Research ArticleInitial Self-Alignment for Marine Rotary SINS Using NovelAdaptive Kalman Filter
Fujun Pei,1,2 Li Zhu,1,2 and Jian Zhao1,2
1School of Electronic Information & Control Engineering, Beijing University of Technology, Beijing 100124, China2Beijing Key Laboratory of Computational Intelligence and Intelligent System, Beijing 100124, China
Correspondence should be addressed to Fujun Pei; [email protected]
Received 24 January 2015; Accepted 15 April 2015
Academic Editor: Oscar Reinoso
Copyright Β© 2015 Fujun Pei et al.This is an open access article distributed under theCreative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The accurate initial attitude is essential to affect the navigation result of Rotary Strapdown Inertial Navigation System (SINS), whichis usually calculated by initial alignment. But marine mooring Rotary SINS has to withstand dynamic disturbance, such as theinterference angular velocities and accelerations caused by surge and sway. In order to overcome the limit of dynamic disturbanceunder the marine mooring condition, an alignment method using novel adaptive Kalman filter for marine mooring Rotary SINSis developed in this paper. This alignment method using the gravity in the inertial frame as a reference is discussed to deal withthe lineal and angular disturbances. Secondly, the system error model for fine alignment in the inertial frame as a reference isestablished. Thirdly, PWCS and SVD are used to analyze the observability of the system error model for fine alignment. Finally, anovel adaptive Kalman filter withmeasurement residual to estimatemeasurement noise variance is designed.The simulation resultsdemonstrate that the proposed method can achieve better accuracy and stability for marine Rotary SINS.
1. Introduction
Because SINS has special advantages, it has been widely usedin aviation, marine, and land vehicle navigation and posi-tioning. Because the bias errors of inertial sensors can causelarge positioning errors accumulated over time, low accuracyship-carrier SINS cannot meet the navigation requirementfor long voyage of the marine [1]. Whereas the high accuracySINS needs strategic inertial sensors, it will increase the costcorrespondingly. With the emergence of rotation modula-tion technique that compensates the constant bias error ofinertial sensors, Rotary SINS plays an important role in themarine field. But because the initial errors in the derived navi-gation parameters cannot be depressed by rotary modulationtechnique, initial alignment is the key technique for RotarySINS. The purpose of initial alignment is to obtain an initialattitude matrix and set the misalignments to zero. Then theaccuracy and stability of initial alignment are critical for highperformance Rotary SINS [2, 3].The initial attitude angle canbe obtained by initial alignment, which is divided into twoprocedures: coarse and fine alignment. The coarse alignmentis used to resolve large misalignment angle rapidly, and then
the fine alignment is used to compensate and correct themisalignment angle further.
A lot of literature has been devoted to coarse alignmentmethods. But marine Rotary SINS has to withstand dynamicdisturbance, such as the interference angular velocities andaccelerations, caused by surge and sway far outweighing theautorotation angular velocity of the Earth. So the conven-tional coarse alignment method cannot be utilized [4]. In2000, the IxSea Company claimed a new alignment algo-rithm,which could be done in 5min under any swinging con-ditions with their new alignment algorithm [5, 6]. Based onthe alignment idea of Octans, relevant investigations aboutthe inertial frame alignment have been carried out [7β9].The inertial frame alignment algorithm can isolate the distur-bance by lineal and angular movements of the vehicle in themarine. But in these coarse alignment algorithms, the headangle error within a few degrees and pitch/roll angle withina few tenths of a degree hence fulfill the requirement for thefine alignment [10].
In the fine alignment procedure, Kalman filter is an opti-mal linear estimator whenmeasurement noise has a Gaussian
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 320536, 12 pageshttp://dx.doi.org/10.1155/2015/320536
2 Mathematical Problems in Engineering
distribution with known measurement and process noisevariance values. In [11], the fine alignment error model in theinertial framewas developed, the velocity differences betweenthe calculated Strapdown INS values and the true valuesalong the inertial frame were considered as measurements,and Kalman filter was used to estimate the error angle. But inthis method, the disturbed acceleration was ignored based onthe integration of the disturbed acceleration which was con-sidered equal to zero. But the disturbed acceleration causedby heavy surge or waves in marine did not periodically vary,and determining an accurate value of this disturbance noisevariance is very difficult in the marine environment. To over-come the uncertainty measurement noise in Strapdown INSfine alignment procedure, several adaptive filter techniqueshave been developed for fine alignment. Most of the workreported for this problem has concentrated on innovation-based adaptive estimation, which utilizes new statisticalinformation from the innovation sequence to correct theestimation of the states. The main idea of these adaptive esti-mation methods is adjusting the π matrix online using fuzzylogic and neurofuzzy logic [12β14], for linear and nonlinearsystems, respectively.This adaptive filter is based on a covari-ance matching technique, where its input is the differencebetween the actual Kalman filter residual covariance and itstheoretical value. But the decision of the fuzzy rule andthe selection of the adjusting step size of π are dependenton experience more or less, and the complex adjustmentalgorithm degrades the real-time performance. In [15], anadaptive Kalman filter was specially presented to estimatemeasurement noise variance directly using the measure-ment residual and employing the Frobenius norm. And theadaptive Kalman filter in [15] has successfully improved thestability of filter in the case of measurement uncertainties.
