Learning Car Speed Using Inertial Sensors for Dead Reckoning Navigation

Abstract

A deep neural network (DNN) is trained to estimate the speed of a car driving in an urban area using as input a stream of measurements from a low-cost six-axis inertial measurement unit (IMU). Three hours of data was collected by driving through the city of Ashdod, Israel in a car equipped with a global navigation satellite system (GNSS) real time kinematic (RTK) positioning device and a synchronized IMU. Ground truth labels for the car speed were calculated using the position measurements obtained at the high rate of 50 [Hz]. A DNN architecture with long short-term memory layers is proposed to enable high-frequency speed estimation that accounts for previous inputs history and the nonlinear relation between speed, acceleration, and angular velocity. A simplified aided dead reckoning localization scheme is formulated to assess the trained model which provides the speed pseudo-measurement. The trained model is shown to substantially improve the position accuracy during a 4 minutes drive without the use of GNSS position updates.

Full text

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Sensor Applications
Learning Car Speed Using Inertial Sensors for Dead Reckoning Navigation
Maxim Freydin, Member, IEEE and Barak Or, Member, IEEE
Maxim Freydin and Barak Or are with ALMA Technologies, Haifa, 3100400, Israel
Manuscript accepted to IEEE Sensors Letters at August 22, 2022; This is a preprint version
Abstract—A deep neural network (DNN) is trained to estimate the speed of a car driving in an urban area using as input
a stream of measurements from a low-cost six-axis inertial measurement unit (IMU). Three hours of data was collected
by driving through the city of Ashdod, Israel in a car equipped with a global navigation satellite system (GNSS) real time
kinematic (RTK) positioning device and a synchronized IMU. Ground truth labels for the car speed were calculated using
the position measurements obtained at the high rate of 50 Hz. A DNN architecture with long short-term memory layers is
proposed to enable high-frequency speed estimation that accounts for previous inputs history and the nonlinear relation
between speed, acceleration and angular velocity. A simplified aided dead reckoning localization scheme is formulated to
assess the trained model which provides the speed pseudo-measurement. The trained model is shown to substantially
improve the position accuracy during a 4 minutes drive without the use of GNSS position updates.
Index Terms—Machine learning, supervised learning, long short-term memory, dead reckoning, real-time kinematic positioning, inertial
measurement unit.
I. INTRODUCTION
Accurate localization is a critical capability for automated driving
[1]–[4]. Existing solutions are based on sensor fusion techniques
that utilize inputs from cameras, radars, global navigation satellite
system (GNSS) receivers, and inertial measurement units (IMUs)
[5]–[7]. However, some environments, like indoor parking lots and
dense urban areas, remain challenging to localize in. Cameras and
radars require line-of-sight toward the object of interest and GNSS
receivers can not operate indoor. IMU sensors, on the other hand,
are immune to those problems and remain available in all settings.
The shortcoming of IMUs for localization is that existing integration
methods accumulate error rapidly over time [8], [9].
In this work, a deep neural network (DNN) [10] model is trained
to predict the speed of a car from IMU measurements which may
then be used for localization (by an additional integration) or used
