Underwater Doppler Navigation with 
Self-calibration 
Xianfei Pan1 and Yuanxin Wu2 
 
1(National University of Defense Technology - College of Mechatronics and Automation, China, 
410073) 
2(Central South University - School of Aeronautics and Astronautics, China, 410083) 
(E-mail: yuanx_wu@hotmail.com) 
 
Precise autonomous navigation remains a substantial challenge to all underwater platforms. Inertial 
Measurement Units (IMU) and Doppler Velocity Logs (DVL) have complementary characteristics and 
are promising sensors that could enable fully autonomous underwater navigation in unexplored areas 
without relying on additional external Global Positioning System (GPS) or acoustic beacons. This paper 
addresses the combined IMU/DVL navigation system from the viewpoint of observability. We show by 
analysis that under moderate conditions the combined system is observable. Specifically, the DVL 
parameters, including the scale factor and misalignment angles, can be calibrated in-situ without using 
external GPS or acoustic beacon sensors. Simulation results using a practical estimator validate the 
analytic conclusions. 
KEYWORDS 
1. Underwater navigation. 2. Doppler velocity log. 3. Inertial measurement unit. 4. Observability. 
 
Submitted: 19 November 2014. Accepted: XX August 2015. 
 
 
1. INTRODUCTION. Current underwater navigation technology enables new emerging 
applications that have been previously considered impossible or impractical, including 
autonomous naval operations, oceanographic studies and under ice surveys. Despite significant 
advances, precise navigation remains a substantial challenge to all underwater platforms (Kinsey 
et al., 2006, Hegrenæs and Berglund, 2009). The Global Positioning System (GPS) provides 
superior three-dimensional navigation capabilities for vehicles above the water surface, but not 
underwater due to water blockage of GPS radio-frequency signals. This limits GPS usage to 
surveying of acoustic transponders or aiding of sensor calibration for underwater applications 
(Kinsey and Whitcomb, 2006). Acoustic navigation is widely used for limited-area scientific and 
industrial underwater vehicles, which requires prior careful placement of beacons fixed or 
moored on the sea floor or on the hull of a surface ship. 
   The development of commercial Doppler sensors and Inertial Measurement Units (IMU, 
consisting of a triad of gyroscopes and a triad of accelerometers) has enabled significant 
improvement to underwater navigation (Kinsey et al., 2006). The multi-beam Doppler Velocity 
Log (DVL) can provide the bottom-track or water-track velocity measurement with a precision 
of 0.3% or less, while the IMU measures the arbitrary three-dimensional (3D) angular velocity 
and translational acceleration which can be integrated to yield navigation information. Combing 
a DVL with IMU makes possible large-scale underwater navigation in unexplored areas. They 
are complementary in characteristics to each other. The DVL velocity aids the IMU in mitigating 
the accumulated navigation errors and in calibrating inertial sensor errors. On the other hand, the 
IMU’s short-time stability helps the DVL to adaptively update parameters that may vary 
significantly due to ambient factors like water temperature and density (Kinsey et al., 2006). 
Their combined navigation accuracy depends on the accuracy of IMU-DVL alignment 
calibration (Whitcomb et al., 1999; Jalving et al., 2004). The alignment calibration problem is 
not easy as IMU and DVL are usually separate units and placed at different locations on the 
vehicle. This mounting arrangement excludes the alignment calibration during manufacture and 
instead requires an in-situ calibration during normal naval operations. Several approaches have 
been reported to attack this alignment calibration using additional external sensors such as GPS 
or Long Baseline (LBL) acoustic navigation beacons (Joyce, 1989; Tang et al., 2013; Kinsey and 
Whitcomb, 2006; 2007). These methods require the vehicle to run on the surface so that a GPS 
signal is available (Joyce, 1989; Tang et al., 2013), or additional navigation beacons to be placed 
at surveyed sites (Kinsey and Whitcomb, 2006; 2007). Troni et al. (2012) and Troni and 
Whitcomb (2010) proposed an in-situ calibration method using only on board DVL, 
gyrocompass and depth sensors, but the gyrocompass is unrealistically assumed to provide exact 
absolute attitude. In fact, precise attitude estimation for gyrocompass is not a solved problem and 
also needs extensive study (Silson, 2011; Li et al., 2013). In view of the inherent characteristics 
of both IMU and DVL, their respective parameter estimation and alignment calibration should be 
accounted for together, so as to achieve autonomous underwater navigation in large unexplored 
areas. Otherwise, the combined navigation capacity of IMU and DVL would be compromised in 
one aspect or another. 
   The content of the paper is organised as follows. Section 2 presents the mathematical 
formulation of the combined IMU/DVL navigation system, and then discusses the state 
estimability from the viewpoint of observability. An ideal observer is derived from the 
observability analysis procedure. Section 3 designs numerical simulations mimicking typical 
underwater vehicle motions and uses both the ideal observer and a practical estimator to validate 
the analytic conclusions. Conclusions are drawn in Section 4. 
 
