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1 INTRODUCTION
The dead reckoning navigation method offers an
innovative approach to localising unmanned aerial
vehicles (UAVs) in environments where GPS signals
are disrupted or unavailable. Utilising data from
inertial measurement units (IMUs) and advanced
mathematical algorithms enables the estimation of
UAV positions and trajectories, eliminating the
reliance solely on external signals. As a result, it
becomes an essential tool in scenarios where traditional
navigation systems fail, such as military operations or
missions in remote and challenging areas.
This study aims to demonstrate the effectiveness of
the dead reckoning navigation method in UAV
applications. The research compares various sensor
data processing algorithms and evaluates their
performance in real-world test scenarios. The
experiments are intended to demonstrate the potential
of the dead reckoning method as a reliable navigation
tool under conditions of limited GNSS access.
Furthermore, the study explores integrating this
method with other navigation techniques to enhance
the accuracy and reliability of autonomous UAV
systems.
2 RELATED WORKS
Hou and Bergmann [1] introduced a pedestrian dead
reckoning (PDR) method designed for head-mounted
sensors, utilising the natural stabilisation of the head
during movement to enhance accuracy. Their solution
achieved an average positional error of 0.88 m,
demonstrating the potential of head-mounted devices
in applications such as sports, rescue operations, and
smart environments.
At the 2024 International Conference on Indoor
Positioning and Indoor Navigation [2], a comparative
analysis of pedestrian dead reckoning algorithms
highlighted the challenges and opportunities of using
inertial data for accurate localisation. The study
underscored the limitations of dead reckoning in
GNSS-denied environments due to cumulative errors
Dead Reckoning Method for an Unmanned Aerial
Vehicle in Conditions of Limited GPS Signal
B. Szykuła & J. Furtak
Military University of Technology, Warsaw, Poland
ABSTRACT: The article presents a dead reckoning navigation method dedicated to unmanned aerial vehicles
(UAVs), enabling position determination under limited GNSS signal access. The method is based on the analysis
of data from inertial measurement units (IMUs) and mathematical algorithms, allowing for precise monitoring of
UAV trajectories. The study outlines the theoretical foundations of dead reckoning navigation, the derivation of
key formulas, and the results of simulation tests of the method based on real inertial sensor data logs. The tests
demonstrated promising effectiveness, highlighting its potential in military and civilian applications. The method
can serve as a complement to traditional navigation systems.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 4
December 2025
DOI: 10.12716/1001.19.04.01
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and explored methods for mitigating these
inaccuracies. The authors emphasised the critical role
of algorithm adaptability in optimising performance
across varying operational conditions. Their findings
underscore the potential of integrating dead reckoning
with complementary technologies to enhance
positioning accuracy in diverse environments, offering
insights highly relevant to UAV navigation.
This study applies the dead reckoning method to
UAV systems, extending its use beyond pedestrian
localisation. Unlike the previous studies, which
emphasise wearable sensors and pedestrian dynamics,
this research employs the Madgwick filter, offering a
computationally efficient solution suitable for UAV
operations under dynamic conditions and limited
GNSS availability.
3 DEAD RECKONING
3.1 Definition of dead reckoning
Dead reckoning navigation is a method for
determining the current position of an object based on
its last known location, the direction of movement, and
the distance travelled, taking into account speed and
time. This process relies on iterative calculations of
successive positions without needing external
navigation systems like GPS.
This navigation model is applicable in mobile
sensor networks, where localization must be
determined autonomously, and navigation data are
calculated using the available motion parameters. This
technique is particularly useful in mobile systems, such
as UAVs, where external signals are disrupted [3].
In dead reckoning navigation for UAVs, it is crucial
to account for the influence of wind on positional
accuracy. Wind can cause the vehicle to drift from its
planned trajectory, accumulating errors in position
estimation. To correct these deviations, velocity vectors
are used. The current position P(t) can be determined
based on the previous position P(t-1), time Δt, and the
sum of these vectors:
( ) ( )
( )
1
aw
tt
P P t v v
−
= + +
(1)
where:
P(t) – current position of the object
P(t-1) – last known position of the object
∆t – time difference between position measurements
a
v
– velocity vector of the object relative to the air
w
v
– velocity vector of the wind
The accuracy of this method depends on precise
measurements of both the velocity relative to the air
and the wind’s speed and direction. Therefore, in
practical navigation, continuously monitoring
atmospheric conditions and updating input data are
essential to ensure the highest possible accuracy of
dead reckoning navigation under variable
environmental conditions.
