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Steering redundancy for self-driving vehicles using differential braking

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This paper describes how differential braking can be used to turn a vehicle in the context of providing fail-operational control for self-driving vehicles. Two vehicle models are developed with differential input. The models are used to explain the bounds of curvature that differential braking provides and they are then validated with measurements in a test vehicle. Particular focus is paid on wheel suspension effects that significantly influence the obtained curvature. The vehicle behaviour and its limitations due to wheel suspension effects are, owing to the vehicle models, defined and explained. Finally, a model-based controller is developed to control the vehicle curvature during a fault by differential braking. The controller is designed to compensate for wheel angle disturbance that is likely to occur during the control event.
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June 27, 2017
Steering Redundancy for Self-Driving Vehicles
using Differential Braking
M. Jonassona,b and M. Thora
aVolvo Cars, Active Safety and Vehicle Dynamics Functions, Gothenburg, SE-405 31, Sweden;
bThe Royal Institute of Technology (KTH), Vehicle Dynamics, Department of Aeronautical and
Vehicle Engineering, Stockholm, SE-10044, Sweden
This paper describes how differential braking can be used to turn a vehicle in the context
of providing fail-operational control for self-driving vehicles. Two vehicle models are devel-
oped with differential input. The models are used to explain the bounds of curvature that
differential braking provides and they are then validated with measurements in a test vehicle.
Particular focus is paid on wheel suspension effects that significantly influence the obtained
curvature. The vehicle behavior and its limitations due to wheel suspension effects are, owing
to the vehicle models, defined and explained. Finally, a model based controller is developed to
control the vehicle curvature during a fault by differential braking. The controller is designed
to compensate for wheel angle disturbance that is likely to occur during the control event.
Keywords: differential braking; vehicle control; autonomous vehicles; redundant steering;
scrub radius
1. Introduction
One of the goals of safety-critical systems is that a fault should not result in system fail-
ures. Since a self-driving vehicle isn’t driven by a human driver, fail-operational ability
must be inherent in the control design of the vehicle. Potential risks, addressed in this
paper, are faults leading to failures with no steering capacity due to for example steering
actuator faults, power black out, communication shut down etc. Those risks could, if
the design is not fail-operational, lead to a failure and a hazard threatening the passen-
gers safety. During, and directly after, a fault occurs, the vehicle should be able to be
controlled into a safe state which is often terminated by a full stop.
The work in this paper is a result from further development of another paper [1]
presented at Proceedings of the 13th International Symposium on Advanced Vehicle
Control. Both papers contain an analysis of the temporary use of differential braking to
regain control of the vehicle in case of faults. Differential braking used for cornering is
described in e.g. [2, 3]. Few descriptions are however found in the literature of how wheel
suspension geometry influences the curvature response during differential braking and
analysis from experiments in real world vehicles are rare. In [1], a simple vehicle model
was presented and used. The simple vehicle model has in this paper been extended to
also include the steering system model with wheel suspension parameters such as scrub
Corresponding author. Email: mats.jonasson@volvocars.com
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June 27, 2017
radius. Particular focus has here been made on modelling the steering friction. The
model validation shows that the new extended model has, compared with the simple
one, a better match to the handling measurements. Owing to the developed vehicle
model in this paper, it has also been possible to analytically express system constraints
and maximum curvature as a function of wheel suspension parameters. This is important
knowledge when designing the vehicle to secure a desired cornering ability.
The outline is as follows; Section 2 describes the underlying problem that seeks a so-
lution by differential braking and Section 3 describes the vehicle model that will be used
throughout the paper. The model is validated by making comparisons of measurement
data from a test vehicle. In Section 4 the vehicle model is used to calculate the maxi-
mum curvature that is possible to obtain during differential braking at steady-state. The
differential braking generates longitudinal tyre forces, and together with the wheel sus-
pension geometry, a steering alignment torque is induced. Section 5 discusses the scrub
radius geometry and how it contributes the resulting curvature. Two different alignments
of scrub-radius is experimentally tested by experiments and evaluated. To explain the
scrub radius effect of curvature and steering angle, a vehicle model including steering
dynamics and scrub radius is developed in Section 6. In Section 7 a differential braking
curvature controller is developed. The controller is developed to compensate for front
wheel angle disturbance induced by the differential braking.
2. VEHICLE AND PROBLEM DESCRIPTION
The vehicle in this work is a conventional passenger car with front axle drive and steering.
The mechanical steering system is provided with a steering actuator, mentioned to as
Electric Power Assisted Steering (EPAS), which overlay an additive steering torque.
Moreover, the vehicle has a friction brake system where brake torque can be applied at
each wheel. See Table 1 for vehicle data.
In general the vehicle controllability during a steering failure depends on:
(1) Character of steering failure
(2) Initial condition of vehicle
(3) Road boundaries ahead
(4) Character of vehicle
(5) Character of failure detection and control algorithms
(6) Disturbances during the control
The character of the steering failure is typically described as too little or much steering
torque. The initial condition of the vehicle are states related to position and orientation
with respect to the road, state of the car itself including the actuators. Road boundaries
ahead are the road curvature, road width and obstacles. The failure detection is typically
characterized by the time to detect the failure and the correctness of the detection.
Disturbances here means eg. rutted road, wind gusts etc.
One special case of a steering failure, among the many dependencies mentioned above,
is when the torque from the steering actuator completely disappears. At the same time,
the driver is assumed to not take part in controlling the vehicle and is not applying any
steering-wheel torque. Figure 1 illustrates the obvious road departure that occurs when
the steering does not provide torque when entering a curve from straight ahead initial
driving. After approximately 20 m the vehicle is outside a 1 meter pre-defined lateral
road margin. A second special case, as illustrated in Figure 2, is when the steering
torque disappears in a curve i.e. the vehicle has cornering as initial motion state. To
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June 27, 2017
simulate the vehicle dynamic behavior after such a failure, an real world experiment
has been done where the steering torque was forced to zero in a curve at 70 km/h at
t= 0 and approx. X= 0. As illustrated in Figure 2, the test vehicle will quickly leave the
orbital motion and departure the road. When the steering torque from EPAS disappears,
the alignment torque generated at the front axle will induce a change of the steering-
wheel angle (SWA) towards zero within approximately 0.8 s. This special case is in the
context of self-driving vehicles most likely an unusual case due to the initial high lateral
acceleration (approximately 7 m/s2) which is outside any conceivable comfort zone. The
case is however selected to demonstrate how quickly the vehicle leaves the road if no
control is performed to reduce the effect of the failure.