 
2. SYSTEM FORMULATION AND STATE OBSERVABILITY 
   2.1. System Formulation. The DVL uses the principle that by reflecting three or more 
non-coplanar radio/sound beams off a surface and measuring the Doppler shifts, the velocity of 
the body with respect to that surface can be obtained (Groves, 2008). Most systems use the Janus 
configuration with four beams. The surface-referenced velocity can be obtained as 
 
d 
v
KΔf                                     (1) 
The superscript d denotes the DVL’s coordinate frame D naturally defined by the beam spatial 
configuration. The vector Δf  is formed by Doppler shifts along each beam and the matrix K  
depends on the transmitted sound wave frequency, the spatial configuration of beams and the 
sound speed in water. The first two factors of K  are fixed for a DVL unit, but the sound speed 
may vary with temperature, depth and salinity by a few percent (Groves, 2008). We rewrite the 
above relationship as 
 
1
1
d
d
s
DVL
DVL
k
k
k





v
KΔf
K Δf
y
y
v                   (2) 
where 
DVL
s
y
K Δf  denotes the DVL output, k is a scalar factor that accounts for the change of 
sound velocity, and 
s
K  is a scaled version of K  that is nearly invariant to water conditions. 
   In contrast to the DVL directly providing the velocity relative to the seabed (bottom-lock), 
the IMU measures the angular velocity and non-gravitational acceleration with respect to the 
inertial space. Numerical integrations must be carried out to derive attitude, velocity or position 
information, which is known as the inertial navigation computation procedure (Groves, 2008; 
Wu and Pan, 2013b). As gyroscopes and accelerometers are subject to errors like bias and noise, 
the computed inertial navigation result is prone to error drift accumulating with time.  
   Denote by N the local level reference frame, by B the IMU body frame, by I the inertial 
non-rotating frame, and by E the Earth frame. The navigation (attitude, velocity and position) 
rate equations in the reference n-frame are well known as (Titterton and Weston, 2004; Groves, 
2008; Savage, 2007) 
 




,
,
n
n
b
b
b
b
n
n
b
b
nb
nb
ib
g
n
ie
en






C
C
ω
ω
ω
b
C
ω
ω
                   (3) 
 



2
n
n
b
n
n
n
n
b
a
ie
en






v
C
f
b
ω
ω
v
g                      (4) 
 
n
c

p
R v                                  (5) 
where 
n
b
C  denotes the attitude matrix from the body frame to the reference frame, 
n
v  the 
velocity relative to the Earth, 
b
ib
ω  the error-contaminated body angular rate measured by 
gyroscopes in the body frame, 
bf
 the error-contaminated specific force measured by 
accelerometers in the body frame, 
n
ie
ω  the Earth rotation rate with respect to the inertial frame, 
n
en
ω  the angular rate of the reference frame with respect to the Earth frame, 
b
nb
ω
 the body 
angular rate with respect to the reference frame, and 
n
g  the gravity vector. The skew symmetric 
matrix 

 is defined so that the cross product satisfies 





a b
a
b  for two arbitrary 
vectors. The gyroscope bias 
g
b  and the accelerometer bias 
a
b  are taken into consideration as 
approximately random constants, i.e., 
g
a


b
b
0. The position 


T
L
h

p
 is described 
by the angular orientation of the reference frame relative to the Earth frame, commonly 
expressed as longitude , latitude L and height h above the Earth’s surface. In the context of a 
specific local level frame choice, e.g., North-Up-East, 


