Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
Adaptive Pid Control With Pitch Moment
Rejection For Reducing Unwanted Vehicle
Motion In Longitudinal Direction
Fauzi bin Ahmad1, Khisbullah Hudha2,
Hishamuddin Jamaluddin3
, Smart Material and Automotive Control (SMAC) Group,
Faculty of Mechanical Engineering,
Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya,
76100 Durian Tunggal, Melaka, MALAYSIA
1 2
Faculty of Mechanical Engineering,
Universiti Teknologi Malaysia (UTM)
81310 UTM Skudai, Johor, Malaysia
3
Email: [email protected],
[email protected], [email protected]
ABSTRACT
This manuscript provides a detailed derivation of a full vehicle model,
which may be used to simulate the behavior of a vehicle in longitudinal
direction. The dynamics of a 14 degrees of freedom (14-DOF) vehicle
model are derived and integrated with an analytical tire dynamics namely
Calspan tire model. The full vehicle model is then validated experimentally
with an instrumented experimental vehicle based on the driver input from
brake or throttle pedals. Several transient handling tests are performed,
namely sudden acceleration and sudden braking test. Comparisons of the
experimental result and model response with sudden braking and throttling
imposed motion are made. The results of model validation show that
the trends between simulation results and experimental data are almost
similar with acceptable error. An adaptive PID control strategy is then
developed on the validated full vehicle model to reduce unwanted vehicle
motions during sudden braking and throttling maneuver. The results show
that the proposed control structure is able to significantly improve the
dynamic performance of the vehicle during sudden braking and sudden
acceleration under various conditions. The proposed controller will be used
to investigate the benefits of a pneumatically actuated active suspension
system for reducing unwanted vehicle motion in longitudinal direction
KEYWORDS: active suspension, dive and squat, 14 D.O.F. vehicle model,
adaptive PID.
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Journal of Mechanical Engineering and Technology
1.0INTRODUCTION
It is deemed necessary and usefull to isolate disturbance elements that
are prevalent in many mechanical systems. A clear example can be seen
in an automotive system in which the passengers of a car should ideally
be isolated from vibration or shaking effects of the car’s body when the
car hits a bump, cornering and braking. Reffering to the characteristic
of the vehicle’s movement in longitudinal direction, the vehicle will
dive forward when brake is applied. This is due to the fact that, inertia
will cause a shift in the vehicle’s center of gravity and weight will be
transferred from the rear tires to the front tires. Similarly, the vehicle
will squat to the rear when throttle input is applied. This is due to
the weight transfer from the front to the rear. Both dive and squat are
unwanted vehicle motions known as vehicle pitching (Ahmad et.al.,
2008a); (Ahmad et.al., 2008b) and (Fenchea, 2008). This motion will
cause the vehicle becomes unstable, lack of handling performance,
out of control and moreover may cause an accident (Bahouth, 2005).
Although this problem is worse, the common passive suspension
cannot be controlled to give energy to suppress the weight transfers that
cause moment. It is because the spring damper elements are generally
fixed and are chosen based on the design requirement of the vehicle
(Priyandoko et.al., 2005).
To solve the problems, a considerable amount of works have been
carried out to solve the problem both theoretically and experimentally
(Alleyne and handrick1995; Lin and Kennelkopoulus, 1997). Through
the combination of mechanical, electrical and hydraulic components,
a wide range of controllable suspension systems have been developed
varying in cost, sophistication and effectiveness. In general, these
systems can be classified into three categories: semi-active suspensions
(Gao et.al., 2006), active anti-roll bars (Sampson et.al, 2000) and
active suspensions (Weeks et.al., 2000; Sam et.al., 2005). A semi active
suspension is a passive system with controlled components usually
the orifice or the fluid viscosity, which is able to adjust the stiffness
of the damper. The active anti-roll bar system consists of an anti-roll
bar mounted in the body and two actuators on each axle to cancel out
the unwanted body motion. While the active suspension system is
a controllable suspension that had the ability to add energy into the
system, as well as store and dissipate it.
Investigation of active suspension systems for car models is recently
increasing mostly because, they offer better riding comfort to passengers
of high-speed ground transportation compared to passive and semiactive suspension systems. The research of active suspension systems
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Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
are derived by optimal control theory on the assumption that the car
model is described by a linear or approximately linear system whose
performance index is given in the quadratic form of the state variables
and the (Donahue, 2001); (Sampson et.al., 2002); (Sampson et.al., 2003a);
(Sampson et.al., 2003b); (Karnopp, 1995); (Sam, 2006) and (Gao et.al.,
2006). However, nonlinear and intelligent active suspension systems
are proposed for complicated models with no negligibly strong nonlinearity and uncertainty. Numerical and experimental results showed
that such active suspension systems give relatively more satisfactory
performance, but need more increasing loads to achieve active control,
compared with the linear active suspension systems (Yoshimura et.al.,
1997a; Yoshimura et.al., 1997b; Yoshimura et.al., 2003); Yoshimura and
Watanabe, 2003; Labaryade et.al., 2004; Kruczek et.al., 2004); (Toshio
and Atsushi, 2004)and (Yoshimura et.al 2001). Some of the studies used
4-DOF vehicle model (Campos et.al., 1999; Vaughan et.al., 2003), and
7-DOF vehicle model has been investigated by Ikanega et.al., (2000),
and Zhang et.al., (2004)
The exploration of the effectiveness active suspension system on real
vehicle has been lead by Lotus Company at early 1980s. The company
proposed a hydraulic active suspension system as a means to improved
cornering in racing cars and come out with a new electro-hydraulic
active suspension that is used in Lotus Excel model in 1985, but this was
never offered to public (Watton et.al., 2001). Six years after that, Nissan
Motor Company produce a new type of luxury car that included two
options joined namely traction control system and the world’s first FullActive Suspension (FAS). It used hydraulic actuators as the controllable
suspension and used 10 sensors to counteract body lean, nose lift and
nose dive, and fore/aft pitch on wavy surfaces (Trevett 2002). Beside
that Mercedes Company used Hydraulic system to eliminate body
roll called Active Body Control system (ABC). In this application the
conventional spring-damper unit is mounted in series with the fluid
chamber. When the car is cornering, the springs on the outer side of the
corner will be compressed and at the same time the fluid chamber will
be filled to compensate for the compression of the spring. In the same
case, BMW company also used a hydraulic system to prevent body roll
namely Dynamic Drive. The hydraulic system is used as the element in
the anti roll bar that can supply a torque to each side of the anti roll bar
(Ficher and Isermann 2004; Kadir 2009).
Since the hydraulic system is widely used as the active suspension,
Citroen Company tries to make a difference by developing an active
suspension base on the combination of the hydraulic and pneumatic
system namely Hydractive Suspension System and then implemented it
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Journal of Mechanical Engineering and Technology
into the Citreon Xantia Model 1989 (Martens 2005). Furthermore, Bose,
known for its high-performance audio products, has developed an
active suspension system which uses linear electric motors that replace
the conventional spring-damper unit to produce the force required for
eliminating body roll and pitch. The improvement in ride quality and
the ability to suppress roll and pitch comes at an expense of energy
consumption. However it contrasts to a hydraulic active suspension
because the linear motor also can function as a generator. This means
that, when the motor has to produce a force in the opposite direction
of its velocity, energy can be generated where normally it would be
dissipated by a conventional damper (Kadir, 2009).
The extensive research on the active suspension have resulted many
control strategies. The control strategies that have been proposed to
control the active suspension system may be loosely grouped into
linear, nonlinear, hybrid and inteligent control approaches. The linear
control strategies are mainly based on the optimal control theory such
as LQR, LQG, LTR and H-infinity and are capable of minimizing a
defined performance index. Application of the LQR method in active
suspension system has been proposed by Hrovat (1988); Tseng and
Hrovat (1990); Esmailzadeh and Taghirad (1996); Sam et.al., (2000); and
Su et.al., (2008). The non-linear control strategies that have been applied
in controlling active suspension are Fuzzy logic control used by Toshio
and Itaru (2005); March and Shin (2007); and Yoshimura et.al., (1999)
and Sliding Mode Control (SMC) by Park and Kim (1998); Decarlo et.al.,
(1998) and Sam et.al., (2005). In terms of hybrid control strategies, PID
Fuzzy Logic Neural Network (NN) and Genetic Algorithm (GA) have
been applied to the active suspension system such as GA based PID
proposed by Feng et.al., (2003) and Sliding Mode Fuzzy control technique
proposed by Ting et.al., (1995). At the side of that, the intelligent control
approaches have also been widely used which resulted many strategy
like Fuzzy reasoning and a disturbance observer (Toshio and Itaru,
2005) and adaptive Fuzzy active force control (Mailah and Priyandoko,
2007).
Sparked with technological sophistication nowadays, a pneumatically
actuated active suspension (PAAS) system is proposed. The proposed
PAAS system is used to minimize the effects of unwanted pitch and
vertical body motions of the vehicle in the presence of braking or
throttle input from the driver. Like those stated, the proposed active
suspension system is the system in which the passive suspension system
is augmented by pneumatic actuators that supply additional external
forces. The system is developed by using four units of pneumatic systems
that are installed between lower arms and vehicle body parallel with
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Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
passive suspension. The reason of using the pneumatic system is due
to the cheap cost for implementing in the real vehicle, the availability
of the air without cost to pay and the compressibility of the air that can
be used as the medium to transmit force. The proposed control strategy
for the PAAS system is the combination of Adaptive PID (APID) based
feedback control and pitch moment rejection based feed forward
control (Stillwel et.al., 1999; Lee et.al., 2001; Sedaghati, 2006; Fialho and
Balas, 2002). Feedback control is used to minimize unwanted body
pitch motions, while the feed forward control is intended to reduce
the unwanted weight transfer during braking and throttle maneuvers.
The forces produced by the proposed control structure are used as the
target forces by the four unit pneumatic actuators. The reason of using
adaptive control based PID controller is because the PID controller has
already been proven effective in many applications where it is easy to
maintain and easy to implement in the Online Famos.
The proposed control structure is implemented on a validated full
vehicle model. The full vehicle model can be approximately described
as a 14-DOF system subject to excitation from braking and throttling
inputs. It consists of 7-DOF vehicle ride model and 7-DOF vehicle
handling model coupled with Calspan tire model (Kasprzak et.al., 2006;
Kadir et.al., 2008; Ahmad et.al., 2008c; Hudha et.al., 2008). MATLABSIMULINK software is chosen as a computer simulation tool used to
simulate the vehicle dynamics behavior and evaluate the performance of
the control structure. In order to verify the effectiveness of the proposed
controller, passive system and active system with PID controller without
pitch moment rejection are selected as the benchmark
This paper is organized as follows: The first section contains the
introduction and the review of some related works, followed by
mathematical derivations of 14-DOF full vehicle model with Calspan
tire model in the second section. The third section presents the proposed
control structure for the pneumatically actuated active suspension
system. The following section explains about the validation of 14-DOF
vehicle model with the data obtained from instrumented experimental
vehicle. The fifth section presents the performance evaluation of the
proposed control structure. The last section contains some conclusion.
2.0
FULL VEHICLE MODEL WITH CALSPAN TIRE MODEL
The full-vehicle model of the passenger vehicle considered in this
study consists of a single sprung mass (vehicle body) connected to four
unsprung masses and is represented as a 14-DOF system. The sprung
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mass is represented as a plane and is allowed to pitch, roll and yaw as
well as to displace in vertical, lateral and longitudinal directions. The
unsprung masses are allowed to bounce vertically with respect to the
sprung mass. Each wheel is also allowed to rotate along its axis and
only the two front wheels are free to steer.
2.1
Modeling Assumptions
Some of the modeling assumptions considered in this study are as
follows: The vehicle body is lumped into a single mass which is
referred to as the sprung mass, aerodynamic drag force is ignored, and
the roll centre is coincident with the pitch centre and is located just
below the body center of gravity. The suspensions between the sprung
mass and unsprung masses are modeled as passive viscous dampers
and spring elements. Rolling resistance due to passive stabilizer bar
and body flexibility are neglected. The vehicle remains grounded at
all times and the four tires never lost contact with the ground during
maneuvering. A 4-degree-tilt angle of the suspension system towards
vertical axis is neglected ( = 0.998 1). The tire’s vertical behavior is
represented as a linear spring without damping, whereas the lateral
and longitudinal behaviors are represented with Calspan model.
Steering system is modeled as a constant ratio and the effect of steering
inertia is neglected.
2.2 Vehicle Ride Model
The vehicle ride model is represented as a 7-DOF system. It consists of
a single sprung mass (car body) connected to four unsprung masses
(front-left, front-right, rear-left and rear-right wheels) at each corner.
The sprung mass is free to heave, pitch and roll while the unsprung
masses are free to bounce vertically with respect to the sprung mass.
The suspensions between the sprung mass and unsprung masses are
modeled as passive viscous dampers and spring elements, while,
the tires are modeled as simple linear springs without damping. For
simplicity, all pitch and roll angles are assumed to be small. A similar
simplicity,
simplicity,
all was
pitch
all pitch
andand
roll
angles
angles
are are
assumed
assumed
to be
to small.
be small.
A similar
A similar
model
model
waswas
usedused
by by
model
used
byroll
Ikanega
(2000).
Ikanega
Ikanega
(2000).
(2000).
Referring to FIGURE 1, the force balance on sprung mass is given as
Referring
Referring
to FIGURE
to FIGURE
1, the
1, the
force
force
balance
balance
on sprung
on sprung
mass
mass
is given
is given
as as
F fl F
flF frFrl FrlFrr F
rrF pfl
FpflF pfr
FpfrF prl
FprlF prr
F prrms Zms s Zs
fr F
where,
where,
Ffl Ffl
where,
Ffr Ffr
Frl Frl
Frr Frr
ms ms
(1) (1)
= suspension
= suspension
force
force
at the
at the
front
front
left left
corner
corner
= suspension
= suspension
force
force
at the
at the
front
front
rightright
corner
corner
= suspension
= suspension
force
force
at the
at the
rearrear
left left
corner
corner
= suspension
= suspension
force
force
at the
at the
rearrear
rightright
corner
corner
= sprung
= sprung
mass
mass
weight
weight
= sprung
= sprung
mass
mass
acceleration
acceleration
at the
at the
body
body
centre
centre
of gravity
of gravity
Zs Zs
F pfl F;32
Fpflpfr; F; F
; F; F
; F=prrISSN:
pneumatic
= pneumatic
actuator
actuator
forces
forces
front
at2front
left,
left,
front
front
right,
right,
rearrear
left left
andand
rearrear
2180-1053
Vol.
2 atNo.
July-December
2010
pfrprl
prlprr
rightright
corners,
corners,
respectively.
respectively.
Referring
to FIGURE
theforce
forcebalance
balance on
on sprung
sprung mass
Referring
to FIGURE
1,1,the
massisisgiven
givenasas
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
F pfl F pfr F prl F prr ms Zs
(1) (1)
F fl F flFfr FfrFrlFrl FrrFrr F pfl
F pfr F prl F prr ms Z s
where,
where,
simplicity,
angles
are
assumed
to be
small.
A similar
model
used
simplicity,
pitch
andand
rollroll
angles
are
assumed
to be
small.
A similar
model
waswas
used
by by
Ffl all all
=pitch
suspension
force
at the
front
left
corner
FflF (2000).
= suspension
force
at
the
front
left
corner
Ikanega
(2000).
Ikanega
= suspension force at the front right corner
fr
Ffr Frl
= suspension
right
corner
= suspensionforce
forceatat the
the front
rear left
corner
Referring
to=FIGURE
1, the
force
balance
on
sprung
mass
is given
Referring
1,
the
force
balance
on
sprung
mass
is given
as as
suspension
force
at
the
rear
left
corner
Frl Frr to FIGURE
= suspension force at the rear right corner
= suspension
force
at the rear right corner
Frr ms
= sprung mass
weight
s Zs
rr FrrFpflmass
FpflFpfrweight
FpfrFprlFprlFprrF prrms Zm
FmflsFZflFfr FfrFrl=F=rlsprung
F
(1) (1)
s
sprung
mass
acceleration
at the body
centre of gravity
s
where,
where,
=
sprung
mass
acceleration
at
the
body
centre
of
gravity
Z
F pfl ; Fspfr ; F prl ; F prr = pneumatic actuator forces at front left, front right, rear left and rear
= suspension
force
at the
front
left
corner
Ffl; F
force
at the
front
leftat
corner
F pfl ; F pfr ; F prl
Ffl == suspension
pneumatic
actuator
forces
front
left, front right, rear left and rear
right
corners,
respectively.
= suspension
force
at the
front
right
corner
Ffr Ffr prr = suspension
force
at the
front
right
corner
right
corners,
respectively.
=
suspension
force
at
the
rear
left
corner
F
= suspension force at the rear left corner
Frl rl
= suspension
force
at the
right
corner
= suspension
force
at the
rearrear
right
corner
Frr Frr
= sprung
mass
weight
ms ms
= sprung
mass
weight
= sprung
mass
acceleration
at the
body
centre
of gravity
= sprung
mass
acceleration
at the
body
centre
of gravity
Zs Zs
; F;pfr
; F;prl
; F prr
pneumatic
actuator
forces
at front
front
right,
F pflF; pfl
F pfr
F prl
F prr
= =pneumatic
actuator
forces
at front
left,left,
front
right,
rearrear
leftleft
andand
rearrear
right
corners,
respectively.
right
corners,
respectively.
FIGURE 1: A 14-DOF Full vehicle ride and handling model
The suspension force at each corner of the vehicle is defined as the sum of the forces
FIGURE
1:14-DOF
A 14-DOF
Full
vehicleforce
rideand
anddamper
handling
model
produced by
suspension
components
namely
spring
force
as the followings
FIGURE
1: A
Full
vehicle
ride
and
handling
model
The suspension
Cats,corner
F
K s, fl Zforce
Z s,the
u, fl Zat
s, fleach
fl Z u, fl of
fl vehicle is defined as the sum of the forces
Theflsuspension
force
each
corner
of the
vehicle
is model
defined
as followings
the sum
produced
by suspension
components
namely
spring
force
and
damper
force
FIGURE
1:14-DOF
A 14-DOF
vehicle
and
handling
modelas the
FIGURE
1: A
Full
vehicle
rideride
and
handling
Full
K s, fr Z u,produced
fr forces
fr Z s, fr Csby
, fr Zsuspension
u , fr Z s, fr of Fthe
components namely spring force
(2)
The
suspension
force
at
each
corner
of
the
vehicle
is defined
forces
The
suspension
force
at
each
corner
of
the
vehicle
is defined
as as
thethe
sumsum
of of
thethe
forces
F
K
Z
Z
C
Z
Z
and
damper
force
as
the
followings
F fl rlK s, fl s,Zrl u, flu,rl Z s, sfl,rl Cs,s,flrlZ uu,,rlfl Zss,rl, fl produced
suspension
components
namely
spring
force
damper
force
as the
followings
produced
by by
suspension
components
namely
spring
force
andand
damper
force
as the
followings
u,rr Zs,rr s,rr Cs,rr Z
F fr FrrK s,Kfrs,rr
Z uZ, fru,rr Z Z
s, fr Cs, fr Z u , fr Z s, fr
K s,rr Z u,rr Z s,rr Cs,rr Z u,rr Z s,rr CZ Z Z Z F F K KZ Z Z Z C
u, flZ Z s, fl
F fl F flK s,K
, flu, Z
, flZ
, flu, Z
fl sZ
fl u
s, Z
fl s,fl C
s, C
flsZ
fl s, fl
Frl K
s,rl Z u,rl Z s,rl Cs,rl Z u,rl Z s,rl F
K
Z
Z
C
F
K
Z
Z Z Z Z
Z
C
Frr
fr fr s, fr s, fru, fru, fr s, fr s, fr s, fr s, fru , fru , fr s, fr s, fr
rl rl s,rl s,rlu,rl u,rl s,rl s,rl s,rl s,rlu,rl u,rl s,rl s,rl
s,rrCsZ,rru,rrZu,rrZs,rrZ s,rr
Z u,rrZs,rrZ s,rrC
Frr FrrK s,K
,rru,rr
rr sZ
where,
where,
Ks,fl
Ks,fr
Ks,rr
Ks,rl
Cs,fr
Cs,fl
Cs,rr
Cs,rl
Z u , fr
= front left suspension spring stiffness
= front right suspension spring stiffness
= rear right suspension spring stiffness
= rear left suspension spring stiffness
= front right suspension damping
= front left suspension damping
= rear right suspension damping
= rear left suspension damping
= front right unsprung masses displacement
Z u , fl = front left unsprung masses displacement
ISSN: 2180-1053
Vol. 2 No. 2 July-December 2010
Z u ,rr = rear right unsprung masses displacement
Z
= rear left unsprung masses displacement
(2)
(2) (2)
27
27
27 27
33
K
front
right
suspension
spring stiffness
s,fr left
= suspension
left suspension
spring
front
right
suspension
spring
stiffness
s,fl
==front
spring
stiffness
KKs,fl
s,fr
=
rear
right
suspension
spring stiffness
stiffness
K
s,rr
= rear
right
suspension
spring
stiffness
K
where,
s,rr right
front
right
suspension
spring
stiffness
rear
suspension
spring
stiffness
s,fr
KKs,fr
==front
right
suspension
spring
stiffness
s,rr
Ks,rl = rear left suspension spring
stiffness
left
suspension
spring
stiffness
front
left
suspension
spring
stiffness
s,rl
=
rear
right
suspension
spring
stiffness
K
s,fl
=Mechanical
rear
left
suspension
spring
stiffness
s,rr right
rear
suspension
spring
stiffness
KKs,rr
s,rl of=
Journal
Engineering
and
Technology
Cs,fr = front right suspension damping
damping
C
= front
right
suspension
spring
stiffness
K
rear
left
spring
stiffness
s,fr
front
right
suspension
damping
s,rl left
KCs,rl
==rear
spring
stiffness
s,fr
=suspension
front
leftsuspension
suspension
damping
C
s,fl
front
left
suspension
damping
=
rear
right
suspension
spring
stiffness
K
s,fl
right
suspension
damping
C
s,rr
=
front
left
suspension
damping
C
s,fr
= front
suspension
damping damping
Cs,fr
s,fl
Cs,rr right
= rear
right suspension
rear
right
suspension
damping
K
left
stiffness
s,rr right
front
leftsuspension
suspension
damping
C
s,rl
rear
suspension
dampingspring
s,fl
==front
left=
damping
CCs,fl
s,rr
= suspension
rear
left
suspension
damping
C
s,rl
=suspension
rear
left
suspension
damping
C
frontright
right
suspension
damping
s,rl right
suspension
damping
s,fr
rear
left
damping
s,rr
CCs,rr
==rear
suspension
damping
s,rl
Z u , fr = front right unsprung masses displacement
right
unsprung
masses
displacement
=suspension
front
left
suspension
damping
C
Z s,fl
rear
left
suspension
damping
s,rl
front
right
unsprung
masses displacement
u , frleft
==rear
damping
CZs,rl
u , fr
Z
=
front
left
unsprung
masses
displacement
C
=
rear
right
suspension
damping
u
,
fl
front
right
unsprung
masses
displacement
s,rr
Z
unsprung
massesdisplacement
displacement
ZZu , fr ==front
Z uu,, frfl right
= front
left unsprung
masses displacement
front
left
unsprung
masses
u , fl
rear
left
suspension
damping
C
s,rl
Z
=
right
unsprung
masses
displacement
Z
= unsprung
front
left unsprung
masses
displacement
ZZu , fl ==front
left=
masses
displacement
Z uu,,rrfl right
rear
right
masses
rear
unsprung
masses
displacement
= front
rightunsprung
unsprung
massesdisplacement
displacement
u ,rr
Z uu,,rr
fr
Z
=
rear
left
unsprung
masses
displacement
rlright
Z
=
rear
right
unsprung
massesdisplacement
displacement
ZZu ,rr ==rear
unsprung
massesdisplacement
displacement
uu
,,,rr
Z
=
rear
left
unsprung
masses
u
rl
rear
left
unsprung
masses
Z u , fl = front left unsprung masses displacement
u ,rl
front
right
unsprung
masses
velocity
Z
Z uu ,,rlfrleft=
=unsprung
rear left
unsprung
masses
displacement
ZZu ,rl ==rear
masses
displacement
=
right
unsprung
masses
velocity
Z
front
unsprung
masses
velocity
Z uu ,,rrfr right
= front
rear
right
unsprung
masses
displacement
u , fr
=
front
left
unsprung
masses
velocity
Z
u
,
fl
=
front
right
unsprung
masses
velocity
Z
right
unsprung
massesvelocity
velocity
ZZu , fr ==front
=
leftunsprung
unsprung
massesdisplacement
velocity
Z
front
unsprung
masses
Z uuu ,,,rlfrfl left
= front
rear
left
masses
u , fl
unsprung
masses velocity
velocity
Z u ,rr left=
= rear
frontright
left unsprung
masses
Z
velocity
ZZu , fl ==front
=
rear
right
unsprung
masses
Z
rear
unsprung
masses
velocity
= unsprung
front
rightmasses
unsprung
massesvelocity
velocity
Z uuu ,,,rrfrfl right
u , rr
=
rear
left
unsprung
masses
velocity
Z u ,rlright
=
rear
right
unsprung
massesvelocity
velocity
Z
unsprung
masses
velocity
ZZu ,rr ==rear
uu ,,rr
=
rear
left
unsprung
masses
Z
rl
rear
left
unsprung
masses
velocity
Z u , fl = front left unsprung masses velocity
u ,rl
rear left unsprung
masses velocity
Z u ,rlleft=unsprung
velocity
Z u ,rl = rear
= rear
rightmasses
unsprung
masses
Z u ,rr mass
The
sprung
position
at
each
corner
canvelocity
be expressed
in terms ofin
bounce,
pitch
The
sprung
mass
position
at
each
corner
be expressed
terms
of
The
sprung
mass
position
at
each
corner
can be can
expressed
in bounce,
terms ofpitch
bounce,
The sprung
mass
at each corner can be expressed
in terms
of
andpitch
roll
given
as;Zposition
=
rear
left
unsprung
masses
velocity
u
,
rl
given
as;
The
sprung
mass
at each
corner
can be expressed
terms ofpitch
bounce,
bounce,
pitch
and
roll corner
given
as;
given
as;
The
sprung
mass
position
atposition
each
can
be expressed
in terms ofinbounce,
and pitch
roll
given as; given as;
T 0.5wat
M corner can be expressed in terms of bounce, pitch
Z s mass
a sinposition
sineach
TheZ
ssprung
, fl
TM
0.5w sin M
Z s , fl given
Z s Zsa, flas;
sin TZ s 0.a5sin
w sin
Z
Z
a
sin
T
0
.
5
w
sin
M
Z ssa,, flfrsin TZZss 0.5aawsin
ZZs , fl ZZs Z
sinTTM 00..55w
sin
wsin
sin M
M
s , fr
s s ,afrsin T s 0.5w sin M
Z
ZZs 0.b5aw
sin
TTM00..55wwsin
MM
(3)
s
,
rl
sin
sin
ZZs , fr ZZs Z
a
sin
T
sin
Zssb,,rlfrsin TZZ
TT M
00..55w
ss 0.ba5sin
w
M
sinsin
wsin
sinM
s ,rl
sZ
s
,
fl
s
(3)
Z
a sinTT 00..55w
w sinMM
Z bss ,,,rr
ZZ s 0
ZZs ,rl ZZs Z
sin
0.5.ba5wsin
sin
TMM 0.5wsin
sin M
rl
rr
sinTTZZss
wsin
sin
s ,rr
s Zss a
, fr
s a sin T 0.5w sin M
Z
Z
a
sin
T
0
.
5
w
sin M
Z s ,rr Z s sa,rrsin T s 0.5w sin M
Z s that
b sinall
T angles
0.5w sin
M small, therefore Eq. (3) becomes;
It isZ assumed
are
s ,rl
It is that
assumed
that all
small, therefore
Eq. (3) becomes;
It is assumed
all angles
areangles
small,are
therefore
Eq. (3) becomes;
Z assumed
athat
sinall
T all
angles
0.angles
5w sin
M small,
It Itisisassumed
are small,
therefore
Eq. (3) becomes;
s ,rr allZ angles
s that
are
therefore
Eq. (3) becomes;
It is assumed
that
are
small,
therefore
Eq.
(3)
becomes;
Z s , fl Z s aT 0.5wM
T
M
Z
Z
a
0
.
5
w
Z s , fl Z s sa, flT 0.s5wM
Z.5swthat
T 00..55angles
wM
M
ItsisZ
fr 0Z
Z
aa
ZZs , fl Z
assumed
sssa,,,aT
fl
MM
aTT all
0.5w
wM are small, therefore Eq. (3) becomes;
frT Z
Z sZ
0.ss5w
s , fr
Z
Z
b
T
0
.
5
w
M
(4)
Zs MbaTT 00..55w
wM
ZZs , fr ZZ s Z
Z ssa,,, rlrlTfrT 0Z
0Z..5ss5s w
w
M
MaT 0.5wM
s , rl
s Zssb
, fl
(4)
Z
Z
a
T
0
.
5
w
M
s
Z bss ,,Trr
0.5wMM
ZZs , rl ZZs Z
Z
MbM
aaTTT rlT0Z
0Z.5.sss5ww
00..55w
wM
s , rr
s Zss ,,arr
fr
Z s , rr Z s Z
sa, rr
T 0Z.5s wMaT 0.5wM
Z s , rl Z s bT 0.5wM
where,
where, where, a
front of vehicle and C.G. of sprung mass
Z s , rr a Z s a=
Tdistance
0.5wM between
distance
between
front ofand
vehicle
and
C.G. of sprung mass
= bdistance=
of vehicle
C.G.and
of sprung
where, a where,
= between
distance front
between
rear of vehicle
C.G. ofmass
sprung mass
distancefront
between
rear ofof
vehicle
and
C.G.
sprung
abdistance=between
front
vehicle
and
C.G.of
of
sprungmass
mass
b
between
rear
of
vehicle
and
C.G.
of
sprung
mass
awhere,
==distance
of
vehicle
and
C.G.
of
sprung
mass
T
= pitch angle at body centre of gravity
where,
T
=
pitch
angle
at
body
centre
of
gravity
bpitch angle
distance
between
rear
of
vehicle
andsprung
C.G. of
sprung mass
bT
==distance
between
rear
of
vehicle
and
C.G.
of
mass
at
body
centre
of
gravity
= roll
anglebetween
at body front
centreofofvehicle
gravityand C.G. of sprung mass
M
aT
distance
= roll
angle
atatbody
M angle
pitch
angle
body
centreofofgravity
gravity
roll
atatbody
centre
ofofcentre
gravity
TM
== pitch
angle
body
centre
gravity
Z
=
front
left
sprung
mass
displacement
bMZ ss ,, flflangle
distance
between
rear
of of
vehicle
and C.G. of sprung mass
roll
angle
at body
centre
gravity
== front
left
sprung
displacement
at body
centre
ofmass
gravity
MZ s , fl = roll
front
left
sprung
mass
displacement
=
front
right
sprung
mass
displacement
T
=
pitch
angle
at
body
centre
of
gravity
Z
Z ss,, frfl left
front
left
sprung
mass
ZZs , fl =
= front
front
sprung
mass
displacement
==front
right
sprung
massdisplacement
displacement
Z
sprung
mass
displacement
s , fr
=rear
roll
angle
at body
centre
of gravity
MZ s , fr right
=
left
sprung
mass
displacement
s
,
rl
= front
right
sprung
mass displacement
Z
right
sprung
mass
displacement
ZZs , fr ==front
Z
rear
sprung
mass
rear
mass
displacement
Z sss,,,rlfrflleft==sprung
frontleft
left
sprung
massdisplacement
displacement
s ,rl
rear
right
sprung
massdisplacement
displacement
Z
Zss,,rrrlleft=
rear
left
sprung
mass
ZZs ,rl == rear
mass
displacement
==sprung
rear
right
sprung
mass
Z
rear
sprung
mass
displacement
=
front
right
sprung
massdisplacement
displacement
s ,rr
Z ss,,rrfrright
= sprung
rear right
sprung
mass displacement
Z s ,rrright
mass
displacement
Z s ,rr = rear
Z s ,rl = rear left sprung mass displacement
Z s ,rr
= rear right sprung mass displacement
28
28
By substituting Eq. (4) and its derivative (sprung mass velocity at each
corner) into Eq. (2) and the resulting equations are then substituted
into Eq. (1), the following equation is obtained ;
34
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
and roll
and roll
and roll
and roll
(3)
(3)
(3)
(3)
(4)
(4)
(4)
(4)
28
28
28
28
By substituting
Eq.its(4)derivative
and its derivative
(sprung
mass atvelocity
at eachinto
corner)
into Eq. (2)
By substituting
Eq. (4) and
(sprung mass
velocity
each corner)
Eq. (2)
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
and the resulting
then substituted
(1), the following
is
and the resulting
equationsequations
are then are
substituted
into Eq. into
(1), Eq.
the following
equation equation
is
obtained
;
substituting
Eq. (4) and its derivative (sprung mass velocity at each corner) into Eq. (2)
obtained ;By
By substituting Eq. (4) and its derivative (sprung mass velocity at each corner) into
and the resulting equations are then substituted into Eq. (1), the following equation is
andEq.the(2)resulting equations
are
substituted into Eq. (1), the following equ
mass velocitymatZeach
then
obtained
aK
mKs Zs,sf ;into
K
bC s , r T
sZ,fs C2smass
2corner)
K2sK
2sC, r s,Zf sC2s,C
aK
2bC
, f s its
, r Z
s s By
,and
r sZ
r(sprung
s , sf velocity
s , rs ,Tf atvelocity
Eq. (4)
each corner)
Eq. (2)
obtained
; and its derivative
substituting
Eq.