Inspired by the adaptive idea in [15], a novel adaptiveKalman filter algorithm is developed in this paper, and sim-ulation test verifies the effectiveness of the proposed method.Similar to [11, 14], utilizing the gravity in the inertial frame as areference, the initial fine alignment errormodel is establishedfor fine alignment procedure of Rotary SINS.Thedifference isthat the PWCS theory and SVD theory are used to analyze theobservability of the fine alignment errormodel before design-ing filter. And the novel calculation method for the adaptivescalar value is obtained and proved based on the systemstability and optimality conditions of the adaptive filter.
This paper is organized as follows. The algorithmapproaches for the principle of rotary modulation techniqueand the coarse alignment algorithm in the inertial frame arepresented in Section 2. Section 3 provides the Rotary SINSfine alignment error model in the inertial frame and theobservability analysis of PWCS theory and SVD theory. Anadaptive filter is developed which is presented in Section 4.The simulation results are illustrated in Section 5. Finally, theconclusion is presented in Section 6.
Coordinate Frame Definitions. The π frame is the bodycoordinate frame. The π₯π axis is parallel to the body lateralaxis and points to the right.The π¦π axis is parallel to the bodylongitudinal axis and points forward.The π§π axis is parallel tothe body vertical axis and points upward.
The π frame is the navigation coordinate frame which isthe local level coordinate frame.The ππ₯ axis points to the East,the ππ¦ axis points to the North, and the ππ§ axis points to thezenith.
The π frame is the Earth-fixed coordinate frame. The ππ§axis is parallel to the Earthβs rotation axis and the ππ₯ axisis in the equatorial plane and points to the meridian of thebody initial position. The ππ¦ axis completes the right-handedcoordinate system.
The π frame is the inertial coordinate frame. It is formedby fixing the π frame at the beginning of the coarse alignmentin the inertial space.
The π frame is the computed inertial coordinate frame.The ππ0 frame is the body inertial coordinate frame. It is
formedby fixing the π frame at the beginning of the alignmentin the inertial space.
The π frame is the inertial measurement unit (IMU)coordinate frame. The origin of this frame is at the IMUβscenter of gravity and it is in accordance with the π frameat first. Besides, ππ₯π runs parallel to the longitudinal gyroβssensing axis and the longitudinal accelerometerβs sensing axis.Similarly, ππ¦π runs parallel to the transverse gyroβs sensingaxis and the transverse accelerometerβs sensing axis, whereasππ§π is vertical toππ₯π π¦π and it is upright.Then the IMU rotatesaround the rotation axis with angular velocity π. Thus, the π frame varied with the changes of IMUβs position is a real-timevariable frame.
2. Realization of Coarse Alignment forRotary SINS in the Inertial Frame
2.1. The Principle of Rotary Modulation Technique. In thissection, the principle of restraining navigation errors will beexplained in detail. First of all, the differential equations ofattitude error and velocity error for Rotary SINS are expressedas
Μππ= βππ
ππΓ ππ+ πΏππ
ππβ CππCππ πΏππ
ππ ,
πΏkΜπ = ππ Γ ππ + CππCππ πΏππ
ππ β (2πΏπ
π
ππ+ πΏππ
ππ) Γ Vπ
β (2ππ
ππ+ ππ
ππ) Γ πΏVπ + πΏππ.
(1)
First of all, the notations used in the equations should beexplained. Subscript denotes a frameβs relative movement toanother frame, and superscript denotes the expression in theframe. For example, ππ is the attitude error expressed in the πframe and ππ
ππis the angular rate of the π frame with respect
to the π frame, expressed in the π frame. Cππ is the transform
matrix from the π frame to the π frame andCππis the transform
matrix from the π frame to the π frame. πΏVπ denotes the vectorerror. In particular, πΏππ
ππ and πΏππ
ππ are the errors caused by the
inertial sensors inaccurate measurement. Vπ is the velocityexpressed in the π frame and ππ is the gravity vector in theπ frame. ππ
ππis the Earth rotation rate vector in the π frame
and ππππis the π frame angular rate in the π frame.
In accordance with IMU rotation which continues, theelement of Cπ
π is changed frequently. The fundamental aspect
of rotation modulation technique is to change the element of
Mathematical Problems in Engineering 3
Cππ periodically so thatmean values ofCπ
πCππ πΏππ
ππ andCπ
πCππ πΏππ
ππ
approximately approach zero. As a result, the navigationsystem errors are reduced and the system performance isimproved owing to the rotational motion.
Adopting dual-axis rotation modulating method in thispaper, three gyros and three accelerometers are used as theinertial sensors. Initially, the π frame is in accordance with theπ frame.Then the system rotates continuously at the uniformrotational angular velocities π1 and π2 (π1 ΜΈ= π2) around theππ₯π and ππ§π axis of the π frame, respectively. We use thetransformmatrixCπ
πto express the relations from the π frame
to the π frame:
Cπ π= Cπ π₯πCπ π§π. (2)
Since the outputs of the inertial sensors are measuredfrom the π frame, the following equations can be obtained:
ππ
ππ = Cπ ποΏ½ΜοΏ½π
ππ+ ππ
ππ + [ππ₯ ππ¦ ππ§]
π, (3)
ππ
ππ = Cπ πππ
ππ+ ππ
ππ + [βπ₯ βπ¦ βπ§]
π. (4)
Among them, it is assumed that the gyroscope drift andaccelerometer bias are ππ₯, ππ¦, and ππ§ and βπ₯, βπ¦, and βπ§. Inaddition, ππ
ππ = βCπ
πππ
π π= [π1 π2 sinπ1π‘ π2 cosπ1π‘]
π isthe angular rate of the π frame with respect to the π frame,expressed in the π frame. ππ
ππ is the specific force of the π
frame with respect to the π frame, expressed in the π frame.ππ
ππ = 0 is the specific force of the π frame with respect to the
π frame, expressed in the π frame. ππππis the measured output
of accelerometer. οΏ½ΜοΏ½πππis the measured output of gyro.
