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Active suspension for single wheel station of off-road track vehicle

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*
Correspondence to: A. Liberzon, Faculty of Agricultural Engineering, Technion
}
Israel Institute of Technology, Haifa32000, Israel.
Copyright
2001 John Wiley & Sons, Ltd.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int
.
J
.
Robust Nonlinear Control
2001;
11
:977
}
999 (DOI: 10.1002/rnc.636)
Active suspension for single wheel station of o
!
-roadtrack vehicle
A. Liberzon
*
, D. Rubinstein and P. O. Gutman
Faculty of Agricultural Engineering, Technion
}
Israel Institute of Technology, Haifa 32000, Israel
SUMMARYThemainobjectiveofgroundvehiclesuspensionsystemsistoisolateavehiclebody(sprungmass)fromroadirregularities in order to maximize passenger ride comfort, control the attitude of the vehicle on the road,andretaincontinuousroad
}
wheelcontactinordertoprovidevehicleholdingquality.Anappropriateactivesuspension design must resolve the inherent tradeo
!
s between ride comfort, road holding quality, andsuspension travel. In this study, a robust controller for the active suspension of an o
!
-road, high-mobilitytracked vehicle is designed, for the
"
rst time, using quantitative feedback theory (QFT). A simulation modelof a single suspension unit of the M-113 armored personnel carrier vehicle was used to achieve a properactive suspension design. Two measured states of the 3-degrees-of-freedom mathematical model were usedas feedback signals in a cascaded SISO control system. The nonlinear dynamics of the tracked vehiclesuspension unit was represented as a set of linear time invariant (LTI) transfer functions, which wereidenti
"
ed with the Fourier integral method. Computer simulations of the vehicle with passive and activesuspension systems over di
!
erent terrain pro
"
les are provided. A signi
"
cant reduction of the verticalaccelerations induced to the sprung mass (e.g., ride comfort improvement) was achieved while keeping theroad-armbetweensuspensiontravellimits (e.g., handlingquality). Copyright
2001JohnWiley&Sons,Ltd.
KEY WORDS
: active suspension; o
!
-road; track vehicle; robust control; non-linear control; QFT
1. INTRODUCTIONThe main objectives of ground vehicle suspension systems is to support a vehicle body, maximizepassenger ride comfort and retain the vehicle stability during turning, breaking and otherhandling actions.It has long been recognized that in passive suspension systems design using springs anddampers, each of these three goals con
#
icts with the others. Long suspension travel is needed todecrease the vertical acceleration of the vehicle body, but this increases the probability of hittingmechanical bump-stops (e.g. suspension travel limits). This causes an extreme vertical acceler-ation of the vehicle body and, therefore, decreases the ride quality. An appropriate activesuspensiondesignmustresolvetheinherenttradeo
!
sbetweenridecomfort,roadholdingquality,and suspension travel.
The design and development of active suspension control algorithms has been undertaken witha wide diversity of control approaches including modal analysis, classical techniques includingpole placement and Bode plots, bond-graph modelling, optimization and optimal control,nonlinear control, and more recently neural networks and fuzzy control [1, 2].The present paper will focus on the question of the extent to which robust linear control issuitable to the inherently nonlinear active suspension control problem. The investigation isconducted by applying quantitative feedback theory (QFT) to the design of a cascaded robustcontroller for the active suspension of an o
!
