Modeling-of-steady-state-performance-of-skid-steering-f_2017_Journal-of-Terr

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Journal of Terramechanics 73 (2017) 25–35
Journal
of
Terramechanics
www.elsevier.com/locate/jterra
Modeling of steady-state performance of skid-steering for
high-speed tracked vehicles
Shouxing Tang a , Shihua Yuan a,⇑ , Jibin Hu a , Xueyuan Li a , Junjie Zhou a , Jing Guo b
a Science and Technology on Vehicular Transmission Laboratory, Beijing Institute of Technology, Beijing 100081, China
b National Key Lab of Vehicular Transmission, China North Vehicle Research Institute, Beijing 100081, China
Received 25 March 2017; received in revised form 9 June 2017; accepted 12 June 2017
Available online 13 July 2017
Abstract
This paper presents a high-fidelity, general, and modular method for lateral dynamics simulation of high-speed tracked vehicle. Based
on classic terramechanics, a novel nonlinear track terrain model is derived. The track terrain model meets the needs of longitudinal and
steering motions, comprehensive track slips, and modular modeling for tracked vehicles with various configurations. The proposed lateral
dynamics model is in reasonably agreement with the available experimental data. Using the lateral dynamics model, the main factors
(normal pressure distribution, position of gravity center, and ratio of track-ground contact length and tread L=B) effecting the steady
state characteristics of skid steering are discussed. The normal pressure distribution is idealized as trapezoid and dual trapezoid distribution
to reflect same common driving situation. The under-steer parameter is introduced in this paper to quantitatively evaluate the
steady-state characteristics of skid steering for tracked vehicle. The results show that the ratio of theoretical speed difference and average
speed of both side tracks Du=u as the steering input is more suitable for the high-speed tracked vehicle. The vehicle with dual trapezoid
normal pressure distribution slightly tending to localize in the middle of track or with slightly rearward position of gravity center has
better handling stability characteristics.
Ó 2017 ISTVS. Published by Elsevier Ltd. All rights reserved.
Keywords: Tracked vehicle; Unmanned vehicle; Skid steering; Lateral dynamics; Steering characteristics
1. Introduction and related research
Tracked vehicles have great mobility, traversability, and
payload carrying capacity on extremely difficult terrain
that wheeled vehicles haven’t (Hohl, 2007). They have been
widely used in many fields such as unmanned/manned military,
agriculture and construction operations (Wong,
2008; Wong et al., 2015; Janarthanan et al., 2011). Whether
wheeled or tracked vehicles, steering characteristics is a primary
research field in their design, manufacture, test, and
control. In recent years, steering characteristics is widely
⇑ Corresponding author at: School of Mechanical and Vehicular
Engineering, Beijing Institute of Technology, Beijing 100081, China.
E-mail address: bityuanshihua@163.com (S. Yuan).
concerned in the unmanned ground vehicles research field,
because it has large impact on the motion planning and
path tracking, especially in high-speed operation
(Urmson et al., 2008; Genya et al., 2007). Researchers have
broadly studied the lateral dynamics of wheeled vehicle and
made a great number of achievements (Gillepie, 1992;
Pacejka, 2006; Vantsevich, 2015). These achievements have
formed a complete theoretical system to support the development
of high-speed wheeled vehicle. In contrast, there
are not enough researches on lateral dynamics of tracked
vehicles to support the development of high-speed tracked
vehicles.
The initial research on steering characteristics of tracked
vehicle was presented by Merritt (1939), who assumed that
the forces generated between the track and terrain obey the
http://dx.doi.org/10.1016/j.jterra.2017.06.003
0022-4898/Ó 2017 ISTVS. Published by Elsevier Ltd. All rights reserved.

