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

㸮

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

ƐůŝƉĂŶŐůĞĚĞŐ

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

ƚŝŵĞƐ

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]

ƚŝŵĞƐ

(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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IEEE Intelligent Transportation System Magazine, Vol.5, No.1, pp.2841, 2013

[13] Suganuma, N., Uozumi, T., “Development of An Autonomous Vehicle

System Overview of Test Ride Vehicle in The Tokyo Motor Show

2011-”, Proc. of SICE Annual Conference 2012, pp.215–218, 2012

[14] Doi M., Zeng, K., Wada, T., Doi, S., Tsuru, N., Isaji, K., Morikawa,

S., ”A Steering Assist Control System in Curve Road Using Carto-car Communicationh, Proc. of IEEE International Conference on

Intelligent Transportation Systems, pp.1–6, 2013

[15] Khatib, O., “Real-Time Obstacle Avoidance for Manipulators and

Mobile Robots”, International Journal of Robotics Research, Vol.5,

No.1, pp.90–98, 1986

[16] N. Noto, H. Okuda, Y. Tazaki, T. Suzuki, ”Steering Assisting System

for Obstacle Avoidance Based on Personalized Potential Field”, IEEE

International Conference on Intelligent Transportation Systems, pp.

1702-1707, 2012

[17] R. Matsumi, P. Raksincharoensak, M. Nagai, ”Pedestrian Collision

Avoidance by Automatic Braking in Intersection Based on Potential

Fields”, Proceedings of 1st International Symposium on Future Active

Safety Technology Toward Zero-Accident, 2011.

[18] S. Akishita, S. Kawamura, T. Hisanobu, ”Velocity Potential Approach

to Path Planning for Avoiding Moving Obstacles”, Advanced Robotics,

Vol.7, No.5, pp.463–478, 1992

[19] S, Sugiyama, S, Akishita, “Path Planning for Mobile Robot at A Crossroads by Using Hydrodynamic Potential”, Journal of the Robotics

Society of Japan, Vol.16, No.6, pp.839–844, 1998 (in Japanese)

[20] Batchelor, G.K., “An Introduction to Fluid Dynamics”, Cambridge

University Press, 1973

[21] T. D. Gillespie, “Fundamentals of Vehicle Dynamicsh, Society of

Automotive Engineers, 1992

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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

㸮

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

ƐůŝƉĂŶŐůĞĚĞŐ

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

ƚŝŵĞƐ

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]

ƚŝŵĞƐ

(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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