weiwen deng, shuqing zeng, qingrong zhao and john daichenyanh/.downloads/ijvd_submission-0… ·...

Int. J. of xxxxxxxx, Vol. x, No. x, xxxx

Copyright © 2009 Inderscience Enterprises Ltd.

Modeling and Simulation of Sensor-Guided Autonomous Driving

Weiwen Deng, Shuqing Zeng, Qingrong Zhao and John Dai Global Research & Development Center

General Motors Company

Warren, MI 48307, USA

Abstract: This paper presents a modeling and simulation method for sensor-

guided autonomous driving. A generic model of range sensing and object

detection in 3D space is discussed first that represents their high-level functions.

The low-level physical characteristics of range sensing and object detection are

further investigated based on frequency modulated continuous wave (FMCW)

radar which is gaining wide popularity in automotive applications. These are the

enablers to modeling and simulation of vehicle interactions with one another

under traffic and with surrounding environment. A closed-loop adaptive cruise

control is then used as an example to demonstrate the vehicle limited

autonomous driving with the proposed model and method that has been shown

to be valid, effective and numerically efficient.

Keyword: Range sensing; Object detection; Modeling and simulation; Vehicle

active safety; Autonomous driving

Biographical notes:

Dr. Weiwen Deng has been working for General Motors R&D Center since

1996. He holds B.Sc. and M.Sc. degrees in Mechanical Engineering from China,

and M.Sc. in Mechanical Engineering and Ph.D. in Electrical Engineering from

U.S. He is currently a Staff Researcher at GM and the twice recipients of the

`Boss' Kettering Award, the most prestigious award in GM for technical

invention and innovation. He holds 17 US patents with another 16 pending, and

is the author of over 30 journal and conference publications. He also serves as

editor and associate editors for several international journals.

Dr. Shuqing Zeng received his PhD degree in Computer Science from Michigan

State University in 2004. He joined General Motors R&D Center in 2004 as a Sr.

Researcher. He was a member of Tartan Racing team who won the first place of

The Defense Advanced Research Projects Agency (DARPA) Urban Challenge

in 2007. He served as a reviewer to IEEE Transactions on Pattern Analysis and

Machine Intelligence and as a judge to Intelligent Ground Vehicle Competition

(IGVC). He served as the newsletter editor (2005-2007) of Autonomous Mental

Development TC, IEEE Computational Intelligence Society. He is currently an

associate editor of International Journal Humanoid Robotics (IJHR).

Dr. Qingrong Zhao is a Sr. Consultant to GM R&D Center from Wipro

Technologies (USA) since 2006 on various projects related to vehicle

autonomous driving and controls on vehicle autonomy. She holds B.Sc. and

M.Sc. degrees from Northwestern Polytechnical University of China and Ph.D.

from University of Cincinnati, all in Electrical Engineering. She is the author of

a dozen peer-reviewed papers of international journals and conferences, and co-

inventor of two filed US patents, and has received numerous honors and awards

from schools and government officials. Currently she serves as an associate

editor of International Journal of Vehicle Autonomous Systems.

Dr. John Dai received his Ph.D. degree in Systems Engineering from Oakland

University in 2002. His research interests include dynamic system modeling and

control methodology. He currently works in General Motors powertrain

engineering as a Sr. Engineer, developing cutting-edge engine control algorithm

and software.

1. Introduction

As new technologies emerge in recent years in the automotive industry, the

demands to extend the traditional human-driving capabilities have been greatly

increased with various active safety and driver-assistance features that enhance

driver comfort, convenience, and safety, which eventually will lead to full

sensor-guided autonomous driving. The trends have been largely driven not only

by the demands, but also by the technology progression and readiness of, in

particular, the low-cost sensors, such as radar, lidar and camera.

It is well known that physical on-road vehicle testing can be very costly, time-consuming, and sometimes even impossible. In addition, physical road

testing is typically unrepeatable and with limited testing scenarios and

conditions. To effectively develop, test and verify the sensor-guided autonomous

driving systems, it has become critically important to properly model and

simulate the system, its surrounding environment and traffic conditions with

adequate fidelity in sensing and object detection. This paper presents a modeling

and simulation approach for sensor-guided autonomous driving, in particular, the

object sensing and detection, to enable the research and development, early

testing and verification on features and functions of active safety and

autonomous driving under a virtual environment.

A sensor-guided autonomous driving system is a complex system that

includes a subject vehicle along with principle other vehicles (POV), road and

infrastructure of the road and traffic, surrounding environment including weather

and light conditions, and many others which may impact or impair autonomous


driving. In addition, autonomous driving is not necessarily driverless, especially

at its early stage; human driver plays an important role. Since vehicle and driver modeling has been extensively discussed in the literature, and is relatively mature,

the focus of this paper is on object sensing and detection modeling, which is one

of the key enabling functions for simulating vehicle interaction with others and

its surrounding environment.

Although modeling object sensing and detection has long been researched in

the literature (Bacso and Bertolini, 1988 and Donohue, 1997), the progression in

senor technology has been greatly advanced during last few years. Thus, those

models with much of the focuses on the physics become less flexible, or even less useful for different sensors used today. In addition, physic model of a sensor

for object detection may not be always feasible practically, due to the complexity

and stochastic nature of the sensor, object, and often unstructured environment.