0 5 10 15 20 25
−1
0
1
2
3
X (m)
Y (m)
vehicle path
road centerline
lateral road margin
Figure 1. Effect of vanishing steering torque before entering a curve with 200 m radius and initially driving
straight ahead (from simulation).
−0.2 0 0.2 0.4 0.6 0.8 1
−40
−20
0
20
40
60
80
time (s)
SWA (deg)
yawrate ωz(deg/s)
−15 −10 −5 0 5 10 15
−50
−48
−46
−44
−42
X (m)
Y (m)
vehicle path
road centerline
lateral road margin
Figure 2. Effect of vanishing steering torque in a curve with 50 m radius and 70 km/h (from vehicle measure-
ments).
To cope with the problems described above, i.e follow the desired path without lateral
tracking error after a steering capacity failure, the work in this paper addresses the use
of differential braking to control the path. The proposed control strategy is intended to
handle all dependencies mentioned above, but the work and the analysis performed are
limited to investigate a complete loss of steering torque from EPAS and driving straight
ahead initial condition.
The approach in this paper is to apply braking along one side of the vehicle to control
the curvature. Since differential braking will influence the total longitudinal tractive force,
they are coupled. It is for example not possible to maximize yaw torque and total brake
force at the same time. The vehicle speed control is not part of the paper.
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June 27, 2017
3. VEHICLE MODEL
3.1. Vehicle model description
The well-established bicycle model with front steering input and constant vehicle speed
assumption is here complemented with longitudinal tyre forces. See for example [2, 4] for
a similar derivation. The longitudinal tyre forces are assumed to be small and limited to
not influence lateral axle forces through combined slip. The model has lateral velocity vy
and yawrate ωzas states and is described by the lateral force and yaw torque equilibrium
equations;
m( ˙vy+vxωz) = (fF L
x+fF R
x) sin(δf)+(fF L
y+fF R
y) cos(δf) + fRL
y+fRR
y(1)
Jz˙ωz= (fFL
yfF R
y) sin(δf)w
2+ (fF L
y+fF R
y) cos(δf)lf(fRL
y+fRR
y)lr
(fF L
x+fF R
x) sin(δf)lf+ (fF L
xfF R
x) cos(δf)w
2+ (fRL
xfRR
x)w
2(2)
where δfis the front wheel angle and vxis the vehicle longitudinal speed. The vehicle
has a mass mand yaw inertia Jz. The lateral distance from center of gravity (CoG) to
front respectively rear axle are denoted lfand lr. Track width is denoted w. The lateral
tyre forces fi
yand longitudinal tyre forces fi
xwith i={F L, F R, RL, RR}are defined
according to Figure 3.
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2
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2
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'(
)*
+
+,
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'!
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,
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,
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,
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,
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CoG
Figure 3. Vehicle model.
The Equations 1 and 2 will now be re-formulated by assuming a small front steering
angle by using sin(δf) = 0 and cos(δf) = 1, merging lateral tyre forces to lateral axle
forces, fF r
yand fRe
y, and by introducing a differential brake force Fb;
m( ˙vy+vxωz) = fF r
y+fRe
y(3)
Jz˙ωz=fFr
ylffRe
ylr+Fb
w
2(4)
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June 27, 2017
where the axle forces are defined
fF r
y=fF L
y+fF R
y, fRe
y=fRL
y+fRR
y(5)
and where the virtual differential brake control signal Fbhas been introduced such that
Fb=fF L
x+fRL
xfF R
xfRR
x(6)
The tyre characteristics are assumed to be linear and tyre side-slip αfand αrto be
small such that
fF r
y=Cfαf=Cfvy+lfωz
vx
δf(7)
fRe
y=Crαr=Crvylrωz
vx,(8)
where the front and rear axle cornering stiffness Cfand Crare parameters.
Further on, actuator dynamics, dead time and tyre relaxation are all lumped into first
order systems with a time constant Tbfor differential braking and Tsfor steering.
˙
Fb=1
Tb
Fb+1
Tb
Freq
b,˙
δf=1
Ts
δf+1
Ts
δreq
f,(9)
where Freq
band δreq
fare actuator requests.
A vehicle’s curvature is in general defined as
ρ=ωz+˙
β
vx
, β = arctan vy
vx,(10)
but for simplicity we will however here define the curvature with ˙
βneglected and for the
calculation of curvature ˙
βis approximated to zero across the paper.
Selecting the state vector xand input vector usuch that
x=
vy
ωz
δf
Fb
, u =
δreq
f
Freq
b
, y =ρ, (11)
the complete vehicle model is then expressed in state-space form as
˙x=Ax +Bu, y =Cx (12)
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June 27, 2017
A=
Cf+Cr
mvxvxlfCflrCr
mvxCf
m0
lfCflrCr
Jzvxl2
fCf+l2
rCr
JzvxlfCf
Jz
w
2Jz
0 0 1
Ts0
0 0 0 l
Tb
(13)
B=
0 0
0 0
1
Ts0
01
Tb
, C =01
vx0 0.(14)
The system in Equation 12 is also described in transfer function form;
y=G(s)u(15)
G(s) = [Gs(s)Gp(s)] = C(sI A)1B, (16)
where the individual transfer functions from steering and differential braking are
ρ=Gs(s)δreq
f, Gs(s) = b2s2+b1s+b0
a4s4+a3s3+· · · +a0
(17)
ρ=Gp(s)Freq
b, Gp(s) = c2s2+c1s+c0
a4s4+a3s3+· · · +a0
,(18)
where the polynomial coefficients are vehicle speed dependent (coefficients are listed for
70 km/h in Table 1). At 70 km/h the poles of the systems are positioned in {−6.5±
3.2i, 3.3,10}.