(4)
(sprung
mass
at eachinto
corner)
into Eq. (2)
d into Eq. By
(1),substituting
the following
equation
isderivative
its
Z mass
substituting
Eq.
and
derivative
(sprung
velocity
atthe
each
corner)(5)
into
(2)
are
into
aCand
bC
are
Z, fu Z
Cs ,Kf Eq.
Z
Kat
Zeach
(5)
and
the resulting
equations
then
substituted
the
equation
is Eq.
By
2 aC
2(4)
bC
(4)
K
K
C
and
the
resulting
into
Eq.
following
equation
is
sEq.
, fTequations
s , rZT
sfthen
, flusubstituted
u , umass
fl,(1),
sf
ufollowing
,(1),
fr corner)
s
,
f
s
,
r
sf
u
,
fl
s
,
fl
sf
fr
By
substituting
By
substituting
Eq.
its
(4)
derivative
and
its
derivative
(sprung
mass
(sprung
velocity
velocity
at
each
into
corner)
Eq.
(2)
into
Eq.
msubstituting
ZEq.
(4)
2Kand
K derivative
Z s its
are
2derivative
C(sprung
Z s velocity
2aK
Eq.
bC
By substituting
By
its
(4)
mass
velocity
each
corner)
atthe
each
into
corner)
Eq. (2)
into
Eq. (2)
(2)
sthe
s resulting
sEq.
,f s , rand
sthen
, f Csubstituted
s ,mass
r(sprung
s , f at
s(1),
, r T
and
equations
into
following
equation
is
obtained
;
obtained
;
m
Z
2
K
K
Z
2
C
C
Z
2
aK
bC
T
s equations
s
s , are
fsubstituted
s ,Z
r Z substituted
s into
sZ
,Eq.
f Z into
s , rC Eq.
s Z following
sthe
, f following
s ,r
and
the
and
resulting
the
resulting
equations
are
then
then
(1),
the
(1),
equation
equation
is
is
C
Z
K
Z
C
K
C
Z
K
Z
C
Z
K
C
ssr
, f uu, ,rlequations
fr aresr sthen
sthen
, r sr u , rl
sEq.
, r ufollowing
, rr
and the resulting
and
equations
(1),
the
(1),
the following
equation equation
is
is
s , the
f u ,;resulting
fr
, ru , rlu ,are
rlsubstituted
usubstituted
, rrintosr sEq.
, rr
, ru , rruinto
obtained
2
aC
bC
T
K
Z
C
Z
K
Z
(5)
s, f
s,r
sf u , fl s , f u , fl
sf u, fr
obtained
obtained
;
;
aK s, f bC obtained
T
s ,r
2
aC
bC
T
K
Z
C
Z
K
Z
;+
obtained
;
F
F
F
F
+
s
,
f
s
,
r
sf
u
,
fl
s
,
f
u
,
fl
sf
u
,
fr
F
F
F
F
pfl
pfr
prl
prr
bC
pflKs Zs,sf pfrKC2s,Kr sZZ,prlfs fr 2KKCssr,prrrsZ,Zf us,rlC2s,CrCZss,,rsf Zu2C,rlaK
m s Z s 2m
T
s , r s,TfC bC
s ,
r sZ
K, fssr
Z2u,aK
rr s , r Z us ,, rrr
m s Zs 2s,Kf s , fu ,(5)
Ks , rCZs ,sf 2ZaK
bC
Z u2,C
K
K
Zs ,ur, rrT C s , r Z u , rr
, fl K sf Z u , fr
s ,fsrZC
srl, r Z C
s s, f fr
u
,
s
,
r
u
,
rl
sr
KKss ZZsss,,,ssfff +2FKKbC
ssKK,,rrs ,ssrsZ,,Z,ffTFsfs pfrKK22bC
ssCC,,sfrrFss,,ZZrfprlf ssuT,flCC22FssK,CC,rrprr
ZZsss,,Kss,fffsfZ22ZubC
,uaK
m s ZZ
22aC
C
2aK
aK
bC
TK
s 2
aC
K
CsfssZZ,,sZ
Z
(5)
f, sfu
, rs ,T
f Z u ,ss ,,frrr T
(5)
m
22m
m
C
aK
sf bC
bC
T
u2,ss
fl,, rr C
, ssfr,
,
fl
fl
s
s
f
s
r
s
,
f
pfl
where,
where,
FKpfr Z
uF, flprlCFs ,prr
+ Fs , pfl
2 aC s , f bC
T
Z u ,fl K sf Z u , fr
(5)
, rr C s , r Z u , rr
r
sf
f
=saC
,TTuurlu,,,of
2
bC
T,fr
K
K
C
C
Kf Zsf,ruuZZ,, rrflu , rr
(5)
(5)
where,
KbC
ZsuT, ,f
ZsZs,ur,body
C
KsZ,Zuf Zu,ZZflof
C
Kss C
C sfsfs , rZ
fT
sf
sZ
, rZ
fl sf
u ,
flC
,K
at
body
centre
gravity
,aC
fpitch
fr
sr, pitch
rl
rrfl
saC
2KbC
aC
rate
bC
K
K
C
C
fr K
K
ZZuuu,, ,frfrrr
(5)
(5)
TC22=
rate
centre
gravity
fat
srssf
uZ
,url
s, f
s ,sr, u,frl
s,,rrZ
fl
sfs ,srZsr
, uf , fl
u,,
,srf sZ
sf u , ufl, fr
C s , f Z u , fr K sr Z u ,rl C s , r Z u , rl K sr Z u , rr
C s , r Z u , rr
Z+swhere,
Z
=
sprung
mass
displacement
at
body
centre
of
gravity
displacement
ofZ
s,+
FC
rlFat
prrsbody
K
C
rr C
C=
Z Fu C
Kmass
Z
K
C
Z
Z,rlu,
C
K
Zsr uZZ, rlcentre
K
C
Zs ,gravity
Z
C s ,r Z
Z u , rr
FC
pfl
uF
FK
FZ
s ,sprung
fZ
frC
swhere,
,srfFZ
, rlfr
sr
s , prl
ruZ
,r Z
u , rr
sr
ruZ
,rru ,
pfl
pfr
prl
prr
pfr
Z
K
C
K
s , f u , fr s sr
f uu,,rlfr
s , ru , rlu , rl
s , r sr u , rlu , rr
sr s , ru , rru , rr
s , r u , rr
F pfr
sr at
F
Fcentre
TZ + Fmass
=pfl, sprung
pitch
rate
body
of of
gravity
prl
prrcentre
=
velocity
atatbody
centre
ofofgravity
Z s+ F=pflsprung
velocity
at
body
gravity
F
F
F
F
Tmass
F
F
F
=
rate
body
centre
gravity
pfr
prl
pfr
prr
prlpitch
prr
F=pfl
F
F
F
F
+ F pfl ZFFs+
+pfr
pfl prl pfr mass
prr prldisplacement
prr
at body centre
of K
gravity
s
Ks,f
= sprung
springZof
stiffness
of front
(K)s,flat=
where, Ks,fwhere,
= spring
stiffness
suspension
(Ks,fl = Ks,fr
s,fr)centre of gravity
= sprung
masssuspension
displacement
body
s front
where,
Z s,r
=
sprung
mass
velocity
at body
centre
of
gravity
s,r = pitch
=
spring
stiffness
of
rear
suspension
(K
=
Ks,rrof
) gravity
K
s
=
spring
stiffness
of
rear
suspension
(K
=
K
)
K
T
rate
at
body
centre
of
gravity
s,rl
T
=
pitch
rate
at
body
centre
of
gravity
s,rl at body
s,rr centre
where, where,
Z s = sprung mass velocity
centre of gravity
where,
where, C
TZ s,f
=s,frpitch
rate
at =body
centre
of
gravity
C
=
C
damping
constant
of
front
suspension
K
=
spring
stiffness
of
front
suspension
(K
=
K
)
C
C
=
C
=
damping
constant
of
front
suspension
ZTss,f ==
sprung
mass
displacement
at
body
centre
of
gravity
s,fl
s,fr
s,fl
s,fr
=
sprung
mass
displacement
at
body
centre
of
gravity
s,fl
pitch
TT s rate
rate
=
at body
body
rate
centre
at
of gravity
centre
gravity
of
gravity
Kmass
spring
stiffness
of
front
suspension
(K = Ks,fr)
of gravity
s,fcentre
T
== pitch
= pitch
pitch
at
at==body
body
of
centre
of
Zs,r
=
sprung
displacement
atgravity
body
centre
of=gravity
C
=rate
C
damping
constant
of
rear
suspension
C
=
stiffness
ofcentre
rear
suspension
(K
Ks,rr) s,fl
K
C
C=
=spring
damping
constant
of
rear
suspension
C
s= mass
s,rl
s,rr
s,rl
Zs,r ==
s,rl
s,rr
ZZ
sprung
velocity
at
body
of
gravity
=
sprung
Z
=
mass
displacement
displacement
at
body
centre
at
body
of
centre
gravity
of
gravity
sprung
mass
velocity
at
body
centre
of
gravity
s
=
spring
stiffness
of
rear
suspension
K
s
s
s,r
= sprung
=
mass
sprung
mass
displacement
at body
centre
at body
of
centre
of gravity(Ks,rl = Ks,rr)
(Ks,fl = Ks,fr)
s
s
CZZs,f
=
Cs,fldisplacement
= Cmass
damping
of gravity
front
suspension
s,fr =velocity
=
sprung
atconstant
body
centre
of) gravity
s stiffness
K
=
spring
of
front
suspension
(K
=
K
C
=
C
=
C
=
damping
constant
K
=
spring
stiffness
of
front
suspension
(K
=
Kof
)front suspension
s,f
s,fl
s,fr
Z
Z
s,f
s,fl
s,fr
=
sprung
=
mass
sprung
velocity
mass
at
velocity
body
centre
at
body
of
centre
gravity
of
gravity
s,f
s,fl
s moment
n (Ks,rl = KSimilarly,
s = sprung
s,rr)
Z sSimilarly,
=
mass
sprung
velocity
mass
velocity
body
centre
atconstant
body
ofpitch
gravity
centre
of
gravity
=
C
=
C
damping
of Trear
suspension
C
TMs,fr
and
roll
, and are
balance
equations
are
derived
for
pitch
s,rlequations
s,rr =at
sbalance
and
roll
,)rear
and
areM given
as ;given as ;
moment
are
for
KZs,r
=
spring
stiffness
ofderived
front
suspension
(K
==K
s,f
s,fl)s,rl
s,fr
=
spring
stiffness
of
rear
suspension
(K
=
K
K
=
C
=
C
=
damping
constant
of
C
=
spring
stiffness
of
rear
suspension
(K
K
K
s,r
s,rl
s,rr
s,r
s,rl
s,rr
s,r
ront suspension K
Ks,f
Ks,f
= spring
spring
stiffness
= spring
spring of
of
stiffness
front suspension
suspension
of front
front suspension
suspension
(Ks,fl
=K
Ks,fr
(K))s,fl
=K
Ks,fr
)s,rr) suspension
s,f =
s,fl =
s,fr
K
=
stiffness
front
of
(K
(K
=
)
s,f stiffness
s,fls,rl
s,fr
=
spring
stiffness
of
rear
suspension
(K
=
K
)
K
s,r
s,rr
= C=s,flC
=
C=s,frC
=spring
damping
constant
ofconstant
front
= Cs,fr
=
damping
of=front
suspension
s,f
s,fl
ear suspension CK
spring
stiffness
=
of
stiffness
rear
suspension
of
rear
suspension
(Ksuspension
K(K
K
) M , and are given as ;
Ks,fs,r
K
s,r
s,rmoment
s,rr2s,rl
2s,rl
2))K =
C
=front
Trf and
balance
equations
are
derived
forK
pitch
stiffness
=
spring
of
stiffness
rear
suspension
of
rear
suspension
K
K
s,r
s,rr
s,rl
s,rr
I=yyC
T=
2= aK
bK
Zs ,=
damping
aC
Z
2(K
absuspension
b 2)roll
Kpitch
=,Similarly,
C
=
damping
constant
of
suspension
I yy T Cs,r
2Similarly,
aK
spring
bK
2C
2bC
Zconstant
bC
2(K
K
K=
Ts,rr
s,f
s,fl
saC
, s,fr
rs,rr
s=f constant
sa
,are
r s,rl
ssderived
ss,,for
s , r T T and
,Zf=
C
suspension
s=
r sC
sdamping
s ,sr, fof
s rear
,rear
f are
sC
, s,rl
fC
=
C
of
rollroll
M , and are given
moment
balance
equations
derived
for
s,rr
s,r
s,rl
Similarly,
moment
balance
equations
pitch
and
C
C
=
=
C
=
C
=
=
damping
C
=
damping
constant
of
constant
front
suspension
of
front
suspension
s,f = C s,fl
s,f
s,fl
s,fr = constant
Cs,f
Cs,f
= C=s,frC
Cs,fl
==
damping
C
damping of
constant
front suspension
of rear
frontsuspension
suspension
s,fl
s,fr
=
C
=
damping
constant
of
C
s,r22 =s,fr
s,rl
s,rr
2
2
=
C
=
C
=
C
=
damping
=
C
=
damping
constant
of
constant
rear
suspension
of
rear
suspension
C
C
,
and
are
given
as
;
or pitch T and rollM
2
a
C
b
C
T
aK
Z
aC
Z
aK
Z
(6)
,
and
are
given
as
;
2
2
s,r
s,rl
s,r
s,rr
s,rl
s,rr
2
a
C
b
C
T
aK
Z
aC
Z
aK
Z
(6)
f s,rl
s,,C
ss,,rZ
uconstant
sof
,f rear
=aK
C
=ss,rr
= bK
damping
=C
constant
damping
C I=yys ,TC
Cs,r
s , f s ,of
rf s,rl
u
, fl aC
f,flrear
u , fl suspension
ususpension
, frK s , f ub, frK
f=
,u f,2fl a
2balance
bC
T, and as
2 ; given as ;
s , rs,rr
s equations
spitch
, r Z s Ts
s ,T
s2, M
r given
s ,If equations
s , f are
balance
and
roll
MZfand
, and
are
Similarly,s,r
moment
are2derived
for
roll
are
Similarly,
moment
derived
for
pitch
T
2
aK
bK
Z
2
aC
bC
2
a
K
b
K
T
yy
s
,
r
s
s
,
r
s
s
,
f
s
,
r
, f bC
s, f Z
Z balance
Z bK
aCmoment
Z u , rrM , and are given as ;
and
Z u, rrTbC
aCSimilarly,
bCZsequations
2 bK
s2bK
, f Zsu, r, fr
ur,Z
rl u , rl bK
s are
, r Zs u,derived
sbC
, r spitch
u, r, rr
, r roll
s , f Z u, fr
u, rl
r, rl u , rr for
2 abalance
C s , f equations
bbalance
C s ,2rs , rT sequations
,are
aK
derived
aC
roll
aK T
Z
Tfor
and
roll
,sand
and
are
M
given
,, and
as ;;given
Similarly,
moment
balance
moment
equations
are
derived
for
pitch
2 a 2 K s , f Similarly,
b 2 K s , r TSimilarly,
, flfor
s , f ZT
u ,and
fl pitch
u , roll
fr are
s2, C
f Z uare
sM
,Mfand
T
and
,
roll
M given
and are
are
as
given as
as ;; (6)
Similarly,
moment
moment
balance
equations
derived
are
derived
pitch
for
pitch
2
a
C
b
T
aK
Z
aC
Z
2
2
sZ,r)
u ,2fl b
f b
u ,2flK aK
pfr
( F2)pfl
F(sFpfr
)2l FZprr
( Fs ,prl
aC
F
l bC
s , f Z u , fr
laK
)fl r2bC
I yy T ( F
2 pfl
aK
2 sa, r sK
faC
I yysT,F
prl
bK
Z,fs , a 2 K s ,srf, T
T
sf, r sZ
s , rprr
s r
f f bK
s , sf s
,
r
s
,
r
,
f
s
,
f
bK
bK sZ, r Zu , rl2 aC
bC s ,
bCsb, r 2ZKu , rr T
s , r2Zau ,2rr
, f Z u, fr (6)
r ZbC
u , rls , r2Z
I yy T aC
2 saK
2Z s , r bC
s , r s , f sZ u , fr bK
s s , r Z u2,22rlK s ,Zf bK
s , fZ
22aC
u , fl aK s , f Z u ,Ifr T
22aK
,srZ
rl
s , r Z u , rr
aK
II yy T
22bK
bK
b
aK
2
bC
aC
bC
u22,aC
ar2aK
Z
K
a
b aK
K
Ksss,, ,ffbK
2 a222aK
f2bK
s
bC
yyT
s , rs ,Z
s
s aC
,aC
r Z
s,f s ,sr,ZfZ
sa
,
ssZ,
f 2
r T
,r T
s
,
f
s
I yy
T
aK
bK
2
Z
2
bC
aC
bC
Z
K
2
a
b
K
K
Zsub
b, r,2frK
Kuss, rr
C
b
C
T
aK
Z
aC
Z
Z
(6)
a
C
C
T
(6)
yy
s ,s
rs, sr, ,F
s fuprl
s ,fprr
f f, fl u , fr ss,s, fr, fT
,r T
s ,sf,f 2( F pfl
, fl sr,sf, uf),lsflr u2, fl s , r s , fsss, ,u
F
, rf (s ,F
f f 2spfr )2l sf , sr
s
,
2
2
.5Fw
s, fs0,Zf.5uZ,wK
l,fsZ
FC
s,M
F
M 2K
.f 5s ,wf K
T0K
M)sC
0, f.u5,(flw
Z0uC),.l5sflr,wK
Z u , fl
20I.xx
f, f
a2s0C
K
b, 2rs(C
aC
aK
(6)
5 w
Mfspfl
s , r Z u , rr bC s , r IZxxu M
,rr
prlsC
prr
,F
s , aK
rpfr
rM
s
,
f
,
r
s
u
,
fr
,
s
,
r
fl
2
2
2
aK
C f, fr
2aC
ba
CZ
aK
C
ZZs ,u
aK
aC
Z
aC
aK
T
(6)
(6)
b
bC
aC
sss,,bC
rrs, ,frT
fl
sbK
u
,,f,,fr fl
fl
,, ffZ
,Z
fru ,rrsbC
u
,, Z
fr
bK
Zs ,uuu
bK
ZZs ,uur
,,bC
ffZ
ssu
,, r,,ff fl
,, ff Z
22 aas , C
absbK
b, rlaK
C
T
Z
aK
Z
aC
Z
aK
Z
(6)
(6)
f Zss,,u
r,,,Z
u,rlssaC
r,,,Z
u
, rrssaK
,C
f ss,s,rr,uffr,TZ
fr u
u
rl
s
rl
rr
s
,
r
u
,
rr
f 2
u
fl
s
,
u
fl
fl
u
fl
,
fr
u
fr
s
,
f
s
,
s
. 5 wC
(7)
5uwK
(7)
0 . 5 wC
.Z5bK
s , 0sf ,.Z
5ZuwC
Z. 5u ,wC
aC
bK
s
uwK
, fls ,sr,Zf0uZ., rl
f Zu ,bC
fr s , r Z u , rr
2u, ,frrl s
, f 0Z
uf ,Zfl u ,2sfr, 0f
,
fr bC
, f0
frr Z su,, rr
s
,
r
s
,
aC
Z
aC
bK
Z
Z
bK
bC
Z
Z
bC
bK
Z
Z
bK
bC
Z
Z
bC
Z
(aC
F pfls ,Ifxx
F
)
l
(
F
F
)
l
M
0
.
5
w
K
K
M
0
.
5
w
C
C
M
0
.
5
wK
Z
(
F
F
)
l
(
F
F
)
l
u
,
fr
s
,
f
s
u
,
,
r
fr
u
,
rl
s
,
r
s
u
,
r
rl
u
,
rl
s
,
r
s
u
,
r
,
rl
u
rr
s
,
r
s
u
,
r
,
rr
u
,
rr
s
,
r
u
,
rr
pfr
f
prl
prr
r
2
2
s
,
f
s
,
r
s
,
f
s
,
r
s
,
f
u
,
fl
f sbC
prl
prr
rZsu, r, rl
aCpfl
ZM
Z
Z
bC
bK
Z
bK
bC
Z
Z
bC
Z
, rlbK
s , f Z u, fr
s ,bK
f Zsu,Ir, pfr
fr
u
,
r
s
u
,
r
,
rl
u
,
rl
s
,
r
u
,
rr
s
,
r
s
u
,
r
,
rr
u
,
rr
s
,
r
u
,
rr
0
.
5
w
K
K
M
0
.
5
w
C
C
M
0
.
5
wK
Z
F.5prlwC
f.5 wCs , r Z
xx
s ,0
s , f u , fl
Fs0pfr
s0F,s.r5,prr
l rs ,rsZ0, ru.5, rrwK
ZfwK
0.5 wKs ,(rFZ0pfl
0, r.5ZwC
, r.5Z))wC
ull,f
rl s
u)), rl
u , rr s ,
u.l5, rlwK
,
r ((Z0
u ,)rll r Z u , rr s , r u , rr
( F pfl F
((pfr
F
F
F
F
ll ru , fr 0 . 5swC
(7)
F
F
0prl
prl
prr
prr
F pfr
F0))pfl
l. 5ff wC
F
F((F
lff
F
)l.r5r wK
F
spfr
, prl
f )Z
u ,
fl (prr
s , f)Z
s , f Z u , fr
pfl
pfr
prl
prr
r
C s , r M 0 .5 wK s , f Z( Fu ,pfl
0
.
5
wC
Z
0
.
5
wK
Z
0
.
5
wC
Z
fl
w
u , fr
s , f u , fr
w K
ws ,2 fC u, fl wC2 M s ,0f .5 wK
2
I xx M (F0pfl
.5xxw
K
M
0
.
5
w
Z
2F
I
M
0
.
5
w
K
K
M
0
.
5
w
C
C
M
0
.
5
wK
Z
(
F
F
)
(
F
F
)
)
(
F
F
)
sZ
, rus,,rlf prr
, f 0 .5u wC
, fl s ,sr, Z
prl0s.5,pflfwK prl
0.5spfr
wC
Z, fu , rl 2 s0, r.5s wK
, r s , rsprr
, f s , r Zs ,ur, rr s
f u , rr
u , fl
pfr
wC s , f Z u , fr
5 wK
0.25wC
u 0, fl.5 wC s , r Z u , rr
I xx M2 02.5 ws22, rK(7)
K s ,2sr, rM
w
C s ,sf, r Z
uC, rls , rM0.5wK
0 .5swK
2Z u, rl0 .5 2
, r Z us, rrf Z
s,2
f0 .
2K
2K0
2M
2CsC, r
II xxM
M 0. 500IIwC
..xx
5
M
w
0
.
5
w
K
M
K
0
.
5
w
C
0
.
5
w
M
0
C
.
5
wK
M
0
Z
.
5
wK
Z
(7)
Z
.
5
wK
Z
0
.
5
wC
Z
(7)
0
.
5
wC
Z
0
.
5
wK
Z
0
.
5
wC
Z
s
,
f
s
,
r
s
,
f
s
,
r
s
,
f
s
,
f
s
,
r
s
,
f
u
,
fl
s
,
f
s
,
f
u
,
fl
s
,
f
u
,
fr
s
,
f
u
,
fr
MFCwss0,, ff.)5ZwuC, frsC, rs,M0f .5 wC
5w
M Ks0, .f5 w sK,K
0K.5s ,w
0C.5s ,wK
fs , rs ,uw
sr ,M
f s
xx xx
fl s , f Z uu ,, fl
fl
Mf, fl
,uf,0Zfr.5u ,wK
(0F.5pflwC
Fs ,prl
( F0pfr
(7)
. r5 wK
r Z u , rr 0 .5 wC s , r Z u , rr
prr
f Z)u (, F
fl s, f (uF, fr F ) ws , f Z u, fr
F
)
(7)
(7)
wCs ,s
,Zf00Z
wC
.Zu55,uwK
wK
5uprl
wK
Z. 2
5s ,uwC
wC
Z.prr
50
2, flpfl
Zu0,u.fr,5rrwC s , r Z u , rr
where, 0where,
Z
pfr
..55rluuwK
Z
5ZwC
Z
0.5srr,,wK
,
fl 0
s.,50
fwC
s0, .f0
, 0sfr,.r5
swK
, f0, rl
,Z
fr
f0
uwC
frZwC
s, f Z
(7)
(7)
00.5..55wK
wC
wC
.. 5
r
,,,.r5
sr, f0uZ.,5
, fl ss, r, 0f .Z
urls, ,flr s,uf,0rlZ
uwK
, fr
s2, u0
f .Z5
ur, fru s,
f0Z. 5uswC
fr 2us,,rrfs ,Z
u , fr
0 .5 wK
Z u , rlacceleration
0at
.5body
wC s , rcentre
Zatu ,body
of0.gravity
5 wK s , rofZ ugravity
0 .5 wC s , r Z u , rr
=
pitch
centre
= T
pitch
acceleration
s
,
r
rl
,
rr
T 0.5 wK
, r ZZ
0..u55w
wK
s0, r..55Z
ZwC
r ZZ00 ..u55, rlw
wC
5Z
ZwK
wK
Z
wKs00, r..5Z
Z
5wC
wC
Z0
0 ..u5
5rrwC
wC s , r Z
Z , rr
ws
ws
rl
u
,, rr..5
u
u
,rr Z
0F.5 wK
wK
ss00F
00..uu55,,rrrrwK
ssM
,roll
r (0F
,,rl
r pfrwC
u ,,rl
rl sF
,,
rat
u), rlwC
u),, rl
rl of
s ,,rr Z
s , r gravity
u ,, rr
rr ss,
u ,,rr
s ,r u
u , rr
M
)u acceleration
prl
(F
=(s0FF,roll
atprr
body
centre
of
(where,
body
gravity
pfl =F
prr
pfl prl )acceleration
pfr centre
w
w
where,
2
2
2
2
w
w
w
w
(
F
F
)
(
F
F
)
pfl
prl
pfr
prr
w
w
w
w
=
pitch
acceleration
at
body
centre
of
gravity
T
= moment
ofacceleration
IF
(( F
(roll
FprlpfrT))axis
F
((moment
F
))inertia
Ixx((FF pfl =
roll
inertia
xxprl
2F
2
pfl prr
of
F
F))axis
F
F
F=
F prr
(F
))pfr
at body centre of gravity
pfl F
prl
pfr prr
pfr2pitch
prr
22axis
22inertia
yy
prl pfl22axis
= roll
acceleration
body
centre of gravity
2 at of
pitch
moment
= IMpitch
moment
of
inertia
Iyywhere,
where,
M
= roll acceleration
at body centre of gravity
of gravity
=
wheel
base
of
sprung
mass
moment
of
w
wheel
base
ofaxis
sprung
mass
where,
== Iw
pitch
acceleration
at
of gravity
= roll
pitch
at inertia
body
centre
gravity
Txx
Ixx acceleration
=body
rollcentre
axis
moment
of of
inertia
where, T where,
f gravity where,
where,
=
pitch
axis
moment
of
inertia
I
yy
acceleration
at
body
centre
of
gravity
TM acceleration
MT
== roll
at
body
centre
of
gravity
=
roll
acceleration
at
body
centre
of
gravity
pitchcentre
axis
moment
ofof
inertia
Iyy acceleration
pitch acceleration
acceleration
=
pitch
at=body
body
centre
at
body
of centre
gravity
gravity
TTpitch
=force
=force
pitch
acceleration
at
at
body
of
centre
gravity
offollowing
gravity
T By performing
balance
analysis
at
the
fourthe
wheels,
the following
are obtained
w
wheel
base
of
sprung
mass
By performing
balance
analysis
at
the
four
wheels,
equationsequations
are obtained
M axis
=
roll
acceleration
at
body
centre
of
gravity
IMxx
=
roll
moment
of
inertia
w
=
wheel
base
of
sprung
mass
I
roll
axis
moment
of
inertia
M
=
M
roll
acceleration
=
acceleration
at
body
centre
at
body
of
centre
gravity
of
gravity
xx
= IMroll acceleration
=
roll
acceleration
at
body
centre
at
body
of
centre
gravity
of
gravity
;
;
= roll
axis
moment
xx
== IIpitch
axis
moment
ofmoment
inertiaof
IIyyxx
=
pitch
axis
ofinertia
yy
roll
axis
=
moment
roll
axis
inertia
of
M0wheels,
xx
=
of
analysis
at0inertia
four
the following equations are obtained
IflxxBym
=force
moment
roll
axis
of
moment
inertia
of
inertia
Kaxis
ofaK
Cbalance
aKss,,balance
Tof
aC
wKthe
mu Zu ,w
Kuperforming
ZZperforming
Tmoment
aC
inertia
.the
5wK
xx
inertia
, IIw
flsroll
s,,base
s
saxis
,sbase
f, fZ
s moment
sanalysis
, f Ts , s, f M
sZ
,=
fuZ
sBy
ff=
spitch
ffT
f .5at
yy C
wheel
sprung
mass
force
four wheels, the following equations are
=
wheel
of
sprung
mass
=
pitch
axis
=
pitch
moment
axis
of
moment
inertia
of
inertia
IIyy
I
yy;
yy
= Iwpitch
axis
=
pitch
moment
axis
of
moment
inertia
of
inertia
yy
=
wheel
base
of
sprung
mass
(8)
0
.
5
wC
M
K
K
Z
C
Z
K
Z
F pfl
;
(8)
0
.
5
wC
M
K
K
Z
C
Z
K
Z
F
s
,
f
s
,
f
t
u
,
fl
s
,
f
u
,
fl
t
r
,
fl
sw
,wheel
f are obtained
u, fl of
s , f u , fl mass
pfl
wheels, the following
w equations
base
=sZ, fwheel
ofCtsprung
sprung
base
mass
sprung
w
==
wheel
of
of
mass
sprung
mass
Zforce
K s , base
aK
aC s ,t f Tr, flthe
0.5following
wK
M equations are obtained
u, w
flwheel
f=
sanalysis
s , base
f Zs s , four
f T at
s , f the
wheels,
By performing
balance
at
the
Bymuperforming
force
balance
analysis
the
four
wheels,
following
equations
are
obtained
m
Z
K
Z
C
Z
aK
T
aC
T
0
.
5
wK
M
u u , fl
,f s
s, f s
s, f
s, f
s, f
By
force
balance
analysis
at
the
four
wheels,
the
following
equations are obtained
K0.s5,swC
(8)
;By performing
0.balance
5wC
Kthe
C
Kfollowing
; performing
s , f Manalysis
f at
t Z ufour
, fl at
s, f Z
u , flthe
t Z r , flthe
pfl
By
performing
force
force
balance
analysis
wheels,
the
four
wheels,
equations
equations
are
obtained
are
Ffollowing
where,
5wK s , f M By
M
K
K
Z
C
Z
K
Z
F pflobtained
performing
By
performing
force
balance
force
analysis
balance
at
analysis
the
four
at
wheels,
the
four
the
wheels,
following
the
following
equations
equations
are
are obtained
obtained
s, f
s, f
t u , fl
s , f u , fl
t r , fl ;
.5wK
M
mus ,Zf uZ, sfl CKss, ,f fZZss aK
C ss, ,f fZTs aC
aKs s, ,f fTT 0aC
T
0
.
5
wK
M
;; mu Z u , fl ;; K
s
,
f
s, f
s, f
K t Z r , fl F pfl
mu Zu , fl K s , f Z s (8)
C s , f Z s aK s , fT aC s , f T 0.5wK s , f M
29
29
rs, ,fl
mu ZZ
K
K
C
Z
C
aK
Tflfs C
aK
aC
Tu,fl, fl
.5K
5swK
M
..F5
wK
(8)
.K
5uwC
M0C
0aC
0
u , fl 0m
,fuZZ
fss, f
KaK
(8)
.
5
wC
M
K
K
C
K
Z
F pfl
sZ
,s ,flsf sK
, fsZ
s,Z
sf,uZ
f,T
sfZ
,Z
fuT
s,wK
,fZfZT
fflM
sr ,, flf M
,
t
s
,
t
pfl
m
m
Z
K
C
Z
aC
aK
0
aC
.
