Dynamic Modelling of Wheel-Terrain Interaction of ... - Semantic Scholar

Proceedings of the 3rd Annual IEEE Conference on Automation Science and Engineering Scottsdale, AZ, USA, Sept 22-25, 2007

MoBPBP.2

Dynamic Modelling of Wheel-Terrain Interaction of a UGV T. H. Tran, N. M. Kwok, S. Scheding, and Q. P. Ha Abstract—Understanding the vehicle-terrain interaction is essential for autonomous and safe operations of skid-steering unmanned ground vehicles (UGVs). This paper presents a comprehensive analysis of the dynamic processes involved in this interaction, using the vehicle kinetics and the theory of terramechanics to derive systematically shear displacement, reaction force, and load distribution for a wheel. The new model is then summarized in the form of an algorithm to allow for computation of characteristic performance of the interaction such as slip ratios, rolling resistance, and moment of turning resistance for a number of terrain types. Given the current state of the vehicle and terrain parameters, the model can be used to estimate its next states and to predict the vehicle running path. The development is illustrated by simulation and verified with experimental data.

I. INTRODUCTION

U

nmanned 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 skid-steering for all-terrain mobility, both in tracked and wheeled platforms. In the context of vehicle-terrain interaction, parametric performance of tracked vehicles are well understood [1], [2]. For wheeled vehicles, except work using heuristic reasoning [3], it seems there has been less analytical work available in the literature for modelling the complicated processes involved. In Weiss [4], lateral forces on wheeled vehicles under skid-steering are used to calculate vertical loads acting on each row of wheels. These loads are then assumed to be distributed equally to all the wheels in a row. Consequently, shear stress developed at each wheel is derived from the load on that wheel. The analysis therein is based on steady manoeuvre of the vehicle. Creedy [5] later extended Weiss’s idea by considering the difference of vertical loads on the wheels in a row due to their positions and torque transitions. However, in their work, shear stress acting on each wheel is assumed to obey the Coulomb friction law.

Manuscript received April 30, 2007. This work was supported by the ARC Centre of Excellence programme, funded by the Australian Research Council (ARC) and the New South Wales State Government, and by a scholarship of the Vietnam Ministry of Education and Training. The authors are with the ARC Centre of Excellence for Autonomous Systems (CAS). T.H. Tran, N.M. Kwok and Q.P. Ha are from the Faculty of Engineering, University of Technology Sydney, Australia (e-mail: {ttran, nmkwok quangha}@eng.uts.edu.au). S. Scheding is from the Australian Centre for Field Robotics, The University of Sydney, Australia (email: [email protected]).

1-4244-1154-8/07/$25.00 ©2007 IEEE.

This implies that the shear stress can change immediately from zero to its maximum value when the wheel begins its motion on the ground. In contrast, experiments have shown that the shear stress follows an exponential function of the shear displacement between the wheel and the ground and only attains its maximum value after a certain shear displacement has occurred in the ground [2]. In addition, the requirement for constant velocities is not always true for an operating unmanned ground vehicle in practice. The vehicle-terrain interaction is addressed in this paper, based on the theory of terramechanics [6], [7], and kinetic equations of an autonomous skid-steering wheeled vehicle, described in [8]-[10]. In general, the analytical relationship between shear stress, shear displacement, and vertical load is used to derive reaction forces. The load on each wheel is determined, depending on the position of each wheel of the vehicle. The shear displacement is obtained by taking integrals of the slip velocity of the contact point of each wheel with the ground. The shear displacement is then used to derive the magnitude and direction of the shear stress acting at the contact point. The shear stress over the contact zone is further integrated to obtain the reaction forces on each wheel by the ground. The vehicle performance during contact with different terrain types can then be predicted by making use of the vehicle kinetics. II. BACKGROUND 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 [11]. The normal stress, acting normal to the wheel-terrain contact point, is related to sinkage z by: ⎞ ⎛k σ (θ ) = ⎜ c + kφ ⎟ z n , (1) b ⎠ ⎝

z (θ ) = r (cosθ − cosθ1 ),

(2)

where n is the sinkage exponent, k c , kφ 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 θ1 is the entry angle at which the considered point on the wheel rim first makes contact with the terrain. The maximum normal stress point, θm, separates the contact zone into front and rear regions (θ1-θm and θm-θ2), where θ2 is the wheel angular location at which the considered point loses contact with the terrain. The normal stresses distributed on these regions are calculated by the following equations [7]: ⎛k ⎞ σ 1 (θ ) = ⎜ c + kφ ⎟r n (cosθ − cosθ1 )n , (3) b ⎝ ⎠