3.2. Vehicle model validation
A validation test for the vehicle model described by Equation 12 with Freq
bas input and
ωzas state for validation has been done at two different vehicle velocities and is shown in
Figure 4. The test was conducted by driving straight ahead at constant speed and high
friction (µ1.0) and then, at t= 0 s, requesting Freq
b=mg
2with steering-wheel angle
fixed to zero. For the real vehicle test, the steering-wheel angle, vehicle velocity, angular
velocities and accelerations were all measured through the CAN bus and filtered by a
30 Hz low pass filter. During the tests of the real vehicle, it was observed that the ABS
was partly activated indicating that friction utilization was maximized along the braked
side of the car. It is also clear from the lower subplot in Figure 4 that real world vehicle
exhibited a linearly decreased speed of approximately 4.5 m/s2. The yawrate response
6
June 27, 2017 Vehicle System Dynamics DiffBrk
−1 −0.5 0 0.5 1 1.5 2
−5
0
5
10
15
20
time (s)
−1 −0.5 0 0.5 1 1.5 2
10
20
30
40
50
60
70
time (s)
measured vehicle velocity vx(km/h)
measured vehicle velocity vx(km/h)
dierential brake input Freq
b(kN)
measured yawrate ωz(deg/s) at 70 km/h
measured yawrate ωz(deg/s) at 50 km/h
modelled yawrate ωz(deg/s) at 70 km/h
modelled yawrate ωz(deg/s) at 50 km/h
Figure 4. Step response validation test of yawrate with differential brake input at two different vehicle velocities.
from the model is however based on constant speed, which explains the deviation of
yawrate at the end of the manoeuvre. vxis not a state in the model since we have
here prioritized simplicity of a linear model. Expected model uncertainties are the axle
cornering stiffnesses and the friction between brake pad and disc, and also, tyre and road.
See Figure 3 for definition of physical entities and Table 1 for values of parameters.
4. STEADY-STATE CORNERING CURVATURE
This section is devoted to investigate and quantify the curvature that differential braking
provides. By deriving the analytic expression of the filter coefficients a0, b0 and c0 in
Equations 17, the steady-state cornering curvature is expressed as
ρ=Gs(0)δreq
f+Gp(0)Freq
b=
CfCrL
CfCrL2+mv2
x(lrCrlfCf)δreq
f+w(Cf+Cr)
2 (CfCrL2+mv2
x(lrCrlfCf))Freq
b(19)
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June 27, 2017
Table 1. Vehicle parameters from [1] extended with steering
parameters.
Parameter Value
vehicle mass (m) 1700 kg
vehicle yaw inertia (Jz) 2600 kgm2
cornering stiffness front, rear (Cf, Cr) 97500,97500 N/rad
front,rear distance to CoG (lf, lr) 1.2,1.5 m
track width (w) 1.5 m
steering gear ratio (i) 16
time constant brakes, steering (Tb, Ts) 0.3,0.1 s
wheel radius (rw) 0.32 m
scrub radius (ly) +0.010 or -0.015 m
caster trail (lx) +0.077 mm
steering inertia (Js) 22 kgm2
steering damping (bs) 7.5 Nm/s
steering Coulomb friction (Mc) 187
steering rest stiffness (σ) 11200
pressure to torque front,rear (kFr , k Re) 24,12 Nm/bar
@ 70 km/h a=(1, 26, 260, 1137, 1757)
@ 70 km/h b=(7.7, 128, 512)
@ 70 km/h c=(0.15, 1.37, 2.92)e-3
Figure 5 shows the upper bound of the linear model to what can be achieved of steady-
state curvature and lateral acceleration for maximum differential brake input and steer-
ing. The maximum differential brake and steering input are assumed to be Fb=mg
2and
δreq
f= 22 deg. When the vehicle’s center of gravity height is large and track width is
small, then the inner wheels decrease their vertical load significantly during lateral ac-
celeration. The maximum differential brake force is then consequently less than mg
2. This
effect is however intentionally ignored since the actual vehicle has a low height of center
of gravity (0.4 m) and we seek expressions with as less parameters as possible. For the
actual vehicle the maximum differential brake force is reduced by 13% during a steady
state lateral acceleration of 5 m/s2. The linear model doesn’t consider friction, which for
example implies that the magnitude of lateral acceleration from differential braking and
steering exceeds gm/s2for vx>30 m/s and vx>8 m/s respectively. Since the vehicle
is understeered (lrCrlfCf), the largest curvature during zero steering angle is found
ρmax = lim
vx0ρ=w(Cf+Cr)
2 (CfCrL2)Fbw(Cf+Cr)µmg
4 (CfCrL2)(20)
In Equation 20 it is assumed that the largest differential brake force is Fb=µmg
2, which
is a conservative estimate due to the intentionally ignored effect of load transfer. From
Equation 20 it follows that the maximum curvature is limited and proportional to friction.
For the vehicle in this study the upper bound of cornering curvature during high friction
(µ= 1) is 0.017 1/m which corresponds to a cornering radius of 59 m. Keeping the
assumption of µ= 1, Figure 5 gives us that the curvature is bounded between 0.011 1/m
and 0.017 1/m. The maximum curvature obtained by differential braking is smaller than
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June 27, 2017
0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
vx(m/s)
ρ(1/m)
Freq
b=mg
2N
δreq
f= 0.52 rad
0 5 10 15 20 25 30
0
5
10
15
vx(m/s)
ay=v2
xρ(m/s2)
Freq
b=mg
2N
δreq
f= 0.52 rad
Figure 5. Linear model according to Equation 19 of steady-state curvature and lateral acceleration for either
maximum differential brake or maximum steering input.
for steering. For vehicle speeds below 50 km/h the lateral acceleration will not exceed 3
m/s2when using differential braking. The corresponding speed for steering is 16 km/h. If
”normal driving” is considered as being able to reach lateral acceleration magnitudes of 3
m/s2, differential braking cannot meet that. The reduced cornering ability for differential
braking will in turn increase the need for a careful path and speed planning.
