Application of Noise Estimator with Limited Memory Index on Flexure
Compensation of Rapid Transfer Alignment
Wei-dong Zhou, Yu-ren Ji
Department of Automation, Harbin Engineering University, Harbin, China
(baziji@163.com)
Abstract - In order to solve the flexure compensation
problem in rapid transfer alignment, the error equations are
simplified by noise compensation method firstly. Due to the
time variant characteristics of flexure process in time
domain, which leads to the fixed noise statistical
characteristics cannot follow the variation of actual
environment, the noise estimator with limited memory index
is proposed. By limiting the memory length of obtained data,
too old historical data is giving up and the accuracy of online
noise estimator is improved. The final simulation verifies
that the method proposed have higher accuracy and faster
convergence speed than conventional methods.
Keywords - Flexural Deformation, Limited Memory,
Noise Compensation, Transfer Alignment
I. INTRODUCTION
Transfer alignment is an important technology of
moving base alignment. In the process of transfer
alignment, the maneuver of carrier is required and the
running environment is getting complicated. As a result,
the flexural deformation is one of the most important error
sources[1-4].
There are mainly two methods for the compensation
of flexural deformation, the model construction method
and noise compensation method. Because of the high
complexity of physical modeling, most researches are
turned to the experimental modeling, which uses the
markov process to describe the random flexural
deformation. The flexure motion is modeled as the thirdorder markov model driven by the white noise through
analysis of experimental data in literature[1]. On the base
of above job, the markov process is further divided into
high frequency mode and low frequency mode by
literature[2]. In literature[4], the elastic deformation of
aircraft wing is modeled as two-order markov process.
The model construction method is realized by computing
the Markov parameters based on the similarity principle
of power spectral density[5-8]. However, modeling of
Markov process will lead to rapid increase of state
dimension and computation burden, so a suboptimal filter
without the third-order markov model is proposed in
literature[1], where the strength of system noise is
enhanced to compensate the uncertainty of flexural
deformation, which is just the noise compensation
method. Comparing to model construction method, the
complexity of noise compensation method is reduced and
the robustness is improved, but the accuracy also declines.
It is mainly attributed to the variation of actual noise
because of the environmental change while the noise
statistics is set to a fixed value in the filter. Adaptive filter
can be used to solve this problem[9-12]. A noise estimation
algorithm based on maximum likelihood is proposed for
linear system in literature[13]. Noise variance estimator is
designed based on EM principle in literature[14]. An
nonlinear noise estimation algorithm is proposed based on
maximum-posterior likelihood in literature[15]. Aiming at
the rapid time-variant characteristics of flexural
deformation, the limited memory index is combined with
the noise estimator. By limitation of memory length, the
old history data is abandoned to improve the precision of
online noise estimator.
In this paper, the system error equation is simplified
by noise compensation method firstly. Then the adaptive
filter with limited memory index is designed. Finally the
effectiveness of the algorithm is verified by simulation.
II. SYSTEM ERROR MODEL
A. Velocity Error Model
The velocity differential equation of master inertial
navigation system(MINS) is given by
Vmn  Cmn f mm   2ien  enn   Vmn  g mn
(1)
The velocity differential equation of slave inertial
navigation system (SINS) is given by
Vsn*  Csn* f ss   2ien  enn   Vsn*  g sn*
(2)
where the relations between the variables can be defined
by
f ss  f ms  fl s  a sf   s
V  V  V  V
n
s*
n
m
n
r
 V  Vsn  Vmn  Vrn
(3)
*
 
n
ie
n
ie
  enn
n
en
g sn*  g mn
By inserting (3) to the difference between (1) and
(2), leading to


Vsn*  Vmn  Csn* f ss  Csn* Cms Csm  f ss  fl s  a sf s    2ien  enn   Vsn*  Vmn (4)
*
The lever-arm velocity and its differential equation
are given by
Vrn  CinCsiiss  r s
Vrn  Cinnii  Vri  Csn iss  iss  r s  iss  r s 
(5)
where the term of iss  iss  r s  iss  r s is lever-arm
acceleration, which can be written as
fl s  iss  iss  r s  iss  r s
(6)
Vrn  Csn f l s  inn  Vrn

n
en
n
s*
V
n
m
(16)
ins is instruction angular velocity computed by
iss  ims   sf   s