369

MoBPBP.2

n

⎞ ⎞ ⎛ θ − θ2 ⎞ ⎛k ⎞ ⎛ ⎛ ⎟⎟(θ1 − θm )⎟ − cosθ1 ⎟ . (4) σ 2 (θ ) = ⎜ c + kφ ⎟r n ⎜ cos⎜⎜θ1 − ⎜⎜ ⎟ ⎜ ⎟ ⎝b ⎠ ⎝ ⎝ ⎝ θm − θ2 ⎠ ⎠ ⎠

vectors on X, Y, Z axis of the vehicle frame, and E [x] is the largest integer that is smaller or equal to x. Z

Experiments have shown that θ2 is very small to be considered as zero and θm is often assumed to be in the middle of the contact zone [11]. In that case, (4) becomes ⎞ ⎛k σ 2 (θ ) = ⎜ c + kφ ⎟r n (cos(θ1 − θ ) − cosθ1 )n . (5) ⎠ ⎝b For loose sand, saturated clay, sandy loam, and most distributed soils, the shear stress, which is a tangential component of the stress at the wheel-terrain contact point, exhibits an exponential relationship with respect to the shear displacement: (6) τ (θ ) = (c + σ (θ ) tan φ ) 1 − e − j / K , 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.

(

z7

b

(7)

Yi = (− 1)i +1 B + y i ,

Z i = −(h − r (1 − cos θ i )), where (Xi, Yi, Zi) are the Cartesian coordinates of Pi on each wheel of the vehicle, i, j, k are respectively the unit

P7

o7

y3 P5

x7

θ1

o1

o3

o5 y5

y7

z1

θ3

z3

θ5

z5

θθ77

θ11

θ13

θ15

θ17 o7

(a)

x5

x1

y1 y1 P1 o1 x1

P3 x3

o3

o5

P1

x1

Y B VY

d

III. WHEEL-TERRAIN INTERACTION ANALYSIS

⎛3 ⎡ i − 1⎤ ⎞ X i = ⎜⎜ − E ⎢ ⎥ ⎟⎟a − d + xi , ⎣ 2 ⎦⎠ ⎝2

h

r

B

A. Shear displacement The vehicle free-body diagram is shown in Fig. 1(a) with its projection on a horizontal plane shown in Fig. 1(b). Assume that the center of mass is displaced at a distance d along the centerline of the vehicle from the centroid and place the origin of vehicle coordinates at the center of mass. The Z-axis is pointing vertically upward and the X-axis is along the vehicle centerline. Let h be the height of the center of mass above the ground, a is the longitudinal distance between successive axles, and 2B is the vehicle track width. Consider an arbitrary point Pi (xi, yi) on the rim of wheel i, ( i = 1,2,...,8 ), which is in contact with the terrain. Positions of these points relative to the vehicle coordinate frame are: Pi (xi , y i ) = ( X i , Yi , Z i ) = X i i + Yi j + Z i k ,

ωL

ωL

)

We consider here the wheel-terrain interaction problem analytically in a Cartesian coordinate frame. The development is applied for eight wheels of an autonomous skid-steering wheeled UGV described in [8]. The following assumptions are made, for the sake of enabling our theoretical analysis: 1. The ground is firm, flat and homogeneous. The wheel tires are hard enough to be considered as rigid wheels. 2. During turning, the bulldozing effect of the wheels during turning is ignored. 3. Shear stress developed at a wheel-terrain contact point is opposite to the direction of the slip velocity.

X

VX

x8

o6

X

VX

Ω y4

y6

y8 o8

O

V

x6

(b)

o4

y2 x4

x2

o2

a

Figure 1. Vehicle free-body diagram on deformable terrain

The value of xi is determined as xi = r sin θ i , i = 1,..,8,

(8)

where θ i is the wheel-terrain contact angle at point Pi, while its lateral variable yi lies arbitrarily in the wheel width interval [ −b / 2, b / 2]. Let V be the velocity of the center of mass with respect to a fixed frame coincident with the vehicle coordinate frame, Ω be the vehicle turning rate, and ω L , ω R be the angular velocity respectively of the left-side (i=1,3,5,7) and right-side (i=2,4,6,8) wheels. They are defined as: (9) V = V X i + VY j, Ω = Ωk , ω L = ω L j, ω R = ω R j . The velocity at the center of each wheel is given by: Vi (xi , y i ) = V X i + VY j + Ωk × ( X i i + Yi j + Z i k )

(

)

= (V X − ΩYi )i + (VY + ΩX i )j = V X i , VYi ,

where V X i = V X − ΩYi , VYi = VY + ΩX i .

