Modification of Vehicle Handling Characteristics via Steer-by-Wire
Paul Yih
Design Division
Dept. of Mechanical Engineering
Stanford University
Stanford, CA 94305-4021
Email: pyih@stanford.edu
Jihan Ryu
Design Division
Dept. of Mechanical Engineering
Stanford University
Stanford, CA 94305-4021
Email: jihan@stanford.edu
J. Christian Gerdes
Design Division
Dept. of Mechanical Engineering
Stanford University
Stanford, CA 94305-4021
Email: gerdes@cdr.stanford.edu
Abstract
This paper presents a physically intuitive method for
altering a vehicle’s handling characteristics through
active steering intervention. A full state feedback
controller augments the driver’s steering command via
steer-by-wire to achieve desired handling behavior.
Accurate estimates of vehicle states are available from
a combination of Global Positioning System (GPS)
and Inertial Navigation System (INS) sensor
measurements. By canceling the effects of steering
system dynamics and tire disturbance forces, the steer-
by-wire system is able to track commanded steer angle
with minimal error. Experimental results verify that
with precise steering control and accurate state
information, a vehicle’s handling characteristics can
be modified to match driver preference or to
compensate for changes in operating conditions.
1 Introduction
As a step toward fully integrated vehicle dynamic
control systems, active steering capability will be
available on select production vehicles within one or
two years. The potential benefits of active steering
intervention, particularly to improve handling
behavior during normal driving, have received
considerable attention from both the automotive
industry and research institutions. As early as 1969,
Kasselmann and Keranen [1] developed an active
steering system based on feedback from a yaw rate
sensor. More recent work by Ackermann [2]
combines active steering with yaw rate feedback to
robustly decouple yaw and lateral motions. This
method is effective in, for example, canceling out yaw
generated when braking on a split friction surface. In
[3], Huh and Kim devise an active steering controller
that eliminates the difference in steering response
between driving on slippery roads and dry roads. The
controller is based on feedback of lateral tire force
estimates derived from vehicle roll motion. Most
recently, Segawa et al. [4] apply lateral acceleration
and yaw rate feedback to a steer-by-wire vehicle and
demonstrate that active steering control can achieve
greater driving stability than differential brake control.
Although feedback of sideslip angle for active steering
control has been proposed theoretically [5], the
difficulty in estimating vehicle sideslip presents an
obstacle to accomplishing this in practice. Stability
control systems currently available on production cars
typically derive slip angle from sensor integration or a
physical vehicle model, but these estimation methods
are prone to uncertainty [6]. Because sideslip is
extremely important to the driver’s perception of
handling behavior, quality of the driving experience
depends strongly on quality of the feedback signal.
While this dependence is less critical for stability
control systems—which tend to engage when the
vehicle is already undergoing extreme maneuvers—to
improve handling behavior during normal driving
requires cleaner and more accurate feedback.
A new sideslip estimation scheme combining GPS and
INS sensor measurements overcomes many of the
drawbacks of previous estimation methods [7]. For
this paper, a test vehicle converted to steer-by-wire is
used to demonstrate that a vehicle’s handling
characteristics may be find-tuned through a
combination of GPS/INS feedback and precisely
controlled active steering. The first part of the paper
briefly discusses the estimation scheme along with a
physically motivated approach for full state feedback
control of an actively steered vehicle. The latter part
of the paper describes the design of the steer-by-wire
system that provides active steering capability to the
test vehicle. Experimental results clearly show the
change in handling behavior achieved with full state
feedback steering control. In addition to matching
handling behavior to driver preference, the system
successfully counteracts handling differences caused
by shifts in weight distribution.
2 Planar Bicycle Model
A vehicle’s handling dynamics in the horizontal plane
are represented here by the single track, or bicycle
model with states of sideslip angle,
β
, at the center of
gravity (CG) and yaw rate, r.
Figure 1: Bicycle model.