5. OBSERVATIONS OF SCRUB RADIUS EFFECT
5.1. Hands-on versus hands-off the steering wheel
The curvature retrieved from differential braking in previous sections assumes zero steer-
ing angle. However, a brake force on a front wheel will, together with a moment arm
mentioned to as scrub radius, induce an alignment wheel torque, which will influence
steering and in turn the cornering curvature. This section seeks to experimentally quan-
tify the scrub radius effects of the resulting curvature.
The scrub radius is a lateral displacement between the tyre’s center of rotation, caused
by the kingpin axis intersection with with the road plane, and the centerline of the tyre.
When the center of rotation is outside of the centerline of the tyre, the scrub radius is
defined as negative, otherwise it is positive. See Figure 6 for clarification. The differen-
tial brake test described in Section 3.2 was conducted with hands-on the steering-wheel,
where the driver fixed the steering-wheel angle to zero. The test is here repeated but now
with hands-off the steering-wheel i.e. no steering-wheel torque is applied during the test
event. As shown in Figure 7, the steering-wheel angle is no longer zero caused by the
(positive) scrub radius. At the end of the manoeuvre, when vehicle speed is lower, the
steering angle is significantly increased up to 65 degrees. This will in turn, contribute to
even higher cornering curvature. The large steering-wheel angle corresponds to approxi-
mately 4 degrees wheel angle, which confirms that the small angle approximation made
in Equations 1 and 2 is relevant. The experiment clearly demonstrate that the scrub
radius effect substantially influences the physical limit of maximum cornering curvature.
It should be mentioned that the scrub radius effect can be modeled and included in the
vehicle model. This would be beneficial since such model predicts the curvature better.
The scrub radius is however uncertain since it for example varies for different wheel hubs,
and consequently, the model becomes sensitive for the scrub radius parameter.
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June 27, 2017 DiffBrk
Kingpin axis
Kingpin
inclination
fx
Scrub radius
Front left wheel,
rear view
Contact patch,
Top view
Upper joint
Lower joint
Center of
rotation
fy
lx
ly
Figure 6. The scrub radius and here illustrated as negative (left). Spacers used to experimentally vary the scrub
radius (right).
−1 0 1 2 3 4
0
0.01
0.02
0.03
0.04
0.05
time (s)
Freq
b/1000 (kN)
curv. ρ(1/m) hands-on
curv. ρ(1/m) hands-o
−1 0 1 2 3 4
−20
0
20
40
60
80
time (s)
SWA (deg) hands-on
SWA (deg) hands-o
Figure 7. Step response test of curvature (left) and steering-wheel angle (right) for zero steering-wheel angle
(hands-on) and zero steering-wheel torque (hands-off).
5.2. Perturbation of the scrub radius
The observation done in Section 5.1 showed that the scrub radius, when releasing the
steering-wheel, increased the steering alignment torque and the cornering curvature sig-
nificantly. The scrub radius was +10 mm. To further test the cornering ability for various
scrub radius, it was changed to -15 mm. The modification of the vehicle has been made
possible due to the use of so called spacers, which is a device mounted inside the rim
which moves the contact patch outwards and hence creating the positive scrub radius.
The hands-off differential brake test done in Section 5.1 was repeated with the two dif-
ferent scrub radii. As seen in Figure 8, the negative scrub radius will generate negative
steering angles, which will counteract the cornering curvature from yaw torque, ending up
in a smaller magnitude of total cornering curvature. The perturbation test demonstrates
the high sensitivity of the cornering curvature with respect to different scrub radius.
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June 27, 2017
Figure 8 demonstrates that the scrub radius in particular influences the curvature for
low vehicle speed.
4 6 8 10 12 14 16
0
0.02
0.04
0.06
vx(m/s)
curv. ρ(1/m) pos. scrub radius
curv. ρ(1/m) neg. scrub radius
curv. ρ(1/m) model
4 6 8 10 12 14 16
0
50
100
vx(m/s)
SWA (deg) pos. scrub radius
SWA (deg) neg. scrub radius
SWA (deg) model
Figure 8. Observations of curvature and steering-wheel angle for positive versus negative scrub radius for Freq
b=
mg
2. The blue dashed line correspond the steady-state model described in Equation 19.
6. VEHICLE MODEL WITH SCRUB-RADIUS
6.1. Vehicle model derivation
In order to get a conceptual understanding of the scrub radius effect observed in the
previous section, e.g understand how curvature and steering angle depend on the wheel
suspension, this section models the vehicle including the steering system. Of special
interest is the curvature capability caused by the wheel suspension design. The steering
system, which can be described by a second order differential equation [5] has been
provided with front tyre longitudinal force input such that
Js¨
δf+bs˙
δf+lxfF r
y+Mf=ly(fF L
xfF R
x).(21)
When there is an asymmetry in the two front tyre longitudinal forces, an alignment
torque is generated due to the scrub radius moment arm ly. In addition to inertia, Js,
and damping, bs, of the steering system, the torque equilibrium contains alignment torque
due to lateral force and a moment arm lx. This arm is the sum of the caster trail and
the pneumatic trail, i.e. the distance along the tyre centerline from the wheel’s center of
rotation at the road to the point where the lateral force acts. This caster trail depends
on the actual wheel suspension design and the pneumatic trail of the tyre. The moment
arms lxand lyare assumed to be constants and are both illustrated in Figure 6. In
Equation 21, the lift effect [6], i.e. the influence from the normal load of the steering
angle has been neglected since it during the work was found to be low. The friction
torque Mfin the entire steering system has been modeled by the Dahl friction model [7]
such that
˙
Mf=σsign 1Mf
Mc
sign ˙
δf˙
δf,(22)
where Mcis the Coulomb friction torque and σis the rest stiffness. In Figure 9, the
steering friction torque is shown for the signal δf(t) = 0.1 sin(2πt) rad.
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June 27, 2017
f
(
)
Figure 9. The hysteresis map from the Dahl friction model.