26 S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35
Nomenclature
Du velocity difference
/ terrain internal friction angle
u heading angle
x yaw velocity
x s rotating angular speed of sprocket
x t yaw angular velocity of track coordinate system
h the angle between the comprehensive absolute
velocity and the longitudinal direction of the
track coordinate system
d longitudinal slips of track
s the shear stress on an element of the trackground
interface
j under-steer parameter
b width of track
B tread of vehicle
c terrain cohesion
dF shear force developed on track element dA
dA area of track element
F y lateral force acting on the track
F x longitudinal force acting on the track
h height of center of gravity
I z rotational inertia about z axis of vehicle
j x shear displacement along the x t direction at
point (x t , y t )
j y shear displacement along the y t direction at point
(x t , y t )
j resultant shear displacement at the point (x t , y t )
K terrain shear deformation modulus
k concentration factor of normal pressure distribution
k 2 front and rear distribution factor
L track-ground contact length
l x longitudinal offset of center of gravity to the
geometrical center of the vehicle
m mass of vehicle
M r turn resistance moment acting on the track
pðx t ; y t Þ normal pressure distribution on the track
r radius of sprocket
R turning radius
sðx t Þ the normal distribution density function on the
track
t duration of track element from contacting the
ground initially to point (x t , y t )
u Actual forward velocity of vehicle
u t longitudinal velocity of the track coordinate system
u r theoretical driving velocity of vehicle
v Actual lateral velocity of C.G.
v t lateral velocity of the track coordinate system
V y relative velocity component of arbitrary point
(x t , y t ) on the track-ground interface in the y t
direction
V x relative velocity component of arbitrary point
(x t , y t ) on the track-ground interface in the x t
direction
W gravity of vehicle
W L normal load on left track
W R normal load on right track
xyz body fixed coordinate system with origin at the
C.G.
x t y t z t track coordinate system is fixed on the center of
track-ground interface and moves with vehicle
XYZ global coordinate system
law of Coulomb friction. Subsequently, many researchers
established kinds of dynamic models and obtained many
innovative conclusions, taking different factors and
assumptions into consideration. Micklethwait (1944) presented
the friction between the track and terrain is anisotropic
in the longitudinal and lateral directions. Hock
(1961) and others introduced some empirical formulas to
characterize the phenomenon that the lateral friction coefficient
decreases with the increase of the turning radius.
Crosheck (1975) introduced the pull-slip equation to
describe the interaction of track and terrain. In order to
find out the adequate vehicle model for the control of
PAISI tracked vehicle test system, Ehlert et al. (1992) used
the tank Jaguar to do some test. Based on test results, he
modified the Hock model, IABG model, and Kitano model
and finally found out an adequate model. Wong and
Chiang (2001) presented a general theoretical steady-state
model of tracked vehicles, considering the shear stressshear
displacement relationship on the track-terrain interface.
The purpose of above-mentioned researches is to predict
steering ability of tracked vehicle and estimate the load
on the steering mechanism of tracked vehicle.
Kitano and Kuma (1977) established a theoretical nonstationary
model based on pull-slip equation to analyze
and predict steady-state and transient steering response
of tracked vehicle on different velocity. In the same way
Janarthanan et al. (2011) developed a 5 DOF nonstationary
model to study the handling behavior at high
or low speeds employing different types of steering input.
Base on the law of Coulomb friction Purdy and Wormell
(2003) established the mathematical model of CVR (Combat
Vehicle Reconnaissance) tracked vehicle to analyze the
steering performance. Similar with tires model in lateral
dynamics of on-road wheeled vehicle, the track-terrain
interaction model is the mechanical foundation of dynamics
model of tracked vehicle, which directly influences the
accuracy of tracked vehicle simulation, particularly the lateral
mechanics. Above-mentioned track-terrain models
have different lateral shear stress distributions along the
longitudinal direction, which may have significant effects

S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35 27
on the steering characteristics of tracked vehicles. To date,
the track-terrain interaction has been well studied in the
research field called Terramechanics. In this field shear
stress-shear displacement relationship is believed to be
the more rational formulation of track-terrain interaction
than others so far (Wong, 2008).
This paper describes a high-fidelity, general, and modular
method for lateral dynamic simulation of high-speed
tracked vehicle. In this method, a novel nonlinear track terrain
model is derived based on classic terramechanics. This
track terrain model meets the needs of longitudinal and
steering motions of vehicle, comprehensive track slips,
and modular modeling for tracked vehicles with various
configurations. This lateral dynamic model is verified by
comparison with the available experimental data. Using
this model, handling and stability characteristics are sufficiently
investigated in high-speed steering situation, considering
the main factors (normal pressure distribution,
position of gravity center, ratio of track-ground contact
length and tread L=B). The results of this study are very
useful for the design and control of unmanned/manned
high-speed tracked vehicles.
The rest of this paper is organized as follows: Section 2
presents the novel track terrain modeling. Section 3
describes the modular lateral dynamic model of tracked
vehicle. Section 4 presents model verification by comparison
with the available experimental data. Section 5 presents
the analysis on handling and stability characteristics
by using this model. Finally, concluding remarks are summarized
in Section 6.