as speedometer. This work is aimed at creating a high-frequency
real-time speed estimator using a single low-cost IMU sensor. For
that, we establish a data-driven approach that consists of two steps:
(1) collecting data of IMU signals from multiple car drives with
corresponding speed measurements as labels, using a real time
kinematic (RTK)-GNSS device, and (2) applying supervised learning
to train a DNN-based model to estimate the car speed in real-time.
The proposed approach may improve existing dead reckoning (DR)
techniques in GNSS-denied areas [11] and enable localization based
only on a single IMU sensor for longer time periods while maintaining
an acceptable positioning error.
Previous attempts to apply supervised learning in estimation
problems with localization applications were focused on learning
the sensor noise models. Mahdi et al. used low-cost sensors as
input together with expensive sensors to generate the true labels for
the measurement noise [12]. Azzam et al. trained long short-term
memory (LSTM) models to improve the trajectory accuracy obtained
using a simultaneous localization and mapping (SLAM) algorithm
Corresponding author: M. Freydin (e-mail: maxim@almatechnologies.com).
Associate Editor: XXXX XXXX.
Digital Object Identifier 10.1109/LSENS.2017.0000000
[13]. However, they used the pose estimated by the SLAM algorithm
as input and not direct sensor measurements. Srinivasan et al. [14]
learned speed estimation for an autonomous racing car. The true
label for the speed was produced using a specially developed mixed
Kalman Filter (KF) [15] to fuse inputs from multiple sensors of
different types. In [16], Reina et al. used a convolutional neural
network (CNN)-based architecture to train a regressor to predict
speed. However, this approach is not able to create memory between
samples. Also, the large window size of the input makes the model
unfit for high-frequency real-time applications. Yanlei et al. used
a similar approach to estimate speed with low-cost IMU as input
[17]. Their solution was based on position measurements from a
common GNSS receiver to calculate the true speed which suffers
from relatively low accuracy and low sampling rate. In addition, the
input was the IMU signal sampled at the low frequency of 10 Hz and
a window size of 4seconds which is not sufficient for most real-time
applications. Karlsson et al. also used a CNN-based architecture to
train a regressor to predict speed from an accelerometer and the yaw
rate [18]. They also considered a large input time window (more than
2 seconds) and an architecture which is not able to establish memory
between samples. This work aims to address the shortcomings of
previous work and apply the trained model in an inertial localization
scheme.
Previous works on DR techniques emphasize the importance of
accurate speed updates to reduce the positioning error. Brossard et
al. proposed special speed updates when specific events are detected
using machine learning or other methods [19]. Key example is the
zero velocity update (ZUPT) when a full stop is detected. Other
examples include nullifying the vertical and horizontal components
of the velocity of a four wheeled vehicle when no slip is detected.
The detection of special events with respective corrections to the
velocity vector are used to nullify accumulated integration errors.
The approach proposed in this work is a generalization of this idea
because the trained model can learn all possible detectable events
with their respective speed updates to automatically reset accumulated
errors.
The rest of the paper is organized as follows: Section II defines
arXiv:2205.07883v2 [cs.LG] 23 Aug 2022