T
0
5
wK
s
t
u
,
t
u u , fl
us , fu , fls
ss,, ff ss
s ,sf, f s
ss,, ff
s , f s , f
s, f M
(8)
0.5wC s , f M K s , f K t Z u, fl C s , f Z u , fl K t Z r , fl F pfl
K
ZZssuu,,,,ffflflKKCCttss,,ZZff ZuuZ,,uuflfl,, flfl CC KssK,, fftt ZZruur,,,flflfl FFKKpflpfltt ZZ rr ,, flfl FF pflpfl
(8)
(8)
wC s , f
M0..55wC
K s ,sf, f M
29
(8)
(8)
00..55wC
wC
K
ttK
K
s , fM0
s ,sf, f MK
Vol. 2
No. 2
29
ISSN: 2180-1053
July-December 2010
29
35
29
29
29
29
29
29
Journal of Mechanical Engineering and Technology
mu Zu, fr muKZsu, f, frZ s KCs,s,f fZZssCaK
aCss, ,f fTTaC
0.5s,wK
s, fsZ
, fsTaK
f T s, f0M.5wKs, f M
aC
0M
CaKZ TaK
mu Zu, fr muK
Zus,, ffrZ s KC
Z
Z
aC
T
T
0
.
5
wK
T
.5wK f M
, sf , K
f s s M fK
s, sfZ Ks, sfC
,f
(9)
(9)
0.5wC 0M
.5s
wC
s,K
Z us, ,frCf s, fsK,ZftuZ, frr , frs,K
tFZpfrr , fr F pfr
s,sf, f aK
t s T
, f u, fraCt ZT
su, f,fr
mu Zu, fr muKZsu,,ffrZss, fKC
Z
0
.
5
wK
M
Z
C
Z
aK
T
aC
T
0
s
,
f
s
s
,
f
s
,
f
s
,
f
s,Cf K
.5wKs, fMF
,f K
s s, f K s Z Ks
,f Z
(9)
(9)
0.5wCs ,0f.M
5swC
s , sf ,M
f K
ts , f u , fr t C
su,, ffrZ
u , fr s , f Z ut ,Zfrr ,fr K
t Z pfr
r , fr F pfr
C K
t Z r , fr K FZpfr F
wC
(9)
0
.
5
wC
M
K
K
Z
C
Z
(9)
0
.
5
M
K
K
Z
Z
s
,
f
s
,
f
t
u
,
fr
s
,
f
u
,
fr
s
,
f
s
,
f
t
u
,
fr
s
,
f
u
,
fr
t
r
,
fr
pfr
0M.5wK M
mu Zu ,rl muKZsu,r,rlZ s KCss,r,rZZss CbK
bCss,r,rTTbC
0.5swK
s ,rsZ,rsT bK
,rT s ,r
s ,r
m
Z
K
Z
C
Z
aK
T
aC
T
0
.
5
wK
M
u
u
,
fr
s
,
f
s
s
,
f
s
s
,
f
s
,
f
s
,
f
mu Z u ,rl muK
Z rlZ s KCs,rs ,Zr Zs sCbK
bK
bCs ,sr,TrTbC
wK
T s ,0r M
.5wK M
s
,
rZ
sK
,sr TZ
r
Ks.srC,MrbK
Z u,0r0,rl.ZC.5s5,uwK
(10) (10)
0.s5u,,rwC
r0M.5wC
,rl Kt C
KZtuZ, rlr ,srl,r K
tFZprl
F
, r ubC
, rl
CZss,0,rrM
,rKTt s
mu Zu ,rl muKZsu,r,rlZs0s..K
C
Z,sfsM
T,frsbC
5
wC
K
K
Z
Kst,Z
r ,Frl pfr F prl
Z
Z
bK
T
T
5wK
s
,
r
s
s
,
r
s
s
,
f
t
u
s
,
f
u
,
, frF
s
,
r
s
,
r
s
s
,
r
s
,
r
r Mr
(10) (9)(10)
0.5 wCs.0r .M5 wC
Ks.rsM, r K
Kts, rZ u,rlKt C
Z us , rrlZu , rlC s, r.frZ
K
ut ,Z
rl r , rl K
t Z prl
r , rl
prl
C K
(10)
0
.
5
wC
M
K
K
Z
C
Z
Z
F
(10)
0
.
5
wC
M
K
K
Z
Z
K
Z
F
s .r
r
, r u , rl t su,,rrl u , rl s , r ut , rlr , rl t prl
prl
s.rs,C
tTs
mu Zu , rr muKZsu,,rrrZ s KCs ,sr,Z
aK
bCs ,sr,TrTbC
0.5s ,wK
.5wK s , rM r , rl
r Zss
sZ,r Z
s
, rs bKaK
r0T.5
swK
, r0M
m
Z
K
Z
C
T
bC
T
M
r s s ,r s s ,r
,r
r
mu Zu , rrmuu K
Zu ,rlrr
Ks ,C
C saK
bC
T, rT sbC
0.s5, rwK
T s,0r.M5s ,wK
M
r Z
s sZ
, rK
sZ sM
ZZKsts,s,rsrTTZ,ruaK
ZuTs,url,r,rrbC
(11)(10)(11)
0.5us,wC
.5s5,wC
rwC
,rK
K
, rr
Ks,trZsC
Z u
Z tuZ,rrr , rr
sK
Z, rtFZprr
r0.M
r,
r
,5
rrC
sZ
,rK
r , rr F prr
.sr,sM
ss ,0K
KbC
C
K
mu Zu , rr muK
Z
C
Z
aK
0
.
wK
M
Z
Z
C
aK
T
T
0
.
5
wK
s
t
s
,
r
u
,
rl
t
r , rl F prl
s
,
r
s
,
r
s
s
,
r
s
,
r
s
,
r
s ,C
s, rK
s , r C
r Z
s , rM
(11) (11)
0.u5rr
wC ,0r .Ms5, rwC
sKs , rsM
Ksts,Z
F rprr
, r raC
u, rrK tT Z
us0, .rrr5ZwK
u , rr
s ,M
r Kut ,Z
rr r ,
rr Kt Z
, rr F prr
mu Zu, fr mKZs, f Z s sKC
Z
aK
T
f s ZaK ss,, ff
aCZs,f TCs, f 0.K
s.s5
,,ffwC
,fM
(11) (11)
u 0.u5, frwCs ,0r M
Z ssKs, rsC
Zrrrs,rr
s,
M
fs,ZK
Kt s ,r u, rrK Tt C
Z
Z5 uwK
Kt F
Z prr
F prr
,
r
s
,
r
u
,
rr
t
u
,
rr
s
,
r
,
r
,
rr
where, where,
K0.Z5wK M
K.5swC
Zt sZu ,aK
TZs ,f bC
(9)
m0u.5ZwC
Kss ,fMC
s ,Kr K
s , C
Z ,sfr,C
u , rrs
,r Z
rT
rF
(9)
, f0M
fr K
frs ,K
pfrr , fr F pfr
s,=
f front
sright
,f tr masses
u , fr u
s , f Zt u , rfr,acceleration
tZ
where, where,
unsprung
unsprung
masses
acceleration
Zu , fr
Z=
u , frfront right
unsprung
(11)
5wC s=
Mfront
K right
Kunsprung
Zmasses
Cmasses
Z u ,rr acceleration
K t Z r , rr F prr
=0.front
where, where,
,
r
s
,
r
t
u
,
rr
s
,
r
right
acceleration
Z
Z
Zu,ufl, fr Zuu,=,flfrfront
left
=bK
front
unsprung
left
unsprung
masses
masses
acceleration
acceleration
0.5wK
s,r0M.5wK
mu Zwhere,
C
Z =right
C
bCunsprung
Tmasses
unsprung
front
right
masses
acceleration
u ,rl Z
,r T
Z
bK
bC s ,masses
Tacceleration
frs ,ur ,Z
uK
sZ u=,K
frfront
u, Z
rlZ
ssfront
,,rr Z ss s ,rsunsprung
s left
ss,,rrT
racceleration
s ,r M
Zm
=
=left
front
unsprung
masses
acceleration
u
,
fl
u
,
fl
rear
right
=
rear
unsprung
right
unsprung
masses
masses
acceleration
acceleration
Zu,rr0.5 wC
Zu ,=
where,
rr
Ks==sleft
sZ,ru ,unsprung
(10)
M 5wC
KKt left
masses
C Z acceleration
Z r , rl KFZprl F
Zu , fl Zus,.=
unsprung
front
(10)
rl K
, rlCs , rKZt uacceleration
fl0.front
.r, rM
t Zs ,ur, rl umasses
, rl
t r , rl
prl
rear=right
rear
unsprung
right
unsprung
masses
masses
acceleration
acceleration
Z
Z frr=
front
right
unsprung
masses
acceleration
Zuu,rl,rr ZZuuu,,,=rr
rear
left
=
rear
unsprung
left
unsprung
masses
masses
acceleration
acceleration
rl
= rear right
unsprung
masses masses
acceleration
Zu ,rr Zu ,rr
=
rear
right
unsprung
acceleration
Z u ,rl ZZuu,,flrl= rear=left
==front
rear
unsprung
left
unsprung
masses
masses
acceleration
acceleration
left
masses
acceleration
profiles
road
at
left,
front
left,
right,
frontrear
right,
right
rear
and
right
rearand
leftrear left
Zmr ,ufrZu , rrZZrm
Zroad
s, rZZr,rsrr, rT=
CaK
unsprung
bC s , rprofiles
Tatfront
0.5wK
Mfront
frZ
flKCZsZ
rrZ
,rZ
rl, rrs r ,rl
,
s
,
s
,
r
Z
aK
T
bC
T
0
.
5wK
Zr,u,fl,uK
u
,
rr
s
,
r
s
s
,
r
s
s
,
r
s
,
r
s , rM
=
rear
left
unsprung
masses
acceleration
rear
left
unsprung
masses
acceleration
,rl Z ===
road
=
profiles
road
profiles
at
front
at
left,
front
front
left,
right,
front
rear
right,
right
rear
and
right
rear
and
leftrear left
Z r , fr Z r , flfrrl ZZZZruru,,,rrrr
Z
right
unsprung
masses
acceleration
fl
rr,,rl
rr rear
r
,
rl
tires
K ,rrMtires
=sZprofiles
(11)left (11)
M.5ZwC
road
respectively
CZat
Z ,rrC
KZt Z
FZprr rear
s , r0
,
rr K
s ,ur,front
r ,left,
rr KKt respectively
K
Fright
right,
and
rear
leftrear
Z r , fr ZZrr,, flfr0.5wC
ZZrr,,rr
, ruroad
t profiles
rr u
s , rfront
ufront
, rr
tfront
r , rr right,
prrrear
atleft,
right
and
Zrr,,rlrrss,=
Z
fl
r
,
rl
tires
respectively
tires
respectively
Z u ,rl
= rear left unsprung masses acceleration
tires respectively
tires respectively
2.3
Vehicle
2.3
Handling
Model
road profiles at front left, front right, rear right and rear left
Z r , fr ZVehicle
Z r ,Handling
Z r ,rl = Model
where, where,
r , fl
rr
2.3
2.3
Vehicle
Vehicle
Handling
Handling
Model
Model
= frontHandling
right
unsprung
masses acceleration
Zu , fr Vehicle
tires
respectively
= Model
front
right
masses
ZHandling
2.3 handling
Vehicle
2.3
Model
u , fr employed
The
The
handling
model
model
employed
in
thisunsprung
paper
in thisispaper
a 7-DOF
is acceleration
a 7-DOF
system system
as shown
as in
shown
FIGURE
in FIGURE
2. It 2. It
account
The handling
The
handling
model
model
employed
employed
in
this
in
paper
this
is
paper
a
7-DOF
isacceleration
athe
7-DOF
system
as
shown
asinshown
in
FIGURE
in
FIGURE
2. It 2. It
Z
=
front
left
unsprung
masses
acceleration
takes
into
takes
into
account
three
degrees
three
degrees
of
freedom
of
freedom
for
the
for
vehicle
vehicle
bodysystem
inbody
lateral
and
lateral
longitudinal
and
longitudinal
Z
u , fl
=
front
left
unsprung
masses
u
,
fl
2.3
Vehicle
Handling
Model
The
model
employed
in
this
paper
isone
a the
7-DOF
system
as
shown
in
FIGURE
2.motion
The
handling
model
employed
in
this
paper
is
a
7-DOF
system
as
shown
in
FIGURE
It
takeshandling
into
takes
account
into
account
three
degrees
three
degrees
of
freedom
of
freedom
for
for
vehicle
the
vehicle
body
in
body
lateral
in
lateral
and
longitudinal
and
longitudinal
motions
motions
as
well
as
well
yaw
as
motion
yaw
motion
(r)
and
(r)
one
and
degree
of
degree
freedom
of
freedom
due
to
the
due
rotational
to
the
rotational
motion
ofIt 2. of
rear right
unsprung
masses acceleration
Z uVehicle
2.3
Handling
Model
, rr
=yaw
rear
right
unsprung
masses
acceleration
Zu=
, rr
takes
into
three
degrees
ofand
freedom
vehicle
body
lateral
and
longitudinal
takes
into
account
three
degrees
freedom
for
the
vehicle
inisthe
lateral
and
longitudinal
motions
motions
as
well
as
as
well
yaw
as
motion
motion
(r)
one
and
degree
one
degree
of
freedom
ofthat
freedom
due
to
the
due
rotational
to
of road.
of
each
tire.
each
Inaccount
tire.
vehicle
In
vehicle
handling
handling
model,
model,
it(r)
isofassumed
itfor
is the
assumed
that
the
vehicle
the in
vehicle
isbody
moving
moving
onrotational
a motion
flat
onroad.
a motion
flat
The
handling
model
employed
inone
thisdegree
paper
is afreedom
7-DOF due
system
as shown
in FIGURE
2. It
as well
Z
=,rlwell
rear
left
unsprung
masses
acceleration
motions
as
yaw
motion
(r)motion
and
of
to
rotational
of
Zas
motions
as
yaw
motion
and
one
degree
ofthat
freedom
due
toisthe
rotational
motion
of
u ,rlInvehicle
=motion
rear
left
unsprung
acceleration
eachvehicle
tire.
each
tire.
vehicle
vehicle
handling
handling
model,
model,
it(r)isthe
assumed
itmasses
isthe
assumed
that
the
thex-axis,
vehicle
isthe
moving
moving
on
amotion
flat
onroad.
athe
flat
The
The
experiences
experiences
along
longitudinal
the
the
y-axis,
lateral
y-axis,
and
androad.
the
uIn
The
handling
model
inlongitudinal
this
paper
isvehicle
a 7-DOF
as
shown
takes
into
account
threeemployed
degrees
of along
freedom
for
the x-axis,
vehicle
bodylateral
insystem
lateral
and
longitudinal
each
tire.
In
vehicle
handling
model,
it
is
assumed
that
the
vehicle
is
moving
on
a
flat
road.
each
tire.
In
vehicle
handling
model,
it
is
assumed
that
the
vehicle
is
moving
on
a
flat
road.
The
vehicle
The
vehicle
experiences
experiences
motion
motion
along
the
along
longitudinal
the
longitudinal
x-axis,
x-axis,
the
lateral
lateral
y-axis,
y-axis,
and
the
and
the
angular
angular
motions
motions
of
yaw
of
around
yaw
around
the
vertical
the
vertical
z-axis.
The
z-axis.
motion
The
motion
in
the
horizontal
in
the
horizontal
plane
can
plane
be
can
be
road
profiles
at front
front
right,
rear
right
andright
rearand
left motion
Z r , fr Zmotions
ZZr ,rlas =yaw
,rfl, fr ZZ
ras
,rrr, flwell
road
profiles
at
front
left,
front
right,
rear
leftof
ZrFIGURE
Z rmotion
(r)account
and
one left,
degree
of
freedom
due
torear
the
rotational
rIt
, rr takes
,rl =into
in
2.
degrees
oflateral
freedom
for
the
The
vehicle
experiences
motion
along
the
longitudinal
x-axis,
the
the
The
vehicle
experiences
motion
along
thethree
longitudinal
x-axis,
lateral
y-axis,
and
angular
angular
motions
of vehicle
yaw
around
yaw
around
the
vertical
the
vertical
z-axis.
z-axis.
The
motion
The
motion
invehicle
the
plane
plane
be
canthe
be
characterized
characterized
bymotions
the
longitudinal
by of
the
longitudinal
and
lateral
and
accelerations,
lateral
accelerations,
denoted
denoted
by horizontal
ainxisand
byhorizontal
ayx respectively,
ay-axis,
and
aand
respectively,
ycan
each
tire.
In
handling
model,
it
is
assumed
that
the
moving
on
a
flat
road.
tires
respectively
tires
respectively
vehicle
body
in of
lateral
and
longitudinal
motions
well
as
yaw
motion
angular
motions
of
yaw
around
the
vertical
z-axis.
The
motion
invas
the
horizontal
plane
can
be
angular
motions
yaw
around
the
vertical
z-axis.
The
motion
in
the
horizontal
plane
can
characterized
characterized
by
the
by
longitudinal
the
longitudinal
and
lateral
and
accelerations,
lateral
accelerations,
denoted
denoted
by
a
by
a
and
and
respectively,
a
respectively,
and
the
and
velocities
the
velocities
in
longitudinal
in
longitudinal
and
lateral
and
direction,
lateral
direction,
denoted
denoted
by
by
and
v
,
and
respectively.
v
,
respectively.
v
x
x
y
y
x
The vehicle experiences motion along the longitudinal x-axis,
they x lateraly y-axis, and the be
characterized
by
the
longitudinal
and lateral
lateral
accelerations,
denoted
by
avxvyand
aof
characterized
byofthe
longitudinal
and
lateral
accelerations,
and
ay respectively,
(r)
and
one
degree
of
freedom
due
to
the
rotational
motion
tire.
and
the
and
velocities
the
velocities
in
in
longitudinal
and
and
direction,
lateral
direction,
denoted
denoted
by
by
and
,by
and
respectively.
vyx yrespectively,
,each
respectively.
vdenoted
x
xhorizontal
angular
motions
yaw
around
the
vertical
z-axis.
The
motion
in
the
plane
can be
2.3
Vehicle
Handling
Model Model
2.3
Vehicle
Handling
and the
velocities
in
longitudinal
and
lateral
direction,
denoted
by
and
v
,
respectively.
v
and
the
velocities
in
longitudinal
and
lateral
direction,
denoted
by
and
v y ,arespectively.
v
x
y x is
Incharacterized
vehicle
handling
model,
itand
isislateral
assumed
vehicle
on a
by longitudinal
the longitudinal
by
ax moving
and
Acceleration
Acceleration
in longitudinal
in
x-axis
isx-axis
defined
defined
as ; accelerations,
as ;that thedenoted
y respectively,
. handling
. thein
.velocities
.vehicle
and
in longitudinal
and
denoted
by
vshown
, respectively.
v x and
Acceleration
Acceleration
longitudinal
longitudinal
x-axis
isx-axis
defined
islateral
defined
as
flat
road.
Thein
experiences
motion
the
longitudinal
The
model
employed
in
this
paper
isaspaper
a; direction,
7-DOF
system
as
shown
2. It 2. It
y FIGURE
employed
in this
is ;a along
7-DOF
system
as in
inx-axis,
FIGURE
v.x aThe
v. xvhandling
ra. longitudinal
v yinmodel
r. longitudinal
(12) (12)
x yin
x three
Acceleration
x-axis
is
defined
as
;
Acceleration
x-axis
is
defined
as
;
takes
into
account
degrees
of
freedom
for
the
vehicle
body
in
lateral
and
longitudinal
the
lateral
y-axis,
and
the
angular
motions
of
yaw
around
the
vertical
three degrees of freedom for the vehicle body in lateral and longitudinal
v. x takes
a x v.xv into
r. x account
v y r.
(12) (12)
ya
motions
as wellr as
motion
(r)
and
one
degree
freedom
due
tocharacterized
the
motion
of
Acceleration
inrlongitudinal
x-axis
defined
as
;
motions
asyaw
well
as yaw
motion
(r)isand
oneof
degree
of
freedom
duerotational
to the rotational
motion
of
motion
in
the
horizontal
plane
can
be
by
the
v xz-axis.
a x vxv yThe
(12)
a
v
(12)
yall
. summing
. thein
By summing
ByIn
all xthe
forces
forces
x-axis,
in longitudinal
x-axis,
longitudinal
acceleration
acceleration
canthebevehicle
can
be isdefined
asmoving
; a flat
as on
; road.
each
tire.
vehicle
handling
model,
itmodel,
is assumed
that the
vehicle
isdefined
moving
on
each
tire.
In
vehicle
handling
it
is
assumed
that
a
flat
road.
and
ayy-axis,
respectively,
longitudinal
and
denoted
bythe
axcan
a x the
v yforces
r thelateral
x all
By summing
Byvsumming
all
in
forces
x-axis,
inaccelerations,
x-axis,
longitudinal
longitudinal
acceleration
acceleration
can
be
defined
be
as
; as
; the(12)
The
vehicle
experiences
motion
along
the
longitudinal
x-axis,
lateral
and
The
vehicle
experiences
motion
along
the
longitudinal
x-axis,
thedefined
lateral
y-axis,
and the
By
summing
all
the
forces
in
x-axis,
longitudinal
acceleration
can
be
defined
as
;
and
the
velocities
in
longitudinal
and
lateral
direction,
denoted
vx can be
By
summing
all
the
forces
in
x-axis,
longitudinal
acceleration
can
be
defined
as
;by
F
Fsin
FxfrFthe
FxrlThe
G around
GFyfl
cos
G Gvertical
cos
Gvertical
GFyfrFsin
G F
motion
Fin
angular angular
motions
of
yaw
z-axis.
The
plane can
be
xfl cos GF
xflFcos
yfl sin
xfr the
yfr sin
xrl
xrr
xrr the horizontal
motions
of
yaw
around
z-axis.
motion
in the horizontal
plane
(13)
(13)
a x By
a
xcos
Fxflv
F
GFxfl
Fcos
Gsin
GFforces
Fsin
Gcos
x-axis,
G xfr
F
cos
Gsin
GF yfr
Fsin
FF
Fxrr can
summing
allby
the
in
longitudinal
defined
as
; a respectively,
and
, respectively.
yfl
yfl
xfrm
yfr
xrl Gacceleration
xrrxrl
characterized
and
lateral
accelerations,
denoted
by abex and
aay xrespectively,
m
the
longitudinal
by
and
y by the longitudinal
y (13)
t
tand lateral accelerations, denoted
(13)
a x characterized
a
Fxfl xcos GFxfl
inFcos
cos
F
sin GF
denoted
Fxrl F
GsinGFyflFsin
GFlateral
cos
Fxrr
yfl
xfr G
yfr G
xrl G
xfr
yfrFsin
andathe velocities
longitudinal
and
byxrrv x andby
v yv, respectively.
m
m
and
in longitudinal
and
lateral
direction,
denoted
t
t direction,
(13) (13)
x and v y , respectively.
a xthe Fvelocities
x
F
F
F
F
F
cos
G
sin
G
cos
G
sin
G
xfl
yfl
xfr m
yfr
xrl
xrr
t
tx-axis
Similarly,
Similarly,
acceleration
lateralinm
y-axis
lateral
isy-axis
defined
is defined
as defined
;
as ; as ;
(13)
aacceleration
Acceleration
ininlongitudinal
is
x
m
t
Similarly,
Similarly,
acceleration
acceleration
in
lateral
in
y-axis
lateral
is
y-axis
defined
is
defined
as
;
as
;
Acceleration
in
longitudinal
x-axis
is
defined
as
;
Acceleration in longitudinal x-axis is defined as ;
..
..acceleration
..
. . in lateral
Similarly,
defined
as ;
Similarly,
acceleration
in y-axis
lateralisy-axis
is defined
as ;
vv.xy aSimilarly,
vv. vyxxyrr. aay acceleration
(12)
v r
(14) (14)
in lateral y-axis is defined as ;
xy (12)
x vx y. r
v. y ay v. yvx r. ay vx r.
(14) (14)
v y ay v. yvx r ay vx.r
(14) (14)
By summing
all the forces in x-axis, longitudinal acceleration can be defined as ;
Byvsumming
(14)
y ay vx rall the forces in x-axis, longitudinal acceleration can be defined as ;
ax
By summing all the forces in x-axis, longitudinal acceleration can be
Fxfl cos GF
F;cos
sin G F sincos
defined
as
yfr sin
G GFxfrFcos
G GFyfrFxrl
sinGFxrrFxrl Fxrr
xfl yfl G F yfl xfr
ax
mt
(13)
mt
Similarly,
acceleration
in lateraliny-axis
defined
as ;
Similarly,
acceleration
lateralisy-axis
is defined
as ;
. 36
ISSN:
2180-1053
. .
.
v y ay vvyx r a v r
y
x
Vol. 2
No. 2
July-December 2010
30
30
30
(14)
(13)
30
30
30
30
(14)
Acceleration
ininlongitudinal
x-axis
defined
asas; ; as
Acceleration
in
x-axis
is
Acceleration
Acceleration
longitudinal
in longitudinal
longitudinal
x-axisisis
x-axis
defined
is defined
defined
as ;;
.
.
..
.
..
.
v vx aax vvvx vy raar x (12)
v r
(12) (12)
(12)
x
x y x v yy r
x Control
Adaptive
Pid
With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
By
all
x-axis,
acceleration
can
asas; ; as
By
all
forces
in
x-axis,
longitudinal
acceleration
can
be
Bysumming
summing
By summing
summing
allthe
theforces
forces
all the
theinin
forces
x-axis,
inlongitudinal
x-axis,
longitudinal
longitudinal
acceleration
acceleration
canbebedefined
can
defined
be defined
defined
as ;;
aax
x
FF
cos
GFxrrFxrl Fxrr
cos
G
sin
G
cos
G
sin
xfl cos
yfl sin
xfr
yfr sin
xrl
xfl
yfl
xfr
yfr
F
Fyfl
FGxfr
F
cosGF
Gxfl
FF
cos
GGF
FF
sin
GF
FF
cos
GF
Gyfr
FF
sin
xfl
yfl Gsin
xfr Gcos
yfrGsin
xrl GFF
xrr
xrl Fxrr
a
a xx
mm
m
t
mtt
t
(13)
(13)
(13)
(13)
(14)
(14)
(14)
(14)
Similarly,
acceleration
ininlateral
isis
defined
asas; ; as
Similarly,
acceleration
in
lateral
y-axis
is
Similarly,
Similarly,
acceleration
acceleration
lateral
iny-axis
lateral
y-axis
y-axis
defined
is defined
defined
as ;;
Similarly,
acceleration
in lateral
y-axis
is defined
as ;
.
.
..
.
.
..
v vy aay vvvyxvr raay v r
y
y y x y vxx r
By summing all the forces in lateral direction, lateral acceleration can
be defined as ;
By summing
all the forces
lateral
lateral acceleration
can be defined
as ;
By summing
all theinforces
in direction,
lateral direction,
lateral acceleration
can be defined
as ;
30
30
30
30
F yflsumming
Fsin
cos
GFthe
Fcos
sin
xfl
cos
GFlateral
Fcos
GF
F yrlacceleration
G the
GF
direction,
Gsinlateral
G F yrr
F yrr
xfl
yfr G
xfr direction,
yrl
yfl
yfr
xfrFsin
By asumming
By
all
forces
all
in
forces
lateral
in
acceleration
lateral
can
be
defined
can
be
defined
as
;
as
;
…………………………….
(15)
……………………………. (15)
ay
y
min
mt direction,
By summing
all the forces
lateral
lateral acceleration
can be defined
as ;
t direction,
By summing
all the in
forces
lateral
lateral acceleration
can be defined
as ;
F yfl cos GFyflFcos
F
F
F
F
F yrl F yrr
sin
G
G
sin
cos
G
G
G GFxfrF sin
G Fyrr
xfl
xfl yfr
yfr cos
xfr sin
yrl …………………………….
…………………………….
(15)
(15)
ay
ycos G F sin G F
Fxijyflaand
Fsin
cos
Gtire
F
sinlongitudinal
Ginxfr
the
denote
the
and
respectively,
where
F
FFxij
xfl F
yfrm
xfr
yrllongitudinal
yrr
Fxfltire
F yrrand directions,
cos
Gyijthe
sin
Gforces
cos
G FF
lateral
denote
the
forces
lateral directions,
respectively,
where
m
yijyfland
yfr in
yrl…………………………….
tF
t G F
(15) (15)
a ywhere
longitudinal
and
lateral
…………………………….
a y Fxij and Fyij denote the tire forces in the
with thewith
index
indicating
frontmt(f)front
ormrear
(r) rear
tires(r)
and
index
indicating
left (l) or
the(i)
index
(i) indicating
(f) or
tires
and(j)
index
(j) indicating
leftright
(l) or right
directions,
respectively,
withtthe
thelongitudinal
index
(i)
indicating
front respectively,
(f) or
rear
and FFyijxijdenote
and Fyijthe
denote
tire forces
the tireinforces
in the longitudinal
and. lateral
and. directions,
lateral directions,
respectively,
where Fwhere
xij
(r) tires.
steering
angle is(j)
denoted
by į, the
yaw
rate
byright
rrate
andby
the
total
vehicle
m
(r)The
tires.
The steering
angle
is denoted
by
į, (l)
the
yaw
and
the
total vehicle
mt denotes
trdenotes
tires
index
indicating
left
or
(r)
tires.
Therespectively,
steering
denote
the
tire the
forces
in
the(r)
longitudinal
andand
lateral
directions,
where(r)F
Fand
xij and
and
tire
forces
inrear
the (r)
longitudinal
and
lateral
directions,
respectively,
where
F
F(i)
yij
with
with
index
the
(i)
index
indicating
indicating
front
(f)
front
or
rear
(f)
or
tires
and
tires
index
(j)
index
indicating
(j)
indicating
left
(l)
or
left
right
(l)
xij
yij denote
mass.the
The
longitudinal
and
lateral
vehicle
velocities
v
and
can
be
obtained
by
theor
mass.isThe
longitudinal
and yaw
lateralrate
vehicle
velocities
vvyx and
cantotal
be obtained
byright
the
vthe
xm
denoted
by front
δ, the
by
r and
and
vehicle
y
. t denotes
.
with angle
thewith
index
(i)
indicating
(f)
or
rear
(r)
tires
index
(j)
indicating
left
(l)
or
right
the
indexsteering
(i) indicating
(f)the
orbyyaw
rear
(r)
tires
and
index
(j)mindicating
lefttotal
(l) or
right
. The
. angle
(r) tires.
(r)
The
tires.
steering
is. angle
denoted
isfront
denoted
bylateral
į,
į, the
rateyaw
byvelocities
rrate
and by
and
denotes
the total
the
vehicle
vehicle
m
.
t rdenotes
tand
mass.
The
longitudinal
and
vehicle
v
v
can
be
.
.slip denoted
x
v y andofvvxangle
They
can
be used
toused
obtain
the
side
angle,
byy Į.vehicle
Thus,
integration
ofsteering
.lateral
They
obtain
the
side
denoted
by
.byThus,
integration
y. and
(r) tires.
The
denoted
bybe
į,
the
rate
rslip
and
thebe
total
mass.
The
mass.
longitudinal
The
longitudinal
andvisxangle
andis can
vehicle
lateral
vehicle
velocities
velocities
vby
vymxcan
and
bevangle,
obtained
by total
theĮvehicle
the
trdenotes
(r)
tires.
The
steering
denoted
byyaw
į,tovthe
yaw
rate
and
denotes
the
my can
vby
t obtained
x and
obtained
by
the
integration
of
v
and
.
They
can
be
used
to
obtain
the
y as
xas ;
the
slip
angle
of
front
and
rear
tires
are
found
;
the
slip
angle
of
front
and
rear
tires
are
found
mass. The
longitudinal
and
lateral
vehiclevehicle
velocities
vx and vvxy can
be
obtained
by
the
. The longitudinal
. .
.
mass.
and
lateral
velocities
and
can
be
obtained
by
the
vy and rear tires
side
slip
denoted
α. be
Thus,
the
slip
ofslip
front
andofv.xv.y They
and
. They
bebyused
can
to used
obtain
tothe
obtain
sideangle
the
slipside
angle,
denoted
angle, denoted
by Į. Thus,
by Į. Thus,
integration
integration
of v .y angle,
v xcan
.
.
are
found
as
;and
§vvangle
·They
Lvfxand
r.can
rear
·They
be
toasobtain
the; side slip angle, denoted by Į. Thus,Į. Thus,
integration
ofof
yand
yfront
the
angle
the
slip
areused
tires
found
are
found
;
as
1
vfxvry.yfront
and
integration
ofLv§¨of
¸
¸ rear
Dslip
tan
G f tires
(16) (16)
D f ¨¨ tan 1
G f can be used to obtain the side slip angle, denoted by
f
¸ vrear
¨
¸
v
the slipthe
angle
of
front
and
tires
are
found
as
;
x ©of front
slip© angle
¹ x and
¹ rear tires are found as ;
§ v y L §f rv ·y L f r ·
D
tanD1f ¨¨ vtan1L¨¨ r¸¸ G f ¸¸ G f
andDf and
v Lf r ·
1 § y v §f v y· tan
f
D f ©¨¨ tan x1 ©¨¨ ¹¸¸ xG f ¹¸¸ G f
v
x
©
¹v
©
x
(16)
(16)
(16)
(16)
(17)
(17)
¹
v L§ fvry · L f r ·
andD and
1 § y
¸
tan
D r ¨¨ tan 1 ¨¨ ¸¸
rand
¸
v x © ¹v x
and and ©
¹
§ v y L§f vr y· L f r ·
¸
(17)
D r tanD r1 ¨¨ vtan 1L¨¨ r¸¸
where,
D f and
are
side
angles
the front
rearand
tires
respectively.
lf and lrlfare
fv yr ·are
v xL the
r ·¸¹side
DDfry and
slip at
angles
at theand
front
rear
tires respectively.
and
D
vx1 §©the
1©§
f slip
¹
¨
¸
(17)
D r where,
tan
¸
D r ¨ tan v ¨¨ ¸
¸
xthe
the distance
between
and
tirerear
to the
gravity
respectively.
