CN110455288A - A kind of posture renewal method based on angular speed high-order moment - Google Patents
A kind of posture renewal method based on angular speed high-order moment Download PDFInfo
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- G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
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Abstract
The present invention discloses a kind of posture renewal method based on angular speed high-order moment.Include the following steps: that numerical relation rule of the multinomial coefficient on cone axis establishes angular speed high order polynomial fitting model under the N number of angle increment 1) continuously exported based on gyro sampling and conical motion environment；2) using the differential equation of obtained angular speed high order fitted polynomial coefficients and equivalent rotating vector, equivalent rotating vector is solved by Taylor series expansion；3) according to the calculated value of equivalent rotating vector, the attitudes vibration of the carrier in the posture renewal period is described in the form of quaternary number.The present invention during Attitude Calculation can noncommutativity error caused by effective compensation Rigid Body in Rotation With, utilize that high-precision attitude calculation result participates in velocity calculated and position resolves, improve carrier navigation accuracy.
Description
Technical field
The invention belongs to technical field of navigation and positioning, and in particular to a kind of posture renewal method is suitable for Strapdown Inertial Units
The attitude algorithm of navigation system (SINS).
Background technique
Posture renewal is the core in the fields such as robot, unmanned plane, vehicle autonomous driving, aircraft precise guidance and control
Problem.Strap-down inertial navigation system (Strapdown Inertial Navigation System, SINS) is using on carrier
Inertia type instrument (gyroscope, accelerometer) measurement posture angular movement and line motion information, and by navigational computer according to leading
Boat algorithm handles inertia type instrument output valve in real time, completes the real-time update of the navigational parameters such as posture, speed and position.Its
In, posture renewal is the key that SINS navigation algorithm, and determines one of the principal element of SINS navigation accuracy.
Carrier angular movement is a continuous process, and angular movement measurement (gyroscope sampling) is a departure process, this
Just bring the noncommutativity error during posture renewal, i.e., the noncommutativity error of so-called finite rotation of rigid body.
Noncommutativity error is the original reason error in attitude updating algorithm, and the compensation of noncommutativity error is to improve posture renewal
One of key of arithmetic accuracy.
Currently, to the main stream approach of noncommutativity error compensation be equivalent rotating vector method and its all kinds of innovatory algorithms,
The core of such algorithm is true based on one group determining of angle movement model (multinomial motion model, conical motion model) acquisition
Fixed can not exchange error penalty coefficient.Since carrier angular movement has randomness and uncertainty, angle movement model precision is straight
Connect the precision for determining the attitude updating algorithms such as equivalent rotating vector algorithm.In general, increasing gyroscope sample frequency can construct
Angle movement model precision out is higher.But under current technical status, sample frequency can not accomplish absolute height.How in determining son
Under sample said conditions, improving angle movement model precision becomes a potential option for improving noncommutativity error compensation precision.
Summary of the invention
Goal of the invention: for the defects in the prior art, the present invention proposes a kind of appearance based on angular speed high-order moment
State update method, can noncommutativity error caused by effective compensation Rigid Body in Rotation With, effectively improve carrier navigation accuracy.
Technical solution: a kind of posture renewal method based on angular speed high-order moment of the present invention, including it is following
Step:
1) it is sampled with multinomial coefficient under conical motion environment on cone axis based on N number of angle increment that gyro continuously exports
Numerical relation rule establish N the or N+1 order polynomial model of fit of angular speed；
2) using the differential equation of obtained angular speed high order fitted polynomial coefficients and equivalent rotating vector, by Taylor's grade
Number expansion solves equivalent rotating vector；
3) according to the calculated value of equivalent rotating vector, the appearance of the carrier in the posture renewal period is described in the form of quaternary number
State variation.
Further, the step 1) includes:
11) within the posture renewal period [0 Nh], gyroscope continuously exports N number of angle increment sampling and is denoted as Δ θi(i=1,
2 ... N), wherein h is the sampling period of gyroscope, and N is sample number；
12) multinomial coefficient is obtained under conical motion environment in cone axis according to the polynomial fitting of carrier movement angular speed
On numerical relation rule, wherein the polynomial fitting of carrier movement angular speed are as follows:
Wherein,For carrier fitting of a polynomial angular speed, (i+1) a 'i+1For the fitting coefficient matrix of (3 × 1) dimension；
The numerical relation rule of multinomial coefficient is approximately as described below:
a′j+1×a′k+1=k0a′l+1×a′m+1, j+k=l+m=2n+1, n=1,2 ...
