This paper presents a sensor fusion algorithm to track pedestrians accurately using measurements from LiDAR and radar sensor. The formulation of standard kalman filter, extended kalman filter and unscented kalman filter is discussed in detail. The objective of this paper is to show the importance of using multiple sensors for estimation purposes in autonomous vehicles. The ability of EKF and UKF to accurately estimate the nonlinear behavior of the pedestrian is studied by com- paring the accuracy of predicted path vs ground truth using performance metrics.

1. Tracking Pedestrians using Extended and Unscented Kalman Filters in
a Sensor Fusion Framework
Kaustav Mondal
Abstract— This paper presents a sensor fusion algorithm to
track pedestrians accurately using measurements from LiDAR
and radar sensor. The formulation of standard kalman filter,
extended kalman filter and unscented kalman filter is discussed
in detail. The objective of this paper is to show the importance
of using multiple sensors for estimation purposes in autonomous
vehicles. The ability of EKF and UKF to accurately estimate
the nonlinear behavior of the pedestrian is studied by com-
paring the accuracy of predicted path vs ground truth using
performance metrics.
I. INTRODUCTION TO SENSOR FUSION
One of the main functions of the sensor fusion module
of an autonomous car is to track it’s environment. The
environment could consist of cars, pedestrians walking on
the road, bikes or even landmarks. In this paper, we have
used sensor fusion techniques to track a pedestrian riding
a bike. Two main sensors have been used in this project
i.e. LiDAR and Radar. Since sensor fusion requires fast
processing time, C++ has been used to program the algorithm
since it’s a high performance language. In this project,
we have used measurement data from a Velodyne LiDAR
and Radar sensor mounted on a prototype autonomous car
designed by Mercedez Benz. The car has 15 sensors mounted
on it. At the front, a stereo camera is installed which gives
image data and distance information. There is a separate
camera installed which has a wide view lens. This is to
ensure that the traffic signals can be detected from far away.
Radar sensors have been installed on the front bumper of
the car. Radars are generally used for adaptive cruise control,
blind spot warning, collision avoidance and collision control.
Radars use Doppler effect to measure the velocity of a
tracked object. The Doppler effect measures the change in
frequency of the radar waves based on relative velocity of the
object w.r.t. the vehicle. Doppler effect is advantageous for
sensor fusion because it calculates velocity as an independent
measurement. Radars can also be used for localization based
on radar waves bouncing off hard surfaces around the car.
It can provide measurements of the distance of the objects
which may not be present in the direct line of sight. Of
all the sensors on the car, the radar is least affected by
rain or fog. It has a wide field of view (eg. about 150◦
)
and a range of 200 meters or more. Compared to LiDARs
and cameras, radars have a low resolution. This can be
particularly problematic when we consider reflection from
the surface of static objects. For example, a man hole cover or
a soda can lying on the road will have high reflectivity even
This work has been done as part of Udacity’s Self Driving Car Nanode-
gree (Term 2), 2017
though they are relatively small. This is called Radar clutter.
This is why current automotive radars usually disregard static
objects.
LiDAR stand for light, detection and ranging. It uses an
infrared laser beam to measure the distance between the
sensor and the object. Most current LiDARs use light in the
900 nanometer wavelength range. There are some LiDARs
which use a longer wavelenth which performs better in rain
and fog. LiDARs have a much greater spatial resolution
than the Radar because of a more focussed laser beam,
larger number of scan layers in the vertical direction and
a high density of LiDAR points per layer. The current
LiDARs used in the automotive industry can’t measure the
velocity of an object independently. They have to rely on
the changing positions over every time step between two
or more scans. LiDAR measurements are more affected by
weather conditions as compared to Radars. It’s measurements
may be erroneous if dirt settles on it. It is required to keep
the LiDARs clean. The prototype car used Velodyne’s new
PUCKT M
(VLP-16) sensor which is the smallest, newest
and most advanced product in Velodyne’s 3D LiDAR product
range.
Velodyne’s PUCKT M
(VLP-16) sensor key features. The
key features of the LiDAR are given in Table I [1].
Key Features
Nominal
Value
Cost 7999 $
Mass 0.84 kg
Number of channels 16
Range 100 m
Spatial Resolution 300,000 points/sec
Field of view(horizontal/azimuth) 360◦
Field of view(Vertical) 30◦(+15◦ to -15◦)
Accuracy ± 3 cm (typical)
Angular resolution(vertical) 2◦
Angular resolution(horizontal/azimuth) 0.1◦ - 0.4◦
Rotation Range 5-20 hz
TABLE I
PUCKT M (VLP-16) KEY FEATURES AND CHARACTERISTICS
LiDARs and radars have strengths and weaknesses of their
own. By combining the data from both of the sensors, we
can precisely estimate the pedestrian location, heading and
speed. Every time we receive a new measurement from
one of the sensors, the estimation function is triggered.
The estimation function consists of two steps. In the
prediction step, we predict the pedestrian’s state( position,
velocity) and covariance while taking into account the
elapsed time ∆t between current and previous observations.