The sideslip angle is defined by the difference
between vehicle heading,
ψ
, and the direction of
velocity,
γ
:
ψ
γ
β
−=
(1)
In Figure 1, δ is the steering angle, u
x
and u
y
are the
longitudinal and lateral components of the CG
velocity, F
yf
and F
yr
are the lateral tire forces front and
rear, respectively, and α
f
and α
r
are the tire slip angles.
Assuming constant longitudinal velocity u
x
=V, the
state equation for the bicycle model can be written as:
(
)
δ
β
β
+
+−
=
−−−
−−−
I
aC
mV
C
IV
bCaC
I
aCbC
mV
aCbC
mV
CC
f
f
rffr
frrf
r
r
22
2
1
&
&
(2)
I is the moment of inertia of the vehicle about its yaw
axis, m is the vehicle mass, a and b are distance of the
front and rear axles from the CG, and C
f
and C
r
are the
total front and rear cornering stiffness that relate
lateral tire force to slip angle:
rryr
ffyf
CF
CF
α
α
=
=
(3)
The model is valid for tires operating in the linear
region and small slip angles.
3 State Estimation
The ability to obtain accurate information on the
vehicle states—yaw rate and sideslip angle—is crucial
to implementing an active handling system with full
state feedback control. Although yaw rate data is
available on many production cars from rate
gyroscopes, sideslip cannot be directly measured and
must be estimated instead. Two common techniques
for estimating this value are to integrate inertial
sensors directly and to use a physical vehicle model.
Some methods use a combination or switch between
these two methods appropriately based on vehicle
states [8]. Direct integration methods can accumulate
sensor errors and unwanted measurements from road
grade and bank angle. In addition, methods based on
a physical vehicle model can be sensitive to changes
in the vehicle parameters and are only reliable in the
linear region.
To overcome these drawbacks, a new method for
estimating vehicle sideslip angle using GPS and INS
sensor measurements is presented in [7]. In this
scheme, GPS measurements from a two-antenna
system are combined with INS sensor measurements
to eliminate errors due to direct integration. Since
both the vehicle heading and the direction of velocity
are directly measured from a two-antenna GPS
receiver, the sideslip angle can be calculated using
Equation (1). INS sensors are integrated with GPS
measurements to provide higher update rate estimates
of the vehicle states and to handle periods of GPS
signal loss. This method is also independent of any
parameter uncertainties and changes because it is
based on purely kinematic relationships.
0 50 100 150 200
-15
-10
-5
0
5
10
15
time (s)
sideslip angle (deg)
0 50 100 150 200
-60
-40
-20
0
20
40
60
time (s)
yaw rate (deg/s)
Figure 2: Sideslip and yaw rate estimation.
Experimental results from the GPS/INS integration are
plotted in Figure 2 on top of simulation results from
the bicycle model for both yaw rate and sideslip angle.
The similarity between estimated and simulated yaw
rates indicates that the bicycle model used in the
comparison is valid and calibrated correctly. The fact
that the sideslip measurement is clean and correlates
with the model makes it suitable for use as a feedback
signal.
4 Full State Feedback Controller
A full state feedback control law for an active steering
vehicle is given by
ddr
KKrK
δ
β
δ
β
++=
(4)
where
δ
d
is the driver commanded steer angle and
δ
is
the augmented angle. A physically intuitive way to
modify a vehicle’s handling characteristics is to define
a target front cornering stiffness as
()
η
+= 1
ˆ
ff
CC
(5)
and the state feedback gains as
)1(
ηηη
β
+=−=−=
dr
K
V
a
KK
(6)
where η is the desired fractional change in the original
front cornering stiffness C
f
. Substituting the feedback
law (4) into Equation (2) yields a state space equation
of the same form as Equation (2) but with the new
cornering stiffness Ĉ
f
:
(
)
d
I
aC
mV
C
IV
bCaC
I
aCbC
mV
aCbC
mV
CC
f
f
rffr
frrf
r
r
δ
β
β
+
+−
=
−−−
−−−
ˆ
ˆ
ˆˆ
ˆˆ
22
2
1
&
&
(7)
Since a vehicle’s handling characteristics are heavily
influenced by tire cornering stiffness, the effect of this
modification is to make the vehicle either more
oversteering or understeering depending on the sign of
η. Clearly, there are many other ways to apply full
state feedback, but the physical motivation behind
cornering stiffness adjustment makes clear through the
bicycle model exactly how the handling characteristics
have been modified. Note that in this formulation, it
is not necessary to know the real cornering stiffness of
the front tire—only vehicle speed and weight
distribution, which are relatively easy to measure—to
achieve the desired handling modification.