T
n
N
U
E
v
v
v
v
 and the local 
curvature matrix is explicitly expressed as 
 


1
0
0
cos
1
0
0
0
1
0
E
c
N
R
h
L
R
h




















R
                            (6) 
where 
E
R  and 
N
R  are respectively the transverse radius of curvature and the meridian radius 
of curvature of the reference ellipsoid.  
   Using Equation (2), the derived velocity from IMU is related to the DVL output by 
 
d
b
n
DVL
b
n
k

y
C C v                                  (7) 
where 
d
b
C  is the misalignment attitude matrix of the d-frame with respect to the b-frame. This 
misalignment matrix and the scale factor are both regarded as random constants. As with 
previous studies (Hegrenæs and Berglund, 2009; Joyce, 1989; Kinsey and Whitcomb, 2006; 
2007; Tang et al., 2013; Troni et al., 2012;, Troni and Whitcomb, 2010), the translational 
misalignment between DVL and IMU will not be considered hereafter. 
   We see from Equations (3), (4) and (7) that in addition to attitude/velocity/position that are of 
immediate interest to us, the combined IMU/DVL navigation also necessitates the finding of  
such parameters as inertial sensor biases (
g
b  and 
a
b ), DVL scale factor k  and misalignment 
matrix 
d
b
C . Insufficient knowledge of these parameters could result in degrading the combined 
navigation accuracy.  
   2.2. State Observability Analysis. Without any other external sensors, is it possible to 
determine the above parameters? From a viewpoint of control system, this question relates to the 
(global) observability of the system state (Chen, 1999; Wu et al., 2012). In such a case, the 
system model is given by Equations (3)-(5) and the observation model is given by Equation (7), 
with the IMU measurement as the system input and the DVL measurement as the system output.  
Hereafter we use y  to replace 
DVL
y
 for notational brevity. From Equation (7), we have 
n
n
b
b
d
k

v
C C y
. Substituting into Equation (4) and using Equation (3) yield 
 








2
n
b
b
n
b
n
b
n
n
n
b
n
b
nb
d
b
d
b
a
ie
en
b
d
k
k








C
ω
C y
C C y
C
f
b
ω
ω
C C y
g   (8) 
or equivalently, 
 






b
n
b
b
b
b
b
n
a
ie
ib
g
d
d
k







C g
f
b
ω
ω
b
C y
C y               (9) 
Troni et al. (2012) and Troni and Whitcomb (2010) used a quite coarse simplification of this 
equation for DVL calibration, for instance, neglecting the term 
b
n
n
C g . 
   On any trajectory segment with constant 
b
n
C , the quantities 
b
ie
ω  and 
b
ib
g

ω
b  keep almost 
unchanged with the moderate speed of an underwater vehicle. As the vector magnitude 
5
7.3 10
b
b
ib
g
ie





ω
b
ω
 rad/s is very small, Equation (9) is reasonably approximated by 
 


b
n
b
b
n
a
d
k



C g
f
b
C y                             (10) 
Time derivative of the above equality is 
 