2. The measurement update step depends on the type of
measurement received at every time step. If the current
measurement is a LiDAR measurement, then we use
the standard kalman filter to update the pedestrian’s
state. However, if the current measurement is a radar
measurement, then we use the Extended Kalman Filter/
Unscented Kalman Filter to update the state. This is because
the radar measurement model is nonlinear in nature. A
standard kalman filter works well only in the case of linear
models. The sensor flow flowshart is illustrated in Fig 1.
Fig. 1. Overview of sensor fusion algorithm
Initially the filter would receive the pedestrian’s initial posi-
tion relative to the car. This measurement could be from a
laser or radar. The state and covariance of the process model
will be initialized by the filter based on the first measurement.
After the filter receives the 2nd measurement at time ∆t, the
algorithm predicts the process model’s state and covariance.
For EKF, we have used a constant velocity model so the
velocity of the pedestrian riding a bicycle is assumed to be
constant during the time period ∆t. Based on the constant
velocity model, the filter predicts that the bicycle has moved
by a distance equivalent to v∆t where v is the velocity of the
pedestrian. In the update step, the filter uses the measurement
from sensor and the predicted state to give a better estimate
of the pedestrian’s location and velocity. This is followed by
another cycle of prediction and update. Hence the kalman
filter is a recursive algorithm. Since the LiDAR uses cartesian
coordinates and the radar uses polar coordinates, we use
different measurement models in the update step, based on
which sensor measurement is received at that particular time
step. If the measurements from LiDAR and radar are received
simultaneously, we just run one prediction step followed by
two update steps instead of running two prediction steps for
each measurement.
II. KALMAN FILTER
In a standard Kalman filter, there are two main steps:
prediction and update. Prediction: In this step, we assume
that we know the current position and velocity of the object.
Assuming that the velocity remains constant, we can predict
the state of the object after a time period ∆t. The equation
x = Fx + Bu + ν represents the predicted state of the
object. But it is very possible that the object didn’t maintain
constant velocity or may be it changed direction, accelerated
or decelerated. This leads to an increase in uncertainty. The
equation P = FPF + Q represents this increase in uncer-
tainty. The process noise is represented by ν. The notation
ν ∼ N(0, Q) represented the process noise as a gaussian
distribution with a mean zero and covariance Q. Please note
that Bu in the prediction step represent the control input
where B is the control input matrix and u is the control
vector. It is to be noted that the autonomous car has no way
of measuring the exact acceleration of the tracked object (
bicycle ) since we cannot model the internal forces of the
object. Hence we will set Bu = 0 and represent acceleration
as a random noise with mean ν. Update: In this step, we use
the sensor data to calculate a better estimate of the object’s
position and velocity. The equation y = z − Hx calculates
the error in the measurement where H is the measurement
matrix, z is the true sensor measurement and ˆz = Hx
is the predicted observation. This equation compares the
measurement data to the predicted position measurement of
the tracked object. The kalman filter gain K is represented
by the equation K = PH (HPH + R)−1
. The kalman
gain combines the uncertainty of where we think the object
is (P ) with the uncertainty of our sensor measurement R.
If the sensor measurements are very uncertain (R is high
relative to P ), then the Kalman gain will give more weight
to where we think we are i.e. x . If the predicted state has a
high uncertainty( P is high relative to R), the kalman gain
will put more weight on the sensor measurement, z. The
notation ω ∼ N(0, R) represents the measurement noise as
a gaussian distribution with mean zero and covariance R.
The sensor measurement uncertainty matrix R is provided by
the manufacturer of the sensor. After calculating the kalman
gain, we update the state vector and the state covariance
matrix. The equations xnew = x + Ky and Pnew =
(I − KH)P represent the new estimated state vector and
state covariance matrix respectively.
Process Model: The state of the moving pedestrian is
represented by position (px,py) and velocity(vx,vy) w.r.t. to
the sensor. The state vector and linear motion model are
defined as follows
x =




px
py
vx
vy







px = px + vx∆t + νpx
py = py + vy∆t + νpy
vx = vx + νvx
vy = vy + νvy
(1)
The linear motion model is expressed in matrix form as




px
py
vx
vy



 =




1 0 ∆t 0
0 1 0 ∆t
0 0 1 0
0 0 0 1








px
py
vx
vy



 +




νpx
νpy
νvx
νvy



 (2)
The model of the bicycle assumes that the velocity is constant
between the time intervals. Hence it is called a constant
velocity model [34]. But in reality, the bicycle’s velocity
may change due to acceleration. Hence this un-modeled
acceleration introduces an uncertainty in the form of a
process noise. Considering the acceleration as uncertain, the
state transition equations can be rewritten as

3. 



px
py
vx
vy



 =




1 0 ∆t 0
0 1 0 ∆t
0 0 1 0
0 0 0 1








px
py
vx
vy



 +




1
2 ax∆t2
1
2 ay∆t2
ax∆t
ay∆t



 (3)
The process noise vector can be redefined as
ν =




νpx
νpy
νvx
νvy



 =




1
2 ax∆t2
1
2 ay∆t2
ax∆t
ay∆t



 . =




∆t2
2 0
0 ∆t2
2
∆t 0
0 ∆t




ax
ay
= Ga
(4)
The random acceleration vector is defined as ν ∼ N(0, Q).
The acceleration in the x and y directions can be defined as
ax ∼ N(0, σ2
ax) and ay ∼ N(0, σ2
ay) respectively. The cross-
correlation between ax and ay is assumed to be zero. But in
real-life situations, this is not considered to be zero. In the
equation of ν, the matrix Q is the measurement covariance
matrix. It is defined as Q = E(ννT
) = GQvGT
where Qv =
σ2
ax σaxy
σaxy σ2
ay
. The noise processes ax and ay are assumed
to be uncorrelated. Hence σaxy = 0. The final expression of
the process covariance matrix Q is
Q =





∆t4
4 σ2
ax 0 ∆t3
2 σ2
ax 0
0 ∆t4
4 σ2
ay 0 ∆t3
2 σ2
ay
∆t3
2 σ2
ax 0 ∆t2
σ2
ax 0
0 ∆t3
2 σ2
ay 0 ∆t2
σ2
ay





(5)
Laser Measurement: In this section, we describe the
measurement model w.r.t. LiDAR. There are 3 main com-
ponents of sensor measurement, the measurement vector z,
the measurement matrix H and the measurement covariance
matrix R. The LiDAR sensor output is a point cloud, but
the data which was provided to us was already analyzed and
we were provided the 2D location of the pedestrian at every
time period. For a LiDAR sensor, we can only measure the
x position and y position of the tracked object relative to
the sensor. Hence the measurement vector z can be defined
as z = px py
T
. Given that our state is a 4D vector
x = px py vx vy
T
, the measurement matrix H is
defined as follows
px
py
= H




