Underwater acoustic positioning based on the robust zero‑difference Kalman filter

Abstract

The accuracy of underwater acoustic positioning is greatly influenced by both systematic error and gross error. Aiming at these problems, the paper proposes a robust zero-difference Kalman filter based on the random walk model and the equivalent gain matrix. The proposed algorithm takes systematic error as a random walk process, and estimates it together with the position parameters by using zero-difference Kalman filter. In addition, the equivalent gain matrix based on the robust estimation of Huber function is constructed to resist the influence of gross error. The proposed algorithm is verified by the simulation experiment and a real one for underwater acoustic positioning. The results demonstrate that the robust zero-difference Kalman filter can control both the effects of systematic error and gross error without amplifying the influence of the observation random noise, which is obviously superior to the zero-difference least squares (LS), the single-difference LS and zero-difference Kalman filter in underwater acoustic positioning.

Full text

Vol.:(0123456789)
1 3
Journal of Marine Science and Technology
https://doi.org/10.1007/s00773-020-00766-x
ORIGINAL ARTICLE
Underwater acoustic positioning based ontherobust zero‑dierence
Kalman lter
JuntingWang1· TianheXu1 · BingshengZhang2· WenfengNie1
Received: 9 March 2020 / Accepted: 5 September 2020
© The Japan Society of Naval Architects and Ocean Engineers (JASNAOE) 2020
Abstract
The accuracy of underwater acoustic positioning is greatly influenced by both systematic error and gross error. Aiming at
these problems, the paper proposes a robust zero-difference Kalman filter based on the random walk model and the equiva-
lent gain matrix. The proposed algorithm takes systematic error as a random walk process, and estimates it together with
the position parameters by using zero-difference Kalman filter. In addition, the equivalent gain matrix based on the robust
estimation of Huber function is constructed to resist the influence of gross error. The proposed algorithm is verified by the
simulation experiment and a real one for underwater acoustic positioning. The results demonstrate that the robust zero-
difference Kalman filter can control both the effects of systematic error and gross error without amplifying the influence of
the observation random noise, which is obviously superior to the zero-difference least squares (LS), the single-difference
LS and zero-difference Kalman filter in underwater acoustic positioning.
Keywords Systematic error· Gross error· Kalman filter· Zero-difference positioning· Robust estimation
1 Introduction
With the development of the national marine strategy and
the marine resource exploration, accurate ocean navigation
and positioning technology are needed to obtain the high-
precision, large-scale marine environmental information
[1–3]. Sound waves, rather than electromagnetic waves or
light waves, are mainly used to estimate the position of the
underwater target. The reason is that sound waves can spread
hundreds of kilometers in the water while electromagnetic
waves and light waves decay quickly [4]. The classical
acoustic-based approaches for underwater target positioning
include long baseline (LBL), short baseline (SBL), ultra-
short baseline (USBL) and underwater global positioning
system (GPS) according to the acoustic baseline range [5,
6]. The shipborne acoustic positioning generally adopts the
voyage positioning mode, which is affected by the geometric
structure of trajectory and the measurement error related to
the time delay as well as the sound speed [7, 8].
For underwater acoustic positioning, there inevitably
exist the gross error, the random error and the systematic
error caused by the marine environment and the observation
instrument. Many studies have been dedicated to improve
the underwater positioning model and the error correction
method. Xu etal. [9] first proposed the underwater differ-
ence positioning algorithm including the single difference
algorithm between the observation epochs and the double
difference algorithm which can greatly improve the accu-
racy of seafloor deformation measurement. Zhao etal.
[10] proposed a ship-board difference positioning method
based on selecting weight iteration. Although the differ-
ence positioning algorithm can weaken the effects of the
systematic errors, it enlarges the influence of random errors,
which decreases the accuracy of the underwater acoustic
positioning. Aiming at the time delay error, a positioning
model considering the apparent time delay error as unknown
parameter is proposed [11]. Yan etal. [12] proposed a long
baseline positioning algorithm for moving buoy by estimat-
ing the uncertain sound speed as an unknown parameter.
However, even though the time delay error or the unknown
sound speed is estimated as a fixed systematic parameter, it
is hard to be accurately estimated since it changes with the
* Tianhe Xu
thxu@sdu.edu.cn
1 Institute ofSpace Science, Shandong University, Weihai,
China
2 College ofGeology Engineering andGeomantic, Chang’an
University, Xi’an, Shanxi, China

