The document describes a method for improving time-to-collision (TTC) estimation in collision avoidance systems. It uses an interacting multiple model (IMM) Kalman filter that dynamically combines the outputs of two Kalman filters, one based on a constant velocity motion model and the other based on a constant acceleration model. The IMM estimates the probability of each model and mixes the filter outputs accordingly. It then calculates TTC based on the distance and velocity estimates from the IMM filter. Experimental results on both simulated and real data show the proposed method improves the accuracy of distance and TTC estimation compared to using a single model filter.

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2009-01-0162
Improving Time-To-Collision Estimation by IMM Based Kalman Filter
Yixin Chen
Delphi Corporation
Oakland University
Copyright © 2009 SAE International
ABSTRACT
In a CAS system, the distance and relative velocity
between front and host vehicles are estimated to
calculate time-to-collision (TTC). The distance estimates
by different methods will certainly include noise which
should be removed to ensure the accuracy of TTC
calculations. Kalman filter is a good tool to filter such
type of noise. Nevertheless, Kalman filter is a model
based filter, which means a correct model is important to
get the good filtering results. Usually, a vehicle is either
moving with a constant velocity (CV) or constant
acceleration (CA) maneuvers. This means the distance
data between front and host vehicles can be described
by either constant velocity or constant acceleration
model. In this paper, first, CV and CA models are used
to design two Kalman filters and an interacting multiple
model (IMM) is used to dynamically combine the outputs
from two filters. In detail, IMM technique is used to
estimate the mode probabilities for CV and CA based
Kalman filters and mix the two filter results based on the
mode probabilities. Then the motion equation is used to
calculate the TTC based on the distance and velocity
estimates obtained from the IMM based filter. The
experimental results on both simulated and real
estimated distance data are found to be satisfactory and
indicate that the proposed algorithm does improve the
signal-to-noise ratio of distance data.
INTRODUCTION
In a CAS system, time-to-collision (TTC) is the most
important output which needs to be estimated in real-
time based on the distance and relative velocity between
front and host vehicles. If TTC is below a threshold, the
correspondent action should be taken, such as giving
driver a sound warning, or automatically applying brake,
etc.
The distance estimates by vision or radar based CAS
system certainly include noise which should be removed
to improve the accuracy of TTC calculations. Kalman
filter is a good tool to filter such types of noise.
Nevertheless, Kalman filter is a model based filter, which
means a correct model is important to get the good
filtering results. Usually, a vehicle is either moving with a
constant velocity (CV) or constant acceleration (CA).
This means the distance between front and host
vehicles can be described by either constant velocity or
constant acceleration model. In this paper, first we
explore the use of an interacting multiple model (IMM)
technique [1] to estimate the mode probabilities for CV
and CA based Kalman filters and mix the two filter
results based on the mode probabilities. Then a simple
motion equation is used to calculate the TTC based on
the distance and velocity estimates obtained from the
IMM based filter.
*9-2009-01-0162*
Manohar Das and Devendra Bajpai

2. ESTIMATION OF TIME-TO-COLLISION (TTC)
In rear-end CAS application, the distance between front
and host vehicles is measured in real-time by the
different methods such as radar or computer vision.
Given the distance )(kx between two cars in frame k ,
the instant speed between two vehicles can be
approximated by the derivative of )(kx , and the time-to-
collision )(kTTC in frame k is defined as
)(
)(
)(
kx
kx
kTTC = (1)
In reality, there are always distance estimation errors
which can affect the accuracy of TTC estimations. Using
the filter techniques to de-noise the distance estimations
can improve the accuracy of TTC estimations.
KALMAN FILTER
Kalman filter is well known for its good performance to
remove Gaussian noise and can be described by the
following state space equations:
⎩
⎨
⎧
+=
++=+
)()()(
)()()()1(
kwkkZ
kvkukk
HX
ΓGFXX
(2)
where )(kv and )(kw are system and measurement
noise, respectively.
Let us denote
])()([
])()([
'
'2'
kwkwER
kvkvEQ v
=
ΓΓ=Γ′Γ= σ
(3)
Then the solution of Kalman filter is given by the
following steps [1]:
1. Predicted state:
)()|()|1(
^^
kukkXkk GFX +=+ (4)
2. State prediction covariance:
QFFPP +=+ '
)|()|1( kkkk (5)
3. Predicted measurement:
)|1()|1(
^^
kkkk +=+ XHZ (6)
4. Measurement prediction covariance:
)1()|1()1( '
+++=+ kkkk RHHPS (7)
5. Filter gain:
1'
)1()|1()1( −
++=+ kkkk SHPW (8)
6. Measurement residual:
)|1()|1()1()1(
~^
kkkkkk +=+−+=+ ZZZv
(9)
7. Updated state estimate:
)1()1()|1(
)1|1(
^
^
++++=
++
kkkk
kk
vWX
X
(10)
8. Updated covariance:
'
)1()1()1()|1(
)1|1(
+++−+=
++
kkkkk
kk
WSWP
P
(11)
A correct model is important while using Kalman filter
described above. In rear-end CAS application, the two
vehicles can travel in different maneuvers such as
cruising in a fixed speed or suddenly acceleration, etc.
This means that the distance between two vehicles can
be described by either constant velocity (CV) model or
constant acceleration (CA) model. A multiple model
approach should be used to estimate the distance.
CONSTANT VELOCITY MODEL (CV)
When both front and host vehicles are moving at a
constant velocity, the distance data between the two
vehicles can be modeled by constant velocity model
which is shown as below:
[ ]
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
+
+
••
•
••
•
••
•
)(
)(
)(
)(
001)(
)(
1
2
)(
)(
)(
000
010
01
)1(
)1(
)1( 2
kw
kX
kX
kX
kZ
kvT
T
kX
kX
kX
T
kX
kX
kX
(12)
where )(kX denotes distance estimate in frame k and
T denotes the time difference between frame 1+k
and frame k .