px
py
vx
vy



 + ω, H =
1 0 0 0
0 1 0 0
(6)
The H matrix projects the belief of the tracked object’s
current 4D state into the 2D measurement space of the
sensor. The measurement covariance matrix R represents
the uncertainty present in the sensor measurements. Since
LiDAR measurements are 2D in nature, the noise vector ω
is 2D as well. The measurement covariance matrix R can be
defined as
R = E(ωωT
) =
σ2
px 0
0 σ2
py
(7)
The parameters of the measurement covariance matrix R is
provided by the manufacturer. It is to be noted that the noise
processes for the sensor are assumed to be uncorrelated.
Radar measurement: Although the LiDAR sensor mea-
sures the relative distance of the tracked object with high
accuracy, it cannot measure the relative velocity of the object
w.r.t the sensor. This is where the radar is advantageous. The
radar measures the radial velocity of the moving object w.r.t
sensor using Doppler effect. The Doppler effect measures the
change in frequency of the radar waves based on the relative
movement of the object w.r.t sensor.
Fig. 2. Overview of Radar Measurement
As shown in Fig 2, the x axis always points in vehicle’s
direction of motion and the y axis points to the left. Instead
of measuring the 2D pose of the object, the radar measures
• Range(ρ), radial distance from origin.
• Bearing(φ), angle between ρ and x.
• Radial Velocity( ˙ρ), rate of change of ρ.
The measurement function for the radar is defined as z =
h(x ) + ω where z = ρ φ ˙ρ
T
and ω ∼ N(0, R) is
the measurement noise. Compared to the LiDAR, there is
no H matrix which will directly map the state vector x to
the polar measurement space of the radar. To convert the
cartesian coordinates to polar coordinates, we will use the
following nonlinear function h(x )


ρ
φ
˙ρ

 h(x )
←−−−




px
py
vx
vy



 , h(x ) =





p 2
x + p 2
y
tan−1
(
py
px
)
pxvx+pyvy
p 2
x +p 2
y





(8)
Assuming that the 3 measurement components are not cross-
correlated, the radar measurement covariance matrix R is
represented as
R =


σ2
ρ 0 0
0 σ2
φ 0
0 0 σ2
˙φ

 (9)
Extended Kalman Filter: In the case of LiDAR measure-
ment, our measurement function was a linear function. Hence
assuming state x to be a Gaussian distribution, the measure-
ment vector z is also a Gaussian distribution. But in the case
of the radar, the measurement function is nonlinear and the
output of a Gaussian distribution passed through a nonlinear
function isn’t Gaussian anymore. Hence in the case of radar,
the standard Kalman Filter update equations aren’t applicable
anymore since Kalman Filter is inherently a Gaussian filter.
In order to fix this problem, a possible solution would be to

4. linearize the nonlinear function h(x )about the mean location
µ which is the best estimate of our predicted distribution.
Using a first order Taylor distribution, we linearize the
function h(x ) as
h(x ) ≈ h(µ) +
δh(µ)
δx
(x − µ) (10)
Similarly the state transition function f(x) may also be
linearized in a similar way if it is nonlinear. By generalizing
this to a higher dimension, we would have to calculate a
4x4 jacobian matrix instead of δh(µ)
δx . For the case of radar
measurement the Jacobian matrix is as follows
Hj =




px√
p2
x+p2
y
py
√
p2
x+p2
y
0 0
−
py
p2
x+p2
y
px
p2
x+p2
y
0 0
py(vxpy−vypx)
(p2
x+p2
y)
3
2
px(vypx−vxpy)
(p2
x+p2
y)
3
2
px√
p2
x+p2
y
py
√
p2
x+p2
y




(11)
The standard Kalman Filter and the Extended Kalman Filter
filter equations are similar. The main differences are
• F matrix is replaced by jacobian F when calculating
P .
• H matrix in the Kalman Filter is replaced by Jacobian
Hj while calculating S,K and P.
Fig. 3. Comparison : Kalman Filter vs EKF
The difference between regular and extended Kalman filter
equations is explained in Fig 3. For this project, we are
using a linear model for the prediction step. Hence we
don’t need to use the jacobian Fj. For our project, since
we have considered a constant velocity model, we use the
standard Kalman filter equations for the prediction step. The
measurement update for LiDAR also uses standard Kalman
filter update equations since LiDAR measurement model is
linear. We use the H matrix for calculating y, S, K and P.
However as explained before, the measurement update for
the radar would require extended Kalman filter equations.
The processing flow of the EKF is summarized in Fig 4.
Summarizing the processing flow of the extended Kalman
Filter, here are a few points.
• The state of the moving pedestrian is represented by a
2D position and 2D velocity. The EKF measurements
are initialized using the data provided by the manufac-
turer.
• Each time we receive a new measurement, the process
measurement module is triggered. On the first iteration,
we initialize the state vector x and the process covari-
ance matrix Q. This is followed by the prediction and
measurement update.
• Before predicting the state and covariance matrix, we
compute the elapsed time between previous and current
observation. We use elapsed time ∆t to calculate the
new state transition matrix F and process covariance
matrix Q. Then we predict the state and covariance of
the process model.
• The measurement update step depends on the sensor
type. If the current observation comes from a radar
sensor, then we linearize h(x ). We compute the new
Jacobian matrix Hj, use the measurement function to
project the predicted state and then call the measurement
update.
• If the current observation comes from a LiDAR, we use
the H matrix instead of linearizing the measurement
function h(x ). We set up the measurement covariance
matrix for LiDAR and then call the measurement up-
date.
Fig. 4. Process flow of Extended Kalman Filter
Root mean square error: In order to check how far the
estimated state are from the true states, we use a metric
called RMSE. In case of tracking, the RMSE gives a measure
of the deviation of the estimated state from the true state.
At a given time, we have two states, xest
t (estimated state)
and xtrue
t (ground truth state). The difference between the
estimated state and the true state is called the residual. The
residual is defined as (xest
t −xtrue
t ). This residual is squared
and averaged over the entire time period and it’s square root
gives the error metric. The lower the error, the higher is the
estimation accuracy. The RMSE is expressed as
RMSE = 1
n
n
n=1(xest
t − xtrue
t )2
EKF Results: Two measurement datasets are used for the
estimation process. Both datasets have a collection of LiDAR
and Radar measurements at each time interval. To give a
sense of what the measurements look like, Figure 5 is a
snapshot of the first 6 measurements of one of the .txt files.
Fig. 5. Measurement data