(10a)

(10b)

The total velocity at the contact point Pi(xi, yi) in the wheel frame of wheel i can be decomposed, as illustrated in Fig. 2, into normal, tangential and lateral components, respectively, as: Vni = V X i sin θ i , Vti = rω i − V X i cosθ i ,

(11)

Vli = VYi ,

⎧ω , if i = 1,3,5,7 where ωi = ⎨ L ⎩ω R , if i = 2,4,6,8. Among these velocity components, Vni tends to compact

370

MoBPBP.2

the terrain to sinkage z (θ i ) while Vti and Vli lead the contact point to shear the soil on tangential and lateral directions. Therefore, they decide the slip velocity of the considered point compared with a coincident point on the ground. Wi

ωi

θi θi

θ1

VXi r

i

V jX i

V jZ i

Vti

(

z(θi ) = r cosθi − cosθ1i

θi

)

VX i sin θi

B. Shear stress, normal stress and reaction force Under the normal and the shear pressures at the contact point, the ground reacts with an amount of shear stress τ i (θ i , yi ) determined by (6), where the shear displacement is given by (15). Thus, components of the shear stress along X, Y, and Z directions can be computed as: τ X i = −τ i cos α i sin β i ,

τ Zi = −τ i cos β i , where the minus sign is to denote the opposite direction to the slip velocity, which is determined by angles α i and β i

(

Thus, by referring to the fixed frame coincident with the vehicle coordinate frame, the slip velocity of the contact point Pi(xi, yi) can be expressed as: V jX i = −Vti cosθ i , V jYi = Vli ,

(12a)

V jZi = −Vti sin θ i .

(

By integrating the stresses acting on a very small contact area around the point Pi(xi, yi) with respect to an area increment rdθ i dy i , the total reaction force of the ground on the ith wheel when the considered point on the wheel is about to leave the contact is obtained as:

Substitution of (7)-(11) into (12a) gives,

(

⎝ ⎠ On the other hand, as the normal pressure is distributed in the vertical plane, the components of the normal stress are determined by: σ X i = −σ i (θ i ) sin θ i , σ Yi = 0, σ Zi = σ i (θ i ) cos θ i . (17)

))

V jX i = V X − Ω (− 1)i +1 B + yi cos 2 θ i − rωi cos θ i , ⎛⎛ 3 ⎞ ⎡ i − 1⎤ ⎞ ⎟ = VY + Ω⎜⎜ ⎜⎜ − E ⎢ ⎥ ⎟⎟a − d + r sin θ i ⎟, 2 2 ⎣ ⎦⎠ ⎝⎝ ⎠

(

(

∫

∫

By integrating these shearing rate components from the position where the wheel first makes contact with the ground to the considered point, shear displacement at the given point along X, Y, Z directions can be obtained. To obtain explicit expressions for these shear displacements one can assume velocities ωi , V X , VY , Ω are slowly time-varying and hence considered as constants in the short interval of the integration. Then,

(18)

Yi dθ i dy i ,

∫ ∫ (τ

Zi

)

+ σ Zi dθ i dy i .

θ 2 i −b / 2

C. Vehicle kinetics In normal operations, the total reaction forces on each wheel shall decide the vehicle acceleration, support the vehicle mass, and overcome the turning resistance to turn the vehicle. During the vehicle-terrain interaction, the following balance equations can be obtained ( θ i = θ 2i ≅ 0 at the end of the contact). ƒ Acceleration: 8 θ1i b / 2

8

)

∑

FX i = r

i =1

∑∫ ∫

8 θ1i b / 2

τ X i dθ i dyi − r

i =1 0 −b / 2

∑ ∫ ∫ σ (θ ) sin θ dθ dy i

i

i

i

i

= ma X ,

i =1 0 −b / 2

(19)