©
¹v x the
the distance
between
front
tirebody
to thecenter
bodyof
center
of gravity
respectively.
© front
¹ rearand
where, Dwhere,
andthe
are the
slipside
angles
slipatangles
the front
at the
and
front
rearand
tiresrear
respectively.
tires respectively.
lf and lr lare
D r side
f and
f and D rf are
where,
D f distance
and
are
side
slip
angles
at
the
front
and
rear
tires
respectively.
l
f and lrlfare
where,
DDfr and
are
the
side
slip
angles
at
the
front
and
rear
tires
respectively.
and
Dfront
the
distance
the
between
between
thethe
the
and
front
rear
and
tire
rear
to
the
tire
body
to
the
center
body
of
center
gravity
of
respectively.
gravity
respectively.
r
where,
a
and
a
are
the
side
slip
angles
at
the
front
and
rear
tires
the distance
between
the
front
and
rear
tire
to
the
body
center
of
gravity
respectively.
f
r
the distance between the front and rear tire to the body center of gravity respectively.
respectively. lf and lr are the distance between the front and rear tire to
the body center of gravity respectively.
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
FIGURE
2: A 7-DOF
VehicleVehicle
handling
model model
FIGURE
2: A 7-DOF
handling
37
(17)
lr are
(17)
lr are
lr are
vx
§ v© y L f r ·v ¹ L r
§ y
1¨
f ·
(17)
D r tan D
¸
tan 1 ¨¨ ¸¸
¨r
v
x
v x slip¸¹ angles
© D r are
¹ side
where,
D fofand
the
at the front and rear tires respectively. lf and lr are
©Engineering
Journal
Mechanical
and Technology
(17)
the distance between the front and rear tire to the body center of gravity respectively.
where, D f where,
and D rDare
the side slip angles at the front and rear tires respectively. lf and lr are
f and D r are the side slip angles at the front and rear tires respectively. lf and lr are
the distance
front and
torear
the body
of gravity
respectively.
thebetween
distancethe
between
therear
fronttire
and
tire tocenter
the body
center of
gravity respectively.
FIGURE 2: A 7-DOF vehicle handling model
FIGURE 2: A 7-DOF Vehicle handling model
FIGURE
2:
Alongitudinal
7-DOF
handling
calculate
the
longitudinal
slip,
component
of the
tire’sbe
To To
calculate
the longitudinal
slip,
component
ofmodel
the tire’smodel
velocity
should
FIGURE
2: Vehicle
Alongitudinal
7-DOF
Vehicle
handling
derived.
The front
and rear
velocity
component
is given
by:
velocity
should
be longitudinal
derived. The
front
and rear
longitudinal
velocity
To calculate
longitudinal
slip, longitudinal
componentcomponent
of the tire’s
be should be
To the
calculate
the longitudinal
slip, longitudinal
of velocity
the tire’sshould
velocity
component
is given
by:
derived.
rear
longitudinal
velocity
component
is
given
by:
v wxf The
V tffront
cos and
DThe
(18)
derived.
front
and
rear
longitudinal
velocity
component
is
given
by:
f
V tf vcos D f V cos D
wxf
tf
v wxf
(18)
31
f
31
where,
the
speed
the
where,
where,
the speed
theofspeed
the front
of of
thetire
front
is,front
tire is,tire is,
where,
theofspeed
of the
where, the
speed
the front
tirefront
is, tire is,
V tf
Vtfv y Lvf ry 2 L vf rx2 2 v x2
where, the
speed
the front
tirefront
is, tire is,
where,
theofspeed
of the
2 L 2r 2 v 2
fvrthe
tfy y V tf where,
where,
theVvspeed
front
is,x tire is,
theLofspeed
ofvf xthetire
front
the rearthe
longitudinal
rear longitudinal
velocity
component
is,
is,
2
2 velocity
2
2 component
wxr
tr
trr V tf
L f vry Lvf xr v x
Vvtfy longitudinal
the
rear
velocity component
2
2 2
the
rear
longitudinal
is,
the V
component
is,
LV
rycos
V
LDfv rxvelocity
v x2 component
tf longitudinal
vrear
vVtfv ycos
Dvf velocity
wxr
r
the rear the
longitudinal
velocityvelocity
component
is,
rear longitudinal
component
is,
Vvtrwxrcos DV rtr cos D r
thev wxr
rearthe
longitudinal
velocity
component
where,
where,
speed
the
ofspeed
the
rear
of the
tire
rear
is, tire
is,is,
rear
longitudinal
velocity
component
is,
v wxr
Vvtrwxrcos DV rtr cos D r
where,
theofspeed
of the
rear
where,
the
thecosrear
is,2 tire is,
2
Vvwxr
cos
D
vspeed
V
D2r rtire
tr trr2
Vv trwxr
V
L
v
r
L
v
v
tr
y
r
y
r
x
the
speed
where,where,
the
speed
the rear
tirethe
is,x rear
where,
theofspeed
of of
the
rear
tire is,tire is,
2
2 L 2r v 2
V
v
tr
y
r
x
the vrear
V tr where,
L r of
rspeed
where,
thev speed
tirerear
is, tire is,
y the
xof the
then, the
then,
longitudinal
the longitudinal
of2ratio
the front
of thetire
front
is, tire is,
2 slip
2 ratio
2 slip
(18)
is,
31
(19)
(19)
(19)
(19)
(19)
(20)
(19)
(20)
(20)
(20)
(21)
(20)
(21)
(21)
(21)
(22)
(19)
(19)
(20)
(20)
(20)
(21)
(21)
vVytr L r rvy Lv rx r v x
V tr
(21)
2 2 slip2ratio of the front tire is,
longitudinal
then,
longitudinal
ratio
of
the
front
tire
is,
vVwxf
V trthethen,
vtrythe
LZ
ryR2slip
v
vrvwxf
L r vx
(21)
f w Zr xf R w
S afthe longitudinal
under
braking
under
braking
conditions
conditions
(22)
then,S afthethen,
longitudinal
slip
ratio
of
the
front
tire
is,
slip
ratio
of
the
front
tire
is,
v v wxf vwxf
Z R w slip ratio of the front tire is,
v wxf
wxf
Zlongitudinal
then,
f R w slipf ratio
S afthe
under
conditions
(22)
then,
longitudinal
ofratio
the braking
front
is, tire is,
then,
the longitudinal
slip
of thetire
front
S af the
under
braking
conditions
(22)
v wxf vwxf
R wvwxf
Zvfwxf
Z f Rw
under
braking
conditions
(22)
theS longitudinal
slip ratio
slip
ofratio
the
rear
of the
tire
rear
is, tire
is,
under
braking
conditions
(22)
af theSlongitudinal
af Z
v wxf
Z f Rw
v wxf v wxf
f Rv
wwxf
S af theSlongitudinal
under
braking
conditions
(22)
under
braking
conditions
(22)
af
slip the
ratiorear
of the
the longitudinal
tire rear
is, tire is,
Zslip
v wxf
Z r of
v wxr v wxf
vrwxr
R wratio
Rw
under braking
under braking
conditions
conditions
(23) (23)
S ar
S ar
the longitudinal
ratio
of the
rear tire is,
the longitudinal
v wxrslip
vwxr
Zslip
v wxr
R wratio of the rear tire is,
r
Z
v wxr
R
rslip
w ratio of the
under
braking
conditions
(23)
ar
theS ar
longitudinal
rear
tirerear
is, tire
under
braking
conditions
(23)
theSlongitudinal
slip ratio
of the
is,
v wxr
vwxr
Z
v
R
Z
v
R
wxr
r
w
wxr
r
w
Ȧr Sand
Ȧrf are
andangular
Ȧf arebraking
angular
velocities
velocities
of the
rear
of the
andrear
front
andtires,
frontrespectively
tires, respectively
and R w and
, is(23)
the
R w , is(23)
the
where,
under
conditions
S ar where,
under
braking
conditions
The
Zvrwxr
v wxrarv wxr
Ryaw
Zmotion
wv
wxr
rR
w braking
F
F
F
F
The
yaw
motion
is
also
dependent
is
also
dependent
on
the
tire
on
the
forces
tire
forces
and
and
as
well
as
as
well
on
as
on
wheel
wheel
radius.
under
conditions
(23)
S ar radius.
xij
xij
yij
yij
under
braking
conditions
(23)
S
Ȧ and Ȧf are angular
velocities
of the
front
tires, respectively
R w , is the
ar Ȧ
velocities
of the rear
andrear
frontand
tires,
respectively
and R w ,and
is the
where, Ȧwhere,
r and
v wxrfrare vangular
wxr
Malso
acting
on
acting
each
tire:
each
tire:
the self-aligning
self-aligning
moments,
moments,
denoted
denoted
bydependent
by
FFxij and
The
yaw
motion
isof
dependent
onon
the
tirerespectively
and RF yij,and
asthe
well
as on
wheel
radius.
38radius.
2180-1053
Vol.
2M
No.
2tire
July-December
2010
zij
zijthe
Fforces
yaw
is
also
therear
forces
as
on
wheel
Ȧthe
ȦȦfrare
angular
velocities
the
rear
and
front
tires,
where,
andISSN:
Ȧmotion
velocities
ofon
and
front
tires,
respectively
R w , is the
where,
r andThe
xij and
yij as well
f are angular
w is
M
acting
each
the
self-aligning
moments,
denoted
bydependent
Fforces
The
yaw
is
also
dependent
the
tire
forces
as well
wheel
radius.
Ȧr and
Ȧmoments,
angular
velocities
the
rear
and
front
tires,
respectively
R wyijand
, as
is on
the
where,
zij
FFxijyijand
F
yaw
motion
is
onon
the
tiretire:
well
as the
on
wheel
radius.
Ȧ
andThe
Ȧmotion
angular
ofon
the
rear
and
front
tires,
respectively
R w , is
where,
Mofalso
on
each
tire:
the
self-aligning
denoted
byvelocities
xij and
rf are
f are
zij acting
..
..
2 2ratio
2 slip
then,
of
then, V
the
thetire
front
tirefront
is, tire is,
where,
the
where,
of
tire
rear
is,ratio
is,the
vthe
rLslip
(21)(21)
vlongitudinal
Vlongitudinal
rthe
rofv xthe
v x2 of
tr
ythe
trspeed
y Lspeed
r rear
where, where,
the
speed
the
rear
tire
is, tire is,
theof
speed
of the
rear
v wxfControl
R Z2 f R
Zv wxf
2w
Adaptive
Pid
2 Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
the
slip
ratio
front
tire
is, is,
vVSthe
Moment
Vthen,
L r rfWith
vy2 2wPitch
L
vunder
ratio
vunder
(21) (22)
(21)
ofbraking
the
front
tire
conditions
Straf then,
conditions
(22)
ylongitudinal
tr
xr 2r slip
x2of the
af longitudinal
2 braking
V tr
v
(21) (21)
v
Vvtry vwxfLrvry L
r
v
wxfr x
x
v
Z R
v wxf f Z wf R w
wxf
then, the
the
longitudinal
slip ratio
slip
of
the
ratio
front
ofconditions
the
tire
front
is, tire
S afthen,
under
braking
Slongitudinal
under
braking
conditions
the
longitudinal
slip
ratio
of
the
rear
tireis,
af
then,
the
slip
ratio
of
the
front
is,
thelongitudinal
longitudinal
slip
ratio
of
the
tire
is, tire
then,
thevslip
longitudinal
slip
ratio
ofrear
thetire
front
is,is,
the
longitudinal
ratio
of
the
rear
tire
is,
wxfv wxf
(22)(22)
v wxf Zvfwxf
Rw Z f Rw
Zunder
Rf R
Sthe
braking
under
braking
conditions
conditions
(22)
(22)
vSwxf
ZvZ
v wxr
Rwxf
vwxr
Zr ratio
wof
af the
af fwR
wratio
r
w under
longitudinal
slip
the
rear
tire
is, conditions
longitudinal
of
the
rear
tire
is,conditions
braking
(23)
SS arv wxf
under
braking
conditions
(23)
SSaraf
vslip
braking
conditions
(22) (22)
under
braking
wxfunder
afv
v
v wxf
vwxr
wxr
wxf
Zr R
v wxrv wxr
Zwr R w
under
braking
conditions
S arthe
braking
(23)
S arlongitudinal
the longitudinal
slip
ratio
ofunder
the
ratio
rear
of the
tire
rear
is,conditions
tireof
is,the rear and front tires, respectively and(23)
Ȧ
Ȧslip
angular
velocities
R w , is the
where,
v wxr
Ȧ
Ȧ
are
angular
velocities
rear
where,
f are
vand
r and
frslip
the longitudinal
ofratio
the
rear
tireof
is,the
wxrratio
the
longitudinal
slip
of the
rear
tire
is, and front tires, respectively and R w , is the
ω yaw
and
ωyaw
are
angular
the
and
front
F yij tires,
The
motion
is also velocities
dependent
onofthe
tire Frear
forces
as well
wheel
radius.
motion
is
also
dependent
on the tire
forces
as well
on as on
wheelwhere,
radius.
f
xij yijand
xij andF F
Z vrrwxr
v wxr The
Rw Z
r Rw
under
braking
under
braking
conditions
conditions
(23)
Swhere,
S
respectively
and
r
,
is
the
wheel
radius.
The
yaw
motion
is
also
Ȧ
and
Ȧ
are
angular
velocities
of
the
rear
and
front
tires,
respectively
and
R
,
is
the
and
are
angular
velocities
of
the
rear
and
front
tires,
respectively
and
R
,
is the (23)
Z
v
R
rarȦ
f Ȧ
ar where,
Z
v
R
r
f
w
w
wxr
r
w
w
M
wxr
r
w
acting
on
each
tire:
the
self-aligning
moments,
denoted
by
M zij acting
the self-aligning
denoted
zijon each tire:
v wxr
under
braking
conditions
(23) (23)
S ar
underby
braking
conditions
S arv wxr moments,
motion
also
dependent
theas
tirewell
forces
as well as on
wheel
radius.
andonFon
asFonand
theFself-aligning
dependent
on
the
tire isforces
Fdependent
yaw
motion
is also
wheel
radius.
v wxrTheThe
vyaw
xij
yij the tire forcesxijF xij andyijF yij as well as on
wxr
moments,
denoted
by
M
acting
on
each
tire:
M
..
acting
on
each
tire:
the
self-aligning
moments,
denoted
by
M
..
acting
on
each
tire:
the
self-aligning
moments,
denoted
by
zij rear
zijof
w angular
w zijvelocities
wwthe
wand
w tires, respectively
and
Ȧf rare
and
Ȧf w
are angular
velocities
of
andrear
front
front
respectively
and R ,and
is the
R w , is the
where,1 Ȧwhere,
w 1Ȧ
w the
w tires,
> rfcos
rr r and
F
Gxrlrear
Fand
FF
G and Rwwand
cos FGxfr
front
sinrespectively
>Ȧ
rwhere, where,
F xfl
Gangular
FGangular
Fxrlxrr
GF yfl
f cos
cos
of the
and
sin
xfl
xrr
Ȧ
are
velocities
of Fthe
tires,
, is the
xfr
yfl
Ȧ
and
Ȧ
are
velocities
rear
front
tires,
respectively
R w , is the
2
2
2
2
2
2 JThe
2
2
2
2
F
F
F
F
yaw
The
motion
yaw
is
motion
also
dependent
is
also
dependent
on
the
tire
on
forces
the
tire
forces
and
and
as
well
as
on
well
as on
wheelJ zradius.
wheel
radius.
z
xij
xijyij
yij
.. .. radius.
F
F
The
yaw
motion
is
also
dependent
on
the
tire
forces
and
as
well
as
on as on
wheel
F
F
The
yaw
motion
is
also
dependent
on
the
tire
forces
and
as
well
wheel
radius.
1 w1 w ww
w w
w w
w w
w w
xij
xij yij
yij
Glsin
GMlGlzijFby
F xrlM
Glcos
Gdenoted
on
G sin
acting
acting
each
tire:
tire:
the rself-aligning
moments,
bycos
r the>self-aligning
Fcos
G F
F xrr
each
xflFcos
xrr
yflFsin
xfl
xfr
xrl
(24)
F
G F2xfr
llrFrFcos
lzijfon
Gyfr
F xfl
Fcos
F >yfrFmoments,
sin
G yfrcos
Fdenoted
Fyrr
lcos
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Gyflyfrsin
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yrl
r2
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r F2
yrl
yrr
f by
f on
f FG
xfl l f G
Jthe
2F
2 each
2yflM
2each
2l f Ftire:
tire:
the self-aligning
denoted
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acting
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denoted
z 2Jself-aligning
z 2 2 moments,
zij acting
2
zij on
w wsin
@ F coswcos
lFf sin
F
sin
M zfr
M zrl
lM
MFzrr
@GMGzrr
G xfrsin
M
M
zfrl zrl
..
f F1
xfrF yfr
GM
w
l rFl yrr
cos
lw
G G (24)(24)
GlzflrGFlyrl
l yfr
l xflFsin sin
1 .. l w
wyfr
wzfl
f F
yfl
fwF
r F
yrl
r Fwyrr
f ww
yfl cos
f
yfr G Gl fF
r ..
G
F
cos
F
G
cos
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G
cos
F
G
F
F
F
F
GF yfl
sinf G xfl
1 >r.. w2F1 xfl2> cos
w
w
w
w
xfl
xfr
xfr
xrl
xrl
xrr
xrryflsin
w
w
w
w
w
2sinFGsin
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2 F
2 FGyfl sin G r J z r> 2 lJ FzFlxfl
G cos
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22FMxrl
>f Fcos
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GxflGcos
M
zfl
zfrxfr
zrl
zrr zrr
zfl 2
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zrl 2 2
2 the
Jf z Jxfr
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is2xfrthe moment
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around
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z-axis.
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pitch motion
where,J zJwhere,
is 2the
moment
of2 inertia
z-axis.
rollThe
androll
depend depend
very very
zw
zw
F yfr sin
F
G yfr
F yrl
G llrrFFyrl
lf F
G yfl lcos
F Gyfr cos
l f FG yfr lcos
F xfl
G sin
l f GF xfl
l sin
l lr fFFyrryfl cos
sin G(24)
(24)
w
r
yrr
f
f
w
on
the
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vehicle’s
body
much onmuch
the
accelerations.
the
2 Flongitudinal
2 FGlongitudinal
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l r and
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Fyrlyrr
Faccelerations.
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l f only
FGSince
Gvehicle’s
lthe
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sin
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(24)
l r lFf yrr
l f FSince
lcos
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body
l f Fdepend
Gvery
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(24)
yfr
yflz-axis.
yfr
f F
yfr
f
yfr
J
where,
is
the
moment
of
inertia
around
the
The
roll
and
pitch
motion
J z is pitch,
where,
the moment
of inertia
around
the z-axis.
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roll
andconsidered
pitch
very the
2z and
roll
the
mass,
denoted
by
has
to be
in depend
determining
roll and
mass,
denoted
tozrr
in motion
determining
the
@ considered
lpitch,
sin
l2f GFsprung
M
sinzflGsprung
M
M zfr
M
M zrl
M
Mmzrr
@ Mm
s be
s has
f F xfr the
xfr
zfl
zfr by
zrl
much
on
the
longitudinal
lateral
accelerations.
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only
thethe
vehicle’s
body
undergoes
@Mthe
l f on
F xfrJ
sin
sin
Mmoment
M
M
M
Maround
much
the
and
lateral
accelerations.
Since
vehicle’s
body
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@ only
FGthe
Gand
pitch
M
of
M
zfl
zfr
zrl
zrr as
is
inertia
z-axis.
The roll
and
pitch
f longitudinal
zfl
zfr
zrl
zrr the
effects
ofzlhandling
on
and
roll
motions
following:
effectswhere,
of
handling
onxfr
pitch
and
roll
motions
as the
following:
rollmotion
and
pitch,
the
sprung
mass,
denoted
by
has
to
be
considered
in
determining
thethe
m
roll and pitch,
the
sprung
mass,
denoted
by
has
to
be
considered
in
determining
m
s
s
depend very much on the longitudinal and lateral accelerations.
J
J
where,
where,
is
the
moment
is
the
moment
of
inertia
of
around
inertia
the
around
z-axis.
the
The
z-axis.
roll
The
and
roll
pitch
and
motion
pitch
depend
motion
depend
very
very
z
z
effects
handling
on on
pitch
andand
roll
motions
asz-axis.
the
following:
. roll
effects
of
handling
pitch
motions
asthe
the
following:
.inertia
J zof
where,
is
of
around
the
The
roll
and
pitch
motionthe
depend
very very
Since
only
the
vehicle’s
body
undergoes
roll
and
pitch,
sprung
iscagc
the
z-axis.
The
roll
and
motion
depend
.. the
.. on
kEIMof
JMmzmoment
Mkmoment
inertia
M
lateral
EM around
mthe
m
where,
on
M lateral
much
much
and
accelerations.
accelerations.
Since
only
Since
theonly
vehicle’s
the pitch
vehicle’s
body
undergoes
body
undergoes
sthe
sgc
Imand
s ca y longitudinal
s y longitudinal
M denoted
M mass,
much
on
the
longitudinal
accelerations.
only
bodyineffects
undergoes
has
tolateral
bedenoted
considered
determining
the
of (25)
byJand
m
much
on
the
longitudinal
accelerations.
Since
only
theinvehicle’s
body(25)
undergoes
. lateral
. and
roll
and
roll
pitch,
and
the
pitch,
mass,
sprung
denoted
mass,
by
bySince
tos in
be
has
considered
tothe
bevehicle’s
considered
determining
determining
the
the
m s has
m
Jsprung
sx s
sx gcthe
.. .. m
ca
m
k
M
M
E
m
ca
m
gc
k
M
M
E
Mdenoted
sand
ys the
s the
roll and
pitch,
mass,
by mas
has
be considered
in determining
the
I
M motions
y sprung
s I sprung
roll
pitch,
mass,
denoted
has
to be considered
in determining
mto
and
roll
the
s by
s following:
M of
(25)
Mhandling
(25) the
effectshandling
effects
of on
handling
onpitch
pitch
on
and
pitch
roll
and
motions
roll
motions
as
the
following:
as
the
following:
J sx onand
effects effects
of handling
onJ sxpitch
rolland
motions
as the following:
. roll motions
of handling
as the following:
.pitch
..
..
Tmsmcas gc
kTT . T E T m s ca
y TkTmsgc
. T E
T.. y
T.. m ca
.
m
m
ca
gc
k
m
gc
k. . M E M
M
M
s y
y sy I s J sy E
.. M
sm
Tys sJm
kTTkIIEMMTEMT. ETEMM mca
T
M
m M
ca
gck M .. .. m
Tgc
s gc
gc
m
ca
Mksm
I
s..cas y sym
s
y
s
T
T M
J sx
J
M
J J sx
J sx
(26) (26)
(25)
(25)
(26)
(26) (25)
(25)
sy syJ sx
.
.
.. y m s ca
Tmsmca
T . kETT T. ET s gc
y TkTmsgc
T .. mTs..ca ymTsca
kmT sgc
TkTETT E T m sygc T
J
J
T
sy
sy
T
J sy
J sy
..
32
32
(26)
(26)
(26)
32 32 (26)
where,
c is themass
height
of the
center
of32gravity
where, c is the height of
the sprung
center
ofsprung
gravitymass
to the
ground,
32 to the
32and 32
,
,
k
and
E
are
the
damping
stiffness
acceleration
and
k
E
T are the damping
is the gravitational acceleration and M M T
is the gravitational
where, c where,
is the height
c is the
of height
the sprung
of themass
sprung
center
mass
of center
gravityoftogravity
the ground,
to thegground,
g is the gravitational
respectively.
The
moments
of
inertia
of
the
sprung
mass ar
and
stiffness
constant
for
roll
and
pitch,
respectively.
The
moments
ofand pitch,
, kTkand
,
E
,
k
are
and
the
E
damping
are
the
damping
and
stiffness
and
stiffness
constant
constant
for
roll
and
for pitch,
roll
acceleration
acceleration
and kM , E Mand
E
T
M
MT T
inertia
of the
sprung
x-axes
and
y-axes
by
jsxy-axes are
Jsprung
denoted
by
sy respectively.
respectively.
respectively.
The
moments
The
moments
of mass
inertia
ofaround
ofinertia
the Jsprung
ofand
the mass
around
mass xare
around
-axesdenoted
and
x-axes
y-axes
andare
sx
and
j
respectively.
Jsysx and
J sxrespectively.
denoted by
denoted
byJ sy
and J sy respectively.
2.4
2.4
Braking and Throttling Torques
2.4
andThrottling
Throttling
Torques
Braking
2.4 Braking
Braking
and Throttling
and
Torques Torques
For
the front
andsum
rearof
wheels,
the sum
of thethe
torque
For the front and rear
wheels,
the
the torque
about
axisabout the axi
For the front
For and
the front
rear wheels,
and rearthe
wheels,
sum of
thethe
sum
torque
of the
about
torque
theabout
axis shown
the axis
inshown
FIGURE
in FIGURE
3 are as 3 are as
follows:
shown
in FIGURE 3 are
as follows:
follows:
follows:
Fxf RZ TFbfxf RTZ af TbfI ZZTf af
I Z Z f Fxf RZ Tbf Taf
I Z Z f
(27)
(27)
Fxr RZ TFbrxrRTZar TbrI ZZTr ar
I Z Z r Fxr RZ Tbr Tar
I Z Z r
(28)
(28)
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
39
Zr f are
Z rangular
where Zwhere
andthe
are thevelocity
angular of
velocity
the front
of the
andfront
rear and
wheels,
rear Iwheels,
ofinertia of
f and Z
Z is the Iinertia
Z is the
where Z f and Z r are the angular velocity of the front and rea
For and
the front
and rearthe
wheels,
sum
of the
torque
about
the
shown
inrear
FIGURE
are
as
rques For the front
rear wheels,
sumFor
ofthethe
torque
about
axis shown
FIGURE
3 are
as 3the
For
theinfront
and
wheels,
theaxis
sum
of thein
the
front
and
rearthe
wheels,
the axis
sum
of
the
torque
about
shown
follows:
follows:
For the front and rear wheels, the sum
of the torque about the axis follows:
shown in FIGURE 3 are as
follows:
Journal
Mechanical
Engineering
and 3Technology
of the
torque
aboutofthe
in FIGURE
are as
follows:
emaxis
shown
in FIGURE
3axis
are shown
as
Fxf RZ TFbfxf RTZaf TbfI ZZTf af I Z Z f
Fxf RZ Tbf Taf (27)
I Z Z f (27)
Fxf RZ Tbf Taf I Z Z f
(27)
Fxf RZ Tbf Taf I Z Z f
(27)
(28)
T
R (27)
TbrI Z Z
I Z Z r
Fxr RZ TF
brxr TZar
r ar
F R T T (28)
I Z
F R T T
I Z
Z
xr
br
ar
Z
r
xr
Z
br
ar
Z
r
(28)
(28)
(28)
I
Z
Z
and
are
the
angular
velocity
of
the
front
and
rear
wheels,
is
the
ofvelocity
I
Z
where Z where
and
are
the
angular
velocity
of
the
front
and
rear
wheels,
is
the
inertia
ofinertia
Z front
f
ZZf and
Z f and
Z r areofthe
wherevelocity
the
angular
ofZwheels,
the
and
rear
wheels, o
r are
where
ωfr fand ωrr are thewhere
angular
velocity
theangular
front
and
rear
I
Z
Z
where
and
are
the
angular
velocity
of
the
front
and
rear
wheels,
is
the
inertia
of
R
T
T
f
Z
r
the
wheel
about
the
is wheel
the
wheel
and
arewheel
thebraking
applied
braking
RZ axle,
Tbrbfare
T radius,
the
about
the
axle,
is the
radius,
and
applied
torques,
Z
br
Rtorques,
RZthe
T
theis
wheel
about
the
axle,
rad
the
about
theaxle,
axle,
is
the
radius,
are wheel
the applie
I Z iswheel
velocity
of wheel
the Ifront
and
rear
wheels,
inertia
of bfthe
ZTis
bf and
br the
isthe
theinertia
inertia
wheel
about
the wheel
radius,
rear wheels,
of of the
Z is
R
T
T
the
wheel
about
the
axle,
is
the
wheel
radius,
and
are
the
applied
braking
torques,
T
T
Z
bf
brand
and
are
the
applied
throttling
torques
for
the
front
and
rear
wheels
and
T
T
and
are
the
applied
throttling
torques
for
the
front
rear
wheels
and
af
ar
Tar are
Taf andtorques
thethe
applied
torq
and
are the applied
throttling
for
the
frontthrottling
and rear whe
and T
applied
braking
and
are
wheel
radius,afTbf braking
andarTbrtorques,
areare
the the
applied
braking
torques,
af and Tartorques,
r are the applied
are the applied
throttling
torques
for the
front
andwheels
rear wheels
and Tapplied
af and Tarthrottling
torques
for
the
front
and
rear
ttling
for the front and rear wheels
nt
and torques
rear wheels
I Z Z r
Fxr RZ Tbr Tar
3: Free
body of
diagram
FIGUREFIGURE
3: Free body
diagram
a wheelof a wheel
FIGURE
3: Free bo
of a wheel
FIGURE 3: Free body diagram ofFIGURE
a wheel 3: Free body diagram
FIGURE 3: Free body diagram of a wheel
3:
Free body diagram of a wheel
a wheel
2.52.5 Simplified
Simplified
Tire
Model
Tire
2.5 Simplified
Calspan Calspan
TireCalspan
Model
Simplified Calspan Tire Model
2.5Model
Simplified
Calspan Tire2.5
Model
2.5 Simplified Calspan Tire Model
The
tire
this
study is
Calspan
model
as
described
in. et al.
The
tiremodel
modelconsidered
considered
ininthis
Calspan
described
in et
Szostak
el
al
The tire
model
considered
in this study
is study
Calspan
model
asmodel
described
in
Szostak
tire
model
considered
in this
study is
The
tire
modelis considered
inThe
thisas
study
is Calspan
model
as described
Szostak
et.al.
(1988).
Calspan
model
is
able
to
describe
the
behavior
of
(1988).
Calspan
model
is
able
to
describe
the
behavior
of
a
vehicle
in
any
driving
scenario
(1988).
Calspan
is ableintothis
describe
behavior
of a isvehicle
anyCalspan
driving
scenario
etisa
alable
.of
The tire
modelmodel
considered
study the
isCalspan
Calspan
model
as
described
in Szostak
(1988).
describe
th
(1988).
model
able
toindescribe
the model
behavior
a to
vehicle
in an
including
inclement
driving
conditions
which
require
severe
steering,
braking,
et
al
. may
is(1988).
Calspan
model
as
inscenario
Szostak
including
inclement
driving
which
require
severe
steering,
braking,
vehicle
inmodel
any
including
inclement
driving
conditions
Calspan
is
to
describe
the
behavior
ofdriving
amay
vehicle
in any
driving
scenario
et described
aldriving
. ableconditions
lstudy
as described
in
Szostak
including
inclement
driving
conditions
including
inclement
conditions
which
may
require
severe w
accelerating,
and
other
driving
related
operations
(Kadir
etsevere
al.,The
2008).
Theother
longitudinal
and
the
ofmay
ascenario
vehicle
in
any
driving
scenario
accelerating,
and
other
driving
related
operations
(Kadir
etrequire
al
., accelerating,
2008).
longitudinal
and
which
require
severe
steering,
braking,
and
other
including
inclement
driving
conditions
which
may
steering,
braking,
aescribe
vehicle
inbehavior
any
driving
accelerating,
and
driving
opera
accelerating,
and
other
driving
related
operations
(Kadir
et alrelated
., 2008).
Th
lateral
forces
generated
by
a
tire
are
functions
of
the
slip
angle
and
longitudinal
slip
of
the
ditions
which
may
require
severe
steering,
braking,
lateral
forces
generated
by
a
tire
are
functions
of
the
slip
angle
and
longitudinal
slip
of
the
tire
accelerating,
andrelated
other driving
related
operations
(Kadir2008).
etby
al.,a tire
2008).
The
longitudinal
andaangle
quire severe
steering,
braking,
lateral
forces
generated
by
tire tire
are
lateral
forces et.al.,
generated
arelongitudinal
functions
of theand
slip
andfunctions
longitud
driving
operations
(Kadir
The
allead
. (1988)
a
road.
The
previous
theoretical
developments
inet
Szostak
ted
(Kadir
et to
al.,the
2008).
longitudinal
and
alto. (1988)
totire
a leadintoSzostak
relative
torelative
the road.