Journal of Marine Science and Technology
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change of the marine environment. In the global navigation
satellite system (GNSS) positioning, Paziewski and Wiel-
gosz [13] used the random walk model to estimate the inter-
system biases as the unknown parameters, which inspires
us to apply this method to estimate the systematic error in
underwater acoustic positioning. As for gross errors, Zhou
[14] proposed the IGG robust estimation method based on
the equivalence weight. On this basis, Yang etal. [15] devel-
oped the bi-factor equivalence weight based on the robust
estimation for correlation observation. Xu etal. [16] pro-
posed a robust estimation method based on the symbol con-
straint. Wang etal. [17] proposed a robust extended Kalman
filter using W-test statistics based on filtering residuals to
eliminate the effect of gross errors on GNSS navigation solu-
tions. Yang etal. [18] proposed the robust M–M unscented
Kalman filtering for GPS/IMU navigation. The applications
of the robust estimation in GPS navigation and position-
ing have been widely adopted and tested [19]. Wang etal.
[20] proposed an adaptive robust unscented Kalman filter
for autonomous underwater vehicle (AUV) acoustic navi-
gation, which constructs the judgment factor and adaptive
factor by the prediction residual to balance the contribution
between the observation information and AUV motion state
information.
The zero-difference (ZD) least squares (LS) is rarely
adopted for the high precision underwater positioning due
to the effects of systematic error as well as gross error. At the
same time, the single-difference (SD) LS of adjacent epoch
enlarges the influence of random errors and even the gross
errors while it suppresses the effects of systematic errors.
To solve the aforementioned problems, this paper proposes
a robust zero-difference Kalman filter based on the random
walk model and the equivalent gain matrix to resist the
effects of systematic errors and gross errors in underwater
acoustic positioning. The proposed method involves a robust
estimation method based on the prediction residual as well
as the observation variance, and an improved KF with sys-
tematic error compensation, which has obvious differences
compared to the method of reference [20].
The paper is organized as follows. We firstly present an
improved zero-difference positioning function model as well
as the zero-difference Kalman filter by estimating the sys-
tematic error as the random walk process in Sect.2. Then
Sect.3 introduces the theoretical derivation and algorithm
implementation of the robust zero-difference Kalman fil-
ter. The robust zero-difference Kalman filter is verified and
analyzed by the simulation experiment and a real one for
underwater acoustic positioning in Sect.4. Finally, we sum-
marize the significant conclusions in Sect.5.
2 Method
2.1 Zero‑dierence positioning function model
The transducer under the survey ship can continuously send
sound waves to the transponder to get the signal propaga-
tion time [21]; therefore, the range between the transducer
and the transponder at the different time and position can be
obtained by the travel time and the sound speed structure
[22].
As shown in Fig.1, assuming that the transducer at posi-
tion
𝐗k
and time
tk
transmits an acoustic signal to the tran-
sponder to get the slant range
𝜌k
, the transponder coordinates
can be obtained through the intersection positioning method,
which can be regarded as a prototype of the underwater zero-
difference positioning, since there are no differential opera-
tions on observations between epochs and transponder sta-
tions. The observation model of underwater zero-difference
positioning can be expressed as
where
𝐗o=(xo,yo,zo)
is the unknown position vector of the
transponder, and
𝐗k=(xk,yk,zk)
is the position vector of
(1)
𝜌k
=f
(
𝐗
k
,𝐗
o)
+𝛿𝜌
dk
+𝛿𝜌
vk
+𝜀
k,
(2)
f
(
𝐗k,𝐗o
)
=
√
(xk−xo)2+(yk−yo)2+(zk−zo)2
,
(3)
𝜌k=ctk,
11
,tX
22
,tX
,
kk
tX,
nn
tX
0
X
Transducer
Transponder
Sea bottom
1
ρ
2
ρ
k
ρ
n
ρ
Fig. 1 The geometric diagram of underwater zero-difference position-
ing