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the DR problem with speed as input, Section III presents the learning
approach, including a highly accurate data collection method and
DNN architecture. Section IV presents the model training results with
hold-out set evaluation and application as input for DR localization,
Section V gives the conclusions.
II. DEAD RECKONING WITH SPEED INPUT
For testing the proposed approach, a two-dimensional DR scheme
is used. The speed and the heading angle of the car (both with respect
to navigation frame) are used in a single integration step to obtain the
position. Consider the following two-dimensional linear kinematic
equation of motion for the car position
𝑝𝑛
𝑥
𝑝𝑛
𝑦[𝑘]=𝑝𝑛
𝑥
𝑝𝑛
𝑦[𝑘−1]+Δ𝑡𝑣𝑛
𝑥
𝑣𝑛
𝑦[𝑘](1)
where 𝑝𝑛
𝑖[𝑘]is the car position and 𝑣𝑛
𝑖[𝑘]is the car velocity in
the 𝑖∈{𝑥, 𝑦}axis, at time step 𝑘. The 𝑛denotes the navigation
coordinate frame and Δ𝑡is the time step size. The car velocity is
provided by
𝑣𝑛
𝑥
𝑣𝑛
𝑦[𝑘]=cos 𝜓
sin 𝜓[𝑘]𝑠𝑏[𝑘](2)
where 𝜓[𝑘]is the car heading angle and 𝑠𝑏[𝑘]is the car speed at
time step 𝑘in the body frame. The car heading angle 𝜓[𝑘]is obtained
using an attitude estimation algorithm [7] which also provides the
roll and pitch angles used to transform IMU measurements from the
body to navigation frame. The car speed 𝑠𝑏[𝑘]can be obtained in
two ways: (1) from the output of the trained DNN model and (2)
by direct integration of the noisy acceleration signal (in navigation
frame).
III. LEARNING APPROACH
A. Motivation
It is widely known that the direct integration of acceleration
measured with a low-cost IMU sensor to obtain speed (or velocity)
results in rapid divergence due to measurement noise [7]. This section
describes the development of a DNN-based car speed estimator with
noisy IMU measurements as input. A supervised learning approach
is presented where the IMU measurements are labeled with highly
accurate speed measurement, using the RTK-GNSS device, which
the DNN model is trained to predict.
B. Dataset collection and generation
Data was collected with a WitMotion BWT61CL IMU sensor [20]
and a STONEX RTK-GNSS device connected to a Windows PC
through serial port connections (Figure 1). Data was recorded at 50 𝐻𝑧
and 100 𝐻𝑧 rates with the RTK-GNSS and IMU sensors, respectively.
The sensors were synchronized to a single clock. Accurate ground
truth labels are critical in supervised learning tasks. The RTK-GNSS
STONEX device provides position measurements with a 2𝑐𝑚 average
positioning error in contrast to the common GNSS receiver which has
an average error on the order of meters [7]. A total of 180 minutes
of driving was recorded in Ashdod, Israel. The recordings were split
into train and validation sets with a ratio of 85:15. Figure 2 shows
the trajectories of the recorded drives for the train and validation sets
(without overlap). The data includes diverse driving with numerous
turns and roundabouts at speeds typical to city driving.
Speed was obtained from the position measurements by taking
the derivative with respect to time for each component and then
calculating the vector norm as shown in Figure 3. Then, the 50 Hz
speed signal was upsampled to match the 100 Hz rate of the IMU.
Train and validation samples were created by matching windows of
20 measurements of the 6 channel IMU with their respective window
of 20 samples of speed. Acceleration measurements were rotated to
navigation frame to remove the gravity component and then rotated
back to body frame.
Fig. 1. STONEX RTK-GNSS device (left) and WitMotion BWT61CL
IMU sensor (right).
Fig. 2. Car trajectory for the train and validation sets in an urban area
(Ashdod, Israel). No overlap between the train and validation data sets.
C. LSTM based-model architecture
A DNN architecture was selected comprising one LSTM layer,
two bi-directional LSTM layers, and a single dense layer. The model
contains a total of 12,860 trainable parameters. The model is of
Fig. 3. Car speed regressor training diagram. The IMU and GNSS-
RTK signals are input to the loss function where the first goes trough
the DNN and the later used to derive the car speed.