b
b
d
k

f
C y                                    (11) 
from which we obtain 
b
k y
f
 whenever 
bf
 is non-vanishing, as an attitude matrix 
does not change the magnitude of a vector. It is trivial to solve the sign ambiguity. For example, 
if the b-frame is roughly aligned with the d-frame, which is often the case in practice, the 
positive sign will obviously be the right option. So the DVL scale factor is now a known quantity. 
Note that Equation (11) is valid for any segment of this type, so if there exit two such segments 
that 
bf  (or equivalently y ) have different directions, the misalignment matrix 
b
d
C  can be 
determined according to Lemma 1 in the Appendix. 
   On any trajectory segment with fast-changing 
b
n
C , the quantity 
b
ib
ω  is much larger in 
magnitude than 
b
ie
ω  or 
g
b  (as far as a quality IMU is concerned). Equation (9) can be 
approximated as 
 




b
n
b
b
b
b
n
ib
d
d
a
k





C g
ω
C y
C y
f
b                       (12) 
Taking the norm on both sides gives 
 
n
a
g 


g
α
b
                               (13) 
which 
is 
a 
quadratic 
equation 
on 
the 
accelerometer 
bias 
a
b
 with 




b
b
b
b
ib
d
d
k



α
ω
C y
C y
f . According to Lemma 2 in the Appendix, we know that 
a
b  will 
be determined if the vectors α , at all times on segments of this type, are non-coplanar, or 
T
αα  is non-singular (Wu et al., 2012). This requirement is naturally met for practical turning 
in water, as shown in the simulation section. 
   Then by the chain rule of the attitude matrix, 
b
n
C  at any time on this segment satisfies 
 











0
0
0
0
0
0
0
b t
n t
n t
b t
b t
b
b
n
n
n t
n t
b t
n t
b t
t

C
C
C
C
C
C
C
                      (14) 
where 

0
b
n t
C
 denotes the initial attitude matrix at the beginning of this segment, and 


0
b t
b t
C
 
and 


0
n t
n t
C
, respectively, encode the attitude changes of the body frame and the reference frame. 
Substituting Equation (14) into Equation (12), 
 






0
0
0
n t
b t
b
n
n
a
n t
b t
t


C
C
g
C
α
b
                          (15) 

0
b
n t
C
 is solvable as there always exist two time instants that 


0
n t
n
n t
C
g  have different directions 
due to the Earth rotation (Wu et al., 2012), so is the attitude matrix 
b
n
C  using Equation (14). 
Then the velocity 
n
v  can be determined by Equation (7), and the gyroscope bias 
g
b  can be 
determined by Equation (3). 
   The above analysis is summarised in the theorem below. 
Theorem 1 (State Observability): If the vehicle trajectory contains segments of constant attitude 
with linearly independent 
bf  (Type-I), as well as turning segments on which the vectors α  as 
defined in Equation (13) are non-coplanar (Type-II), then the system of Equations (3)-(5) and (7) 
is observable in attitude, velocity, inertial sensor biases, and DVL scale factor and misalignment 
matrix. 
   Here are some explanations on the condition of linearly independent 
bf , the rate of the 
specific force (sum of external forces except gravitation).  
Remark 1: For a multiple-thruster-propelled underwater vehicle as in Kinsey and Whitcomb 
(2006; 2007), it is not difficult to fulfil this condition, for example by thruster switching. This 
will normally create driving forces in different directions in the IMU body frame, resulting in 
linearly independent 
bf .  
Remark 2: For an underwater vehicle with a single thruster, it is tricky to fulfil this condition 
while keeping constant attitude. No matter the thruster force or the water resistance, its direction 
is fixed relative to IMU or DVL. If the vehicle moves at the same depth, linearly independent 
bf  
might only occur instantaneously at both start and end times of Type-I segments (a situation 
much like that in Tang et al. (2009)). This excitation is in practice not sufficient to produce a 
good estimate. Consider an example where 
0
0
T
b
b
xf




f
 for all Type-I segments, which 
means the thruster force aligns with x-axis of the IMU body frame. Suppose the rotation 
sequence from d-frame to b-frame is first around y-axis (yaw angle, ), followed by z-axis 
(pitch angle, ) and then by x-axis (roll angle, ), 
b
d
C  can be re-parameterised in Euler 
angles as 
 
cos cos
sin
cos sin
sin sin
cos cos
sin
cos cos
cos
sin
cos sin sin
.
cos sin
cos
sin sin
cos sin
cos cos
sin sin sin
b
d














