Journal of Marine Science and Technology
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transducer under the ship, which can be directly calculated
from the kinematic GNSS. f
(
𝐗
k
,𝐗
o)
is the theoretical range
between the transponder and the transducer.
𝛿𝜌dk
is the sys-
tematic error due to the time delay in re-transmitting the
received signal from the transponder back to the transducer,
𝛿𝜌
v
k
is the systematic error due to the spatial and temporal
variation in the sound speed structure,
𝜀k
is the random rang-
ing error.
tk
is the travel time between the transducer and the
transponder, and
c
is the sound speed.
In the actual underwater acoustic positioning, the sound
speed error is affected by the ocean internal wave and has a
periodic variation. In addition, the systematic error related
to time delay for the same transponder is approximately
equal [23, 24]. Therefore, the systematic error related to
the time delay and the sound speed can be estimated as
an unknown parameter and the observation Eq.1 can be
rewritten as
where
𝛿𝜌k
is the estimated parameter of the systematic error.
Equation4 is linearized as
where
𝐗0
o
,
𝛿𝜌0
k
are approximate values for
𝐗o
and
𝛿𝜌k
.
d𝐗o
and
d𝛿𝜌k
are the unknown coordinate correction vector and
the systematic error correction vector to be estimated with
𝐗o
=𝐗
0
o
+d𝐗
o
and
𝛿𝜌k
=𝛿𝜌
0
k
+d𝛿𝜌
k
, respectively.
ak
and
bk
are the first-order partial derivatives with respect to
𝐗o
and
𝐗k
, respectively, and
𝜀𝐗k
is the random errors of the survey
ship positions.
When combining all the measurements, the linear
observation equation of underwater zero-difference posi-
tioning can be expressed as:
(4)
𝜌k
=f
(
𝐗
k
,𝐗
o)
+𝛿𝜌
k
+𝜀
k,
(5)
𝜌
k−f
(
𝐗k,𝐗
0
o)
−𝛿𝜌
0
k
=akd𝐗o+d𝛿𝜌k+𝜀k+bk𝜀𝐗
k,
(6)
𝐙=
𝐇
d
𝐗
�
o+
𝐕
,
(7)
𝐇
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
𝜕f
1
𝜕x
𝜕f
1
𝜕y
𝜕f
1
𝜕z1
𝜕f2
𝜕x
𝜕f2
𝜕y
𝜕f2
𝜕z1
𝜕f3
𝜕x
𝜕f3
𝜕y
𝜕f3
𝜕z1
⋮ ⋮ ⋮⋮
𝜕fn
𝜕x
𝜕fn
𝜕y
𝜕fn
𝜕z
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(8)
𝐙
=







𝜌1−f

𝐗1,𝐗0
o

−𝛿𝜌0
1
𝜌2−f𝐗2,𝐗0
o−𝛿𝜌0
2
𝜌3−f𝐗3,𝐗0
o−𝛿𝜌0
3
⋮
𝜌n−f

𝐗n,𝐗0
o
−𝛿𝜌0
n







,
where
𝐗�o
=(x
o
,y
o
,z
o
,𝛿𝜌
k)
,
𝐙
is the constant term,
𝐇
is
the coefficient matrix of observation equation, and
𝐕
is the
observation residual vector.
2.2 Zero‑dierence Kalman lter
The observation and state equations using zero-difference
Kalman filter for underwater acoustic positioning can be
expressed as [25]:
where
𝐗�o,k
=(x
o,k
,y
o,k
,z
o,k
,𝛿𝜌
k)
denotes the estimated
parameter vector of the transponder position and the sys-
tematic error at time
tk
,
𝐙k
is the observation vector with
covariance matrix
𝐑k
, assumed to be white,
𝐇k
is the coef-
ficient matrix of observation equation and
𝐕k
is the obser-
vation residual vector.
𝛗k,k−1
is the state transition matrix
from epoch
tk−1
to
tk
.
𝛚k−1
is the process noise vector with
covariance matrix
𝐐k
, assumed to be white.
The discrete first-order Gauss–Markov process [26]
describes the epoch state changes of related parameters, and
the mathematical expression is as follows:
where
Λt=tk−tk−1
,
𝜏
is the time constant, and
𝛿
is Dirac
function.
When
𝜏→∞
, the state of Eq.11 is the random walk
process
When
𝜏→0
, the state of Eq.11 is the white noise
where
𝜎2
𝐰
is the variance of the state parameters.
The unknown parameters of the zero-difference Kalman
filter are the transponder position and the systematic error.
Since the transponder position parameters are constants and
the systematic error parameter changes regularly with time,
the state transition matrix and the state noise matrix of Eq.10
are given by
(9)
𝐙k
=𝐇
k
d𝐗
�
o,k
+𝐕
k,
(10)
𝐗�
o,k
=𝛗
k,k−1
𝐗
�
o,k−1
+𝛚
k−1,
(11)
⎧
⎪
⎨
⎪
⎩
𝐗�o,k=e−Λt
𝜏𝐗�o,k−1+𝛚k−1,𝛚k−1=
tk
∫
tk−1
e−Λt
𝜏𝛚dt
E(𝛚k𝛚k−1)= 1
2
𝜏(1−e−2Λt
𝜏)𝛿
,
(12)
{
𝐗
�
o,k=𝐗
�
o,k−1+𝛚k−1
E(𝛚
k
𝛚
k−1
)=𝜎2
𝐰
𝛿
.
(13)
{
𝐗
�
o,k=𝛚k−1
E(𝛚
k
𝛚
k−1
)=𝜎2
𝐰
𝛿
,
(14)
𝛗k,k−1=𝐈4×4,