This paper presents a generic modeling approach with combined functional

and physic models of sensors to abstract range sensing and object detection: in a

higher level, the functions of object sensing and detection are to identify the

range, direction and speed of both moving and stationary objects. In a lower level,

such functions can include physical phenomena such as surface curvature,

material absorption and reflectivity, and be represented by capturing the

stochastic nature of the physics with physically interpretable parameters, such as

noise.

The higher-level geometric and functional representation of range sensor and

object detection in 3D space is briefly discussed in Section 2. More detail can be

found in a separate paper (Deng, et al, 2009). Section 3 presents an enhanced

modeling of range sensing and object detection with a radar model, as an

example, of a popular kind in automotive industry with frequency modulated

continuous wave (FMCW). This model takes the input of the scatters from the

geometric models discussed in Section 2, and generates object tracks with

superimposed noises from various sources. A closed-loop simulation of an

autonomous driving with range sensing and object detection, and control has

been conducted in Section 4. This further demonstrates that the proposed model

and algorithm are valid and effective in enabling the simulation of vehicle active

safety systems with multiple vehicle interactions under traffic. The paper

concludes in Section 5.

2. Functional Modeling of Range Sensing and Object Detection

To abstract the generic functions of range sensing and object detection, a geometric approach is first taken to represent the sensor beam to be a cone and the object to be a cuboid. Then the idealized sensing model is to generate the discretized scatters from the object(s). Noise will then be superimposed to the scatters and generates a more realistic object tracks with range, range rate and

z

x β

x α

y

R

sensor beam

x⋅tan(β/2)

y x⋅tan(α/2)

z

azimuth angle, to reflect stochastic nature of sensing physics.

A. Geometric Representation of Sensor Beam, Vehicle and Objects

A sensor beam is assumed to have attributes of maximal range R , azimuth

angle α and elevation angle β . The cone that simulates the sensor view volume

(SVV) is expressed in (1) where ( , , )x y z denote the coordinates of a 3D point.

Figure 1 shows the three orthographic projections of the sensor beam.

≤++

≤+

2222

22

2

22

2

1

)2

(tan)2

(tan

Rzyx

x

z

x

y

βα

(1)

Figure 1: Sensor beam (cone) in 3D space

An object is geometrically represented by a rectangular cuboid, as shown in

Figure 2, with 6 rectangles (faces), 12 edges, and 8 vertices. Its geometric

attributes are described by {L W H a b c}.

z3

y3 y3

x3

L

a

b

W

H c


Z3

Y3

X3

OBJECT

Z2

Y2

X2

X1

Z1

Y1

Y0

X0

Z0

VEHICLE

SENSOR BEAM

WORLD

Figure 2: Geometric representation of object (a cuboid)

The axis and coordinate systems used in range sensing and object detection

are shown in Figure3 for world, vehicle, sensor, and object respectively. Under

world frame, denoted by o0x0y0z0, vehicle position and orientation (as yaw-pitch-

roll angles) can be defined as:

[ ]T

vzvyvx ppp=vp, [ ]T

vvv φθψ=vo

Similarly in vehicle frame, denoted by o1x1y1z1, sensor mounting position sp

and position error s∆p , and sensor mounting orientation so (as yaw-pitch-roll

angles) and orientation error s∆o are defined, respectively, as:

[ ]T

szsysx ppp=sp , [ ]T

szsysx ppp ∆∆∆=s∆p

[ ]T

sss φθψ=so , [ ]T

sss φθψ ∆∆∆=s∆o

Figure 3: Axis and coordinate systems for world (o0x0y0z0), vehicle (o1x1y1z1),

sensor, and object (o3x3y3z3)

z3

x3

y3

6

5

8

7

3

2

1

4

object cuboid

Under the world frame, the origin of the object coordinate system, its

orientation in yaw-pitch-roll sequences are defined, respectively, as:

[ ]T

ozoyox ppp=0p , [ ]T

ooo φθψ=0o

Range sensing is to compute the sensing attributes, such as range, range rate,

object azimuth angle and object elevation angle, while object detection is to

determine if the object is falling into the SVV.

In order to compute the sensing attributes, the position and orientation of the

object (a cuboid), under the world frame, need to be first converted to the sensor

frame, under which the sensor beams have explicit mathematical expression.

To transform the coordinates of a point from world frame to vehicle frame,

the consecutive rotations are Euler angles (Gillespie, 1992), or yaw, pitch, and

roll angles [ψ,θ,φ].