C
   (16) 
Substituting into Equation (11) 
 


cos cos
sin
cos sin
T
b
x
k f






y                  (17) 
which is irrelevant to the roll angle  around x-axis. The other two angles can be computed by 
element comparison, while the angle  is not estimable. This case is right what we encountered 
in land vehicle navigation subject to the non-holonomic constraint (Wu et al., 2009). It is the 
angle around the thruster force that is inestimable, so if the thruster force was in a general 
direction, it can be imagined that all three Euler angles would be affected. 
Remark 3: Even if 
bf  (or equivalently y ) on the segments of constant attitude are linearly 
dependent, all states in Theorem 1 are still observable except the angle around the direction of 
bf . This interesting conclusion can be obtained from the proof of Theorem 1. We only need to 
show that the products 
b
d
C y  and 
b
d
C y  are known quantities in this scenario, in spite of the 
undetermined 
b
d
C
. Denote by η  the unit direction of linearly dependent y . For a 
single-thruster underwater vehicle, both the DVL velocity y  and the velocity rate y  almost 
certainly align with the direction of η. From Equation (11), for some time 
1t  on this segment 



1
1
1
b
b
b
d
d
k
t
t
t


f
C y
y
C η
 where 

1t
y
 is 
non-zero. 
Therefore, 


1
1
b
b
b
d
d
k
t
t


C y
y C η
y f
y
 and 


1
1
b
b
b
d
d
k
t
t


C y
y C η
y f
y
 are both functions 
of known quantities. 
Remark 4: Ascending or descending motion makes it possible for a single-thruster underwater 
vehicle to have linearly independent 
bf . The water resistance due to ascending/descending 
velocity change will incur non-zero 
bf  that deviates in direction from the thruster force. The 
ascending/descending motions are realised by unloading/loading the ballast tank. 
   2.3. Derived Ideal Observers. The above observability analysis not only tells a ‘yes or no’ 
answer to state estimability, but provides us valuable insights on designing observers. This is an 
additional advantage of global observability analysis (Wu et al., 2012). Specifically, Equation 
(11) allows us to construct an ideal observer to estimate DVL parameters, using IMU and DVL 
measurements on Type-I segments. No additional external sensors like GPS or LBL beacons are 
required. Suppose the time interval 
j
j
s
e
t
t



 corresponds to the j-th Type-I segment 
(
1,2,
j 
 ). For any 
j
j
s
e
t
t
t



, integrating (11) twice over the subinterval 
j
st
t



 
 


b
d
k
t
t

β
C γ
                               (18) 
where 



j
s
t
b
b
j
j
s
s
t
t
d
t
t
t



β
f
f
 and 





j
j
j
s
s
s
t
t
t
t
t
t



γ
y
y
y
. Compared with 
Equation (11), this integral form spares the differentiation of the measurements that would be 
subject to noise amplification (Wu and Pan, 2013a; Wu et al., 2014). The above equation 
obviously applies to all Type-I segments. Denote 
j
j
s
e
j
t
t




, we have the following 
observer to calibrate the DVL parameters. 
Ideal Observer – DVL Calibration (IO-DVLC):  
1) Compute 


k
t
t
γ
β
 for each t ; 
2) Obtain 
b
d
C  by solving the constrained optimization problem 


2
(3)
min
SO
k
t
t
dt




C
β
Cγ
 as 
done in (Wu and Pan, 2013a). 
The “ideal observer” will be used to verify the analytic conclusions above. 
 
Figure 1. Moving trajectory of the underwater vehicle (3D run: in blue, 2D run: in red). 
 