3.2 Inertial sensors
To ensure the precision of dead reckoning navigation
under limited GPS signal conditions, using accurate
measurement data provided by onboard systems is
crucial. Inertial sensors play a fundamental role,
enabling the monitoring of motion and orientation of
UAVs in space.
The inertial measurement unit (IMU), an integral
component of the dead reckoning navigation system,
consists of several key elements, each providing
unique data that supports the estimation of an object’s
position and trajectory:
− A gyroscope measures angular velocities around
axes, allowing the determination of changes in the
UAV’s orientation in space.
− An accelerometer records linear accelerations along
three axes, enabling the calculation of the object’s
velocity and displacement.
− A magnetometer measures the Earth’s magnetic
field, facilitating the determination of spatial
direction and orientation relative to the magnetic
pole.
− A barometer measures atmospheric pressure,
aiding in altitude determination based on pressure
variations.
The data provided by these sensors are
subsequently integrated and processed using
advanced mathematical algorithms, allowing precise
tracking of UAV trajectories even under dynamically
changing environmental conditions.
4 METHODOLOGY
4.1 Sensor fusion
The sensor fusion process is a critical component of
data processing in inertial systems, enabling consistent
and precise information about an object’s state in space.
Each sensor generates data with distinct characteristics
and noise levels; thus, their integration ensures mutual
compensation of imperfections and enhances result
quality.
In dead reckoning navigation, this process involves
filtering and integrating data from the gyroscope,
accelerometer, magnetometer, and barometer to
determine parameters such as orientation, linear
acceleration, and angular velocities. Complementary
filtering, a widely used approach, combines
measurements from various sensors, reducing the
impact of noise and drift.
This study employs the Madgwick filter, known for
its rapid convergence and stable system state
estimations. This algorithm is based on gradient
optimisation of the object’s orientation using inertial
data. Although its implementation is more complex
than traditional methods, it offers significant
advantages in systems requiring high precision, such
as UAV navigation [4].
Data fusion allows for precisely determining an
object’s tilt angles, enabling accurate trajectory control
and navigation in challenging conditions.
4.2 Linear acceleration
The linear acceleration of the platform is calculated
using data from inertial sensors. These sensors provide
raw acceleration data along the XYZ axes, which are
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trans-formed into a one-dimensional linear
acceleration value using mathematical formulas. The
coordinate system (Fig. 1) is centred at the UAV’s
centre of mass, with the X-axis aligned with the drone’s
movement direction, the Y-axis perpendicular in the
horizontal plane, and the Z-axis aligned with
gravitational acceleration. This system moves with the
platform as the reference for all acceleration
measurements.
Figure 1. Coordinate system for UAV and IMU
One of the fundamental methods involves using the
sum of the squares of acceleration components along
each axis:
222
x y z
a a a a= + +
(2)
where:
a – linear acceleration
ax, ay, az – acceleration along the individual axes of the
XYZ coordinate system
This value represents the resultant acceleration of
the object in space; however, accelerometer data also
include a component related to gravitational attraction.
To accurately determine the drone’s displacement
relative to the Earth’s surface, it is necessary to
compensate for the effect of gravity on accelerometer
readings. This is achieved by using a gyroscope and
magnetometer to determine the object’s orientation
relative to gravity’s direction, separating motion-
induced acceleration from the gravitational force [5].
This process involves transforming accelerometer
readings along the XYZ axes using appropriate
corrections:
( )
( )
( )
( )
( )
sin
sin cos
cos sin
xc x pitch
yc y roll pitch
zc z roll pitch
a a g
a a g
a a g
= −
= +
= +
(3)
where:
axc, ayc, azc – compensated accelerations along the XYZ
axes
g – gravitational acceleration constant
θpitch – pitch angle in the XY plane
θroll – roll angle in the YZ plane
This compensation eliminates the effect of gravity,
which is crucial for accurately determining motion-
induced acceleration. As a result, the input data for
dead reckoning navigation algorithms are more
precise, leading to more accurate trajectory
estimations. Moreover, this approach in UAV systems
enhances their reliability in environments with
dynamically changing conditions, such as wind gusts
or rapid manoeuvres.
4.3 Determination of displacement
To apply the dead reckoning navigation method,
precise determination of linear acceleration a (Formula
2) is essential, including compensation for the effect of
gravitational acceleration on the XYZ axes (Formula 3).
Linear velocity v is calculated through discrete
integration of linear acceleration a. The navigation
system reads inertial sensor data at equal time intervals
∆t, necessitating an iterative integration process
expressed by the formula:
( ) ( ) ( )
1
k k k
v t v t a t t
−
= +
(4)
where:
v(tk) – velocity at time tk
a(tk) –acceleration at the time tk
∆t – time interval between date readings
It is important to emphasise that onboard inertial
sensors provide information only about the object’s
velocity relative to the surrounding medium, which in
the case of aerial vehicles refers to airspeed. The
omission of the wind velocity vector in such
calculations leads to the accumulation of measurement
errors, especially during long-duration operations.