2. Track-terrain modeling
The various ways that a tracked vehicle could be steered
are skid-steering, articulated-steering, and curved-steering
(Kar, 1987; Liu and Liu, 2009). In general, an independent
track-terrain model can be proposed for lateral dynamic
modeling for tracked vehicles with various configurations.
The track-terrain model is able to calculate the track forces
and moment under the normal pressure distribution
pðx t ; y t Þ and the track motions inputs, as illustrated in
Fig. 1. The track motions are described as longitudinal
velocity component u t , lateral velocity component v t , and
yaw angle velocity x t of the track coordinate system in
track coordinate system, and the sprocket rotational speed
x s , as shown in Fig. 2. The track coordinate system is
defined moving with the vehicle, and its origin is at the center
of track-terrain interface, as shown in Fig. 2. The forces
and moment are longitudinal force component F x , lateral
force component F y and turn resistance moment M r acting
on the track. The track-terrain model is derived from the
track-terrain interaction model and contains the track
parameters and terrain parameters.
Various track-terrain interaction models have been
described in the introduction. The model selected for a
given application may consider corresponding factors. This
study aims at steering characteristics of high-speed tracked
vehicle. Therefore, the lateral shear stress distribution
along the longitudinal direction on the track terrain interface
is of prime importance. In classic terramechanics, the
theoretical relationship was confirmed through experiment.
For typical terrain, the shear stress initially increases
rapidly with increase of shear displacement, and then
approaches a constant value with a further increase in
shear displacement (Wong, 2010). This type of shear
stress-shear displacement relationship may be described
by an exponential function of the following form
s ¼ s max 1 e j K ¼ ðc þ r tan / Þ 1 e j K
ð1Þ
where s is the shear stress, J is the shear displacement, K is
the shear deformation modulus, s max is the maximum shear
stress, r is the normal stress, c is the terrain cohesion, and /
is the angle of internal shearing resistance of the terrain.
Therefore, the shear stress at arbitrary point on the trackground
interface is related to the shear displacement at that
point. In order to avoid complex curvilinear integral and
simplify calculate process, the curve trajectory of track pad
moving on the ground can be approximated by the straight
line. And the direction of the shear stress at a point on
track–ground interface is assumed to opposite to the direction
of the relative sliding velocity of the track with respect
to the ground at that point (Wong and Chiang, 2001;
Muro and O’Brien, 2006). Then the shear displacement
can be obtained by integrating the relative shear velocity
between the track and the ground. Relative shear velocity
components of arbitrary point (x t , y t ) on the track-ground
interface in the x t and y t direction can be expressed by
Fig. 1. Track-terrain model schematic diagram.

28 S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35
Fig. 2. Single track motions on terrain and forces and moment applied.
V x ¼ u t y t x t rx s ð2Þ
V y ¼ v t þ x t x t
ð3Þ
where r is the radius of the sprocket, u t is the longitudinal
velocity of track coordinate system, v t is the lateral velocity
of track coordinate system, and x t is the yaw velocity of
track coordinate system.
Then the duration of track element from contacting the
ground initially to point (x t ,y t ) can be given by
Z t
Z L
2
dx t
t ¼ dt ¼ ¼ L=2 x t
ð4Þ
0
x t
rx s rx s
where L is the track-terrain contact length.
Consequently, The shear displacement component along
the x t direction at point (x t , y t ) is derived by integrating the
relative shear velocity component along the x t direction.
Z t
Z L
2
J x ¼ V x dt ¼ ðu t y t x t rx s Þ dx t
0
x t
rx s
L
x
2 t
¼ ðu t y t x t rx s Þ
ð5Þ
rx s
And the shear displacement component along the y t
direction at point (x t ,y t ) is derived by integrating the relative
shear velocity component along the y t direction.
Z t
Z L
2
J y ¼ V y dt ¼ ðv t þ x t x t Þ dx t
0
x t
rx s
h i
L
x
2 t vt þ 1 L 2
2 2
x t2
x t
¼
ð6Þ
rx s
The comprehensive shear displacement at point (x t , y t )
can be expressed by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j ¼ J 2 x þ J 2 y
ð7Þ
As mentioned above, the relationship between the shear
stress and the shear displacement satisfies the Eq. (1).