3. NUMERICAL STUDY. In this section, we carry out numerical simulations to study the 
validity of the analysis. We will see that the above analysis can explain quite well what we will 
encounter in practical state observers or estimators. The simulations are designed to mimic the 
typical motions of a single-thruster propelled underwater vehicle. The 3D trajectory of the 
vehicle is illustrated in Figure 1 and the motion sequences with time are listed in Table 1. The 
thruster force is arranged along the vehicle’s longitudinal direction (x-axis of the vehicle frame, 
v-frame, defined as forward-upward-rightward). On board the vehicle is an IMU equipped with a 
triad of gyroscopes (bias 0.01 / h

, noise 0.1 /
/
Hz
h

) and accelerometers (bias 50 g
, noise 
10
/
Hz
g

), and a DVL with measurement noise 2 cm/s (1σ). The vehicle v-frame is assumed 
to align perfectly with the IMU (b-frame), from which the DVL (d-frame) misaligns in attitude 
by -0.1° (roll), -0.5° (yaw) and -0.2° (pitch). The DVL scale factor is set to 0.9998. The initial 
attitude of the IMU is set to 3° (roll), 10° (yaw) and zero (pitch). 
 
Table 1. Motion Sequences with Time. 
Time (s) 
Motions 
Segment Type 
3D Run 
2D Run 
0-600 
0-600 
Static 
/ 
600-660 
600-800 
Level motion with constant attitude but varying specific 
force rate 
I 
660-720 
/ 
Descending with constant attitude but varying specific 
force rate 
I 
-3000
-2000
-1000
0
1000
2000
-3000
-2000
-1000
0
1000
2000
3000
-400
-350
-300
-250
-200
-150
-100
-50
0
50
South-North (m)
West-East (m)
Height (m)
720-750 
/ 
Level motion with constant attitude but varying specific 
force rate 
I 
750-1970 
800-2040 
Running along a square shape, with tilted turning and 
constant speed 
I & II 
1970-2000 
/ 
Level motion with constant attitude but varying specific 
force rate 
I 
2000-2060 
/ 
Ascending with constant attitude but varying specific 
force rate 
I 
 
   The DVL output profile and IMU output profile generated by 3D motion sequences in Table 
1 are respectively plotted in Figures 2 and 3. As the vehicle moves forward over 600-660 s and 
downward over 660-720 s, and the vectors of specific force rate (Figure 3) or DVL acceleration 
on these two segments are linearly independent (Figure 2), the condition of Theorem 1 for 
Type-I segments is satisfied. The derived IO-DVLC observer in Section 2.3 is used to estimate 
the DVL scale factor (Figure 4) and misalignment angles (Figure 5). We see that the scale factor 
is computable at either of two segments, while the misalignment angles, especially the roll angle, 
cannot converge until the downward segment is carried out at 660 s. These observations have 
been well predicted already by Remarks 2 and 4, although the IO-DVLC observer is not accurate 
enough in the misalignment angles. 
 
Figure 2. Profile of DVL outputs with specific time tag. 
 
600660720
1970 2060
-10
-5
0
5
10
Time(s)
DVL output
 
Figure 3. Profile of IMU gyroscope/accelerometer outputs with specific time tag. 
 
Figure 4. DVL scale factor estimate by ideal observer IO-DVLC. 
 
750
1005
1260
1615
-0.5
0
0.5
x axis
Gyroscopes (deg/s)
600720
1970
-1
-0.5
0
0.5
1
Accelerometers (m/s2)
750
1005
1260
1615
0
1
2
3
y axis
660
2000
9
9.5
10
10.5
805
1060
1315
1670
-0.04
-0.02
0
0.02
Time (s)
z axis
750
1005
1260
1615
-0.6
-0.4
-0.2
0
Time (s)
580
600
620
640
660
680
700
720
740
0
0.2
0.4
0.6
0.8
1
1.2
Time(s)
DVL scale factor
600
620
640
660
680
700
720
0.9995
0.9996
0.9997
0.9998
0.9999
1
 
Figure 5. DVL misalignment angle estimate by ideal observer IO-DVLC (unit: degree). 
 