Journal of Marine Science and Technology
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where
𝐈4×4
is a unit array of four rows and four columns, and
𝜎2
𝜔
𝛿𝜌
is the variance of the systematic error parameter.
The covariance propagation equation is given by
The solutions for the estimated state vector, the Kalman
filter gain matrix and the covariance matrix of the estimated
state can be obtained as [25]
3 Robust zero‑dierence Kalman lter
In underwater acoustic positioning, when the observation
contains the gross error, the measurement equation should
be
where
𝐁k
is the interference matrix of gross error, and
𝚫k
is
the gross error.
If the standard Kalman filter is still used for calculation,
the prediction residual with the gross error is as follows:
where
Vk
and
̃
Vk
are the prediction residual vector without
the gross error and with the gross error respectively.
The gross error is fully reflected in the prediction resid-
ual, then the state vector is
According to Eq.22, the gross error in the observation
affects the state vector
𝐗′
k
through the gain matrix
𝐊k
. To
resist the influence of gross error, a robust zero-difference
Kalman filter is adopted based on the equivalent gain matrix.
By using the prediction residual and the observation vari-
ance to construct the judgment factor
Sk
, the gross error in
the observation equation can be efficiently detected. Based
on the idea of equivalent gain matrix [20] and equivalent
weight function of Huber [27, 28], the equivalent gain
(15)
𝐐
k=
⎡
⎢
⎢
⎢
⎢
⎣
0
0
0
𝜎2
𝜔𝛿𝜌
⎤
⎥
⎥
⎥
⎥
⎦
,
(16)
𝐏k,k−1
=𝛗
k,k−1
𝐏
k−1
𝛗
T
k,k−1
+𝐐
k−1.
(17)
𝐗�
o,k
=𝐗
�
o,k−1
+𝐊
k
[𝐙
k
−𝐇
k
d𝐗
�
o,k−1
]
,
(18)
𝐊k
=𝐏
k,k−1
𝐇
T
k
[𝐇
k
𝐏
k,k−1
𝐇
T
k
+R
k
]
−1,
(19)
𝐏k=[𝐈−𝐊k𝐇k]𝐏k,k−1.
(20)
𝜌k
=f
(
𝐗
k
,𝐗
o)
+𝛿𝜌
k
+𝐁
k
𝚫
k
+𝜀
k,
(21)
̃
Vk
=𝜌
k
−f
(
𝐗
k
,𝐗
o)
−𝛿𝜌
k
=V
k
+𝐁
k
𝚫
k,
(22)
𝐗�
k
=𝐗
�
k,k−1
+𝐊
k
̃
V
k.
matrix related to the constant
k0
is constructed to reduce
the influence of gross error. The equivalent gain matrix is
expressed as
where
k0
is a constant, generally,
k0=1∼2
.
where
Vk
is the prediction residual vector, and
𝐃V(k)
is the
covariance matrix of observation vector.
When there exists gross error in the observations, the cor-
responding
𝐊k
will be decreased and the influence of gross
error on KF will be reduced.
To resist the effects of both systematic error and gross
error, a robust zero-difference Kalman filter based on the
random walk model and the equivalent gain matrix is pro-
posed. The flowchart of the proposed algorithm is shown
in Fig.2. The detailed steps of the algorithm are explained
as follows:
(23)
𝐊
k=