From Equation 21 it is obvious that front wheel braking can be used to control the
steering angle. Apart from controlling the steering angle, an asymmetry in front and rear
brake forces generates a yaw torque. We will now reformulate the vehicle model to also
include the effect that the wheel steering angle changes during differential braking.
To capture the difference of braking the front and rear wheels, the lumped brake force
Fbcannot be used any longer. Ideally, all four individual brake forces should be used, but
to keep complexity lower, the braking is here assumed to be done only at one side of the
vehicle. To simplify the notation, the left side is selected for the derivation. The brake
actuator first order system dynamics is here neglected to keep complexity low. Selecting
the state vector xsand input vector ussuch that
xs=
vy
ωz
δf
˙
δf
, us=
fF L
x
fRL
x
Mf
, ys=
ρ
δf
.(23)
The complete vehicle model is then expressed in state-space form as
˙xs=Asxs+Bsus, ys=Csxs.(24)
As=
Cf+Cr
mvxvxlfCflrCr
mvxCf
m0
lfCflrCr
Jzvxl2
fCf+l2
rCr
JzvxlfCf
Jz0
0 0 0 1
lxCf
Jsvx
lxlfCf
JsvxlxCf
Jsbs
Js
(25)
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June 27, 2017
Bs=
0 0 0
w
2Jz+lylf
lxJzw
2Jz0
0 0 0
ly
Js01
Js
, Cs=
01
vx0 0
0 0 1 0
.(26)
When braking at the right side of the vehicle, the sign of Bsshould be changed. Note
that the friction torque, due to its nonlinear nature, is an input to the linear system.
6.2. Model validation
Figure 10 shows the steady-state solution of system 24, which now includes friction
torque, caster trail and scrub radius, together with the measured data. As seen from the
figure, the model outputs resemble the results from the measurements, i.e. the steering
angle and curvature for various vehicle speed. It is seen from both the model output and
measurement that the steady-state steering angle is approximately doubled when de-
creasing vehicle speed from 15 m/s to 10 m/s. It was however observed during the work
that the modeling of the friction was important to get a good match with the measured
data. Caster trail and scrub radius are parameters that can be retrieved relatively easy
from e.g Adams modeling tools, drawings etc. Friction torque in the steering is signifi-
cantly harder to predict. The parameters of the friction model was identified to get as
good match as possible, but Figure 10 shows also the result from a perturbed friction
with a 50% increase of the Coulomb friction torque. High Coulomb friction torque when
there is a positive scrub radius results in smaller magnitude of curvature since the friction
resists the front wheels to be steered in the intended direction. On the contrary, high
Coulomb friction torque when there is a negative scrub radius results in larger magni-
tude of curvature. From the figure it is evident that a correct modeled steering friction
is important in order to get a valid vehicle model. Noteworthy, it was subjectively ob-
served during the measurements that just tiny friction between driver’s hand and the
steering-wheel did reduce or stop the rotation of the steering-wheel.
4 6 8 10 12 14 16
0.01
0.02
0.03
0.04
0.05
vx(m/s)
ρ(1/m)
model
model with pertubed friction
measurement
4 6 8 10 12 14 16
−20
0
20
40
60
vx(m/s)
SWA (deg)
model
model with pertubed friction
measurement
Figure 10. Steady-state curvature and steering-wheel angle from system 24 and measurement for +10 mm scrub
radius for fF L
x=fRL
x=mg
4. The model is simulated with two different parameterizations of steering friction.
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June 27, 2017
6.3. Maximum curvature capability
The non-linearity of friction in the steering model make it more difficult to express steady
state solutions. For example; the steady state solution of the steering angle depends on
previous states, which complicates the interpretation and assessments of the steady state
solution. Further on in this section, we will neglect the steering friction to be able to derive
simple analytical expressions. The analytical expressions will be used to understand, at
a conceptual level, the physics that determines the cornering during differential braking.
The steady-state curvature from Equation 24 when Mf= 0 is derived
ρ=4lylf+ 2lylr+lxw
2lxlrmv2
x
fF L
x+w
2lrmv2
x
fRL
x.(27)
Note that the models steady-state curvature does not depend on the axle’s cornering
stiffness. The lateral velocity is however dependent on cornering stiffness (not shown in
this paper).
The expression for the steady-state steering angle δ(ρ) is long and not suitable to be
presented, but it can through Equation 21 and 7 be expressed as δ(vy, ω) instead
δf=ly
Cflx
fRL
x+vy+lfωz
vx .(28)
Note that if the FL wheel is not braked, the FL wheel will be pointed in the front axle
slip direction.
As seen from Equation 27 the curvature will be increased for the front (left) wheel
braking, if the vehicle’s design parameters is selected such that (4lylf+ 2lylr+lxw)>0
. Otherwise the curvature will decreased. If the curvature is to be maximized, (4lylf+
2lylr+lxw) must be greater than zero together with maximum brake forces on the (left)
side. Assuming that maximum brake forces are applied such that fF L
x=fRL
x=mg
4the
maximum curvature is expressed as
ρmax =µg (ly(2lf+lr) + lxw)
4lxlrv2
x
.(29)
6.4. System design constraints
The ratio between scrub radius and the caster trail is now defined as
ξ=ly
lx
.(30)
From Equation 29, together with the fact that the steady-state lateral acceleration is
quadratic proportional to vehicle speed, it is clear that the steady-state lateral accelera-
tion capability is independent from vehicle speed such that
ay,max =µg (ξ(2lf+lr) + w)
4lr
.(31)
Hence, it can be concluded that the following items contribute to a large lateral acceler-
ation capability:
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June 27, 2017
(1) Large vehicle width
(2) Center of gravity close to the rear axle
(3) Large positive scrub radius
(4) Small positive caster trail
Note that when caster trail approaches 0+, the linear vehicle model gives infinitely
high lateral acceleration capability and steering angle, which of course is not possible
since tyre forces in practice are limited.