   Next we implement an Extended Kalman Filter (EKF) to estimate the states of the combined 
IMU/DVL system, of which the system dynamics are given by Equations (3)-(5) and the 
measurement is given by Equation (7). The EKF is a nonlinear state estimator widely used in 
numerous applications. In addition to those states in Theorem 1, the EKF also estimates the 
position. The position is unobservable, but it is correlated with other states through the system 
dynamics and the determination of other states will help mitigate the position error drift. The first 
600 s static segment is used for IMU initial alignment. Additional angle errors of 0.1° (1σ, for 
yaw) and 0.01° (1σ, for roll and pitch) are added to the final alignment result so as to mimic the 
influence of non-benign alignment conditions underwater. Figures 6 and 7 respectively present 
the DVL scale factor estimate and misalignment angle estimate, as well as their standard 
variances. The scale factor in Figure 6 converges swiftly from the initial value 0.8 to the truth 
once the vehicle starts to move at 600 s, as are the two misalignment angles, yaw and pitch, in 
Figure 7. The roll angle approaches its true value when the descending motion starts at 660 s. 
Apparently, EKF is more accurate than the IO-DVLC observer in estimating the DVL 
parameters. The inertial sensor bias estimates and their standard variances are presented in 
Figures 8 and 9. We see that turning on the square trajectory after 750 s drives the accelerometer 
bias estimate to convergence (Figure 9). The gyroscope bias in Figure 8 is relatively slower in 
convergence, especially that in vertical direction (y-axis), due to weaker observability. Figures 
10 and 11 give the attitude error and positioning error, as well as their standard variances. The 
normalised standard variances for attitude, gyroscope/accelerometer biases, DVL scale factor 
and misalignment angles are plotted in Figure 12. It is the DVL scale factor and misalignment 
angles (yaw and pitch) that have the strongest observability in this simulation scenario. The 
interaction among states are quite obvious from Figure 12; for example, yaw angle and the DVL 
roll angle (at 660 s), and roll/pitch angles and the x-axis accelerometer bias (at 750 s). All of the 
EKF’s behaviours accord with Theorem 1. 
   A 2D trajectory is also examined to verify the analysis of Remark 3. This planar run is 
similar with the above 3D run, but excludes the descending/ascending segments (as seen in 
Figure 1 and Table 1). The DVL scale factor and misalignment angle estimates, attitude error 
580
600
620
640
660
680
700
720
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
DVL misalignment angle
 
 
roll
yaw
pitch
and position error by EKF are respectively plotted in Figures 13-16. As predicted by Remarks 2 
and 3, all states but the DVL roll angle are estimable. Although the DVL roll angle does not 
converge (Figure 14) in this case, other states are estimated quite well. 
 
Figure 6. DVL scale factor estimate and standard variance by EKF. 
 
Figure 7. DVL misalignment angle estimate and standard variance by EKF (unit: °). 
500
1000
1500
2000
0.8
0.85
0.9
0.95
1
1.05
1.1
DVL scale factor
500
600
1000
1500
2000
0
0.2
0.4
0.6
0.8
1
Time (s)
Standard variance
500
1000
1500
2000
0.9995
0.9998
1
600
1,000
1,500
2,000
-1
-0.5
-0.2
0
0.5
DVL misalignment angle
 
 
500
660
1,000
1,500
2,000
0
1
2
3
4
5
6
Time (s)
Standard variance
 
 
roll
yaw
pitch
 
Figure 8. Gyroscope bias estimate and standard variance by EKF (unit: °/h) 
 
 
Figure 9. Accelerometer bias estimate and standard variance by EKF (unit: micro g). 
600
750
1,000
1,500
2,000
-0.02
-0.01
0
0.01
0.02
0.03
Gyroscope bias
600
720
805
1,000
1,500
2,000
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (s)
Standard variance
 
 
x
y
z
600
750
1,000
1,500
2,000
-50
0
50
100
Accelerometer bias
600
750
1,000
1,500
2,000
0
50
100
150
200
250
Time (s)
Standard variance
 
 
x
y
z
 
Figure 10. Attitude error and standard variance by EKF (unit: °). 
 