Wherein, a 'j+1, a 'k+1, a 'l+1With a 'm+1Respectively t in the polynomial fitting of angular speedj, tk, tlAnd tmBe
Number, j, k, l, m are respectively a nonnegative integer, they meet this numerical relation of j+k=l+m=2n+1, k0For a non-zero constant；
13) N the or N+1 order polynomial model of fit of angular speed is established.
Further, numerical value of the multinomial coefficient on cone axis under conical motion environment is obtained in the step 12) to close
It is that regular process is as follows:
According to the polynomial fitting of carrier movement angular speed:K >=N,
And under conical motion carrier angular velocity of satellite motion ω (t) expression formula:
Wherein, α is the semi-cone angle of conical motion, ω0For circular cone frequency；
Two formula both sides obtain expiring in x-axis between angular speed coefficient multiplication cross item simultaneously to the derivation of time t and the value at t=0
The following numerical relation of foot:
(a′j+1×a′k+1)x=k0(a′l+1×a′m+1)x, j+k=l+m=2n+1, n=1,2 ...
Since the aperiodicity error of conical motion appears in x-axis, only occurs periodic error in y-axis and z-axis, therefore
Consider the numerical relation in x-axis between angular speed coefficient multiplication cross item, the numerical relation rule of multinomial coefficient is obtained by above formula approximation
Are as follows: a 'j+1×a′k+1=k0a′l+1×a′m+1, j+k=l+m=2n+1, n=1,2 ....
Further, the step 13) includes:
It is derived by the numerical relation between corresponding multinomial coefficient according to the value of sample number N, and is combined corresponding
Angle increment Δ θiNumerical relation between (i=1,2 ... N) and angular speed multinomial coefficient, is derived by under the conditions of corresponding sample number
Angular speed polynomial fitting.
Further, the step 2) includes:
Equivalent rotating vector differential equation concrete form are as follows:
Wherein, φ is equivalent rotating vector, φ=| φ |, ω is angular speed, × indicate multiplication cross operation, ignore the high order of φ
Item is simultaneously approximate with angle increment by the Section 2 φ in above formula, and the equivalent rotating vector differential equation is simplified are as follows:
Equivalent rotating vector, solution of Taylor series of the equivalent rotating vector at moment Nh are solved using Taylor series expansion method
It is as follows:
Initial value φ (0)=0, the N of angular speed or N+1 polynomial fitting, which are substituted into above formula, can calculate φ (Nh).
Further, the step 3) includes:
The attitudes vibration quaternary number constructed by equivalent rotating vector φ (Nh) are as follows:
The posture renewal of period [0 Nh] interior carrier can be completed according to attitudes vibration quaternary number q (Nh).
The utility model has the advantages that compared with the prior art, the present invention has the following beneficial effects:
1) present invention makes full use of gyroscope measured value under conditions of not changing sample number N, and building angular speed high order is more
Item formula model of fit reduces multinomial to the error of fitting of the practical angular movement of carrier；
2) in high-order moment coefficient solution procedure, using conical motion as constraint condition, multinomial coefficient is determined
Numeric value analysis relationship provides constraint condition for multinomial coefficient solution, improves the utilization efficiency of gyroscope measured value；
3) solution that equivalent rotating vector and its corresponding quaternary number are completed based on high-order moment and Taylor expansion, is improved
The attitude algorithm accuracy of SINS is completed attitude of carrier information and is resolved.
Detailed description of the invention
Fig. 1 is the posture renewal method flow diagram based on angular speed high-order moment according to the embodiment of the present invention；
Fig. 2 is under conical motion environment, and the present invention and classical more increment algorithms, the more increment algorithm circular cones of optimization float when N=3
Shift error comparison diagram；
Fig. 3 is under conical motion environment, and the present invention and classical more increment algorithms, the more increment algorithm circular cones of optimization float when N=4
Shift error comparison diagram；
Fig. 4 is under conical motion environment, and the present invention and classical more increment algorithms, the more increment algorithm circular cones of optimization float when N=5
Shift error comparison diagram；
Fig. 5 is under conical motion environment, and the present invention and classical more increment algorithms, the more increment algorithm circular cones of optimization float when N=6
Shift error comparison diagram；
Fig. 6 is α=90 ° under conical motion environment, and the present invention and classical more increment algorithms, the more increments of optimization are calculated when N=4
The non-axis of cone angular error comparison diagram of method；
Fig. 7 is α=0.1 ° under conical motion environment, and the present invention and classical more increment algorithms, the more increments of optimization are calculated when N=4
The non-axis of cone angular error comparison diagram of method.
Specific embodiment
Technical solution of the present invention is described further with reference to the accompanying drawing.