It is necessary to transform the inertial sensor measure-ments from the π frame to the π frame, so we have
ππ
ππ= (Cπ π)πππ
ππ + ππ
π π, (5)
ππ
ππ= (Cπ π)πππ
ππ + ππ
π π, (6)
where ππππis the angular rate of the π frame with respect to
the π frame, expressed in the π frame. οΏ½ΜοΏ½πππand ππ
ππdenote the
actual and measured value of the gyro output, respectively.Substituting (3) and (4) using (5) and (6), respectively, the
drift-bias errors of gyros and accelerometers by the rotationalmotion are
[[[
[
ππ
π₯
ππ
π¦
ππ
π§
]]]
]
=[[[
[
ππ₯ cosπ2π‘ β ππ¦ cosπ1π‘ sinπ2π‘ + ππ§ sinπ1π‘ sinπ2π‘ππ₯ sinπ2π‘ + ππ¦ cosπ1π‘ cosπ2π‘ β ππ§ sinπ1π‘ cosπ2π‘
ππ¦ sinπ1π‘ + ππ§ cosπ1π‘
]]]
]
,
[[[
[
βπ
π₯
βπ
π¦
βπ
π§
]]]
]
=[[[
[
βπ₯ cosπ2π‘ β βπ¦ cosπ1π‘ sinπ2π‘ + βπ§ sinπ1π‘ sinπ2π‘βπ₯ sinπ2π‘ + βπ¦ cosπ1π‘ cosπ2π‘ β βπ§ sinπ1π‘ cosπ2π‘
βπ¦ sinπ1π‘ + βπ§ cosπ1π‘
]]]
]
.
(7)
Above all, we can find that the drift-bias errors of gyrosand accelerometers on three axes in the π frame have beenchanged periodically. Consequently, it is confirmed that thedual-axis modulating rotationmethod is useful in restrainingthe drift-bias errors of the inertial sensor on three axes.
2.2. The Coarse Alignment Algorithm in the Inertial Frame.The coarse alignment for Rotary SINS is to determine thecoordinate transformmatrix from the π frame to the π frame,which is also called the attitudematrix and could be describedas follows:
Cππ (π‘) = Cπ
πCππ(π‘)Cπππ0Cππ0π (π‘) , (8)
where
Cππ=[[
[
0 1 0
β sin πΏ 0 cos πΏcos πΏ 0 sin πΏ
]]
]
, (9)
Cππ(π‘) =
[[
[
cosπππ (π‘ β π‘0) sinπππ (π‘ β π‘0) 0β sinπππ (π‘ β π‘0) cosπππ (π‘ β π‘0) 0
0 0 1
]]
]
, (10)
where πΏ is latitude, πππ is the Earth rotation rate, and π‘0 isthe start time of the coarse alignment. Cππ0
π (π‘) represents the
transformmatrix from the π frame to the ππ0 frame, which canbe updated by using the gyro output in the π frame throughthe attitude quaternion algorithm (its initial value is the unitmatrix).
Thus, the key of the coarse alignment based on the inertialframe is to compute the transformmatrixCπ
ππ0.Cπππ0represents
the transform matrix from the ππ0 frame to the π frame and isa constantmatrix. According to dual-vector attitude determi-nationmethod,Cπ
ππ0can be calculated by using the integration
of the gravity vector in the π frame and the ππ0 frame at twodifferent moments.
The accelerometer outputs in the π frame consist of theprojection of the gravity vector ππ , the projection of theinterference acceleration πΏππ
π·, the induced acceleration due
to the lever-arm ππ LA, and the constant bias error of theaccelerometer β, namely,
ππ = βππ + πΏππ
π·+ ππ
LA + β. (11)
As the constant bias error of the accelerometer β isreduced to zero after every rotation period, the integration of
4 Mathematical Problems in Engineering
the accelerometerβs measured specific force projected in theππ0 frame can be described as follows:
Vππ0π(π‘π) = β«
π‘π
π‘0
Cππ0πCππ ππ ππ‘
= β«
π‘π
π‘0
Cππ0πCππ (βππ + πΏππ
π·+ ππ
LA) ππ‘
= βCππ0πβ«
π‘π
π‘0
ππππ‘ + πΏVππ0
π·+ Vππ0LA,
(12)
Vππ0LA = β«π‘π
π‘0
πππ0
LAππ‘ = Cππ0π (ππ
ππ Γ ππ ) . (13)
As the integration of the πΏππ π·approaches zero approximately
after every rotation period, (12) becomes
Vππ0 (π‘π) = Vππ0
π(π‘π) β V
ππ0
LA (π‘π) = βCππ0
πβ«
π‘π
π‘0
ππππ‘
= Cππ0πVπ (π‘π) ,
(14)
where ππ is the gravity vector in the π frame. Integrating ππ
from π‘0 to π‘π yieldsΞπ‘π = π‘πβπ‘0, andVπ(π‘π) could be computedas
Vπ (π‘π) =
[[[[[[[[
[
π cos πΏπππ
sinπππΞπ‘π
π cos πΏπππ
(1 β cosπππΞπ‘π)
Ξπ‘ππ sin πΏ
]]]]]]]]
]
. (15)
Substituting (15) in (14), the following equations can beobtained:
Vππ0 (π‘π1) = Cππ0
πVπ (π‘π1) ,
Vππ0 (π‘π2) = Cππ0
πVπ (π‘π2) ,
(16)
where π‘π2 is the end time of coarse alignment and π‘0 < π‘π1 <π‘π2.