5 Steer-by-Wire System
A production model 1997 Chevrolet Corvette is
modified for full steer-by-wire capability by replacing
the steering shaft with a brushless DC servomotor
actuator. The stock hydraulic power assist unit and
rack and pinion mechanism in the test vehicle are
retained as part of the steer-by-wire system, since the
incorporation of the power assist unit eliminates the
need for extensive modifications to the existing
steering system and allows the use of a much smaller
actuator. A rotary position sensor measures the lower
steering shaft angle, which is equal to the front wheel
steer angle scaled by the steering ratio. An identical
sensor attached to the upper steering shaft measures
the handwheel angle.
Figure 3: Steer-by-wire schematic.
The servomotor actuator specifications are chosen
based on the maximum torque and speed necessary to
steer the vehicle under typical driving conditions
including moderate emergency maneuvers. On
average, steering torque required at the handwheel
during normal driving ranges from 0 to 2 Nm, while
emergency maneuvers can demand up to 15 Nm of
torque [9]. The actuator installed in the test vehicle
provides a maximum steering torque of 17.1 Nm with
a maximum steer rate of 700 degrees per second.
The differential equation describing the steering
system dynamics is as follows:
ττθθθ
=+++
aac
kFbJ
&&&&
sgn
(8)
θ is the pinion angle, J is the total moment of inertia of
the system, b is viscous damping, F
c
represents
coulomb friction, k
a
is a scale factor, τ
a
is the tire self-
aligning moment, and τ is the actuator torque.
The purpose of the steer-by-wire controller is to track
commanded steer angle with minimal error; the
control effort consists of three components:
aligningdfeedforwarfeedback
τ
τ
τ
τ
+
+
=
(9)
The proportional derivative (PD) feedback component
is given by
(
)
(
)
θθθθτ
&&
−+−=
dddpfeedback
KK
(10)
pinion an
g
le senso
r
steering actuato
r
handwheel angle senso
r
belt drive
feedback moto
r
power assist unit
pinion
rack
where θ
d
is the desired steer angle, K
p
is the
proportional feedback constant, and K
d
is the
derivative feedback constant. The feedback gains K
p
and K
d
are selected to give a fast closed loop system
response without oscillatory behavior. Because the
system is second order, however, PD control alone
results in some steady state error when tracking the
type of command shown in Figure 4 (steering angle is
given at the front wheels). To obtain these
measurements, the front wheels are raised off the
ground so as to isolate the influence of J, b and F
c
from static friction at the tire-ground interface. The
addition of feedforward compensation,
(
)
dcdddfeedforwar
FbJ
θθθτ
&&&&
sgn++=
(11)
to the PD controller cancels any tracking errors
associated with the system dynamics and internal
friction (Figure 5). J, b and F
c
are determined through
closed-loop identification of the steering system.
6 8 10 12 14 16 18 20 22 24
-10
-5
0
5
10
time (s)
steering angle (deg)
actual
commanded
6 8 10 12 14 16 18 20 22 24
-0.5
0
0.5
time (s)
steering angle error (deg)
Figure 4: Feedback control only.
6 8 10 12 14 16 18 20 22 24
-10
-5
0
5
10
time (s)
steering angle (deg)
actual
commanded
6 8 10 12 14 16 18 20 22 24
-0.5
0
0.5
time (s)
steering angle error (deg)
Figure 5: Feedback with feedfoward compensation.