When designing the self-driving vehicle Equation 31 can be used to conceptually find
vehicle parameters to get a desired lateral acceleration capability. If the scrub radius
should be designed in order to reach a lateral acceleration capability of ay,max = 3 m/s2
then a vehicle with parameters according to Table 1 would require ξ >= 0.09 and in turn
a positive scrub radius ly>= +7 mm.
Looking at the poles in Figure 11, the vehicle is stable for the positive scrub radius
tested (+10 mm). However when the caster trail switches sign from positive to negative,
the system becomes unstable due to poles in the right half complex plane. Since unstable
system is more difficult to control, negative caster trail is normally avoided.
Pole−Zero Map
Real Axis (seconds
−1
)
Imaginary Axis (seconds−1)
−25 −20 −15 −10 −5 0
−20
−10
0
10
20
Figure 11. Poles and zeros from system 24 for +10 mm scrub radius and +77 mm caster trail at 6 m/s (blue),
12 m/s (green), 18 m/s (red).
7. CURVATURE CONTROLLER DESIGN
In order to tackle the steering fault described in the Introduction, we will here design an
on-board curvature controller to be used after the occurrence of a fault when the vehicle
is driven in autonomous mode. Note again that during the control event, the steering
actuator nor the driver are assumed to steer the vehicle. When the fault is detected, the
curvature controller starts to request differential brake force.
7.1. Selection of control design
In this work, two vehicle models have been developed. This first one (Equation 12) has
no model of the wheel suspension mechanism that has shown to be important during
control of curvature and hands-off the steering wheel. The model contains cornering
stiffness parameters which are uncertain. The second model (Equation 24) has the sus-
pension mechanism modeled, but suffers from that the steering system is hard to model
due to steering friction uncertainty. Variation in steering friction will make it hard to
15
June 27, 2017
design robust controller, therefor the control design will be based on the first model
(Equation 12).
7.2. Control design
The initial steering angle and its evolution, e.g. depending on the uncertain scrub radius,
is here considered as a disturbance. By rearranging Equation 19, the differential brake
request for static conditions is formulated as
Freq
b=1
Gp(0)ρreq Gs(0)
Gp(0)δf,(32)
where the first term in Equation 32 will be the feedforward compensator for the requested
curvature ρreq and the second term the feedforward compensator for rejecting the dis-
turbance δf. The rejection of the steering disturbance is possible since δfis measured
(through steering-wheel angle sensor) and the influence of the curvature is modeled.
Due to model uncertainties and neglected dynamics for both Gp(0) and Gs(0), there
will be a control error and disturbance will not be completely eliminated. Therefore a
feedback loop is needed along with the two compensators. See [8] where the feedforward
and feedback control design are separated. Equation 32 is now expressed
Freq
b(s) = 2CfCrL2+mv2
x(lrCrlfCf)
w(Cf+Cr)ρreq 2CfCrL
w(Cf+Cr)δf+C(s)e(33)
C(s) = Kp1 + 1
sTi
+sTd
1 + sTd/N (34)
e=ρreq ρ=ρreq ωz
vx
.(35)
To limit the high frequency gain of the derivative term, a low pass filter has been incor-
porated in the design of C(s) which gives an upper bound of KpN. Finally, the unfiltered
curvature request is passed through a set-point rate-limiter filter, which protects for un-
desired derivative kicks during abrupt step changes in the curvature requests. See [10] for
PID filter design to avoid over shoots. The control structure in its entirety is illustrated
in Figure 12. The control Fbis virtual and needs to be allocated to wheel individual
brake pressures. The sign of Freq
bdetermines which side should be actuated. There is a
freedom in how Freq
bis distributed among the two tyres. The distribution is here done to
preserve equal friction utilization and hence provide equal lateral force margins for good
robustness. The lateral force component is here for simplicity neglected.
Neglecting the wheel dynamics, the brake pressure requests preq =
[pF L,req pF R,req pRL,req pRR,req]Tto wheels are finally defined as
preq =0lrrw
LkF r 0lfrw
LkRe T
Freq
bif Freq
b0 (36)
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June 27, 2017
1/Gp(0)
C(s) Gp(s)
+
-
+
+
Gs(0)/Gp(0)
-
!
"#
$%&,'()*+-
Gs(s)
++
$%&
Vehicle plant
Controller
Figure 12. Curvature control structure consisting of two feedforward compensators, one feedback controller and
a set-point filter.
preq =lrrw
LkF r 0lfrw
LkRe 0T
Freq
bif Freq
b>0,(37)
where rwis the wheel radius and kF r and kRe are conversion factors from brake pressures
to wheel torques depending on wheel geometry and hydraulic brake cylinders. The brake
pressure request is send to a brake control unit, which is provided with a slip controller
to prevent from wheel lock and limit longitudinal slip to approximately 10%.
The tuning of the PID controller was done with the Ziegler-Nichols method [11] where
frequency information of the controlled system was used to determine the controller
coefficients.
7.3. Test of the control design
To test the controller, we will now revisit the special case discussed in Section 2 and shown
in Figure 1. The vehicle is driving straight ahead at 70 km/h and a steering failure occurs
before a 200 meter road radius. When the curve is entered, a constant curvature request,
corresponding to the actual road curvature ahead, is sent to the curvature controller.
The curvature request is not updated during the control, which is a simplification in the
test set up. The step response with the developed controller is shown in Figure 13. The
time to reach 63% of the final curvature is by the closed loop controller reduced from
approximately 0.4 to 0.3 s compared with the open loop response, but at the expense
of an overshoot. The level of overshooting is a tuning issue, but here fast response has
been prioritized to compensate for delays. The control error converge to zero and the
remaining noise is caused by the wheel slip controller which work close to the slip limit.
The resulting path is shown in Figure 14. The vehicle stays within the pre-defined
one meter lateral road margin. Due to the response time of the differential braking, the
path is shifted to the right in the figure and a constant tracking error remains during
17
June 27, 2017
−1 −0.5 0 0.5 1 1.5
0
5
10
15 x 10−3
time (s)
requested curvature ρref (1/m)
measured curvature ρ(1/m)
dierential brake input Fb/1000 (kN)
Figure 13. Close loop control of curvature starting at t= 0 s.
the control event. This strengthen the arguments that the curvature request should be
updated during the control, i.e. close the loop upon position and orientation relative to
the road, if that is possible.