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type "many-to-many", i.e., each prediction expects 20 samples of a
6 channel IMU as input and returns 20 samples of speed prediction.
The architecture and input dimension are summarized in Figure 4.
The LSTM layers were configured to pass the hidden state between
samples to allow long term memory learning between predictions.
Fig. 4. DNN architecture with many-to-many input-output relation,
receives 20 samples of a 6-channel IMU as input and returns 20
samples of speed prediction.
D. Loss function and training
The following mean square error (MSE) loss function was
minimized in the training stage
L=1
2𝑁
𝑁

𝑗=1s𝐺𝑇
𝑗−ˆ
s𝑗(a, 𝜔;W)2
(3)
Here, ˆ
sis a function of a,𝜔, and the network weights W.aand 𝜔are
the accelerometer and gyroscope batch readings. The training data
was divided into 4 drives each about 40 minutes long. Each drive
was broken into batches ordered by acquisition time to allow the
model learn long term memory. Each training batch consisted of 4
windows of 6×20 IMU measurements (with the gravity component
subtracted) that followed the previous batch and passed forward the
LSTM hidden states. Shorter drives were padded with zeros (which
the loss function was modified to ignore) to match the drive time. At
the start of each epoch the LSTM hidden states were reset to zeros.
IV. RESULTS
A. Training results
The model was trained until convergence for 200 epochs. The
root MSE (RMSE) reached a value of 0.83 𝑚/𝑠on the train set
and 1.21 𝑚/𝑠on the validation set. Figure 5 shows a representative
example of speed prediction from the validation set. The above RMSE
on the validation set may be interpreted as a 12 meter positioning
error during a 10 𝑠drive. For a drive at an average speed of 15 𝑚/𝑠,
the car passes a total 150 ±12 𝑚. In this case the positioning error
is 8% of the driven distance.
B. Model application in aided DR navigation
The trained DNN model is tested on a two-dimensional navigation
problem using a hold-out dataset. A 4 minute drive in the Matam
high-tech park, Haifa was recorded using the same equipment and car.
The train/validation and the hold-out sets were recorded in relatively
flat urban areas. Position is obtained by integrating the car speed
after decomposition to velocity vector using the heading angle as
shown in Equations 1 and 2. For the integration, speed was obtained
in two ways: (1) using the trained model and (2) by direct integration
of the acceleration in navigation frame (used as reference).
Fig. 5. Estimated (DNN model) and true car speed vs. time for a
single drive from the validation set (RMSE = 0.90 𝑚/𝑠).
Figure 6 shows the speed of the car versus time obtained by direct
integration of acceleration (in navigation frame), DNN prediction,
and the ground truth (RTK-GNSS). The car starts stationary and the
DNN prediction takes several seconds to nullify despite starting with
hidden states set to zero. The first acceleration starts near 𝑡=10 𝑠
and the DR speed diverges rapidly from the ground truth. The DNN
speed follows the ground truth with good agreement until 𝑡=110 𝑠
(RMSE (0𝑠 < 𝑡 < 110𝑠)=1.08 𝑚/𝑠). At that point the road quality
was observed to be low and significant external vibrations were
introduced to the signal (RMSE (110𝑠 < 𝑡 < 150𝑠)=3.30 𝑚/𝑠).
At 𝑡=150 𝑠the car finished a roundabout and returned to the high
quality asphalt which resulted in better speed prediction accuracy
(RMSE (150𝑠 < 𝑡 < 233𝑠)=1.72 𝑚/𝑠).
Fig. 6. Car speed vs. time (test set): dead reckoning (DR), trained
DNN model, and ground truth (GT).
Figure 7 shows the car trajectory obtained using the two different
methods and the ground truth. The plain (without DNN aid) DR
fails to follow the car true trajectory at the drive start, which was
expected due to the speed error shown in Figure 6. At later time
steps the shape of the DR curve becomes more like the ground
truth. However, this is mainly because of the heading angle accuracy
which defines the shape but not the length of the segments. The
aided DR (DL-DNN aided) shows good agreement with the ground
truth and a substantial improvement over plain DR. The significant
deviation from true speed is observed near the roundabout located
at the southern point of the trajectory. Figure 8 shows the error
versus time for the two position solutions. The error is defined as

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the distance between the position solution and the ground truth. The
plain DR diverges rapidly over time while the DNN aided scheme
maintains less than 20 m error for more than a minute and less than
120 m during the whole 4 minute drive.
Fig. 7. Car trajectory calculated using plain and aided DR vs. ground
truth.
Fig. 8. Position error vs. time using plain and aided DR.
V. CONCLUSIONS
A deep neural network model was trained to estimate the speed
of a car from noisy IMU measurements. A new method for
collecting accurate speed measurements at high frequency was
presented using an RTK-GNSS device. The trained model is light
(less than 13,000 trainable parameters) and suitable for real-time
applications. A simplified aided dead reckoning localization scheme
was used to asses the performance of the trained model. The DNN
provided speed pseudo-measurements which substantially improved
the positioning accuracy over the direct integration approach. The
proposed methodology can be adapted and extended to other ground
vehicle and pedestrians as well as aerial and water vehicles. In future
work, more data will be collected to introduce additional driving
scenarios, rich terrain properties, and non-flat roads. Additionally,
real-time implementation for GNSS-free navigation is currently under
development.
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