Therefore, the shear force developed on a track element
dA in contact with the ground can be expressed by
dF ¼ sdA ¼½c þ px ð t ; y t Þtan /Š 1 e j K
dA
ð8Þ
As shown in Fig. 2, the direction of shear force element
is opposite to the direction of relative velocity at the point
(x t , y t ) on the track-ground interface. The component of the
shear force element along the x t direction constitutes tractive
or braking force F x . The component of the shear force
element along the y t direction constitutes the lateral force
F y . The moment of the shear force element about the origin
of the track coordinate system constitutes the moment of
turning resistance. The angle between the comprehensive
relative velocity and the longitudinal direction of the track
coordinate system can be defined by the following
h ¼ arctan
V y
V x
The longitudinal force F x and the lateral force F y acting
on the track can be expressed by
Z Z b
2
Z L
2
F x ¼ dF x ¼ ½c þ px ð t ; y t Þtan / Š 1 e j K cos hdx t dy t
b L
2 2
Z
F y ¼
ð9Þ
ð10Þ
Z b
2
Z L
2
dF y ¼ ½c þ px ð t ; y t Þtan / Š 1 e j K sin hdx t dy t
b L
2 2
ð11Þ
where b is the width of the track.
The moment of the turning resistance M r about the center
of track-ground interface acting on the track can be
expressed by
Z
M r ¼ x t dF y
¼
Z b
2
Z L
2
½c þ px ð t ; y t Þtan /Šx t 1 e j K sin hdx t dy t
b L
2 2
ð12Þ
The proposed track terrain model doesn’t contain any
configuration information of tracked vehicle. It makes it
possible to support the modular lateral dynamic simulation
for high-speed tracked vehicles with various configurations.
To improve the accuracy in aggressive maneuvering conditions,
the combination of the longitudinal shear and the

S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35 29
lateral shear is used to calculate the shear stress. In the
model proposed by Wong and Chiang (2001), the yaw
velocity x t is retained in the denominator. So this model
(Wong and Chiang, 2001) is not applicable to the longitudinal
driving (x t ¼ 0). But this problem is solved in this
study.
3. Vehicle dynamic model
The object of this study is the skid-steering tracked vehicle,
which employs the double-differential steering mechanism
with two input shafts. By using the input shafts,
driving and steering can be controlled, independently. So
the forward velocity may be kept constant during steadystate
turning operation. To improve the modeling efficiency,
we adopt the popular modular method. The topology
of the vehicle system model is illustrated in Fig. 3. The
input of overall simulation model is from the driver, who
controls the powertrain. In this paper, the dynamics of
powertrain is neglected. The powertrain module just contains
the simplified kinematics relationship between the
command and the sprocket speeds. The track terrain model
is proposed in Section 2, which calculates the track forces
and moment under the normal pressure distribution and
the track motion inputs. The vehicle body module calculates
the motions of vehicle under the track forces and
moment inputs. And the normal pressure distribution module
calculates the dynamic normal pressure distribution
under the accelerations inputs.
In deriving the equations of vehicle dynamic model, the
basic assumptions for this model are as follows:
(1) The xyz coordinate system is fixed on the vehicle’s
center of gravity and moves along with the vehicle.
The x tL y tL z tL is the track coordinate system of left
track, and the x tR y tR z tR is the track coordinate system
of right track.
(2) The vehicle is symmetric with respect to xz-plane.
(3) The vehicle drives on the horizontal ground. Track
sinkage and the bulldozing effect in the lateral direction
are neglected.
(4) The rolling resistance and aerodynamic resistance are
neglected. Because that turn resistance is usually much
larger than rolling resistance (Ehlert et al., 1992).
3.1. Kinematics of vehicle
Fig. 4 depicts the model of the tracked vehicle planar
motions with three degrees of freedom on horizontally flat
terrain. The three degrees of freedom are longitudinal
velocity u, lateral velocity v and yaw velocity x. Then the
left and right tracks motion can be expressed:
u tL ¼ u x B 2
v tL ¼ v þ xl x
u tR ¼ u þ x B 2
v tR ¼ v þ xl x
x tL ¼ x tR ¼ x
ð13Þ
where l x is the longitudinal offset of the vehicle center of
gravity to the geometrical center of the vehicle.
The steering input is the theoretical velocity difference
between the relative velocity of right track and the relative
velocity of left track with respect to vehicle body.