Thus the rotation matrix is given by φx,θy,ψz,

1

0 RRRR = :

−

=

100

0)cos()sin(

0)sin()cos(

ψψ

ψψ

ψz,R

−

=

)cos(0)sin(

010

)sin(0)cos(

θθ

θθ

θy,R

−=

)cos()sin(0

)sin()cos(0

001

φφ

φφφx,R

Consider both rotational and translational transformation, a homogeneous

transformation matrix from the world frame to the vehicle frame is defined

as 44×∈ R1

0H :

=

× 131

v

1

01

00

pRH

(2)

Similarly, the transformation from the vehicle frame to the sensor frame can

be represented by the homogeneous transformation matrix 2

1H :


+=

× 131

ss

2

12

10

∆ppRH

wheressssss ∆φφx,∆θθy,∆ψψz,

2

1 RRRR +++= , and

∆+∆+

∆+−∆+

=+

100

0)cos()sin(

0)sin()cos(

ssss

ssss

ψψψψ

ψψψψ

ss ∆ψψz,R

∆+∆+−

∆+∆+

=+

)cos(0)sin(

010

)sin(0)cos(

ssss

ssss

θθθθ

θθθθ

ss ∆θθy,R

∆+∆+

∆+−∆+=+

)cos()sin(0

)sin()cos(0

001

ssss

ssss

φφφφ

φφφφss ∆φφx,

R

Further, we can define homogenous transformation matrix from the world

frame to the sensor frame as 2

0H , and from the sensor frame to the object frame

as 3

2H , respectively:

;1

2

02

0

=

×31

2

0

oRH

=

× 131

2

3

23

20

dRH

where

3

0

T1

0

T2

1

3

0

11

0

12

1

3

0

0

1

1

2

3

2 R)(R)(RR)(R)(RRRRR ==⋅⋅= −− (3)

2

1

1

0

2

0 RRR ⋅= , )o(p)(Rd 2o

T2

02 −= , )4,3:1(2

02 Ho =

where o2 is the origin of the sensor frame projected to the world frame, and d2 is

the origin of the target frame projected to the sensor frame.

Under the object frame, all the vertices of a cuboid can be expressed as simple

functions of its geometric size, and location with respect to the origin of the

cuboid:

8,,1),,,,,,( ⋯== iHWLcbaf ioi )(p

With properly defined coordinate transformation, the vertices of an object

(cuboid) can be defined now with respect to sensor frame as:

8,,1,11

3

2 ⋯=

=

i

oii )(pH

p

B. Abstracted Range Sensing and Object Detection

The problem of object detection can be abstracted geometrically as the

intersection(s) of a cone (sensor beam) with one or more cuboids (objects) in 3D

space, that is, if an object falls into sensor view volume (SVV), then the sensing attributes can be calculated accordingly. Furthermore, the intersection between a

cuboid and a cone can be regarded as the intersection of one or more faces of the

cuboid with the cone. Therefore, the problem becomes to detect which face(s)

and how they intersect with the cone, and then to determine the minimum

distance from the sensor origin to the face(s), and the associated azimuth and

elevation angles.

The method and algorithm described below are with respect to the sensor

frame (denoted as o2x2y2z2), into which the coordinates of the object vertices, planes, and their norms are all transformed.

Denote eight vertices of an object as821 p,,p,p ⋯ , along with their

coordinates defined under the sensor coordinate system as:

[ ] .8,,1, ⋯== izyxT

iiiip

Among six faces of a rectangular cuboid, the “visible” face(s) to the sensor

beam are those who are facing the sensor beam emission direction. To determine

if a face is “visible”, denote the four sequenced points of the face

as{ }4321 pppp . The positive normal vector of the surface can be calculated

and normalized as:

[ ]11)(

)(−∈

−×−

−×−=

=2312

2312

p(p)pp

p(p)ppn

z

y

x

n

n

n

The face is “visible” to sensor beam only if

0≤• nn2X

(4)

where T]001[=2xn denotes the unit vector of the X axis in the sensor coordinate


system. Therefore, the criterion (4) can be alternatively expressed as .0≤xn

For each of the “visible” faces, the next step is to determine if it intersects with the cone, a solution of which is usually difficult to obtain directly, as the

face-and-cone intersection very often causes ill-shaped boundary shapes and

therefore complicates the analysis. In this paper, we propose an effective method

that simplifies the face-and-cone intersection problem into a problem with

intersection between a line and a cone. A “visible” face is first discretized into a

number of parallel lines, as illustrated in Table 1 and Figure 4. If any of the lines

on the face intersects the cone, then intersection between the face and the cone is detected. This way, the points of intersection can be calculated more easily and

accurately.

Figure 4: Illustration of discretization of a face into lines

Table 1. Discretization of a face plane into lines

Line

number 1

st vertex (q1) 2

nd vertex (q2)

L1 p1 p4

L2 1

1•

−

−+

N

12

1

ppp 1

1•

−

−+

N

43

4

ppp

L3 2

1•

−

−+

N

12

1

ppp 2

1•

−

−+

N

43

4

ppp

… … … LN p2 p3

Denote q1, q2 as the two vertices of a line. The intersection between the line

and the cone can be discussed in two cases as below.

Case A: both q1 and q2 are inside the cone

L2

L3

L1

LN

p4

O

Z

Y

X

p3

p1

p2

In this case, the face-and-cone intersection is given by the fact that both

vertices of the line are inside the cone. Then the minimum distance from the sensor origin to the line may exist at one of the two vertices, or a point (denoted

as q) in between. For the latter, the minimum distance must also be the shortest

distance. The point q can be calculated with the following fomula:

)( 121 qqqq −+= λ

where2

12

121

qq

)q(qq

−

−•=λ and λ∈(0, 1) .

