Time-Optimal Vehicle Posture Control to Mitigate Unavoidable

Time-Optimal Vehicle Posture Control to Mitigate Unavoidable
2013 American Control Conference (ACC)
Washington, DC, USA, June 17-19, 2013
Time-Optimal Vehicle Posture Control to Mitigate Unavoidable
Collisions Using Conventional Control Inputs
Imon Chakraborty, Panagiotis Tsiotras, and Ricardo Sanz Diaz
Abstract— This paper analyzes the mitigation of an unavoidable T-bone collision, where an “intelligent” vehicle executes
an aggressive time-optimal rotation to achieve a favorable
relative orientation with another vehicle prior to impact. The
current paper extends the previous work by the authors on this
problem, by modeling additional vehicle dynamics (neglected
in the prior work) and by utilizing conventionally available
control commands (that is, steering, braking, handbrake) for
the maneuvering vehicle. The commands can either be applied
directly by a trained driver, or (as in the majority of cases)
can be executed with the help of a combination of an Active
Front Steering (AFS) and an Electronic Stability Control (ESC)
system onboard the vehicle. The optimal yaw rotation maneuver
is analyzed for different initial speeds on both dry and wet
asphalt. The results confirm the existence of an “option zone”
for some cases, within which such an aggressive maneuver may
be possible and perhaps even preferable to straight line braking.
I. I NTRODUCTION
Collisions of automobiles either with other automobiles or
with fixed objects present a significant threat to the life and
well-being of the vehicle occupants. While passive safety
systems such as three-point seatbelts, airbags, frontal crumple zones, etc, significantly improve accident survivability,
a closer review reveals that the majority of these features
are aimed at mitigating the effect of a frontal collision.
Unlike front airbags and engine-bay crumple zones, side
impact protection in the form of side airbags and side impact
beams are not yet standard features on all currently produced
commercial passenger vehicles. As a result, a significant
number of them remain vulnerable to side impact collisions.
A particularly dangerous collision scenario, commonly referred to as a “T-bone” collision (Fig. 1), occurs when one
vehicle (referred to as the ‘bullet’ vehicle) drives into the side
of another vehicle (which is said to have been T-boned) [1],
[2]. The inherent risk of injury or fatality for the occupants of
a T-boned vehicle is immediately apparent from its structural
deformation seen in Fig. 1 if there is inadequate side impact
protection. Corroboration of these observations can be found
in the accident statistics published by the National Highway
Traffic Safety Administration (NHTSA) [2].
A push for even higher automobile safety has led to the
development of active safety systems, starting with ABS
to the more recent TCS, ESP, AFS, RSC, etc. ([3], [4],
[5], [6], [7]). These systems are all aimed at stabilizing
Imon Chakraborty, Graduate Student, Aerospace Engineering,
Institute of Technology, Email: [email protected]
Panagiotis Tsiotras, Professor, Aerospace Engineering, Georgia
of Technology, Email: [email protected]
Ricardo Sanz Diaz, Aerospace Engineering undergraduate
Universidad Politécnica de Valencia, Valencia, Spain,
[email protected]
978-1-4799-0176-0/$31.00 ©2013 AACC
Georgia
Institute
student,
Email:
Fig. 1.
T-bone collision.
the vehicle during “abnormal” driving conditions (skidding,
understeer/oversteer). Such conditions are characterized by
nonlinear tire dynamics and a reduced stability envelope. The
primary goal of these stability systems is therefore to restrict
the operation of the vehicle within a linear, well-defined,
stable regime.
While natural from the stability perspective, the current
design of active safety systems leads to an overly conservative approach as far as vehicle maneuverability is concerned.
As noted in Chakraborty et al [8], mechanical, physical and
human factors can all contribute towards scenarios where
a collision is unavoidable. In these cases, the effect of
collisions may be alleviated by deliberately operating in
the nonlinear region of the tires, and executing controlled
aggressive maneuvers. This observation opens the possibility
for the design of more proactive safety systems for passenger
vehicles, which will take advantage of the whole operational
envelope of the vehicle to avoid or mitigate the result of a
collision. In [9], for instance, an Emergency Steering Assist
(ESA) feature is proposed, which assists the driver to steer
away from an obstacle at high speed when the distance to
the obstacle is less than what can be handled by a normal
driver. Along the same lines, in this paper, we consider the
mitigation of a T-bone collision by intentionally operating the
vehicle in the nonlinear tire regime. The proposed collision
mitigation maneuver involves a rapid yaw rotation (analyzed
in this work) that brings the longitudinal axes of the two
vehicles into a near parallel alignment, in order to distribute
the residual kinetic energy of the collision over a larger
surface area, thus mitigating its effects.