From (16), Cπππ0could be calculated by the following
equation:
Cπππ0=
[[[[
[
(Vπ)π
(π‘π1)
(Vπ)π
(π‘π2)
[Vπ (π‘π1) Γ Vπ (π‘π2)]π
]]]]
]
β1
β
[[[[
[
(Vππ0)π
(π‘π1)
(Vππ0)π
(π‘π2)
[Vππ0 (π‘π1) Γ Vππ0 (π‘π2)]π
]]]]
]
.
(17)
After deriving Cπππ0, the attitude matrix Cπ
π can be calcu-
lated through (8).
3. Fine Alignment Error Model andObservability Analysis
The coarse initial attitude angle can be obtained by the coarsealignment method in Section 2. The marine heading angleerror is estimated to within a few degrees and pitch/roll angleto within a few tenths of a degree, which allow the fine align-ment filter to operate within it in the linear region. In this sec-tion, the fine alignment errormodel of Rotary SINS in inertialframe is developed. Moreover, observability analysis of theerror model is discussed using PWCS and SVD theory beforedesigning the filter.
3.1. Fine Alignment Error Model. Using (8) for reference, theattitude matrix Cπ
π (π‘) which remains to be calculated with
high accuracy at the end of the fine alignment procedure canbe represented as
Cππ (π‘) = Cπ
πCππ(π‘)Cππ (π‘) , (18)
whereCππandCπ
π(π‘) are derived from (9) and (10), respectively.
Cππ (π‘) can be described as follows:
Cππ (π‘) = [Cπ
π(π‘)]
β1
Cπ
π (π‘) , (19)
whereCπ
π (π‘) represents the transformmatrix from the π frame
to the computed inertial frame π and is given by
Cπ
π (π‘) = Cπ
π (0) ΞCπ
π , (20)
where the attitude matrix Cπ
π (0) relates the π frame to the π
frame at the beginning of the fine alignment, which is derivedfrom the coarse alignment. Through the attitude quaternionupdating algorithm, the matrix ΞCπ
π can be computed recur-
sively with the gyro and accelerometer outputs. Furthermore,there exist misalignment angles ππ between the inertial frameπ and the computed inertial frame π owing to componenterrors; the transitionmatrixCπ
πcan be constructed as follows:
Cπ
π= πΌ β [π
πΓ] =
[[[
[
1 βππ
π§ππ
π¦
ππ
π§1 βπ
π
π₯
βππ
π¦ππ
π₯1
]]]
]
. (21)
By substituting (20) in (19), the transform matrix Cππ (π‘)
can be determined. So themain problemof the fine alignmentbecomes the estimation of misalignment angles ππ. Systemerror equations should be established firstly, which includevelocity and attitude error equations. The measured specificforce projected from accelerometer in the π frame can bewritten as
Cπ
π ππ
= Cπ
π [(πΌ + πΏπΎπ΄) (πΌ + πΏπ΄) (βπ
π + ππ
LA + ππ
π·) + β]
β βππ+ ππΓ ππ+ Cππ β + Cπ
π ππ
LA + Cπ
π πΏππ ,
(22)
Mathematical Problems in Engineering 5
where πΏππ is the interference acceleration in the π frame andaccelerometer bias β includes constant drift βπ and Gaussianwhite noise βπ€, namely, β = βπ + βπ€. So (22) can be writtenas
Cπ
π ππ + ππβ Cπ
π ππ
LA = ππΓ ππ+ Cππ βπ + C
π
π βπ€ + πΏπ
π. (23)
Integrating (23) from π‘1 to π‘ yields
VΜπ = β«π‘
π‘1
Cπ
π ππ ππ + β«
π‘
π‘1
ππππ β β«
π‘
π‘1
Cπ
π ππ
LAππ
= β«
π‘
π‘1
ππΓ ππππ + β«
π‘
π‘1
Cππ βπππ + β«
π‘
π‘1
Cππ βπ€ππ
+ β«
π‘
π‘1
πΏππππ,
(24)
where VπLA = β«π‘
π‘1Cπ
π ππ
LAππ = Cπ
π (ππ
ππ Γ ππ ), πΏVππ·= β«π‘
π‘1πΏππππ,
Vππ= β«π‘
π‘1ππππ, and Vπ
π= β«π‘
π‘1Cπ
π ππ ππ. Consider
πΏVπ = β«π‘
π‘1
ππΓ ππππ + β«
π‘
π‘1
Cππ βπππ + β«
π‘
π‘1
Cππ βπ€ππ. (25)
Measurement vectors could be described as follows:
Z = VΜπ = Vππ+ Vππβ VπLA = πΏV
π+ Vπ€ + πΏV
π
π·. (26)
In interference angular velocities and accelerations envi-ronment, the velocity error equation in the inertial frame canbe defined as
πΏVΜπ = βππ Γ ππ + Cππ βπ + C
π
π βπ€. (27)
In accordance with the differential equations of Cππ and
Cπ
π , themisalignment angle equation in the inertial frame can
be defined as
οΏ½ΜοΏ½π= βCππ ππ β C
π
π ππ€, (28)
where ππ and ππ€ are gyro constant drift and Gaussian whitenoise.