The
previous
theoretical
developments
in previous
Szostak
lateral
forces
generated
by
aThe
tire are
functions
of
theroad.
slip
angle
andof
longitudinal
slip
of
the
al.,operations
2008).
The
longitudinal
and
relative
theet
road.
The
previous
theoretical
et a
relative
to
the
The
theoretical
developments
lateral
forces
generated
by
a the
tire
are
functions
slip
angle
and
complex,
highly
non-linear
composite
as
a of
function
of the
composite
It
istoconvenient
functions
of the
slip
angle
and
longitudinal
slip
of
tire
complex,
highly
non-linear
composite
force
asdevelopments
aforce
function
composite
slip.
is slip.
convenient
et alforce
.It(1988)
a composite
relative
to
the
road.
The
previous
theoretical
in
Szostak
le
and longitudinal
slip
of
the
tire
complex,
highly
non-linear
forcesl
complex,
highly
non-linear
composite
as
alead
function
of composite
longitudinal
slip
of the
tire
the road.
The
previous
theoretical
definelead
aSzostak
saturation
f(ı),
obtain
a composite
force
any
and
alfunction,
. (1988)
lead as
to
ato
heoretical
developments
innon-linear
tocomplex,
define
a(1988)
saturation
function,
f(ı),
to relative
obtain
composite
force
with
any
normal
load
and loadforce
highly
force
function
of
composite
slip.
It
is
convenient
et al.to
to
a etcomposite
in Szostak
to define
aobtain
saturation
function,
f(ı), with
to obta
to
define
aa asaturation
function,
f(ı),
towith
anormal
composite
an
developments
in
Szostak
et.al.
(1988)
lead
to
a
complex,
highly
noncoefficient
of
friction
values
(Singh
et
al
.,
2002).
The
polynomial
expression
of
the
saturation
ite
force
as
a
function
of
composite
slip.
It
is
convenient
coefficient
of
friction
values
(Singh
et
al
.,
2002).
The
polynomial
expression
of
the
saturation
to define
function, f(ı),coefficient
to obtain of
a composite
force(Singh
with any
load
and
f composite
slip. Ita issaturation
convenient
coefficient
of2002).
friction
values
(Singh et
al., 20
friction values
et al.,normal
The
polynomial
expressio
is presented
by:normal
,force
to obtain
aany
composite
force
withforce
any
load
function
isfunction
presented
by:
linear
asfunction
of composite
slip. Itisispresented
coefficient
of composite
friction
values
(Singh
etaalfunction
., 2002).
The
polynomial
expression
ofconvenient
the saturation
with
normal
load
and
function
by:
isand
presented
by:
et
al
.,
2002).
The
polynomial
expression
of
the
saturation
to of
define
a saturation
function, f(σ), to obtain a composite force with
function
is
presented
by:
omial expression
the saturation
33
33
any normal load and coefficient of friction values (Singh et.al., 2002).
33
The polynomial
expression of the 33
saturation function is presented by:
33
f (V )
C1V 3F C2VC21V3( 4 C)2VV 2 ( 4 )V
Fc
S
S
f (V ) 3 c
PFz C1V PFzC2VC2 V3C4VC V12 C V 1
1
2
(29)
(29)
4
, C2, C3Cand
constant
fixed to the
specific
Thetires.
tire’sThe
contact
where, C1where,
, 4Care
and
C are parameters
constant parameters
fixed
to thetires.
specific
tire’s contact
1, C2C
where,
Ccalculated
, C2are
, C3calculated
and4C4 using
are constant
parameters
fixed to the specific
1
3 using the
patch lengths
following
two equations:
patchare
lengths
the following
two equations:
tires. The tire’s contact patch lengths are calculated using the following
two equations:
0 . 0768
0F. 0768
F z F ZT
z F ZT
(30)
ap
40 ap
0
ap 0
Tw T
p
T5 w T
p
5
§
K a F§x · K a F x ·
¨¨ 1 ap ISSN:
¨ 1 ¸2180-1053
¸
F z ¨© ¸¹
F z ¸¹
©
Vol. 2
No. 2
July-December 2010
(31)
(30)
(31)
C1VP3Fz C2VC12V3C4C
V 2V12 C4V 1
PFz
C V 3 C VC2 V 3( 4 C)VV 2 ( 4 )V
3Moment
3 2Rejection
1 Pitch
2CCV123V
222 Fc and
Adaptive
Pid Control
With
Vehicle
Motion
InThe
Longitudinal
Direction
4constant
44 Reducing
c and
C3F
V
are
C
4S2(V
Vparameters
)V((For
)S
Cf1where,
, V4C
are
constant
fixedUnwanted
to the
fixed
specific
to the
tires.
specific
tires.
tire’sThe
contact
tire’s contact
where,
3 f1(,FVC
1VC
2 4
F
(,VC)2, CFC
)2C
C
(C
)V (29)
(29)
)VVparameters
c13 CC
21V
2 3 2S S 2 SS
cf ( Vc) 1F3P
f)( VP)F
(29)
(29)(29)
cF
f
(
V
(29)
C
V
C
V
C
V
C
V
C
V
1
C
V
1
f (calculated
) 1are
zVPF
z3using
3 the
2 4 using
2
patch lengths
patchare
lengths
calculated
following
the
following
two
equations:
two
equations:
2
1
2
4
P
F
3
2
C zV CCV3V C
CV V
1 V 1
2 C
z
PFz
C1PVFz1 CC21V1V2 CC422VV4 1 C44V 1
2
330and
2 4 3are
0768
F ZT
C1where,
,C
C0and
,0768
C2C2C
C3are
constant
parameters
fixed
fixed
to the
tires.
specific
The
tire’s
contact
tire’s
contact
where,
where,
2, C
4V
zF
ZT
zconstant
Cparameters
( 4parameters
F.C
VCC
(4Fconstant
2)Vparameters
F
C
C
,.ap
C
, ,1C
C
and
constant
fixedto
fixed
tothe
thespecific
to
specific
the
specific
tires.
tires.
Thetires.
tire’s
The The
tire’s
contact
contact
where,
(30)
(30)
23F,c11
3 and
41Vare
2C
ap
S )Vfixed
c4 are
Sparameters
C
,( VC
C10)2,are
,C
are
constant
to
the
specific
tires.
The
tire’s
where,
C
,fand
C
and
C
are
constant
parameters
fixed
to the specific
tires.
Thecontact
tire’s
where,
1fpatch
4
( VT
13calculated
2,)0C
3
4
(29)
(29) contact
patch
lengths
lengths
are
calculated
using
the
using
following
the
following
two
equations:
two
equations:
TP3calculated
5following
ware
pF T5using
w2T3p using
patchpatch
lengths
lengths
are
calculated
the
the
following
two
equations:
two
equations:
2
P
F
C
V
C
V
C
V
1
z
C
V
C
V
C
V
1
1
2
4
z
patch lengths
are
calculated
using
the
following
two
equations:
patch lengths are
using
the following two equations:
1 calculated
2
4
0 . 0768
. z0768
F
F
0K. 0768
F ZT
FF
z FFZT
z Fz ZT ZT
§C
§0x30.Fand
·0768
(30)
(30)
K
(30)
ap
ap
a, F
aZT4 are
x ·
ap
01
.ap
F
FC
,0768
C
C
parameters
fixed
to the tires.
specific
tire’s
contact
. 0768
F
Fconstant
0 02C
,apC02ap
,C
and
constant
parameters
fixed to the
specific
Thetires.
tire’sThe
contact
where, Cwhere,
zT
¨¨013ap
¸
1ap
4¨0are
1
(31)
(31)
z¸5
ZT
T
T
5
T
5
(30)
T
T
T
T
5
(30)
wp ¸ p w wF p p ¸
0
0 wF ¨
patchare
lengths
two equations:
zT© pcalculated
zp the
T w are
© calculated
¹Tusing
patch lengths
two equations:
the following
¹using
5following
w5T
§ KaK
§§xa 0F·.xK
·a aFF
§ ap
· TF patch,
F
K
x x· F
is ap
the
is
contact
width,
treadand
width,
Tp isand
the Ttire
the tire (31)
pressure.
The(31) (30)
The
Where apWhere
. 10768
¨¨ap
¨¨contact
ap
w is the
p ispressure.
¸¸ z wZTis the Ttread
z ¸F patch,
·FK0768
ap
1 0tire’s
(31)
(31)
K0 a§the
F¨¨1F1x¸¸tire’s
(30)
¸aFFZTx ·¸¸contact
ap
ap 0 ¨¨§¨ap
Where
is
the
tire’s
patch,
Twpatch
is the
tread
width,
andand
T(31)
is (31)
F
F
¸
© F
ap
¨
¸
z¹the
ztire’s
©
¹
¹
ap
1
T
T
5
z
z
©
¹
p
TZT
T
5
values ofvalues
FZT ©and
K
are
and
K
are
the
contact
tire’s
patch
contact
constants.
constants.
The
lateral
The
and
lateral
longitudinal
longitudinal
w
p
¨ 1of
¸
w
p
¨
¸
ĮF
FĮ values
z
¹ The
¹ K , respectively)
Kαa of
arethethe
tire’s
contact
the
tire© pressure.
of are
FZTaand
stiffness
coefficient
stiffness
coefficient
(K©s and K
(cK,zsrespectively)
and
function
are
function
tireof
contact
the tire
patch’s
contactpatch
length
patch’s length
c
ap
is
ap
the
is
tire’s
the
tire’s
contact
contact
patch,
patch,
T
is
T
the
is
tread
the
tread
width,
width,
and
T
and
is
T
the
is
tire
the
pressure.
tire and
pressure.
The
Where
Where
w
w
p
p
constants.
The
lateral
and
longitudinal
stiffness
coefficient
(K
Kc, The The
apWhere
isload
the§ of
ap
tire’s
is
the
contact
tire’s
patch,
contact
T
patch,
is
the
T
tread
is
the
width,
tread
and
width,
T
is
and
the
tire
T
is
pressure.
the
tire
pressure.
The
Where
and normal
and
normal
the
load
tire
of
as
the
expressed
tire
as
expressed
as
follows:
as
follows:
w
w
p
p
§
·
K
F
s
· contact
Kthe
Ftire’s
apatch,
x ¸ T
a contact
x
ap
is
the
tire’s
is
the
tread
width,
and
T
is
the
tire
pressure.
The
Where
ap
is
patch,
T
is
the
tread
width,
and
T
is
the
tire
pressure.
The
Where
w
p
¨
ap
1
(31)
w
p
¨
¸
values
values
of
of
F
and
F
K
and
are
K
the
are
tire’s
the
tire’s
contact
contact
patch
patch
constants.
constants.
The
lateral
The
lateral
and
longitudinal
and
longitudinal
ap
1
(31)
ZT
Į¨ a
Į tire’s
function
of
thepatch
tire contact
length
and
normal
values respectively)
ofvalues
FZT and
FZTĮZTFare
are
and
the
contact
tire’s
contact
constants.
patch patch’s
constants.
The lateral
Theand
lateral
longitudinal
and longitudinal
¨ of K
¸K
ĮFare ¸the
z, respectively)
© s and
¹ c,contact
z
© coefficient
¹
stiffness
stiffness
coefficient
(
(
K
K
K
and
K
respectively)
are
a
are
function
a
function
of
the
of
tire
the
contact
tire
contact
patch’s
patch’s
length
length
values
of
F
and
K
are
the
tire’s
patch
constants.
The
lateral
and
longitudinal
values
of
F
and
K
are
the
tire’s
contact
patch
constants.
The
lateral
and
longitudinal
s
c
2
2
ZT
Į
ZT
Į§(cK
stiffnessload
coefficient
stiffness
(§tire
Ks and
Kexpressed
, respectively)
,· respectively)
are
a· function
are a of
function
the tireofcontact
the tirepatch’s
contactlength
patch’s length
s and
of the
as
A1 FK
A1follows:
F
2 coefficient
2 the
z c ¸ as
zas ¸follows:
¨(sK
¨as
and stiffness
normal
and
normal
load
load
tire
expressed
follows:
stiffness
coefficient
,tire
respectively)
are
a function
of the tire
contact
patch’s patch’s
length
K load
Kof
Athe
A21(K
Fexpressed
A
ofof
0expressed
as
as
(32) length
(32)
coefficient
and
K
,expressed
respectively)
are a function
of the
tire contact
s0 and
czsA
1cF
s normal
zas
and
normal
and
of
load
tire
as
the
tire
as
follows:
as
follows:
2the
¨
¨
¸
¸
A
apof
ap 0the
2 as
ap
tire’s
contact
Ttread
is the
treadand
width,
the tire pressure.
The
Where
p is pressure.
ap
is
the
contact
patch,
Tas
isfollows:
the
width,
Tp isand
the Ttire
The
Where
and
normal
load
the
tire
as ©expressed
0tire’s
©is the
¹wpatch,
¹wfollows:
and
normal
load
of
tire
asA2expressed
2tire’s
2 ·contact patch constants. The lateral and longitudinal
§
§
·
values
of
F
and
K
are
the
A 2Fcontact
2 the
values of FZT and §2KZT
are
constants. The lateral and longitudinal
z· A¸1 Fz patch
§AĮ0Ftire’s
AA11FFz1K
A1 F2z¸2 ·¸
K 2 K ¨s 2Į ¨ A022¨ A
zand
(32) (32)
z 2¸, respectively)
are a function
the tirepatch’s
contactlength
patch’s
length
¸· a function
2A1§¨FK¨(zc1K
K s s 222coefficient
K§(sAK0s¨ap
A
A
F
2and
stiffness stiffness
coefficient
,
are
of the tireofcontact
(32)
(32)
s0respectively)
cz ·A¸AF
A
A
F
1
ap
2
2
1/ FZ
z ¸¸ ¹1 z ¹¸ ¸
2
0
0
©
©
(33)
(33)
K
K
F
CS
/
FZ
F
CS
¨
¨
¨
¨
A
A
cload
z 0 ap
z
K csload
A
A
F
(32)
K22sthe
A
A
F
(32)
apof
2
2
2
1
z
and
normal
of
the
tire
as
expressed
as
follows:
0
1
z
and normal
tire
as
expressed
as
follows:
0
0
ap 0 ©¨ ap20 ¨ ©
A ¹¸ A ¸ ¹
ap 0 © ap 0 ©
2
2
2
¹
2
¹
(33) (33)
CS A/ FZ
2 · A F 2 ·
K c 2 K c§ 2 Fz 2CS
F/z FZ
2ap
1 stiffness
AzA0CS
,ap
A012/2,Aof
AF§¨F
and
A1CS/FZ
AzF2 and
stiffness constants
and can be
and
found
can be
in Table
found
2.
in Table
2.
Where theWhere
the
values
z ¸ are constants
1,/FFZ
2A
0,CS
(33)
(33)
KKvalues
K
F
FZ
¨
¸ areCS/FZ
2of
cs
c
z
K
A
A
0
(32)
2
(32)
2 s 0
21 z 0
1 z 0 ap / 2F
1 ¨zCS / FZ ¸
s
¸
2 F
(33) (33)
K
¨slip
Abecomes:
Then, theThen,
composite
the
calculation
c ap
z CS
Abecomes:
z slip calculation
2 ¹
0K2ccomposite
0 FZ
ap
2 ¹
0 © apap20 ©
ap
0
0
and CS/FZ
are stiffness
are stiffness
constants
constants
and can
andbecan
found
be found
in Table
in Table
2. 2.
Where
Where
the values
the values
of A0,ofA1A, 0A, 2Aand
1, A2CS/FZ
A0, slip
A12, of
A
and
ACS/FZ
and
arebecomes:
CS/FZ
stiffness
stiffness
constants
and can beand
found
can in
be Table
found2.in Table 2.
Where
theThen,
Where
values
the
ofcomposite
2 areconstants
2
Then,
the
composite
the
calculation
2A
0,calculation
1, A2becomes:
2values
2slip
2
apvalues
ap
Sthe
Sof
s stiffness
· CS/FZ
§ sstiffness
·are stiffness
Where
ofF20slip
, CS/FZ
A
,A
and
and
can
2/ 2FZ
22§2are
A ,CS
A
constants
and canconstants
be found
Table
Wherethe
the
,zand
A
A
CS/FZ
constants
and
can beinfound
in2.Table(34)
2.
Where
the
(33)
CS
12,/ A
KVvalues
K
F
FZ
(33)
2A
1and
Then,
Then,
composite
theof
slip
calculation
c z0K
(34)
K
K
tan
K¸c 2are
D10,calculation
Vvalues
c
¨
¸
2composite
s 2tan
s becomes:
c ¨D becomes:
8ap
F z slip
8ap
s ¹ © 1 s ¹slip calculation becomes:
P 0composite
Pcalculation
Then, the
composite
becomes:
Then,
the
calculation
becomes:
© 1 composite
be
found
Table
2.
the
0in
z Then,
00 Fslip
2
2
Sap 2 Sap 2 2
§ s §· s ·
(34) (34)
K s2 tan
K s22 Dtan 2KDc 2¨ K c 2 2¨¸
V
¸ 2
2
ap
ap
S
S
A
,
A
,
A
and
CS/FZ
stiffness
constants
and
can
be
found
in Table0.3
2.
Where
the
values
of
8
F
8
F
stiffness
ss¹ of
P
P
§
·asand
2 ofand
©s1
©2value
¹21§are
2and
2 are
constants
and
canfor
be
found
in
Table
2.
the
of 0A z, Acoefficient
is a coefficient
nominal
has
has
0.85
valuefor
of
normal
0.85
road
normal
condition,
road
condition,
0.3
µWhere
µVovalues
zA2friction
0212, of
0K
1CS/FZ
2 2friction
2a·
o is a nominal
(34)
(34)
K
tan
K
D
¨
¸
¨
¸
Sap 2V 0SKap
s
s 2 tan 2 D
s
c
c
§
·
s
2
2calculation
2 § 1 ·s
2
8
F
8
F
1
s
P
P
Then,
the
composite
slip
becomes:
(34)saturation
K
tan
K icy
V
Then,
composite
calculation
becomes:
for
wettheroad
for
wet
condition,
road
condition,
and
for
condition.
Given theGiven
polynomial
the polynomial
saturation
(34)
KD
K c¹¸icy
D©¨ road
¨ © road
¸ ¹ condition.
z slip
z 0.1
0V
0 tan
sand
c0.1
s for
has
s ¹ a© has
8 Pcoefficient
F z of
1value
the
sa ¹value
© 1stiffness,
z longitudinal
0F
0 and
afunction,
is a8 Pnominal
coefficient
friction
of friction
and
and
of
0.85
of for
0.85
normal
forlongitudinal
normal
road
road
condition,
0.3
0.3 are
µo is µ
function,
lateral
and
lateral
longitudinal
stiffness,
the
normalized
normalized
lateral
and
lateral
and condition,
longitudinal
forces
areforces
onominal
for wet
for
road
wet
road
condition,
condition,
and
for
0.1
icy
for
road
icy
road
condition.
condition.
Given
Given
theangle
polynomial
the
polynomial
saturation
2value
by
derived
by
the
composite
the0.1
composite
force
into
force
into
the
side
angle
and
longitudinal
and
longitudinal
slip saturation
ratio
is
a coefficient
nominal
coefficient
ofand
of
and
friction
has
athe
and
value
hasof
aslip
0.85
for
ofslip
normal
0.85
for
road
normal
condition,
road
condition,
0.3 slip ratio
0.3
µderived
µo resolving
2 side
2 friction
o is a nominal
2 resolving
ap
Sand
§the snormalized
· 0.85
aplateral
Scoefficient
sthe
§Dhas
· value
2stiffness,
2 has
2
2 oflongitudinal
22stiffness,
a nominal
friction
and
a
of
for
normal
road
condition,
0.3saturation
µo iswet
function,
longitudinal
normalized
lateral
lateral
and
longitudinal
and
longitudinal
forces
forces
are
are0.3
is alateral
nominal
coefficient
of
friction
and
a
value
of
0.85
for
normal
road
condition,
µroad
ofunction,
components:
components:
(34)
K
tan
K
Vand condition,
K
tan
K
D
V
for
for
wet
condition,
road
and
0.1
for
and
icy
0.1
road
for
icy
condition.
road
condition.
Given
the
Given
polynomial
the
polynomial
saturation
¨
¸
(34)
¸c
s
s
c ¨
iswet
a condition,
coefficient
of©friction
has
aslip
value
of
0.85
forforces
normal
8the
1and
the
s condition.
Pand
8nominal
1road
into
sicy
Pand
derived
by
resolving
by
resolving
composite
force
force
the
into
side
slip
side
angle
angle
and
longitudinal
and
longitudinal
slip
ratio
slip
ratio are
zthe
0 Fcomposite
©road
¹normalized
for derived
wetµfor
0.1and
for 0.1
icy
Given
the
polynomial
saturation
¹ condition.
road
condition,
for
Given
the
polynomial
saturation
zlongitudinal
0F
oroad
function,
function,
lateral
lateral
and
longitudinal
stiffness,
stiffness,
the
normalized
the
lateral
and
lateral
longitudinal
and
longitudinal
are
forces
components:
components:
function,
lateral
and
longitudinal
stiffness,
the
normalized
lateral
and
longitudinal
forces
are
road
condition,
0.3
for
wet
road
condition,
and
0.1
for
icy
road
condition.
function,
lateral
and
longitudinal
stiffness,
the
normalized
lateral
and
longitudinal
forces
F yresolving
Ffy resolving
the
K composite
derived byderived
by
the side
into slip
the angle
side slip
andangle
longitudinal
and longitudinal
slip ratio slip are
ratio
tan D intoforce
V Kcomposite
f D
Vthe
s tan
s force
(35)
(35)
Y
J
Y
J
derived
by
resolving
the
composite
force
into
the
side
slip
angle
and
longitudinal
slip condition,
ratio
derived
by
resolving
the
composite
force
into
the
side
slip
angle
and
longitudinal
slip ratio
J
J
is
a
nominal
coefficient
of
friction
and
has
a
value
of
0.85
for
normal
road
0.3
µ
o
is
a
nominal
coefficient
of
friction
and
has
a
value
of
0.85
for
normal
road
condition,
0.3
µ
Given
polynomial
saturation
function,
lateral
and
longitudinal
o
components:
2
2
Pcomponents:
Fz F the
P
F
2 z f 2V Kf 2V
' KD2 2tan D '
2
tan
ycondition,
y tan
components:
tan
KF
D
K
K
S
D
K
S
components:
for wet
road
condition,
and
0.1
for
icy
road
condition.
Given
the
polynomial
saturation
s
s
for wetstiffness,
road
and
0.1
for
icy
road
condition.
Given
the
polynomial
saturation
s
s
c
c
the normalized lateral
longitudinal forces are derived(35)
by (35)
YJ J Yand
JJ
2 2 ' 2 2 stiffness,
PFz and
PF
Flateral
longitudinal
the normalized
lateral
and longitudinal
function,
lateral
longitudinal
the
normalized
lateral
and
longitudinal
forces areforces are
2K and
22 f D
2K
'stiffness,
Ffunction,
tan
tan
fzyKV
V
D
y
resolving
the
composite
force
into
the
side
slip
angle
and
longitudinal
tan
tan
K
D
K
D
S
K
S
s
s
s
c composite
(35)
force
YcJ J intoforce
Ythe
derived
into slip
the side
and longitudinal
F y resolving
Fbyf resolving
Vsthe
K scomposite
derived by
angleslip
andangle
longitudinal
slip (35)
ratioslip ratio
VDthe
ftan
K s tan
D
J J side
Pcomponents:
Fz ratio Pycomponents:
F
(35) (35)
YJ 'J2 2 YJ J
slip
2z
2 ' 2 22 2
'
'
S D
components:
PFFxz
DKK2cs SKKtan
f V2D
K SK2c S
cf'22V
PKFsFz 2x tan
K s tan K
S 2 cK c' S 2
(36)
(36)
c D
s tan
' 2
'
PFz Fx P
F
2 fcS
2f V K
' 2K2c S
'2 2
V
x2 z
KFF
D
K
K
tan
S
D
K
S
Fy
(36)
(36)
tan
f
V
K
D
s yf tan
s
c
c
V K tan D 2s 2
(35)
YJ J
PFz PFz 2 s 2'2
(35)
Y J
tan
K sc D
f '2K
PFF2K
2VD2c' DK2'S c'2KSKc' J'S22S 2
F
Stan
PF
xz
xz fs V 2K
'2 K
tan
K
(36)
(36)
tan
K
D
S
c component
Vhas
K csan
Fx has
Sf Vcadditional
san f additional
Kcomponent
Fforce
Lateral P
force
Lateral
due to the
duetire’s
to the
chamber
tire’s chamber
angle, Ȗ, angle,
which
Ȗ,is which is
cS
Fz
Px 2F
(36)
(36)
2 ' 2 22
'2 2
z 2
K
tan
D
K
K
tan
S
D
K
S
2
modeled
modeled
as
a
linear
as
effect.
a
linear
Under
effect.
significant
Under
significant
maneuvering
maneuvering
conditions
conditions
with
large
with
lateral
large
and
P
F
2
s
s
c
c
P
F
Lateral
Lateral
has
component
due to
duetheto tire’s
the tire’s
chamber
chamber
angle,angle,
Ȗ, which
Ȗ, which
islateral
is and
2 additional
2an additional
zforce force
z2 an
Khas
K c'2 DS 2component
K c' S 2
s tan KD
s tan
modeled
modeled
as a linear
as a linear
effect.effect.
UnderUnder
significant
significant
maneuvering
maneuvering
conditions
conditions
with large
with large
laterallateral
and and
Vcomponent
K c' S component
F
Vhas
K c'an
Fx hasforce
S f additional
Lateral force
Lateral
anx fadditional
due to the
duetire’s
to the
chamber
tire’s chamber
angle, Ȗ, angle,
which(36)
Ȗ34
is, which
is
34
(36)
Lateral Lateral
force
has
aneffect.
additional
component
due to due
the
tire’s
angle,
Ȗangle,
, which
force
has
anUnder
additional
component
to
the chamber
tire’s
chamber
Ȗand
,34is
which
is
2 significant
Pas
F
modeled
as
modeled
maneuvering
conditions
conditions
with
largewith
lateral
large
lateral
34 and
P
Fza linear
2 effect.
'
2 maneuvering
2z a linear
2
' 22 significant
2 Under
K c Ssignificant
K s aeffect.
tan
DK sUnder
Ktan
modeledmodeled
as a linear
maneuvering
conditions
with large
as
linear
effect.
Under
maneuvering
conditions
withlateral
large and
lateral and
c SDsignificant
V
34
34
Lateral
has an additional
component
duetire’s
to the
tire’s chamber
,iswhich
Lateral force
hasforce
an additional
component
due to the
chamber
angle, Ȗ, angle,
whichȖ34
34is
Lateral
force
has Under
an
additional
component
due toconditions
the
chamber
modeled
as effect.
a linear
effect.
Under significant
maneuvering
with
largeand
lateral and
modeled
as a linear
significant
maneuvering
conditions
withtire’s
large
lateral
angle, γ, which is modeled as a linear effect. Under significant
34
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
41
34
Journal of Mechanical Engineering and Technology
maneuvering conditions with large lateral and longitudinal slip, the
force converges to a common sliding friction value. In order to meet
this criterion,
the
longitudinal
stiffness
coefficient
is modified
at high
longitudinal
slip, the slip,
force
converges
to a common
sliding
friction
order In
to order
meet
this
longitudinal
the
force converges
to a common
sliding value.
frictionInvalue.
to meet this
longitudinal
slip,
the
force
converges
to
a
common
sliding
friction
value.
In
order
to
meet
this
slips
to
transition
to
lateral
stiffness
coefficient
as
well
as
the
coefficient
longitudinal
slip,
the
force
converges
to
a
common
sliding
friction
value.
In
totomeet
this
criterion,criterion,
the longitudinal
stiffness stiffness
coefficient
is modified
at high slips
to transition
toorder
lateral
the longitudinal
coefficient
is modified
at high
slips to transition
lateral
criterion,
the
longitudinal
stiffness
coefficient
is
modified
at
high
slips
to
transition
to
lateral
criterion,
the
longitudinal
stiffness
coefficient
is
modified
at
high
slips
to
transition
to
lateral
of
friction
defined
by
the
parameter
K
.
Kµ.
stiffness stiffness
coefficient
as well as
coefficient
of friction
the parameter
Kµ.
coefficient
asthe
well
as the coefficient
of defined
friction by
defined
by the parameter
µ
Kµ.
stiffness coefficient
as well asasthe
coefficient
of frictionofdefined
the parameter
Kµ.
stiffness coefficient
well
as the coefficient
frictionby
defined
by the parameter
2
K c' ' K c K
'K sKc KcK ssin
D S 2 2cos 22 DS 2 2cos 2 2D
(37)
K
2 c sin2 2D K c K c Kcc'K s K
cKc Ksin
Dc SsincosD DS cos D
(37)
K
s
longitudinal slip, the force
converges
to
a
common
sliding
friction
value.
In
order
to
meet
this
2
2
Pcriterion,
P 0 1Pthe
Klongitudinal
sin
S 222 cos
(38)
PP0 1 sin
K2PDstiffness
2 SD 2 2cosis2 2D
modified at high slips to transition to lateral
2Dcoefficient
P
P
1
K
sin
D
S
cos
(38)
P PP 0 1as well
K P as sin
D DS cos
D defined by the parameter Kµ.
stiffness0 coefficient
the coefficient
of friction
2.6
2.6
'
2
2
K ssimulation
Kofc the
ofK the
Dmodel
S 2 cosmodel
D
Description
c
c sin
2.6 KDescription
simulation
2.6Description
simulation
Description
of the simulation
model
2.6
Description
ofof
thethe
simulation
modelmodel
(37)
(37)
(38)
(38)
(37)
2
2
The vehicle
dynamics
model
ismodel
developed
onbased
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equationsequations
from
dynamics
P P 0dynamics
1
K P sin
D Sis2 developed
cos based
D
(38)thefrom the
The vehicle
vehicle
on
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The
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based
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The vehicle
dynamics
model
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equations
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The
vehicle
dynamics
model
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based
on SIMULINK
the on
mathematical
equations
the
previous
vehicle
handling
equations
by
using
MATLAB
software.
The
previous vehicle handling equations by on
using
MATLAB
SIMULINK
software.
equations
from
the
previous
vehicle
handling
equations
by using
previous
vehicle
handling
equations
by
using
MATLAB
SIMULINK
software.
The
previous
vehicle
handling
equations
by
using
MATLAB
SIMULINK
software.
The
relationship
between
handling
model,
ride
model,
tire
model,
slip
angle
and
longitudinal
slip
relationship between handling model, ride model, tire model, slip angle and longitudinal slip
relationship
between
handling
model,
model,
tire
model,
angle
and
longitudinal
slip
relationship
between
model,
ride
model,
tireslip
model,
slip
angle
slip
MATLAB
SIMULINK
The
relationship
handling
are
clearly
described
in
FIGURE
4. software.
In ride
this
model
there
arethere
two
inputs
that
can
be and
used
in
2.6
Description
of the
simulation
model
are
clearly
described
inhandling
FIGURE
4.
In this
model
are twobetween
inputs
that
canlongitudinal
be the
used in the
are
clearly
described
in
FIGURE
4.
In
this
model
there
are
two
inputs
that
can
be
used
in
the
are
clearly
described
in
FIGURE
4.
In
this
model
there
are
two
inputs
that
can
be
used
in
the
model,
tire
model,
slip
angle
andsteering
longitudinal
slip
are
dynamic
analysisride
of themodel,
vehicle
torque
and
steering
input
which
from
thefrom the
dynamic
analysis
of the namely
vehicle
namelyinput
torque
input
and
inputcome
which
come
dynamic
analysis
of
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vehicle
namely
torque
input
and
steering
input
which
come
from
the
dynamic
analysis
of
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namely
torque
input
and
steering
input
which
come
from
the
The
vehicle
dynamics
model
is
developed
based
on
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mathematical
equations
from
the
driver. clearly
Itdriver.
simply
thatFIGURE
the that
model
created
is able there
to able
perform
the inputs
analysis
for
described
in
4. In
this model
are
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simply explains
the
model
created
is
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perform
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analysis
for
driver. previous
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simply
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the
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iscreated
able to
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analysis
for
vehicle
handling
equations
bycreated
using MATLAB
software.
It explains
simply
explains
that
the
model
isSIMULINK
able to perform
theTheanalysis
for
longitudinal
andused
lateral
direction.
longitudinal
and
lateral
can
be
in
the
dynamic
analysis
of
the
vehicle
namely
torque
input
relationship
between
handling
model, ride model, tire model, slip angle and longitudinal slip
longitudinal
and lateral
direction.
longitudinal
and
lateral
direction.
and
steering
input
which 4.come
It simply
that
are clearly
described
in FIGURE
In thisfrom
model the
theredriver.
are two inputs
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analysis
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torque input
andanalysis
steering input
come from the
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model
created
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for which
longitudinal
and
driver. It simply explains that the model created is able to perform the analysis for
lateral
direction.
longitudinal and lateral direction.