The present invention is difficult to really reflect carrier angular movement for tradition using the angle movement model that gyro output N increment is established
The problem of, under conditions of output sample number N is determined, N increment and conical motion is utilized to constrain inferior horn velocity fitting system of polynomials
N the or N+1 order polynomial model of fit of numerical relation building angular speed between number multiplication cross items, then using Taylor series expansion and
The fitting of angular speed solves equivalent rotating vector and its corresponding attitude quaternion, completes posture renewal and calculates.Attitude Calculation mistake
Cheng Zhong, can noncommutativity error caused by effective compensation Rigid Body in Rotation With, utilize high-precision attitude calculation result to participate in velocity solution
It calculates and position resolves, improve carrier navigation accuracy.
Referring to Fig.1, the posture renewal method proposed by the present invention based on angular speed high-order moment the following steps are included:
Step S1, multinomial coefficient is in circle under the N number of angle increment sampling continuously exported based on gyro and conical motion environment
Numerical relation rule in the axis of cone establishes N the or N+1 order polynomial model of fit of angular speed, and N is more than or equal to 3.
Polynomial fitting model specifically comprises the following steps:
Consider in posture renewal period [0 Nh] (h is the sampling period of gyroscope, and N is increment), gyroscope is continuously defeated
N number of angle increment out is sampled as Δ θi(i=1,2 ... N).In the case where carrier movement angular speed is the assumed condition of polynomial form,
Remember the polynomial fitting of angular speed are as follows:
Wherein,For carrier fitting of a polynomial angular speed, k is polynomial number, (i+1) a 'i+1For (3 × 1) dimension
Fitting coefficient matrix.
Under conical motion, carrier movement angular velocity omega (t) be may be expressed as:
Wherein, α is the semi-cone angle of conical motion, ω0For conical motion angular frequency, abbreviation circular cone frequency.
To formula (1) and formula (2) both sides, value can obtain angular speed coefficient multiplication cross to the derivation of time t and at t=0 simultaneously simultaneously
Meet following numerical relation in x-axis between:
(a′j+1×a′k+1)x=k0(a′l+1×a′m+1)x, j+k=l+m=2n+1, n=1,2 ... (3)
Wherein, k0For a non-zero constant, j, k, l, m are respectively a nonnegative integer, they meet j+k=l+m=2n+1 this
Kind numerical relation.
For the conical motion of formula (3) statement, aperiodicity error is appeared in x-axis, the period only occurs in y-axis and z-axis
Property error.Therefore the main numerical relation considered in x-axis between angular speed coefficient multiplication cross item, can be obtained by formula (3) approximation:
a′i+1×a′j+1=k0a′i+1×a′j+1, j+k=l+m=2n+1, n=1,2 ... (4)
When N=3, obtained by formula (4):
Convolution (5) and angle increment Δ θiNumerical relation between (i=1,2,3) and angular speed multinomial coefficient can obtain, N=
When 3, angular speed polynomial fitting are as follows:
Wherein,
Wherein, subscript+representing matrix generalized inverse, a1, a2, a3It is solved by formula (8) in N=3.
tj=jh (j=1,2 ..., N)
Formula (8) is to be equal to the integrated value direct derivation of angular speed according to angle increment and obtain.
When N=4, obtained by formula (4):
Convolution (9) and angle increment Δ θi(i=1,2 ... 4) numerical relation between angular speed multinomial coefficient can obtain, N
When=4, angular speed polynomial fitting are as follows:
Wherein,
In formula, a1, a2, a3, a4It is solved by formula (8) in N=4.
Similarly when available N >=5, the polynomial fitting of angular speed in formula (1) can be solved.Wherein, when N is odd number,
The n times polynomial fitting of angular speed, which can be constructed, can construct N+1 polynomial fitting of angular speed when N is even number.When such as N=5,
A ' can be obtained by formula (4)1×a′6=1/5 (a '3×a′4), it is contemplated that angle increment Δ θi(i=1,2 ... 5) with angular speed system of polynomials
Several numerical relations can obtain 5 polynomial fittings of angular speed；When N=6, a ' can be obtained by formula (4)2×a′7=2/7 (a '4×
a′5) and a '1×a′6=-1/14 (a '4×a′5), it is contemplated that angle increment Δ θi(i=1,2 ... 5) with angular speed multinomial coefficient
Between numerical relation can obtain 7 polynomial fittings of angular speed.
Step S2 utilizes obtained angular speed high order fitted polynomial coefficients (i+1) a 'i+1It is micro- with equivalent rotating vector
Divide equation, equivalent rotating vector is solved by Taylor series expansion.
