Accurate vehicle position estimation using a Kalman filter and neural network-based approach

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Accurate Vehicle Position Estimation Using a
Kalman Filter and Neural Network-based Approach
Stanley Baek, Chang Liu, Paul Watta, and Yi Lu Murphey
Department of Electrical and Computer Engineering
University of Michigan - Dearborn
Dearborn, MI 48128
{stanbaek, liuchan, watta, yilu}@umich.edu
Abstract—Accurate detection of vehicle position plays an
important role in many intelligent transportation systems,
especially vehicle-to-vehicle applications. In this paper, we
propose an Extended Kalman Filter (EKF) based method to detect
Global Positioning System (GPS) errors for such vehicle-based
applications. A machine learning methodology is presented for
Kalman filter parameter tuning with application to GPS error
correction in vehicle positioning. We also present a model free
neural network that is trained on past vehicle GPS trajectories to
predict the current vehicle position. Experimental results on real-
world data show that the proposed system is effective for detecting
and reducing GPS errors. The machine learning algorithm for
EKF parameter tuning can be implemented through in-vehicle
learning, and the proposed GPS error detection method can be
implemented for in-vehicle applications.
Keywords—Kalman filter; Global Positioning System (GPS);
GPS error
；
online detection; neural network
I. INTRODUCTION
Accurate detection of vehicle position plays an important
role in many intelligent transportation systems (ITS), especially
in applications involving vehicle-to-vehicle (V2V), vehicle-to-
pedestrian (V2P) and vehicle-to-other vulnerable road users
(V2X), such as pre-crash detection, collision avoidance and
autonomous driving. V2V, V2P and V2X communications all
require a vehicle on-board communication unit, such as
dedicated short-range radio communication (DSRC) devices or
cellular networks, or other means of communication, to transmit
messages about a vehicle’s position, speed, heading, brake
status, and other information to other vehicles (and receive such
information, as well). According to the National Highway
Traffic Safety Administration, V2V applications have the
potential to address 80% of unimpaired crashes. However, most
GPS devices used in such V2V systems are low-cost grade, and
therefore, prone to errors resulting from a number of sources,
such as receiver error, satellite error, multi-path error, and
random error. Studies have shown that typical GPS errors range
between 1 - 10 meters [1, 17, 18], which is not acceptable in
applications related to driving safety systems.
GPS error is caused by a number of factors, including:
time delays (troposphere and ionosphere), orbital errors due to
inaccuracies in satellite position, multipath signal errors
resulting from the GPS signal reflecting off structures in the
environment, such as buildings and other vehicles, and timing
errors due to the receiver clock inaccuracies [1, 3, 17].
A large amount of research has been conducted in correcting
GPS errors. One type of approach is to use other sensors in the
vehicle or roadside units as reference to estimate vehicle true
position and compare it with the GPS measured position [17,
19]. In [17], Bhawiyuga, Nguyen, and Jeong presented a
vehicular positioning accuracy enhancement (VITAE)
algorithm that reduces the vehicle position error by taking into
account GPS estimates, as well as estimates of nearby vehicles
using a vehicular radar sensor. They first used a greedy data-
association (GDA) method to find a minimal-weight matching
between a GPS estimate and a sensing estimate based on a
Euclidian distance. The algorithm then constructs two polygons
of position estimates: a GPS polygon and a radar sensing
polygon. The GPS polygon is constructed by connecting the
GPS measures for remote vehicles provided by V2V
communications. The sensing polygon is constructed by
connecting the relative locations of remote vehicles provided by
the radar sensor mounted on the host vehicle. The position of the
host vehicle is adjusted by the difference between the mass
center of the GPS polygon and that of the sensing polygon.
The most popular approach to correcting GPS errors is to
formulate a system model of the vehicle dynamics, which
involves not only GPS position, but also velocity and heading,
and then use a Kalman Filter (KF) to smooth the trajectory [1, 6,
8, 9, 11, 12, 15, 16].
Xu et al used an improved Kalman filter approach to correct
GPS errors in map-matching a vehicle location to the underlying
road map [1]. Their method consists of a Kalman filter with a
novel method to minimize the biased error of GPS after the
vehicles makes a turn. The authors used real-world driving trips
to evaluate their method. The result shows that it handles the
biased error and the random error of the GPS signals reasonably
well in both the along-road and cross-road directions. The
authors showed that the raw GPS data along-track have a mean
error of 21 meters, and the corrected GPS data along the same
track have a mean error of 1.6 m.
Considerable research has been conducted on selecting
proper Kalman Filter tuning parameters. Although the Kalman
filter and the Extended Kalman Filter (EKF) are well established
techniques for state estimation, the choice of the filter tuning
parameters still poses a major challenge [11, 12, 16]. Saha et al.
recently presented two new metrics for determining the filter
tuning parameters on the basis of the innovation covariance [11].
Among all the tuning parameters, they identified the tuning of
the process noise covariance matrix Q as being the most critical,