FIGURE
4: Full Vehicle
Matlab
FIGURE
4: Full Model
VehicleinModel
inSIMULINK
Matlab SIMULINK
FIGUREFIGURE
4: Full Vehicle
in Model
MatlabinSIMULINK
4: FullModel
Vehicle
Matlab SIMULINK
3.0
System 4:
Design
3.0Active Control
Active
Control
System
Design
FIGURE
Full
Vehicle
Model in
SIMULINK
FIGURE
4: Full
Vehicle
Model
inMatlab
Matlab
SIMULINK
3.0
Active Control
System
Design
3.0
Active
Control
System
Design
3.0
Active Control
System
The controller
structure
consists
of anDesign
inner
loop
controller
to rejecttothereject
unwanted
weight weight
The controller
structure
consists
of an
inner
loop controller
the unwanted
The controller
structure
consists
inner
loop
controller
toresponses
reject
the
unwanted
weight
Theancontroller
structure
consists
oftoan
inner
loop
controller
reject
the
weight
transfer
and
outer
loop
controller
toanstabilize
heave
and
pitch
due
to pitch
torque
transfer
and an
outer
loop of
controller
stabilize
heave
and
pitchto
responses
dueunwanted
to pitch torque
transfer
and
an driver.
outer
loop
controller
toofstabilize
heave
andheave
pitch
responses
due
to
pitch
torque
The
controller
structure
consists
an
inner
loop
controller
to
reject
the
unwanted
weight
transfer
and
an
outer
loop
controller
to
stabilize
and
pitch
responses
due
to
pitch
input
from
the
An
input
decoupling
transformation
is
placed
between
the
inner
and
input from the driver. An input decoupling transformation is placed between the innertorque
and
transfer
and
anthe
outer
loop
controller
to System
stabilize
heave
and pitch
responses
due The
to between
pitch
torque
3.0
Active
Control
Design
input loop
from
theloop
driver.
An
input
decoupling
transformation
isouter
placed
the
inner
and
input
from
driver.
An
input
decoupling
transformation
isbetween
placed
the
innerloop
and
outer
controllers
that
blend
theblend
inner
loop
and
outer
loop
controllers.
outer
loop
outer
controllers
that
the
inner
loop
and
loop
controllers.
The
outer
input
from
thecontrollers
driver.
An input
decoupling
transformation
is outer
placed
between
the inner
and
outer
loop
controllers
that
blend
the
inner
loop
and
outer
loop
controllers.
The
outer
loop
outer
loop
that
blend
the
inner
loop
and
loop
controllers.
The
outer
loop
controller
provides
the ridethat
control
thatinner
isolates
theouter
vehicle’s
body from
vertical
and
controller
provides
the
ride consists
control
that
isolates
the
vehicle’s
body
from
vertical
and
outer
loop controllers
blend
loop
and
loop
controllers.
The
outer
loop
The
controller
of
inner
loop
controller
to
reject
controller
provides
the structure
ride by
control
that
isolates
the
vehicle’s
body
from
vertical
and
controller
provides
the
ridethe
control
thatan
isolates
the
vehicle’s
body
from
vertical the
and
rotational
vibrations
induced
pitch
torque
input.
The
inner
loop
controller
provides
the
rotational
vibrations
induced
by
pitch
torque
input.
The
inner
loop
controller
provides
controller
providesweight
the ridetransfer
control that
isolates
the vehicle’s
body from vertical
and
the
unwanted
and
an
outer
loop
controller
to
stabilize
rotational
vibrations
induced
by
pitch
torque
input.
The
inner
loop
controller
provides
the
rotational
vibrations
induced
by
pitch
torque
input.
The
inner
loop
controller
provides
the
weight transfer
rejection
control
maintains
load-leveling
distribution
during
weight transfer
rejection
control
that
maintains
and
load
distribution
during
rotational
vibrations
induced that
by pitch
torque
input.
Theload-leveling
innerand
loopload
controller
provides
the
weightheave
transfer
rejection
control
that
maintains
load-leveling
and
load
distribution
during
weight
transfer
rejection
control
that
maintains
load-leveling
and
load
distribution
during
and
pitch
responses
due
to
pitch
torque
input
from
the
driver.
An
vehicle
maneuvers.
The
proposed
control
structure
is
shown
in
FIGURE
5.
weight
rejection
that control
maintains
load-leveling
and in
load
distribution
vehicletransfer
maneuvers.
Thecontrol
proposed
structure
is shown
FIGURE
5. during
vehicle vehicle
maneuvers.
The proposed
control
structure
is is
shown
ininshown
FIGURE
5.
vehicle
maneuvers.
proposed
control
structure
in FIGURE
5.
maneuvers.
The The
proposed
control
structure
shownis
FIGURE
5.
42
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
35
3535
35
35
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
input decoupling transformation is placed between the inner and outer
loop controllers that blend the inner loop and outer loop controllers.
The outer loop controller provides the ride control that isolates the
vehicle’s body from vertical and rotational vibrations induced by pitch
torque input. The inner loop controller provides the weight transfer
rejection control that maintains load-leveling and load distribution
during vehicle maneuvers. The proposed control structure is shown in
FIGURE 5.
3.1
FIGURE 5:
5: The
The Proposed
Control
Structure
for Active
Suspension
System
FIGURE
Proposed
Control
Structure
for Active
Suspension
System
Decoupling Transformation
3.1Decoupling
Transformation
The
outputs of the outer loop
controller are vertical forces to stabilize body bounce Fz and
moment to stabilize pitch M T . These forces and moments are then distributed into target
The outputs of the outer loop controller are vertical forces to stabilize
forces of the four pneumatic actuators produced by the outer loop controller. Distribution of
) into
and
moment
toStructure
stabilize
(M
) . These
forces
and
bounce
(FThe
thebody
forcesFIGURE
andFIGURE
moments
target
forces
of
the
four
pneumatic
actuators
is System
performed
z 5:
o Suspension
5:
Proposed
Control
for pitch
Active
Suspension
The
Proposed
Control
Structure
for Active
Systemusing
moments
are then distributed
into outputs
target forces
the fourtransformation
pneumatic
decoupling
transformation
subsystem. The
of the of
decoupling
subsystem
namely
the
target
forces
for outer
the fourloop
pneumatic
actuators Distribution
are then subtracted
3.1
Transformation
3.1Decoupling
Decoupling
Transformation
actuators
produced
by the
controller.
of with
the
theforces
relevantand
outputs
from the into
inner target
loop controller
the pneumatic
ideal target forces
for the
moments
forces to
ofproduce
the four
actuators
four outputs
pneumatic
actuators.
Decoupling
transformation
subsystem
requires
an body
understanding
Fz and
The
of the
loop
controller
are vertical
forces
to stabilize
body
bounce
Fof
The outputs
ofouter
the outer
loop
controller
are vertical
forces
to
stabilize
bounce
z and
issystem
performed
using
decoupling
transformation
subsystem.
The
outputs
the
dynamics
in
the
previous
section.
From
equations
(5),
(6)
and
(7),
equivalent
transformation
forcesforces
and moments
are then
distributed
into target
moment
to stabilize
pitchpitch
M T .MThese
and moments
are then
distributed
into target
to stabilize
T . These
ofmoment
the
decoupling
subsystem
forces
and moments
for heave,
pitch and roll can
be defined as:namely the target forces
forcesforces
of theoffour
actuators
produced
by the
loop loop
controller.
Distribution
of of
the pneumatic
four pneumatic
actuators
produced
byouter
the outer
controller.
Distribution
theand
four
pneumatic
actuators
are
then
subtracted
with
the
relevant
thefor
forces
moments
into target
forces
of the
actuators
is performed
usingusing
the forces
and
moments
into
target
forces
offour
the pneumatic
four pneumatic
actuators
is performed
"
"
"
"
F z transformation
F
Fthe
Fsubsystem.
F prr The
outputs
from
loop
controller
to
the ideal
target
(39)
decoupling
outputs
of produce
the
transformation
decoupling
subsystem.
The
outputs
of decoupling
the decoupling
transformation
pfltransformation
pfr inner
prl subsystem
namely
thefour
target
forcesforces
for the
actuators
are then
subtracted
with with
subsystem
the target
forfour
the pneumatic
four pneumatic
actuators
are
then
subtracted
forces
for namely
the
pneumatic
actuators.
Decoupling
transformation
thesubsystem
relevant
outputs
the"an
inner
loop
controller
produce
the ideal
targettarget
forcesforces
for the
the
theMrelevant
outputs
inner
controller
to
the dynamics
ideal
for the
" from from
" loop
" toof
understanding
theproduce
system
in
Frequires
F pfr lthe
(40)
T
pfl l f Decoupling
f F transformation
prl l r F prr l r subsystem requires an understanding
four pneumatic
actuators.
of of
four pneumatic
actuators.
Decoupling
transformation subsystem requires an understanding
previous
section.
From
equations
(5),
(6)equations
andequations
(7),(5),
equivalent
forces
and
the system
dynamics
in thein previous
section.
FromFrom
(6)
the system
dynamics
the previous
section.
(5),and
(6) (7),
and equivalent
(7), equivalent
moments
and
roll
be
w
wcan
w as:
§ w ·pitch
§ can
· be
§defined
·
" § heave,
"pitch
"and
"bedefined
forces
and
for·¸ heave,
pitch
and roll
forces
andFfor
moments
heave,
as:as:
(41)
M M moments
for
F pfr
Fdefined
¨ ¸ F prl
¨ roll
¸ can
¸
pfl ¨
prr ¨
"
FFz pfl
© 2¹
" "
pfl
F pfr
F
© 2¹
" "
pfr
F prl
F
" "
prl
F prr
F
© 2¹
"
F prr
© 2¹
Fz
(39) (39)
"
"
"
"
are the pneumatic forces which are produced by the outer loop
where F pfl
, F pfr
, F prl
and F prr
controller in the front
left,
right, rear
left
"
"
"
" rear
" front
" and
" right corners, respectively. In the case
M T M T F pfl
lFf pfr
F
(40) (40)
lFf pfl
lF " lFf prl
lF prl
l r prr
lFr prr
l
of the vehicle
input that
comesf pfr
from ther brake
torque, r the roll moment can be neglected.
Equations (39), (40) and (41) can be rearranged in a matrix format as follows:
" § w" · § w ·" § w
" · § w ·" § w
" · § w ·"
M M MF
F¸pfr ¨F pfr
¨F pfl¸ ¨2180-1053
¸ ¨ F
¸prl ¨2F prl¸No.
¨ F2
¸prr
M pflISSN:
Vol.
© 2 ¹© 2 ¹ © 2 ¹© 2 ¹ © 2 ¹© 2 ¹
§w
" ·§ w ·
¨FJuly-December
¸
prr¸ ¨
2010
© 2 ¹© 2 ¹
(41)43(41)
the
dynamics
in the
section.
FromFrom
equations
(5), requires
(6)
and
(7),understanding
equivalent
the
system
dynamics
in previous
the
previous
section.
equations
(5), an
(6)understanding
and
(7), equivalent
four
pneumatic
actuators.
Decoupling
transformation
subsystem
an
foursystem
pneumatic
actuators.
Decoupling
transformation
subsystem
requires
of of
forces
and
moments
forinheave,
pitch
andsection.
roll
can
be
as: as:
forces
and
moments
for
pitch
andsection.
rollFrom
candefined
beequations
defined
the
system
dynamics
inheave,
the
previous
From
equations
(6) and
equivalent
the system
dynamics
the
previous
(5), (5),
(6) and
(7), (7),
equivalent
forces
and
moments
for heave,
and can
roll be
candefined
be defined
forces
and
forEngineering
heave,
pitchpitch
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roll
as: as:
Journal
of moments
Mechanical
and
"
"F " " F " " F "
"
F z FFz pfl
(39) (39)
Fpfl
F pfr prl
F prl prr
F prr
pfr
" "
" "
" "
"
"
Fpfr
Fprl
F prr
F z FF
F prl
F prr
(39) (39)
z pfl FpflF pfr
"
" F " l " F " l " F " l "
M T M T F pfl
lFf pfl
l f pfr
Ffpfr l f prl
Fr prl l r prr
Frprr l r
"
"
" F " l " F " l " F " l
F
M T M T F pfl
l fpfll F
l fpfr F
Fr prr l rprr r
f pfr
f prl l rprl
" § w
" ·§ wF· " § w
" ·§ w F· " § w
" ·§ wF· " § "w ·§
M M M MF pfl
¨F pfl
¸¨ ¸ pfr
¨ pfr
¸¨ ¸ prl
¨ prl
¸¨ ¸ prr
F
F
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w
w
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§
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2
2
w
w
§ " ¹·© 2 ¹ " F
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¨ prl ¸¨ F¸ prr
¨ prr ¸¨
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(40) (40)
(40) (40)
w
w
2
2
·
¸
·¹
¸
¹
(41) (41)
(41) (41)
"
"
"
"" and
"
" the pneumatic forces which are produced by the outer loop
are
where
where
F pfl
, F pfl
and
pfr , F pfr
prl , F prl F
prr F
prr are the pneumatic forces which are produced by the outer loop
"
"
"
"
" F " ,F " ,F
" Fare
are
the pneumatic
forces
which
arerespectively.
produced
byIn
the
outer
where
the
pneumatic
forces
which
arecorners,
produced
by the
outer
loop
where
and
F
,
,
F
and
F
controller
left,
right,
rear
left
rear
right
corners,
theIn
case
pfl
pfr
prl front
prr
controller
the
left,
right,
rearand
left
and
rear
right
respectively.
the loop
case
pflin the
pfrinfront
prl front
prr front
"
"
"
"
of
the
vehicle
input
that
comes
from
the
brake
torque,
the
roll
moment
can
be
neglected.
of theinvehicle
input
that
comes
from
theand
brake
torque,
thecorners,
rollforces
moment
canInbethe
neglected.
controller
infront
the
front
left,
front
right,
rear
left
and
rear
right
respectively.
In
the case
controller
the
left,
front
right,
rear
left
rear
right
corners,
respectively.
case
where
are
the
pneumatic
which
are
F
,
F
,
F
and
F
pfl
pfr
prl
prr
Equations
(40)input
and
(41)
can
be
a matrix
format
as follows:
Equations
(39),
(40)
and
(41)
canrearranged
bethe
rearranged
intorque,
a matrix
format
as follows:
of
the(39),
vehicle
that
comes
from
the in
brake
the
roll
moment
be neglected.
ofproduced
the
vehicle
input
thatouter
comes
from
brake
torque,
the
roll
moment
can can
be
neglected.
by
the
loop
controller
in
the
front
left,
front
right,
rear
Equations
and (41)
canrearranged
be rearranged
in a matrix
format
as follows:
Equations
(39),(39),
(40) (40)
and (41)
can be
in a matrix
format
as follows:
left and rear right corners, respectively. In the case of the vehicle input
that comes from the brake torque, the roll moment can be neglected.
Equations (39), (40) and (41) can be rearranged in a matrix format as
follows:
ª f z t º
«
»
« M T t »
« M (t ) »
¬ M ¼
a linear
ª
ª
«1
ª f z1t º 1«1
«
«
«
»
« L f « M T Lt f » L«r L f
«
« M w(ªt ) » w «ª w
M
ǻ w
««1 f z t ¬ºªf z «t1 ¼º
«¬« 2
»« 2« » 2 «¬ 2
"
"
º ª F pfl º
º ª F pfl º
«
»
«
»
»
»
1 1 « " 1»
1
" »
» F pfr
» « F pfr
»
L r»L «f
L»
Lr » «
" »r
"
" º
" «F
» « F prl
ª F pfl
ª F» pfl
º prl »
º
º
w w
w
« w»
«
»
11 » « 11" »» «« "1 »»» «« » " « »» " »
F
F
»
»
2 2¼ ¬ prr
2 ¼ « pfr 2»» F»¼ pfr¬ F prr ¼
36 36
36 36
(42)
(42)
»
LrL f
LLrr »
L » «
« M T t »« M T« t L» f « LLf f
" r» « " »
«
F
»
«
F
»
«
prl
« M (t ) »« M (wt ) »
ww
w
w » « w »» « prl »
¬ M ¼¬ M« ¼ « w
mxn "
" » mxn
« F2prr
» full
«F
system
equations
if 2C2ƒ22y=Cx,
row rank,
rightrank,
inverse
Forofa linear
system
has then
a fulla row
then
«¬of
2 equations
¼» has
¬« 2 y=Cx,
¬ ifa¼»¼C
¬ prrƒ¼
mxn
(42)
(42)
a right inverse
For
a linear system
of equations y=Cx, if C ∈ ℜ
has a1full row
rank,
T
T 1 1
1
mxm
1mxm
C. C T CC T 1 .
C CC
canright
be computed
using
such as C C C I suchstill
C The
I right
stillinverse
exist. The
inverse can
be Ccomputed
using
as Cexist.
−1 mxn
mxm
−1
mxn
suchcan
asequation
still
exist.
The
right
inverse
right
inverse
C
I ƒ acan
Cbe
 expressed
ƒ
For athen
linear
system
of
y=Cx,
ify=Cx,
full
row
rank,
then
inverse
has
Foraathe
linear
system
of Cequations
ifCC=(42)
has
aexpressed
full
row
rank,
a right inverse
s, the inverse
relationship
of equations
equation
(42)
as
;be
Thus,
inverse
relationship
of
asa right
;then
1
1
1
mxm
1
T 1 (42)
1 can be
1
mxm
1 computed
using
.
Thus,
the
inverse
relationship
of
equation
C as IC CstillI exist.
C CCC TC T .CC T .
right
inverse
can be computed
using C using
C suchCas C
stillThe
exist.
The
right inverse
can be computed
such
be
as
Thus,can
inverse
relationship
(42)
be expressed
Thus,
the inverse
of1 equation
can1be expressed
as ;
l expressed
1 ;lof equation
º 1can(42)
ªthe
º as ;
ª relationship
1
" º
ª F pfl
« " »
« F pfr »
«F" »
« prl »
" »
« F prr
¼
¬
r
r
»
« 2( l l ) 2(« l2( l l )l ) 2 w
2»( l f l r )
2w »
f
r
f
r r
f
«
«
lr ª
1
ª"
lr
1 1 º
1 º »
1
l r ª F«pfl º «« 1 lr
1
«
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2( l l )
2( l l ) » 2 w »
« 2(ª l " º«l )«" " »f 2(««l2r2(( llf ll)lr )f) 2r2w(2l»(f lª lFr z)l» º ) 2 w »2 w » ª F z º
1«f 1 »r» 1
«pfrpfl»ºl r «« f flr r r 1
« « Ffpfl »« rFª« F
» «ºM T »
» « M T» » ª F z º»» ª F
"
»
l
«
" 2"( »l » ««l 21() l l 1
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l
l
)
w
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« « F pfrf » F« F«prl
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)
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l
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)
w
f
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r
f
r
«
»
f
r
f
r
pfr »
» » M T » « M «»M »
« 2(« l " »«l « )««" " »»l2f( «l««2(l l f ) l 1) 2 w2(»l1 «¬ M
M
T
»
«
»
¬» M ¼
1 l r» ¼) 1 2» w«
r F prl»
»
»f
« « Ffprl »«¬ F
« «prr ¼» ««f f r r
» M »¼
» «¬ M
M ¼» «¬ M
"
2
2
2
(
l
l
)
(
l
l
)
w
"
2
2
2
(
l
l
)
(
l
l
)
w
l
« «¬ F prrf »¼ «¬ F« prr »¼f «« r 1 f l f r f r 1 f» 1r »
1 »
»» »
l f ««
lf
« 2( l l )«
1 2 w »1 1 »
1
2
(
l
l
)
2f ( l f r l r )
2(¼l f l r» ) 2w ¼
f
r «
¬
2( l ¬«l 2() l 2(l l ) l 2() l l2 w
2w »
)
¬
f
¬r
f
r f
r
f
r
¼
(43)
(43)
(43)
(43)
¼
Pitch3.2
Moment
Rejection
Loop Rejection
3.2Pitch
Pitch
Moment
Loop
Rejection
Loop Loop
3.2 Moment
Pitch Moment
Rejection
3.2Pitch Moment Rejection Loop
he outerInloop
controller,
APID
control
is
applied
foris suppressing
bothboth
body
vertical
In outer
thetheouter
loop
APID
is
for
suppressing
bothvertical
body vertical
the
loop
APID control
is control
applied
for applied
suppressing
body
vertical
In
outercontroller,
loop controller,
controller,
APID
control
applied
for suppressing
both
body
In
the
outer
loop
controller,
APID
control
is
applied
for
suppressing
lacementdisplacement
and
body
pitch
angle.
The
inner
loop
controller
of
pitch
moment
rejection
and
body
pitch
angle.
The
inner
loop
controller
of
pitch
moment
rejection
displacement
pitchangle.
angle.TheThe
inner
controller
of moment
pitch moment
displacement and
and body
body pitch
inner
looploop
controller
of pitch
rejection rejection
both
body
vertical
displacement
body
angle.
The
inner
control
isas
described
as during
follows:
during throttling
and
braking,
a vehicle
will
a loop
force,produce
control
described
as
follows:
during
throttling
and
braking,
a vehicle
willaproduce
a force, a force,
trol is described
follows:
throttling
andand
braking,
a pitch
vehicle
will
produce
force,
control
isisdescribed
as
follows:
during
throttling
and
braking,
aproduce
vehicle
will
namely
throttling
force
and
braking
force
respectively
at
the
body
center
of
gravity.
The
controller
of
pitch
moment
rejection
control
is
described
as
follows:
namely
throttling
force
and
braking
force
respectively
at
the
body
center
of
gravity.
The
mely throttlingnamely
force and
braking
force
at the
body center
of gravity.
The of gravity.
throttling
force
andrespectively
braking force
respectively
at the
body center
The
throttling
force
generates
a
pitch
moment
causing
the
body
center
of
gravity
to
shift
backward
throttling
force
generates
a
pitch
moment
causing
the
body
center
of
gravity
to
shift
backward
during
throttling
and
braking,
a
vehicle
will
produce
a
force,
namely
ttling force generates
pitchgenerates
moment causing
the bodycausing
center of
to shiftofbackward
throttlingaforce
a pitch moment
thegravity
body center
gravity to shift backward
as shown
FIGURE
6 and
vice
versa
when
braking
input is input
applied.
Shifting
the center
bodythe
center
asin
shown
FIGURE
6 braking
and
vice
versa
when
braking
is applied.
body
center
throttling
force
and
force
respectively
at
the
body
of the
hown in of
FIGURE
6 and in
vice
versa
braking
input
is
applied.
Shifting
theShifting
body
center
as of
shown
FIGURE
6when
and from
vice an
versa
input
isdefining
applied.
body center
gravity
causesinacauses
weighta transfer
axlewhen
to axle
thebraking
other
Byaxle.
theShifting
distance
gravity
weight transfer
from
an
to theaxle.
other
By defining
the distance
gravity.
The
throttling
force
generates
a
pitch
moment
causing
the
body
ravity causes
a
weight
transfer
from
an
axle
to
the
other
axle.
By
defining
the
distance
of
gravity
causes
a
weight
transfer
from
an
axle
to
the
other
axle.
By
defining
the
distance
betweenbetween
the bodythe
center
gravity
the and
pitchthe
pole
is H
pitch
is defined
by:
bodyofcenter
of and
gravity
pitch
pole
, pitch moment
is defined
by:
H pcmoment
pc , is
ween the bodybetween
center of
gravity
and
the
pitch
pole
is
,
pitch
moment
is
defined
by:
H
the body center of gravity and thepcpitch pole is H pc , pitch moment is defined by:
Mp
Mp
MM
(apx ) HMpc(a x ) H pc
M44
(a x ) H pc M p ISSN:
M (a x2180-1053
) H pc
Vol. 2
No. 2
July-December 2010
(44)
(44)
(44)
In case of
the two the
pneumatic
actuatorsactuators
installedinstalled
in the front
axle
have
to have
produce
the
In braking,
case of braking,
two pneumatic
in the
front
axle
to produce
the
(44)
» « » » «M »
«
l f«« 2l ( l)l f l2( )l 1 2l ( l) 1 l 2)w
1 »» ¬2M
1 M»»¼ ¬ M ¼
w
»
»
»
2( l l f¬«2l(r l)f l f l2r()l f 12l(r l)f 1 l r 2) 1w ¼» 21w »¼»
Adaptive Pid Control¬« Withf Pitch
Moment Rejection For Reducing
Unwanted
Vehicle Motion In Longitudinal Direction
« 2( l « 2l ( l) l2( )l 2l ( l) l 2) w » 2 w »
In theloop
outer
loop
APID
control
for suppressing
bothvertical
body vertical
f APID
r f control
ris applied
In the outer
for suppressing
both body
f ¬ r controller,
r f
¼ is ¼applied
¬ controller,
¬«
«prl ¼»
prr
prl ¼»¬« prr
««" 2( l
"
r f
r f Rejection
»«¬Pitch
« F prr Moment
F » f Rejection
r
3.2
Pitch
Loop r fLoop
¼ ««prr ¼ Moment
¬
««
3.2
3.2 3.2
Pitch
Moment
Pitch
Moment
Rejection
Rejection
Loop
Loop
displacement
and
body
pitch
angle.
Theloop
innercontroller
loop controller
pitch moment
displacement
and body
pitch
angle.
The
inner
of pitchofmoment
rejectionrejection
control
is
described
as
follows:
during
and braking,
a vehicle
produce
control
is
described
as
follows:
during
throttling
andasbraking,
a vehicle
will produce
a force,
3.2
Pitch
Moment
Loop throttling
3.2 center
Pitch Moment
Rejection
Loop
of
gravity
toRejection
shift
backward
shown
in FIGURE
6will
and
vicea force,
In theInouter
the outer
loop
controller,
loop and
controller,
APID
control
is applied
is applied
forthe
suppressing
for
both
body
both of
vertical
body
namely
throttling
force
and APID
braking
force
respectively
at suppressing
thecenter
body
center
gravity.
The
namely
throttling
force
braking
force control
respectively
at
body
of
gravity.
Thevertical
versa
when
braking
input
isThe
applied.
Shifting
the body
center
of
gravity
displacement
displacement
and
body
and
pitch
body
angle.
pitch
angle.
inner
The
inner
loop
controller
loop
controller
of
pitch
of
moment
pitch
moment
rejection
rejection
throttling
force
generates
a
pitch
moment
causing
the
body
center
of
gravity
to
shift
backward
throttling
force
generates
a
pitch
moment
causing
the
body
center
of
gravity
to
shift
backward
In
the
outer
loop
controller,
APID
control
is
applied
for
suppressing
both
body
vertical
In the outer loop controller, APID control is applied for suppressing both body vertical
causes
a weight
transfer
from
an
axle
tobraking,
the
other
axle.
By
defining
the
control
control
isindescribed
is
described
as
asvice
follows:
during
during
throttling
throttling
and
and
a vehicle
aShifting
vehicle
will
produce
will
produce
acenter
force,
a force,
as
shown
inbody
FIGURE
6 and
vice
versa
when
braking
input
is
Shifting
the
body
center
as
shown
FIGURE
6 follows:
and
versa
when
braking
input
is braking,
applied.
the
body
displacement
and
body
pitch
angle.
The
inner
loop
controller
of moment
pitch
moment
rejection
displacement
and
pitch
angle.
The
inner
loop
controller
ofapplied.
pitch
rejection
, force,
distance
between
the
body
center
of
gravity
and
the
pitch
pole
is
H
namely
namely
throttling
throttling
force
and
force
braking
and
braking
force
respectively
force
respectively
at
the
at
body
the
center
body
center
of
gravity.
of
gravity.
The
The
of
gravity
causes
a
weight
transfer
from
an
axle
to
other
axle.
By
defining
the
of
gravity
causes
a weight
transfer
from
an axle
to the
other
axle.
By
defining
theproduce
distance
control
is described
as
follows:
during
throttling
and
braking,
a vehicle
will
pc adistance
control
is described
as follows:
during
throttling
and
braking,
a vehicle
will produce
a force,
throttling
throttling
force
generates
force
generates
a
pitch
a
moment
pitch
moment
causing
causing
the
body
the
center
body
center
of
gravity
of
gravity
to
shift
to
backward
shift
backward
between
the
body
center
of
gravity
and
the
pitch
pole
is
,
pitch
moment
is
defined
by:
H
pitch
moment
is
defined
by:
between
the
body
center
of
gravity
and
the
pitch
pole
is
,
pitch
moment
is
defined
by:
H
throttling
forcebraking
and braking
force respectively
body center
of gravity.
namelynamely
throttling
force and
force respectively
at
body
of gravity.
The The
pcthe center
pc the at
as
shown
as
shown
in
FIGURE
in
FIGURE
6generates
and
vice
6 and
vice when
versa
braking
when
braking
input
applied.
iscenter
Shifting
Shifting
body
the
center
body
center
throttling
force
aversa
pitch
moment
causing
theisinput
body
of gravity
tobackward
shift
backward
throttling
force
generates
a pitch
moment
causing
the
body
center
ofapplied.
gravity
tothe
shift
of
gravity
of
causes
aFIGURE
weight
a weight
transfer
transfer
from
an
from
axle
antobraking
axle
the
to
other
the
axle.
other
By
axle.
defining
By
defining
the
distance
the
distance
asMgravity
shown
in
6
and
vice
versa
when
input
is
applied.
Shifting
the
body
as shown
incauses
FIGURE
6
and
vice
versa
when
braking
input
is
applied.
Shifting
the
body
center
(44)
(44) center
MM
(apx ) HMpc (a x ) H pc
p
between
between
body
thecauses
center
center
gravity
of gravity
and
theand
pitch
thean
pole
pitch
ispole
is, pitch
pitch
moment
is defined
is defined
by:
by:
H
Haxle.
of the
gravity
aofweight
transfer
from
to
other
axle.
By
defining
the distance
pcthe
pc ,moment
of gravity
causes
abody
weight
transfer
from
an
axle
toaxle
the
other
By
defining
the
distance
between
thecenter
body center
of gravity
thepole
pitchispole
pitch moment
is defined
by:
H pc ,moment
between
body
of gravity
and theand
pitch
is
defined
by:
H pcis, pitch
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axle
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pneumatic
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axle
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axle
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necessary
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¨ corner
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r
r
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r
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M
( apcx ) H pc '
x )pH
''
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¨ p ¨ x p pcx ¸ pc ¸
37 (46)37
and Fand
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37
where,
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F pfr FFFpfl
F pfl
pfr FFpfl
pfr
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"
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F prr "F prr ISSN:
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Vol. 2
No. 2
July-December 2010
(50) 45
(49) (49)
controller
The
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FIGUREFIGURE
6: Free body
6: Free
diagram
bodyfor
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pitch motion.
for pitch motion.
FIGUREFIGURE
6: Free body
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3.3
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3.3
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PID FIGURE
Control
PIDFIGURE
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FIGURE
6:
Free
body
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3.3
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FIGURE 6: Free body diagram for pitch motion.
3.3 many
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but
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isequations
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in the suspension system is shown in FIGURE 6 and can mathematically
38
be described as the following equations ;
38
38
38
38
38
d
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dt k d ( t )
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ktgain,
tgain
)acceleration
longitudinal
The
proportional
gain
proportional
as
the
function
the
longitudinal
acceleration
as
the
function
ofacceleration
the
ktthe
kofplongitudinal
(the
tof
)of
derivative
gain,
are
functions
of
the
position
longitudinal
)p (gain
A
inalofacceleration
Alongitudinal
xbe
p (t))k as
function
of
the
longitudinal
(tbe
) be
can be
proportional
function
the
longitudinal
A(Atx)x((tt)can
proportional
(t )the
k(proportional
derivative
functions
position
longitudinal
acceleration
.xcan
The
)Aacceleration
kthe
x (t ) . The
x (t )k d. (The
p the
p)(tare
dgain
as
the
function
of
the
longitudinal
acceleration
A
(
t
)
can
be
gain
k
(
t
)
function
of
the
longitudinal
acceleration
can
be
expressed
expressed
mathematically
as follows;
nction
ofp the longitudinal
Afollows;
(t ) of
canthebelongitudinal
expressed
mathematically
as
xacceleration
as
expressed
mathematically
xthe
proportional
gain
as follows;
function
Ax (t ) can
kmathematically
(acceleration
tgain
)mathematically
acceleration
Aexpressed
canmathematically
proportional
as follows;
function
of as
thefollows;
longitudinal
acceleration
Axbe
(t ) can be
as follows;
kthe
expressed
pbe
x (t )expressed
p (t ) as
mathematically
as
follows;
athematically
as
follows;
expressedexpressed
mathematically
as
follows;
as follows;
k p (t ) 162k.3pA(mathematically
2231
(52)
k p (t ) 162.3 Ax (t ) 2231
t )(t ) 162
.3 Ax (t ) 2231
(52)
) 162.3 Ax (t ) 2231
(t ) .3 A
162
Ax2231
(tk)p(t2231
k p (t )x k p162
(52) (52)
x (t.)3
162
.
3
A
(
t
)
2231
(52)
(52)
x
d
k p (t ) 162k.3p(52)
A(tx)(t ) 162
2231
(52)
.