5. R stands for radar data and L stands for LiDAR data. In the
rows representing the radar measurement, the first 3 values
are v, ρ and ˙ρ of the tracked object relative to the sensor.
In the rows representing the LiDAR measurement, the first
two measurements are positions px and py of the tracked
object relative to the sensor. The last 4 measurements in
every row represents the ground truth values of px, py,vx and
vy. The big number is the timestamp. Timestamps are used
for logging a sequence of events, so that we know exactly,
for example, when the measurements were generated. Fig 6
shows how the timestamp works.
Fig. 6. Measurement data (timestamp)
The timestamp values are used to calculate the elapsed time
between two observations. The difference is divided by 106
to calculate ∆t in seconds rather than microseconds.
∆t =
timestamp(k + 1) − timestamp(k)
1000000.0
(12)
Figure 7 and Figure 12 illustrate the estimated trajectory
(px and py) of the pedestrian, measurement data and the
ground truth data. In Figure 7, it is interesting to note that
the extended Kalman Filter does not estimate the position
of the bicycle accurately where there is a sharp turn. This
is because Extended Kalman Filter is using a constant
velocity linear process model. Hence it is unable to estimate
a highly nonlinear maneuver performed by the bicycle. Later
we would delve into Unscented Kalman Filter which helps
circumvent this problem by using a nonlinear process model.
4 5 6 7 8 9 10 11 12
-14
-12
-10
-8
-6
-4
-2
0
2
Fig. 7. Estimated trajectory using EKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
4
5
6
7
8
9
10
11
12
Fig. 8. Estimated px using EKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-14
-12
-10
-8
-6
-4
-2
0
2
Fig. 9. Estimated py using EKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-4
-3
-2
-1
0
1
2
3
Fig. 10. Estimated vx using EKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-4
-3
-2
-1
0
1
2
3
4
5
Fig. 11. Estimated vy using EKF : Dataset 1
0 50 100 150 200 250
-5
0
5
10
15
20
25
30
35
40
Fig. 12. Estimated trajectory using EKF : Dataset 2
In Table II, the root mean square error (RMSE) values
are listed for two cases. In the first case, the RMSE was
calculated by using just LiDAR data to estimate the states.

6. 1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
0
50
100
150
200
250
Fig. 13. Estimated px using EKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
-5
0
5
10
15
20
25
30
35
40
Fig. 14. Estimated py using EKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 15. Estimated vx using EKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
-3
-2
-1
0
1
2
3
Fig. 16. Estimated vy using EKF : Dataset 2
In the second case, both LiDAR and Radar data have been
used to estimate the states using sensor fusion techniques.
We clearly see that the RMSE values are less for the second
case which means that the sensor fusion is advantageous
for estimating the state of a tracked object instead of just
using one sensor. In Table II, it is particularly interesting
to note that the RMSE values for vx and vy show a large
dip when both sensors are used as compared to the case
of just single sensor( LiDAR) usage. This is because the
LiDAR can only measure the position of the object. But the
radar can measure the radial velocity of the tracked object.
Hence a fusion of both Radar and LiDAR data is particularly
advantageous when it comes to estimating the velocity of a
tracked object.
State LiDAR LiDAR & Radar
Data 1
px 0.115 0.065
py 0.095 0.060
vx 0.794 0.531
vy 0.694 0.544
Data 2
px 0.218 0.185
py 0.194 0.190
vx 0.937 0.476
vy 0.834 0.814
TABLE II
ROOT MEAN SQUARE ERROR FOR EKF : DATASET 1 AND DATASET 2
III. UNSCENTED KALMAN FILTER
We know that EKF has two major drawbacks[26]:
• Linearisation may produce highly unstable filters if the
assumption of local linearity is violated
• The derivation and calculation of Jacobian matrices may
be nontrivial and computationally expensive.
Need for UKF: The central operation which the Kalman
Filter performs is the propagation of a gaussian random
variable (GRV) through the system dynamics (process model
and measurement model). In the EKF, the state distribution
is approximated by a GRV which is then propagated ana-
lytically through the first order linearization of a nonlinear
system [25, 26]. For example, while implementing the EKF
in the previous section, we linearized the nonlinear measure-
ment model for Radar (process model was already linear)
and used the first order approximations of the model to
perform the update step. These approximations can introduce
errors in the true posterior mean and covariance of the
predicted and/or updated GRV which may lead to sub-
optimal performance and sometimes divergence of the filter.
The UKF circumvents this problem by using a deterministic
sampling approach [25, 26]. In UKF, the state distribution
which is approximated by a GRV, is represented by a set of
carefully chosen sample points. These points are propagated
through the nonlinear system. The transformed points capture
the mean and covariance accurately to 3rd
order( Taylor
series expansion) for any nonlinear system. In contrast, EKF
only achieves first-order accuracy. Also the computational
complexity of the UKF is the same order as EKF[25, 26].
CTRV Model: In the EKF section, we used a constant
velocity (CV) model for pedestrian tracking. In the literature,
we can find various implementations of motion models [34]
for tracking such as
• Constant turn rate and velocity magnitude (CTRV)
model
• Constant turn rate and acceleration model (CTRA)
model
• Constant steering angle and velocity (CSAV) model