3. CONSTANT ACCELERATION MODEL (CA)
If either front or host vehicle is moving with a constant
acceleration and the other vehicle is moving at a
constant velocity or a different acceleration, the distance
data between the two vehicles can be modeled by
constant acceleration model (CA) which is shown as
below:
[ ]
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
+
+
••
•
••
•
••
•
)(
)(
)(
)(
001)(
)(
1
2
)(
)(
)(
100
10
2
1
)1(
)1(
)1( 22
kw
kX
kX
kX
kZ
kvT
T
kX
kX
kX
T
TT
kX
kX
kX
(13)
THE INTERACTING MUULTIPLE MODEL (IMM)
ESTIMATOR
The CV and CA models can be used to model the
distance between front and host vehicles, but we don’t
know when a specific model should be used. The
interacting multiple model (IMM) estimator [1] is an
algorithm which can be used to handle such case. In
IMM algorithm, at time k the previous estimates from the
multiple models are mixed based on the mixing
probabilities to generate different mixed initial conditions
for different filters. Then, based on the multiple filter
outputs, the likelihood and then model probabilities for
each model can be calculated. Lastly, the final IMM
based filter outputs are calculated based on the
individual filter outputs and the model probabilities.
One cycle of the algorithm consists of the following
steps:
a. Calculation of the mixing probabilities (the probability
that mode iM was in effect at 1−k given that jM is
in effect at k conditioned on
1−k
Z ):
rjikp
c
ZkMkMPkk
iij
j
k
jiji
,...,1,)1(
1
}),(|)1({)1|1( 1
|
=−=
−=−− −
Δ
μ
μ
(14)
where r is the number of filters, and
}),1(|)({ 1−
−= k
ijij ZkMkMPp (15)
}|)1({)1( 1−
−=− k
ii ZkMPkμ (16)
∑
=
−
=−=
=
r
i
iij
k
jj
rjkp
ZkMPc
1
1
,...,1)1(
}|)({
μ
(17)
b. Mixing initial conditions for the filter matched to
)(kM j :
rjkkkkx
kkx
ji
r
i
i
j
,...,1)1|1()1|1(ˆ
)1|1(ˆ
|
1
0
=−−−−
=−−
∑
=
μ
(18)
[ ]
[ ] rjkkxkkx
kkxkkx
kkPkk
kkP
ji
ji
r
i
i
ji
j
,...,1})1|1(ˆ)1|1(ˆ
)1|1(ˆ)1|1(ˆ
)1|1(){1|1(
)1|1(
0
0
1
|
0
=
′
−−−−−
•−−−−−+
−−−−
=−−
∑
=
μ
(19)
c. Mode-matched filtering:
The likelihood functions corresponding to the r filters are
given by
]),(|)([)( 1−
=Λ k
jj ZkMkzpk (20)
which is the probability of measurement )(kz given the
model )(kM j and previous measurement outputs. The
measurement prediction errors and measurement
prediction covariance from )(kz for each filter can be
used to find the likelihood )(kjΛ as below:
rjkkPkS
kkxkkzkzk
jj
jj
j
,...,1)]]1|1(;[
)],1|1(ˆ;1|[ˆ);([)(
0
0
=−−
−−−Ν=Λ
(21)
where )(ˆ •j
z denotes the estimate of )(kz by filter j
shown in Eq. (6), )(•j
S denotes the measurement
prediction covariance by filter j shown in Eq. (7), and
)(•N denotes the normal distribution such as