⎤ ⎛⎛ 3 ⎞⎞ 1 ⎡⎛⎜ ⎡ i − 1⎤ ⎞ ⎟⎟ ⎢ VY + Ω⎜⎜ ⎜⎜ − E ⎢ ⎥ ⎟⎟a − d ⎟ ⎟(θ1i − θ i ) + rΩ(cos θ i − cos θ1i )⎥, 2 2 ⎥⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎠

(

∫ ∫τ

θ1i b / 2

∫

ω i ⎢⎜⎝ ⎣

(

)

+ σ X i dθ i dy i ,

θ 2 i −b / 2

FZi = dFZi = r

⎤ 1⎡ 1 ⎛1 ⎞ VX −Ω (−1)i+1B + yi ⎜ (θ1i −θi ) + (sin2θ1i −sin2θi )⎟ − rωi (sinθ1i −sinθi )⎥, ωi ⎢⎣ 4 ⎝2 ⎠ ⎦

j Zi =

Xi

θ1i b / 2

FYi = dFYi = r

The longitudinal and lateral slip ratios, relative to the ground, are defined as [9] V jX i V jY S Xi = − , SYi = − i , (13) rω i rω i

jYi =

∫ ∫ (τ

θ 2 i −b / 2

(12b)

))

(

θ1i b / 2

FX i = dFX i = r

V jZi = V X − Ω (− 1)i +1 B + yi sin θ i cosθ i − rωi sin θ i .

(

)

2 2 α i = tan −1 V jYi / V jX i , β i = tan −1 ⎛⎜ V jX / V jZi ⎞⎟. (16b) + V jY i i

Figure 2. Velocity components at a contact point on wheel i

jXi =

(16a)

in spherical coordinates:

rωi

V jYi

(15)

τ Yi = −τ i sin α i sin β i ,

V X i cos θ i

Pi

j X2 i + jY2i + j Z2i , i = 1,2,..., 8 .

ji =

8 θ1i b / 2

8

∑F

))

1 ⎡1 ⎤ V X − Ω (− 1)i +1 B + yi (cos 2θ i − cos 2θ1i ) + rωi (cosθ1i − cosθ i )⎥. ωi ⎢⎣ 4 ⎦

(14) Hence, magnitude of the shear displacement at the wheel-terrain contact point Pi(xi, yi) is

i =1

Yi

=r

∑ ∫ ∫τ

i =1 0 −b / 2

Yi dθ i dy i

= maY ,

(20)

where m=W/g, W is the vehicle weight, g is the gravitational acceleration, aX and aY are respectively the vehicle longitudinal and lateral accelerations on X and Y directions.

371

MoBPBP.2

ƒ Rolling resistance:

8 θ1i b / 2

In (19), the second term involving the normal stress indicates a resistive component against the tractive effort. Since the grade resistance is zero on flat ground, if the aerodynamic resistance and longitudinal drawbar load are negligible, the rolling resistance at a wheel-terrain contact point is determined by: 8 θ1i b / 2

Rr = r

∑ ∫ ∫σ (θ )sinθ dθ dy . i

i

i

i

(21)

i

i =1 0 −b / 2

M r = −r

i =1

D. Vertical load distribution The vehicle weight is in general distributed on each wheel according to its position and the vehicle acceleration. For simplicity, it is assumed that all stresses concentrate on the origin of each wheel frame (xi = yi = 0). Therefore, rolling moments (25) of the vehicle on flat terrain becomes:

ƒ Weight balance:

4

MX = B

The vertical load distributed on wheel i is: θ1i b / 2

∫ ∫ (− τ

FZi = r

i

cos β i + σ i (θ i ) cos θ i )dθ i dy i = Wi ,

M Y = −h

∑ i =1

∑

Wi = W = mg.

8

i

Zi

i =1

8

MZ =−

8

8

∑ ∫ Y dF − ∑ ∫ Z dF , M = ∑ ∫ Z dF − ∑ ∫ X dF

∑ ∫ Y dF i

i

Yi

i =1

Y

i

i =1

8

Xi

i =1

+

∑ ∫ X dF i

Xi

Yi

the

,

i

i =1

rolling

Zi ,

and

moments

i =1

around X and Y axis can be derived as: 8 θ1i b / 2

MX =r

∑ ∫ ∫ ((− 1)

i +1

i =1 0 −b / 2

)(

)

(23a)