Case B: all other conditions

In this case, the line may either intersect the cone surface, or the spherical-

frontal surface, or both, or none of them. Any point q on the line from q1 to q2

can be expressed as

)( 121 qqqq −+= λ (5)

where λ∈[0, 1]

If q is an intersection point on the cone surface, it must satisfy the following

constraints

<<

≤++

=+

>

πβαβα ,0,1

)2

(tan)2

(tan

0

2222

22

2

22

2

Rzyx

x

z

x

y

x

(6)

If q is an intersection point on the spherical-frontal surface, it must satisfy the

following constraints

<<

=++

≤+

>

πβαβα ,0,1

)2

(tan)2

(tan

0

2222

22

2

22

2

Rzyx

x

z

x

y

x

(7)

Combining (5) & (6), or (5) & (7), we can calculate if there is a solution for q,

and then determine if the line intersects with the cone. If intersection occurs, we

can seek the point of minimum distance to the sensor origin from the line section

inside the cone. We can similarly find that point as Case A.


With the above analysis, the minimum distance from the sensor origin to the

intersection face can be determined as the least of minimum distances from the sensor origin to each of the lines on the face. As defined by sensing attributes,

range is the least of minimum distances from the sensor origin to the object

“visible” face(s), while the range rate can be simply obtained by taking its

derivative. Given minimum distance point T

zyx ],,[=p in sensor frame, the

azimuth angle aξ and elevation angles

eη of the object, as shown in Figure 5, can

be computed respectively as:

( )0 0atan2( , ) 180 / , 90 90a a

y xξ π ξ°= • ∈ −

( )0 0atan2( , ) 180 / , 90 90e e

z xη π η°= • ∈ −

Numerical simulation under Matlab has demonstrated, as shown in Figure 6

and 7 with some of the results, that the proposed method and algorithm has taken into account all possible cases, and can effectively determine if a cuboid

intersects with a cone under full 3D space, or if an object is fallen into SVV

(sensor view volume).

Figure 5: Object azimuth angle ξa and elevation angle ηe

ξa

ηe

(x, 0, z)

(x, y, 0)

p (x, y, z)

z2

x2

y2

sensor beam

Figure 6: Object detection: cuboid surface is normal to the cone center axis

Figure 7: Object detection: the cuboid intersects the frontal surface of cone

3. Physical Modeling of Range Sensing and Object Detection

The physical mechanism of range sensing and object detection with a range sensing (or finder) device is in principle through the emission of a particular wave, such as electromagnetic, sound or light wave, that is reflected by the object and detected by a receiver followed by the recording of its echo. In addition to the reflection, the wave can also be absorbed or scattered during this process.

0

10

20

30

40

50 60

70-20

-10

0

10

20

-20 -15 -10

-5

0

5

10

15

20

Y axis

X axis

Z axis

-10

0 10

20

30

40

50

60

-20 -10

0 10

20

-20

-10

0

10

20

X axis

Y axis

Z axis


In this section, a basic radar signal processing method is presented (Winkler,

and Schoor, 2007). Based on the method, a high-fidelity model is built that models the measurement process of the range, range rate, and azimuth angle of a

target for frequency-modulated continuous wave (FMCW) radars which have

gained significant popularity in automotive applications. Thus, without loss of

generality, FMCW radars are chosen for this study.

The key principle for FMCW radar is homodyne, i.e., detecting frequency-

modulated radiation by nonlinear mixing with radiation of a reference frequency.

Figure 8 shows the FMCW radar block diagram. The CW signal is modulated in

frequency to produce a linear chirp which is radiated toward a target through the TX antenna. The return radiation is collected by four receive channels with

identical distances between two adjacent receive antennas, forming a uniform

linear array (ULA). This array enables direction of arrival (DOA) estimation

using electronically scanned scheme or high-resolution digital beam-forming.

The return signals are mixed with the TX reference signal and fed into the low-

pass filter. The yielded beat signals are digitized by the analog-to-digital circuit

and fed into the signal processing module.

Figure 8: FMCW radar system diagram

3.1. Range and Doppler Processing

The basic idea in FMCW is to generate a linear frequency ramp (chirp). The

transmission frequency for one ramp with bandwidth B and duration T between

[0, ]T can be written as

( )T c

Bf t f t

T= + (8)

CW

Mu

ltiple p

ow

er splitter

TX

RX1

Low-pass

filter ADC

Signal

processing RX2

RX3

RX4

Chirp

where cf is the carrier frequency. We apply frequency modulation (FM)

principle, and after integration, the phase ( )T tϕ of the transmitted signal

cos( ( ))T tϕ becomes

0

2

02

1)(2)(2)( ϕπττπϕ −

+== ∫ t

T

Btfdft c

t

TT (9)

where 0ϕ is the initial phase. The phase of the down-converted signal (beat)

( )tϕ∆ from the mixed signal cos( ( ))cos( ( ))T Tt tϕ ϕ τ− is:

2( ) ( ) 2 ( )

2c

B Bt t f t

T Tϕ ϕ ϕ τ π τ τ τ∆ = − − = + − (10)

where τ is the delay between the transmitted and received signal of a target. For

ASDAS application, / Tτ is sufficiently small, and the last term in (10) can be

neglected. If the target at distance R with constant velocity v is assumed, this

leads to 2( )R vt

cτ

+= , where c is the speed of light in vacuum Ignoring the

high-order term, thus (10) becomes

2 2 2

( ) 2 c cf R f v BR

t tc c Tc

ϕ π

∆ = + +

(11)

The received beat signal B cos( ( ))s tϕ= ∆ is sampled with an interval

AT .

