2008 10th Intl. Conf. on Control, Automation, Robotics and VisionHanoi, Vietnam, 17–20 December 2008
Fast Algorithm for UGV WheelTerrain Interaction Analysis
T. H. Tran
Dept. of Automation College of Engineering, Can Tho University Can Tho, Vietnam tthung@ctu.edu.vn
Q. P. Ha
Faculty of Engineering University of Technology, Sydney Sydney, Australia quangha@eng.uts.edu.au
Abstract
—Online prediction of unmanned ground vehicle (UGV) behaviour during interactions with terrain is essential for autonomous and safe operations of skidsteering UGVs. This paper presents a fast and accurate algorithm for a new interaction model to predict performance of a UGV running on a particular terrain. By approximating nonlinear relations involved in the interaction between the vehicle and terrain, a closed form of the interaction model is obtained to enable online computation. The minimum absolute error criteria are applied to secure accuracy for the proposed method. The development is compared with the other models in terms of both computation speed and accuracy.
Keywords—
UGV, terramechanics, wheelterrain interaction.
I.I
NTRODUCTION
Unmanned ground vehicles (UGVs) have many potential applications, both in military and civil areas, such as reconnaissance, surveillance, target acquisition, and rescue. Most UGVs are currently teleoperated machines which require human intervention, thus, the range of applications is limited. Therefore, knowledge of the interaction between UGVs and terrain plays an important role in increasing the autonomy of UGVs and securing the safety for their locomotion. Many UGVs, especially in military applications, use skidsteering for allterrain mobility, both in tracked and wheeled platforms. Behaviour of a skidsterring UGV operating on offroad areas depends strongly on interactions between the vehicle and terrain. For performance analysis, the theory of terramechanics has been applied to for several offroad vehicles [1, 2]. In general, experiments are used to derive normal stresses and shear stresses beneath wheels or tracks. The stresses are then integrated to yield reaction forces and moments of terrain acting on the vehicle. For track vehicles, a comprehensive theory for their skidsteering mechanics was developed by Wong and Chiang [3]. For skidsteering wheeled vehicles, a general approach is recently proposed to analyse their mobility performance [4]. By using the interaction model described therein, behaviour of vehicles operating on deformable terrain can be predicted. However, the prediction is timeconsuming as the integated equations of stresses are not obtained in closed forms. Meanwhile, predicting online the vehicle performance is essential to avoid dangereous situations that may be encountered in autonomous navigation. Therefore, a fast approach to vehicleterrain interaction analysis is desirable. With regard to this particular problem, there has been little research effort reported in the literature, see e.g., [5, 6]. Therein, based on the observation that the normal and shear stresses under a rigid driven wheel are nearly linear, a modified version of the basic wheelterrain interaction equations is developed. By approximating the normal stress and shear stress by straight lines going through their start and end points, as shown in Fig. 1a and Fig. 1c, solutions to the integral equations can be obtained and easily computed. Figure 1 indicates, however, that the distribution of normal and shear stresses is generally nonlinear and the method [5] can lead to large errors. In addition, this approach can be applied only if lateral forces on the wheels are ignored. In this paper, the idea proposed by Shibly
et al.