It specifically includes:
Equivalent rotating vector differential equation concrete form are as follows:
Wherein, φ is equivalent rotating vector, φ=| φ |, ω is angular speed, × indicate multiplication cross operation.Ignore the high order of φ
Item is simultaneously approximate with angle increment by the Section 2 φ in formula (12), and formula (12) can simplify are as follows:
Equivalent rotating vector, solution of Taylor series of the equivalent rotating vector at moment Nh are solved using Taylor series expansion method
It is as follows:
Initial value φ (0)=0.The N of angular speed or N+1 polynomial fitting, which are substituted into formula (14), can calculate φ (Nh).
When such as N=3, shown in angular speed polynomial fitting such as formula (6), at this point,
Step S3 describes the load in the posture renewal period according to the calculated value of equivalent rotating vector in the form of quaternary number
The attitudes vibration of body.
It specifically includes:
The attitudes vibration quaternary number constructed by equivalent rotating vector φ (Nh) are as follows:
The posture renewal of period [0 Nh] interior carrier can be completed according to attitudes vibration quaternary number q (Nh), complete carrier
Posture information resolves.
Beneficial effects of the present invention are verified below by an emulation experiment.Inertia type instrument number is simulated using Matlab
According to for attitude updating algorithm, conical motion is worst environmental condition, it can induce the serious drift of mathematical platform
It moves.Conical motion mathematical model carrier movement angular velocity omega (t) are as follows:
Wherein, α is the semi-cone angle of conical motion, ω0For circular cone frequency.
Obtain inertial navigation instrument gross data by above-mentioned emulation digital simulation, inertial navigation to the instrument actual acquired data into
Row sampling, is used for navigation calculation, sampling period 10ms.
The relevant parameter of emulation includes: circular cone frequencies omega0=1Hz；"~90 ° of semi-cone angle variation range α=0.05；Increment
Number N=3~6.
Coning error drift calculation and the verifying that non-axis of cone angular error calculates are as follows:
Proof of algorithm is carried out in ordinary PC.It emulates and carries out 3s, during simulation process, (1) generates instrumented data；(2)
N the or N+1 order polynomial of angular speed is constructed according to instrumented data；(3) method that the present invention is mentioned, classical more increments are utilized respectively
Algorithm optimizes the solution that more increment algorithms carry out equivalent rotating vector；(4) error rotating vector, analysis coning error drift are calculated
It moves and non-axis of cone angular error；(5) change sample number N, repeat the above steps.Fig. 2~Fig. 5 successively shows corresponding sample number N=3
When~6, method (New), classical more increment algorithms (Classical1) and the more increment algorithms of optimization that the present invention is mentioned
The drift comparison of (Classical 2) coning error.Fig. 6 and Fig. 7 is set forth in α=90 °, N=4 and α=0.1 °, N=4 feelings
Under condition, method (New), classical more increment algorithms (Classical1) and the more increment algorithms of optimization that the present invention is mentioned
The angle of (Classical 2) in the non-circular axis of cone updates application condition.
When Fig. 2~Fig. 5 shows small semi-cone angle, method precision of the invention is lower than on the whole better than classical more increment algorithms
Optimize more increment algorithms and with sample number N increase, in semi-cone angle very little, method precision of the invention be higher than more increments optimization
Algorithm.When such as N=3, for method of the invention when semi-cone angle is not less than 10 °, precision is higher than more increment optimization algorithms.And in N=6
When, method of the invention is not less than 10 in semi-cone angle-2° when, precision be higher than more increment optimization algorithms.When big semi-cone angle, the present invention
Method precision better than optimizing more increment algorithms, and not less than classical more increment algorithms.Therefore it can obtain, with classical, optimization mostly son
Sample algorithm is compared, and the method that the present invention is mentioned has certain global advantage within the scope of entire cone angle.
Fig. 6~Fig. 7 shows the high quantity of the max value of error for optimizing more increment algorithms method error more of the invention
Grade.This explanation, if aperiodic conical motion occurs in carrier, the attitude algorithm error of method of the invention is more much smaller than optimizing
Increment algorithm.And classical more increment Algorithm Error curves are almost overlapped with method error curve of the invention, the two is in non-conical
Error precision on axis is suitable.
Claims (6)
1. a kind of posture renewal method based on angular speed high-order moment, which is characterized in that the described method comprises the following steps:
1) it is sampled and number of the multinomial coefficient on cone axis under conical motion environment based on N number of angle increment that gyro continuously exports
Value relationship schedule establishes N the or N+1 order polynomial model of fit of angular speed；
2) using the differential equation of obtained angular speed high order fitted polynomial coefficients and equivalent rotating vector, by Taylor series exhibition
Open solution equivalent rotating vector；
3) according to the calculated value of equivalent rotating vector, the posture that the carrier in the posture renewal period is described in the form of quaternary number becomes
Change.