Du ¼ x sR r x sL r ð14Þ
where x sR r is the relative velocity of right track with respect
to vehicle body, x sL r is the relative velocity of left track
with respect to vehicle body, r is the radius of sprocket,
x sL is the rotating angular speed of left sprocket, and x sR
is the rotating angular speed of right sprocket.
Fig. 3. Simulation model architecture.

30 S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35
W R ¼ W 2 þ h B m v x
ð18Þ
Fig. 4. Kinematics of the tracked vehicle planar motions.
Because the rolling resistance and aerodynamic resistance
are neglected, the driving input theoretical velocity
u r can be considered equivalent to the actual longitudinal
velocity u.
u u r ¼ x sRr þ x sL r
ð15Þ
2
For given driving input and steering input, the rotating
angular speeds of sprockets can be obtained
2u þ Du
x sR ¼ ð16Þ
2r
x sL ¼ 2u Du
ð17Þ
2r
3.2. Normal pressure distribution
The flat terrain is assumed, so the static normal loads on
two tracks are equivalent. Due to the lateral centrifugal
force, the normal load on right track W R and the normal
load on left track W L can be expressed by
W L ¼ W h
2 B m v x
ð19Þ
where W is the gravity of the vehicle, m is the mass of the
vehicle, h is the height of center of gravity, and m v x is
the lateral centrifugal force ma y , as shown in Fig.5a.
The distribution of normal pressure on the track-ground
interface is a significant factor for maneuverability of
tracked vehicle. In this paper, the tread of vehicle B is far
greater than track width. So the difference of normal pressure
along the y t direction can be neglected (Wong and
Chiang, 2001; Muro and O’Brien, 2006). For analyzing different
situations, the normal distribution density functions
are defined as sðx tL Þ and sðx tR Þ, respectively. When the normal
pressure distribution are uniform, sðx tR Þ = sðx tL Þ¼1.
The normal pressure distribution on right track p R ðx tR Þ
and the normal pressure distribution on left track p L ðx tL Þ
can be expressed by [see Fig.5b]
p R ðx tR Þ ¼ W R
bL sx ð tRÞ ð20Þ
p L ðx tL Þ ¼ W L
bL sx ð tLÞ ð21Þ
3.3. 3DOF vehicle dynamic model
Fig.4 shows the forces and the moments acting on the
tracked vehicle during a steady state turning operation.
The dynamic equations can be obtained from dynamic balance
among all forces, moments, inertial forces and inertial
moments.
B
F yR þ F yL lx F xL
2 þ F B
xR
2 þ M rL þ M rR ¼ I z _x ð22Þ
F yR þ F yL ¼ m ð_v
uxÞ ð23Þ
F xR þ F xL ¼ m ð_u þ vxÞ ð24Þ
where _x is the angular acceleration about the z axis, _v is the
lateral acceleration along the y direction,_u is the longitudi-
Fig. 5. Normal pressure distribution on the track-ground interface.

nal acceleration along the x direction, I z is the rotational
inertia of the vehicle about the z axis.
In this paper, the vehicle is in steady turn, the longitudinal
velocity u, the yaw rate x, and lateral velocity v are constant.
The acceleration _x, _u, and _v equal to zero.
4. Model verification
S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35 31
In this section, the proposed model has been verified by
comparison with available data reported by Rui et al.
(2015). The experimental platform is a modified tracked
armored vehicle. The vehicle parameters are shown in table
1. The experiment is in progress on sandy terrain, whose
terrain parameters are shown in table 2. The speed and torque
sensors are attached on both output shafts of transmission.
The speed and torque of sprockets can be translated
from the measurement values of the speed and torque sensors.
The differential GPS system with base station is also
equipped. This system can provide real-time accurate data
of position, velocity, and orientation. The actual turning
Table 1
Vehicle parameters.
Item Units Value
m kg 20,380
L m 4.51
B m 2.84
b m 0.38
h m 1.11
r m 0.26
l x m 0.358
Table 2
Terrain parameters.
Item Units Value
K m 0.12
c kPa 0
/ ° 39
Fig. 7. Longitudinal slips of both tracks with the actual turning radius.
radius, longitudinal velocity, lateral velocity, and yaw rate
of the vehicle are obtained based on GPS data.