s
 m  Cms Csmnm
  sf   s  ims  ˆ ins
*
(8)
n
r
When physical misalignment  a and measurable
misalignment  m are small, their direction cosine matrix
can be written as
*
Cms  I  m 
(9)
C  I  a 
m
s
Using (7) (8) and (9), reducing two-order small terms,
(4) can be given by
 V  Csn*  m  a   f ss   2ien  enn    V  Csn*  a sf   s  (10)
B. Attitude Error Model
Differential equation of direction cosine matrix of
 m can be written as
s
Csm*  Csm* ms
* 
*
s
ms
  m 
*
*
C  C  C  C C  
n
mn
n
s*
m
n
n
s*
s
ns
s
ims  ˆ ins  nm
 nss
(19)
According to (9), by inserting (19) to (18) leading to
the attitude error model
 m   m  a  nss   sf   s
(20)
C. Inertial Instrument Error Model
The error model of accelerometer and gyro are
composed by constant drift and white noise, which can be
written as
  c  a ,  c  0
   c  g ,  a  0
(21)
(22)
(12)
(13)
 xk   k 1 xk 1  wk 1

 zk  H k xk  vk
(23)
where xk is the state of n  1 vector and zk is the
observable variable of m1 vector. Process noise wk and
*
*
where
According to the error model of small
misalignments, the state equation and measurement
equation are linear equation with additive noise, whose
discrete general formula can be given by
By inserting (11) (12) to (13) leading to
n
 m   Cns mn
 Csn  nss 
(18)
D. Analysis of System Error Model
Expansion and differentiation of (11) results in
m
n
(17)
(11)
where
thus
nss  iss  ins
Using (16) (17), (15) becomes
  V
 V   V 
 2
n
ie
m
s*
(15)
where
(7)
The Coriolis term is given as

*
SINS,  iss is measured angular rate which can be written
as
So (5) is given by
n
in
s
 m  Cms Csmnm
 nss
(14)
measurement vk are zero mean white noise and
uncorrelated, whose prior statistical characteristics can be
expressed as
T

 E[ wk , w j ]  Qk  kj ,

T

 E[vk , w j ]  Rk  kj
Pk ,k 1  k 1Pk 1Tk 1  Qk 1
(24)
zˆk , k 1  H k xˆk , k 1
where  kj is the function of kronecker-  . The flexure
 
af
xk  k 1 xk 1  k 1  wk 1
(26)
where  k 1 can be written by
 k 1
C n* a sf 
  ss 
 f 
(27)
So the process noise is adjusted as
wk*1   k 1  wk 1
K k  Pk ,k 1H kT  H k Pk ,k 1H kT  Rk 
Pk   I  Kk H k  Pk ,k 1
 f  (25)
When Markov model is used to describe the flexure
process, the dimension of state will increase rapidly. If the
east channel and west channel are considered and the twoordered Markov model is adopted, the required state
variables will be 8. If all the three channels are described
by three-ordered Markov model, the required state
variables will be 18. So the system model needs to be
simplified and the state equation can be written by
(28)
In time domain, the uncertainty caused by flexure
process is presented as zero mean oscillation curve, which
means the process noise obeys the statistical
characteristics of zero mean and unknown variance. So
the compensated noise needs to be estimated online.
III. DESIGN OF ADAPTIVE FILTER BASED ON
NOISE ESTIMATOR WITH LIMITED MEMORY
INDEX
(34)
(35)
B. Noise Estimator Based On Limited Memory Length
According to the analysis of section II, the variance
of compensated noise is unknown, so the noise estimator
will be designed in this section. The process noise
estimator based on maximum likelihood principle can be
given by
k


T
1
Qˆ k   Ki  zi  hi xˆ i ,i 1  zi  hi xˆ i ,i 1   KiT  i 1Pi 1Ti 1  Pi (36)


k i 1
Because the observability needs to be increased by
adopting required maneuvers, the running environment
will become complicated and the statistical characteristics
of compensated noise vary rapidly, as a result the
effectiveness of too old history data will become weak
and even negative. So the length of history data is limited
to improve the accuracy of process noise estimator, which
will be expressed as follows. Firstly the weighted
coefficient i  is given by
m

i 1
i
 1,  i   i 1b
(37)
where m is the memory length, b is forgetting factor.
According to (37),  i can be further deduced as
i 
bi 1  bi
1  bm
(38)
By inserting (38) to (36), leading to

k

T
1
Qˆ k    k 1i Ki  zi  hi xˆ i ,i 1  zi  hi xˆ i ,i 1   KiT  i 1Pi 1Ti 1  Pi (39)


k i 1
(29)

(30)

T
1 b
Qˆ k  bQˆ k 1  m K k  zk  H k xˆ k ,k 1  zk  H k xˆ k ,k 1   K kT   k 1Pk 1Tk 1  Pk