𝐊k

Sk

<k0
𝐊k
k0

S
k
Sk

≥k0
,
(24)
S
k=Vk∕
√
𝐃V(k)
,
(25)
𝐃V(k)=[𝐈−𝐇k𝐊k]Rk,
Fig. 2 The flowchart of the robust zero-difference Kalman filter

Journal of Marine Science and Technology
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1. The initial state vector
𝐗�
o,1 =(xo,1,yo,1 ,zo,1,
𝛿𝜌
1)
includ-
ing the transducer position and the systematic error is
given.
2. The state vector
𝐗′
o,k
and the corresponding covariance
matrix
𝐏k,k−1
are calculated by Eqs.10 and 16.
3. The gain matrix of the Kalman filter
𝐊k
is computed by
Eq.18. The equivalent gain matrix
𝐊k
based on robust
estimation is calculated by Eq.23.
4. The state vector and the error covariance matrix are esti-
mated by Eqs.26 and 27.
(26)
𝐗
�
o,k
=𝐗�
o,k−1
+𝐊
k
[Z
k
−𝐇
k
d𝐗�
o,k−1
]
,
(27)
𝐏k
=[𝐈−𝐊
k
𝐇
k
]𝐏
k,k−1.
5. The steps (2)–(4) are repeated until filter convergence.
4 Simulation andreal experimental analysis
4.1 Simulation analysis
Simulation analysis on the acoustic positioning based on
the proposed method is conducted in this paper. As shown
in Fig.3, the four transponders are located at the positions
of the asterisk symbol with the different underwater depth
of 30m, 100m, 500m and 3000m. The trajectories of the
ship are circles with the radius of 100m, 200m, 800m and
3000m as well as the linear track with the grid shape. The
sampling interval is 2s and the speed of the survey ship is
Fig. 3 The diagram of simulated ship and transponder

Journal of Marine Science and Technology
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Fig. 4 The observation residuals of LS, SD and KF at different depths

Journal of Marine Science and Technology
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Fig. 5 The results of the estimated systematic errors by KF at different depths

Journal of Marine Science and Technology
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Fig. 6 The underwater positioning results of different algorithms at different depths

Journal of Marine Science and Technology
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about 1m/s. The measured sound speed profile of 3000m
is adopted for the calculation of sound speed. The layered
ray acoustic tracking algorithm is used to simulate the travel
time, and the systematic error [29] is simulated based on
Eq.28 proposed in Xu etal. [9]. The slant range error caused
by the random error is 0.1m; therefore, the initial measure-
ment noise variance is
R=0.01
m2. The initial system noise
matrix is
𝐐=diag[0 0 0 0.001]
m2.
where the constant term is
c1=0.1
m, the short-period
internal wave error term is
c2=0.12
m, the long-period
error term is
c3=0.3
m, the term related to the measure-
ment range is
c4=0.02
m, the short period of internal wave
is
TS=15 ∗60 s
(equal to 15min) and the long period of
internal wave is
TL=12 ∗3600 s
(equal to 12h).
‖
‖
𝐗
o
−𝐗
‖
‖
is the distance between the transducer and the transponder.
The zero-difference least squares (LS), the single-dif-
ference least squares (SD), the zero-difference Kalman
filter (KF), the robust zero-difference least squares(R-LS),
the robust single difference least squares (R-SD) and the
robust zero-difference Kalman filter (R-KF) are conducted
and compared for underwater acoustic positioning. Firstly,
the proposed algorithm is validated in the case without
gross error. The observation residuals of LS, SD and KF
as well as the estimated systematic errors by KF are shown
in Figs.4 and 5. Monte Carlo experimental simulation with
100 times is conducted, the root mean squares (RMS) of the
transponder position calculated at the different depth and
the different algorithm is shown in the Fig.6. In calculating
the formula of RMS,
Xo,k
and
̂
X
o,k
are the calculation value
(28)
𝛿𝜌
v=c1+c2sin