4. ))()(
2
1
exp(|2|
),;(
12/1
zzPzzP
PzzN
−′−−
=
−−
π
(22)
d. Mode probability update:
rj
ck
ck
ZkMPk
r
j
jj
jj
k
jj
,...,1
)(
)(
}|)({)(
1
=
Λ
Λ
==
∑
=
Δ
μ
(23)
where jc is given by Eq. (17).
e. Estimate and covariance combination:
}])|(ˆ)|(ˆ)][|(ˆ)|(ˆ[
)|(){()|(
)()|(ˆ)|(ˆ
1
1
′−−
+=
=
∑
∑
=
=
kkxkkxkkxkkx
kkPkkkP
kkkxkkx
jj
r
j
j
j
r
j
j
j
μ
μ
(24)
EXPERIMENTAL RESULTS
SIMULATIONS OF VEHICLE DISTANCE DATA – First,
the simulated vehicle distance data is used to verify the
model described above. The simulated maneuver is as
follows:
• The initial distance between front and host
vehicles is 80 meters;
• First, the host vehicle is approaching the front
vehicle with a relative speed of 0.25 m/frame (27
KPH or 7.5 m/s assuming frame rate is 30
frames per second);
• When the distance between the two vehicles is
60 meters, the host vehicle starts to accelerate
with acceleration 0.001 m/s2
until the two
vehicles collide (the distance is 0).
The distance changes in the above maneuver are given
by:
⎪⎩
⎪
⎨
⎧
≥−−
≤≤−
=+
81
2
1
)(
800)(
)1( 2
kaTvTkx
kvTkx
kx (25)
where )(kx denotes the distance in frame k , T is
frame time. The true distance data is shown in Figure 1.
0
10
20
30
40
50
60
70
80
90
0 20 40 60 80 100 120 140 160
Frame Number
Distance(meter)
Distance
Figure 1: The true distance data between front and
host vehicles
TTC estimated by Eq. (1) is shown in Figure 2.
Time-To-Collision
0
2
4
6
8
10
12
0 20 40 60 80 100 120 140 160
Frame Number
TTC(seconds)
Time-To-Collision
Figure 2: True time-to-collision data plot
In reality, there are always distance estimation errors.
Assuming the estimation errors can be described by an
additive Gaussian noise ),0(~)( 2
σNkn , Figure 3
shows one example of the original and noisy distance
data plots. The purpose of the simulation experiment is
to verify the performance of the IMM based filter
quantitatively.

5. Figure 3: Original and noisy distance data
SYSTEM AND MEASUREMENT NOISE
CONSIDERTATIONS IN KALMAN FILTERS - While
implementing CV or CA based Kalman filters,
the system and measurement noise need to be
estimated. From Eq. (3), the system noise Q
for Eq. (12) or Eq. (13) is as below
2
2
23
234
2
1
2
1
2
1
2
1
2
1
4
1
vv
TT
TTT
TTT
Q σσ
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=Γ′Γ= (26)
For a constant velocity model, vσ should be of the
order of the maximum acceleration magnitude Ma . A
practical range is MvM aa ≤≤ σ5.0 [1]. For constant
acceleration model, vσ should be of the order of the
magnitude of the maximum acceleration increment over
a sampling period, MaΔ . A practical range is
MvM aa Δ≤≤Δ σ5.0 . In this study, the acceleration
a and the acceleration increment aΔ are simply
estimated based on the raw distance data as follows
)1()()(
)1()()(
)1()()(
−−=Δ
−−=
−−=
kakaka
kvkvka
kxkxkv
(27)
where )(kx is the distance average value in sample k
to improve the estimation accuracy of )(ka and )(kaΔ .
The measurement noise )(kw in Eq. (3) should be
reasonably true to ensure the Kalman filter accuracy. In
this study, we estimate the measurement noise as
shown below:
)(ˆ)()(ˆ kxkxkw −= (28)
where )(ˆ kx denotes the estimate of true distance,
which is calculated using polynomial curve fitting based
on the raw (noisy) distance data. Matlab function
“polyfit(x,y,m)” is used to find interpolation polynomial
p(x) described above. We have found that the IMM filter
gives the satisfactory results for 4≥m .
SIMULATED VEHICLE DISTANCE DATA FILTERING
SIMULATION RESULTS - Figure 4 shows an example
of the vehicle data filtering simulation results. The
simulation results shown in Figure 4(a) indicate that
noise is dramatically reduced by using an IMM based
filter. The velocity estimates shown in Figure 4(b) are
fluctuating around the true velocity is -0.25 m/s for
80≤k , and same are true for TTC estimates shown in
Figure 4(c). The mode probabilities shown in Figure 4(d)
are dynamically changing to adjust the contributions to
the final distance estimates from CV and CA models.
Specifically, the solved model probability for CA is
getting higher than that for CV while 125>k which
means the distance is changing more like a constant
acceleration model.
(a) Noisy data (red): SNR = 31 dB, IMM filtered data
(green): SNR = 42 dB