2
since all the model uncertainties and inaccuracies as well as the
noise affecting the process are incorporated quantitatively into
Q. They presented a metric based offline method usable for
predicting the actual filter RMSE performance for a particular
application. The metric can be used to find the proper
combination of filter tuning parameters.
In this paper we present a machine learning methodology for
Kalman filter parameter tuning with application to GPS error
correction in vehicle positioning, and a model free neural
network that is trained on past vehicle GPS trajectories to predict
the current vehicle position. This paper is organized as follows.
Section II gives an overview of the EKF approach for detecting
vehicle positioning error. Section III presents the machine
learning approaches we proposed for vehicle positioning error
detection. Section IV presents experimental results conducted
on real world driving trips and Section V concludes the paper.
II. OVERVIEW OF THE EXTENDED KALMAN FILTER
The displacement of a ground vehicle during the duration of
time, , can be written as




where  and  are, respectively, the lateral and longitudinal
displacement of the vehicle,  is the speed of the vehicle,  is
the heading of the vehicle, and  is the constant acceleration of
the vehicle during , the sampling step size. In the discrete-
time domain with  and , we can
rewrite these equations as


(1)


(2)

(3)

(4)
where  and  are, respectively, the lateral and longitudinal
position of the vehicle at time ,  and  are,
respectively, the lateral and longitudinal predicted position of
the vehicle at time  based on the position (, ), speed
, acceleration, heading , and angular velocity  of the
vehicle at time .
The state variables of interest in this work are the position,
heading, and speed of the vehicle. Thus, the state vector at time
 is defined by 
For the optimal estimate of the states, we have formulated an
extended Kalman filter as follows


where  is the state transition matrix at time ,  is the
observation matrix,  is the sensor measurements,
 is a zero mean Gaussian random vector with
covariance , representing system uncertainty, and
 is a zero mean Gaussian random vector with
covariance , and which represents sensor noise The state
transition matrix  can be obtained from (1) - (4) and is given
by
  
  
   
   
The acceleration and angular velocity  of the system
in (1) - (4) are unknown, and therefore we consider them as
unmodeled system parameters (or model uncertainty). Thus, the
system uncertainty  can be defined by



















where  and  are uncorrelated zero mean Gaussian random
variables with variance of  and 
, respectively. The system
uncertainty covariance  can be formulated by

where  denotes expectation of a random variable, and the
elements of  are shown below:














Since we directly measure the state  contaminated with
sensor noise, the observation matrix  becomes the identity
matrix independent of time, i.e.,  . The
measurement noise covariance matrix is defined by

3






  
 
 
  
   






(5)
where , 
,  
 are the variances of sensor noise of the
lateral position, longitudinal position, speed, and heading,
respectively.
With this formulation, the Kalman prediction and update
steps are 



 

where  is the estimate of 
given the observations , the quantity  
is the estimate of  given the
observations  the matrix 
 is the a posteriori covariance
matrix (a measure of uncertainty of state estimate
 is the a priori
covariance matrix (a measure of uncertainty of state estimate
, and  is the optimal Kalman gain.
The most arduous step in the implementation of a Kalman
filter is finding the elements in the  and  matrices, which are
the most influential in the estimation performance. The
variances of sensor noise in (5) can be measured from GPS and
an inertial measurement unit (IMU) whereas the variances of
system noise ( 
 and 
 in the  matrix are not easy to
determine. In the next section, we develop a robust method to
compute these tuning parameters to correct GPS errors.
III. MACHINE LEARNING APPROACHES FOR GPS ERROR
DETECTION
In this paper, we assume that the following vehicle data are
available:
: Latitude : Longitude : Vehicle speed
: Vehicle acceleration : Vehicle heading
These measures are commonly available on GPS and On-board
Diagnostics (OBD) devices. Note that these are time-based
signals with a sampling period of . Angular velocity can be
computed from the vehicle heading as follows:


The UTM coordinate system was used to covert latitude and
longitude (in degrees) to Cartesian coordinates:  and 
(in meters).
The EKF outlined in the previous section requires that we
choose two important parameters: angular velocity variance 
,
and acceleration variance: . The most straightforward
approach to computing the needed parameters is to use an
available data set of recorded trips and simply compute each
using the standard variance formula; for example:




 (6)
where  are the recorded angular velocities. Note
that angular velocity is a zero mean signal, so the  term can be
dropped.
Fig. 1. Calculation of 
. The histogram shown in red represents normalized
count values:  of the measured angular velocity, while the blue curve
shows the best fit Gassian, with samples: . To determine the best fit
Gaussian, we minimize  or 
 over . A similar historam and
optimization problem are formulated to determine the variance of acceleration:
 , as well.
We propose a machine learning approach that is based on
tuning parameters to get a best-fit Gaussian curve to the data.
The algorithm consists of the following steps. First, we compute
a histogram of the tuning parameter values from training trip
data. Figure 1 shows such a (normalized) histogram for the
angular velocity (in red). Let  represent the normalized
count (y-axis) from the histogram for the given value of .
Next, we want to find the best fitting Gaussian curve, of the
form: 



(7)
For a given value of  and , we can compute the
corresponding value of the Gaussian curve: . Hence, we
can formulate the following optimization problem:
Minimize 
s.t. 
where the objective function  is the sum-squared error
between the f and h samples:

(8)
In the next section, we will show that the following weighted
objective function yields slightly better results:

4


 (9)
This objective function puts more weight on fitting the tails of
the distribution.
Similar fitting functions: , , and objective
functions: , and  were formulated to compute the
variance of acceleration , as well.
We propose a model free neural network-based method for
vehicle position error detection based on the past trajectory of
vehicle positions (measured GPS coordinates). We used a
multi-layer perceptron (MLP) with architecture shown in Figure
2. We found that this simple structure with one hidden layer was
sufficiently powerful for the present error detection problem.
The inputs to the neural network are measurements of four
variables: vehicle position, speed, and heading, over the time
period: , where w is a parameter which
specifies the size of the temporal sliding window. The outputs of
the network:  and  occur at time  and represent the
amount of error in position. A data set of inputs and target
outputs is needed to train the neural network.
Fig. 2. Architecture of GPS error-correcting neural network.
IV. EXPERIMENTS
To evaluate the effectiveness of the proposed approaches,
we used real GPS data collected over several trips, each about
50 km long in Dearborn, MI, and contains both city and highway
driving. Figure 3 shows a map of the route (a roundtrip) taken
by the driver. GPS data was collected from 17 such trips with 5
different drivers. The vehicle position, velocity, heading, and
acceleration were collected at a sampling rate of 100 Hz.
We are interested in how the parameter values and prediction
accuracy are affected by road/traffic conditions. From the entire
trip, 5 different segments:  through  were identified as
having interesting traffic characteristics. The 5 segments are also
shown in Figure 3, and the associated traffic characteristics are
listed below:
: Intersection and left turn
: Straight section
: Curve
: Stop sign and U-turn 
: Slight turn
Fig. 3. The 50km route (roundtrip) taken by each driver. Five different
segments of the trip are indicated.
In the following subsections, we will corrupt the recorded
GPS data with various amounts of additive white Gaussian
noise. Figure 4 shows a short segment of the trip along with the
original GPS data (in red) and then with added noise set to 0.5m,
1.5m, and 2.5m.
Fig. 4. Illustration of the trip data along with added noise. (a) Original GPS
data. (b) The trip with 0.5m additive white Gassuian noise (c) 1.5m noise and
(d) 2.5 m noise.
x(t– 1)
x(t–w)
y(t– 1)
y(t–w)
v(t– 1)
v(t–w)
θ(t– 1)
θ(t–w)
E(t)
1
2
J
x
E(t)
y
(a)
(b)
(c)
(d)