The integrated linear velocity allows for the
subsequent calculation of the distance travelled by the
object over a specified period. For aerial vehicles, a
critical aspect is the separation of horizontal
displacement from altitude changes, eliminating the
influence of altitude variation on the total distance. The
horizontal distance can be calculated using the relation
derived from the Pythagorean theorem:
( )
2
2
s v t h= −
(5)
where:
s – horizontal distance travelled
∆h – altitude difference
Separately addressing horizontal and vertical
components in calculations increases the precision of
trajectory estimation, particularly in dynamically
changing flight conditions. However, uncertainties
stemming from the lack of wind velocity vector
information remain a challenge.
4.4 Angular calculations
The displacement in meters is converted into
geographic coordinates by transforming the heading
into radians and calculating the divisors for latitude
and longitude. In dead reckoning navigation, this
process requires accounting for the variability of the
metric length of a degree of latitude and longitude
depending on the Earth’s position. This consideration
is critical because the Earth’s shape as a geoid causes
the metric length of a degree of longitude to decrease
with increasing distance from the equator [6].
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For the calculation of divisors of longitude and
latitude, you can use the following formula:
2
360
90
2
360 90
M
ss
d
E
d
R
DP
P
R
D
=
−
=
(6)
where:
Ds, Dd – divisors for latitude and longitude,
respectively
RM, RE – radii of the meridian and equator, respectively
Ps, Pd – previous latitude and longitude, respectively
Based on these values, you can determine
geographic coordinates for the final location with the
formulas:
( )
( )
cos
sin
ss
s
dd
d
s
LP
D
s
LP
D
= +
= +
(7)
where:
Ls, Ld – current location (latitude and longitude,
respectively)
θ – heading expressed in radians
These formulas allow for the precise determination
of the object’s current position, considering the impact
of local geographic conditions and trajectory.
5 EXPERIMENT
5.1 Procedure
To evaluate the effectiveness of the dead reckoning
algorithm, test flights were con-ducted using a DJI
Mini 3 drone with an IMU and GPS module. The
experiment simulated diverse navigation conditions to
ensure the GPS signal stability, including
environmental variations and flight dynamics, in areas
with minimal terrain obstacles.
The recorded flight data included IMU readings,
altitude information from the barometer, and GPS
coordinates as a reference for accuracy evaluation.
The algorithm processed real-time data, integrating
acceleration to calculate velocity and position.
Simulations covered ideal conditions and disruptions,
such as wind effects and sensor errors. For data
analysis, a Python script was developed to integrate
sensor data and process it using the following libraries:
− NumPy - numerical operations,
− pandas - managing and analysing log data,
− Matplotlib - visualising results, including flight
trajectories, deviations from reference coordinates,
and error histograms.
5.2 Test cases
The experiment included two main scenarios to
examine the impact of different data processing
strategies on the accuracy of dead reckoning
navigation:
1. Scenario without GPS updates: The navigation
algorithm operated solely based on inertial sensor
data, starting from known initial coordinates. In this
case, calculations were performed without
subsequent GPS updates, simulating a complete
loss of GNSS signal. This scenario assessed the error
accumulation rate due to sensor drift and its effect
on final accuracy.
2. Scenario with periodic GPS updates: The algorithm
started from the same initial coordinates but
updated its position every 15 seconds using GPS
data to correct deviations from the actual trajectory.
This scenario aimed to evaluate the effectiveness of
GPS data integration in minimising navigation
errors.
Three experiments were conducted for each
scenario. Fig. 2, Fig. 3 and Fig. 4 illustrate the results for
Scenario 1, while Fig. 5, Fig. 6 and Fig. 7 present the
results for Scenario 2.