3
A
(
t
)
2231
(52)
(51)
x
e (t ) The integral
in equation
(53)
asgain
a (53)
function
position
longitudinal
k i (t ) gain
equation
a of
function
of(53)
position
Thegain
integral
k iThe
(t ) inintegral
(53)longitudina
as a fun
The integral
gain
k i (tacceleration
) in equation
dt
equation
as
a longitudinal
function
of acceleration
position
k i (tas
) in
in equation
(53)
a function
of
position
longitudinal
acceleration
The integral
equation
(53)
as
a as
function
of position
longitudinal
acceleration
The integral
gain kgain
i (t )
i (t ) kin
can
be
expressed
mathematically
as
follows:
A(tx)(function
t integral
)in equation
can
be
expressed
mathematically
aslongitudinal
follows:
Ax (of
t ) position
as
ak function
of
position
gainaskThe
can
be
mathematically
as follows
Ax (t ) acceleration
inbe
equation
(53)
as
a (53)
function
of
position
longitudinal
acceleration
k(53)
t ) gain
53)
longitudinal
acceleration
ia
be
expressed
as expressed
follows:
A (tmathematically
) can
i (can
equation
amathematically
function
of
position
longitudinal
acceleration
on longitudinal
expressed
mathematically
as
follows:
i (t )x in
Aintegral
can
expressed
asasfollows:
AThe
(t )gain
x (t .)2be
xt))acceleration
tional
gain,
,
integral
gain,
and
(
)
k
t
k
(
t
(53)
k
(
131
A
(
t
)
720
.
6
(53)
The
integral
gain
in
equation
(53)
as
a
function
of
position
k
(
t
)
131
.
2
A
(
t
)
720
.
6
ellyexpressed
follows:
xasmathematically
p i be expressed
ki (t ) 131.2 Ax (t ) 720.6
i be
xi
can
follows:
Ax (t )mathematically
as follows:
expressed
Ax (t ) kcan
k(it ()tas
131
(t ) 720.6
(53) (53)
t ) .
.6 .2asAxfollows:
2 131
Ax (t.2)mathematically
Ax720
.)6 720
i (t ) k i(131
osition
.
The
(
)
A
t
(53)
131.2longitudinal
Ax (t ) ki720
(53)
(t ) .6acceleration
131
.
2
A
(
t
)
720
.
6
x x (t ) 720.6
(53)
(53)
ki(53)
(t ) x 131.2 A
The derivative
gain k d (t )gain
in equation
as a function
longitudinal
acceleration
(t )equation
can Ax (t )(53)
The derivative
k d (t ) in (53)
equation
(53) as aof
function
of longitudinal
can as a fun
The derivative
gain
(tA) xin
kacceleration
he longitudinalThe
acceleration
Ax (kThe
tgain
) (tcan
be
derivative
gain
((53)
k d as
t ) ainfunction
equation
(53)
aslongitudinal
a function
of
longitudinal
accelera
d acceleration
The
derivative
(
)
k
t
in
equation
as
a
function
of
acceleration
derivative
gain
)
in
equation
(53)
of
longitudinal
Ax (t )Acan
d
x (t ) can
d
be
expressed
mathematically
as
follows:
be
expressed
mathematically
as
follows:
46
ISSN:
2180-1053
Vol.
2
No.
2
July-December
2010
The
gain
k(53)
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
ve
gain
(derivative
kad function
t ) in The
equation
asinaequation
function
of
longitudinal
acceleration
A
(
t
)
can
be
expressed
mathematically
as
follows:
derivative
(
)
k
t
in
equation
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
d (t )gain
x
(53)
as
of
longitudinal
acceleration
A
(
t
)
can
bed expressed
mathematically
as follows:x
x
x follows:
udinal acceleration
Axexpressed
(t ) can
be
mathematically
as
be expressed
mathematically
as follows:
be expressed
mathematically
as follows:as follows:
mathematically
follows: mathematically
beasexpressed
ws:
(54)
( t ) ³ e ( t ) dt k d ( t ) F z (et()t ) K
t ) ( tk)i e( t()t³) e( tk) dt
ek (dt ()tdt) ke ( t()t ) e ( t )
(51)
F p((tt)) e (K
pkas
i ( t )functions
dlongitudinal
³of
derivative
are
the
the
position
longitudinal
The
)t ).Aand
t()pt )(gain
t(can
dt
dt
derivative
are
the
the
position
longitudinal
acceleration
The
)functions
A
ofthethe
Ax (A
t )xk(i gain,
proportional
tZ) sZ
d of
and
the
proportional
gain,
, integral
gain,
Where
eWhere
(t(gain,
) gain
Z kzref
t()t(the
klongitudinal
)acceleration
dt
ds (k
as
function
of
the
(be
t )kx i((tcan
be
t) eZfunction
and
proportional
, integral
(dt((t)kgain,
ref
(the
t )the
t)). and
kacceleration
p (tgain,
d
x
p)t)()and
(51)
s
p (t ) acceleration
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e
(
t
)
k
(
t
)
dt
k
(
t
)
e
(
t
)
proportional
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,
integral
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)
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(tt)) ZeKFse(setzp(ref
(
t
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Z
(
t
k
t
k
(
t
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integral gain,
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i
p gain,kk p(t()t ), , integral
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)Z
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proportional
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ref
ktki)(i t()tcan
s ref
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pas
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sfunction
p
dt
proportional
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t
)
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t
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as
the
function
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(
be
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(
t
)
expressed
mathematically
follows;
x Ax (t )Direction
dt
derivative
gain,
of For
theReducing
position
longitudinal
acceleration
. The
k d (gain,
tmathematically
)p are
xAx (t ) . The
p are the
expressed
as follows;
functions
of the
position
longitudinal
acceleration
kZdthe
( Moment
t t))functions
Adaptive
Rejection
Motion
In Longitudinal
and
the
proportional
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gain,
and
ederivative
(Pid
tgain,
) Control
Zgain,
k pVehicle
(longitudinal
tlongitudinal
) , integral
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,t(the
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and
(of
)the
) and the Where
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tproportional
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theUnwanted
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acceleration
The
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tskd)ref
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Ax x(kt()it().t.)The
and
the
gain,
,
integral
and
Where
(
)
)
Z
(
t
)
k
(
t
)
dtgain,
sekref
s)t((integral
i t()t ). A
derivative
gain,
the
functions
position
acceleration
d
(
s
p
expressed
mathematically
askfollows;
expressed
as the
follows;
proportional
the
of the longitudinal
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kmathematically
(gain
t ) Zas
proportional
as
function
of gain,
the longitudinal
acceleration
(be
t ) can be
(and
t ) function
p)
the
proportional
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gain,
and
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Z
ref
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tA
k p (gain,
tlongitudinal
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proportional
gain
as
function
longitudinal
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Ax A
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kep(t(gain,
162
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2231
(52)
derivative
the
functions
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.)x(xThe
)A
tktgain
skek
sk()(ptacceleration
proportional
function
of
the
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)x k(can
be
t(t)) as
and
, integral
and
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Zt)162
tx(the
)are
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functions of
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The
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2231
(52)
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tthe
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tican
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t(A
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x
x
expressedexpressed
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mathematically
as
follows;
derivative
gain,
are
the
functions
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the
position
longitudinal
acceleration
.
The
(
)
k
(
t
)
162
k
.
3
(
A
t
)
(
t
)
2231
A
t
(52)
162
.3 Afollows;
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function
2231
(52)
expressed
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as
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the
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proportional
gainkmathematically
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expressed
as
follows;
dmathematically
p mathematically
pgain,
proportional
the
of
longitudinal
acceleration
Ax (A
tbe
)x (tcan
be
derivative
the
functions
of longitudinal
the
position
longitudinal
acceleration
) . The
kacceleration
expressed
follows;
the function
of the
longitudinal
Axfunction
(of
t ) can
can
be
longitudinal
acceleration
bethe
expressed
mathematically
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d (kt p) x(tare
in
equation
(53)
as
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function
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position
longitudinal
acceleration
The integral
gain
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the
function
of
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longitudinal
acceleration
A
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t
)
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proportional
gain
in
equation
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as
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function
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position
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integral
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expressed
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x
follows:
kexpressed
(t ) 162
A p(gain
t 162
) 2231
as
follows;
proportional
as
the function of the longitudinal
acceleration
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(t ) can(52)
be
d
k .p3.mathematically
3kAixp (tt)) (51)
2231
ollows;
d
kppThe
(tbe
) gain
162
3)(tt)A)(xxt162
tgain
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3(..t3A
2231
(52)
t ) e ( t ) The
A
2231
(52)
of
acceleration
)(162
expressed
Ax (t integral
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kkpcan
t(()ttK
(52)
p(t(k
F
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e ( t as
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( tfunction
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eas
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t )adt function
kposition
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in
position
longitudinal
acceleration
integral
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))p2231
i be
expressed
mathematically
A
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t
)
z.mathematically
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expressed mathematically as follows;
dt
kkpi ((Attbe
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expressed
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))x (gain
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t162
) k131
xA
.2231
3equation
A720
t ).x6in
(tmathematically
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(53)
x expressed
.(x2t()A
)2231
720.as
6asfollows:
pkk
1
(52)
in)131
(53)
a(53)
function
of position
longitudinal
acceleration
The
i i ()t )gain
equation
as
a
function
of
position
longitudinal
acceleration
The
integral
k
i
in
equation
(53)
as
a
function
of
position
longitudinal
acceleration
The integral
gain
k
(
t
)
in
equation
(53)
as
a
function
of
position
longitudinal
acceleration
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integral
gain
k
(
t
)
i
k
(
t
)
162
.
3
A
(
t
)
2231
(52)
in
equation
(53)
as
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function
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integral
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(
)
i t720
(53)
p
x
k
(
t
)
131
.
2
A
(
t
)
.
6
i
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k
(
t
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131
.
2
A
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t
)
720
.
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ortional gain,
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)
k
t
k
(
t
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k
(
t
)
162
.
3
A
(
t
)
2231
(52)
i be
x emathematically
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the proportional gain, k p (t ) , integral gain,
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(t ) and
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can
mathematically
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i
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can
expressed
mathematically
as
follows:
AxThe
equation
(53)
as
a
function
of
position
longitudinal
acceleration
The
kbe
)gain
The
derivative
gain
(in
)mathematically
t gain
in
equation
(53)
as
a
function
longitudinal
acceleration
A
(
t
)
can
can
be
mathematically
as
follows:
A
)
x(tbe
i (ktexpressed
in
equation
(53)
as
a
function
of
position
longitudinal
acceleration
The
integral
k
(
t
)
derivative
(
)
k
t
in
equation
(53)
as
a
function
of
longitudinal
acceleration
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(
t
)
can
d
x
ation
(53)longitudinal
as a function
of
acceleration
(53) x (53)
ki (t ) acceleration
derivative
131
) gain,
gain
position
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A.i720
ki (.t2position
) Ax(t131
2xlongitudinal
720
.6 equation
are
the
functions
of
the
position
longitudinal
acceleration
(
(
t
)
A
t) .
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derivative
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(53)
as
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function
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131
(131
tgain
720
in
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of
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t integral
)derivative
k (kt .)i 2(ktA
x131
.2A
)t)720
720
.6asas
The
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)) .2kin
tmathematically
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asa(53)
afunction
function
longitudinal
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follows:
derivative
in
(53)
asa afunction
function
longitudinal
acceleration
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t ) canx
kgain
.equation
6as
expressed
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()texpressed
)integral
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be
follows:
equation
ofofposition
longitudinal
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The
(A
txkas
)(xdt(.)6t(in
i (it ) be
x
ematically
as follows:
the longitudinal
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Amathematically
(ti) gain
can be
acceleration
can
expressed
mathematically
as follows:acceleration
the function
of the longitudinal
Ax (t ) can
kbe
) as
can
expressed
follows:
A
(
t
)
p (t.as
k
(
tbe
)(texpressed
proportional
131
2kAx
(t131
t))x in.2720
.(6tfollows:
be
expressed
mathematically
be
mathematically
x
i
ki (t.)be
Aas
)) in720
6asfollows:
can
expressed
mathematically
as(53)
follows:
A
)
The
derivative
gain
(
equation
(53)
as
a (53)
function
of longitudinal
acceleration
Ax (t )(53)
canAx (t(53)
x
The
derivative
gain
(
k
t
equation
as
a function
of longitudinal
acceleration
) can
x
d
0 .6
d (t ) in equation
(54)
k
(
t
)
106
.
0
A
(
t
)
1660
The derivative
gain
(
)
k
t
in
equation
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
The
derivative
gain
k
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
(54)
x ) 106
k (t )dgain
.0d(tA
t ) equation
1660 as (53)
x (53)
mathematically
follows;
The
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as a function of longitudinal acceleration
Ax x(t ) (53)
can
ki d(t )derivative
expressed
131
k.720
ki (dt.2) Axmathematically
(t131
2dA
720
.6follows:
be expressed
mathematically
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be
expressed
as
(54)
k dexpressed
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106
k1660
be expressed
mathematically
(54)
k d .k(0tmathematically
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.equation
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Atx)follows:
(tin
) as
1660
be
mathematically
as
follows:
The
derivative
gain
(xgain
t()t )106
in
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
be
follows:
The
(
equation
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
x
d
quation (53)The
as derivative
athefunction
ofother
longitudinal
acceleration
Afunction
t ) can
On
other
hand
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same
function
is2231
used
stabilizing
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pitch
andpitch
it (can
beitx can be
(52)
On
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hand
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body
and
x (for
k
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t
)
162
.
3
A
(
t
)
gain
(
)
k
t
in
equation
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of
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t
)
can
p
x
be expressed
mathematically
as
follows:
kbe
) derivative
106
t106
) 1660
x (54)A (t )(54)
expressed
as follows:
The
equation
(53) as a function of longitudinal acceleration
can
k d (.0t )dAmathematically
(gain
.k0 dfollows:
A(xt()t ) in
1660
d (tmathematically
expressed
(54)
(t ) the
Ax106
tsame
) .0as
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.
7
be
expressed
mathematically
as
follows:
The
derivative
gain
k
t
in
equation
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
i
x
derivative
gain
(t (52)
) in equation
kequation
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as a function
of longitudinal
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kThe
56
133
.6kd)dAgain
) in
The derivative
gain
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t())t133
(53)
a(53)
function
of longitudinal
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Ax (t(57)
)(58)
can
(7A)tgain
)(tin
kd1261
asaafunction
function
longitudinal
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Ax x(xt()t )can
can
(58) as
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equation
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k243
ofoflongitudinal
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A
are
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kt1261
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kThe
the
27
d (tA
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ortional(53)
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The
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uation
as expressed
akfunction
of
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A((53)
(:and
t )kas
can
k)d, (txintegral
equation
p7
i (,t )derivative
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(57)(53)
the
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k
(
t
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56
.
27
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)
243
.
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t
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(
)
t
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k (t
iZ((ttfollows:
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(
t
p (t )be
The
derivative
gain
(
)
t
in
equation
(53)
as
a
function
acce
x
be
mathematically
as
expressed
mathematically
as
follows:
i
x
s
s
p
d
(57)
3.7
(58)
k
(
t
)
133
.
6
A
(
t
)
1261
be
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mathematically
as
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k
(
t
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133
.
6
A
(
t
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1261
bedexpressed
expressed
as
follows:
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(gain
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as a(53)
function
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xt ) inkequation
be
as
follows:
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in(52)
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t ) can
dmathematically
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kof
t ) acceleration
106.mathematically
0 Axacceleration
(t ) A
1660
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follows:
be
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as
d (longitudinal
position
longitudinal
acceleration
A
be
mathematically
as
follows:
derivative
gain,
are
theatofunctions
position
longitudinal
Thederivative
constant
values
the
(51)
(58)
are
obtained
from
linearization
of
the of the
t ) in
Ax (t ) .
x equations
ngitudinal
A
(t ) can
The
derivative
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equation
asthe
aare
function
offrom
longitudinal
d (the
The
constant
values
in.k2(56)
equations
(51)(53)
toofof
(58)
obtained
linearization
function ofacceleration
longitudinal
acceleration
The
(in
tgain
equation
(53)
as
function
longitudinal
acceleration
Ax (acceleration
t )(58)
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expressed
mathematically
as
kbe
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xgain
133
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)value
121
Alongitudinal
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tas
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2015
expressed
mathematically
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k61261
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k dwith
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d (tgained
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x 1261
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controller
acceleration
as
discussed
in
Appendix.
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(58)
the longitudinal
acceleration
A
(
t
)
can
be
k
(
t
)
133
.
6
A
(
t
)
controller
gained
with
the
value
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longitudinal
acceleration
as
discussed
in
Appendix.
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quation
(53)
as
a
function
of
longitudinal
acceleration
A
(
t
)
can
(58)
The
constant
values
in
the
equations
(51)
to
(58)
are
obtained
from
linearization
of
the
k
(
t
)
133
.
6
(
t
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1261
The
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the
equations
(51)
to
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the
d
x x .6 A
function
of other
the longitudinal
acceleration
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proportional
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k p (expressed
t ) as the
(58)
k d(t ) 133
) be
1261
x (54)
acceleration
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mathematically
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ows:
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the
the
0a function be
khand
106
.all
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1660 is xused fo
becontroller
expressed
mathematically
as
follows:
linearization
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it
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necessary
parameters
ofsame
kto
) oflongitudinal
106
.the
0identify
AxPID
(t ) On
controller
1660
of
position
longitudinal
acceleration
linearization
is
made,
itthe
necessary
toacceleration
theas
PID
controller
parameters
of
all theBefore
input
d (tidentify
controller
gained
with
the
value
ofislongitudinal
discussed
in Appendix.
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gained
with
value
acceleration
asd discussed
in xAppendix.
(58)
k
(
t
)
133
.
6
A
(
t
)
1261
as
follows:
(58)
d
x
k
(
t
)
133
.
6
A
(
t
)
1261
expressed
mathematically
as
follows;
constant
values
in
the
(51)
toand
(58)
are
obtained
from
linearization
ofacceleration
the of the A (t
expressed
mathematically
asthe
follows:
function ofThe
longitudinal
acceleration
can
)of
Aequations
d integral
x(t(53)
The
constant
values
in
the
equations
(51)
to
(58)
are
obtained
from
linearization
condition
in simulation
of
sudden
acceleration
sudden
braking
test.
The
gain
in
equation
(53)
as
a
function
of
longitudinal
xin
k
(
t
)
condition
in
simulation
sudden
acceleration
and
sudden
braking
test.
linearization
is
made,
it
is
necessary
to
identify
the
PID
controller
parameters
of
all
input
linearization
is
made,
it
is
necessary
to
identify
the
PID
controller
parameters
of
all
the
input
i
The
constant
values
in
the
equations
(51)
to
(58)
are
obtained
from
linearization
of
the
llows:
The
constant
values
the
equations
(51)
to
(58)
are
obtained
from
linearization
of
the x
(54)
The
in the
thelongitudinal
equations
(51)
to (58)
arediscussed
obtained
linearization
ofBefore
the
(58) function
kcontroller
t )constant
133
.t6) Avalues
t ) value
body
the
of
acceleration
as
infrom
Appendix.
Before
d (gained
x(133
gained
with
of hand
longitudinal
acceleration
asis
discussed
inthe
Appendix.
(58)
k dwith
(gained
.1261
6On
Athe
) value
value
1261
unction is controller
used
for
stabilizing
the
pitch
and
itascan
besame
On
the
other
hand
same
u
x (tthe
condition
in
simulation
of
sudden
acceleration
and
sudden
braking
test.
other
the
function
used
for
stabilizing
the
bodyis pi
condition
in
simulation
of
sudden
acceleration
and
sudden
braking
test.
controller
gained
with
the
value
of
longitudinal
acceleration
as
discussed
in
Appendix.
Before
controller
with
of
longitudinal
acceleration
as
discussed
in
Appendix.
Before
be
expressed
mathematically
follows:
controller
gained
the
value
acceleration
as
discussed
Appendix.
Before
(58)
(52)
The constant
values
in
the
(51)
to
(58)
are
obtained
from
linearization
thethe
61
linearization
is
made,
itvalues
iswith
to
identify
the
PID
controller
parameters
ofinall
theof
input
The
constant
init is
(51)
to
(58)
are controller
obtained
from
linearization
ofinput
the
k pmade,
(tnecessary
) equations
162
.the
3(53)
Anecessary
(equations
t )of
longitudinal
2231
linearization
is
to
identify
the
PID
parameters
of
all
x
llows:
expressed
mathematically
as
follows:
function
ofcontroller
longitudinal
acceleration
Aexpressed
(is
tof
)iscan
linearization
is
made,
itmade,
isvalue
necessary
tomathematically
identify
the PIDthe
controller
parameters
as
follows:
Mcontroller
( t ) in
Kparameters
e (all
t ) the
kBefore
(input
t ) ³39
e ( tinput
)input
dt 39
k d (t )
linearization
isthe
necessary
toand
identify
the
PID
parameters
ofi all
all
the
x it
Ttest.
p ( t )of
linearization
issimulation
it
necessary
toto
identify
PID
controller
of
the
gained
with
longitudinal
acceleration
as discussed
Appendix.
g the body
pitch
and
itconstant
can
condition
in
simulation
ofthe
sudden
acceleration
sudden
braking
The
constant
values
inmade,
equations
(51)
(58)
are
obtained
from
linearization
of the
controller
gained
with
the
value
of acceleration
longitudinal
acceleration
as discussed
Appendix.
Before
condition
inbe
of(57)
sudden
and
sudden
braking
test.
The
values
in
the
equations
(51)
to
(58)
are braking
obtained
from in
linearization
of the
condition
in
simulation
of
sudden
acceleration
and
sudden
braking
test.
condition
in
simulation
of
sudden
acceleration
and
sudden
braking
test.
k
(
t
)
56
.
27
A
(
t
)
243
.
7
condition
in
simulation
of
sudden
acceleration
and
sudden
test.
39
linearization
is
made,
necessary
to
the
controller
parameters
of all theBefore
input
i is
x identify
controller
gained
with
value
of
longitudinal
astodiscussed
in parameters
Appendix.
linearization
isitthe
made,
is linearization
necessary
to acceleration
identify
the
PID
controller
offrom
all theBefore
input39
(51)
to The
(58)
are
obtained
from
of PID
the
constant
values
the
(51)
(58)
obtained
aquations
functionof
of
longitudinal
acceleration
Aitthe
)value
can
function
position
longitudinal
acceleration
dcontroller
gained
with
ofequations
longitudinal
acceleration
as are
discussed
in Appendix.
x (tin
(53)
as
a function
integral
gain
kacceleration
i (t ) in equation
(55)
( t )of
elongitudinal
( t ) dt condition
k d ( t )acceleration
e (simulation
tis
) The
M Tof
(kttest.
) position
Kdall
(et(the
)t )elongitudinal
(input
t ) k i ( t ) ³ e acceler
( t ) dt in
of
sudden
and
sudden
braking
test.
p
linearization
made,
it
is
necessary
to
identify
the
PID
controller
parameters
of
condition
in
simulation
of
sudden
acceleration
and
sudden
braking
M
(
t
)
K
(
t
)
e
(
t
)
k
(
t
)
e
(
t
)
dt
(
t
)
³
ue
as
discussed
in
Appendix.
Before
T
p
i the
d(t )longitudinal
the
proportion
Where
e(value
t ) T refof
dtT (t ) and
linearization
ofbethe
gained
with
³controller
linearization
is
made,
it(tcontroller
is) can
necessary
to
identify
the
PID
parameters
of all39
the input
dt
39
ows:
can
expressed
mathematically
as
follows:
A
(
t
)
function
of
longitudinal
acceleration
A
39
(54)
39 x (t
x derivative
The
) in
k d (in
tof
(53)
as
a function
oftest.
longitudinal
acceleration
xgain
condition
inPID
simulation
of discussed
sudden
andinput
sudden
braking
test.
essary to identify
the
controller
parameters
allequation
the
condition
in simulation
ofacceleration
sudden
acceleration
and
sudden
braking
(55)
acceleration
as
Appendix.
Before
linearization
is made,
it of39the Aposi
ki (t ) test.
131(53)
.2 Ax (t ) 720
.6follows:derivative gain, k d ( t ) are the function
39
den
acceleration
and
sudden
braking
be
expressed
mathematically
as
39 prop
d the proportional
gain, k p (t )to
, identify
integral(54)
gain,PID
k i (controller
t )(tand
the
Where
e(t )of Tall
(the
t ) gain,
input
T (t )kand
is necessary
the
parameters
the proportional
) T the
(t ) and
p (t ) , integral
proportional
gain kref
for stabilizing the body pitch
and itWhere
can be e(t ) T ref
p (t ) are : 39
39 of th
condition
in simulation
ofgain,
acceleration
and
sudden
unction
position
longitudinal
acceleration
A) sudden
(1261
tequation
) k, dwhere
k d ( t )braking
are
the function
derivative
gain,
functionofofthe
longitudinal
acceleration
Axgain
(t )(58)
can
( t ) are
theas
function
of the
position
longitudinal
accelerat
derivative
xin
kThe
derivative
(
)
k
t
(53)
a
function
of
longitudinal
acceleration
Ax (t
k
(
t
)
133
.
6
A
(
t
p (t ) , integral gain,
i (t ) and
d
x
d
39
ed for stabilizing test.
the body pitch and it can be
:dinal acceleration A (t ) be
are
:
the
proportional
gain
k
(
t
)
proportional as
gain
k p (t ) are : k p (t ) 121.2 Ax (t ) 2015p
expressed the
mathematically
follows:
, where
x
d
ot ) (58)
linearization
of(55)
thein the equations (51) to (58) are obtained from linearization of
The
constant values
e ( tare
) obtained from
dt
l acceleration
as discussed
in Appendix.
(54)
controller
with
value
longitudinal
acceleration
in (53)
Appendix.
Be
(56)
k d (gained
t ) Before
106
.0 kAxthe
(t ) 1660 of
k p (t ) k i121
(t ) 2015
d
The integral
gain
inAx equation
as a fun
(as
t ).2discussed
p (t ) 121.2 Ax (t ) 2015
(55)
t ) PID
e ( t )controller parameters
yd (the
of
all
the
input
linearization is made, it is necessary to identify the PID controller parameters of all the i
dt
be expressed mathematically as follows:
(56)
gain,braking
, test.
integral
gain,
kiti (can
t ) and
k p (t )the
and
condition
in 2180-1053
simulation
of
sudden
acceleration
and sudden
braking
test.
dional
forsudden
stabilizing
body
pitch
and
be
Vol.
2function
No. 2 is July-December
2010
47
Onlongitudinal
theISSN:
other
hand
the
same
used
stabilizing
body
pitch
and
it as
ca
Thefor
integral
tion
(53)
as a function
of
acceleration
can
in longitudinal
equation
(53)acce
(
)
A
t
kthe
i (t ) of
The
integral gainx k i (t ) in equation
(53)
as again
function
osition
longitudinal
acceleration
A
(
t
)
,
where
expressed
mathematically
as
follows:
x
ortional
k i (t ) and
kexpressed
Ax (t ) 243.7 as follows:
follows: gain, k p (t ) , integral gain,
mathematically
i (t ) 56.27
be expressed
mathematically as be
follows:
Journal of Mechanical Engineering and Technology
In these conditions, the simulations have been made with 1Mpa, 3Mpa
and 6Mpa brake pressure in sudden braking test, while 0.2, 0.6 and full
In these conditions, the simulations have been made with 1Mpa, 3Mpa and 6Mpa brake
step throttle for the sudden acceleration test. The controller parameters
pressure in sudden braking test, while 0.2, 0.6 and full step throttle for the sudden
obtained
from
the testsparameters
are described
infrom
Table
acceleration
test.
The controller
obtained
the1.
tests are described in Table 1.
TableTable
1: PID
parameters
1:controller
PID controller
parameters
Controller Parameter
Driver Input
1Mpa brake
3Mpa brake
6Mpa brake
0.2 step
throttle
0.6 step
throttle
full step
throttle
4.0
Longitudinal
Acceleration
(G)
Body Displacement
(controller for F z )
Body Pitch
(controller for M T )
Kp
800
1000
2000
Ki
700
1000
2000
Kd
1600
2000
3000
Kp
600
1000
1852
Ki
300
500
7000
Kd
1500
2000
2251
-1.36
-2.72
-8.2
3000
100
500
2500
50
700
0.96
1000
900
750
2700
57
800
2.89
5000
100
2500
30000
60
8000
4.8
Validation Of 14 Dof Vehicle Model Using Instrumented Experimental Vehicle
Dof Vehicle
Model
Using
To 4.0Validation
verify the full vehicle ride Of
and 14
handling
model that have
been derived,
experimental
works are performed
using an instrumented
experimental vehicle.
This section provides the
Instrumented
Experimental
Vehicle
verification of ride and handling model using visual technique by simply comparing the trend
To verifyresults
the full
ridedata
andusing
handling
that have
beenor
of simulation
with vehicle
experimental
the samemodel
input signals.
Validation
verification
is
defined
as
the
comparison
of
model’s
performance
with
the
real
system.
derived, experimental works are performed using an instrumented
Therefore,
the validation
does not
that the
fitting of the
simulated
data is exact
as the
experimental
vehicle.
Thismean
section
provides
verification
of ride
measured data, but as gaining confidence that the vehicle handling simulation is giving
and handling model using visual technique by simply comparing the
insight into the behavior of the simulated vehicle reference. The tests data are also used to
trend
of simulation
results
experimental
data using
the same
check
whether
the input parameters
forwith
the vehicle
model are reasonable.
In general,
model
input signals.
Validation
or verification
is defined
the comparison
validation
can be defined
as determining
the acceptability
of aasmodel
by using some
statistical
tests forperformance
deviance measured
or the
subjectively
using visual
techniques
of model’s
with
real system.
Therefore,
thereference.
validation
does not mean that the fitting of simulated data is exact as the measured
Vehicle Instrumentation
data,
but as gaining confidence that the vehicle handling simulation
is
giving
insight
into
the isbehavior
of the
simulatedvehicle
vehicle
reference.
The data acquisition
system
(DAS)
installed into
the experimental
to obtain
the real
The
tests
data
are
also
used
to
check
whether
the
input
parameters
vehicle reaction as to evaluate the vehicle’s performance in terms of longitudinal
acceleration,
body vertical
and pitch
The DAS
usesvalidation
several types
for the vehicle
modelacceleration
are reasonable.
In rate.
general,
model
canof
transducers
such as
axis accelerometer
to measure the
massby
andusing
unsprung
mass
be defined
assingle
determining
the acceptability
ofsprung
a model
some
accelerations
for
each
corner,
tri-axial
accelerometer
to
measure
longitudinal,
vertical
and
statistical tests for deviance measured or subjectively using visual
lateral accelerations at the body center of gravity, tri-axial gyroscopes for the pitch rate and
techniques
reference.
wheel
speed sensor
to measure angular velocity of the tire. The multi-channel µ-MUSYCS
4.1
system Integrated Measurement and Control (IMC) is used as the DAS system. Online
FAMOS
software as the
real time data processing and display function is used to ease the
4.1Vehicle
Instrumentation
data collection. The installation of the DAS and sensors to the experimental vehicle can be
data acquisition
system (DAS) is installed into the experimental
seenThe
in FIGURE
7.
vehicle to obtain the real vehicle reaction as to evaluate the vehicle’s
performance in terms of longitudinal acceleration, body vertical
40
48
ISSN: 2180-1053
Vol. 2
No. 2
July-December 2010
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
acceleration and pitch rate. The DAS uses several types of transducers
such as single axis accelerometer to measure the sprung mass and
unsprung mass accelerations for each corner, tri-axial accelerometer
to measure longitudinal, vertical and lateral accelerations at the body
center of gravity, tri-axial gyroscopes for the pitch rate and wheel
speed sensor to measure angular velocity of the tire. The multi-channel
µ-MUSYCS system Integrated Measurement and Control (IMC) is
used as the DAS system. Online FAMOS software as the real time data
processing and display function is used to ease the data collection. The
installation of the DAS and sensors to the experimental vehicle can be
seen in FIGURE 7.
a) Tri-axis
accelerometer
a) Tri-axis
accelerometer
a) Tri-axis accelerometer
d) DAS
d) DAS
d) DAS
sensor
sensor
Gyrosensors
sensors
b)b)Gyro
b) Gyro sensors
c) Speed
sensor
c) Speed
sensor
c) Speed sensor
Frontspeed
wheelsensor
speed sensor
f) Rear
wheel
speed
e) Fronte)wheel
f) Rear wheel
speed
sensor
e) Front
wheel speed sensor
f) Rear
wheel
speed
FIGURE 7: In-vehicle instrumentations
4.2Experimental Vehicle
An instrumented experimental vehicle is developed to validate the full
vehicle model. A Malaysian National car is used to perform sudden
braking and sudden acceleration test, defined in (SAE J266, 1996). Note
that the vehicle is 1300 cc and uses manual gear shift as the power
terrain systems. The technical specifications of the vehicle are listed in
Table 1.
g) Instrumentation inside the vehicle
h) Experimental vehicle
g) Instrumentation inside the vehicle
h) Experimental vehicle
FIGURE 7: In-vehicle instrumentations
FIGURE 7: In-vehicle instrumentations
4.2
4.2
Experimental Vehicle
Experimental Vehicle
An instrumented experimental vehicle is developed to validate the full vehicle model. A
An instrumented experimental vehicle is developed to validate the full vehicle model. A
Malaysian National car is used to perform sudden braking and sudden acceleration test,
Malaysian National car is used to perform sudden braking and sudden acceleration test,
defined in (SAE J266, 1996). Note that the vehicle is 1300 cc and uses manual gear shift as
defined in (SAE J266, 1996). Note that the vehicle is 1300 cc and uses manual gear shift as
the power terrain systems. The technical specifications of the vehicle are listed in Table 1.
the power terrain systems. The technical specifications of the vehicle are listed in Table 1.