1 b
m
m 1
T
(40)
b b

K k m  zk m  H k m xˆ k m,k m1  zk m  hk m xˆ k m,k m1   K kTm
m


1 b
 k m1Pk m1Tk m1  Pk m 

(2) Time and measurement update
xˆk , k 1   k 1 xˆk 1
(33)
So the process noise estimator based on limited
memory length is given by
xˆ0  E  x0 
T
P0  E  x0  xˆ0  x0  xˆ0  


1
xˆk  xˆk , k 1  K k  zk  zk , k 1 
A. Classic Kalman filer
When error model is built accurately and the system
noise can be obtained correctly, the classic Kalman filter
can be written by
(1) Set of initial value
(32)
(3) State update
process a f and  f are chosen as the state variables
X   V  m  a
(31)
m can be adjusted according
IV. SIMULATION AND ANALYSIS
The rapid transfer alignment of shipboard aircraft is
simulated, where the swing maneuver of ship is driven by
sea wave. To obtain better observability, Velocity/attitude
matching is selected whose measurement equation is
given by
z  Hx  v
(41)
is used in Scheme 2, but the statistical characteristics of
noise is set a fixed value, in which the compensation
coefficient is set 1.5. The process noise estimator with
limited memory index is used in Scheme 3, the memory
length m is set 10 and forgetting factor b is set 0.3. The
filter frequency is set 5Hz. Estimation results of three
schemes are shown from Fig.1-3.
20
Scheme 1
15
Scheme 2
10
Scheme 3
5
x/'
where the memory length
to actual environment.
0
-5
-10
According to section II, after compensation of
process noise, the state can be given by
X  V
m a


-15
-20
(42)
0
10
20
30
40
50
Time/s
60
70
80
90
100
Fig.1 Estimation error of misalignment  x
So the observing matrix can be written as
10
5
028 
038 
023
I 33
0
(43)
-5
y/'
I
H   22
032
-10
-15
The initial position of ship is 45.6 north latitude and
the velocity is Vn  10m / s . The ship move northward and
the swinging model driven by the sea wave can be
expressed as
Scheme 3
x
y
,
accelerometer are set respectively
are
 0.001 / h 
2
and
. Misalignments  x ,  y ,  z are 15' , 30' , 1 .
Simulation time is 100s.
The two-ordered Markov model is adopted for the
true flexural deformation and the model coefficients of 3
channels are set  x  0.1 ,  y  0.2 ,  z  0.4 ,
simultaneously the variance of white noise are
 0.05 / h 
2
40
50
Time/s
60
70
80
90
100
-20
-30
-40
Scheme 1
-50
z
0.05 / h and accelerometer offset is set 10 3 g . The
variance(variances) of white noise of gyro and
2
30
0
 z/'
,
 z 0 are all set 0 . The gyro constant drift of SINS is set
4
20
-10
0.18Hz , 0.13Hz , 0.06Hz and initial angle  x 0 ,  y 0 ,
10 g 
10
10
The swinging amplitude  xm ,  ym ,  zm are 5 , 4 ,
frequency
0
Fig.2 Estimation error of misalignment  y
(44)
 z   zm sin z t    z 0
the
Scheme 2
20
 y   ym sin  y t   y 0 
,
Scheme 1
-25
-30
 x   xm sin x t   x 0 
2
-20
and 103 g  . Scheme 1 takes the same
2
model but with different coefficients, which are set
 x  0.2 ,  y  0.3 ,  z  0.5 . Noise compensation method
Scheme 2
-60
Scheme 3
-70
0
10
20
30
40
50
Time/s
60
70
80
90
100
Fig.3 Estimation error of misalignment  z
There are slight deviations in model coefficients
between true model and Scheme 1. However, it can be
seen from the simulation results that in Scheme 1 the
standard deviations of three misalignments are
0.32' , 0.67' , 2.19' . But limited to the high computation
burden, the convergence speed declines. In Scheme 2,
because of the reduction of state, the convergence speed is
improved. However, the cost is the declining of filter
accuracy since the fixed noise statistical characteristics
cannot follow the variation of actual environment. The
standard deviations of three misalignments are 1.43' ,
3.56' , 12.53' . On the basis of Scheme 2, Scheme 3 uses
noise estimator with limited memory index to real-time
track the system noise. The standard deviations of three
misalignments are 0.28' , 0.76' , 2.55' , which are better than
Scheme 2 and faster than Scheme 1.
V. CONCLUSION
After model building of rapid transfer alignment, the
error equations are simplified by noise compensation
method. Aiming at the time variant characteristics of
flexure process in time domain, the noise compensation
problem of flexural deformation is transformed to the
problem of online estimation of system noise, which is
dealt with the noise estimator based on limited memory
length. The final simulation shows that, when external
interference cannot be obtained accurately, the method
proposed by this paper can provide a new idea for the
compensation of flexural deformation.
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