[5] is refined to yield an efficient algorithm without losing accuracy. The stress is also approximated by a straight line going through its maximum stress point. But the other point of intersection is chosen so that effects of the approximation error on the vehicle performance are minimised (Fig. 1b and Fig. 1d). The method is applied for all components of normal stress and shear stress in the longitudinal, lateral, and vertical directions and new equations for the reaction forces are obtained from that linearisation. The new vehicleterrain interaction model, that does not involve integral equations, results in a fast algorithm for wheelterrain interaction analysis. The development is verified by comparison with the srcinal model given in [4] and with the approximate model [5], in terms of both computation speed and accuracy. II.A
NALYTICAL
B
ACKGROUND
Consider a driven rigid wheel running on a firm, deformable ground. Under the action of the vertical load and driving torque, the wheel compresses the soil to a sinkage
z
[1]. The normal stress, acting normal to the wheelterrain contact point, is related to sinkage
z
by:
( )
,
nc
z k bk
+=
φ
θ σ
(1)
( ) ( )
,coscos
1
θ θ θ
−=
r z
(2)
6749781424422876/08/$25.00 c
2008 IEEE ICARCV 2008
where
n
is the sinkage exponent,
c
k
,
φ
k
are the pressuresinkage moduli of the terrain,
b
is the wheel width,
r
is the wheel radius,
θ
is the contact angle at a considered point, and
θ
1
is the entry angle at which the considered point on the wheel rim first makes contact with the terrain. The maximum normal stress point,
θ
m
, separates the contact zone into front and rear regions (
θ
1

θ
m
and
θ
m

θ
2
), where
θ
2
is the wheel angular location at which the considered point loses contact with the terrain [7]. The normal stresses distributed on these regions are calculated by the following equations [8]:
( ) ( )
,coscos
11
nnc
r k bk
θ θ θ σ
φ
−
+= (3)
( ) ( )
.coscos
112212
nmmnc
r k bk
−
−
−−−
+=
θ θ θ θ θ θ θ θ θ σ
φ
(4)
Experiments have shown that
θ
2
is very small to be considered as zero and
θ
m
is often assumed to be in the middle of the contact zone [7]. In that case, (4) becomes
( ) ( )( )
.coscos
112
nnc
r k bk
θ θ θ θ σ
φ
−−
+= (5)
For loose sand, saturated clay, sandy loam, and most distributed soils, the shear stress, which is a tangential component of the stress at the wheelterrain contact point, exhibits an exponential relationship with respect to the shear displacement:
( ) ( )( )
( )
,1tan
/
K j
ec
−
−+=
φ θ σ θ τ
(6)
where
K
is the shear deformation modulus,
j
is the shear displacement,
c
and
φ
are respectively cohesion and internal friction angle of the terrain. The above equations were extended to account for lateral motion and applied to all wheels of a skidsteering UGV [9], as shown in Fig. 2. Reaction forces acting on the
i
th
wheel (
i
=1,2,…,8) along longitudinal, lateral and vertical directions are derived by integrating components of normal and shear stresses on these directions as
( )
,
1
02/2/
−
+=
iiii
bbii X X X
dyd r F
θ
θ σ τ
(7)
,
1
02/2/
−
=
iii
bbiiY Y
dyd r F
θ
θ τ
(8)
( )
,
1
02/2/
−
+=
iiii
bbii Z Z Z
dyd r F
θ
θ σ τ
(9)
0 5 10 15010203040506070contact angle (deg)
n o r m a l s t r e s s ( k P a )
normal stressApproximation
a)
0 5 10 15010203040506070contact angle (deg)
n o r m a l s t r e s s ( k P a )
normal stressApproximation
b)
0 5 10 15012345contact angle (deg)
s h e a r s t r e s s ( k P a )
shear stressApproximation
c)
0 5 10 15012345contact angle (deg)
s h e a r s t r e s s ( k P a )
shear stressApproximation
d)
Figure 1. Stress approximation used in the paper: a), c) method by Shibly
et al
. [5]; b), d) our method
θ
7
x
1
o
13
o
35
o
57
o
7
x
3
x
5
x
7
x
22
o
24
o
46
o
6 8
o
8
x
4
x
6
x
8
Y X Oa
7
1
θ
V
X
V
Y
Ω
x
1
1
1
θ
θ
1
3
1
θ
θ
3
5
1
θ
θ
5
B Bd b X Z
1
ω
L
ω
L
θ
7 7
r h
(a)(b)
x
1
o
1135
o
3
o
5
o
7
P
1
P
1
P
3
P
5
V
P
7
χ
O
Center of mass Center of mass Geometric center
Figure 2. Vehicle freebody diagram on deformable terrain
Similarly, the vehicle’s turning moment is obtained as
( )
( )
( )
,sin21231
8102/2/8102/2/1
11
=−=−+
+−
−−+++−−=
ibbiiY i
ibbii X X i
i Z
iiiii
dyd r d a
i E r dyd y Br M
θ θ
θ τ θ θ σ τ
(10)
where
E
(
x
) gives the integer part of
x
. Solving integrals in (7)(10) is timecomsuming due to the nonlinearity of normal and shear stresses. Their closed forms are to be sought for speeding up the computation.
