CN103604442A - Observability analysis method applied to online calibration of strapdown inertial navitation system - Google Patents

Abstract

The invention discloses an observability analysis method applied to online calibration of a strapdown inertial navitation system. The method comprises the following steps: establishing an online calibration model of a single-axial rotation strapdown inertial navitation system; analyzing the observability of parameters of an inertial device and errors of a gyroscope and an accelerometer by utilizing an analytical method; analyzing the influence of the rotation of a single-axial rotation mechanism on the observability of the errors of the parameters of the unobservable inertial device; analyzing the influence of zigzag maneuver of a ship on improvement of the observability and observation degree of the constant drift of the gyroscope and the scale factor error; analyzing the influence of variable motion of the ship on improvement of the observability and observation degree of the zero offset of the accelerometer and the scale factor error; analyzing the influence of high-precision reference information provided by external equipment on improvement of the observability and observation degree of the error of the parameters of the inertial device. The error of the parameters of the calibrated inertial device is separated, the convergence rate and the estimated accuracy are improved, and the effective estimation of the error of the parameters of the inertial device is guaranteed.

Description

Be applied to the Observability analysis of power system of strapdown inertial navitation system (SINS) on-line proving

Technical field

The invention belongs to inertial navigation technology field, relate in particular to a kind of Observability analysis of power system that is applied to strapdown inertial navitation system (SINS) on-line proving.

Background technology

Single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving technology is in the unseparated situation of inertial navigation system and naval vessel, according to the revolving property of the maneuvering characteristics on naval vessel and single-shaft-rotation mechanism, carry out Error Excitation, thereby utilize filtering algorithm to carry out the demarcation of inertia device parameter error.While utilizing filtering algorithm to estimate inertia device parameter error, the observability of inertia device parameter error and observability degree have determined its speed of convergence and estimated accuracy, only take measures to improve observability and the observability degree of inertia device parameter error, could further improve the precision of demarcating.

Therefore, in on-line proving technology, must study the observability problem of inertia device parameter error, to know the estimation degree of estimable parameter error, imponderable parameter error and estimable parameter error, i.e. the observability of parameter error and observability degree in advance.

Summary of the invention

The object of the embodiment of the present invention is to provide a kind of Observability analysis of power system that is applied to strapdown inertial navitation system (SINS) on-line proving, is intended to solve that the existing observability of inertia device parameter error being estimated to the Observable parameter error that exists is low, inertia device parameter error is unobservable and the speed of convergence of inertia device parameter error and the low problem of estimated accuracy.

The embodiment of the present invention is achieved in that a kind of Observability analysis of power system that is applied to strapdown inertial navitation system (SINS) on-line proving, and this carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving Observability analysis of power system comprises the following steps:

Step 1, sets up single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving model, and strapdown inertial navitation system (SINS) medium velocity error equation is:

Wherein, δ V=[δ V _eδ V _n] ^tfor horizontal velocity error, V=[V _ev _n] ^tfor the speed of navigation calculation,

for the value of earth rotation angular speed in navigation system,

the value that the angle of rotation speed that with respect to the earth is for navigation system in navigation is,

with

for the error of calculation, for naval vessel is tied to the transition matrix that navigation is; ${\mathrm{\Δf}}^n=\left[\begin{array}{ccc}\mathrm{\Δ}{f}_E& \mathrm{\Δ}{f}_N& \mathrm{\Δ}{f}_U\end{array}\right]=C_b^n\mathrm{\Δ}{f}^b=C_b^n(A_0+\mathrm{\Δ}{S}_a{f}^b)$ For accelerometer output error is fastened projection at navigation coordinate; $f^b={\left[\begin{array}{ccc}{f}_x^b& f_y^b& f_z^b\end{array}\right]}^T$ For the measured value of accelerometer, subscript x, y, z represents respectively the x of gyro and acceleration, y, tri-axles of z, Δ S _a=diag[Δ S _axΔ S _ayΔ S _az] be accelerometer scale factor error, A ₀=[A _0xa _0ya _0z] ^tfor accelerometer bias, subscript a represents accelerometer;

In strapdown inertial navitation system (SINS), attitude error angle is used

form represent, subscript E, N, U represent respectively east, north, day direction, attitude error equations is:

Wherein, subscript n representative navigation system, i represents inertial system, e represents earth system,

for gyro output error is fastened projection at navigation coordinate, ${\hat{w}}^b={\left[\begin{array}{ccc}{w}_x^b& w_y^b& w_z^b\end{array}\right]}^T$ For the measured value of gyro, Δ S _g=diag[Δ S _gxΔ S _gyΔ S _gz] be the scale factor error of gyro, D ₀=[D _0xd _0yd _0z] ^tconstant value drift for gyro;

Inertia device parameter error to be calibrated is gyro scale factor error, gyroscope constant value drift, accelerometer scale factor error and accelerometer bias; Meanwhile, also velocity error and attitude error are extended for to parameter to be calibrated;

Step 2, utilize analytical method to analyze the observability of inertia device parameter error to be calibrated:

On the basis of quiet pedestal analysis, naval vessel is added to the impact on inertia device parameter error observability of maneuvering characteristics that speed and angular velocity analyzes naval vessel;

Under quiet pedestal condition, velocity error equation and attitude error equations can be write as:

$\mathrm{\δ}\stackrel{\·}{V}=-w_{\mathrm{ie}}^n\×\mathrm{\δV}+f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(3\right)$

$\stackrel{\·}{\mathrm{\φ}}=-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n---\left(4\right)$

In formula, f ⁿ=[0 0 g] ^t, ${\mathrm{\ω}}_{\mathrm{ie}}^n={\left[\begin{array}{ccc}0& {\mathrm{\ω}}_{\mathrm{ie}}\cos L& {\mathrm{\ω}}_{\mathrm{ie}}\sin L\end{array}\right]}^T,$ L is local latitude,

The noise of supposing velocity error is zero, and note horizontal velocity error is measuring value y:

y=δV ⁿ (5)

In on-line proving process, inertia device output error Δ f ⁿand ε ⁿbe all normal value, have

to the repeatedly differentiate simultaneously of formula (5) both sides, have:

$\stackrel{\·}{y}=\mathrm{\δ}{\stackrel{\·}{V}}^n---\left(6\right)$

$\stackrel{\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·}{V}}^n---\left(7\right)$

$\stackrel{\·\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·\·}{V}}^n---\left(8\right)$

Formula (3) and formula (5) substitution formula (6) are obtained:

$\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y=f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(9\right)$

Order $z^{\′}=\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y$ And $z^{\′}=\left[\begin{array}{c}{z}_1^{\′}\\ z_2^{\′}\end{array}\right],$ Above formula is launched, obtains:

-gφ _N+Δf _E=z′ ₁ (11)

gφ _N+Δf _N=z′ ₂ (12)

To formula (3) two ends differentiate, and by formula (6) and formula (7) substitution, can obtain:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×\stackrel{\·}{\mathrm{\φ}}---\left(12\right)$

By formula (4) substitution above formula, have:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×(-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n)---\left(13\right)$

Order $z^{\′\′}=\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y},$ $z^{\′\′}=\begin{array}{}\end{array}\left[\begin{array}{c}{z}_1^{\′\′}\\ z_2^{\′\′}\end{array}\right],$ Above formula is launched, obtains:

gΩ _uφ _E+gε _N=z″ ₁ (14)

gΩ _uφ _N-gΩ _nφ _U-gε _E=z″ ₂ (15)

Wherein, Ω _n=w _iecosL, Ω _u=w _iesinL;

To the differentiate simultaneously of formula (12) both sides, and by formula (4) and formula (8) substitution, obtain:

$f^n\×(-{\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·}{\mathrm{\φ}})=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(16\right)$

By formula (5) substitution above formula, can obtain:

$f^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×\mathrm{\φ}-{\mathrm{\ϵ}}^n))=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(17\right)$

Order $z^{\′\′\′}=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}$ And $z^{\′\′\′}=\left[\begin{array}{c}{z}_1^{\′\′\′}\\ z_2^{\′\′\′}\end{array}\right],$ Above formula further expands into:

-gΩ _u(-Ω _uφ _N+Ω _nφ _U+ε _E)=z′″ ₁ (18)

-gΩ _u(Ω _uφ _E+ε _N)-gΩ _n ²φ _E+gΩ _nε _U=z′″ ₂ (19)

Under static condition, velocity error is the speed that accelerometer resolves, and the velocity error of strapdown inertial navitation system (SINS) is periodically propagated in time, propagates periodic packets containing Schuler period, earth rotation period and Foucault cycle, and can pluridifferentiation, this expression: the measuring value y of velocity error also exists single order differential, second-order differential and three rank differential, therefore, formula (11,13), formula (15,16) and formula (18,19) the right function z relevant to y be known quantity, i.e. Observable; Simultaneously, formula (11,13), formula (15,16) and formula (18,19) to comprise attitude error angle and inertia device parameter error be also observable in interior linear condition combination on the left side, and speed measurement equation formula (5) is included, and has the equation that seven available quantity measured value y represent, these seven incoherent equations can solve seven quantity of states or linear condition combination, are respectively: horizontal velocity error delta V _e, δ V _n, attitude error angle φ _e, φ _n, φ _uand gyro output error ε _n, ε _u;

Remove velocity error δ V _e, δ V _n, the estimated value of remaining 5 Observable quantity of states can be expressed as:

${\widehat{\mathrm{\φ}}}_N={\mathrm{\φ}}_N-\frac{{\mathrm{\Δf}}_E}{g}---\left(20\right)$

${\widehat{\mathrm{\φ}}}_E={\mathrm{\φ}}_E+\frac{\mathrm{\Δ}{f}_N}{g}---\left(21\right)$

${\widehat{\mathrm{\φ}}}_U={\mathrm{\φ}}_U+\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(22\right)$

${\widehat{\mathrm{\ϵ}}}_N={\mathrm{\ϵ}}_N+{\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(23\right)$

${\widehat{\mathrm{\ϵ}}}_U={\mathrm{\ϵ}}_U-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(24\right)$

From formula (20) and formula (21), can draw horizontal attitude error angle φ _eand φ _nevaluated error:

${\mathrm{\Δ\φ}}_N=-\frac{{\mathrm{\Δf}}_E}{g}---\left(25\right)$

${\mathrm{\Δ\φ}}_E=\frac{{\mathrm{\Δf}}_N}{g}---\left(26\right)$

From formula (22) and formula (23) formula, can obtain sky to attitude error angle φ _uwith north gyro output error ε _nevaluated error:

${\mathrm{\Δ\φ}}_U=\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(27\right)$

${\mathrm{\Δ\ϵ}}_N={\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(28\right)$

From formula (24) formula, can obtain sky to gyro output error ε _uevaluated error:

${\mathrm{\Δ\ϵ}}_U=-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(29\right)$

First derive horizontal misalignment φ _eand φ _n, draw φ _uand ε _nestimated value, then derive ε _uexpression formula, reflected the mutual relationship of above-mentioned state estimation and the size of observability degree;

By above analysis, static lower strapdown inertial navitation system (SINS) is incomplete observability system, horizontal velocity error delta V _e, δ V _n, misalignment φ _e, φ _n, φ _uand gyroscope constant value drift ε _n, ε _useven quantity of states or its linear condition combination Observable, observability degree is descending to be respectively: δ V _e, δ V _n, φ _e, φ _n, φ _u, ε _n, ε _u;

Step 3, the rotation of single-shaft-rotation mechanism can improve the observability of Unobservable variable:

β is defined as to the angular velocity rotating with angular velocity w in the rotating mechanism t time, β meets β ∈ (0 °～360 °), now, and attitude matrix

for:

$C_s^b=\left[\begin{array}{ccc}\cos \mathrm{\β}& -\sin \mathrm{\β}& 0\\ \sin \mathrm{\β}& \cos \mathrm{\β}& 0\\ 0& 0& 1\end{array}\right]---\left(30\right)$

Unobservable linear condition combines ε _e, Δ f _e, Δ f _ncan be expressed as:

ε _E=ε _xcosβ-ε _ysinβ (31)

${\mathrm{\Δf}}_E={\▿}_x\cos \mathrm{\β}-{\▿}_y\sin \mathrm{\β}---\left(32\right)$

${\mathrm{\Δf}}_N={\▿}_x\sin \mathrm{\β}+{\▿}_y\cos \mathrm{\β}---\left(33\right);$

Step 4, utilize the zigzag manoeuvre on naval vessel can improve the observability of gyroscope constant value drift and scale factor error: the angular velocity on the responsive naval vessel of gyro, by

know: the movement environment at gyro scale factor error and place, naval vessel has much relations, naval vessel angular motion Shaoxing opera is strong, the equivalent error being brought by scale factor error is larger, caused attitude error is more serious on navigation accuracy impact, when naval vessel is during in static state, naval vessel angular speed is earth rate, and gyro scale factor error is difficult to can be intensified; When naval vessel is during in dynamic environment, naval vessel angular speed is larger, and gyro scale factor error can be intensified, with the form of attitude error, shows;

Step 5, naval vessel carries out the observability that variable motion can improve accelerometer bias and scale factor error: the output of accelerometer is relevant with the motion of the line on naval vessel, by

know: accelerometer scale factor error is relevant with the line motion on naval vessel on the impact of naval vessel velocity error, be that accelerometer scale factor error is larger, line motion Shaoxing opera is strong, the velocity error causing is more obvious, when the relative earth accelerated motion in naval vessel, accelerometer scale factor error can be intensified, with the form of velocity error, shows;

Step 6, naval vessel is under motion state, and position, speed and the attitude in each moment of inertial navigation system all change, and can utilize the high precision reference information that external equipment provides to improve inertia device parameter error observability and observability degree.

Further, in step 3, unobservable linear condition combination ε _e, Δ f _e, Δ f _nrelevant with the angle that single-shaft-rotation mechanism rotates, rotating mechanism rotates different angles, can make all linear condition combination Observables, therefore, on the basis of Analysis on Observability, set up the relation between inertia device attitude and attitude error and velocity error, the attitude that regularly changes the rotation of single-shaft-rotation mechanism can improve observability and the observability degree of Unobservable variable, and then improves estimated accuracy and the estimating speed of inertia device parameter error.

Carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving Observability analysis of power system provided by the invention, by analytical method, analyze the observability of inertia device parameter error in carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving technology, estimable parameter error and imponderable parameter error have been told, improved the observability of unobservable parameter error, improve speed of convergence and the estimated accuracy of Observable inertia device parameter error, guaranteed effective estimation of inertia device parameter error in single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving technology.

Accompanying drawing explanation

Fig. 1 is the carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving Observability analysis of power system process flow diagram that the embodiment of the present invention provides;

Fig. 2 is the anglec of rotation schematic diagram of the single-shaft-rotation mechanism that provides of the embodiment of the present invention.

Embodiment

In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.

Below in conjunction with drawings and the specific embodiments, application principle of the present invention is further described.

As shown in Figure 1, the carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving Observability analysis of power system of the embodiment of the present invention comprises the following steps:

S101: the on-line proving model of setting up single-shaft-rotation formula strapdown inertial navitation system (SINS);

S102: utilize analytical method to analyze inertia device parameter, the observability of gyro and accelerometer error;

S103: the impact of the rotation of analysis single shaft rotating mechanism on unobservable inertia device parameter error observability;

S104: analyze the zigzag manoeuvre on naval vessel to improving gyroscope constant value drift and the observability of scale factor error and the impact of observability degree;

S105: analyze naval vessel and carry out variable motion to improving accelerometer bias and the observability of scale factor error and the impact of observability degree;

S106: the high precision reference information that analysis external equipment provides is on improving the impact of inertia device parameter error observability and observability degree.