∑ ∫ ∫ (h − r (1 − cosθ ))τ i

i =1 0 −b / 2

Yi dθ i dy i ,

8 θ1i b / 2

M Y = −r

∑ ∫ ∫ (h − r (1 − cos θ ))(τ i

Xi

)

i =1 0 − b / 2

8 θ1i b / 2

−r

⎞ ⎛⎛ 3 ⎞ ⎜ ⎜ − E ⎡ i − 1 ⎤ ⎟ a − d + r sin θ i ⎟ τ Z + σ Z dθ i dy i , ⎢ 2 ⎥⎟ i ⎟ i ⎜⎜ 2 ⎣ ⎦⎠ ⎠ 0 −b / 2 ⎝ ⎝

(

∑∫ ∫ i =1

)

(23b) and similarly for the turning moment: 8 θ1i b / 2

M Z = −r

∑ ∫ ∫ ((− 1) i =1 0 −b / 2

i +1

)(

)

B + yi τ X i + σ X i dθ i dyi

⎛⎛ 3 ⎞ i − 1⎤ ⎞ & ⎜⎜ − E⎡ ⎟ ⎢ 2 ⎥ ⎟⎟a − d + r sin θ i ⎟τ Yi dθ i dyi = I Z Ω, ⎜⎜ 2 ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ 0 −b / 2

∑∫ ∫ i =1

(

)

)

(

(24) where in normal conditions of the vehicle on a flat ground, the right hand sides of rolling moments (23a, b) are zero, and IZ is the vehicle moment of inertia around Z axis. ƒ Moment of turning resistance: It is noted that while the first term in (24) denotes the total thrust required in skid steering to create a turning moment at the vehicle track width, the second term in (24) thus represents the moment of turning resistance due to skidding as per the lateral component of the shear pressure:

)

(26b)

)

While at the end of the contact θ 2i ≅ 0 [11], the entry

(27)

where u, v, and w are to be determined. Thus, the vertical loads acting on each wheel can be determined explicitly from (27) as [12]: m ((5a + 6d )Bg − 6Bha X 40aB m ((5a + 6d )Bg − 6Bha X W2 = 40aB m W3 = ((5a + 2d )Bg − 2 Bha X 40aB m W4 = ((5a + 2d )Bg − 2Bha X 40aB W5 =

8 θ1i b / 2

+r

(

angle θ1i for each wheel at the beginning of the contact is of particular interest when interpreting the wheel-terrain interaction. To determine this angle, one would make use of the vehicle weight balance (22) to solve a set of independent equations. One way to do this is to assume that the reaction force on Z direction on each wheel is linearly related to its current position (Xi, Yi) on the vehicle frame by:

W1 =

+ σ X i dθ i dy i

(26a)

= 0,

FZ i = u + vX i + wYi ,

B + y i τ Z i + σ X i dθ i dy i

8 θ1i b / 2

+r

i

i =1

⎛3 ⎞ ⎛1 ⎞ − ⎜ a − d ⎟ FZ1 + FZ 2 − ⎜ a − d ⎟ FZ 3 + FZ 4 ⎝2 ⎠ ⎝2 ⎠

(

i =1

8

MX =

Xi

) ∑ FY

− FZ 2 i + h

⎛1 ⎞ ⎛3 ⎞ + ⎜ a + d ⎟ FZ 5 + FZ 6 + ⎜ a + d ⎟ FZ 7 + FZ8 = 0. ⎝2 ⎠ ⎝2 ⎠

(22b)

ƒ Rolling and turning moment:

As

∑F i =1

8

FZi =

Z 2 i −1

8

(22a)

8

∑ (F i =1

0 −b / 2

8

⎛⎛ 3 ⎞ ⎞ ⎜ ⎜ − E ⎡ i − 1⎤ ⎟ a − d + r sin θ i ⎟τ Y dθ i dy i .(25) ⎢ 2 ⎥⎟ ⎜⎜ 2 ⎟ i ⎣ ⎦⎠ ⎠ 0 −b / 2 ⎝ ⎝

∑∫ ∫

− 5ahaY ), + 5ahaY ),

− 5ahaY ), + 5ahaY ),

m ((5a − 2d )Bg + 2 Bha X − 5ahaY ), 40aB

(28)

m ((5a − 2d )Bg + 2Bha X + 5ahaY ), 40aB m W7 = ((5a − 6d )Bg + 6Bha X − 5ahaY ), 40aB m ((5a − 6d )Bg + 6 Bha X + 5ahaY ). W8 = 40aB W6 =

These loads, in turn, are distributed over the wheelterrain contact zones in terms of the normal stress as given in (1) or (3)-(5), and compact the ground to wheel sinkages zi, i = 1,2,...,8 . Angle θ1i , at which each wheel first enters in contact with the terrain can be determined by solving (22a) and the right hand side is weight distributions given in (28), e.g. by a search algorithm.