The samples are multiplied with a window function ( )w n , and zero padding is

performed, before a fast Fourier transform (FFT). A peak at frequency IFf can be

detected:

IF

2 2cf v BRf

c Tc= + (12)

Neglecting the Doppler shift term (the first term) in (12), it can be easily seen

that the range resolution is determined by the bandwidth B and the sweep

durationT :

2

A Z

TcR

BT N∆ =

where ZN is the size of the FFT window.

We note that (12) is ambiguous in a sense that the target's range and range rate cannot be determined simultaneously with one frequency ramp. Multiple ramps

are needed to resolve the range-Doppler ambiguity. Figure 9 shows L identical

end-to-end frequency ramps. The ramp repetition interval is denoted by RRIT .

The phase drift φ∆ across contiguous sweep cycles can be derived using the


following 2D-FFT frequency analysis.

Figure 9: Ramp generation for 2D-FFT

Let B,2Ds denote the 2D temporal-spatial signal generated by the target (R,v) at

different chirp sweep cycles denoted by the variable l :

RRI RRIB,2D

22

RRI

2 ( ) 2 2 ( )exp 2

2 2 2exp 2

c

c c

f Ri

c cc

f R vT l f v B R vT ls A i t

c c Tc

vT f f v BRAe i l t

c c Tc

π

π

π

+ + = + +

= + +

(13)

where A denotes the amplitude, and the frequency increase can be neglected,

because the movement during the measurement is short compared to the distance

R. It can be easily seen from (13) that RRI2 cvT f

cφ∆ = where φ∆ is defined in

Figure 9.

The following 2D Fourier transformation is performed:

1 1

2D ,2

0 0

2 1 12RRI

0 0

( , ) ( , ) exp 2

2 2 2exp 2 exp 2

2exp

c

L N

B D

l n z Z

f R L Nic cc

A

l n Z

Z

lp nkS p k s l n i

L N

vT f f v BR nkAe i l i nT

c c Tc N

i lp

L

π

π

π π

π

− −

= =

− −

= =

= − +

= + −

−

∑∑

∑ ∑

where zL and

ZN are 2D FFT window size after zero-padding. We note a peak

appears at the following position:

φ∆ φ∆ φ∆ t

f

B T

L

TRRI

RRIRRI

2 2

2

cA Z

cD Z

f v BRk T N

c Tc

vf T Lp f T L

c

= +

= =

(14)

In order to fulfill the sampling theorem, the following constraint for the

maximum Doppler frequency D,maxf must be met:

D,max

RRI

1

2f

T<

Thus the maximum range rate is limited by max

RRI4 c

cv

T f< which can be used

to design the interval time of the sweep cycle.

It can be easily seen from (13) that the velocity resolution v∆ can be

determined by the overall measurement time, RRIT L :

RRI2

c

cv

f T L∆ =

3.2. Direction-of-Arrival Estimation

In the method outlined in Section 3.1, each ( , )R v cell corresponds to a

baseband sinusoid signal Bs and may be contributed by multiple targets. To

correctly distinguish the targets, we need to estimate direction of arrival (DOA).

The common method to estimate DOA is electronically scanned scheme. By

varying the signal phases of the RX antennas, the receiving antenna becomes

directional and is a function of the azimuth angle. Figure 10 shows that the beam

pattern is steered from the right to left at different time slices. In beam-forming,

both the amplitude and phase of each antenna are controlled. The combined

amplitude a kand phase shift

kθ is called a complex weight and is represented by

a complex constantkw , for the k-th antenna. As shown in Figure 11, a beam-

former applies the complex weight to the signals from RX antennas, then sums all the signals into one that has the desired directional pattern [Haynes,1998].


Figure 10: Electronically scanned scheme

Figure 11: Receive digital beam-forming

One important drawback of the beam-former method shown in Figure 11 is

the lack of angular resolution due to the limited aperture size of the antenna, i.e.,

to resolve two or more closely spaced targets in the same (R,v) cell. However, the

DOA of the incident wavefront can be improved with much higher accuracy than

the beam width of the RX antenna. Figure 12 outlines the basic idea called

phase-comparison mono-pulse, or phase-interferometry. A wavefront from

direction θ arrives at antenna 1 first. Then, after travelling an additional path

distance sinL d θ∆ = , it arrives at antenna 2 where d is the distance between

two adjacent elements. Therefore, the refined DOA can be computed

as1sin

2 d

φλθ

π− ∆

=

where φ∆ is the measured phase shift.

