mathrm{\ϵ}}_z^n=\cos L\cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)(\cos \mathrm{\λ}\cos \mathrm{\ωt}-\sin \mathrm{\λ}\sin \mathrm{\ωt}+\cos L){\mathrm{\ϵ}}_x^s+\sin L{\mathrm{\ϵ}}_z^s+\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}\cos \mathrm{\ωt}{\mathrm{\ϵ}}_x^s-\sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\cos \mathrm{\λ}\sin \mathrm{\ωt}{\mathrm{\ϵ}}_x^s+\\ \cos (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\sin \mathrm{\ωt}-\sin L\cos \mathrm{\ωt}){\mathrm{\ϵ}}_y^s+\\ \sin (\mathrm{\λ}+{\mathrm{\ω}}_{\mathrm{ie}}t)\sin \mathrm{\λ}(\cos L\cos \mathrm{\ωt}+\sin L\sin \mathrm{\ωt}){\mathrm{\ϵ}}_y^s\end{array}$

Horizontal gyro is often worth the component that deviation fastens at navigation coordinate after the rotation in Inertial Measurement Unit relative inertness space to be modulated completely, is zero through the action effect after complete cycle integration; Gyroscope on azimuth axis is on azimuth axis, to have produced normal value deviation at navigation coordinate after being often worth the latitude coupling of deviation and carrier position;

(4) gyroscope scale factor error and alignment error in the modulation type inertial navigation system of spatial stability are analyzed to the attitude error that in calculating IMU coordinate system and inertial system transfer process, gyroscope scale factor error and alignment error cause;

1) in Inertial Measurement Unit forward continuous rotation process, the attitude error causing due to the existence of scale factor error is transformed into inertial coordinates system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{\mathrm{\δK}}_{\mathrm{gx}}& 0& 0\\ 0& {\mathrm{\δK}}_{\mathrm{gy}}& 0\\ 0& 0& {\mathrm{\δK}}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ {\mathrm{\δK}}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]$

In like manner can obtain during Inertial Measurement Unit rotates backward, the attitude error that gyroscope scale factor error causes is at the component of inertial system:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{\mathrm{\δK}}_{\mathrm{gx}}& 0& 0\\ 0& {\mathrm{\δK}}_{\mathrm{gy}}& 0\\ 0& 0& {\mathrm{\δK}}_{\mathrm{gz}}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ {-\mathrm{\δK}}_{\mathrm{gz}}\mathrm{\ω}\end{array}\right]$

Suppose that a rotation period is T '=2T in lasting rotating scheme, attitude error angle being projected as in inertial system that the output error causing due to gyroscope scale factor error so produces through integration in a complete positive and negative continuous rotation period:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Adopt the continuous positive and negative rotation of Inertial Measurement Unit, disappeared by positive negative with the scale factor error of angular velocity of rotation coupling, owing to adopting the spatial stability method of relative equatorial plane, the namely spatial stability type inertial navigation system of four framed structures, do not have the coupling of rotational-angular velocity of the earth and gyroscope scale factor error, the attitude error that the attitude error under inertial system obtains the lower carrier of navigation system through transfer process is zero;

2) Inertial Measurement Unit relative inertness space is rotated in the forward in process continuously, and the component that the gyroscope output error that gyroscope alignment error causes is fastened at inertial coordinate is:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& -\sin \mathrm{\ωt}& 0\\ \sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ \mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}{K}_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\sin \mathrm{\ωt}+K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]$

In like manner can obtain in the continuous reverse rotation process of Inertial Measurement Unit, because the gyroscope that alignment error causes is exported:

$\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}=C_s^i\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^s}^{\′\′\′}=\left[\begin{array}{ccc}\cos \mathrm{\ωt}& \sin \mathrm{\ωt}& 0\\ -\sin \mathrm{\ωt}& \cos \mathrm{\ωt}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& K_{\mathrm{gxy}}& K_{\mathrm{gxz}}\\ K_{\mathrm{gyx}}& 0& K_{\mathrm{gyz}}\\ K_{\mathrm{gzx}}& K_{\mathrm{gzy}}& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ -\mathrm{\ω}\end{array}\right]=\left[\begin{array}{c}{-K}_{\mathrm{gxz}}\mathrm{\ω}\cos \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\sin \mathrm{\ωt}\\ K_{\mathrm{gxz}}\mathrm{\ω}\sin \mathrm{\ωt}-K_{\mathrm{gyz}}\mathrm{\ω}\cos \mathrm{\ωt}\\ 0\end{array}\right]$

Adopt forward and reverse rotation angle to be the lasting rotating scheme of 360 °, a complete rotation period elapsed time is T '=2T, and wherein T represents the cycle of unidirectional complete rotation; The attitude error causing due to gyroscope alignment error is:

${\∫}_0^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}}^i}^{\′\′\′}\mathrm{dt}={\∫}_0^{T^{\′}/2}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}+}^i}^{\′\′\′}\mathrm{dt}+{\∫}_{T^{\′}/2}^{T^{\′}}\mathrm{\δ}{{\mathrm{\ω}}_{\mathrm{is}-}^i}^{\′\′\′}\mathrm{dt}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Adopt in the continuous positive and negative scheme of rotation in Inertial Measurement Unit relative inertness space, gyroscope alignment error can not cause attitude of carrier error.