Collision Avoidance Control With Steering

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Collision Avoidance Control with Steering
using Velocity Potential Field*
Naoki Shibata1 , Seiji Sugiyama1 and Takahiro Wada1
Abstract— It is expected that a collision avoidance system
based on steering control could help avoid collisions even
in cases where a collision cannot be avoided by braking
only. For realizing steering-based collision avoidance, accurate
environmental recognition and rapid motion planning in a
highly dynamic environment are needed. The velocity potential
field control method, which was proposed for mobile robot
motion planning, has the advantages of a very low computation
cost, even in dynamic environments with moving obstacles
without any deadlock. In addition, the method generates the
desired velocity vector to be followed but vehicle trajectory.
With this feature, avoidance velocity vector is generated in
realtime even in the case that the obstacles are moving and
their movements cannot be anticipated. Thus, the present
study proposes a collision avoidance method with steering
control by generating a trajectory for obstacle avoidance using
local environmental recognition based on application of the
velocity potential field approach. In addition, a parameter
determination method for the velocity potential function is
derived. A preview steering control method for following the
derived velocity vector is proposed. The simulation results
obtained using vehicle dynamics software demonstrate that the
proposed method can generate appropriate obstacle avoidance
trajectories, even for moving obstacles, and the derived velocity
vector can be tracked by an automobile.

I. I NTRODUCTION
Active safety technologies that prevent collisions, including warning systems and advanced driver assistance systems,
have been developed. Among such technologies, it is expected that collision avoidance systems would dramatically
improve road traffic safety. Consequently, such systems are
already available in the market. In fact, braking-based collision avoidance systems such as the Advanced Emergency
Braking System (AEBS) have contributed to reducing the
number of collisions[1][2][3][4].
Recently, collision avoidance systems that use steering
control for avoiding collisions have drawn attention because
such systems have the potential to further reduce the number
of collisions[5],[6]. Several studies on collision avoidance
using steering control have been conducted. Trajectorygeneration methods for specific and simple situations have
been proposed. For example, Hayashi et al. proposed a
rigorous path-generation method for avoiding a single lead
vehicle by connecting circular arcs based on obstacle motion
prediction [7] [8]. In addition, a few studies have attempted
to derive a methodology for generating a collision avoidance
*This work was not supported by any organization
1 Naoki Shibata, Seiji Sugiyama, and Takahiro Wada are with School
of Information Science and Engineering, Ritsumeikan University, 1-11, Noji-higashi, Kusatsu, Shiga, 525-8577, Japan {[email protected],

[email protected]}.ritsumei.ac.jp

path in complex situations through iterative calculation for
obtaining rigorous solutions. For example, a method that
chooses to avoid a collision using either braking or steering
was investigated based on detailed calculations of multiple
vehicles’ collision courses [9]. Literatuge [10] proposes a
control method to prove safety for various scenarios by
reachability analysis. Most of these approaches require long
calculation time. More recently, fully automated vehicles
have been studied based on environmental recognition methods using external sensors such as LIDARs and cameras
[11] [12] [13]. These studies focus on environment recognition under highly complex and dynamic situations. A
study [14] proposed an automated lateral control method for
vehicle passage through a curve using C2C communication
for reducing the environment recognition load. Currently,
it is thought that these methods focus on stationary driving rather than on crash-imminent situations. Thus, a fast
path-generation method for collision avoidance in a crashimminent situation that can be applied to dynamic and
complex environments such as one with multiple vehicles
is required.
The potential field approach was proposed for path planning to a target position for mobile robots[15]. This approach
is suitable for navigating a robot to a target position even
in complex environments. Recently, a few studies applied
the potential field approach to automobile control[16],[17].
However, there is a concern that in complex situations, it is
difficult to guarantee no deadlock using the potential field
approach. In contrast, the velocity potential field approach is
capable of fast trajectory generation even in dynamic environments with moving obstacles without any deadlock[18],
[19]. In addition, the method generates the desired velocity
directly by differentiating the potential function, which is an
advantage from the automotive application viewpoint.
Thus, the present study proposes a collision avoidance
method with steering control by the velocity potential approach for generating an obstacle-avoidance velocity vector
with local environmental recognition. This methd does not
generate any trajectory to be followed but generates a velocity vector of the host vehicle at every moment of the
driving for avoiding collision. With this feature, avoidance
velocity vector is generated in realtime even in the case that
the obstacles are moving and their movements cannot be anticipated. A parameter determination method for the velocity
potential function is derived as well. In addition, a preview
steering control method for following the derived velocity
vector is proposed. Numerical simulations are conducted
with vehicle dynamics software considering both static and