-road high-mobility tracked vehicle. A model of a single suspension unit of the M-113 armored personnel carrier vehicle was used to get a properactive suspension design. Two measured states of the 3 degrees-of-freedom mathematical modelwere used as feedback signals in a cascaded SISO control system. The nonlinear dynamics of thetracked vehicle suspension unit is represented as a set of linear time invariant (LTI) transferfunctions [3], which were identi
"
ed with the Fourier integral method or Correlation method[4, 5].The performanceof theactivesuspensionsystemis evaluated by running simulationsof thesystem subjected to discrete and realistic road inputs. The discrete road input models a channeland half-circular bump disturbances. The realistic road input is represented by a measured roadpro
"
le.Section 2 describes the suspension model development. The control design is described inSection 3, including the uncertain linear transfer function identi
"
cation technique, which ispresented in section 3.1. The case study is described in Section 4 and followed by conclusions inSection 5.2. SUSPENSION MODEL
2.1. Physical model
The model for which the QFT control scheme was developed is shown in Figure 1. This modelrepresents one of the ten suspension stations and about one-tenth of the hull (vehicle centralbody) mass of an M-113 armored personnel carrier vehicle, for which detailed physical data weretaken from the literature [6, 7]. The road
}
wheel is mounted on a trailing arm (road-arm) and thecentral body (sprung mass) is allowed to move in the horizontal (
x
) and vertical (
x
) directions.The road
}
wheel can move freely between two mechanical bump-stops with respect to the groundand the sprung mass (
x
). Since the central body of the model is constrained to move only invertical and horizontal directions, the model can be noted to be similar to the well-known
carmodel for a four-wheeled vehicle. The 3-degrees-of-freedom model is depicted in Figure 1. Thenumerical values are listed in Table I.In this paper we deal with the case in which the suspension unit components includea controlled active actuator that is designed to work in parallel to the existing passive compo-nents (the torsionalrod and the translationaldamper). The case of fully active suspension withoutpassive components is not provided in this paper. The details of both cases are provided inLiberzon [8].
2.2. Mathematical model
The mathematical model of the single-wheel station represents the dynamic behaviour of thevehicle over a rough terrain. The 3-degrees-of-freedom model is depicted in Figure 1, where the978
A. LIBERZON, D. RUBINSTEIN AND P. O. GUTMANCopyright
2001 John Wiley & Sons, Ltd.
Int
.
J
.
Robust Nonlinear Control
2001;
11
:977
}
999
Figure 1. Model of a single-wheel suspension station, where 1
"
vehicle body; 2
"
lineardamper; 3
"
road-arm; 4
"
wheel; 5
"
torsion spring; 6
"
mechanical bump-stop; 7
"
bump-stop that models track-wheel interaction; and 8
"
torsion actuator model.Table I. Suspension model data.Parameters ValuesMass of the body 1600 kgMass of the road-arm 9.98 kgMass of the wheel 40.82 kgTorsion spring coe
$
cient 7900 Nm/radLinear damping coe
$
cient 18
10
Nm/rad/sWheel
}
soil sti
!
ness 2
10
N/mWheel
}
soil damping 100 N/m/sBump-stop sti
!
ness 7.3
10
N/mMaximum angle in bump
!
0.2 radMaximum angle in rebound 0.45 rad
states of the model are de
"
ned as:
x
: horizontal coordinate of the central body, measured at its centre of mass [m];
x
: vertical coordinate of the central body, measured at its centre of mass [m];
x
: relative angle between road-arm axis and horizontal axis, passing through road-arm bearingpoint [rad];
x
"
x
R
: horizontal velocity of the central body [m/s];
x
"
x
R
: vertical velocity of the central body [m/s];
x
"
x
R
: relative angular velocity of the road-arm [rad/s].
ACTIVE SUSPENSION FOR SINGLE WHEEL STATION
979
Copyright
2001 John Wiley & Sons, Ltd.
Int
.
J
.
Robust Nonlinear Control
2001;
11
:977
}
999
The equations of motion are derived by Lagrange formulation in its classic form [9, 10]:dd
t
¹
x
R
!