Compared to our previous results in [8], where the same
problem was analyzed, this paper offers the following enhancements: (a) First, in [8] the maneuver was made possible with the help of a Torque-Vectoring (TV) technology,
which allows a direct yawing moment to be generated in
order to complement the moment generated by front-wheel
2168
steering [10]. This technology may not be available for most
current passenger vehicles, although it is currently installed
in some high-end vehicles [11], [12]. In the current paper we
demonstrate that the same (almost identical) maneuver can be
performed using more conventional inputs (steering, braking,
handbrake) thus extending the applicability of this maneuver.
(b) Second, a more accurate model of the friction circle is
utilized in this paper to more accurately take into account
the coupling between longitudinal and lateral tire forces.
(c) Third, the load transfer during acceleration/deceleration
owing to the inertia effects was neglected in [8]; load transfer
changes the normal forces at the tires, thus modifying the
applied friction forces, even if the slip ratios and the surface
area friction coefficient remain the same. Modulation of
the friction forces at the front and rear tires by carefully
choreographing acceleration and braking commands is a
common technique used by expert human drivers when
operating over surfaces with low friction coefficient (loose
gravel, snow, ice, etc) [13]. (d) Finally, in [8] the wheel
dynamics were neglected and the control design was done
at the longitudinal force level. Here the control input is the
actual torque applied at the wheels instead.
II. DYNAMIC M ODEL OF V EHICLE
In this work we use a simplified single-track “bicycle”
model for the vehicle dynamics [14], [15], [16], [17]. The
equations of motion are then given by
1
(Fxf cos δ − Fyf sin δ + Fxr ) + vr, (1a)
u̇ =
m
1
v̇ =
(Fxf sin δ + Fyf cos δ + Fyr ) − ur, (1b)
m
1
(ℓf (Fxf sin δ + Fyf cos δ) − ℓr Fyr ), (1c)
ṙ =
Iz
ψ̇ = r,
(1d)
ẋ = u cos ψ − v sin ψ,
(1e)
ẏ = u sin ψ + v cos ψ,
(1f)
1
ω̇f =
(Tbf − Fxf R),
(1g)
Iw
1
ω̇r =
(Tbr − Fxr R),
(1h)
Iw
where the state vector is x = [u, v, r, ψ, x, y, ωf , ωr ]T , and
where u and v are, respectively, the body-fixed longitudinal
and lateral velocities, r is the yaw rate, ψ is the vehicle’s
heading, x and y are, respectively, the Earth-fixed coordinates
of the vehicle CG, and ωf ≥ 0 and ωr ≥ 0 are the angular
speeds of the front and rears wheel, respectively. In (1), m
and Iz are respectively the mass and yaw moment of inertia
of the vehicle, Iw is the rotational inertia of each wheel
about its axis, R is the effective tire radius, and ℓf , ℓr are
the distances of the front and rear axles from the vehicle CG,
respectively.
The control inputs entering the system are the front and
rear wheel torques, denoted by Tbf and Tbr respectively,
and the road-wheel steering angle δ. Note that when the
wheel has nonzero angular velocity, i.e. ωj > 0, Tbj is equal
to the torque applied by the braking mechanism. However,
when a wheel “locks”, i.e. ωj = 0, Tbj is independent of
the applied brake pressure and self-adjusts to balance the
moment generated by the road force. For the front tire, we
therefore have
(
− (1 − γb )Tb , ωf > 0,
Tbf =
(2)
Fxf R,
ωf = 0.
A similar relationship holds for the rear wheel, but also
includes the handbrake torque, as follows
(
− γb Tb − Thb , ωr > 0,
Tbr =
(3)
Fxr R,
ωr = 0.
In the previous expressions Tb is the braking torque of the
regular brakes (henceforth, “braking torque”), Thb is the
braking torque due to handbrake application (henceforth,
“handbrake torque”) and γb is the front-to-rear torque distribution ratio. Note that all-wheel braking with the additional
application of a handbrake is equivalent to independent
front/rear wheel braking, which can be easily implemented
via an on-board ESC system. Similarly, the steering input can
be commanded by an (expert) human driver, or via the use of
an AFS. In the sequel, we will not distinguish between the
two possible modes of operation (manual or semi-automatic).
Rather, we will focus solely on the optimization results. We
leave the actual implementation of the optimal commands to
be determined by the particular user and/or application, as
the case might be.