Figure 11. Horizontal position error and standard variance of position estimate by EKF (unit: metre). 
 
500
660
750
1,000
1,500
2,000
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Attitude error
 
 
600660
750
1,000
1,500
2,000
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Standard variance
roll
yaw
pitch
500
1000
1500
2000
0
0.5
1
1.5
2
Horizontal position error
500
1000
1500
2000
0
1
2
3
4
5
6
Time (s)
Standard variance of position
 
 
west-east
south-north
height
 
Figure 12. Normalised standard variances for attitude, inertial sensor bias, DVL scale factor and misalignment 
angles. 
 
 
Figure 13. DVL scale factor estimate by EKF in 2D run. 
600
660
750
805
1000
1100
1200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Normalized standard variances
 
 
Roll
Yaw
Pitch
Gyro-x
Gyro-y
Gyro-z
Acc-x
Acc-y
Acc-z
DVL-Roll
DVL-Yaw
DVL-Pitch
SF
600
1000
1500
2000
0.8
0.85
0.9
0.95
1
1.05
1.1
DVL scale factor
600
1000
1500
2000
0
0.2
0.4
0.6
0.8
1
Time (s)
standard variance
500
1000
1500
2000
0.9995
0.9998
1
 
Figure 14. DVL misalignment angle estimate by EKF in 2D run (unit: °). 
 
 
Figure 15. Attitude error and standard variance by EKF in 2D run (unit: °). 
 
600
1000
1500
2000
-5
-4
-3
-2
-1
0
DVL misalignment angle
 
 
600
1000
1500
2000
0
1
2
3
4
5
6
Time (s)
Standard variance
 
 
roll
yaw
pitch
500
1000
1500
2000
-1
-0.5
-0.2
0
 
 
600
800
1,000
1,500
2,000
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Attitude error
 
 
600
800
1,000
1,500
2,000
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Standard variance
roll
yaw
pitch
 
Figure 16. Horizontal position error and standard variance of position estimate by EKF in 2D run (unit: metre). 
 
4. CONCLUSIONS. The development of commercial gyroscope/accelerometer and Doppler 
sensors has enabled significant improvements to underwater navigation, especially in unexplored 
areas without artificial beacons. This paper addresses combined IMU/DVL underwater 
navigation from the viewpoint of the control system. We show by analysis that the combined 
system is observable under moderate motion conditions. The DVL parameters, such as the scale 
factor and misalignment angles, especially can be in-situ calibrated without relying on additional 
external GPS or acoustic beacons. The IMU bias errors can also be effectively estimated, aided 
by the DVL measurement. These benefits are promising to enable a fully autonomous 
underwater navigation. We carried out numerical simulations and used a practical EKF to 
integrate IMU and DVL information. The simulation results accord very well with our analytic 
conclusions. The essential result also applies to IMU/Doppler laser airborne applications. The 
analytic result is important and a necessary step to ultimately solve the challenging problem of 
in-situ IMU/DVL self-calibration. It also provides useful guidance for field test planning and 
implementation. High quality field test data will be collected in the future and used to verify the 
proposed IMU/DVL integration scheme in this paper. 
 
FINANCIAL SUPPORT 
This work was supported in part by the Fok Ying Tung Foundation (grant number 131061) and National 
Natural Science Foundation of China (grant number 61174002, 61422311). 
 
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APPENDIX 
 
Lemma 1 (Black, 1964; Shuster and Oh, 1981): For any two linearly independent vectors, if their 
coordinates in two arbitrary frames are given, then the attitude matrix between the two frames can be 
determined. 
Lemma 2 (Wu et al., 2012): Given 
 known points 
, 
, in three-dimensional space 
satisfying 
, where 
 is an unknown point, 
 is a positive scalar. If points 
 do not lie in 
any common plane, 
 has a unique solution. 
 
m
ia
1,2,
,
i
m

i
r


a
x
x
r
ia
x