Y

Obstacle

Vehicle

X



Fig. 1.

Scenario

gyroscope

Steering
servo motor

Steering angle
sensor

C. Velocity Potential

Range Sensor

It is known that the velocity of an incompressible inviscid
fluid can be described by v = ∇φ and the following equation
is satisfied[20].

Wheel encorder

Fig. 2.

between the HV and obstacles on the road boundary can be
obtained by the HV-mounted sensors.
An overview of the proposed collision avoidance system is
shown in Fig. 3. A velocity potential function is constructed
based on the sensor information, and it is used to calculate
the desired velocity vector for collision avoidance. The
desired vehicle sideslip angle βd and the desired vehicle
velocity Vd are determined using the velocity vector and its
norm, respectively (see Section II-G for the detail).
The desired vehicle sideslip angle is converted to the
steering wheel angle using a bicycle model. Finally, an
appropriate feedback controller for steering wheel angle
and vehicle velocity realizes collision avoidance using the
abovementioned variables’ calculated values.

Microelectric vehicle

div v = ∇2 φ = 0
slip angle

Obstacle position
Roadside
State of vehicle

βd

Obstacle avoidance
algorithm
(Hydrodynamic
potential)

Generation of
steering angle

Apedald
Bpedald

velocity

Vd

β

δd

Vehicle

Speed controller

V

Map information

Fig. 3.

Block diagram of proposed system

moving obstacles for validating the proposed control method.
II. C OLLISION AVOIDANCE C ONTROL USING V ELOCITY
P OTENTIAL F IELD
A. Scenario
As the first step in this study, we assume a scenario in
which the host vehicle (HV) is driven on a two-lane road in
a single direction and the HV encounters a lead vehicle (LV)
that is either driving slowly or stationary, as shown in Fig.1.
The present study aims to propose a collision avoidance
control system that avoids collisions with a LV using steering
control as well as braking when the driver does not react to
an imminent crash with the LV.
It is assumed that the HV’s coarse lateral position within
the lane markers is known based at least on the distances
from the lane markers, which are determined using HVmounted sensors such as a laser range finder (LRF) and
cameras. In addition, it is assumed that the relative position
to the LV and the LV’s approximate size are determined by
the car-mounted sensors. It should be noted that only local
information such as LV width was available in the first stage
of avoidance because the car-mounted sensors are capable of
measuring local information only.
B. System Overview
In this study, COMS, a micro electric vehicle (Toyota Auto
Body Co., Ltd.) (Fig. 2), is assumed as the HV. In addition,
it is assumed that the HV status and the relative distance

(1)

where φ is referred to as the velocity potential function. The
function φ is a harmonic function and is continuous in all
regions except for singularity points such as a source and a
sink. This means that there is no deadlock in the region. In
the case of a flow on a two-dimensional plane, the velocity
potential can be expressed as the complex velocity potential,
eq. (2), by introducing the complex plane.
f (z) = φ(x, y) + iψ(x, y)