372

MoBPBP.2

E. Interaction modelling procedure The above analysis provides a general framework for characterising the wheel-terrain interaction. The proposed model is incorporated in a computer algorithm to predict velocity profiles and interaction characteristics of the vehicle, given its geometry and terrain supplied with terrain parameters. The input entries for the algorithm are the wheel left and right angular velocities from an encoder or output of the vehicle skid steering control system [10]. Initial conditions are obtained either from experimental data (e.g., from GPS) or from calculations of the vehicle velocities for the no-slip case.

IV. SIMULATION AND EXPERIMENTAL VERIFICATION Terrain parameters used in our simulation are provided in [2] as shown in Table I. All terrains are assumed to behave as plastic media. Geometrical parameters of the UGV used [8] are given in Table II. Given initial conditions of the vehicle and readings of its angular velocities of the right and left wheels, our objective is to use the proposed dynamic model of the interaction to estimate all velocity profiles in the next states, and obtain the vehicle running path, particularly its performance during skid-steering turning, then to validate the simulation with field test results.

Terrain Dry sand Sandy loam Clayed soil Dry clay

c (kPa)

φ

(o)

TERRAIN PARAMETERS kc (kN/mn+1)

kφ (kN/mn+2)

n

K (m)

1.04

28

0.99

1528.43

1.1

0.01

1.72

29

5.27

1515.04

0.7

0.025

4.14

13

13.19

692.15

0.5

0.006

68.95

34

12.70

1555.95

0.13

0.006

TABLE II.

Several experiments were conducted on Marulan field, in the Southern Tablelands region of New South Wales on the 150 meridian, with sensing data logged from differential GPS. These data are used to validate our interaction model. One hundred future states of the vehicle on different terrains in five seconds later are estimated and compared with GPS data recorded during the UGV operation on the dry, brown land in Marulan. Results are given in Fig. 4. It can be seen in Fig. 4(b) that with the same pattern of wheel velocities (Fig. 4(a)), the predicted trajectory of the vehicle is rather coincident with the experimental running path, and the terrain is more likely of the clayed soil type. Vehicle position 0

VEHICLE PARAMETERS

m Iz (Kg) (Kg.m2)

r (m)

b (m)

a (m)

B (m)

d (m)

h (m)

490

0.25

0.246

0.61

0.61

0.17

0.35

309.43

B. Comparison with experimental data

North (m)

TABLE I.

distances are much less than the theoretical value, calculated when no slip occurs, and decrease from dry sand, dry clay, clayed soil to sandy loam because of the wheel sliding effect (Fig. 3(b)). In all terrains, when the vehicle makes a turn, the inner wheels (right wheels in a right turn) slow down, its longitudinal slip is reduced in magnitude and changing in direction. Meanwhile, the outer wheels tend to slide faster longitudinally (Fig. 3(b)). On dry sand, the slip ratio of the outer wheels significantly increases as compared with the other terrain types. Fig. 3(c) shows that rolling resistance slightly increases when turning, is smallest on dry clay, with clayed soil coming next, then sandy loam, and last is dry sand. Likewise, the turning resistance moment of dry clay is smallest, next is sandy loam, then clayed soil, and dry sand coming last (Fig. 3(d)).

-2 -4 -6

A. Simulation results

373

0

2

4

(a)

6 East (m)

8

10

Longitudinal slip ratio

60 40 20 %

The simulation presents a scenario when the vehicle is controlled first to accelerate from zero in a straight line and then make a right turn using skid steering for the vehicle driveline model [8]. The set of angular velocity data of the left-side and right-side wheels obtained to steer the vehicle in the desired pattern is supplied to the above interaction modelling procedure to predict the vehicle performance on the four terrain types. The vehicle running path is shown in Fig. 3(a), constructed from output data using the proposed procedure. The wheel slip, rolling resistance and moment of the turning resistance can also be obtained according respectively to (13) and (21), as shown in Fig. 3(b), (c) and (d). The results indicate that running paths of the vehicle on various terrains appear to be different under the same steering commands. The running