2. the posture renewal method according to claim 1 based on angular speed high-order moment, which is characterized in that the step
It is rapid 1) to include:
11) within the posture renewal period [0 Nh], gyroscope continuously exports N number of angle increment sampling and is denoted as Δ θi(i=1,2 ... N),
Wherein h is the sampling period of gyroscope, and N is sample number；
12) multinomial coefficient is obtained under conical motion environment on cone axis according to the polynomial fitting of carrier movement angular speed
Numerical relation rule, wherein the polynomial fitting of carrier movement angular speed are as follows:
Wherein,For carrier fitting of a polynomial angular speed, (i+1) a 'i+1For the fitting coefficient matrix of (3 × 1) dimension；
The numerical relation rule of multinomial coefficient is approximately as described below:
a′j+1×a′k+1=k0a′l+1×a′m+1, j+k=l+m=2n+1, n=1,2 ...
Wherein, a 'j+1, a 'k+1, a 'l+1With a 'm+1Respectively t in the polynomial fitting of angular speedj, tk, tlAnd tmThe coefficient of item, j,
K, l, m are respectively a nonnegative integer, they meet this numerical relation of j+k=l+m=2n+1, k0For a non-zero constant；
13) N the or N+1 order polynomial model of fit of angular speed is established.
3. the posture renewal method according to claim 2 based on angular speed high-order moment, which is characterized in that the step
It is rapid 12) in obtain numerical relation rule of the multinomial coefficient on cone axis under conical motion environment process it is as follows:
According to the polynomial fitting of carrier movement angular speed:
And under conical motion carrier angular velocity of satellite motion ω (t) expression formula:Wherein, α
For the semi-cone angle of conical motion, ω0For circular cone frequency；
Two formula both sides obtain meeting such as in x-axis between angular speed coefficient multiplication cross item simultaneously to the derivation of time t and the value at t=0
Lower numerical relation:
(a′j+1×a′k+1)x=k0(a′l+1×a′m+1)x, j+k=l+m=2n+1, n=1,2 ...
Since the aperiodicity error of conical motion appears in x-axis, only occurs periodic error in y-axis and z-axis, therefore consider
Numerical relation in x-axis between angular speed coefficient multiplication cross item obtains the numerical relation rule of multinomial coefficient by above formula approximation are as follows:
a′j+1×a′k+1=k0a′l+1×a′m+1, j+k=l+m=2n+1, n=1,2 ....
4. the posture renewal method according to claim 3 based on angular speed high-order moment, which is characterized in that the step
It is rapid 13) to include:
It is derived by the numerical relation between corresponding multinomial coefficient according to the value of sample number N, and corresponding angle is combined to increase
Measure Δ θiNumerical relation between (i=1,2 ... N) and angular speed multinomial coefficient is derived by corresponding sample number condition inferior horn speed
Spend polynomial fitting.
5. the posture renewal method according to claim 2 based on angular speed high-order moment, which is characterized in that the step
It is rapid 2) to include:
Equivalent rotating vector differential equation concrete form are as follows:
Wherein, φ is equivalent rotating vector, φ=| φ |, ω is angular speed, × indicate multiplication cross operation, ignore the high-order term of φ simultaneously
By the angle increment approximation of the Section 2 φ in above formula, the equivalent rotating vector differential equation is simplified are as follows:
Equivalent rotating vector is solved using Taylor series expansion method, solution of Taylor series of the equivalent rotating vector at moment Nh is such as
Under:
Initial value φ (0)=0, the N of angular speed or N+1 polynomial fitting, which are substituted into above formula, can calculate φ (Nh).
6. the posture renewal method according to claim 5 based on angular speed high-order moment, which is characterized in that the step
It is rapid 3) to include:
The attitudes vibration quaternary number constructed by equivalent rotating vector φ (Nh) are as follows:
The posture renewal of period [0 Nh] interior carrier can be completed according to attitudes vibration quaternary number q (Nh), complete attitude of carrier
Information resolves.
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| CN112747773A (en) * | 2020-12-30 | 2021-05-04 | 中建八局第二建设有限公司 | Method for improving precision of gyroscope based on Allan variance and random polynomial |
| CN112747773B (en) * | 2020-12-30 | 2022-12-06 | 中建八局第二建设有限公司 | Method for improving precision of gyroscope based on Allan variance and random polynomial |
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Application publication date: 20191115 |