The control vector is therefore given by u = [δ, Tb , Thb ]T .
It is assumed that the controls are bounded in magnitude
between upper and lower bounds as follows:
δmin ≤ δ(t) ≤ δmax ,
0 ≤ Tb (t) ≤ Tb,max ,
0 ≤ Thb (t) ≤ Thb,max .
(4a)
(4b)
(4c)
In addition, we will assume that the brakes have a fixed
front-to-rear distribution ratio as follows
1 − γb
Tfront
=
,
γb ∈ (0, 1).
(5)
Trear
γb
In (1), Fij (i=x, y; j=f, r) denote the longitudinal and
lateral force components developed by the tires defined in a
tire-fixed reference frame. These forces depend on the normal
loads on the front and rear axles, Fzf and Fzr . as well as on
the normalized longitudinal and lateral velocity components
(namely, the longitudinal and lateral slip). Since aggressive
maneuvers involve large slip angles, the well-known Pacejka
“Magic Formula” (MF) [18] is used, which models the tire
forces as transcendental functions of the slips. In particular,
µj
=
µij
=
Fij
=
sin(C arctan(Bsj )),
sij
− µj ,
sj
µFzj µij , i = x, y; j = f, r,
(6)
where µ is the tire-road coefficient of friction, sij (i =
x, y; j = f, r) are the slip ratios given by
Vxj − ωj R
Vyj
,
sxj =
,
syj = (1 + sxj )
ωj R
Vxj
q
s2xj + s2yj , j = f, r.
(7)
sj =
2169
TABLE I
V EHICLE AND TIRE DATA .
with
Vxf
Vyf
Vxr
=
=
=
u cos δ + v sin δ + rℓf sin δ,
−u sin δ + v cos δ + rℓf cos δ,
u, Vyr = v − rℓr
Variable
m
Iz
Iw
ℓf
ℓr
h
R
(8)
and Fzf , Fzr = mg − Fzf are the axle normal loads. These
are computed using geometry, force and moment balance in
the x − z plane, and the assumed force - axle load linearity
shown in (6). The front axle normal load is given by
Value
1245
1200
1.8
1.1
1.3
0.58
0.29
Unit
Kg
Kg.m2
Kg.m2
m
m
m
m
Variable
B
C
δmax =-δmin
Tb,max
Thb,max
γb
g
Value
7
1.4
45
3000
1000
0.4
9.81
Unit
deg
Nm
Nm
m/s2
IV. D ISCUSSION OF R ESULTS
Fzf
mgℓr − µhmgµxr
=
ℓf + ℓr + µh(µxf cos δ − µyf sin δ − µxr )
(9)
For more details on the nonlinear friction model used in this
paper, see [14], [15], [16].
III. O PTIMAL C ONTROL P ROBLEM F ORMULATION AND
N UMERICAL S OLUTION
The T-bone collision mitigation maneuver is analyzed for
three different initial speeds and for two different fiction
coefficients (high and low). The road-tire friction coefficient
µ is set to 0.8 (which approximately corresponds to average
tires on a dry road) for the high µ case, and to 0.5 (which
roughly corresponds to wet road) for the low µ case. The
initial speeds where chosen as V0 = 40, 55, and 70 km/h in
both cases.
The proposed maneuver is formulated as a constrained
minimum-time optimal control problem (OCP), i.e.,
Vehicle Ground Trajectory
16
14
tf
dt = tf ,
12
(10)
0
10
via
an
optimal
control
input
u(t)∗
=
∗
∗
∗ T
[δ(t) , Tb (t) , Thb (t) ] , subject to the control constraints
(4), the differential constraints (vehicle dynamics) given by
(1), and the force modeling conditions given by (6)-(9).
The initial condition corresponds to straight-line motion
with front and rear wheels in pure rolling condition, i.e.,
x0 = [V0 , 0, 0, 0, 0, 0, V0 /R, V0 /R]T . The final condition is
ψ(tf ) = ψf = π/2, with other state variables free.
The optimal control problem was solved numerically using
the Gauss Pseudospectral Optimization Software (GPOPS)
code [19]. GPOPS requires the user to provide initial guesses
for the state and control vector elements at the initial and
final time (and optionally at intermediate times as well) and
then iterates towards the optimal solution.