Differential braking could lead to instability when braking hard on a rear wheel, which
is not investigated in this paper. It is assumed that there exist a stability system moni-
toring and acting when margins to instability is below a threshold.
0 5 10 15 20 25
−1
0
1
2
3
X (m)
Y (m)
vehicle path
road centerline
lateral road margin
Figure 14. Resulting path from the closed loop control of curvature starting at X= 0 m.
8. CONCLUSIONS
Analysis of the developed vehicle models and tests in real world vehicle has shown that
differential braking could be used as an alternative to front axle steering for fault-tolerant
control of self-driving vehicles. There are however physical limits on how large curvature
and lateral acceleration could be achieved compared with steering. Compared with steer-
ing, differential braking can in general not provide as much lateral acceleration for lower
speeds. Large curvature needs large longitudinal differential brake forces, which makes
the curvature directly road friction dependent. Having said that, differential braked vehi-
cles must plan and adapt their speed to not exhibit too large lateral accelerations. There
are two cases of differential braking; either hands-off (torque free) the steering wheel
or hands-on (torque from human driver or steering actuator is engaged). The hands-off
case results in a curvature which is sensitive to wheel suspension parameters since an
unsymmetrical front wheel braking induces a change in front steering angle.
18
June 27, 2017
For the hands-off case, the scrub radius is an important wheel suspension parameter.
Negative scrub radius gives unacceptable cornering capabilities, while a relative large
positive scrub radius is an appropriate design alterative for acceptable cornering capabil-
ities. The paper has shown how the scrub radius should be designed to meet curvature
requirements. Apart from the scrub radius, the friction in the steering system has a key
role for the curvature capability of the vehicle. Large friction will reduce the steering
angle. Large friction together with positive scrub radius will also reduce the curvature
capability.
When selecting the control method it is crucial to get insight about which parameters
are uncertain. Due to the uncertainty in the steering system model, and in particular the
steering friction, the selected control concept do not rely on the steering system.
Future work is suggested to be devoted on the evaluation of the many different combina-
tions of manoeuvres that may occur, e.g. the aggressive special case that was introduced
in Section 2.
References
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[2] Pilutti P, Ulsoy G, Hrovat D. Vehicle steering intervention through differential braking. Journal of
Dynamic Systems, Measurement, and Control; 1998. Vol. 120(3). p. 314(8).
[3] Moshchuk NK. et al. Collision avoidance maneuver through differential braking. Patent US
20130030651 A1; 2013.
[4] Rajamani R. Vehicle Dynamics and Control. Springer US; 2012.
[5] Yih P, Ryu J, Gerdes JC. Vehicle state estimation using steering torque. American Control Confer-
ence; 2004. Vol. 3. p. 2116–2121.
[6] Katzourakis DI. Driver steering support interfaces near the vehicle’s handling limits. PhD Thesis.
Netherlands: TU Delft; 2012.
[7] Drincic B. Mechanical models of friction that exhibit hysteresis, stick-slip, and the stribeck effect.
PhD Thesis. The US: The University of Michigan; 2012.
[8] Brosilow C, Joseph B. Techniques of model based control. Prentice Hall PTR; 2002.
[9] MacAdam C, Ervin, R. Differential braking for limited-authority lateral maneuvering. IDEA Pro-
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This milestone-oriented approach can be divided into seven steps: i) Study the state-of-the-art driver support systems and identify the potential space for improvement. ii) Develop the means (driving simulators, vehicular instrumentation and data analysis methods) to aid the driver steering support interface research. iii) Study the driver steering interface without any support. iv) Utilize the gathered knowledge to develop steering support interfaces, assess them in simulation level, v) and adapt the simulation support controllers into real vehicles and test them. vi) Evaluate the influence of the support interface with the real vehicle results. vii) Based upon the assessment, make a road-map for the commercial implementation of the support interface; if it is fruitful promote its further development with ultimate goal the adoption into production vehicles. he aforementioned milestone-oriented approach has been followed for the development of the driver steering support interfaces presented in this thesis. The current summary substantiates the milestones into the distinct goal addressed in Chapters 2 – 7. The goal to develop the hardware and performance evaluation-control methods in order to engineer realistic haptic cues on the steering wheel of our driving simulator is addressed in Chapter 2. A relatively low-cost solution for hardware is deployed, consisting of a velocity-controlled three-phase brushless servomotor, whose high bandwidth control allows for a realistic representation of forces. To test the system, different inertia-spring-damper systems were simulated and evaluated in time and frequency domain. We concluded, that the designed system allowed reproduction of a large range of steering wheel dynamics and forces, comparable to those found in actual cars. Our target to systematically adjust the steering systems properties of the driving simulator so that it matches the steering feedback and vehicle response of a certain vehicle is addressed in Chapter 3. To do so, we employed the steering sensitivity and steering torque gradient, which are two important metrics describing on-centre vehicle dynamics response and steering feedback. We acquired the steering metrics of real cars during double-lane change tests and indicated the key parameters of the vehicle that determine these steering metrics. We instrumented and tested five modern passenger cars, and used a vehicle dynamics model to extract the metrics for multiple vehicular parameterizations (steering ratio, power assist level, etc.) and test speeds. Sensitivity analysis showed that steering sensitivity was mainly influenced by the components that determine the steering ratio whereas the steering torque gradient was also affected by power assist steering settings. By completing this work, we had the foundation to easily assess the realism of our simulated vehicles’ response as well as to easily adapt the vehicular settings to achieve a realistic steering feedback in our driving simulator. Lane departure appears relevant in 179,000 crashes per year and is related to the greatest number of fatal crashes; up to 7,500 fatal crashes per year in the United States. Infiniti predicts that if lane departure prevention (LDP) were fitted to all vehicles, some 12% of all road fatalities could be prevented annually. The problem is that although numerous studies have shown the potential of lane keeping and LDP systems, there are few studies related to their effects during emergency manoeuvres. Thus, Chapter 4 aims to investigate a road-departure prevention (RDP) system during an emergency manoeuvre. We present a driving-simulator experiment which