2

t−t0

TS
𝜋

+c3sin

t−t0

TL
𝜋

+c4

1−exp

−1
2


𝐗o−𝐗


∕(2km)2
,
and the real value of the transponder respectively.
N
is the
number of the transponder.
As shown in Figs.4 and 5, the SD can effectively reduce
the effects of systematic errors compared with the LS. How-
ever, the SD produces larger random errors compared with
the KF. The KF can effectively estimate systematic error
parameters, together with position parameter without enlarg-
ing the influence of random errors. Therefore, the KF can
significantly improve the underwater positioning accuracy.
As shown in Fig.6 and Table1, when there exist random
errors, systematic errors and no gross errors in the acous-
tic observations, the LS cannot resist the effects of the sys-
tematic errors on the positioning result, especially in the Z
direction of coordinates. The SD and KF can both reduce the
influence of the systematic errors and greatly improve the
positioning accuracy. For the case of underwater 30m depth,
the three-dimension (3D) RMSs of SD and KF are 0.065m
and 0.048m, respectively, compared to LSs 0.455m, with
the improvement of 85.7% and 89.4%. For the case of 100m
depth, they are 0.087m and 0.044m, respectively, compared
to LSs 0.380m, with the improvement of 77.1% and 88.4%.
For the case of 500m depth, they are 0.090m and 0.040m,
respectively, compared to LSs 0.406m, with the improve-
ment of 77.8% and 90.1%. For the case of 3000m depth,
they are 0.110m and 0.065m, respectively, compared to
LSs 0.504m, with the improvement of 78.2% and 87.1%. In
addition, the KF can further enhance the positioning accu-
racy with about 5–12% improvement compared to the SD.
Secondly, to further verify the performance of LS, SD
and KF in the case of the ship trajectories with irregular
curve, the transponders with the different underwater depth
of 30m, 100m, 500m and 3000m are positioned by the
simulated trajectories as shown in Fig.7. Table2 presents
the RMSs of different algorithms and depths. For the case
of underwater 30m depth, the 3D RMSs of SD and KF
are 0.079m and 0.057m compared to the 0.362m of LS,
with the improvement of 78.2% and 84.3%. For the case of
100m depth, they are 0.175m and 0.082m compared to the
0.304m of LS, with the improvement of 42.4% and 73.0%.
For the case of 500m depth, they are 0.127m and 0.061m
compared to the 0.373m of LS, with the improvement of
66.0% and 83.6%. For the case of 3000m depth, they are
0.223m and 0.165m compared to the 0.442m of LS, with
the improvement of 47.3% and 62.7%. Therefore, the per-
formance of KF is also better than that of LS and SD in the
case of the ship trajectories with irregular curve.
Table 1 The positioning result statistics of different algorithms
Depth (m) Method Mean
RMS-X
(m)
Mean
RMS-Y
(m)
Mean
RMS-Z
(m)
3D-RMS (m)
30 LS 0.061 0.059 0.447 0.455
SD 0.014 0.013 0.062 0.065
KF 0.011 0.012 0.045 0.048
100 LS 0.032 0.065 0.373 0.380
SD 0.014 0.015 0.084 0.087
KF 0.010 0.013 0.041 0.044
500 LS 0.034 0.038 0.403 0.406
SD 0.016 0.020 0.086 0.090
KF 0.012 0.011 0.036 0.040
3000 LS 0.039 0.051 0.500 0.504
SD 0.029 0.018 0.105 0.110
KF 0.014 0.014 0.062 0.065

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Finally, the time delay observation is added to the gross
errors based on normal distribution with zero mean and
standard deviation of 0.05s, and the gross errors are added
in the acoustic observations every 50s. Figure8 shows the
result of the estimated systematic error by the R-KF. At the
same time, the positioning results of the non-robust estima-
tion and the robust estimation are shown in Figs.9, 10, 11
and 12.
Figure8 shows that the R-KF can resist the influence
of gross errors on the position and systematic error param-
eter based on the equivalent gain matrix. As shown in the
Figs.9, 10, 11 and 12, the accuracy of the SD is significantly
reduced compared with the LS and the KF due to the effects
of the gross errors. The R-SD can also resist the influences
of the gross errors by the robust estimation and reduce the
influence of the systematic errors to improve the accuracy
of the underwater positioning compared with the R-LS.
However, the R-SD has also the disadvantage of enlarging
the effects of random errors, which leads to the position-
ing accuracy lower than that of the R-KF. The R-KF can
estimate the systematic errors by the random walk process
without enlarging the influence of the random errors, and
provide robust solutions by using the equivalent gain matrix,
which has higher precision and stability than those of the
other two algorithms.
Fig. 7 The diagram of the simulated ship and transponder in different depths