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III.F
AST ALGORITHM FOR VEHICLE

TERRAIN INTERACTION ANALYSIS
In this section, normal stress and shear stress beneath a wheel are linearised so that their integrals can be obtained in a closed form. Criteria are first chosen to minimize the approximation error, and then methods for normal and shear stress approximation are developed. The approximate stress is then substituted into equations of the vehicleterrain interaction model developed in [4] to derive a new version for predicting behaviour of the vehicle on deformable terrain.
A. Normal stress approximation
The distribution of the stress under the
i
th
wheel is dicomposed in front and rear regions as given in equations (47). As the stress is normal to the wheel rim, its components on the X and Z directions are
( )( )
.
iii Z
iii X
ii
θ θ σ σ
θ θ σ σ
cos,sin
=−= (11)
In most cases, the contact angle
i
θ
is small, and as a result,
i
X
σ
is very small compared with
i
Z
σ
. The largest influence of the normal stress is on the reaction force in the Z direction, as given in (9). If the normal stress is approximated so that its effect on the component
i
Z
F
does not change, then the absolute error due to the approximation will be minimized. Applying this criterion, our proposed method is described in the following. A linear curve of the normal stresses in the front and rear regions can be generally expressed by ,ˆ,ˆ
222111
iiiiii
ck ck
ii
+=+=
θ σ θ σ
(12)
where
iiii
cck k
2121
and,,,are constants to be determined.
As the stress and its approximate lines intersect at the maximum stress point, (12) becomes
( )( ) ( )( )
,ˆ,ˆ
22222
22211111
111
iiiiiii
iiiii
iiiiiii
iiiii
ck k k
k ck k k
k
immi
mmiimmi
mmi
+=+−=
+−=+=+−=
+−=
θ θ σ θ θ
θ σ θ θ σ
θ θ σ θ θ
θ σ θ θ σ
(13)
where
( )
.,
222111
iiiii
iiiii
mmmm
k ck c
θ σ θ θ σ θ
+−=+−= (14)
The angular coefficients of the approximations are chosen so that
( ) ( )( ) ( )
,coscosˆ,coscosˆ
20202111
11
iimiimiiiimiiimi
Ad d Ad d
iiiiii
iiiiii
====
θ θ θ θ θ θ
θ θ θ σ θ θ θ σ
θ θ θ σ θ θ θ σ
(15)
where
ii
A A
21
and are constants assigned for the expressions in the right hand side of (15). Substitution of (13) into (15) gives
( )
( ) ( )
[ ]
( )
( ) ( )
[ ]
.coscosˆ,coscosˆ
22221111
2211
iimiiiii
imiiiiimiiii
iimi
Ad k d Ad k d
iimmiiii
iimmiiii
=+−==+−=
θ θ θ θ θ θ θ θ
θ θ θ σ θ θ θ θ θ σ
θ θ θ σ θ θ θ θ θ σ
(16)
The angular coefficients can then be derived from (16) as
( )( )( )( )
.coscossinsinsin,coscossinsinsin
22222221111111
iiiii
iiiii
iiiiii
iiiii
i
mmmmmmmm
Ak Ak
θ θ θ θ θ
θ θ θ σ
θ θ θ θ θ
θ θ θ σ
−+−−−=−+−−−= (17)
Let
i
x
1
θ
and
i
x
2
θ
be the contact angles at which the normal stress intersects its approximations in the front and rear regions respectively. Simulations using the proposed method with experimental wheel data collected from a field test show that on a given terrain, the ratios
ii
x
11
/
θ θ
and
ii
x
12
/
θ θ
are almost constant with different values of
i
1
θ
. This is a great advantage because it is not necessary to recalculate angles
i
x
1
θ
and
i
x
2
θ