(2)

where x, y denote Cartesian coordinates of the position in
the plane, i denotes the imaginary unit, and z = x + iy.
ψ(x, y) and φ(x, y) are the stream function and the velocity
potential, respectively.
The conjugate complex velocity is obtained by differentiating the velocity potential with respect to z (Eq. (3)).
df (z)
= u(z) − iv(z)
(3)
dz
where u(z) and v(z) represent the velocity in x and y
directions, respectively. An object can move along a flowline
if this velocity is achieved. It should be noted that the
principle of superposition can be employed in the case of
multiple elements. Thus, a complex potential field can be
constructed by superimposing multiple basic flow elements,
as shown in Fig. 4 [20].
The flow of a fluid whose potential function is given by
harmonic function is suitable for vehicle motion generation
because it guarantees no collisions with obstacles and no
deadlock. In addition, the desired velocity is generated in real
time by differentiating the potential function. This is highly
advantageous for vehicle control. Therefore, the harmonic
velocity potential field of an incompressible inviscid fluid is
used for automobile collision avoidance. Please note again
that the proposed method does not generate any vehicle
trajectory but does generate velocity vector at every moment
¯ (z) =
v

y

y

y
U

(a) Uniform flow

0

x

a

x

0

y

α

(b) Cylinder

Fig. 4.

x

(c) Vortex

0

x

(d) Source

P (z) = U e−iα z

Flow elements

3
4
5

(8)

where α denotes the angle of flow from x axis, and U denotes
the uniform flow velocity.
Finally, the velocity potential field of the scenario is given
by eq.(9)

Points where potential function located
1

The locations and numbers of points at which the potential
functions were located, were determined by trial and error.
In addition, the uniform flow shown in Fig.4-(a) is used for
describing the travel direction or road direction. The potential
function of the uniform flow is given by eq.(8).

6

2

f (z) = P (z) +

6


{Gj (z) + C j (z) + D j (z)}

(9)

j=1

Fig. 5.

Potential Field Design

E. Parameter Determination Method

for realizing collision avoidance. Thus, the proposed method
is robust to changes in environment.
D. Potential Field Design
In this section, we propose a method for designing the
velocity potential f (z) of the scenario described in Fig. 1.
The potential function elements are set to six points on the
obstacle and the road, as shown in Fig. 5. The velocity
potential function at point j is defined by eq.(4).
f j (z) = Gj (z) + C j (z) + D j (z)

(4)

where z (= x+iy) is the position of interest on the plane, and
the functions Gj (z), C j (z), and D j (z) denote the velocity
potential functions describing flow around a cylinder, vortex,
and source, respectively, and are given by eqs.(5), (6), and
(7) (Fig.4).

C j (z) =

a2j
z − zj
ikj log(z − zj )

D j (z) =

mj log(z − zj )

Gj (z) =

In this section, a method for determining the potential
functions’ parameters is derived. A set of parameters was
determined so that it reproduces the driving pattern corresponding to a given driving data as closely as possible. A
fixed-base driving simulator (DS) was used to measure the
collision avoidance trajectories of a driver. The subject drove
the DS from the different three initial positions after getting
used to the simulator.
Please note that the cylinder parameters were fixed as ai =
0.01 for i = 1, 2, 3, 4 and ai = 0.55 for i = 5 and 6 by trial
and error. In addition, magnitude of the uniform flow, U ,
is determined as the initial HV value. Thus, the parameters
of the potential functions of the vortex and the source are
determined by solving the following optimization problem.
find
ms , k s ,

s ∈ {1, 2, · · · , 6}

(10)

which minimize
(5)

J =−

(6)
(7)

where zj (= xj + iyj ) denotes the position of point j
or cylinder, vortex, and source. Scalars aj , kj , and mj
represent the radius of the cylinder, intensity of the vortex,
and intensity of the source, respectively. Please note that the
flowline described in Fig.4-(b) is generated by the potential
function Gj (z) in eq.(5) with the uniform flow given in
eq.(8).
The six points’ locations are as follows (Fig.5):
Points 1 and 2 are set to both the road boundary points
that are closest to the HV and move along with the HV.
Points 3, 4, and 5 are fixed to both ends and center of
the LV’s back.
Point 6 is set on the right side of the LV, which is the
closest from the HV, and this point moves along with the
HV.