5
A. Computation of Kalman Filter Parameters
To study the computation of the parameters  and 
needed by the Kalman filter, we performed a preliminary
experiment of computing both parameters using the data from
Trip #1. Three different methods were used to compute the
variances: equation 6, and the two curve fitting methods outlined
in (7) – (9). To test the performance of each variance setting,
Trips #1 - #3 were corrupted with 1.5m GPS noise, and the
Kalman filter was applied to the noisy trip data. A mean-squared
error measure was used to assess performance:




where   is a ground truth position vector
taken from the original (noiseless) positions and  is the
corresponsing position vector after the Kalman filter has been
applied to the noisy signal.
Figure 5 shows a plot of the two curve fitting error functions:
 or 
 Also shown are the minimizers of the
respective functions: 
 and 
 which give the best
Gaussian fit.
Fig. 5. A plot of the curve fitting errors: , and 
 as a function of
. In each case, we choose the value of  which minimizes the error.
The three values of : (i) the one computed from Equation
5, (ii) 
 and (iii)  were used to filter the noisy
signal. Table I shows the mean-squared error tested over Trips
#1 ˗ #3. The best performance was with the curve fitting
approach using the weighted objective function. In the
experiments that follow, this is the method that will be used to
compute  and .
The variance values, of course, depend on the data set used
to compute them. Figure 6 shows a scatter plot of  vs. 
computed over all 12 trips (shown in blue).
TABLE I. ERROR OF KALMAN FILTERED SIGNAL TESTED ON 3 TRIPS AND
WITH 3 DIFFERENT SETTINGS OF THE VARIANCE PARAMETERS.
Fig. 6. A scatter pot of  vs. from the first 7 trips (shown in blue). The
average values:  and  (averaged sigmas over the first 7 trips) are shown in
red. The sigmas resulting from using the data from all 7 trips: 
 and  are
shown in green.
In light of Figure 6, there are two different ways to obtain a
reasonable estimate of the variance. The first is to compute the
center of mass of individual 
 and  from the various trips
This will be denoted by 


 and 


and shown in red in Figure 6. The second method is to compute
the variance of measured data (including all trips) which will be
denoted by 
 and  and they are shown in green in Figure
6.
B. Evaluation of Performance on Fixed Route
In the following experiments, data from the first 12 trips
were used to compute the parameters:  and . Several
different ways of computing these statistical measures were
explored. In the first approach, the standard deviations were
computed over each trip separately: 
 and  , .
With these values, the Kalman filter was used to filter the noisy
data on the corresponding trip data with 3 levels of additive
white noise: 0.5m, 1.5m, and 2.5m. So, for example, 
 and 
were used to filter the noisy Trip #1 data. The results are shown
in Figure 7.
Next, to determine how robust the standard deviation is, we
used 
 and  and evaluated the Kalman filter on all the other
11 noisy trips (2 through 12). The process was repeated for 

and  through 
 and . The results are shown in Figure 8.
Kalman Filter
Error (m)
Trip 1
Trip 2
Trip 3
Variance Formula
1.4003
1.2195
1.2387
Curve Fitting
1.3370
1.1598
1.2047
Weighted Curve Fit
1.3115
0.9942
1.1108




6
Note that the results here represent the average error over the 11
other trips.
Next, we tested system performance using the average
parameters:  and . The results of this experiment are shown
in Figure 9.
Fig. 7. 
 and were computed on trip i and evelauted on trip i.
Fig. 8. 
 and  were computed on trip i and evelauted against all 11 other
trips.
Fig. 9.  and  were computed by averaging the 12 individual  and 
and evaluated on all 12 trips.
Finally, we tested system performance using the parameters
fitted by all 12 trip data: 
 and . The results are shown in
Figure 10. A summary of the average errors (averaged over all
the test trips) for each approach is given in Table II.
Fig. 10. 
 and  were computed by using all of the data in the 12 trips.
and evaluated on all 12 trips.
TABLE II. SUMMARY OF KALMAN FILTER ERROR.
Average
Kalman Filter Error
(m)
Input Noise
0.5m
Input Noise
1.5m
Input Noise
2.5m
 and  computed
and tested on same trip
0.4012
1.2218
2.1612
 and  computed
on one trip and tested
on all 11 others
0.4097
1.2452
2.2739

0.4090
1.2301
2.1640


0.4088
1.2293
2.1627
From the results of this set of experiments we observe the
following properties:
(1) the EKF using the parameters fitted on the same trip data
as the test data gives the best performance. However, this can
only serve as a measure of performance upper bound. It cannot
be used for online error prediction.
(2) The performances given by the EKF that uses 
 and
 and the EKF that uses  and  are very close. However,
unlike 
 and   and  can be updated when new trip
data are available for learning without referring to the previous
training data. Therefore, we recommend use of  and.
Next, we investigate how road/traffic features affect system
performance. Figure 11 shows GPS data for several trips (with
different drivers) over segments  and . The variance
parameters were computed on each segment of the 12 trips and
then tested on the same 12 trips. Table III shows a comparison
of the average Kalman filter error. The results in the table are in