6. (b) Velocity estimation )(kx by IMM (true value: -0.25
m/frame for 80≤k )
(c) TTC estimations by IMM (the red curve is true TTC)
(d) CV (black) and CA (red) mode probability by IMM
Figure 4: Example for IMM based vehicle distance
data filtering
COMPARISONS BETWEEN SINGLE MODEL BASED
KALMAN FILTER AND IMM ESTIMATOR - Figure 5
shows the filtering results in a simulated mixed
maneuver including a constant velocity course first and
then a high constant acceleration course. From Figure
5(a), we can see that CV based Kalman filter fails to
track the rapid distance change which is caused by high
constant acceleration. But IMM based filter is able to
capture the rapid distance change by adjusting the mode
probability during maneuver changes.
(a) Filtered data by constant velocity model based
Kalman filter
(b) Filtered data by CV- and CA- based IMM filter

7. (c) IMM mode probability in mixed CV and CA maneuver
Figure 5: Kalman filter versus IMM on a mixed CV
and CA maneuver data set
REAL DISTANCE DATA FILTERING RESULTS - Figure
6 shows one example of the IMM based filtering results
on noisy vehicle distance data solved using the method
described in [2].
From Figure 6(a), we see that the noisy distance data is
smoothed by IMM filter, although we don’t have a
quantitative result because the real distance data is
unknown. The estimated velocity and TTC are shown in
Figure 6(b), 6(c) which indicate somewhat stable velocity
and TTC estimates after the 25th
frame. The mode
probabilities shown in Figure 6(d) indicate that the filter
is dynamically adjusting the output based on mode
probabilities of CV and CA models.
(a) Noisy (red) and IMM filtered (green) distance data
(b) Estimated velocity )(kx by IMM
(c) TTC estimates using IMM
(d) CV (black) and CA (red) mode probability by IMM
Figure 6: Example of IMM based real vehicle
distance data filtering

8. CONCLUSION AND FUTURE WORK
To improve the accuracy of time-to-collision (TTC)
estimation, this study applies Kalman filter to remove
noise from the distance data which is used to estimate
TTC. By analyzing the scenarios that arise in real road
driving, this study proposes to use two different motion
models: constant velocity (CV) and constant
acceleration (CA), to describe the distance changing
dynamics. An interacting multiple mode (IMM) algorithm
is used to dynamically merge the outputs from CV and
CA based Kalman filters. The experimental results on
both simulated and real estimated distance data are
found to be satisfactory and indicate that the proposed
algorithm does improve the signal-to-noise ratio of
distance data.
The proposed algorithm is tested using the distance data
which is estimated using the multiple image features
based vehicle tracking and distance estimation method
described in [2]. Nevertheless, the algorithm is also
applicable to the distance data filtering in other type of
CAS systems such as a radar based CAS system.
REFERENCES
1. Yaakov Bar-Shalom and X.-Rong Li, Thiagalingam
Kirubarajan, Estimation with Applications To
Tracking and Navigation, John Wiley & Sons, INC.,
2001.
2. Yixin Chen, Manohar Das, Devendra Bajpai, Vehicle
Tracking and Distance Estimation Based on Multiple
Image Features, The Fourth Canadian Conference
on Computer and Robot Vision (CRV2007),
Montreal, Canada, May 28-30, 2007, pp. 371-378.
CONTACT
Yixin Chen, PhD
Senior Electronics Systems Engineer
Delphi Corporation
yixin.chen@delphi.com