7. • Constant curvature and acceleration (CCA) model
While using constant velocity (CV) model in the EKF
implementation in the previous section, we were simplifying
the way vehicles actually move. This was a major drawback
in the implementation process because a process model
assuming constant velocity will predict moving vehicles and
pedestrians incorrectly. Hence, we chose the constant turn
rate and velocity magnitude (CTRV) model to implement
the Unscented Kalman Filter as shown in Fig 17. In the
CTRV model , the state vector and the nonlinear motion
model xk+1 = f(xk, νk) is defined as
x =






px
py
v
ψ
˙ψ






, xk+1 = xk +
tk+1
tk






˙px(t)
˙py(t)
˙v(t)
˙ψ(t)
¨ψ(t)






dt + νk (13)
where (px, py) is the position of the tracked object, v is
the velocity magnitude, ψ and ˙ψ are the yaw and yaw rate
respectively and νk represents the process noise. The motion
can be further rewritten as
xk+1 = xk +







vk
tk+1
tk
cos(ψk + ˙ψk(t − tk))dt
vk
tk+1
tk
sin(ψk + ˙ψk(t − tk))dt
0
˙ψk∆t
0







+νk (14)
Solving it further, we obtain the final CTRV motion model
[34].
xk+1 = xk +







vk
˙ψk
(sin(ψk + ˙ψk∆t) − sin(ψk))
−vk
˙ψk
(cos(ψk + ˙ψk∆t) − cos(ψk))
0
˙ψk∆t
0







+ νk
(15)
There may be situations when tracked vehicle is driving on
a straight road. In this case, the yaw rate ( ˙ψ) is zero. Since
division by zero is unacceptable, we modify equations of the
CTRV model as
xk+1 = xk +






vk cos(ψk)∆t
vk sin(ψk)∆t
0
˙ψk∆t
0






(16)
The vector νk in the process model is the noise vector and
it is defined as νk = νa,k ν ¨ψ,k
T
. The noise process νa,k
represents the longitudinal acceleration noise. It influences
the longitudinal speed of the tracked object and randomly
changes it’s value at every time step k. It is expressed as
νa,k ∼ N(0, σ2
a). It is normally distributed white noise
with zero mean and variance σ2
a. The noise process ν ¨ψk
∼
N(0, σ2
¨ψ
) is also a normally distributed white noise with
Fig. 17. Constant turn rate and velocity magnitude(CTRV model)
zero mean and variance σ2
¨ψ
. In Fig 17, it is interesting to
notice that the yaw acceleration in the form of ν ¨ψ,k will
change the radius of the tracked vehicle, so it would have
some effect on the position of the tracked object. However
we will assume that the effect of yaw acceleration noise on
the position is negligible and we will neglect it here. In the
CTRV model, we have assumed that the noise processes are
uncorrelated. The process covariance matrix Q is defined
as Q = E(vkvT
k ) =
σ2
a 0
0 σ2
¨ψ
. Considering the effect of
process noise νa,k and ν ¨ψ,k on the states, the process noise
vector νk is defined as
νk =






1
2 (∆t)2
cos(ψk).νa,k
1
2 (∆t)2
sin(ψk).νa,k
∆t.νa,k
1
2 (∆t)2
.ν ¨ψ,k
∆t.ν ¨ψ,k






(17)
Unscented Transformation: The unscented
transformation(UT) is a method for calculating the
statistics of a random variable which undergoes a nonlinear
transformation[25, 26]. The procedure is illustrated in Fig 18.
Fig. 18. The principle of unscented transformation
Let us consider propagating a GRV x(dimension nx) through
a nonlinear function y = f(x). Let us assume that x has
mean ¯x and covariance Pxx. To calculate the mean ¯y and
covariance Pyy of y, we form a matrix X of 2nx+1 sigma
vectors Xi( with corresponding weights Wi). The sigma
vectors are chosen around the mean ¯x and in certain relation
to the standard deviation Pxx. This is shown as below:

8. X0 = ¯x
Xi = ¯x + ( (nx + λ)Pxx)i, i = 1, . . . , nx
Xi = ¯x − ( (nx + λ)Pxx)i−nx
, i = n + 1, . . . , 2nx
W0 =
λ
nx + λ
Wi =
1
2(nx + λ)
, i = 1, . . . , 2nx
(18)
where ( (nx + λ)Pxx)i is the ith row or column of the
matrix square root of (nx + λ)Pxx, Wi is the weight
associated with the ith point and λ is a scaling parameter
which determines the spread of the sigma points around the
mean ¯x. For our project, we have considered λ = 3−nx. The
Unscented Transformation procedure is as follows[25, 26] :
• Instantiate each sigma vector through the function to
yield a set of transformed sigma vectors,
yi = f(Xi)
• The mean of the transformed sigma vectors is given by
the weighted average of the transformed sigma vectors,
¯y =
2nx
i=0
Wiyi (19)
• The covariance of the transformed sigma vectors is
given by the weighted outer product of the transformed
sigma vectors,
Pyy =
2nx
i=0
Wi(yi − ¯y)(yi − ¯y)T
(20)
To capture the importance of Unscented transformation, a
simple example is shown in Fig 19 using a 2D system
[25, 26]. The left plot shows the true mean and covariance
propagation using Monte-Carlo Sampling which we will talk
later in particle filter. The centre plot shows the results using
a linearization approach like EKF. The right plot shows the
application of unscented transformation(UT). It is clear that
the UKF outperforms EKF.
Fig. 19. Comparison: EKF vs UKF
UKF roadmap: The UKF is a straightforward extention
of the unscented transformation technique.The various steps
involving the application of the Unscented Kalman Filter is
as follows
• Prediction
1) Generate Sigma points
2) Predict Sigma points
3) Predict Mean and Covariance
• Update
1) Predict measurement
2) Update state
At the beginning of the prediction step, we have the posterior
state xk/k, the mean ¯xk/k and the posterior covariance Pk/k
from the last iteration. They represent the state distribution of
the current state. The number of sigma points to be generated
depends of the dimension of the state. For our CTRV model,
we have 5-D state, hence nx = 5. We would generate nσ =
2nx + 1 = 11 sigma points. The sigma points are generated
as follows:
Xk|k = ¯xk|k ¯xk|k + (nx + λ)Pk|k ¯xk|k − (nx + λ)Pk|k
(21)
The square root of the matrix (nx+λ)Pk|k is calculated using
Cholesky decomposition. To take into account the non-linear
effect of the process noise νk =
νa,k
ν ¨ψ,k
of CTRV model, we
include the process noise in the state vector. This is called
augmentation. Also recall that for the CTRV model
Q = E(vkvT
k ) =
σ2
a 0
0 σ2
¨ψ
(22)
The state vector xa,k|k and state covariance matrix after
augmentation Pa,k|k is defined as
xa,k|k =










px
py
v
ψ
˙ψ
νa,x
ν ¨ψ,x










, Pa,k|k =
Pa,k|k 0
0 Q
(23)
The number of sigma points after augmentation would be
nσ = 2nx +1 where nx = 7. The 4 extra sigma points would
represent the uncertainty caused by the process noise. We
calculate the augmented sigma points Xa,k/k using Equation
21 but with xk|k replaced by the augmented state vector
xa,k|k. In the prediction step , every column of the augmented
sigma point matrix Xa,k|k is passed through the CTRV
process model. We calculate Xk+1|k = f(Xa,k|k) where
Xk+1|k is the predicted sigma point matrix. Please note that
Xa,k|k is a 7x15 matrix and Xk+1|k is a 5x15 matrix since
each state is a 5D vector. The prediction step of UKF is
illustrated in Figure 20.