With gyro drift and accelerometer bias added to statevectors, the state equation in the dynamic disturbance envi-ronment is described as follows:
XΜ (π‘) = A (π‘)X (π‘) + B (π‘)W. (29)
The state vectors and system noise in (29) are
X (π‘) = [πΏπππ₯, πΏππ
π¦, πΏππ
π§, ππ
π₯, ππ
π¦, ππ
π§, πππ₯, πππ¦, πππ§, βππ₯,
βππ¦, βππ§]π
W = [βπ€π₯, βπ€π¦, βπ€π§, ππ€π₯, ππ€π¦, ππ€π§, 0, 0, 0, 0, 0, 0]π
,
(30)
where six vectors in π(π‘) are the noise of accelerometersand gyros in the π frame, which is normally distributed whitenoise with zero mean andπ(π‘) variance.The state matrix and
system noise matrix of Rotary SINS in the inertial frame areas follows:
A =[[[[[[
[
03Γ3 β (ππ(π‘) Γ) 03Γ3 Cππ (π‘)
03Γ6 βCππ (π‘) 03Γ303Γ12
03Γ12
]]]]]]
]
,
B =[[[[[
[
Cππ (π‘) 03Γ3 03Γ3 03Γ3
03Γ3 βCππ (π‘) 03Γ603Γ12
03Γ12
]]]]]
]
.
(31)
Furthermore, velocity errors are selected as measurementvectors and the measurement equation is
Z (π‘) = HX (π‘) + Vπ€ + πΏVπ
π·. (32)
The measurement matrix in (32) is
H = [I3Γ3 03Γ9] , (33)
where πΏVππ·is uncertainty measurement disturbance caused
by the marineβs rock and sway. Vπ€ is system measurementnoise, which is white noise.
3.2. Observability Analysis Using PWCS and SVD. For thedesigning of optimal filter, the observability of the systemmodel is an important issue and requires careful investiga-tion, as it is crucial for the stability and convergence of the fil-ter. In this section, the observability analysis of the fine align-ment model for Rotary SINS is discussed using twomethods,which are piecewise constant system (PWCS) [16, 17] methodand singular value decomposition (SVD) [18, 19]. First, thePWCS analysis method is used to discover the number of theobservable state value inMATLAB. Second, the SVDmethodis used to calculate the observable degree of every state value,because it is an efficient method to analyze the observabledegree of every state value for completely observable and theincompletely observable system.
Assume that the matrix π π is the observability matrix ofa dynamic system; it can be expressed by
π π = πΞ£ππ, (34)
where π = [π’1, π’2, . . . , π’π] and π = [V1, V2, . . . , Vπ] are bothorthogonal matrixes. Ξ£ = [π 0(πβπ)Γπ]
π is a matrix withπΓπorder, π = diag{π1, π2, . . . , ππ} is a diagonal matrix, and π1 β₯π2 β₯ β β β β₯ ππ > 0 are called the singular value of the matrixπ π, π = rank(π΄).
The measurement matrix is given by
π ππ0 = π, (35)
6 Mathematical Problems in Engineering
Table 1: The comparison of observable degree for Rotary SINS.
Time πΏππ₯
πΏππ¦
πΏππ§
ππ₯
ππ¦
ππ§
ππ₯
ππ¦
ππ§
βπ₯
βπ¦
βπ§
1 1.0135 1.1031 1.0324 0.6251 13.8570 13.8636 13.7323 13.9108 2.2515 1.4198 1.4641 1.39462 1.1089 1.2246 1.1567 0.6785 13.8836 13.8975 13.7635 13.9438 2.4624 1.4856 1.5358 1.42983 1.2452 1.3244 1.2346 0.7037 13.9104 13.9044 13.8015 13.9704 2.6584 1.5203 1.5747 1.4742
where π0 is the initial state vector and π is measurementvector. By substituting (34) into (35), we can obtain
π =
π
β
π=1
ππ (Vπ
ππ0) π’π (36)
π0 = (πΞ£ππ)β1
π =
π
β
π=1
(π’π
ππ§
ππ
) Vπ. (37)
According to (37), initial state vector π0 describes anellipsoid when measurement vector π is constant norm, andthe equation is given by
π
β
π=1
(Vπππ0π’π
ππ
)
2
=π¦
2, (38)
where ππ = 1/ππ represents the principal axis length of theellipsoid. Obviously, the volume of the ellipsoid is determinedby singular value. The volume reduces as well as the statevector π0 when the singular value rises up. Therefore,singular value represents the ellipsoid radius formed by theestimated initial state. The ellipsoid radius becomes smalleras the singular value becomes larger, and the estimationperformance of the initial state is better.
For Rotary SINS, we use dual-axis rotation schemepresented above; the initial state vector corresponding to 12singular values is found by (34)β(38), and the rotating angularvelocity is π1 = 3π2 = 18
β/s. Table 1 shows the comparison
of observable degree for Rotary SINS in interference angularvelocities and accelerations environment during three succes-sive time periods.
Table 1 demonstrates that the system is completelyobservable and every state value can be estimated with theSVD theory. Particularly, ππ¦, ππ§, ππ¦, ππ₯, and ππ§ which have alarger observable degree can reach a better alignment effect.The observable degree of πΏππ₯, πΏππ¦, and πΏππ§ is almost 1.Compared with the other eleven states, the observable degreeof ππ₯ is a little smaller. But it is large enough to be observable.On the basis of observability theory, the filter has beendesigned to estimate the system state parameters.
4. Novel Adaptive Kalman Filter Designed forFine Alignment
In the fine alignment procedure, Kalman filter is an optimallinear estimator when the measurement noise has a Gaussiandistribution with known covariance. However, when RotarySINS is under the large dynamic disturbance environment,the conventional Kalman filter becomes not an optimalestimator, which may tend to diverge. In this section, a novel
adaptive Kalman filter for fine alignment that utilizes themeasurement residual to estimate measurement noise vari-ance in real time is investigated. The novel adaptive filteris developed according to the optimality condition of theadaptive Kalman filter presented in [20, 21].