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The means of RMS for 100 times of each algorithm
are shown in Table3. From Table3, it can be seen that
the positioning accuracy of the LS, SD and KF is greatly
decreased by gross errors, especially for the SD due to the
enlarged gross errors. R-LS, R-SD and R-KF can obvi-
ously improve the positioning by using robust estimation
to resist the influence of the gross errors. For the case of
underwater 30m depth, the three-dimension (3D) RMSs
of R-SD and R-KF are 0.076m and 0.045m, respectively,
compared to R-LSs 0.463m, with the improvement of
83.5% and 90.3%. For the case of 100m depth, they are
0.103m and 0.051m, respectively, compared to LSs
0.378m, with the improvement of 72.7% and 86.5%. For
the case of 500m depth, they are 0.099m and 0.040m,
respectively, compared to LSs 0.406m, with the improve-
ment of 75.6% and 90.1%. For the case of 3000m depth,
they are 0.137m and 0.062m, respectively, compared to
LSs 0.504m, with the improvement of 72.8% and 87.7%.
In addition, the R-KF can further enhance the position-
ing accuracy with about 7–15% improvement compared
to R-SD.
4.2 Real experiment analysis
The insitu data were collected from an experiment con-
ducted at Lingshan Island in Dec. 2017. Lingshan Island
is located in Qingdao City, Shandong Province in China
with longitude and latitude about 120°13′ 02″ E, and
35°46′53″N, respectively. The single transponder is located
at the ocean bottom, and the trajectory of the voyage move
is centered around the transponder with a radius of about
50m. The ultrashort baseline response mode is used for
underwater positioning, and GPS receiver, attitude sensor
and sound velocity profiler are auxiliary installed.
After preprocessing the measured data, LS, SD, KF,
R-LS, R-SD and R-KF are used for the positioning cal-
culation and then compared. The observation noise and
the system noise of Kalman filter are set as
R=1
m2 and
Q=diag[0 0 0 0.001]
m2, respectively. Since the position
of the underwater transponder in the experimental area is
unknown, the positioning accuracy cannot be directly evalu-
ated. To verify the accuracy of the algorithm, as shown in
Fig.13, the observations that are not involved in positioning
calculation on the trajectory are selected to calculate the
residuals, namely the observation ranges minus computa-
tion/theory ranges (O–C).
As shown in Table4, the RMS of the validated residu-
als of the SD is lower than that of the LS and the KF. The
reason may be that: (1) since the experiment is conducted
in shallow sea, the influence of the systematic errors is
relatively small; (2) the systematic error between the adja-
cent epochs are not exactly equal in the actual observa-
tions, and the SD cannot totally eliminate the systematic
errors; (3) SD enlarges the influence of random errors.
When using robust estimation, all the RMSs of the three
methods decrease, which indicates that robust estima-
tion can efficiently control the influence of gross errors.
Compared with the R-LS and the R-SD, the RMS of the
validated residuals of the R-KF is greatly reduced from
1.63m and 1.81m, respectively, to 0.85m, which proves
the higher precision of R-KF. From Fig.14, it can be seen
that both the KF and the R-KF need some certain epochs
to make the filtering solution convergence. There is a bias
between the solutions of R-KF and KF, since the former
uses robust estimation to reduce the influence of the gross
errors on the systematic error parameter, while the latter
has no action on gross errors and inevitably brings the
deviation of solution for underwater positioning.
5 Conclusion
To reduce the effects of the systematic error and the gross
error on the underwater positioning, this paper proposes a
robust zero-difference Kalman filter based on the random
walk model and the equivalent gain matrix. After the vali-
dation from the simulation experiment and a real example,
the following conclusions can be drawn.
1. Compared with the zero-difference LS, the single-dif-
ference LS between the observation epochs can reduce
the influence of the systematic error. However, it also
enlarges the influence of the random errors and the
gross errors. Although the robust single-difference LS
can eliminate the influence of the gross errors by the
robust estimation, its accuracy of underwater position-
Table 2 The positioning result statistics of different algorithms
Depth (m) Method Mean
RMS-X
(m)
Mean
RMS-Y
(m)
Mean
RMS-Z
(m)
3D-RMS (m)
30 LS 0.047 0.066 0.353 0.362
SD 0.020 0.017 0.074 0.079
KF 0.015 0.013 0.054 0.057
100 LS 0.022 0.045 0.299 0.304
SD 0.021 0.019 0.172 0.175
KF 0.014 0.014 0.079 0.082
500 LS 0.037 0.037 0.369 0.373
SD 0.027 0.023 0.122 0.127
KF 0.019 0.012 0.056 0.061
3000 LS 0.039 0.049 0.437 0.442
SD 0.023 0.017 0.221 0.223
KF 0.018 0.014 0.163 0.165