When driving a vehicle over the road, however, an
additional disturbance acts on the system causing a
steering error (Figure 6) that is directly attributable to
tire self-aligning moment. The total aligning moment
is given by
(
)
(
)
fyfmpa
Ftt
α
τ
+
=
(12)
where t
p
and t
m
are the tire pneumatic and mechanical
trails, respectively. Front tire slip angle, α
f
, can be
calculated from the following relationship involving
estimated sideslip and other measurable parameters:
δβα
−+=
x
f
u
ar
(13)
Aligning moment may also be directly approximated
as an empirical function of tire slip angle [10]. This
approximation of aligning moment is added to the
feedback and feedforward control as
(
)
faaaligning
k
α
τ
τ
ˆ
=
(14)
where k
a
is a scale factor to account for torque
reduction by the steering gear.
10 15 20 25 30 35 40 45 50 55
-20
-10
0
10
20
time (s)
steering angle (deg)
actual
commanded
10 15 20 25 30 35 40 45 50 55
-1
-0.5
0
0.5
1
time (s)
steering angle error (deg)
Figure 6: Error due to aligning moment.
10 15 20 25 30 35 40 45 50 55
-20
-10
0
10
20
time (s)
steering angle (deg)
actual
commanded
10 15 20 25 30 35 40 45 50 55
-1
-0.5
0
0.5
1
time (s)
steering angle error (deg)
Figure 7: Steering controller with aligning moment
compensation.
From a comparison between Figures 6 and 7, the
addition of τ
aligning
to the actuator effort effectively
eliminates most of the steering disturbances that arise
when turning at speed.
6 Experimental Results
Figure 8: Steer-by-wire test vehicle.
The steer-by-wire test vehicle is equipped with
multiple-antenna GPS configured to provide absolute
velocity and heading information. INS sensors
measure lateral and longitudinal acceleration, yaw
rate, and roll rate. The experimental setup for vehicle
state estimation is same as described in [7]. In Figure
9, the measured yaw rate from a sinusoidal steering
input while driving at 13.4 m/s (30 mi/hr) compare
well to simulation results from the bicycle model.
5 10 15 20 25
-40
-30
-20
-10
0
10
20
30
40
time (s)
yaw rate (deg/s)
simulation
experiment
Figure 9: Comparison between bicycle model and
experiment with normal cornering stiffness.
Next, handling modification is implemented on the
test vehicle. Changes in handling behavior under full
state feedback control are evaluated by comparing
measured vehicle response to the nominal case shown
in Figure 9. In Figure 10, the effective front cornering
stiffness is reduced 50% by setting the parameter η to
-0.5. The experimental results exhibit lower peak yaw
rate and sideslip values than the nominal case. This
behavior is expected since reducing the front
cornering stiffness causes the vehicle to tend toward
understeer. Figure 11 confirms that test results for the
reduced case match bicycle model simulation.
5 10 15 20 25
-40
-30
-20
-10
0
10
20
30
40
time (s)
yaw rate (deg/s)
normal
reduced
Figure 10: Comparison between normal and
effectively reduced front cornering stiffness.
5 10 15 20 25
-40
-30
-20
-10
0
10
20
30
40
time (s)
yaw rate (deg/s)
simulation
experiment
Figure 11: Comparison between bicycle model and
experiment with reduced cornering stiffness.
Experimental data show a corresponding but opposite
change in handling behavior when the effective front
cornering stiffness is increased such that the vehicle
tends toward oversteer.
For the final series of tests, 182 kg (400 lbs) of weight
are added to the rear of the vehicle so that 57% of the
total vehicle weight lies over the rear axle with 43%
over the front axle. The unloaded vehicle has a
weight distribution balanced equally front to rear. As
seen in Figure 12, the loaded vehicle exhibits slightly
more oversteering behavior than the unloaded vehicle.
However, with active handling modification, a 20%
reduction in front cornering stiffness returns the
controlled vehicle to the near neutral handling
behavior of the unloaded vehicle (Figure 13). While
the difference in handling behavior may seem small