9. Fig. 20. Generation of predicted sigma points
The predicted mean xk+1|k and covariance Pk+1|k of the
sigma points are as follows:
xk+1|k =
nσ
i=1
ωiXk+1|k,i
Pk+1|k =
nσ
i=1
ωi(Xk+1|k,i − xk+1|k)(Xk+1|k,i − xk+1|k)T
ωi =
λ
λ + na
, i = 1
ωi =
1
2(λ + na)
, i = 2 . . . σ
(24)
In the measurement step, we use different approaches based
on the the type of sensor measurement received. If we
receive a LiDAR measurement, we use the same steps
as used in EKF to predict the measurement. However,
when we receive radar measurement, we use unscented
transformation to predict the measurement since the
measurement model pertaining to the radar is nonlinear. To
predict the measurement, we could use the same sequence
of steps as used in the prediction step but in order to
save computation time, we reuse the sigma points Xk+1|k
generated in the prediction step instead of generating new
sigma points. We also note that the measurement noise
doesn’t have a nonlinear effect on the measurement model.
Since the effect of the measurement noise is purely additive(
zk+1 = h(xk+1) + ωk+1 ), we can skip the augmentation
step as well. The transformed measurement sigma points
are generated the same way as in the prediction step. It is
illustrated in Fig 21.
Fig. 21. Generation of measurement sigma points
The predicted mean zk+1|k and the covariance Sk+1|k of the
measurement sigma points are as follows:
zk+1|k =
nσ
i=1
ωiZk+1|k,i
Sk+1|k =
nσ
i=1
ωi(Zk+1|k,i − zk+1|k)(Zk+1|k,i − zk+1|k)T
+ R
(25)
where R = E(ωωT
) is the measurement noise covariance.
In UKF, we need the cross-correlation Tk+1|k of sigma
points in state space and measurement space to calculate
the kalman gain. The equations of Tk+1|k(cross-correlation),
Kk+1|k(Kalman Gain), xk+1|k+1(Updated State) and
Pk+1|k+1(Updated Covariance) are as follows:
Tk+1|k =
nσ
i=1
ωi(Xk+1|k,i − xk+1|k)(Zk+1|k,i − zk+1|k)T
Kk+1|k = Tk+1|kS−1
k+1|k
xk+1|k+1 = xk+1|k + Kk+1|k(zk+1 − zk+1|k)
Pk+1|k+1 = Pk+1|k − Kk+1|kSk+1|kKT
k+1|k
(26)
Noise Parameters: In this section, we will describe how
we chose the noise parameters. This section applies to both
EKF and UKF. The process noise and the measurement
noise vectors and matrices are illustrated in Figure 22.
Fig. 22. Overview of Noise Parameters
Measurement noise vector ωk is an indication of how precise
is the measurement of the Radar sensor. These values are
usually given in the sensor manual. For example the sensor
manual tells us that the radar measures the distance ρ with
a standard deviation of σρ = 0.3 meters. This given us the
value of σ2
ρ = 0.09 in the measurement noise matrix R.
The process noise parameters are hard to determine since
we have to depend on approximation. Based on intuition,
we decide the maximum acceleration at which the tracked
object could possibly move. For example, in our project,
we are tracking a pedestrian riding a bicycle. We assume
that the bicycle can’t accelerate more than 1 m/sec2
. For
calculating the standard deviation (σa) of the process noise,
we use a rule of thumb. We consider the standard deviation to
be half of the maximum acceleration. So we consider σ2
a =
0.25 m2
/sec4
. To calculate σ ¨ψ, we consider the maximum
angular acceleration of the bicycle to be 1 rad/sec2
. So
using the rule of thumb, we obtain σ ¨ψ = 0.25 rad2
/sec4
.
The process noise really depends on the application. If we
want smoother estimates, we consider process noise to be