each time when the entry angle changes. Instead, the average values of these ratios can be used. As the intersections of the normal stress and its approximations can be derived from the angle ratios, the approximating lines can now be simply described as
( ) ( )( )
,,where,ˆ
11111111111
iiiii
iiiiii
iiii
mmm xm xi
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (18)
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( ) ( )( )
.,where,ˆand
22222222222
iiiii
iiiiii
iiii
mmm xm xi
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (19)
From the observation that the shapes of normal stress and its components in the X and Z directions are nearidentical, the same intersection points in (18) and (19) can be used to approximate the stress components as follows
( ) ( )( )
,,where,ˆ
11111111111
iiiii
iiiiii
iiii
m X m X X
m xm X x X
X X i X X
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (20)
and similarly,
( ) ( )( )
,,,ˆ
22222222222
iiiii
iiiiii
iiii
m X m X X
m xm X x X
X X i X X
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (21)
( ) ( )( )
,,,ˆ
11111111111
iiiii
iiiiii
iiii
m Z m Z Z
m xm Z x Z
Z Z i Z Z
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (22)
( ) ( )( )
.,,ˆ
22222222222
iiiii
iiiiii
iiii
m Z m Z Z
m xm Z x Z
Z Z i Z Z
k ck ck
θ σ θ θ θ θ σ θ σ
θ σ
+−=−−=+= (23)
B. Shear stress approximation
Let us consider the shear stress under the
i
th
wheel. Components of the shear stress along X, Y, and Z direction are given as ,sincos,sin,coscos
iii Z
iiY iii X
iii
θ ϕ τ τ
ϕ τ τ θ ϕ τ τ
=== (24)
where
ϕ
i
is the angle between the shear stress and its projection on the XZ plane (Fig. 3). As noted above, the contact angle
i
θ
is small so that
i
Z
τ
isvery small compared with
i
X
τ
. When the vehicle runs in a straight line, the angle
ϕ
i
is zero and so is
i
Y
τ
. In most cases, therefore, the largest effect of the shear stress is on its component along the X direction
i
X
τ
. This component, in turn, is the main contribution to the longitudinal reaction force, or drawbar pull, expressed in (7). Again, our criterion is that if the shear stress is approximated so that its effect on
i
X
F
does not change, then the absolute error of the approximation is the smallest. In general, the linear shear stresses can be written as ,ˆ,ˆ
442331
iiiiii
ck ck
ii
+=+=
θ τ θ τ
(25)
where
iiii
cck k
4343
and,,,are constants to be calculated.
Because the stress and its approximate lines intersect at the maximum stress point, (25) becomes,
( )( ) ( )( )
,ˆ,ˆ
44244
24233133
131
iiiiiii
iiiii
iiiiiii
iiiii
ck k k
k ck k k
k
immi
mmiimmi
mmi
+=+−=
+−=+=+−=
+−=
θ θ τ θ θ
θ τ θ θ τ
θ θ τ θ θ
θ τ θ θ τ
(26)
where
( )
.,
244133
iiiii
iiiii
mmmm
k ck c
θ τ θ θ θ
+−=+−= (27)
To satisfy the suggested criterion, the angular coefficients of the approximations are chosen so that
( ) ( )( ) ( )
.coscosˆ,coscosˆ
422311
2211
iimiiimiiiiimiiimi
Ad d Ad d
iiiiii
iiiiii
====
θ θ θ θ θ θ θ θ
θ θ θ τ θ θ θ τ
θ θ θ τ θ θ θ τ
(28)
Substitution of (26) into (28) gives,
( )
( ) ( )
[ ]
( )
( ) ( )
[ ]
.coscosˆ,coscosˆ
42423131
2211
iimiiiii
imiiiiimiiii
iimi
Ad k d Ad k d
iimmiiii
iimmiiii
=+−==+−=
θ θ θ θ θ θ θ θ
θ θ θ τ θ θ θ θ θ τ
θ θ θ τ θ θ θ θ θ τ
(29)
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