B1sB2s B3s

B4s

Beam

-form

er

FMCW processing

Power Combiner

t-1 t-2

t-7

t

Figure 12: Phase-comparison mono-pulse (courtesy of wikipedia.com)

3.3. FMCW Radar Model

We have outlined the basic signal processing. In this section, we will outline

an efficient radar model.

High-fidelity radar model is difficult to implement due to considerable

number of variables and conditions to be taken into account. However, with the

new advances in modeling technology and computer speed, such a model is

feasible in Matlab-Simulink environment. For example, many off-the-shelf

blocks from the standard signal processing and communication blocksets can be

used to model the major components of a radar system.

To execute the simulation in real-time, only baseband subsystem of the radar

is modeled in this paper. We need a target model to represent the received signal

if the geometric shapes of targets and clutters are given. The simple way of

implementing is to represent the target model as a collection of point scatters.

Each scatter, therefore, will be characterized by distance from radar (and, thus, a

path loss associated with that distance), radial velocity, incident angle, and the

strength of reflection (i.e., RCS). Let K scatters be denoted by their distance,

radial velocity, azimuth angle, and RCS, i.e., { }( , , , ) | 1,...,k k k kR v k Kθ σ = . We

use the following baseband approximation for the arrival signal of the j-th RX antenna, holding for most practical cases:

22

RRI, ,

1

, 42

2 2 2exp 2

( ) 1

2(4 )

c k

j

f RK ik c c k kc

B j k j

k

it t k eff

k j

k

v T f f v BRs A e i l t

c c Tc

PG AA e

R

π

φ

π

σ θ

π

=

∆

= + +

= ⋅

∑ (15)


In (15), the complex amplitude ,k jA is proportional to the receiving signal

power and its phase shifts a constant increment ( φ∆ ) depending on the incident

angle, i.e., sin

2d θ

φ πλ

∆ = . tP is the transmitted power (in the transmitted

pulse); R is the distance from the sensor to the object, tG is the directional gain

of the antenna measured in the direction of the object, and ( )eff

A θ is the

effective area of the sensor receiving antenna.

Figure 13: Block schematic of the FMCW model

Figure 13 illustrates the main building blocks of the FMCW radar model.

Based on the method described in Section 2, a target is defined geometrically in a

3D space, which is then converted into scatters, as shown above at the start of the

simulation. The remaining blocks are outlined as follows:

Baseband RX Signal: The baseband receiving signal is represented by (15),

given the K scatters.

Chirp Non-Linearity: The chirp slope B

bT

= is a constant chosen by the

design of the radar. However, if the chirp slope b is not linear, then the beat

frequency for a point target will not be constant, and the range accuracy will be

reduced. b can be modeled as a function of the distance, the deviation from the

Scatters Baseband

RX Signal

Chirp

Non-Linearity

Spread

Convolution Noise

Temperature

Ambient

Environment

Attenuation

Clutter

Noise

2D FFT

Beam-

forming &

DOA

Estimation

Phase Comparison

Mono-pulse

Peak Clustering-

Association

Tracking

AGWN

nominal constant will create a bias in range measurement. A lookup table is used

to model the non-linearity in the chirp slope in this block.

Spread Convolution: The two main noise sources covered in this block are

quantization noise at analog to digital (ADC) and phase noise by the non-ideal mixing process, frequency synthesizer, and other component. A 2D kernel

function was convoluted into the receiving signal. We note that the signal in (15)

can be represented as the sum of infinite spikes (Dirac function) in frequency

domain. The convolution blurs the spikes. In addition, an additive Gaussian white

noise (AGWN) is superposed on the center position of the kernel function.

Noise Simulation: Three additional sources of RF impairment are modeled

as: Noise Temperature, Ambient Environment Attenuation, and Clutter Noise.

The noise temperature block allows us to select effective radar system noise temperature. Three typical values can be chosen: 0 K (no noise), 40 K (very low

noise level) and 290 K (typical noise level). The ambient environment

attenuation block models the attenuation effects experienced by EM waves from,

such as atmospheric gases, rain, and fog. The clutter noise block superposes

noises contributed from clutter background (e.g., ground).

2D FFT: A Fourier transformation is performed on the 2D input signal. Then

a constant false-alarm rate (CFAR) threshold is applied to the yielded frequency

spectrum. The block outputs a list of range-Doppler bins whose signal-noise-ratio (SNR) is larger than the threshold.

Beam-forming & DOA Estimation: The block can resolve two or more

closely spaced targets in the same range-Doppler bin. We iterate through each

electronically steerable beam pattern. For each beam, we multiply the input

signal by the complex weight w and sum all weighted signals together. The

beam index of the amplitude of a summed signal that is larger than a threshold

indicates the DOA angle of a target. Therefore, the block outputs a list of range-

Doppler-beam bins, and each of those bins contains a target.

Phase Comparison Mono-pulse: We refine the DOA angle of a target

detected in the proceeding block using the phase-interferometry process outlined

in Section 3.2.

Peak Clustering-Association: The resulted bins are clustered and associated

with the track database (e.g., nearest neighbor).