s.t.

n
L 


ˆ (xlj , yjl )
v(xlj , yjl )T v

l=1 j=1

||v(xlj , yjl )||||ˆ
v (xlj , yjl )||

m6 = m2 and k6 = −k2

(11)

(12)

where v(xli , yil ) denotes the observed velocity vector at
ˆ (xlj , yjl ) denotes the velocity vector
position (xli , yil ) and v
calculated using the velocity potential function. Scalars L
and n denote the number of driving trials and the number
of observed points during each driving trial, respectively;
L = 3 and n = 20 were selected in our experiments. Fig.6
shows the measured three driving trajectories. The twenty
observed points for each trial were determined by dividing
its trajectory into x-direction. A quasi-Newton method was
used for determining the optimal parameter values.
Please note that parameter optimization of the potential
field is much simpler than that of the force potential because
the cost function includes only the velocity vector. When the
force potential is used, integration is required for calculating
velocity.

F. Parameter Identification Results
Table I lists the parameters identified from the driving
data. Fig.7 shows the flowlines calculated using the potential
function with the identified parameters, as well as one of
the three observed vehicle trajectory for comparison. This
figure indicates that the potential function with the identified
parameters can generate a smooth trajectory within the road
boundaries while avoiding any collision with the LV. In
addition, it is found that the flowline generated by the model
agrees well with the observed driving data.
TABLE I
I DENTIFIED PARAMETERS OF POTENTIAL FUNCTION
parameter
kj
mj

1
-14.08
52.02

2
32.74
10.00

3
148.18
220.50

4
160.97
213.83

5
169.00
203.31

6
-32.74
10.00

The desired sideslip angle of the vehicle βd (t) is determined using the velocity vector.
βd (t) = arg v¯ (z(t + τ ))

where τ denotes the preview time, τ = 0.15s in the present
paper, and z(t + τ ) = x(t + τ ) + iy(t + τ ) is defined in
eq.(14).
z(t + τ ) = z(t) + V τ ei(θ(t)+β(t))

1
0

y[m]

-1

v(x20,y20)
v(x21,y21)
v(x22,y22)
v(x0,y0)
v(x1,y1)
v(x2,y2)
v(x41,y41)
v(x40,y40)

-2

Driving data1
Driving data2
Driving data3

v(x42,y42)

-3

v(x57,y57) v(x58,y58)v(x59,y59)

-4

v(x17,y17) v(x18,y18) v(x19,y19)

-5

v(x39,y39)
v(x37,y37) v(x38,y38)

-6
0

1

2

3

4

5

6

Fig. 6.

7

8

9

10
x[m]

11

12

13

14

15

16

17

18

19

20

(14)

where θ(t) denotes the vehicle’s yaw angle at time t.
The desired steering angle of a vehicle that realizes the
desired sideslip angle βd is calculated using eq.15 under the
assumption that the vehicle has a steady turning circle[21].

δd (t)
2

(13)

=

2
1 + AV(t)

1−

l

M lf
2 lr
2l lr Kr V(t)

βd (t)

(15)

where A is the stability factor. Scalars lr and lf represent
the distances of the rear and front wheel axles from the
vehicle’s center of gravity, respectively, and l denotes vehicle
wheelbase. Scalars Kr , M , V denote the cornering power,
vehicle weight, and vehicle velocity, respectively.
The desired steering angle given by eq.(15) is realized with
a simple position feedback scheme such as PD feedback of
the steering angle δ.
III. C OLLISION AVOIDANCE S IMULATION

Observed vehicle trajectory

A. Simulation condition
2
Flow line

0
y[m]

Driving data 2
-2
-4
-6
0

2

4

Fig. 7.