10. lower. If we expect the tracked object to behave erratically,
then we would choose a higher process noise.
Filter Consistency: The goal of filter consistency is to check
whether the filter has an incorrect estimate [2]. The validity
of the estimate can be determined based on whether the
estimate of the filter agrees with the observed measurements.
A filter is consistent if it provides a realistic estimation un-
certainty. The Normalized Innovations Squared (NIS) check
for Kalman Filters is based on the assumption that , the
normalized innovations of a Kalman Filter will have unit
variance under nominal conditions[2]. The innovation is the
difference between the predicted measurement and the actual
measurement. We normalize the innovation by putting it in
relation to Sk+1|k as shown in Equation 27. The NIS follows
a chi-squared(X2
) distribution. It is expressed as:
= (zk+1 − zk+1|k)T
S−1
k+1|k(zk+1 − zk+1|k) (27)
Based on the X2
distribution table and the fact that radar
measurement is 3 dimensional, we choose the 5% statistic.
The NIS plot for both dataset 1 and dataset 2 are shown in
Figure 29 and Figure 36 respectively.
UKF Results : Table III and Table IV compares the
performance of EKF and UKF for datasets 1 and 2
respectively.
State RMSE(EKF) RMSE(UKF)
LiDAR
px 0.115 0.131
py 0.095 0.118
vx 0.794 0.731
vy 0.694 0.669
LiDAR
& Radar
px 0.065 0.077
py 0.060 0.082
vx 0.531 0.590
vy 0.544 0.574
TABLE III
COMPARISON OF EKF AND UKF : DATASET 1
Based on the RMSE values, it is seen that UKF and EKF
performs the same for Dataset 1. However, UKF outperforms
EKF for Dataset 2. For both dataset 1 and 2, we notice that
the RMSE values are less if both LiDARs and radars are
used as compared to using just LiDAR. Thus we confirm
that sensor fusion techniques are advantageous in tracking
applications irrespective of the type of estimation filters used.
State RMSE(EKF) RMSE(UKF)
LiDAR
px 0.218 0.207
py 0.194 0.193
vx 0.937 0.480
vy 0.834 0.585
LiDAR
& Radar
px 0.185 0.180
py 0.190 0.187
vx 0.476 0.262
vy 0.814 0.463
TABLE IV
COMPARISON OF EKF AND UKF : DATASET 2
In Fig 23 and Fig 30, the estimated trajectory of the bicycle
for both the datasets are plotted. We notice that UKF does a
good job estimating the nonlinear trajectory of the bicycle.
By comparing Fig 23 and Fig 7, we can conclude that UKF
estimates the trajectory better around sharp corners than EKF.
The estimated vx, vy and φ for Dataset1 are plotted in Fig 26,
Fig 27 and Fig 28 respectively. In Fig 24 and Fig 25, we plot
the estimates px and py respectively.
4 5 6 7 8 9 10 11 12
-14
-12
-10
-8
-6
-4
-2
0
2
Fig. 23. Estimated trajectory using UKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
4
5
6
7
8
9
10
11
12
Fig. 24. Estimated px using UKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-14
-12
-10
-8
-6
-4
-2
0
2
Fig. 25. Estimated py using UKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-4
-3
-2
-1
0
1
2
3
Fig. 26. Estimated vx using UKF : Dataset 1
The estimated vx, vy and φ for Dataset 2 are plotted in
Fig 33, Fig 34 and Fig 35 respectively. Based on the NIS plot

11. 1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-4
-3
-2
-1
0
1
2
3
Fig. 27. Estimated vy using UKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
-150
-100
-50
0
50
100
150
200
Fig. 28. Estimate φ using UKF : Dataset 1
1.47701044 1.47701045 1.47701046 1.47701047 1.47701048 1.47701049 1.4770105 1.47701051
10 15
0
2
4
6
8
10
12
NIS(5%)
5% (7.815)
Fig. 29. Normalized Innovation Squared(NIS) using UKF : Dataset 1
0 50 100 150 200 250
-5
0
5
10
15
20
25
30
35
40
Fig. 30. Estimated trajectory usign UKF : Dataset 2
(Fig 29 and Fig 36), we can conclude that the uncertainties
have been accurately modeled.
IV. SUMMARY
In this paper, a sensor fusion algorithm was presented to
track a pedestrian riding a bike by fusing the measurements
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
0
50
100
150
200
250
Fig. 31. Estimated px using UKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
-5
0
5
10
15
20
25
30
35
40
Fig. 32. Estimated py using UKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 33. Estimated vx using UKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 34. Estimated vy using UKF : Dataset 2
from a Velodyne LiDAR and a radar. Two bayesian filters,
EKF and UKF were implemented to estimate the state of
the pedestrian. In both filter implementations, it was noticed
that estimation error was less while using both sensors as
compared to using just LiDAR measurements. Hence it was
concluded that sensor fusion is advantageous. In future work,