The optimality of the obtained solution was verified from
the time histories of the Hamiltonian and the co-states of
the problem, also computed by GPOPS, but not shown here
owing to space limitations. Conformity with (4) was used to
ensure control feasibility. The feasibility of the optimal state
trajectory computed by GPOPS was also verified by running
a MATLAB model of the vehicle with the computed optimal
control inputs in an open-loop fashion, thus validating the
GPOPS solution. The resulting solutions were also validated
against a four-wheel, full dynamic vehicle model that includes suspension dynamics, roll and pitch motion using
CarSim. For an animation of the optimal T-Bone mitigation
maneuver using CarSim, please see http://www.ae.gatech.
edu/labs/dcsl/movies/TargetAcceleratingBraking.avi.
The numerical values for all vehicle and tire parameters
used in the optimization and the numerical simulations are
shown in Table I.
X Position (m)
min J =
Z
8
6
4
2
0
−2
0
5
10
15
20
Y Position (m)
Fig. 2. Vehicle trajectories for the three initial velocities: 40 km/h (green),
55 km/h (red), and 70 km/h (blue) for µ = 0.8.
Figure 2 shows the vehicle trajectories and the vehicle
posture for the case with µ = 0.8. The aggressive nature of
the rotation essentially decouples the much faster rotational
dynamics from the translational dynamics. Therefore, while
the forward distance traveled naturally increases with higher
initial speed, the lateral deviation (sideways movement) of
the vehicle is small for all three cases. The vehicle trajectories for µ = 0.5 exhibit similar pattern and are not shown
here for the sake of brevity.
Figure 3 shows the time-history of the vehicle states. For
all cases, the vehicle reaches and maintains a maximum yaw
rate for the duration of the maneuver. Since the terminal
condition is determined by the time to rotate clockwise by
90 deg, it turns out that the duration of the maneuver is
independent of the initial speed. As a result, the time history
of the heading angle is essentially identical for all cases. This
is clearly shown in the middle right plot in Fig. 3.
Figure 4 shows the control histories over time. The top
plot shows the road-wheel steering angle in degrees. The
maneuver is initiated with a pro-steering (steering into the
turn) input, gradually decreasing somewhat to apply counter-
2170
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Yaw Rate
200
100
Case 1
Case 2
Case 3
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Front wheel speed
600
Case 1
Case 2
Case 3
r
F. wheel speed, ω (RPM)
150
500
400
300
200
100
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
Fig. 3.
0.7
0.8
0.9
=
κj
=
where Fj =
q
Steering Angle (deg.)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Footbrake input vs. Time
Case 1
Case 2
Case 3
0.8
0.6
0.4
0.2
0
−15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.9
Time (s)
Heading Angle
100
80
60
40
Case 1
Case 2
Case 3
20
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Front tire long. grip usage
200
100
0
0.5
0.6
0.7
0.8
0.9
50
0.8
0.9
Rear tire long. grip usage
0
Case 1
Case 2
Case 3
50
0
−50
−50
−100
−100
Time (s)
0
j = f, r,
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
Time (s)
Front tire lateral grip usage
State histories (µ = 0.8).
Fij
, i = x, y,
µFzj
Fj
j = f, r,
µFzj
0.7
100
Case 1
Case 2
Case 3
300
0.4
0.6
Control histories (µ = 0.8).
Fig. 4.
100
0.3
0.9
Time (s)
400
0.2
0.8
Case 1
Case 2
Case 3
0.9
Case 1
Case 2
Case 3
0.1
0.7
1
600
0
0.6
0.8
Time (s)
Rear wheel speed
500
0.5
Time (s)
Handbrake input vs. Time
0
The middle plot in Fig. 4 shows the normalized brake
input. The brake input is moderate in each case, occurring
mainly during the initial and final part of the maneuver.
The handbrake input, whose time history is shown in the
bottom plot of Fig. 4, plays a vital role during this maneuver.
Initially, full handbrake input is applied to saturate the rear
tires through a sudden reduction in their angular velocity (in
fact, notice from Fig. 3 that for each case, the rear tires are
either locked, or nearly locked, for a significant portion of
the maneuver). This results in a deliberate loss of traction
at the rear tires, which is a well-known technique (often
used by expert drivers) to initiate and maintain yaw rotation
of the vehicle. The deliberate saturation of the rear tires
removes the stabilizing influence of the rear axle forces on
the vehicle dynamics, and a rapid yaw rotation is facilitated
by the instability thereby produced.
Figure 5 shows the operating conditions of the tires
during the maneuver. The following quantities are defined
as suitable metrics for tire saturation status, and in each case
the denominator represents the dynamically varying radius
of the friction circle owing to longitudinal load transfer.