Tracking: The tracks are updated by Kalman filtering. Those tracks

(including range, range rate, azimuth angle) are outputted as the sensor

measurements.


3.4. Simulation Results

In this subsection, we show the simulation results of the proposed FMCW

model. As shown in Figure 13, the simulation results presented here are based on

the radar with the following key parameters: radar RF frequency is 24GHz; field-

of-view is 30°; the designed sensing range is [ 50,50]− m/s for range rate and

[0.1,150] m for range; the bandwidth B =1GHz; the ramp repetition interval

-56.2457 10RRI

T = × ; the number of range bins 1024ZN = ; the number of

range rate bins 256ZL = ; 32 receiving antenna elements are placed with the

space between two adjacent elements 5d = mm; 32 electronic steerable beams

are simulated; and transmit antenna power is 10 Watts. The simulated signal-to-

noise ratio is 6dB. The effective area of the sensor receiving antenna

1eff

A = 2m .

Three trihedral corner reflectors with radar cross section (RCS) of 1 2m are

simulated with following parameters:

• Target 1: range = 20 m, range rate = 9 m/s, azimuth angle = -15°

• Target 2: range = 32 m, range rate = -10 m/s, azimuth angle = 2°

• Target 3: range = 32m, range rate = -10 m/s, azimuth angle = -2°

We note that the Targets 1 and 2 can be easily separated by range-Doppler

bin. However, Targets 2 and 3 are in the same range and range rate and cannot be

separated by range-Doppler bin.

Figure 14 is the real part of the synthesized base band signal (c.f., (15)) for the

abovementioned three targets during a chirp sweep by a receiving antenna.

Figure 14: The synthesized baseband signal of a chirp sweep

Figure 15 is the power spectral density among the range-Doppler. The result is

derived by applying 2D FFT to the synthesized 2D temporal-spatial signal ,B js

defined in (15), for the given j-th receiving antenna. We note two detected peaks

that correspond to the simulated three targets. The left peak corresponds to

Target 1 while the right FFT peak corresponds to the lumped Targets 1 and 2,

which cannot be separated by range-Doppler bin. Given an FFT peak is detected

at bin ( , )n l , the corresponded range and range rate can be computed by

( 1) ( 1),

2 2 Z RRI c

n c l cR V

B L T f

− −= =

To handle negative range rate, we shift the zero component to the center of

spectrum.


Figure 15: 2D FFT power spectrum among range-Doppler bins

Figure 16 illustrates the slices along the detected two peaks in 2D FFT power

spectrum shown in Figure 15. The upper plot is the spectrum density with

respect to the range, which can be derived using a 1D FFT on the signal ,B js in a

chirp sweep window (i.e., a row of ,B js ), giving the receiving antenna j. The

lower plot is the spectrum density with respect to the range rate, which can be

derived using a 1D FFT on a column of ,B js . One can easily verify that the two

peaks in the upper plot are the ranges 20 m and 32 m, respectively, and the two

peaks in the low plot are the range rates 9 m/s and -10 m/s, respectively.

Figures 15 and 16 illustrate that Targets 2 and 3 cannot be separated only by

range-Doppler bin. However, the electronically scanned digital beams can

discriminate the two targets. Figure 17 shows the response of the all the 32

beams for the two detected range-Doppler peaks. The green curve corresponding

to Target 1 shows there is a maximum peak at -15°, which indicates the azimuth

of the target. The blue curve corresponding to Targets 2 and 3 has two maximum

peaks at about -2° and 2°, respectively. This means the response of the digital

beams can separate targets with the same range and range rate bin.

Figure 16: 1D FFT power spectrum vs. range bins and range rate bins,

respectively.

Figure 17: The amplitude of the detected range-Doppler bins (clusters) as a

function of the steerable RX beams.


4. Integrated Simulation with Sensing Model

To simulate vehicles on traffic, a traffic vehicle model is designed that is

much simplified to achieve efficient numerical computation, but with sufficient

fidelity to capture basic vehicle dynamics under both linear and nonlinear operating regions.

The traffic vehicle model is used to calculate the vehicle trajectory in terms of

its position ( , )x y and orientation in yaw plane ( Φ ), and its velocity in both

lateral and longitudinal directions, xv and yv , respectively. Denote the output

vector as [ ]Tyx vvYXy Φ=

Assume steering angle fδ . To simplify modeling on vehicle longitudinal

dynamics, a speed controller is used to represent vehicle powertrain and brake

systems. Assume a set of desired vehicle speed [ ]nx vvvv ˆˆˆˆ21 ⋯= is given,

which can be from driver’s input via set speed, but can be overridden by other

control strategies, such as adaptive cruise control based upon on-board sensors.

Figure 18 shows the block diagrams of the traffic vehicle model.

Figure 18: Simple traffic vehicle model

A simple PID control can be used for the speed controller as:

( )xxdxxixxpx vv

dt

dkdvvkvvkF −+−+−= ∫ ˆ)ˆ()ˆ( τ

which is subject to actuator saturation as shown in the figure.