7
the format: Error from using 
 and  / error from using
 and . From the table, we see that the two methods give
very similar values. This is important because it indicates that it
is possible to use our algorithm in an online learning mode. That
is,  and  can be continually updated online, as more trips
are made by the driver.
(a)
(b)
Fig. 11. Illustration of a few trips over segments: (a)  (b) .
TABLE III. KALMAN FILTER ERROR USING TWO DIFFERENT SETTINGS OF
VARIANCE PARAMETERS: 
 AND  /  AND 
KF Error
(m)
Input Noise
0.5m
Input Noise
1.5m
Input Noise
2.5m
S1
0.2536 / 0.2587
0.6281 / 0.6456
1.7809 / 1.7845
S2
0.2190 / 0.2165
0.5892 / 0.5859
1.3315 / 1.3278
S3
0.3981 / 0.4007
1.1244 / 1.1287
2.0993 / 2.1019
S4
0.4136 / 0.4029
1.2331 / 1.2245
2.1898 / 2.1697
S5
0.5009 / 0.4891
1.4895 / 1.4710
2.4076 / 2.4031
C. Comparison of EKF and Neural Network
A neural network with 10 hidden nodes and architecture
shown in Figure 2 was trained using data from the first 12 trips
and tested the other 5 trips. Figure 12 shows the error of the
neural network approach, as well as the two EKF methods with
two parameters settings: 
 and  , and  and .
Fig. 12. Performance comparisons between the EKF-based approach and the
neural network-based approach using five test trips.
The neural network gives the best performance. The two
EKF based methods show almost identical performance.
However, we want to point out that the neural network requires
supervised learning (with known target values), which is not
easy to implement for in-vehicle applications.
D. Testing on a New Route
In the above experiments, the system was trained and tested
on data collected over a fixed route, but with multiple drivers. In
this section, we utilize the same Kalman filter parameters as
computed in Section IV.B, but we test on an entirely new route.
The route was local driving for approximately 9 miles and 12
minutes as shown in Figure 13. Two data acquisition systems
were used. The first provided high quality GPS measurements
with 100 Hz sampling rate. The signals captured with this device
were considered ground truth for the trip. The second was a low
quality device with a sampling rate of 1 Hz. The signals in this
device were considered the noisy input signal that was sent to
the Kalman filter.
Fig. 13. A new route consisting of 9 miles of city driving and approximately 12
minutes.
Fig. 14. A plot of Kalman filter error vs. time when tested on the new route.
A plot of the Kalman filter estimation error as a function of
time is shown in Figure 14. Notice that there is a large amount
of error at the beginning of the trip, but then after some time, the
Kalman filter is very effective in removing noise. Figure 15
shows what was occurring at the beginning of the trip: the driver
was exiting a parking lot and making several starts and stops.
Part1
Part2

8
Figure 16 show a later portion of the trip where the Kalman filter
provides effective error correction even with a low cost sensor.
Fig. 15. At the beginning of the trip, the driver was in a parking lot and the input
signal (red dots) is very noisy. The blue line is the ground truth.
Fig. 16. A section of the trip which shows the the effect of Kalman filtering.
The red dots are the noisy signal, the green dots are the Kalman-filtered signal,
and the blue line is the ground truth.
V. CONCLUSION
In this paper, we proposed several methods for improving
vehicle position estimation. A curve fitting approach was
proposed for tuning important Kalman filter parameters. A
neural network was proposed for estimating position errors. The
results show that the neural network can give slightly better
results that the Kalman filter. However, the neural network
requires a training set containing ground truth, which makes it
less appealing for in-vehicle applications. The Kalman filter
approach is more amenable to an online and in-vehicle
implementation, where the tuning parameters can be continually
updated as new data is obtained. In future work, we will apply
these approaches to larger data sets, consisting of many drivers
and different driving scenarios. In addition, we will explore its
use as a component in other intelligent vehicle systems design
problems, such as autonomous vehicle design.
ACKNOWLEDGMENT
This work is support in part by grants from Mobility
Transformation Center and Michigan Institution for Data
Science at the University of Michigan.
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