Fig. 6 shows the variations of the longitudinal forces
acting on both tracks with the actual turning radius. The
results show that theoretical longitudinal force curves are
in agreement with the corresponding experimental data
with relatively good accuracy. And it also shows that the
longitudinal forces acting on the both tracks decrease with
increasing actual turning radius.
Fig. 7 shows the variations of longitudinal slips d for
both tracks with the actual turning radius. The results show
that theoretical longitudinal slips curves are in agreement
with the corresponding experimental data with relatively
good accuracy. It also shows that the longitudinal slips acting
on both tracks decrease with increasing actual turning
radius.
Based on above comparisons, the proposed vehicle
dynamic model demonstrates adequate accuracy in predicting
steering performance.
5. Simulation results and discussions
This tracked vehicle is modeled in Section 2 and 3. And
the proposed vehicle dynamic model is verified by comparison
with available data in Section 4. In this section, the
steady-state characteristics are investigated under different
factors, which include types of steering input, distribution
of normal pressure, the position of gravity center, and
the ratio of track-ground contact length and tread L=B.
The object of this paper is high-speed tracked vehicle,
whose top speed is more than 70 km/h so far. So the speed
in simulation is in the range from 40 km/h to 70 km/h. The
simulation parameters are reported in Tables 1 and 2.
5.1. Types of steering input
Fig. 6. Longitudinal forces acting on both tracks with the actual turning
radius.
The typical types of steer input are the velocity difference
Du and the velocity difference rate Du=u. In the traditional
steering theory, the tracked vehicle with doubledifferential
steering mechanism satisfied the kinematics
Eq. (25) (neglecting track slips). For a given Du, the turning
radius R increases with increasing forward velocity u. The

32 S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35
Du is selected as the steering input, which makes the vehicle
has the inherent under-steer characteristic. Zhang et al.
(2014) derived that the velocity difference rate Du=u is
equivalent to the steering angle of front wheels for the
Ackermann-steer vehicle by establishing an analytical
dynamic model of skid steering for wheeled vehicle. And
selecting the Du=u as the steering input, the vehicle may
has under-steer, neutral-steer or over-steer characteristics.
R ¼ B u
ð25Þ
Du
Fig. 8 shows the comparison of turning radius at different
speeds using two types of steering input. The center of
gravity is assumed to locate at the geometrical center of the
vehicle. The distribution of normal pressure is assumed as
uniform. The result shows this vehicle has under-steer characteristic
using both types of steering input. The vehicle
adopted Du as the steering input has more under-steer characteristic
tendency. It means that large compensatory angle
of steering wheel is needed during acceleration at constant
turning radius. The velocity has significant influent on the
turning radius. Therefore, this type of steering input
Du=u is more suitable for the high-speed tracked vehicle.
This type of steering input Du=u is adopted in the following
analysis.
5.2. Distribution of normal pressure
The distribution of normal pressure on the track-ground
interface is a significant factor for maneuverability of
tracked vehicle. In the traditional steering theory, the distribution
of normal pressure is idealized as triangle, trapezoid,
sine etc. The turning resistance moment with these
types of normal pressure distribution was analyzed. In general,
distribution of normal pressure tending to localize in
the middle of track results in lower turning resistance
moment. On the contrary, distribution of normal pressure
tending to localize in the ends of track results in larger
turning resistance moment. These results indicate that the
tracked vehicle is harder to turn on the concave ground
than the convex ground. In this paper, the influence of normal
pressure distribution on the steady state characteristics
of skid- steering is studied.
Firstly, normal pressure distribution is idealized as dual
trapezoid, as shown in Fig.9. The density function of the
normal pressure distribution sðx t Þ can be expressed by
Eq. (26). The slope of the diagonal line k is defined as the
concentration factor, which reflects the degree of concentration.
When k ¼ 0, the normal pressure distribution is
uniform. When k > 0, the normal pressure distribution
tends to localize in the ends of track. When k < 0, the normal
pressure distribution tends to localize in the middle of
track.