6

8

10

12

14

x[m]

16

18

20

22

24

26

28

30

Flowlines by velocity potential and driving data

G. Preview Steering Control for Following Desired Direction

Ff

v

δ

β

Fr
lf
lr

Fig. 8.

Bicycle model

In this study, the kinematics of a bicycle model shown in
Fig. 8 is used for determining the desired steering angle from
the calculated velocity vector.

The validity of the proposed collision avoidance method
is examined by computer simulations. The proposed control
method was implemented in CarSim, a vehicle motion simulation software (MSC Co., Ltd.). The scenario described in
section II was used for the simulation. The vehicle parameter
of the micro electric vehicle COMS were used. The test
course was a two-lane road with 4 m wide for each lane.
The vehicle was driven on the center of the left lane at about
20 [km/h], approaching the LV.
The activation timing of the collision avoidance function
was determined as time-to-collision TTC = 1.4 s. The
LV location was changed in four manners as follows for
examining the robustness of the method to changes in
obstacle location:
Condition 1: The LV is parked throughout the simulation.
The LV’s center of mass (COM) in the lateral direction is 1
m left of the HV’s lateral position.
Condition 2: The LV is parked throughout the simulation.
The LV’s COM in the lateral direction is same as the HV’s
lateral position.
Condition 3: The LV is parked throughout the simulation.
The LV’s COM in the lateral direction is 1 m right of the
HV’s lateral position.
Condition 4: First, the LV is parked in the same location
as in condition 1. Then, the LV is moved in the −45 deg
direction at a speed of 3.6 km/h.

ϱ
ϰ
ϯ
Ϯ
ϭ
Ϭ
Ͳϭ
ͲϮ
Ͳϯ
Ͳϰ
Ͳϱ

ƐůŝƉĂŶŐůĞ΀ĚĞŐ΁

slip angle [deg]

3
2
1
0
-1
-2
-3
-4
-5

Desired slip angle
Resultant slip angle
1

3

time[s]

5

ĞƐŝƌĞĚƐůŝƉĂŶŐůĞ
ZĞƐƵůƚĂŶƚƐůŝƉĂŶŐůĞ
ϭ

7

ϯ

(a) Vehicle slip angle in condition 1

25
20
Desired velocity
Resultant velocity

15
10
3

time[s]

5

25
20
Desired velocity
Resultant velocity

15

7

1

3

time[s]

5

7

(b) Vehicle velocity in condition 3
Vehicle position y [m]

Vehicle position y [m]

(b) Vehicle velocity in condition 1
2
0

-2

2
0

-2

Desired trajectory
Avoidance trajectory in condition 1

-4

Desired trajectory
Avoidance trajectory in condition 3

-4

-6

-6

5

10

15

20
Vehicle position x [m]

25

30

35

5

10

15

(c) Trajectory in condition 1
Simulation results under condition 1

4
3
2
1
0
-1
-2
-3
-4
-5

Fig. 11.

Desired slip angle
Resultant slip angle
1

3

ƚŝŵĞ΀Ɛ΁

20
Vehicle position x [m]

25

30

35

(c) Trajectory in condition 3

5

Simulation results under condition 3

5
4
3
2
1
0
-1
-2
-3
-4
-5

slip angle [deg]

Fig. 9.

slip angle [deg]

ϳ

10
1

7

Desired slip angle
Resultant slip angle
1

3

(a) Vehicle slip angle in condition 2

time[s]

5

7

(a) Vehicle slip angle in condition 4

30

30

25

velocity [km/h]

velocity [km/h]

ϱ

30
velocity [km/h]

velocity [km/h]

30

20
Desired velocity
Resultant velocity

15
10
1

3

time[s]

5

25
20
Desired velocity
Resultant velocity

15
10

7

1

3

(b) Vehicle velocity in condition 2

time[s]

5

7

(b) Vehicle velocity in condition 4

2
0

-2

10

15

0

Desired trajectory
Avoidance trajectory in condition 4

-4

-6
5

2

-2

Desired trajectory
Avoidance trajectory in condition 2

-4

Vehicle position y [m]

Vehicle position y [m]

ƚŝŵĞ΀Ɛ΁

(a) Vehicle slip angle in condition 3

20
Vehicle position x [m]

25

30

35

-6
5

10

(c) Trajectory in condition 2
Fig. 10.