evaluated various steering interfaces of a road-departure prevention (RDP) system in an emergency situation. The interfaces were: 1) haptic-feedback (HF) where the RDP provided advisory steering torque; 2) drive-by-wire (DBW) where the RDP automatically corrected the front-wheel angle; and 3) DBW & HF, which combined both setups. The RDP system intervenes by applying haptic (guidance) feedback torque and/or correcting the angle of the front wheels (drive-by-wire) when road departure is likely to occur. Thirty test drivers tried to avoid an obstacle (a pylon-confined area) while keeping the vehicle on the road. The results showed that HF without DBW had a significant impact on the measured steering torque, but no significant effect on steering-wheel angle or vehicle path. DBW prevented road departure and reduced mental workload, but lead to inadvertent human-initiated counter-steering. It was concluded that a low level of automation, in the form of HF, does not prevent road departures in an emergency situation. A high level of automation, on the other hand, is highly effective in preventing road departures. Chapter 5 has been divided into three parts (A, B, C), all related to real vehicle testing. Our goal to construct a versatile low-cost instrumentation suitable to be fitted on race cars and develop the methods for processing from raw measurements to user-friendly data suitable for driver behaviour studies is addressed in part A. Through a case study on driving behaviour, during the execution of high speed skid-pad manoeuvres, we could easily notice the markedly different driving behaviours between an expert and a novice driver. The experienced driver could learn quickly how to perform repeatable trajectories, unlike the novice driver. The consistently high performance of the expert driver was realized by relatively small correcting inputs (steering wheel angle, throttle). The experienced driver was able to quickly learn how to generate the correct inputs to the vehicle, to yield repeatable vehicle behaviour and consistently perform well. Our aim to investigate driver control actions during high speed cornering with a rear wheel drive vehicle is depicted in Chapter 5, part B. Six drivers were instructed to perform the fastest manoeuvres possible around a marked circle, while trying to retain control of the vehicle and constant turning radius. The data reveal that stabilization of the vehicle is achieved with a combination of steering and throttle regulation. The results show that the drivers used steering control to compensate for disturbances in yaw rate and sideslip angle. Vehicle accustomed drivers had the most consistent performance resulting in reduced variance of task metrics and control inputs. Our target to design controllers that can stabilize the vehicle as an expert driver would is approached in part C of Chapter 5. There, we present data of driver control commands and vehicle response during the execution of cornering manoeuvres at high sideslip angles (drifting) by an expert driver using a RWD vehicle. The data reveal that stabilization of the vehicle with respect to such cornering equilibria requires a combination of steering and throttle regulation. A four wheel vehicle model with nonlinear tire characteristics is introduced and the steady-state drifting conditions are solved numerically to derive the corresponding control inputs. A sliding mode control is proposed to stabilize the vehicle model with respect to steady-state drifting, using steering angle and drive torque inputs. The performance of the controller is validated in a high fidelity simulation environment; the controller can stabilize the vehicle similarly to an expert driver. We also conceptually describe how the proposed controller can motivate a driver steering support drifting interface in the by-wire sense. Our goal to objectively evaluate vehicular steering systems through detailed driver models is substantiated in Chapter 6. It presents a driver model that consists of a preview controller part that responds to visual feedback and a neuromuscular component that reacts to force-feedback. The developed model is sensitive to steering wheel systems with different dynamics, and can predict both goal-directed steering wheel movements, as well as neuromuscular feedback. To provide evidence, we simulated different parameterizations of a steering system and tested them in conjunction with the developed driver model. We concluded that the developed model could predict the expected response for different steering setups. Our milestone goal to propose haptic steering wheel support when driving near the vehicle’s handling limit (Haptic Support Near the Limits: HSNL) is addressed in Chapter 7. The rationale behind the HSNL, derives from the vehicle’s property to reduce the steering “stiffness” (the steering feedback torque as a function of the steering wheel angle) before the vehicle reaches its handling limits and starts to understeer. The HSNL exaggerates the reduction of the steering “stiffness” and makes it profound to the driver, so he/she avoids excessive steering angle inputs which will result in increased tire slip and consequently lateral force loss. Chapter 7 is divided into two parts (A, B). Part A of Chapter 7 studies the influence of the HSNL in (a) driver-in-the-loop simulation and in (b) real track testing with a vehicle (Opel Astra G/B) equipped with a variable steering feedback torque system. In the simulator study (a) 25 drivers attempted to achieve maximum velocity, on a dry skid-pad while trying to retain control of the simulated vehicle parameterized as the Astra. In (b) 17 drivers attempted to achieve maximum velocity, around a wet skid-pad while trying to retain control of the Astra. Driving aids (ABS and traction control) were disabled during testing. Both the driving simulator and the real vehicle tests led to the conclusion that HSNL assisted the test subjects to drive closer to the designated path while achieving effectively the same speed. In the presence of HSNL, the drivers operated the tires in smaller slip angles and hence avoided saturation the front wheels’ lateral forces and excessive understeer. Finally, the support reduced their mental and physical demand. Part B of Chapter 7, studies the influence of HSNL during high speed cornering in a test-track. 17 test subjects drove around a narrow-twisting tarmac circuit, the aforementioned Opel Astra equipped with a variable steering feedback torque system. The drivers were instructed to achieve maximum velocity through corners, while receiving haptic steering feedback cues related to the vehicle’s cornering potentials. Driving aids (ABS and traction control) were disabled during testing. The test-track tests led to the conclusion that HSNL reduced drivers’ mental and physical demand. One of the primal goals of automotive manufacturers is to reduce the driver’s mental and control effort (c.f. Chapter 7); the work that will be presented in this thesis revealed that steering support near the vehicle’s handling limits can reduce the drivers’ mental and physical demand and can potentially promote safety. We can therefore conclude that certain of the developed support interfaces can be implemented into production vehicles.
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