(
kL
kLx t þ 1 2
ðx 4 t 0Þ
sðx t Þ¼
ð26Þ
kL
kLx t þ 1 2
ðx 4 t < 0Þ
The ratio of the yaw rate and the steering input is
defined as the yaw velocity gain. Fig. 10 shows the yaw
velocity gain versus forward speed for tracked vehicles with
different dual trapezoid normal pressure distribution while
the steering input is kept constant Du=u ¼ 0:05. For the
cases of the dual trapezoid normal pressure distribution
k ¼ 0:2; 0; 0:2, the yaw velocity gain increases with the
increment of the forward speed to the maximum yaw velocity
gain. The result shows that the vehicle has under-steer
characteristic. And the yaw velocity gain increases with
the increment of the degree of the normal pressure distribution
tending to localize in the middle of track. When
k ¼ 0:4, the yaw velocity gain increases with the forward
speed at an increasing rate. The vehicle with this dual
trapezoid normal pressure distribution has over-steer
characteristic.
The steady state characteristics can be evaluated using
the under-steer parameter for Ackermann-steer vehicle.
The under-steer parameter is defined as the difference
between the reference steer angle gradient and the Ackermann
steer angle gradient for Ackermann steering vehicle
(Riede et al., 1984). Riede tested 400 production vehicles.
The under-steer parameter of the vehicles ranged from
0.7 deg/g to 8.2 deg/g with lateral acceleration level at
0.15 g. The average under-steer parameter was 3.8 deg/g.
For tracked vehicle, the under-steer parameter j can also
be defined as the ratio of the difference between the reference
steering input and the steering input required by the
kinematics with no track slips and the corresponding lat-
Fig. 8. Comparison of turning radius at different speeds using two types of
steering input.
Fig. 9. Idealized dual trapezoid normal pressure distribution.

S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35 33
Fig. 10. Yaw velocity gain versus speed with different dual trapezoid
normal pressure distributions.
eral acceleration, which is shown as Eq. (27). For the cases
of the dual trapezoid normal pressure distribution
k ¼ 0:4; 0:2; 0; 0:2, the under-steer parameter j is
4.1, 5.57, 9.23, 11.38, respectively. So the vehicle with
slightly tending to localize in the middle of track has better
handling characteristics.
j ¼
Du
u
u 2
R
B
R
ð27Þ
The normal pressure distribution can also be idealized as
trapezoid distribution, as shown in Fig. 11. It represents
the effect of longitudinal component of inertial force
applied on the vehicle, when the vehicle accelerates/brakes
or drives on slope. The density function of the normal pressure
distribution sðx t Þ can be expressed by Eq. (28). The
slope of the diagonal line k 2 is defined as the front/rear distribution
factor, which reflects the degree of forward or
rearward weight transfer. When k 2 > 0, the normal pressure
distribution tends to localize in the front of track.
When k 2 ¼ 0, the normal pressure distribution is uniform.
When k 2 < 0, the normal pressure distribution tends to
localize in the rear of track.
sðx t Þ¼ 2k 2 x t
þ 1
ð28Þ
L
Fig. 12 shows the yaw velocity gain versus forward
speed for tracked vehicles with different trapezoid normal
pressure distribution while the steering input is kept constant
Du=u ¼ 0:05. For the cases k 2 ¼ 0:6; 0, the normal
pressure distributions are uniform or tends to localize in
the rear of track, the yaw velocity gain increases with the
increment of the forward speed to the maximum yaw velocity
gain. And the vehicle has under-steer characteristic. For
the case k 2 ¼ 0:6, the yaw velocity gain increases with the
forward speed at an approximately constant rate. The vehicle
with this trapezoid normal pressure distribution has
slightly under-steer characteristic. When k 2 ¼ 0:8; 1, the
normal pressure distribution tends to localize in the front
of track, the yaw velocity gain increases with the forward
speed at an increasing rate and reaches infinite value if
the forward speed exceeds the critical speed. The vehicle
with this trapezoid normal pressure distribution has oversteer
characteristic.
5.3. Position of gravity center
In traditional steering theory, the projection of the center
of gravity on the ground should coincide with the projection
of the geometrical center of the vehicle on the
ground. For skid-steering wheeled vehicle, Zhang et al.
(2014) derives the under-steer parameter from a 2DOF
model, which indicates that the steady-state characteristics
is influenced by the position of gravity center and the
cornering stiffness of front tire and the cornering stiffness
of rear tire. As mentioned above, when the projection of
the center of gravity on the ground coincides with the projection
of the geometrical center of the vehicle on the
ground, the vehicle has under-steer characteristic. Fig. 13
shows the yaw velocity gain versus speed with different
positions of gravity center. The yaw velocity gain increases
with the increasing rearward displacement of position of
gravity center. For the cases l x ¼ 0; 0:2, the yaw velocity
gain increases with the increment of the forward speed to
the maximum yaw velocity gain. The vehicles have understeer
characteristics, whose under-steer parameter j are
9.23 and 7.5, respectively. So the vehicle with slightly rearward
position of gravity center has better handling characteristics.