5.3 Experiment results
5.3.1 Scenario without GPS updates - Flight No. 1
Figure 2. Comparison of the actual and computed trajectory
for Flight No. 1 in the scenario without GPS updates
Table 1. MSE, MAE, and Maximum Error for Flight No. 1 in
the Scenario 1
Error
Value
MSE
0.000704 [°]
78.32 [m]
MAE
0.000592 [°]
65.95 [m]
Maximum Error
0.001209 [°]
134.59 [m]
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5.3.2 Scenario without GPS updates - Flight No. 2
Figure 3. Comparison of the actual and computed trajectory
for Flight No. 2 in the scenario without GPS updates
Table 2. MSE, MAE, and Maximum Error for Flight No. 2 in
the Scenario 1
Error
Value
MSE
0.001188 [°]
132.24 [m]
MAE
0.000988 [°]
110.04 [m]
Maximum Error
0.002209 [°]
245.95 [m]
5.3.3 Scenario without GPS updates - Flight No. 3
Figure 4. Comparison of the actual and computed trajectory
for Flight No. 3 in the scenario without GPS updates
Table 3. MSE, MAE, and Maximum Error for Flight No. 3 in
the Scenario 1
Error
Value
MSE
0.000704 [°]
78.32 [m]
MAE
0.000592 [°]
65.95 [m]
Maximum Error
0.001209 [°]
134.59 [m]
5.3.4 Scenario with periodic GPS updates - Flight No. 1
Figure 5. Comparison of the actual and computed trajectory
for Flight No. 1 in the scenario with periodic GPS updates
Table 4. MSE, MAE, and Maximum Error for Flight No. 1 in
the Scenario 2
Error
Value
MSE
0.000191 [°]
21.24 [m]
MAE
0.000144 [°]
16.05 [m]
Maximum Error
0.000499 [°]
55.58 [m]
5.3.5 Scenario with periodic GPS updates - Flight No. 2
Figure 6. Comparison of the actual and computed trajectory
for Flight No. 1 in the scenario with periodic GPS updates
Table 5. MSE, MAE, and Maximum Error for Flight No. 2 in
the Scenario 2
Error
Value
MSE
0.000196 [°]
21.86 [m]
MAE
0.000158 [°]
17.59 [m]
Maximum Error
0.000489 [°]
54.42 [m]
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5.3.6 Scenario with periodic GPS updates - Flight No. 3
Figure 7. Comparison of the actual and computed trajectory
for Flight No. 3 in the scenario with periodic GPS updates
Table 6. MSE, MAE, and Maximum Error for Flight No. 3 in
the Scenario 2
Error
Value
MSE
0.000106 [°]
11.82 [m]
MAE
0.000085 [°]
9.51 [m]
Maximum Error
0.000268 [°]
29.87 [m]
5.4 Analysis of results
The studies that were conducted revealed significant
limitations of the dead reckoning navigation system,
particularly in scenarios without GPS updates. Based
on the results, including MSE, MAE, and maximum
deviations, noticeable differences in positional
estimation accuracy were observed depending on the
availability of external reference data. For Flight No. 1
in the scenario without GPS updates, the mean squared
error (MSE) was 78.32 m (Table 1), whereas, in the
analogous scenario with periodic updates, it decreased
to 21.24 m (Table 4), representing a reduction of over
70%.
The trajectory comparison plots (Fig. 2, …, Fig. 7)
clearly show that navigation errors increase
significantly over time in scenarios without GPS
updates. In contrast, cases with periodic GPS updates
demonstrate much greater alignment between
computed and actual trajectories. This approach
indicates that reliance on dead reckoning without
external reference data, such as GPS, is infeasible.
While dead reckoning is helpful for short durations, it
does not provide sufficient accuracy for long-term
operations without regular support from external
systems like GNSS.
Notably, maximum errors in scenarios without GPS
updates (Table 1, …, Table 3) pose significant risks in
operations requiring high precision. During test flights,
maxi-mum deviations reached 291.33 m in Flight No. 3
(Table 3), which could negatively impact tasks
demanding precise positioning.
The circular test trajectories may have partially
compensated for wind effects, as the drone alternated
between moving with and against the wind,
potentially distorting results and producing lower
average errors than expected for longer linear flights.
6 CONCLUSION
The dead reckoning navigation method is moderately
effective for UAV position estimation under limited
GPS conditions, performing well over short distances
and timeframes where error accumulation is minimal.
It is a valuable complement to traditional systems
during temporary GNSS signal loss.
However, a key challenge remains the impact of
wind, which prevents the complete independence of
dead reckoning from external data sources. Due to the
limitations of IMU data, which only provides airspeed,
additional solutions are needed to determine ground
speed. In this context, optical flow technology, which
analyses images to determine relative motion
concerning the Earth’s surface, is worth considering.
Alternatively, other instruments, such as Doppler
radar or advanced weather models, could be utilized to
estimate the relative wind vector.
In conclusion, despite its limitations, the method
shows potential for development. Integrating it with
other measurement systems could enhance precision
and resilience, benefiting industrial and military
applications under challenging conditions.
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Comparative Analysis of Pedestrian Dead Reckoning
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International Conference on Indoor Positioning and
Indoor Navigation (2024)
[3] Hu L., Evans D.: Localisation for mobile sensor networks.
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