Concrete steps of the present invention are:

Step 1, set up single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving model:

Strapdown inertial navitation system (SINS) medium velocity error equation is:

Wherein, δ V=[δ V _eδ V _n] ^tfor horizontal velocity error, V=[V _ev _n] ^tfor the speed of navigation calculation,

for the value of earth rotation angular speed in navigation system,

the value that the angle of rotation speed that with respect to the earth is for navigation system in navigation is,

with

for

the error of calculation,

for naval vessel is tied to the transition matrix that navigation is; ${\mathrm{\Δf}}^n=\left[\begin{array}{ccc}\mathrm{\Δ}{f}_E& \mathrm{\Δ}{f}_N& \mathrm{\Δ}{f}_U\end{array}\right]=C_b^n\mathrm{\Δ}{f}^b=C_b^n(A_0+\mathrm{\Δ}{S}_a{f}^b)$ For accelerometer output error is fastened projection at navigation coordinate; $f^b={\left[\begin{array}{ccc}{f}_x^b& f_y^b& f_z^b\end{array}\right]}^T$ For the measured value of accelerometer, subscript x, y, z represents respectively the x of gyro and acceleration, y, tri-axles of z, Δ S _a=diag[Δ S _axΔ S _ayΔ S _az] be accelerometer scale factor error, A ₀=[A _0xa _0ya _0z] ^tfor accelerometer bias, subscript a represents accelerometer;

In strapdown inertial navitation system (SINS), attitude error angle is used

form represent, subscript E, N, U represent respectively east, north, day direction, attitude error equations is:

Wherein, subscript n representative navigation system, i represents inertial system, e represents earth system,

for gyro output error is fastened projection at navigation coordinate, ${\hat{w}}^b={\left[\begin{array}{ccc}{w}_x^b& w_y^b& w_z^b\end{array}\right]}^T$ For the measured value of gyro, Δ S _g=diag[Δ S _gxΔ S _gyΔ S _gz] be the scale factor error of gyro, D ₀=[D _0xd _0yd _0z] ^tconstant value drift for gyro;

Inertia device parameter error to be calibrated is gyro scale factor error, gyroscope constant value drift, accelerometer scale factor error and accelerometer bias; Meanwhile, also velocity error and attitude error are extended for to parameter to be calibrated;

Step 2, utilize analytical method to analyze the observability of inertia device parameter error to be calibrated:

According to strapdown inertial navitation system (SINS) medium velocity error equation, know: it is mainly caused velocity error by inertia device parameter error, attitude error equal error source, can be expressed as simply δ V=f (ε ^b, 0f ^b, φ), otherwise, also can represent inertia device parameter error and attitude error equal error source with the velocity error δ V resolving, first of velocity error equation (1) is irrelevant with δ V, and it, can be ignored without impact the observability of inertia device parameter error; Meanwhile, in attitude error equations,

with

also on the observability of inertia device parameter error without impact, can ignore, while utilizing analytical method to carry out Analysis on Observability, can start with from the most basic quiet pedestal condition, on the basis of quiet pedestal analysis, naval vessel be added to the impact on inertia device parameter error observability of maneuvering characteristics that speed and angular velocity analyzes naval vessel;

Under quiet pedestal condition, velocity error equation and attitude error equations can be write as:

$\mathrm{\δ}\stackrel{\·}{V}=-w_{\mathrm{ie}}^n\×\mathrm{\δV}+f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(3\right)$

$\stackrel{\·}{\mathrm{\φ}}=-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n---\left(4\right)$

In formula, f ⁿ=[0 0 g] ^t, ${\mathrm{\ω}}_{\mathrm{ie}}^n={\left[\begin{array}{ccc}0& {\mathrm{\ω}}_{\mathrm{ie}}\cos L& {\mathrm{\ω}}_{\mathrm{ie}}\sin L\end{array}\right]}^T,$ L is local latitude,

The noise of supposing velocity error is zero, and note horizontal velocity error is measuring value y:

y=δV ⁿ (5)

In on-line proving process, inertia device output error Δ f ⁿand ε ⁿbe all normal value, have

to the repeatedly differentiate simultaneously of formula (5) both sides, have:

$\stackrel{\·}{y}=\mathrm{\δ}{\stackrel{\·}{V}}^n---\left(6\right)$

$\stackrel{\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·}{V}}^n---\left(7\right)$

$\stackrel{\·\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·\·}{V}}^n---\left(8\right)$

Formula (3) and formula (5) substitution formula (6) are obtained:

$\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y=f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(9\right)$

Order $z^{\′}=\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y$ And $z^{\′}=\left[\begin{array}{c}{z}_1^{\′}\\ z_2^{\′}\end{array}\right],$ Above formula is launched, obtains:

-gφ _N+Δf _E=z′ ₁ (11)

gφ _N+Δf _N=z′ ₂ (12)

To formula (3) two ends differentiate, and by formula (6) and formula (7) substitution, can obtain:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×\stackrel{\·}{\mathrm{\φ}}---\left(12\right)$

By formula (4) substitution above formula, have:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×(-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n)---\left(13\right)$

Order $z^{\′\′}=\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y},$ $z^{\′\′}=\begin{array}{}\end{array}\left[\begin{array}{c}{z}_1^{\′\′}\\ z_2^{\′\′}\end{array}\right],$ Above formula is launched, obtains:

gΩ _uφ _E+gε _N=z″ ₁ (14)

gΩ _uφ _N-gΩ _nφ _U-gε _E=z″ ₂ (15)