Dry sand Sandy loam Clayed soil Dry clay when no slip

0 -20 -40 -60 0

(b)

Dry sand: Left wheels Dry sand: Right wheels Sandy loam: Left wheels Sandy loam: Right wheels Clayed soil: Left wheels Clayed soil: Right wheels Dry clay: Left wheels Dry clay: Right wheels

1

2

3 Time (s)

4

5

6

MoBPBP.2

V. CONCLUSION Rolling resistance

1000 800

N

600 400

Dry sand Sandy loam Clayed soil Dry clay

200 0 0

2

Time (s)

4

6

(c) Turning moment resistance

700

Dry sand Sandy loam Clayed soil Dry clay

600

Nm

500 400

REFERENCES [1] J. Y. Wong and C. F. Chiang, "A general theory for skid steering of tracked vehicles on firm ground," Proc. Inst. Mechanical Engineers, Pt. D: J. Automobile Engineering, vol. 215, 2001, pp. 343-355.

300

[2] J. Y. Wong, Theory of Ground Vehicle, 3rd ed: John Wiley & Son, Inc., 2001.

200 100 0 0

1

2

(d)

3 Time (s)

4

5

6

Figure 3. Interaction performance on various terrains

Wheel angular velocities

20

Left wheels Right wheels

rad/s

0 791

792

793 794 Time (s)

(a)

795

796

Vehicle position 10 Dry sand Sandy loam Clayed soil Dry clay GPS position

8 North (m)

[4] K. R. Weiss, "Skid-steering," Automobile Engineer, 1971, pp. 22-25. [5] A. P. Creedy, Skid steering of wheeled and tracked vehicles: analysis with Coulomb friction assumptions: Maribyrnong, Engineering Development Establishment, 1984.

[7] J. Y. Wong and A. R. Reece, "Prediction of Rigid Wheel Performance Based on Analysis of Soil-Wheel Stresses, Part 1 Performance of driven rigid wheels," J. of Terramechanics, vol. 4, 1967, pp. 81-98.

10

5

6 4 2

(b)

[3] J. T. Economou and R. E. Colyer, "Fuzzy-hybrid modelling of an Ackerman steered electric vehicle, " Int. J. Approximate Reasoning, Vol. 41, 2006, pp. 343-368.

[6] M. G. Bekker, Theory of land locomotion: University of Michigan Press, 1956.

15

0

We have presented a comprehensive analysis for the wheel-terrain interaction for a skid-steering UGV. The novel interaction model takes into account the vehicle kinetics and the relationship between shear stress and shear displacement in firm, deformable ground to characterize analytically performance of the vehicle during contact with the terrain, such as reaction force, slip ratios, rolling and turning moments. A procedure is proposed for computation of these interaction parameters. The model was tested in simulation with four typical terrain types and compared with field test data. Future positions of the vehicle predicted by the model from the current state are shown close to the experimental trajectory, which verifies its validity. Work is in progress to incorporate this interaction model in the vehicle speed control system.

-8

-6

-4

-2 0 East (m)

2

4

[8] T. H. Tran, Q. P. Ha, R. Grover, and S. Scheding, "Modelling of an autonomous amphibious vehicle," Proc. Australian Conference on Robotics and Automation, 2004 (CD-ROM). [9] Q. P. Ha, T. H. Tran, S. Scheding, G. Dissanayake, and H. F. Durrant-Whyte, “Control Issues of an Autonomous Vehicle”, Proc. of the 22nd Int. Symp. on Automation and Robotics in Construction (ISARC2005), Ferrara Italy, September 11-14, 2005. [10] T. H. Tran, N. M. Kwok, M. T. Nguyen, Q. P. Ha, and G. Fang, "Sliding Mode-PID Controller for Robust Low-level Control of a UGV," Proc. of the IEEE Conf. Automation Science and Engineering, Shanghai, China, 2006, pp. 684689. [11] K. Iagnemma, S. Kang, H. Shibly, and S. Dubowsky, "Online Terrain Parameter Estimation for Wheeled Mobile Robots With Application to Planetary Rovers," IEEE Transactions on Robotics, vol. 20, no.5, 2004, pp. 921-927.

[12] K. Waldron, "Argo Ground Interaction Equations," Internal Document, unpublished.

Figure 4. Comparison with experimental data

374