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Fig. 8 The results of the estimated systematic errors by R-KF at different depths

Journal of Marine Science and Technology
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Fig. 9 The positioning results at the depth of underwater 30m
Fig. 10 The positioning results at the depth of underwater 100m

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ing is greatly reduced by the enlarged random errors and
the remained systematic errors. The accuracy of zero-
difference Kalman filter can be significantly improved
compared to the zero-difference LS and the single-differ-
ence LS. At the same time, the zero-difference Kalman
filter has the better performance in the case of the ship
trajectories with irregular curve.
2. The proposed robust zero-difference Kalman filter
can estimate the systematic error by the random walk
model without enlarging the influence of the random
Fig. 11 The positioning results at the depth of underwater 500m
Fig. 12 The positioning results at the depth of underwater 3000m

Journal of Marine Science and Technology
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errors, and resist the influence of the gross error by the
equivalent gain matrix. In the simulating experiment,
the positioning accuracy of the proposed algorithm is
obviously superior to that of robust zero-difference LS
and robust single-difference LS with the improvement
of about 86–90% and about 5–15%, respectively. In
the real data experiment, the RMS of the validated
residuals of the robust zero-difference Kalman filter
Table 3 The positioning result
statistics of different algorithms Depth (m) Method Mean RMS-X
(m)
Mean RMS-Y
(m)
Mean RMS-Z (m) 3D RMS (m)
30 LS 0.399 0.382 0.662 0.862
SD 1.382 1.013 6.183 6.416
KF 0.554 0.480 2.249 2.365
R-LS 0.058 0.062 0.455 0.463
R-SD 0.017 0.015 0.073 0.076
R-KF 0.008 0.011 0.043 0.045
100 LS 0.400 0.467 0.562 0.833
SD 1.099 1.463 4.077 4.469
KF 0.491 0.652 2.463 2.595
R-LS 0.033 0.067 0.370 0.378
R-SD 0.015 0.018 0.100 0.103
R-KF 0.007 0.017 0.048 0.051
500 LS 0.442 0.442 0.490 0.794
SD 1.920 1.458 5.563 6.063
KF 0.547 0.543 2.588 2.600
R-LS 0.036 0.039 0.402 0.406
R-SD 0.018 0.027 0.093 0.099
R-KF 0.008 0.011 0.038 0.040
3000 LS 0.554 0.477 0.558 0.920
SD 1.892 0.138 12.070 12.218
KF 0.617 0.584 2.511 2.651
R-LS 0.039 0.051 0.500 0.504
R-SD 0.031 0.018 0.133 0.137
R-KF 0.014 0.011 0.059 0.062
Fig. 13 The trajectory of ship and checkpoint
Table 4 The residuals statistics of the 20 epochs
Method RMS (m) Max (m) Min (m)
LS 1.73 2.16 0.99
SD 1.91 2.59 1.39
KF 1.24 2.12 0.51
R-LS 1.63 2.17 0.92
R-SD 1.81 2.55 1.26
R-KF 0.85 1.69 0.04

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is about 0.8m, and obviously less than that of robust
zero-difference LS and the robust single-difference LS,
which proves that the proposed algorithm has higher
accuracy and stability.
Acknowledgements The study was funded by National Key Research
and Development Program of China (2016YFB0501701), National
Natural Science Foundation of China (41931076, 41874032 and
41731069).
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Fig. 14 The positioning result of epoch by epoch (blue line represents
KF and red line is R-KF)