When l x ¼ 0:4, the yaw velocity gain increases
with the forward speed at an approximately constant rate.
This vehicle has neutral-steer characteristics. For the cases
Fig. 11. Idealized trapezoid normal pressure distribution.
Fig. 12. Yaw velocity gain versus speed with different trapezoid normal
pressure distributions.

34 S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35
Fig. 13. Yaw velocity gain versus speed with different positions of gravity
center.
Fig. 14. Yaw velocity gain versus speed with different ratios L/B.
l x ¼ 0:6; 0:8, the yaw velocity gain increases with the forward
speed at an increasing rate and reaches infinite value
if the forward speed exceeds the critical speed. The vehicle
has over-steer characteristic.
5.4. The ratio of L and B
In traditional steering theory, the ratio of track-ground
contact length and tread L=B has significant effect on the
steerability, which is recommended to range from 1.2 to
1.8. In this paper, the effect of ratio L=B on the steady state
characteristics is analyzed. Fig. 14 shows the result of yaw
velocity gain versus speed with different ratios L=B. For the
cases L=B ¼ 1; 1:2; 1:59; 1:8; 2, the yaw velocity gain
increases with the forward speed at an decreasing rate
and reaches its maximum yaw velocity gain. This result
indicates the vehicle has under-steer characteristics, and
the ratios L=B within the recommended range are unlikely
to change the steady state characteristics essentially. Smaller
ratio L=B can raise the yaw velocity gain response.
6. Conclusions
This study proposed a high-fidelity, general, and modular
method for lateral dynamic simulation of high-speed
tracked vehicle on deformable terrain. In this method a
novel nonlinear track terrain model is derived. This track
terrain model meets the need of longitudinal and steering
motions, comprehensive track slips, and modular modeling
for tracked vehicles with various configurations. The lateral
dynamic model is in reasonably agreement with the
available experimental data. Based on the comparisons,
the proposed vehicle dynamic model demonstrates adequate
accuracy in predicting steering performance. Using
this vehicle dynamic model, the main factors (normal pressure
distribution, position of gravity center, ratio L/B)
effecting the steady state characteristics of skid steering
are analyzed. In order to quantitatively evaluate the steady
state characteristics of skid steering for tracked vehicle, the
under-steer parameter is introduced in this paper referring
to the Ackermann-steering wheeled vehicle. Several conclusions
can be drawn on the basis of this research.
(1) The steering input Du=u is more suitable for the highspeed
tracked vehicle. Because that the tracked vehicle
with this type has better handling characteristic
when turning at high speeds.
(2) The vehicle with dual trapezoid normal pressure distribution
tending to localize in the ends of track or
slightly tending to localize in the middle of track has
under-steer characteristic. The vehicle with dual trapezoid
normal pressure distribution tending to localize in
the middle of track has over-steer characteristic. The
vehicle with dual trapezoid normal pressure distribution
slightly tending to localize in the middle of track
k ¼ 0:2 has better handling characteristics.
(3) The vehicle with trapezoid normal pressure distribution
tending to localize in the rear of track has
under-steer characteristic tendency. The vehicle with
trapezoid normal pressure distribution tending to
localize in the front of track has over-steer characteristic
tendency.
(4) When the projection of the center of gravity on the
ground coincides with the projection of the geometrical
center of the vehicle on the ground, the vehicle has
under-steer characteristic. The vehicle with rearward
position of gravity center has over-steer characteristic
tendency, and the critical speed is obtained.
(5) The vehicle with the ratio L/B in the recommend
range always has under-steer characteristic. And the
vehicle with smaller ratio L/B has bigger yaw velocity
rate response.
In this study, it is assumed that the tracked vehicle conduct
steady-state turning maneuvers to analyze the steady
state characteristics. However, the transient characteristics
are also significant for the high speed tracked vehicles. This
model is worth to be modified for the transient performance
in the future work.
Acknowledgement
This work was supported by the Technology Research
and Development Program of the Scientific Research Base
of China (Grant number VTDP-3301).

S. Tang et al. / Journal of Terramechanics 73 (2017) 25–35 35
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