TABLE 1: Experimental vehicle Parameter
TABLE 1: Experimental vehicle Parameter
Parameter
Value
Parameter
Value
Vehicle mass
920kg
Vehicle mass
920kg
Wheel base Vol. 2 No. 2 2380mm
ISSN: 2180-1053
July-December 2010
49
Wheel base
2380mm
Wheel track
1340mm
Wheel track
1340mm
Spring rate: Front:
30 N/mm
An instrumented experimental vehicle is developed to validate the full vehicle model. A
Journal ofNational
Mechanical
Technology
Malaysian
carEngineering
is used to and
perform
sudden braking and sudden acceleration test,
defined in (SAE J266, 1996). Note that the vehicle is 1300 cc and uses manual gear shift as
the power terrain systems. The technical specifications of the vehicle are listed in Table 1.
TABLE
1: 1Experimental
Parameter
TABLE
: Experimental vehicle
vehicle Parameter
Parameter
Value
Vehicle mass
920kg
Wheel base
2380mm
Wheel track
1340mm
Spring rate: Front:
30 N/mm
Rear:
30 N/mm
Damper rate : Front:
1000 N/msecʛ¹
Rear:
1000 N/msecʛ¹
Roll center
100
Center of gravity
550mm
Wheel radius
285mm
4.3Validation Procedures
The dynamic response characteristics of a vehicle model that include
longitudinal acceleration, longitudinal slip in each tire and pitch rate
can be validated using experimental test through several handling
test procedures namely sudden braking test and sudden acceleration
test. Sudden braking test is intended to study transient response of the
vehicle under braking input. In this case, the tests were conducted by
accelerating the vehicle to a nominal speed of 60kph and activating the
instrumentation package. The driver then applies the brake pedal hard
enough to hold the pedal firmly until the vehicle stopped completely
as shown in FIGURE 8(a). On the other hand, sudden acceleration test
is used to evaluate the characteristics of the vehicle during a sudden
increase of speed. In this study, the vehicle accelerated to a nominal
speed of 40kph and activated the instrument package. The driver then
manually applied the throttle pedal full step as required to make the
vehicle accelerated immediately as shown in FIGURE 8(b).
4.4Validation Results
FIGUREs 9 and 10 show a comparison of the results obtained
using SIMULINK and experimental. In experimental works, all the
experimental data are filtered to remove out any unintended data. It is
necessary to note that the measured vehicle speed from the speed sensor
is used as the input of simulation model. For the simulation model, tire
parameters are obtained from Szostak et.al. (1988) and Singh et.al. (2002).
The results of model verification for sudden braking test at 60 kph are
shown in FIGURE 9. FIGURE 9(a) shows the vehicle speed applied for
the test. It can be seen that the trends between simulation results and
experimental data are almost similar with acceptable error. The small
difference in magnitude between simulation and experimental results
is due to the fact that, in an actual situation, it is indeed very hard for
50
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July-December 2010
41
then manually applied the throttle pedal full step as required to make the vehicle accelerated
immediately as shown in FIGURE 8(b).
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
4.4
Validation Results
the driver
to maintain
vehicleofinthe
a perfect
speed as
compared
to and
the
FIGUREs
9 and
10 show athe
comparison
results obtained
using
SIMULINK
experimental.
In experimental
works, all the experimental data are filtered to remove out any
result obtained
in the simulation.
unintended data. It is necessary to note that the measured vehicle speed from the speed sensor
is used as the input of simulation model. For the simulation model, tire parameters are
In terms
of Szostak
both longitudinal
and The
pitch
rateofresponse,
it can
et al. (1988) andacceleration
Singh et al. (2002).
results
model verification
obtained
from
for
braking
testare
at 60
kph are
shown
in FIGURE 9.during
FIGUREthe
9(a)initial
shows the
vehicle
be sudden
seen that
there
quite
good
comparisons
transient
speed
for the
test. It can
be following
seen that thesteady
trends between
simulation
results and
phaseapplied
as well
as during
the
state phase
as shown
in
experimental data are almost similar with acceptable error. The small difference in magnitude
FIGUREs
9(b)
and
(c)
respectively.
Longitudinal
slip
responses
of
the
between simulation and experimental results is due to the fact that, in an actual situation, it is
front very
tireshard
also
satisfactory
matches
only
small
deviation
in
indeed
forshow
the driver
to maintain the
vehicle inwith
a perfect
speed
as compared
to the
result
obtained in the
simulation.
the transition
area
between transient and steady state phases as shown
in FIGURE 9(d) and (e). It can also be noted that the longitudinal slip
In terms of both longitudinal acceleration and pitch rate response, it can be seen that there are
responses
to all tires
in the
experimental
dataas are
than
quite
good comparisons
during
the initial
transient phase
wellslightly
as during higher
the following
the longitudinal
obtained
simulation
responses
steady
state phase as slip
showndata
in FIGUREs
9(b) from
and (c)the
respectively.
Longitudinal
slip
responses
of
the
front
tires
also
show
satisfactory
matches
with
only
small
deviation
in
particularly for the rear tires as seen in FIGUREs 9(f) and (g). Thistheis
transition area between transient and steady state phases as shown in FIGURE 9(d) and (e). It
due to the fact that it is difficult for the driver to maintain a constant
can also be noted that the longitudinal slip responses to all tires in the experimental data are
speed higher
during
maneuvering.
simulation,
is also
that the
slightly
than
the longitudinalIn
slip
data obtainedit from
the assumed
simulation responses
particularly
for
the
rear
tires
as
seen
in
FIGUREs
9(f)
and
(g).
This
is
due
to
the
fact
that it
vehicle is moving on a flat road during step steer maneuver. In fact,
it isis
difficult
for
the
driver
to
maintain
a
constant
speed
during
maneuvering.
In
simulation,
it
is
observed that the road profiles of test field consist of irregular surface.
also assumed that the vehicle is moving on a flat road during step steer maneuver. In fact, it is
This can
source
deviation
longitudinal
observed
thatbe
theanother
road profiles
of testoffield
consist of on
irregular
surface. Thisslip
can response
be another
of
the
tires.
source of deviation on longitudinal slip response of the tires.
a) Vehicle speed
b) Longitudinal acceleration
c) Pitch rate
d) Longitudinal slip front right
e) Longitudinal slip front left
ISSN: 2180-1053
42
f) Longitudinal slip rear left
Vol. 2
No. 2
July-December 2010
51
Journal of Mechanical Engineering and Technology
e) Longitudinal slip front left
f) Longitudinal slip rear left
g) Longitudinal slip rear right
FIGURE 9: Response of the Vehicle for sudden braking
test at constant speed of 60 kph
FIGURE 9: Response of the Vehicle for sudden braking test at constant speed of 60 kph
The results of Sudden Acceleration test at constant speed of 40kph indicate that measurement
data
and results
the simulation
results agree
with a relativelytest
goodataccuracy
as shown
in FIGURE
The
of Sudden
Acceleration
constant
speed
of 40kph
10. FIGURE 10(a) shows the vehicle speed which is used as the input for the simulation
indicate
that
measurement
data
and
the
simulation
results
agree
a
model. In terms of pitch rate and longitudinal acceleration, it can be seen clearly that with
the
relatively
good
accuracy
as
shown
in
FIGURE
10.
FIGURE
10(a)
shows
simulation and experimental result are very similar with minor difference in magnitude as
shown
FIGUREspeed
10(b) and
(c). The
minor difference
in magnitude
small fluctuation
the invehicle
which
is used
as the input
for theand
simulation
model.
occurred on the measured data is due to the body’s flexibility which was ignored in the
In
terms
of
pitch
rate
and
longitudinal
acceleration,
it
can
be
seen
simulation model. In terms of tire’s longitudinal slip, the trends of simulation results show
clearly
that
the
simulation
and
experimental
result
are
very
similar
close agreement for both experimental data and simulation are shown in FIGURE 10(d), (e),
(f) with
and (g).
Closelydifference
similar to the
resultsas
obtained
from
acceleration
test, (c).
minor
invalidation
magnitude
shown
in sudden
FIGURE
10(b) and
theThe
longitudinal
responses
of
all
tires
in
experimental
data
are
smaller
than
the
longitudinal
minor difference in magnitude and small fluctuation occurred on
slip data obtained from the simulation. Again, this is due to the difficulty of the driver to
the
measured
due sudden
to theacceleration
body’s flexibility
which
was ignored
maintain a constant data
speed is
during
test maneuver.
Assumption
in
in the model
simulation
model.
In terms
longitudinal
slip,
thevery
trends
simulation
that the vehicle
is moving
on aof
flattire’s
road during
the maneuver
is also
difficult
to realize in practice.
fact, road
irregularities
of the for
test field
may
cause the changedata
of simulation
resultsInshow
close
agreement
both
experimental
in tire properties during the vehicle handling test. Assumption of neglecting steering inertia
and simulation are shown in FIGURE 10(d), (e), (f) and (g). Closely
may possibility lower down the magnitude of tire longitudinal slip in simulation results
similar
tothe
themeasured
validation
compared
with
data. results obtained from sudden acceleration test,
the longitudinal responses of all tires in experimental data are smaller
than the longitudinal slip data obtained from the simulation. Again,
43
this is due to the difficulty of the driver to maintain a constant speed
during sudden acceleration test maneuver. Assumption in simulation
model that the vehicle is moving on a flat road during the maneuver
is also very difficult to realize in practice. In fact, road irregularities of
the test field may cause the change in tire properties during the vehicle
handling test. Assumption of neglecting steering inertia may possibility
lower down the magnitude of tire longitudinal slip in simulation results
compared with the measured data.
a) Vehicle speed
52
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c) Pitch rate
b) Longitudinal acceleration
Vol. 2
No. 2
July-December 2010
d) Longitudinal slip front left
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
a) Vehicle speed
b) Longitudinal acceleration
c) Pitch rate
d) Longitudinal slip front left
e) Longitudinal slip front right
f) Longitudinal slip rear left
g) Longitudinal slip rear right
FIGURE 10: Response of the vehicle for sudden acceleration test at
constant speed 40 kph
FIGURE 10: Response of the vehicle for sudden acceleration test at constant speed 40 kph
5.0
Performance Evaluation Of Gspid Control
The performance of GSPID controller is examined through simulation studies using
5.0
Performance
Evaluation
Of For
Gspid
Control
SIMULINK
toolbox of the MATLAB
software package.
comparison
purposes, the
performance of the GSPID is compared with both the conventional PID control approach and
The
performance
of GSPIDofcontroller
through
passive
system. The performance
the controlleris
is examined
examined through
vehiclesimulation
translational
studies
usingbody
SIMULINK
toolbox
of the
MATLAB
package. For
motion namely
pitch angle, pitch
rate, body
acceleration
andsoftware
body displacement.
comparison purposes, the performance of the GSPID is compared with
both the conventional PID control approach and passive system. The
performance
ofParameters
the controller is examined through vehicle translational
5.1
Simulation
motion namely body pitch angle, pitch rate, body acceleration and
44
body displacement.
5.1
Simulation Parameters
The simulation study was performed for a period of 10 seconds using
Heun solver with a fixed step size of 0.01 second. The controller
parameters are obtained using trial and error technique. The numerical
values of the 14-DOF full vehicle model parameters and Calspan tire
model parameters as well as the controller parameters are shown in
Table 2:
ISSN: 2180-1053
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July-December 2010
53
The simulation study was performed for a period of 10 seconds using Heun solver with a
Journal
Mechanical
Technology
fixed
stepofsize
of 0.01 Engineering
second. Theand
controller
parameters are obtained using trial and error
technique. The numerical values of the 14-DOF full vehicle model parameters and Calspan
tire model parameters as well as the controller parameters are shown in Table 2:
TABLE
TABLE2:2:Tire
Tireparameter
parameter
Parameter
FWD radial
Tire designation
P185/70R13
7.3
Tw
24
Tp
980
FZT
1.0
C1
0.34
C2
0.57
C3
0.32
1068
11.3
2442.73
0.31
-1877
0.05
17.91
0.85
C4
Ao
A1
A2
A3
A4
Ka
CS/FZ
µo
5.2
Step Change Test Driver Input
5.2
Step Change Test Driver Input
To predict the performance of APID controller, a step function driver input test is applied.
predict
performance
APID
controller,
stepIn function
driver
TheTo
tests
consist the
of sudden
braking testofand
sudden
accelerationatest.
sudden braking
test,
is applied.
tests consist
of sudden
sudden
the input
vehicletest
is accelerated
andThe
maintained
to a nominal
speed braking
of 70 kph,test
thenand
6MPa
brake is
applied
to hold the test.
pressure
untilbraking
the vehicle
stopped
completely.
For the remainder
acceleration
In firmly
sudden
test,
the vehicle
is accelerated
andof
the maintained
simulation, theto
steering
is
maintained
constantly
at
zero
degree
appropriately.
In
sudden
a nominal speed of 70 kph, then 6MPa brake is applied
acceleration test, the same condition is applied to the vehicle, and then full step throttle input
to hold the pressure firmly until the vehicle stopped completely. For
is applied to make the vehicle accelerate immediately. In order to fulfill the objective of
the remainder
the simulation,
maintained
constantly
designing
the activeofsuspension
system, the
theresteering
are fouris parameters
observed
in the
at zero The
degree
appropriately.
In sudden
acceleration
the same
simulations.
four parameters
are the vehicle
body acceleration,
bodytest,
displacement,
pitch
ratecondition
and pitch angle.
The solid
lines
are defined
the full
active
suspension
is applied
to the
vehicle,
and as
then
step
throttleunder
inputAPID
is
controller,
theto
dashed
lines
arevehicle
describe accelerate
as the active immediately.
suspension underIn
PID
controller
and the
applied
make
the
order
to fulfill
dotted lines are label of the passive system response.
the objective of designing the active suspension system, there are four
parameters
observed
the
simulations.
four controller
parameters
are better
the
From
FIGUREs 11(a)
and (b),inthe
pitch
behavior for The
the APID
indicates
vehicle reduced
body acceleration,
body displacement,
pitch rate
pitchin
performance
dive during the maneuver
compared to conventional
PID. and
It is shown
the angle.
vertical The
acceleration
plot. are
The defined
APID made
the vehicle
the momentum
the
solid lines
as the
active lose
suspension
underduring
APID
maneuvers
and
reduced
the
vehicle
weight
transfer
to
the
front.
FIGURE
11(c)
illustrates
controller, the dashed lines are describe as the active suspension under
clearly
the APID and
can effectively
absorb
the vehicle’s
vibration
in comparison
to active
PIDhow
controller
the dotted
lines
are label
of the
passive system
suspension under PID controller and the passive suspension system. The oscillation of the
response.
body
acceleration using the APID system is much reduced significantly, which guarantees
better ride comfort and reduce of body vertical displacement as shown in FIGURE 11(d).
From FIGUREs 11(a) and (b), the pitch behavior for the APID controller
indicates better performance reduced dive during the maneuver
compared to conventional PID. It is shown in the vertical acceleration45
plot. The APID made the vehicle lose the momentum during the
maneuvers and reduced the vehicle weight transfer to the front. FIGURE
11(c) illustrates clearly how the APID can effectively absorb the vehicle’s
vibration in comparison to active suspension under PID controller and
54
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July-December 2010
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
the passive suspension system. The oscillation of the body acceleration
using the APID system is much reduced significantly, which guarantees
better ride comfort and reduce of body vertical displacement as shown
in FIGURE 11(d).
a) Pitch angle
b) Pitch rate
c)
d) b)
Body
displacement
a)Body
Pitch acceleration
angle
Pitch
rate
FIGURE
11:
Performance
of
GSPID
at
step
function
brake
6
MPa
FIGURE 11: Performance of GSPID at step function brake
6 MPa
The results of the sudden acceleration test showed that the proposed controllers, APID are
reasonably
efficient
methods
in enhancing
vehicle stability
in term of that
reducing
In this
The
results
of the
sudden
acceleration
test showed
thesquat.
proposed
case, the input APID
of the vehicle
is changed from
braking methods
to throttling in
input.
FIGURE 12
show
controllers,
are reasonably
efficient
enhancing
vehicle
that performance of the APID controller has a good tracking performance with good transient
stability
term 12
of (a)
reducing
case, from
the input
of the vehicle
response. in
FIGURE
shows thesquat.
responseIn
of this
the vehicle
sudden acceleration
test
ismaneuver
changed
braking
to throttling
12 the
show
that
whichfrom
is input
for the simulation
model. Ininput.
terms ofFIGURE
the pitch rate,
proposed
controller gives the
a morecontroller
stable outputhas
whena itgood
is compared
to PID
structures as
BodyAPID
acceleration
d) tracking
Body displacement
performance
ofc)system
the
performance
shown in FIGURE
12 11:
(b).Performance
The result of the
bodyat acceleration
and
body
displacement are
GSPID
step(a)
function
brake
6 MPa
with
goodFIGURE
transient
response. of
FIGURE
12
shows
the
response of the
shown in FIGURE 12 (c) and FIGURE 12 (d) respectively. The FIGURE explains that the
vehicle
from
acceleration
test
maneuver
which
is input
forare
the
APID
controller
gives
performance
in terms
time
and
reducing
the
The
results
of thesudden
suddenbetter
acceleration
test showed
that of
the settling
proposed
controllers,
APID
magnitude efficient
as compared
toIn
thein
counterparts..
simulation
model.
terms
of the
pitch
rate,
the ofproposed
controller
reasonably
methods
enhancing
vehicle
stability
in term
reducing squat.
In this
case, thethe
inputsystem
of the vehicle
is changed
from
braking to
throttling
input.
FIGURE 12to
show
gives
a more
stable
output
when
it is
compared
PID
that performance of the APID controller has a good tracking performance with good transient
structures
as
shown
in
FIGURE
12
(b).
The
result
of
the
body
acceleration
response. FIGURE 12 (a) shows the response of the vehicle from sudden acceleration test
and
body
displacement
shownmodel.
in FIGURE
12the
(c)pitch
andrate,
FIGURE
12 (d)
maneuver
which
is input for theare
simulation
In terms of
the proposed
controller gives the
a moreexplains
stable output
when
is compared
to PID structures
as
respectively.
Thesystem
FIGURE
that
theitAPID
controller
gives better
shown in FIGURE 12 (b). The result of the body acceleration and body displacement are
performance
in terms of settling time and reducing the magnitude as
shown in FIGURE 12 (c) and FIGURE 12 (d) respectively. The FIGURE explains that the
compared
to
the
APID controller givescounterparts..
better performance in terms of settling time and reducing the
magnitude as compared to the counterparts..
a) Pitch angle
b) Pitch rate
a) Pitch angle
b) Pitch rate
46
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Vol. 2
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July-December 2010
55
Journal of Mechanical Engineering and Technology
c) Body acceleration
d) Body displacement
FIGURE
12: PerformanceofofGSPID
GSPID at
at step
full full
throttle
FIGURE
12: Performance
stepfunction
function
throttle
5.3
5.3
Response Of The Controller For Unbounded Signal
Response Of The Controller For Unbounded Signal
To verify the robustness of the controller, a similar test is conducted for unstable driver input
Toofverify
the robustness
theunstable
controller,
a similar
out
predetermined
range. The testofused
brake pressure
as thetest
inputisofconducted
the sudden
braking
test and unstable
the predetermined
input of the sudden acceleration
test.test
FIGURE
for unstable
driverthrottle
inputpush
outas of
range. The
used
13 (a) shows the braking input which is used as the input for the sudden braking test. The
unstable brake pressure as the input of the sudden braking test and
result is clearly explained that the simulation under APID controller is yielded very
unstabletothrottle
pushstability
as theandinput
of the
sudden
acceleration
effectively
enhance vehicle
ride quality
in terms
of reducing
vehicle divetest.
as
Body
acceleration
d)which
Bodyand
displacement
shown
in FIGURE
13 shows
(b).
The major
difference in
magnitude
small
fluctuation
occurred
FIGURE
13c) (a)
the braking
input
is used
as the
input
FIGURE
12: flexibility
Performance
of
at
fullcondition
throttle held on the
in
graph
is due tobraking
the
theGSPID
APID
tostep
adapt
with the
forthethe
sudden
test.of The
result
is function
clearly
explained
that the
system which shows in pitch rate plot in FIGURE 13 (c). In terms of body acceleration as
simulation
under
APID
is yielded
very
effectively
to enhance
5.3
Response
Of13
The
Controller
For
Signal
shown
in FIGURE
(d),
the controller
trends
ofUnbounded
simulation
results
show
the APID significantly
vehiclewith
stability
ride quality
in terms
reducing PID.
vehicle
improve
better and
performance
as compared
to theofconventional
Due dive
to theas
To
verify the
of13
the(b).
controller,
a similar
test of
is conducted
for unstable driver
input
improvement
of the body
acceleration,
magnitude
the body
displacement
is reduced.
shown
in robustness
FIGURE
The the
major
difference
in magnitude
and
small
out
of predetermined
range.
The test used
unstable
pressure
the input
theunstable
sudden
FIGURE
13 (e) shows
the evidence
on the
vehiclebrake
remains
steadyaseven
whenofthe
fluctuation
occurred
in the graph is due to the flexibility of the APID
braking
test and
unstable throttle push as the input of the sudden acceleration test. FIGURE
driver input
is applied.
to(a)adapt
the condition
theinput
system
in pitch
13
shows with
the braking
input which held
is usedon
as the
for thewhich
sudden shows
braking test.
The
result
clearly
explained that
the simulation
APID acceleration
controller is yielded
very
rate isplot
in FIGURE
13 (c).
In termsunder
of body
as shown
effectively
to enhance
vehicle
and of
ridesimulation
quality in terms
of reducing
vehicle
as
in FIGURE
13 (d),
thestability
trends
results
show
thedive
APID
shown in FIGURE 13 (b). The major difference in magnitude and small fluctuation occurred
with ofbetter
performance
as condition
compared
the
insignificantly
the graph is dueimprove
to the flexibility
the APID
to adapt with the
held to
on the
conventional
PID.
Duerate
to plot
the in
improvement
body
acceleration,
system
which shows
in pitch
FIGURE 13 (c).ofInthe
terms
of body
acceleration the
as
shown
in FIGURE
13 (d),
the displacement
trends of simulation
results show
the APID13significantly
magnitude
of the
body
is reduced.
FIGURE
(e) shows
improve
with better
as remains
compared steady
to the conventional
PID.the
Due
to the
the evidence
onperformance
the vehicle
even when
unstable
improvement of the body acceleration, the magnitude of the body displacement is reduced.
a) shows
Braking
input
b) Pitch
driver 13
input
is applied.
FIGURE
(e)
the
evidence on the vehicle remains steady
evenangle
when the unstable
driver input is applied.
c) Pitch rate
d) Body acceleration
a) Braking input
b) Pitch angle
47
56
c) Pitch rate
ISSN: 2180-1053
Vol. 2
No. 2
d) Body acceleration
July-December 2010
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
a) Braking input
b) Pitch angle
c) Pitch rate
d) Body acceleration
47
e) Body displacement
FIGURE
Performance ofofGSPID
at unbounded
brakingbraking
input
FIGURE
13:13:Performance
GSPID
at unbounded
input
The performance of APID controller is then observed using unstable throttle input. The
The performance
of APID
controller
is then
using
simulation
test needs the vehicle
to accelerate
in a constant
speedobserved
of about 20kph,
andunstable
then
the throttle
inconstant throttle
applied as shown
in FIGURE
14(a).
In this case,
the reference in a
input.input
Theissimulation
test
needs the
vehicle
to accelerate
position
and pitch
are remained
at zero
degreeand
respectively.
Theinconstant
responses of the
vehicleinput
are
constant
speed
of about
20kph,
then the
throttle
is
described in FIGUREs 14(b) to (e). FIGUREs 14(b) and (c) explain on the performance of
applied
as shown
FIGURE
In this
reference
APID
in reducing
the pitch in
angle
and pitch14(a).
rate which,
as a case,
result, the
is able
to decreaseposition
the
vehicle
In addition,
the magnitude
of body
acceleration
(FIGURE 14(d))
the body of
andsquat.
pitch
are remained
at zero
degree
respectively.
Theandresponses
displacement
(FIGURE
(e)) of the vehicle
are visibly14(b)
reduced.
it can be said
that and
the vehicle
are 14
described
in FIGUREs
toOverall,
(e). FIGUREs
14(b)
the proposed controller has a good performance, stable and has good transient response in
(c) explain
onand
the
of APID
the the
pitch
tracking
of the system
canperformance
adapt with the stated
range andinthereducing
value in between
statedangle
and
ratebraking
which,
result,
is able test.
to decrease
the vehicle
squat.
range
heldpitch
on sudden
testas
or asudden
acceleration
The comparison
between the
APID
the conventional
PID controller of
presents
thatacceleration
the proposed schemes
are much
more and
Inand
addition,
the magnitude
body
(FIGURE
14(d))
effective in improving the stability of the vehicle during sudden acceleration and braking
the body displacement (FIGURE 14 (e)) of the vehicle are visibly
condition. As a result, the controller is verified to be able to enhance better vehicle
reduced.
it can
be said
the proposed
performance
andOverall,
riding quality
as shown
in thethat
simulation
responses. controller has a good
performance, stable and has good transient response in tracking of the
system and can adapt with the stated range and the value in between
the stated range held on sudden braking test or sudden acceleration
test. The comparison between the APID and the conventional PID
controller presents that the proposed schemes are much more effective
in improving the stability of the vehicle during sudden acceleration
and braking condition. As a result, the controller is verified to be able
to enhance better vehicle performance and riding quality as shown in
a) Throttle input
b) Pitch angle
the simulation
responses.
c) Pitch rate
ISSN: 2180-1053
Vol. 2
d) Body acceleration
No. 2 July-December 2010
57
range held on sudden braking test or sudden acceleration test. The comparison between the
APID and the conventional PID controller presents that the proposed schemes are much more
effective in improving the stability of the vehicle during sudden acceleration and braking
Journal
of Mechanical
Engineering
and Technology
condition.
As a result,
the controller
is verified to be able to enhance better vehicle
performance and riding quality as shown in the simulation responses.
a) Throttle input
c) Pitch rate
b) Pitch angle
d) Body acceleration
48
e) Body displacement
FIGURE 14: Performance of GSPID at unbounded throttle input
FIGURE 14: Performance of GSPID at unbounded throttle input
6.0
Conclusion
An adaptive
controller with pitch moment rejection for vehicle dive and squat to enhance
6.0 PID
Conclusion
vehicle stability and ride quality has been evaluated. The proposed controller which includes
the proportional,
integral
andcontroller
derivative gains
are allowed
vary withinrejection
predetermined
An adaptive
PID
with
pitch tomoment
forrange
vehicle
of the sudden braking and sudden acceleration tests. Simulation studies for an active
dive
and
squat
to
enhance
vehicle
stability
and
ride
quality
has
suspension with validated full vehicle model are presented to demonstrate the effectiveness ofbeen
evaluated.
The proposed
includes
proportional,
using
the APID controller.
Two types controller
of simulationwhich
tests namely
sudden the
braking
test and
sudden
acceleration
have been gains
performed
data gathered
fromwithin
the testspredetermined
were used as
integral
and test
derivative
areand
allowed
to vary
the benchmark of the proposed verification. Some of the vehicle’s behaviors observed in
range of the sudden braking and sudden acceleration tests. Simulation
these works are pitch rate, pitch angle, body acceleration and body displacement responses.
for an
active suspension
with validated
model
Thestudies
performance
characteristics
of the controller
are evaluatedfull
andvehicle
compared
with are
conventional
PID.toThe
result shows the
that the
use of the proposed
APID
technique
presented
demonstrate
effectiveness
of using
thecontrol
APID
controller.
proved
be effective
in controlling
pitchsudden
and vibration
and test
achieve
Twototypes
of simulation
testsvehicle
namely
braking
andbetter
sudden
performance than the conventional PID controller.
acceleration test have been performed and data gathered from the tests
of the proposed verification. Some of the
vehicle’s behaviors observed in these works are pitch rate, pitch angle,
This work is supported by the Ministry of Science Technology and Innovation (MOSTI)
body
and body
displacement
The (MoHE)
performance
through
its acceleration
scholarship and financial
support
and Ministry ofresponses.
Higher Education
of
characteristics
theentitled
controller
are evaluated
and Actuated
compared
Malaysia
through FRGSof
project
“Development
of a Pneumatically
Activewith
Stabilizer
Bar to Reduce
Motion
In that
Longitudinal
lead by Dr.
conventional
PID. Unwanted
The result
shows
the useDirection”
of the proposed
APID
Khisbullah Hudha at Universiti Teknikal Malaysia Melaka. This financial support is
control technique proved to be effective in controlling vehicle pitch and
gratefully acknowledged.
7.0 were
Acknowledgement
used as the benchmark
8.0
Reference
58
ISSN: 2180-1053
Vol. 2 No. 2 July-December 2010
Ahmad, F., Hudha, K., Said. M. R. and Rivai. A. (2008a). Development of Pneumatically
Actuated Active Stabilizer Bar to Reduce Vehicle Dive. Proceedings of the
Adaptive Pid Control With Pitch Moment Rejection For Reducing Unwanted Vehicle Motion In Longitudinal Direction
vibration and achieve better performance than the conventional PID
controller.
7.0
Acknowledgement
This work is supported by the Ministry of Science Technology and
Innovation (MOSTI) through its scholarship and financial support
and Ministry of Higher Education (MoHE) of Malaysia through FRGS
project entitled “Development of a Pneumatically Actuated Active
Stabilizer Bar to Reduce Unwanted Motion In Longitudinal Direction”
lead by Dr. Khisbullah Hudha at Universiti Teknikal Malaysia Melaka.
This financial support is gratefully acknowledged.
8.0Reference
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APPENDIXES
APPENDIXES
Controller Parameters for Body Bounce (Fz )
Controller Parameters for Body Bounce Fz Ͳ5
0
5
Ki
Kp
Ͳ10
2500
6000
5000
4000
3000
2000y=162.3x+2231.
1000
0
2000
y=Ͳ131.2x+720.6
1500
1000
500
0
10
Ͳ10
longitudinalacceleration
Ͳ5
0
5
10
longitudinalacceleration
(a): Kp value
ISSN: 2180-1053
(b): Ki value
Vol. 2
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July-December 2010
52
63
Journal of Mechanical Engineering and Technology
4000
4000
3000
3000
3000
2000
y=Ͳ106.0x+1660.
2000
2000
y=Ͳ106.0x+1660.1000
y=Ͳ106.0x+1660.
1000
1000
0
0 0
Ͳ10
Ͳ5
5
0
Ͳ105
Ͳ510
0
5
Ͳ10
Ͳ5
0
longitudinalacceleration
longitudinalacceleration longitudinalacceleration
Kd
Kd Kd
4000
10
10
(c): Kd Value
(c): Kd Value
(c): Kd Value
Controller
Parameters
for (Body
Controller Parameters
for Body
Pitch
M q )Pitch M T Controller
Parameters
Controller Parameters for
Body Pitch
M T for Body Pitch M T 800
4000
800
4000
800
3000
600
3000
600
3000
600
2000
y=Ͳ56.27x+243.7
400
2000
y=Ͳ56.27x+243.7
400
2000
y=Ͳ56.27x+243.7
400
1000
y=121.2x+2015.
200
1000
y=121.2x+2015.
200
1000
0
y=121.2x+2015.
200
0
0 0
Ͳ10
Ͳ5
5
10
0
0 0
Ͳ10
Ͳ5
5
0
Ͳ200
Ͳ10 5
Ͳ5 10
0
5
10
Ͳ5
0
Ͳ10 5
Ͳ5 10
0
5
longitudinalacceleration
longitudinalacceleration
Ͳ10
Ͳ5
0
Ͳ200
Ͳ200
longitudinalacceleration longitudinalacceleration longitudinalacceleration longitudinalacceleration
Ki Ki
Ki
Kp
Kp Kp
4000
Ͳ10
(a): Kp value
Kd Kd
2500
y=Ͳ133.6x+1261
2000
Kd
1500
1000
500
0
Ͳ10
Ͳ5
0
Ͳ10
Ͳ10 5
(a): Kp value
(a): Kp value
(b): Ki value
2500
y=Ͳ133.6x+1261
2500
2000
y=Ͳ133.6x+1261
2000
1500
1500
1000
10
10
(b): Ki value
(b): Ki value
1000
500
5000
0 0
Ͳ5
5
Ͳ5 10
0
5
Alongitudinalacceleration
10
10
Alongitudinalacceleration Alongitudinalacceleration
(c): Kd value
(c): Kd value
(c): Kd value
53
64
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5
53