¹
x
#
;
x
"
Q
(1)where
¹"
kinetic energy,
;
"
potential energy,
x
"
generalized coordinate
i
,
Q
"
generalizedforce
i
,
i
"
1,2,3. The kinetic energy
¹
of the suspension model is given by the followingexpression:
¹"
M
(
x
#
x
)
#
M
(
x
R
#
y
R
)
#
I
x
(2)where
M
is the mass of the body,
M
"
M
#
M
, the mass of the suspension unit (theroad-arm and the wheel),
I
*
moment of inertia of the suspension unit, and
x
,
y
*
coordi-nates of the common centre of mass of the road-arm and the wheel. Furthermore,
x
"
x
!
r
cos(
x
) [m]
y
"
x
!
r
sin(
x
) [m]
r
"
(
l
/2
M
#
l
M
)
M
[m]
I
"
I
#
M
r
!
l
2
#
M
(
r
!
l
)
[kgm
]where
l
"
0.375 m
*
length of the road-arm and
I
*
moment of inertia of the road-arm. Thepotential gravitational energy
;
[
J
] is
;
"
(
M
x
#
M
y
)
g
(3)where
g
is the gravity acceleration.The kinetic energy in Equation (2) explicitly depends on the generalized coordinates and theirtime derivatives. Therefore, the time derivatives of the Lagrange equations can be written asdd
t
¹
x
R
"
¹
x
R
x
R
x
(
#
¹
x
R
x
R
x
R
(4)A general representation of the equations of motion (1) can be arranged in the form
Ax
(
#
Bx
R
!
C
"
Q
(5)where
A
and
B
are 3
3 matrices and
C
is a vector of length 3, with the following elements:
a
"
¸
x
R
x
R
b
"
¸
x
R
x
(6)
c
"
¸
x
where
¸
denotes the Lagrangian
¸"¹!
;
(7)980
A. LIBERZON, D. RUBINSTEIN AND P. O. GUTMANCopyright
2001 John Wiley & Sons, Ltd.
Int
.
J
.
Robust Nonlinear Control
2001;
11
:977
}
999
Applying the values given in Table I and above, we get the following matrices:
A
"
1650.8 0 14.5 sin(
x
)0 1650.8
!
14.5 cos(
x
)14.5 sin(
x
)
!
14.5 cos(
x
) 4.7
B
"
0 0 14.5 cos(
x
)
x
0 0 14.5 sin(
x
)
x
0 0 14.5 cos(
x
)
x
#
14.5 sin(
x
)
x
(8)
C
"
0
!
1.6
10
14.5 cos(
x
)
x
x
#
14.5 sin(
x
)
x
x
#
142.7 cos(
x
)The generalized forces
Q
of all forces and torques, except the gravitational forces, are obtainedfrom the virtual work principle
Q
"
F
M
r
N
x
#
M
M
M
x
(9)where
F
M
and
M
M
are forces and torques, respectively. These include internal and external forcesand torques, which are shown in Figure 1. The internal forces and torques are those that arecreated by the suspension components:
¹
"
torsion spring torque, applied to the road-arm at its pivot [Nm];
F
"
nonlinear damper force, acting along the line between the two end points of the damper[N];
F
"
mechanical bump-stop force, which is produced while the road-arm hits the bump-stop[N];
M
"
torque, which models the interaction between the track and the road-arm [Nm];The external forces are created by the soil, the track, the drive element, and the control system:
F
"
soil
}
wheel interaction forces, acting on the wheel at its centre, in the vertical direction [N];
F
"
soil
}
wheel interaction force, acting on the wheel at its centre, in the horizontal direction[N];
F
"
driving force, applied by the spring
}
damper model [N];
F
"
torque of the torsional actuator, which is provided by the control system [Nm].The vector
Q
of generalized forces is
Q
"
F
#
8.3
10
!
1.5
10
x
F
0.3
F
sin(
x
)
!
0.3
F
cos(
x
)
#
F
which includes implicitly the driving force
F
"
1.5
10
(
<
!
x
) [N], where
<
"
10 m/s is thedesired velocity of the vehicle during the presented simulation.
ACTIVE SUSPENSION FOR SINGLE WHEEL STATION
981
Copyright
2001 John Wiley & Sons, Ltd.
Int
.
J
.
Robust Nonlinear Control
2001;
11
:977
}
999

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