A nonlinear bicycle model is defined as:

Vehicle

Dynamics

Model

fδ

xv̂

xv

xv∆ xF

Vehicle

Kinematics

Model

Φ

y

x

v

v

Y

X

Speed

Controller

Saturation

r

v

v

y

x

xF

yrfyffxz

yrfyffxxy

bFFFarI

FFFrvvm

−+=

++=+

)cossin(

cossin)(

δδ

δδ

ɺ

ɺ

where the front and rear tire lateral slip angles are defined respectively as below:

x

y

r

x

y

ffV

brV

V

arV −−=

+−= −− 11

tan,tan αδα

Thus, the tire lateral force at front or rear tire is a non-linear function of normal

force of front or rear axle, and slip angle of front or rear tire, as defined below

respectively:

ffzfffyf FfCF ααα ⋅== ),(

rrzrrryr FfCF ααα ⋅== ),(

The traffic vehicle model thus can be represented by a set of nonlinear

differential equations: ),,( tuxfx =ɺ , with state variables to be [ ]Ty rvx = , and

input variables to be [ ]TfxFu δ= .

With the simulated range sensing and object detection capability, the proposed

sensor model and algorithms can be applied to the simulation studies of many

xv̂

xv

xv∆Desired

speed

Distance

Controller

Desired

range

Range rate

Range

Desired

range rate

Radar

sensor

model

Vehicle

Longitudinal

Dynamics

Target

speed

xv

Speed

Controller

Figure 19: Simulation on an adaptive cruise control


active safety, driver assistance, and autonomous driving features and functions

such as adaptive cruise control, collision warning and prevention, etc. As an application example, a closed-loop simulation of an adaptive cruise control

(ACC) system, shown in Figure 19, is presented in this section to demonstrate

that the proposed sensor model and algorithm enables the simulation of multiple

vehicles interacting with one another under traffic. The radar sensor model

simulates the in-lane target’s relative range and target’s speed. The distance

controller monitors the desired range from the driver’s input and the in-lane

target’s relative range rate, and outputs the desired range rate. The speed

controller monitors the error from the desired vehicle speed and the error from

the desired range rate, and generates either throttle or brake force command such

that the desired following distance between the vehicle and the target is

maintained.

Figure 20 shows an ACC simulation scheme created under iSim® and

Matlab/Simulink environment, with a scenario of a host vehicle (white color)

interacting with two target vehicles (yellow color) on a road. The animation is

shown in Figure 21. A range sensor, simulated with the proposed sensor model,

is mounted in front of the host vehicle with a forward-sensing range of 50 meters

to detect if any of the target vehicles is falling into its scanning zone. Based on

the sensor outputs such as range, range rate, detection flag, etc, the ACC module

generates either throttle or braking force command to adjust the host vehicle’s

speed such that the host vehicle maintains a desired following distance from the

target vehicle.

The simulation results are illustrated through the animation shown in

Figure 22. In the simulation, the host vehicle initially travels at a set speed

of 60kph, as shown in Figure 22(a). Within a close distance at the second

left lane, a target vehicle (marked as target vehicle 1) travels at a constant

speed of 50kph. Around time t=2 second, the target vehicle 1 changes its

lane and cuts in front of the host vehicle inside the sensing range, as

shown in Figure 22(b). The host vehicle quickly slowed down in order to

keep a pre-defined following distance around 15 meters. Shortly the target

vehicle 1 changes back to its previous lane and gets out of the sensing

zone, as shown in Figure 22(c), the host vehicle then accelerates back to

its pre-set speed of 60kph until another slower-traveling vehicle ahead

(marked as target vehicle 2) falls into its sensing range, then the host

vehicle slowed down again with adjusted speed to follow the target

vehicle 2 and maintain the desired following distance of 15 meters, which

lasts till the simulation ends. The simulation results further verify the

effectiveness of the range sensing model and object detection algorithms

proposed in this research.

Figure 20: Simulation on adaptive cruise control

(a) Host vehicle travels at set speed of 60 kph


(b) Target vehicle 1 cuts in front of the host vehicle

(c) Target vehicle 1 gets out of sensing range

(d) Host vehicle accelerates to the set speed of 60 kph

(e) Target vehicle 2 falls into the sensing range

Figure 21: Animation for ACC Simulation

(a) ACC set speed and speed profiles of host vehicle and two target vehicles

(b) Desired range, actual range and detection flag


(c) Range rate and detection flag

Figure 22: Simulation Results of ACC

5. Conclusions

This paper presents a generic approach in modeling range sensing and object

detection, both functionally and physically. This is to support the emerging

simulation of multiple vehicles interacting with one another under traffic and

with surrounding environment. The mathematical derivation behind the described

method is discussed in detail, along with algorithm for numerical implementation.

The proposed model and algorithm have been implemented and verified under Matlab with simulation under various scenarios. The results demonstrate that the

proposed models and object detection algorithm are valid, effective and

numerically efficient. A closed-loop simulation with an adaptive cruise control is

further presented to demonstrate an integrated modeling and simulation of

limited autonomous driving with the proposed sensing model and object

detection algorithm under iSim®, an integrated vehicle simulation environment.

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