Simulation results under condition 2

The HV’s velocity was controlled to the velocity calculated from the velocity potential function, while the upper
limit of velocity was set to the HV’s initial velocity. The parameters of the potential functions identified in the previous
section were used here.
B. Simulation Results
Figs. 9–12 show the simulation results under all conditions. The blue lines, which is written as desired trajectory in
Figs. 9–12(c) denotes the trajectories calculating by integrating the desired velocity from the current position to 0.5s later.
Overall, the resultant trajectories under all these conditions
successfully avoided collision with the LV, while driving

15

20
Vehicle position x [m]

25

30

35

(c) Trajectory in condition 4
Fig. 12.

Simulation results under condition 4

a smooth trajectory and not departing from the road. In
addition, it is confirmed that the resultant vehicle trajectories
are similar to the generated trajectory by integrating the
desired velocity vector. As can be seen in Figs.9–11, the
proposed method yields similar collision avoidance results
regardless of the difference in LV position, even with the
same control parameters. A resultant collision avoidance
trajectory with a larger inclination toward the right was
automatically generated for condition 3 than for conditions
1 and 2. Figure 12 shows that the proposed control scheme
avoids collisions even if the LV moves by changing the
desired path in real time.

From Figs.9–11-(c), it is found that the vehicle drives
almost along the calculated desired flowline even existence of
the vehicle dynamics. It is thought that the preview control
leads the results. In contrast, some delay is found in the
case with a moving obstacle (Fig.12). However, it should be
noted again that even in this case, no collision with the LV
occurred in the simulation. Moreover, Fig. 13 shows that a
larger lateral acceleration was required for avoidance action
in tighter situations, i.e. condition 4.
lateral acceralation [g]

0.3
0.2
0.1
0

-0.1

condition 1

-0.2

condition 3

condition 2
condition 4

-0.3
1

3

Fig. 13.

time[s]

5

7

Lateral acceleration

These results indicate that the proposed method automatically generates collision avoidance trajectories corresponding
to the LV’s position without changing the velocity potential
function parameters. In addition, stable collision avoidance
can be achieved even in the case that the HV cannot
realize the desired velocity vector because another velocity
vector is generated at the next moment using the measured
environmental information at the moment.
IV. C ONCLUSION
A steering-based collision avoidance method for automobiles was proposed using a velocity potential approach.
Desired velocity vecotors at each moment were generated
in realtime. A driving data–based parameter determination
method was proposed. In addition, a preview steering control
method was proposed using the desired velocity vector. The
results of a simulation that considered vehicle dynamics
showed that collision avoidance was successfully realized
using the proposed method. Furthermore, it was shown that
the parameters of the same velocity potential can be used
in different vehicle locations for collision avoidance even
with the moving obstacle. The results indicate the robustness
of the velocity potential approach against spatial errors
because the proposed method can automatically generate
desired velocity vectors that realizes similar flowlines near
the original trajectory for collision avoidance. This feature
can realize smooth changes of the desired trajectory in the
case of a delay in the vehicle’s or the obstacle’s motion.
Simulation conditions should be added to prove the collision avoidance ability of the proposed method. Experimental
verification of the proposed method using a vehicle is a
future work. Furthermore, expansion of the proposed method
to situations with multiple vehicles is important as a future
study. Finally, a method to prove safety is necessary like
[10]. It is expected that proof of the collison free is easily
realized because the resultant trajectory can be calculated by
integrating the desired velocity.

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