Wherein, Ω _n=w _iecosL, Ω _u=w _iesinL;

To the differentiate simultaneously of formula (12) both sides, and by formula (4) and formula (8) substitution, obtain:

$f^n\×(-{\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·}{\mathrm{\φ}})=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(16\right)$

By formula (5) substitution above formula, can obtain:

$f^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×\mathrm{\φ}-{\mathrm{\ϵ}}^n))=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(17\right)$

Order $z^{\′\′\′}=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}$ And $z^{\′\′\′}=\left[\begin{array}{c}{z}_1^{\′\′\′}\\ z_2^{\′\′\′}\end{array}\right],$ Above formula further expands into:

-gΩ _u(-Ω _uφ _N+Ω _nφ _U+ε _E)=z′″ ₁ (18)

-gΩ _u(Ω _uφ _E+ε _N)-gΩ _n ²φ _E+gΩ _nε _U=z′″ ₂ (19)

Under static condition, velocity error is the speed that accelerometer resolves, and the velocity error of strapdown inertial navitation system (SINS) is periodically propagated in time, and it propagates periodic packets containing Schuler period, earth rotation period and Foucault cycle, and can pluridifferentiation, this expression: the measuring value y of velocity error also exists single order differential, second-order differential and three rank differential, therefore, formula (11,13), formula (15,16) and formula (18,19) the right function z relevant to y be known quantity, i.e. Observable; Simultaneously, formula (11,13), formula (15,16) and formula (18,19) to comprise attitude error angle and inertia device parameter error be also observable in interior linear condition combination on the left side, and speed measurement equation formula (5) is included, and has the equation that seven available quantity measured value y represent, these seven incoherent equations can solve seven quantity of states (or linear condition combination), are respectively: horizontal velocity error delta V _e, δ V _n, attitude error angle φ _e, φ _n, φ _uand gyro output error ε _n, ε _u;

Remove velocity error δ V _e, δ V _n, the estimated value of remaining 5 Observable quantity of states can be expressed as:

${\widehat{\mathrm{\φ}}}_N={\mathrm{\φ}}_N-\frac{{\mathrm{\Δf}}_E}{g}---\left(20\right)$

${\widehat{\mathrm{\φ}}}_E={\mathrm{\φ}}_E+\frac{\mathrm{\Δ}{f}_N}{g}---\left(21\right)$

${\widehat{\mathrm{\φ}}}_U={\mathrm{\φ}}_U+\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(22\right)$

${\widehat{\mathrm{\ϵ}}}_N={\mathrm{\ϵ}}_N+{\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(23\right)$

${\widehat{\mathrm{\ϵ}}}_U={\mathrm{\ϵ}}_U-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(24\right)$

From formula (20) and formula (21), can draw horizontal attitude error angle φ _eand φ _nevaluated error:

${\mathrm{\Δ\φ}}_N=-\frac{{\mathrm{\Δf}}_E}{g}---\left(25\right)$

${\mathrm{\Δ\φ}}_E=\frac{{\mathrm{\Δf}}_N}{g}---\left(26\right)$

From formula (22) and formula (23) formula, can obtain sky to attitude error angle φ _uwith north gyro output error ε _nevaluated error:

${\mathrm{\Δ\φ}}_U=\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(27\right)$

${\mathrm{\Δ\ϵ}}_N={\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(28\right)$

From formula (24) formula, can obtain sky to gyro output error ε _uevaluated error:

${\mathrm{\Δ\ϵ}}_U=-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(29\right)$

From above-mentioned observability derivation, can find, first lead-out level misalignment φ _eand φ _n, draw on this basis φ _uand ε _nestimation, then derive ε _uexpression formula, this has also reflected the mutual relationship of above-mentioned state estimation and the size of observability degree from the side;

By above analysis, static lower strapdown inertial navitation system (SINS) is incomplete observability system, horizontal velocity error delta V _e, δ V _n, misalignment φ _e, φ _n, φ _uand gyroscope constant value drift ε _n, ε _udeng seven quantity of states (or its linear condition combination) Observable, observability degree is descending to be respectively: δ V _e, δ V _n, φ _e, φ _n, φ _u, ε _n, ε _u;

Step 3, the rotation of single-shaft-rotation mechanism can improve the observability of Unobservable variable:

For single-shaft-rotation formula inertial navigation, itself possesses rotating mechanism system, can facilitate the attitude that at random changes inertia device, and around sky, the rotation to axle can change warship ship's head to rotating mechanism, and the change of course angle is to unobservable linear condition combination ε _e, Δ f _e, Δ f _nobservability tool have a significant impact;

β is defined as to the angular velocity rotating with angular velocity w in the rotating mechanism t time, with reference to figure 2, β meets β ∈ (0 °～360 °), now, and attitude matrix for:

$C_s^b=\left[\begin{array}{ccc}\cos \mathrm{\β}& -\sin \mathrm{\β}& 0\\ \sin \mathrm{\β}& \cos \mathrm{\β}& 0\\ 0& 0& 1\end{array}\right]---\left(30\right)$

Unobservable linear condition combines ε _e, Δ f _e, Δ f _ncan be expressed as:

ε _E=ε _xcosβ-ε _ysinβ (31)