12. 1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
timestamp ×1015
-80
-60
-40
-20
0
20
40
60
φ
Estimated φ vs Ground Truth (Unscented Kalman Filter) : Data 2
Ground Truth
Estimated φ
Fig. 35. Estimate φ using UKF : Dataset 2
1.47701044 1.47701046 1.47701048 1.4770105 1.47701052 1.47701054 1.47701056
10 15
0
2
4
6
8
10
12
14
NIS(5%)
5% (7.815)
Fig. 36. Normalized Innovation Squared(NIS) using UKF : Dataset 2
several other sensors like stereo cameras would be used
to further improve the accuracy of position and velocity
estimates of tracked objects.
REFERENCES
[1] PUCK, Velodyne LiDAR, PUCKT M Idf, National Conference,
2015.
[2] Sarah E Houts, Shandor G Dektor, Stephen M Rock. A robust frame-
work for failure detection and recovery for terrain-relative navigation.
Unmanned Untethered Submersible Technology,2013
[3] Jesse levinson, Michael Montemerlo, Sebastian Thrun Map-Based
Precision Vehicle Localization in Urban Environments. Robotics:
Science and Systems, Volume 4, pages 1, Citeseet 2007.
[4] Neil Getz. Control of balance for a nonlinear nonholonomic non-
minimum phase model of a bicycle. American Control Conference,
Vol 1, Pages 148-151, IEEE 1994.
[5] Brian Yamauchi. All-weather perception for man-portable robots
using ultrawideband radar. 2010 IEEE International Conference on
Robotics and Automation(ICRA), pages 3610–3615, IEEE 2010.
[6] Nico Kaempchen, Kay Ch Fuerstenberg, Alexander G Skibicki, Klaus
CJ Dietmayer. Sensor fusion for multiple automotive active safety
and comfort applications. Advanced Microsystems for Automotive
Applications 2004, pages 137–163, Springer 2004.
[7] Tarak Gandhi, Mohan Manubhai Trivedi. Pedestrian protection sys-
tems: Issues, survey, and challenges IEEE Transactions on intelligent
Transportation systems, Vol 8, No.3, pages 413-430, IEEE 2007.
[8] Daniel Olmeda, Arturo De LA Escalera, Jos M Armingol. Detection
and tracking of pedestrians in infrared images 2009 3rd International
Conference on Signals, Circuits and Systems (SCS), pages 1-6, IEEE
2009.
[9] Klaus CJ Dietmayer, Jan Sparbert, Daniel Streller. Model based object
classification and object tracking in traffic scenes from range images.
Proceedings of IV IEEE Intelligent Vehicles Symposium, Vol.200,
2001.
[10] Daniel Streller, Klaus Dietmayer. Object tracking and classification
using a multiple hypothesis approach. Intelligent Vehicles Symposium,
IEEE 2004.
[11] Cristiano Premebida, Nunes Urbano. A multi-target tracking and
GMM-classifier for intelligent vehicles. Intelligent Transportation
Systems Conference (ITSE), pages 313-318, IEEE 2006.
[12] Samuel Gidel, Paul Checchin Christophe Blanc, Thierry Chateau,
Laurent Trassoudaine. Pedestrian detection method using a multilayer
laserscanner: Application in urban environment IEEE/RSJ Interna-
tional Conference on Intelligent Robots and Systems(IROS), pages
173-178, IEEE 2008.
[13] Uwe Franke, Armin Joos. Real-time stereo vision for urban traffic
scene understanding. Proceedings of the IEEE Intelligent Vehicles
Symposium, pages 273-278, IEEE 2000.
[14] Mirko Mahlisch, Roland Schweiger, Werner Ritter, Klaus Dietmayer.
Sensor fusion using spatio-temporal aligned video and LiDAR for
improved vehicle detection 2006 IEEE Intelligent Vehicles Symposium,
pages 424-429, IEEE 2006.
[15] Raphael Labayrade, Cyril Royere, Dominique Gruyer, Didier Aubert.
Cooperative fusion for multi-obstacles detection with use of stereo-
vision and laser scanner Autonomous Robots, Vol.19, No.2, pages
117-140, Springer 2005.
[16] Jeoung Woo, Young-Joong Kim, Jeong-on Lee, Myo-Taeg Lim. Lo-
calization of mobile robot using particle filter International Joint
Conference(SICE-ICASE), pages 3031-3034, IEEE 2006.
[17] Fredrik Gustafsson, Fredrik Gunnarsson, Niclas Bergman, Urban Fors-
sell, Jonas Jansson, Rickard Karlsson, P-J Nordlund. Particle filters
for positioning, navigation, and tracking. IEEE Transactions on Signal
Processing,Vol 50, No.2, Pages 425-437, IEEE 2002.
[18] Sebastian Thrun, Wolfram Burgard, Dieter Fox. Probabilistic Robotics.
MIT press,2005
[19] Sebastian Thrun. Particle filters in robotics. Proceedings of the
Eighteenth conference on Uncertainty in artificial intelligence, pages
511-518, Morgan Kaugmann Publishers Inc. 2002.
[20] Dieter Fox, Sebastian Thrun, Wolfram Burgard, Frank Dellaert. Par-
ticle filters for mobile robot localization. Sequential Monte Carlo
methods in practice, pages 401-428, Springer 2001.
[21] Fengbing Luo, Bianjing Du, Zhen Fan. Mobile robot localization
based on particle filter. 11th World Congress onIntelligent Control
and Automation(WCICA), pages 3089-3093, IEEE 2014.
[22] Dongsheng Wang, Jianchao Zhao, Wei Wang. Particle filter based
robust mobile robot localization. International conference on Mecha-
tronics and Automation(ICMA), pages 817-821, IEEE 2009.
[23] Mirko Meuter, Uri Iurgel, Su-Birm Park, Anton Kummert. The
unscented Kalman filter for pedestrian tracking from a moving host.
2008 IEEE Intelligent Vehicles Symposium, pages 37-42, IEEE 2008.
[24] Daniel Olmeda, Arturo De La Escalera, Jos M Armingol. Detection
and tracking of pedestrians in infrared images. 2009 3rd International
Conference on Signals, Circuits and Systems(SCS),pages 1-6, IEEE
2009.
[25] Eric A Wan, Rudolph Van Der Merwe. The unscented Kalman Filter
for Nonlinear Estimation. Adaptive Systems for Signal Processing,
Communications and Control Symposium 2000, pages 153-158, IEEE
2000
[26] Simon J Julier, Jeffrey K Uhlmann. New Extension of the Kalman
Filter to Nonlinear Systems. AeroSense’97, International Society for
Optics and Photonics. pages 182-193, 1997
[27] Ehab I Al Khatib, Mohammad A Jaradat, Mamoun AbdelHafez, Milad
Roigari. Multiple sensor fusion for mobile robot localization and
navigation using the Extended Kalman Filter 2015 10th International
Symposium on Mechatronics and its Applications(ISMA), pages 1-5,
IEEE 2015.
[28] John Shackleton, Brian VanVoorst, Joel Hesch. Tracking people with
a 360-degree lidar. 2010 7th IEEE International Conference on
Advanced Video and Signal Based Surveillance(AVSS), pages 420-426,
IEEE 2010.
[29] Osama Masoud, Nikolaos P Papanikolopoulos. A novel method for
tracking and counting pedestrians in real-time using a single camera.
IEEE transactions on vehicular techonology, Vol.50, No.5, pages 1267-
1278, IEEE 2001.
[30] Hyunggi Cho, Young-Woo Seo, BVK Vijaya Kumar, Rahunathan Raj
Rajkumar. A multi-sensor fusion system for moving object detection
and tracking in urban driving environments. 2014 IEEE International
Conference on Robotics and Automation(ICRA), pages 1836-1843,
IEEE 2014.
[31] Christophe Blanc, Laurent Trassoudaine, Jean Gallice. EKF and
particle filter track-to-track fusion: a quantitative comparison from
radar/lidar obstacle tracks. 2005 8th International Conference on
Information Fusion, Vol.2, pages 7–pp, IEEE 2005.
[32] Fadi Fayad, Veronique Cherfaoui. Tracking objects using a laser
scanner in driving situation based on modeling target shape. 2007
IEEE Intelligent Vehicles Symposium, pages 44-49, IEEE 2007.
[33] Leonardo Marin, Marina Valles, Angel Soriano, Angel Valera, Pedro
Albertos. Event-based localization in ackermann steering limited
resource mobile robots. IEEE/ASME Transactions on Mechatronics,
Vol.19, No.4, pages 1171-1182, IEEE 2014.

13. [34] Robin Schubert, Eric Richter, Gerd Wanielik. Comparison and
evaluation of advanced motion models for vehicle tracking. 2008
11th International Conference on Information Fusion,pages 1–6, IEEE
2008.