The system error model which we discussed above can beconsidered as the following state-space system:
ππ = Ξ¦π,πβ1ππβ1 + Ξπβ1ππβ1, (39)
ππ = π»πππ + ππ. (40)
The measurement residual is given by
πΏππ = ππ β π»πππ,πβ1. (41)
By substituting (40) in (41), themeasurement residual canbe written as
πΏππ = π»π (ππ β ππ,πβ1) + π1π + πΌπ2π
+ π (ππ, ππ,πβ1) ,
(42)
where πΌ is a real number, πΌ β₯ 0, π1π is the measurementnoise with a fixed variance, and π2π is the time-varyingmeasurement noise. It is assumed that the two types of mea-surement noises are uncorrelated. π(ππ, ππ,πβ1) is the higherorder item in the estimation error. If this item is ignored, (42)can be simplified to
πΏππ = π»π (ππ β ππ,πβ1) + π1π + πΌπ2π. (43)
The covariance of the measurement residual is expressedas
Sπ,πβ1 = πΈ [πΏZππΏZπ
π] = HπPπ,πβ1H
π
π+ R1π + π½R2π, (44)
where π½ = πΌ2 is scalar value that accommodates the filterβsstability and optimality, R1π = πΈ[V1πVπ1π], and R2π =πΈ[V2πVπ2π].
The covariance of state estimation error is defined as
ππ,πβ1 = Ξ¦π,πβ1ππβ1Ξ¦π
π,πβ1+ ππ. (45)
Substituting (45) into (44) results in
Sπ,πβ1 = Hπ (Ξ¦π,πβ1ππβ1Ξ¦π
π,πβ1+ ππ)H
π
π+ R1π
+ π½R2π = πΏ1 + π½πΏ2,(46)
where πΏ1 = Hπ(Ξ¦π,πβ1ππβ1Ξ¦ππ,πβ1+ππ)Hπ
π+R1π and πΏ2 = R2π.
Mathematical Problems in Engineering 7
Thus we can get the optimal gain of the adaptive Kalmanfilter as follows:
πΎπ = ππ,πβ1π»π
ππβ1
π,πβ1. (47)
To obtain the optimal gain, in [15], the cost equation isdefined in terms of minimizing the Frobenius norm to calcu-late π½ value in real time. In this paper, we employ the opti-mality condition of the adaptive Kalman filter to obtain theoptimal gain. The following equation shows the optimalitycondition, which means that the sequence of measurementresiduals is uncorrelated:
πΆπ (π) = E [πΏZπ+ππΏZπ
π] = HπPπ,πβ1H
π
π+ R1π + π½R2π
= 0 (π = 1, 2, . . . ; π = 0, 1, . . .) .
(48)
The autocovariance of the measurement residuals is
πΆπ (π) = E [πΏZπ+ππΏZπ
π]
= π»π+πππ+1,π+πβ1 [πΌ β πΎπ+πβ1π»π+πβ1]
β β β ππ+2,π+1 [πΌ β πΎπ+1π»π+1]
β ππ+1,π [ππ|πβ1π»π
πβ πΎππΆ0 (π)] , βπ = 1, 2, 3, . . . .
(49)
In reality, the real covariance of the measurement residu-als is different from a theoretical one owing to model errorsor unknown external disturbances. Thus, πΆπ(π) may not beidentical to zero. From (49), we know that if the last termof πΆπ(π) which is the only common item of πΆπ(π) for allπ = 1, 2, 3, . . . is zero,
ππ|πβ1π»π
πβ πΎππΆ0 (π) = 0, (50)
then the gain πΎπ is optimal. In other words, if the gain πΎπis optimal, (50) holds. Equation (50) forms the basis fordetermining scalar value π½. In particular,πΆ0(π) in (50) whichis computed from measured data in real time, according to πsuccessive residuals, can be shown as
πΆ0 (π) =1
π β 1
π
β
π=1
πΏπππΏππ
π. (51)
The optimal scalar value π½ can be obtained by (52) whenthe system model satisfies Assumptions 1 and 2, which is theoptimality condition of the adaptive Kalman filter presentedin [20, 21].
Assumption 1. π(π), π (π), and π(0) are all positive definitesymmetrical matrix.
Assumption 2. Themeasurement matrixπ»(π) is full ranked:
π½ =tr [πΆ0 (π) β πΏ1]
tr [πΏ2], (52)
where if π½ β€ 0, π½ = 0 and if π½ β₯ maxπ½, π½ = maxπ½.
Proof. Substituting (47) into the optimality condition (50)gives
ππ|πβ1π»π
π[πΌ β (πΏ1 + π½πΏ2)
β1πΆ0 (π)] = 0. (53)
Since ππ|πβ1 and πΆ0(π) are full-ranked symmetric matrixaccording to Assumptions 1 and 2 and (51), left multiplyingπβ1
π|πβ1and right multiplying πΆ0(π)
β1 on both sides of (53), itis obvious that (53) implies the following relation:
π»π
ππΆ0 (π)
β1β π»π
π(πΏ1 + π½πΏ2)
β1= 0; (54)
then
π»π
ππΆ0 (π)
β1(πΏ1 + π½πΏ2) = π»
π
π. (55)
Since π»π is full-ranked matrix according toAssumption 2, (55) is simplified as
π½πΏ2 = πΆ0 (π) β πΏ1. (56)
To avoid inversionmanipulation, trace is directly taken inboth sides of (56):
tr (π½πΏ2) = tr (πΆ0 (π) β πΏ1) (57)
then we can get the scalar value π½ in (52). Proof ends.