${\mathrm{\Δf}}_E={\▿}_x\cos \mathrm{\β}-{\▿}_y\sin \mathrm{\β}---\left(32\right)$

${\mathrm{\Δf}}_N={\▿}_x\sin \mathrm{\β}+{\▿}_y\cos \mathrm{\β}---\left(33\right)$

By above analysis, known: unobservable linear condition combination ε _e, Δ f _e, Δ f _nrelevant with the angle that single-shaft-rotation mechanism rotates, rotating mechanism rotates different angles, can make all linear condition combination Observables, therefore, on the basis of Analysis on Observability, set up the relation between inertia device attitude and attitude error and velocity error, the attitude that regularly changes the rotation of single-shaft-rotation mechanism can improve observability and the observability degree of Unobservable variable, and then improves estimated accuracy and the estimating speed of inertia device parameter error;

Step 4, utilize the zigzag manoeuvre on naval vessel can improve the observability of gyroscope constant value drift and scale factor error:

The angular velocity on the responsive naval vessel of gyro, by know: the movement environment at gyro scale factor error and place, naval vessel has much relations, naval vessel angular motion Shaoxing opera is strong, the equivalent error being brought by scale factor error is larger, caused attitude error is more serious on navigation accuracy impact, when naval vessel is during in static state, naval vessel angular speed is earth rate, and gyro scale factor error is difficult to can be intensified; When naval vessel is during in dynamic environment, naval vessel angular speed is larger, and gyro scale factor error can be intensified, form with attitude error shows, therefore, by the mode that strengthens naval vessel angular motion, encourage gyro scale factor error, improve its observability and observability degree;

Step 5, naval vessel carries out the observability that variable motion can improve accelerometer bias and scale factor error:

The output of accelerometer is relevant with the motion of the line on naval vessel, by

know: accelerometer scale factor error is relevant with the line motion on naval vessel on the impact of naval vessel velocity error, be that accelerometer scale factor error is larger, line motion Shaoxing opera is strong, the velocity error causing is more obvious, when the relative earth accelerated motion in naval vessel, accelerometer scale factor error can be intensified, shows, therefore with the form of velocity error, by the mode that strengthens naval vessel line motion, carry out excitation acceleration meter scale factor error, improve its observability and observability degree;

Step 6, naval vessel is under motion state, and each position, speed and attitude constantly of inertial navigation system all changes, and can utilize high precision reference information that external equipment provides to improve observability and the observability degree of inertia device parameter error.

The present invention utilizes Observability analysis of power system in single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving technology, the separable inertia device parameter error that goes out can demarcate, can improve its estimating speed and estimated accuracy again, and guarantee normally carrying out of on-line proving technology simultaneously.

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any modifications of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., within all should being included in protection scope of the present invention.

Claims (2)

1. an Observability analysis of power system that is applied to strapdown inertial navitation system (SINS) on-line proving, is characterized in that, this carrier-borne single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving Observability analysis of power system comprises the following steps:

Step 1, sets up single-shaft-rotation formula strapdown inertial navitation system (SINS) on-line proving model, and its strapdown inertial navitation system (SINS) medium velocity error equation is:

Wherein, δ V=[δ V _eδ V _n] ^tfor horizontal velocity error, V=[V _ev _n] ^tfor the speed of navigation calculation,

for the value of earth rotation angular speed in navigation system,

the value that the angle of rotation speed that with respect to the earth is for navigation system in navigation is,

with

for

the error of calculation,

for naval vessel is tied to the transition matrix that navigation is; ${\mathrm{\Δf}}^n=\left[\begin{array}{ccc}\mathrm{\Δ}{f}_E& \mathrm{\Δ}{f}_N& \mathrm{\Δ}{f}_U\end{array}\right]=C_b^n\mathrm{\Δ}{f}^b=C_b^n(A_0+\mathrm{\Δ}{S}_a{f}^b)$ For accelerometer output error is fastened projection at navigation coordinate; $f^b={\left[\begin{array}{ccc}{f}_x^b& f_y^b& f_z^b\end{array}\right]}^T$ For the measured value of accelerometer, subscript x, y, z represents respectively the x of gyro and acceleration, y, tri-axles of z, Δ S _a=diag[Δ S _axΔ S _ayΔ S _az] be accelerometer scale factor error, A ₀=[A _0xa _0ya _0z] ^tfor accelerometer bias, subscript a represents accelerometer;

In strapdown inertial navitation system (SINS), attitude error angle is used form represent, subscript E, N, U represent respectively east, north, day direction, attitude error equations is:

Wherein, subscript n representative navigation system, i represents inertial system, e represents earth system,

for gyro output error is fastened projection at navigation coordinate, ${\hat{w}}^b={\left[\begin{array}{ccc}{w}_x^b& w_y^b& w_z^b\end{array}\right]}^T$ For the measured value of gyro, Δ S _g=diag[Δ S _gxΔ S _gyΔ S _gz] be the scale factor error of gyro, D ₀=[D _0xd _0yd _0z] ^tconstant value drift for gyro;

Inertia device parameter error to be calibrated is gyro scale factor error, gyroscope constant value drift, accelerometer scale factor error and accelerometer bias; Meanwhile, also velocity error and attitude error are extended for to parameter to be calibrated;

Step 2, utilize analytical method to analyze the observability of inertia device parameter error to be calibrated:

On the basis of quiet pedestal analysis, naval vessel is added to the impact on inertia device parameter error observability of maneuvering characteristics that speed and angular velocity analyzes naval vessel;

Under quiet pedestal condition, velocity error equation and attitude error equations can be write as:

$\mathrm{\δ}\stackrel{\·}{V}=-w_{\mathrm{ie}}^n\×\mathrm{\δV}+f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(3\right)$

$\stackrel{\·}{\mathrm{\φ}}=-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n---\left(4\right)$

In formula, f ⁿ=[0 0 g] ^t, ${\mathrm{\ω}}_{\mathrm{ie}}^n={\left[\begin{array}{ccc}0& {\mathrm{\ω}}_{\mathrm{ie}}\cos L& {\mathrm{\ω}}_{\mathrm{ie}}\sin L\end{array}\right]}^T,$ L is local latitude.

The noise of supposing velocity error is zero, and note horizontal velocity error is measuring value y:

y=δV ⁿ (5)

In on-line proving process, inertia device output error Δ f ⁿand ε ⁿbe all normal value, have to the repeatedly differentiate simultaneously of formula (5) both sides, have:

$\stackrel{\·}{y}=\mathrm{\δ}{\stackrel{\·}{V}}^n---\left(6\right)$

$\stackrel{\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·}{V}}^n---\left(7\right)$

$\stackrel{\·\·\·}{y}=\mathrm{\δ}{\stackrel{\·\·\·}{V}}^n---\left(8\right)$

Formula (3) and formula (5) substitution formula (6) are obtained:

$\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y=f^n\×\mathrm{\φ}+{\mathrm{\Δf}}^n---\left(9\right)$

Order $z^{\′}=\stackrel{\·}{y}+{2w}_{\mathrm{ie}}^n\×y$ And $z^{\′}=\left[\begin{array}{c}{z}_1^{\′}\\ z_2^{\′}\end{array}\right],$ Above formula is launched, obtains:

-gφ _N+Δf _E=z′ ₁ (11)

gφ _N+Δf _N=z′ ₂ (12)

To formula (3) two ends differentiate, and by formula (6) and formula (7) substitution, can obtain:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×\stackrel{\·}{\mathrm{\φ}}---\left(12\right)$

By formula (4) substitution above formula, have:

$\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y}=f^n\×(-w_{\mathrm{ie}}^n\×\mathrm{\φ}+{\mathrm{\ϵ}}^n)---\left(13\right)$

Order $z^{\′\′}=\stackrel{\·\·}{y}+{2w}_{\mathrm{ie}}^n\×\stackrel{\·}{y},$ $z^{\′\′}=\begin{array}{}\end{array}\left[\begin{array}{c}{z}_1^{\′\′}\\ z_2^{\′\′}\end{array}\right],$ Above formula is launched, obtains:

gΩ _uφ _E+gε _N=z″ ₁ (14)

gΩ _uφ _N-gΩ _nφ _U-gε _E=z″ ₂ (15)

Wherein, Ω _n=w _iecosL, Ω _u=w _iesinL;

To the differentiate simultaneously of formula (12) both sides, and by formula (4) and formula (8) substitution, obtain:

$f^n\×(-{\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·}{\mathrm{\φ}})=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(16\right)$

By formula (5) substitution above formula, can obtain:

$f^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×({-\mathrm{\ω}}_{\mathrm{ie}}^n\×\mathrm{\φ}-{\mathrm{\ϵ}}^n))=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}---\left(17\right)$

Order $z^{\′\′\′}=\stackrel{\·\·\·}{y}+{2\mathrm{\ω}}_{\mathrm{ie}}^n\×\stackrel{\·\·}{y}$ And $z^{\′\′\′}=\left[\begin{array}{c}{z}_1^{\′\′\′}\\ z_2^{\′\′\′}\end{array}\right],$ Above formula further expands into:

-gΩ _u(-Ω _uφ _N+Ω _nφ _U+ε _E)=z′″ ₁ (18)

-gΩ _u(Ω _uφ _E+ε _N)-gΩ _n ²φ _E+gΩ _nε _U=z′″ ₂ (19)

Under static condition, velocity error is the speed that accelerometer resolves, and the velocity error of strapdown inertial navitation system (SINS) is periodically propagated in time, propagates periodic packets containing Schuler period, earth rotation period and Foucault cycle, and can pluridifferentiation, this expression: the measuring value y of velocity error also exists single order differential, second-order differential and three rank differential, therefore, formula (11,13), formula (15,16) and formula (18,19) the right function z relevant to y be known quantity, i.e. Observable; Simultaneously, formula (11,13), formula (15,16) and formula (18,19) to comprise attitude error angle and inertia device parameter error be also observable in interior linear condition combination on the left side, and speed measurement equation formula (5) is included, and has the equation that seven available quantity measured value y represent, these seven incoherent equations can solve seven quantity of states or linear condition combination, are respectively: horizontal velocity error delta V _e, δ V _n, attitude error angle φ _e, φ _n, φ _uand gyro output error ε _n, ε _u;

Remove velocity error δ V _e, δ V _n, the estimated value of remaining 5 Observable quantity of states can be expressed as:

${\widehat{\mathrm{\φ}}}_N={\mathrm{\φ}}_N-\frac{{\mathrm{\Δf}}_E}{g}---\left(20\right)$