κij
0
κxr (%)
0.2
−10
(11)
(12)
2 + F 2 . For the friction circle constraint
Fxj
yj
κyf (%)
0.1
−5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Rear tire lateral grip usage
100
100
50
50
0
Case 1
Case 2
Case 3
−50
0
Case 1
Case 2
Case 3
−50
−100
−100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
Time (s)
Front tire total grip usage
κf (%)
0
−40
κyr (%)
0
0
−20
100
100
50
50
0
Case 1
Case 2
Case 3
−50
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Rear tire total grip usage
κr (%)
5
Brake input
10
Case 1
Case 2
Case 3
20
1
Case 1
Case 2
Case 3
0
Handbrake input
15
Lateral Velocity
Steering Input vs. Time
40
5
κxf (%)
Case 1
Case 2
Case 3
R. wheel speed, ωr (RPM)
Heading Angle, ψ (°) Lateral Velocity, v (m/s)
Longitudinal Velocity
20
°
Yaw Rate, r ( /s)
Long. Velocity, u (m/s)
steering towards the end of the maneuver. Note that this
counter-steer command is much less aggressive than the one
reported in [8] where full counter-steer is necessary to arrest
the vehicle and stop the rotation completely at the final time.
0
Case 1
Case 2
Case 3
−50
−100
−100
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Fig. 5.
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
Tire operating points (µ = 0.8).
to be satisfied, we must have κij ∈ [−1, 1] and κj ∈ [0, 1].
From Fig. 5, it is seen that the front tire operates on the
boundary of the friction circle for almost the entire maneuver
in all three cases. The majority of the available road grip
is utilized to generate lateral (cornering) forces, while the
remaining available grip is used during the initial and final
stages for generating longitudinal (braking) forces as well.
The results for the low friction coefficient case (µ = 0.5)
are very similar and thus are not repeated here for the sake
of brevity. The only important observations were that the
steering and footbrake commands were more gradual and
2171
Speed
(Km/h)
40
55
70
SLB dist
(m)
8
15
24
TBM dist Window
(m)
(m)
8
0
12
3
15TABLE III4
SLB vel
(Km/h)
24
43
Longitudinal Separation (m)
TABLE II
O PTION W INDOWS FOR C ASES 1, 2, AND 3 (µ = 0.8)
TBM vel
(Km/h)
21.6
36
54
O PTION W INDOWS FOR C ASES 1, 2, AND 3 (µ = 0.5)
Speed
(Km/h)
40
55
70
SLB dist
(m)
13
24
39
TBM dist
(m)
11
15
20
Window
(m)
2
9
19
SLB vel
(Km/h)
14
33
48
TBM vel
(Km/h)
25
39
54
40
40
30
30
Z-3
20
Longitudinal Separation (m)
40
40
40
30
30
30
Z-3
Z-3
Z-3
20
20
20
Z-2
10
Z-1
Z-2
10
10
Z-1
40 Km/h
Fig. 6.
Z-1
55 Km/h
70 Km/h
Decision making options (µ = 0.8).
Tables II and III show (for the three initial speeds and two
friction coefficients considered) the SLB stopping distances
and the forward distance traveled during the TBM. From
these results, the scenarios depicted in Fig. 6 and Fig. 7
arise. In Zone Z-1 neither successful braking nor successful
Z-3
30
20
Z-2
20
10
Z-1
10
Z-2
10
Z-3
Z-2
Z-1
Z-1
40 Km/h
the majority of the activity was from the handbrake. This
makes sense since over low surfaces, steering the vehicle
by redirecting the front tires is less effective and using the
handbrake to modulate the lateral tire forces becomes a better
option.
In order to compare the proposed T-Bone mitigation maneuver (TBM) with normal, straight line braking (SLB), we
also computed the minimum distances for a straight-braking
vehicle to come to a complete stop from the same initial
conditions. Note that the straight line distances reported in
[8] included the driver reaction time (about 1.5 sec) since
the objective of that paper was to compare a completely
autonomous active safety system against a human-operated
vehicle. In this work, the driver reaction time is not taken
into account for either the SLB or the TBM, allowing a
direct comparison of the maneuver characteristics only. The
most optimistic SLB scenario results in maximum possible
deceleration of |amax | = (1/m) µ (Fzf + Fzr ) = µg,
achieved at the maximum of the slip-force curve of the
MF [20], and in general this would require ABS to prevent
wheel-lock.
40
Fig. 7.
55 Km/h
70 Km/h
Decision making options (µ = 0.5).
rotation is possible, while a second vehicle inside zone Z-3
poses no real threat, as simple SLB will allow a full-stop in
time.