As shown in [20, 21], instability issues may arise when noupper bound is set for the scalar value. Thus the upper andlower bound are defined for the measurement noise varianceas [0 π½max], and the rule is as follows: if π½ β€ 0, π½ = 0 and ifπ½ β₯ π½max, π½ = π½max.The π½max value can be defined by the useraccording to (44).
Substituting π½ into (46) during the initial alignmentprocess, the proposed adaptive filter for the system stateestimation is as follows:
ππ,πβ1 = Ξ¦π,πβ1ππβ1,
ππ,πβ1 = Ξ¦π,πβ1ππβ1Ξ¦π
π,πβ1+ ππ,
πΏ1 = Hπ (Ξ¦π,πβ1ππβ1Ξ¦π
π,πβ1+ ππ)H
π
π+ R1π,
πΏ2 = R2π,
πΆ0 (π) =1
π β 1
π
β
π=1
πΏπππΏππ
π,
π½ =tr [πΆ0 (π) β πΏ1]
tr [πΏ2],
ππ,πβ1 = π»πππ,πβ1π»π
π+ π 1π + π½π 2π,
πΎπ = ππ,πβ1π»π
ππβ1
π,πβ1,
ππ = [πΌ β πΎππ»π] ππ,πβ1,
ππ = ππ,πβ1 + πΎπ [ππ β π»πππ,πβ1] .
(58)
For (32) it consists of three measurements. A scalar valueπ½ is required for each measurement, as every measurement
8 Mathematical Problems in Engineering
0 1000 2000 3000 4000
20
0
β20
β40
β60
β80
β100
100
80
60
40
20
0
β20
t (s)0 1000 2000 3000 4000
t (s)0 1000 2000 3000 4000
t (s)
πΏπx()
πΏπz()
50
0
β50
β100
πΏπy()
(a) Estimation of attitude errors
0
5
10
15
0204060
β5
β10 β60
β40
β20
0204060
β60
β40
β20
0 1000 2000 3000 4000t (s)
0 1000 2000 3000 4000t (s)
0 1000 2000 3000 4000t (s)
πΏVx
(m/s
)
πΏVy
(m/s
)
πΏVz
(m/s
)
(b) Estimation of velocity error
0 1000 2000 3000 4000
1
0.5
0
β0.5
β1
β1.5
β2
2
0
β2
β4
β6
β8
β10
1
0
β1
β2
β3
β40 1000 2000 3000 40000 1000 2000 3000 4000
t (s) t (s) t (s)
Γ10β4 Γ10β5
Γ10β3
π x(β/h)
π y(β/h)
π z(β/h)
(c) Estimation of gyro drifts
0
50
0
20
40
60
80
050
β50
β100
β20
β50
β100
β150
β200
β250β1500 1000 2000 3000 4000 0 1000 2000 3000 4000 0 1000 2000 3000 4000
t (s) t (s) t (s)
βx
(πg)
βy
(πg)
βz
(πg)
(d) Estimation of accelerometer biases
Figure 1: The simulation results of NAKF.
has unique characteristics. Similar to [15], this paper solvesthis problem using a scalar measurement updating π/π·covariance factorization filter, whereπ is an upper triangularand unitary matrix and π· is a diagonal matrix. It is alsoadvantageous to utilize the π/π· covariance factorizationfilter when designing and implementing the novel adaptivefilter.
5. Simulation Results
The parameters of the simulation are set as follows.
The marine is rocked by the surf and wind. The pitch π,roll πΎ, and heading π resulting from the marine rocking arechanged periodically and can be described as follows:
π = 7β cos(2π
5π‘ +
π
4)
πΎ = 10β cos(2π
6π‘ +
π
7)
π = 30β+ 5β cos(2π
7π‘ +
π
3) .
(59)
Mathematical Problems in Engineering 9
0 100 200 300 400 500 600
SHAFAKFNAKF
t (s)
100
80
60
40
20
0
β20
β40
Hea
ding
erro
r( )
(a) Tilt misalignment angle
0 100 200 300 400 500 600t (s)
Pitc
h er
ror(
)
40
20
0
β20
β40
β60
β80
β100
β120
SHAFAKFNAKF
(b) Eastern misalignment angle
0 100 200 300 400 500 600t (s)
Roll
erro
r( )
50
0
β50
β100
SHAFAKFNAKF
(c) Northern misalignment angle
Figure 2: The results depending on the filter with mean misalignment angles.
The velocity caused by surge, sway, and heave is as follows:
ππ·π = π΄π·πππ·π cos (ππ·ππ‘ + ππ·π) , (60)
where π = π₯, π¦, π§,π΄π·π₯ = 0.02m,π΄π·π¦ = 0.03m,π΄π·π§ = 0.3m,ππ·π = 2π/ππ·π, ππ·π₯ = 7 s, ππ·π¦ = 6 s, and ππ·π§ = 8 s. ππ·π obeysthe uniform distribution on the interval [0, 2π].
The IMU errors are set as follows: the gyro constant drift:0.01β/h; the gyro random noise: 0.001β/h; the accelerator bias:1Γ10
β4 g; and the accelerator measurement noise: 1Γ10β5 g.Rotary SINS location: north latitude is 40β and east longitudeis 118β. According to the rule shown in the previous section,double-axis rotation rate is given by π1 = 3π2 = 18
β/s.