${\widehat{\mathrm{\φ}}}_E={\mathrm{\φ}}_E+\frac{\mathrm{\Δ}{f}_N}{g}---\left(21\right)$

${\widehat{\mathrm{\φ}}}_U={\mathrm{\φ}}_U+\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(22\right)$

${\widehat{\mathrm{\ϵ}}}_N={\mathrm{\ϵ}}_N+{\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(23\right)$

${\widehat{\mathrm{\ϵ}}}_U={\mathrm{\ϵ}}_U-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(24\right)$

From formula (20) and formula (21), can draw horizontal attitude error angle φ _eand φ _nevaluated error:

${\mathrm{\Δ\φ}}_N=-\frac{{\mathrm{\Δf}}_E}{g}---\left(25\right)$

${\mathrm{\Δ\φ}}_E=\frac{{\mathrm{\Δf}}_N}{g}---\left(26\right)$

From formula (22) and formula (23) formula, can obtain sky to attitude error angle φ _uwith north gyro output error ε _nevaluated error:

${\mathrm{\Δ\φ}}_U=\frac{{\mathrm{\ϵ}}_E}{{\mathrm{\Ω}}_n}-\frac{{\mathrm{\Δf}}_E}{g}\tan L---\left(27\right)$

${\mathrm{\Δ\ϵ}}_N={\mathrm{\Ω}}_u\frac{{\mathrm{\Δf}}_N}{g}---\left(28\right)$

From formula (24) formula, can obtain sky to gyro output error ε _uevaluated error:

${\mathrm{\Δ\ϵ}}_U=-{\mathrm{\Ω}}_n\frac{{\mathrm{\Δf}}_N}{g}---\left(29\right)$

First derive horizontal misalignment φ _eand φ _n, draw φ _uand ε _nestimated value, then derive ε _uexpression formula, reflected the mutual relationship of above-mentioned state estimation and the size of observability degree;

By above analysis, under static condition, strapdown inertial navitation system (SINS) is incomplete observability system, horizontal velocity error delta V _e, δ V _n, misalignment φ _e, φ _n, φ _uand gyroscope constant value drift ε _n, ε _useven quantity of states or its linear condition combination Observable, observability degree is descending to be respectively: δ V _e, δ V _n, φ _e, φ _n, φ _u, ε _n, ε _u;

Step 3, the rotation of single-shaft-rotation mechanism can improve the observability of Unobservable variable:

β is defined as to the angular velocity rotating with angular velocity w in the rotating mechanism t time, β meets β ∈ (0 °～360 °), now, and attitude matrix

for:

$C_s^b=\left[\begin{array}{ccc}\cos \mathrm{\β}& -\sin \mathrm{\β}& 0\\ \sin \mathrm{\β}& \cos \mathrm{\β}& 0\\ 0& 0& 1\end{array}\right]---\left(30\right)$

Unobservable linear condition combines ε _e, Δ f _e, Δ f _ncan be expressed as:

ε _E=ε _xcosβ-ε _ysinβ (31)

${\mathrm{\Δf}}_E={\▿}_x\cos \mathrm{\β}-{\▿}_y\sin \mathrm{\β}---\left(32\right)$

${\mathrm{\Δf}}_N={\▿}_x\sin \mathrm{\β}+{\▿}_y\cos \mathrm{\β}---\left(33\right);$

Step 4, utilize the zigzag manoeuvre on naval vessel can improve the observability of gyroscope constant value drift and scale factor error: the angular velocity on the responsive naval vessel of gyro, by

know: the movement environment at gyro scale factor error and place, naval vessel has much relations, naval vessel angular motion Shaoxing opera is strong, the equivalent error being brought by scale factor error is larger, caused attitude error is more serious on navigation accuracy impact, when naval vessel is during in static state, naval vessel angular speed is earth rate, and gyro scale factor error is difficult to can be intensified; When naval vessel is during in dynamic environment, naval vessel angular speed is larger, and gyro scale factor error can be intensified, with the form of attitude error, shows;

Step 5, naval vessel carries out the observability that variable motion can improve accelerometer bias and scale factor error: the output of accelerometer is relevant with the motion of the line on naval vessel, by know: accelerometer scale factor error is relevant with the line motion on naval vessel on the impact of naval vessel velocity error, be that accelerometer scale factor error is larger, line motion Shaoxing opera is strong, the velocity error causing is more obvious, when the relative earth accelerated motion in naval vessel, accelerometer scale factor error can be intensified, with the form of velocity error, shows;

Step 6, naval vessel is under motion state, and position, speed and the attitude in each moment of inertial navigation system all change, and can utilize the high precision reference information that external equipment provides to improve inertia device parameter error observability and observability degree.

2. a kind of Observability analysis of power system that is applied to strapdown inertial navitation system (SINS) on-line proving as claimed in claim 1, is characterized in that, in step 3, and unobservable linear condition combination ε _e, Δ f _e, Δ f _nrelevant with the angle that single-shaft-rotation mechanism rotates, rotating mechanism rotates different angles, can make all linear condition combination Observables, therefore, on the basis of Analysis on Observability, set up the relation between inertia device attitude and attitude error and velocity error, the attitude that regularly changes the rotation of single-shaft-rotation mechanism can improve observability and the observability degree of Unobservable variable, and then improves estimated accuracy and the estimating speed of inertia device parameter error.

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