What is interesting to note from these figures is that
for moderate-to-high speeds, the forward distance traversed
during the aggressive rotation is less than the SLB stopping
distance. This results in the creation of an “option window”
(Zone Z-2), such that if the second vehicle is spotted within
this window, a successful 90 deg rotation is possible, whereas
braking to a full stop using straight-line braking is not. It
is in this zone that the aggressive yaw rotation maneuver
(TBM) may allow the effects of a collision to be mitigated
by allowing a relative reorientation of the vehicles even if a
collision is ultimately inevitable.
It is evident that the option to use a TBM maneuver
becomes more attractive at higher speeds and for lower
friction coefficient surfaces. This makes sense since the
maneuver does not alter too much the total velocity of the
vehicle. It rather re-distributes the velocity between the body
x and y axes. For low friction coefficient surfaces the velocity
of the vehicle is not going to change significantly using
the brakes so the only reasonable option is to re-distribute
the velocity to the lateral vehicle direction; this is exactly
what the maneuver does. Another interesting observation
from Tables II and III is that the translational velocity of
the vehicle at the end of the TBM maneuver is almost the
same regardless of the road friction. This implies that such
a maneuver can be reliably (i.e. robustly) executed over a
variety of road condition surfaces.
Considering the physics of a collision between two automobiles, and the results shown in Tables II and III and
Figures 6 and Fig. 7 the following may be reasoned:
• For a case where collision avoidance is impossible
(Zone Z-2), if the projected terminal pre-collision velocity is sufficiently low, then it may be wise to select
SLB. In that case, the maximum possible deceleration
of the bullet vehicle would result, and even though
the collision would ultimately be T-bone in nature,
it would nevertheless be at a lower velocity. In that
case, the TBM “option”, though available, would not
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be exercised.
If the projected pre-collision velocity is high, then a
failure to perform the TBM would result in a T-bone
impact at high velocity. In that case, exercising the TBM
“option” might allow a mitigated impact, in which the
pre-impact velocity would be higher than that which
would result from SLB, but in which the relative “sideon” orientation of the vehicles would distribute the
residual kinetic energy of the impact through the vehicle
frames and make possible a more favorable outcome for
occupants of both vehicles.
It would be interesting to investigate whether there exist
crit
a critical projected pre-impact velocity Vrel
above which
the TBM option is recommended, while below it SLB is
always preferable. The investigation of the existence and
crit
magnitude of Vrel
requires a detailed consideration of the
structural deformation during the collision, the actual load
paths for the energy dissipation for both vehicles, etc. Albeit
such an investigation would provide extremely interesting
insight into the practical viability of the TBM maneuver
(whose feasibility was demonstrated here), it is nonetheless
a problem pertaining to the structural dynamics of collisions
between deformable bodies and is therefore beyond the scope
of the current paper.
•
V. C ONCLUSION
Results are presented for a time-optimal aggressive maneuver aimed at mitigating the effect of an unavoidable
collision between two vehicles at a traffic intersection. The
aggressive maneuver is posed as an optimal control problem
and solved numerically. The solution is validated using a
nonlinear model of the vehicle. It is shown that the existence
of an option zone, where a T-Bone mitigation maneuver may
be beneficial, depends on the velocity of the incoming vehicle
and it becomes more favorable at high speeds and over roads
with low friction coefficient.
In contrast to our previous work [8], which was based
on torque-vectoring (TV), the proposed collision mitigation
maneuver utilizes only conventional control inputs. It thus
offers an alternative methodology to execute the same type
of maneuver, either by a trained driver using a conventional
automobile, or (most likely) through the use of active front
steering (AFS) together with individual brake controls like
ESC. The ESC, in particular, can mimic the expert driver’s
use of handbrake to saturate the rear wheel, thus inducing a
large yaw rotation of the vehicle about its z-axis.
It should be clear that the purpose of this paper, along with
[8], was to show the feasibility of a time-optimal aggressive
collision mitigation maneuver. The use of optimal control
does not allow a real-time implementation of this technique
for the time being, and we have not made any attempt
towards this direction in this paper. However, the results show
that the option is there. Natural extensions therefore include
the real-time command generation along with superposition
of a controller to track the maneuver in the presence of
system uncertainties (friction coefficient, mass, moment of
inertia, CG position, etc.), and the linking of the aggressive
rotation with an optimal straight-line deceleration segment
preceding it to yield a complete optimal collision mitigation
maneuver.
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