Simulation 1. The coarse alignment lasts 120 s. The values ofπ‘π1 and π‘π2 in (17) are set to 50 s and 120 s, respectively. Thesimulation for the coarse alignment runs 50 times.The pitch,roll, and heading errors at the end of the coarse alignmentsare shown in Table 2.
From Table 2, it is obvious that the level attitude errors ofthe coarse alignment are less than 0.2 degrees and the headingattitude error is less than 1.5 degrees.The attitude errors calcu-lated by the proposed coarse alignment algorithm can fulfill
the requirement for the fine alignment. But it only accom-plished the coarse estimation of attitude errors in initialalignment. Thus 600-second fine alignment is followed.
Next, the mean and the maximum of misalignmentangles in Table 2 are used, respectively, as input for finealignment to validate the proposed adaptive filter in initialalignment under dynamic disturbance condition. The meanofmisalignment angles is used in Simulations 2 and 3, and themaximum of misalignment angles is used in Simulation 4.
Simulation 2. The simulation results are shown in Figure 1.It is clear that all the states are observable and the resultconfirms the former analysis. According to Figure 1(a), con-vergence time of level misalignment angle is within 60 s, andprecision is 1, while azimuth misalignment angle conver-gence time iswithin 80 s, and precision is about 1.3.The threevelocity errors as observed variables depicted in Figure 1(b)are certainly observable. As seen from Figure 1(c), gyro driftsconverge not so rapidly, but a rather good filtering result canbe obtained. Figure 1(d) shows that the horizontal accelerom-eters constant biases βπ₯ and βπ¦ converge to a constantvalue in short time. The vertical converges soon afterwardswith good estimation performance. The simulation results
10 Mathematical Problems in Engineering
0 100 200 300 400 500 600
120
100
80
60
40
20
0
β20
Hea
ding
erro
r( )
t (s)
SHAFAKFNAKF
(a) Tilt misalignment angle
0 100 200 300 400 500 600t (s)
SHAFAKFNAKF
Pitc
h er
ror(
)
20
0
β20
β40
β60
β80
β100
β120
(b) Eastern misalignment angle
20
0
β20
β40
β60
β80
β120
β100
Roll
erro
r( )
0 100 200 300 400 500 600t (s)
SHAFAKFNAKF
(c) Northern misalignment angle
Figure 3: The results depending on the filter with maximum misalignment angles.
Table 2: Statistics for coarse alignment results.
Parameter item Max. Min. MeanPitch error (β) 0.0740 β0.0986 0.0287Roll error (β) 0.1563 β0.2668 β0.0419Heading error (β) 1.4611 β0.1044 0.1506
demonstrate that all the system state parameters can be esti-mated by the presented filter. It can be seen that the presentedalignment method can effectively depress the random distur-bances under dynamic environment.
Simulation 3. Consideringmeasurement uncertainties underthe dynamic disturbing conditions, Sage-Husa adaptive filter(SHAF) used in [22], adaptive Kalman filter (AKF) pre-sented in [15], and the novel adaptive Kalman filter (NAKF)proposed in this paper are employed to evaluate the filterperformance. The simulation results are shown in Figure 2and Table 3.
The simulation results demonstrate that the performanceand convergence speed of NAKF which can estimate the
variance of the measurement noise in real time are betterthan those of AKF and SHAF.This implies that, in the case ofdynamic disturbance, NAKF functions better than SHAF andAKF and provides the optimal estimation.Then Simulation 3with the maximum initial figures is made.
Simulation 4. The simulation results show that the proposedNAKF is efficient with large initial attitude error. The resultsare shown in Figure 3 and Table 4. Compared with SHAF andAKF, NAKF provides higher performance and better stabilityin estimating themisalignment angles when large errors existin the measurement values. The result demonstrates thatNAKF can effectively depress the random disturbances inmeasurements noise variance caused by the ship rocking.Taking into account the unknown outside disturbances andthe complicated working environment, the proposed NAKFwith the optimality of filter is applicable in implementinginitial alignment of Rotary SINS.
6. Conclusions
Based on the alignment mechanism of tracing the grav-itational apparent motion in inertial frame, an initial
Mathematical Problems in Engineering 11
Table 3: The mean misalignment angle errors depending on thefilter.
Parameter item SHAF AKF NAKFPitch error () β1.43 β1.17 1.02Roll error () β1.62 β1.29 1.07Heading error () 1.85 1.54 β1.26
Table 4: The maximum misalignment angle errors depending onthe filter.
Parameter item SHAF AKF NAKFPitch error () 2.93 2.59 β1.89Roll error () 3.09 β2.29 1.80Heading error () 3.81 β2.14 1.84
self-alignment method based on the novel adaptive Kalmanfilter for Rotary SINS is developed. Before the filter isdesigned, observability is analyzed using PWCS method andSVD method. The theoretical analysis results show that thefine alignment error model is completely observable and allthe system state parameters can be estimated. For the uncer-tainties measurement noise, the novel adaptive Kalman filterthat estimates measurement noise variance in real time usingmeasurement residual is designed.
Simulation results proved the accuracy and validity ofthe proposed method, and the estimation results of thesystem states can approach the theoretical analysis results. Inaddition, this method is fully self-aligned with no externalreference information under dynamic disturbances condi-tion. But in this paper, only simulation test is finished. Moretests, such as turntable and shipboard test, should be studiedbefore they can be widely used in engineering.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
This work was supported by the Program for the Top YoungInnovative Talents of Beijing CITTCD201304046.
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