Model Parameter Extraction From Contact Geometry and Compliance in Tire–Soil Interaction

Classical soil mechanics-based standard test methods for direct shear test of soils under consolidated drained conditions are commonly used to extract soil strength parameters [13]. In some cases, cone penetrometer data are used to characterize soil strength as a means to estimate model parameters.

Such test methods require quasi-static rate (controlled strain rate) of shearing to be low enough to ensure drained conditions. Parameters determined using small plates, and small displacements do not scale to full tire–soil contact surface [10] and associated large displacements. Similarly, cone penetrometer and shear vane measurements are not scalable to tire and soil interaction. Moreover, these parameters are soil centric and do not account for tire distortion or kinematics.

El Hariri et al. [16] presented a good summary of soil shear strength measuring methods including translational shear box, shear vane test, triaxial compression test, and bevameter technique. Parameters obtained through these methods, to the extent that they are useful for traction modeling represent deformation in soil with tire mostly represented as rigid wheel.

Wheel sinkage is a key variable in estimating and predicting wheel–terrain interaction phenomena [17]. Depending on tire pressure setting, sinkage is due to both tire and soil deformation. If the soil medium is loose initial sinkage is dominated by soil deformation. As soil becomes firm both tire and soil deform. Tire deformation that occurs after penetration into the soil will exert lateral forces on the walls of rut formed. Elongation and lateral bulge of the tire increases available shear surface area for traction. Consequently, sinkage and the coupled lateral and longitudinal tire and soil distortion become key factors/variables in traction modeling.

Gee-Clough [12] and others articulate the following persistent questions:

• Can quasi-static analysis ever give reasonable prediction of off-road vehicle performance even for slow-moving vehicles such as agricultural tractors?

• Which soil properties should be used in terramechanics analyses and how these be measured?

While significant computational advances and incremental improvements in model parameter identification and estimation have been made over the past several decades, there are no indications in the published literature that these questions have been fully addressed. This paper attempts to address these questions and presents a different method for identification and estimation of parameter(s) that explicitly incorporate tire–soil interface geometry, soil water content, and slip. Soil moisture is affected by porosity, permeability, compaction, and bulk density and therefore has a dominant influence on the contact geometry and slip. The existence of solid particles with various shapes and sizes and the complex pore fabric formed among adjacent particles control the interaction between solid particles [18]. Due to the differences in shapes and sizes of the air–water–solid interface, pockets of air and water may exist. The discontinuities between the air and water pockets add to the variability in the intergranular friction and alignment of the soil particles under tire load.

He et al. presented a comprehensive literature review conducted on the parameterization of terramechanics models [19]. Their work summarizes evolution of terramechanics research since the 1960s discussing limitations and incremental improvements achieved over the years.

Maclaurin [20] pointed to the difficulty in developing reasonably accurate traction models for pneumatic tires in soft soils and that researchers frequently resort to empirically based methods for vehicle–tire design and performance assessment. Maclaurin [20] used empirical “mobility numbers” based on dimensional analysis methods developed in the 1960s [21].

Lopez-Arreguin and Montenegro [22] reported a concise survey of existing methodologies for detecting wheel forces and torque, slip-sinkage, and what they refer to as soil strength constants. Although their work is focused on investigating machine learning in planetary rovers, they present a good classical terramechanics overview in their paper. Empirical and semi-empirical models are predominant in traction modeling. The viability of the discrete element method and finite element method is yet to be proven. The high computational cost and convergence problems are difficulties that have not yet been overcome [23].

The basic contributions to soil strength are frictional resistance between soil particles in contact and interface kinematic constraints of soil particles associated with changes in the soil fabric [24]. Depending on the liquidity of the soil, the friction between soil particles can be significantly reduced. When large, rapid deformation/detachment of soil occurs, the contribution of cohesion to resist the shear force applied by rotating tire is substantially reduced or is negligible.

The aim of this paper is to address the unscalability of parameters and model assumptions that do not take tire deformation and large soil displacement/detachment into account in traction models by proposing a different paradigm to obtain model parameters.

The assumption of rigid wheel interaction with soil in traction models is getting more and more divergent from practice as tire pressure management systems are used increasingly to adjust tire compliance to terrain conditions to serve as part of spring damper suspension system and for better flotation. Flotation results from the increases in contact surface area and less digging of the tire into the soil.

Parameter identification methodology presented here makes use of observations of compliance in tire–soil interaction, soil mechanics principles, and mathematical formulations to develop functional relationships of variables influencing traction.

During driving, as slip increases with an increase in angular velocity of the wheel, this can cause vibrations of the bulk soil and loosen the soil material and reduce its strength. This occurs largely because more rapidly spinning wheels remove material from behind the wheel, which reduces the pressure in the rear zone, thereby weakening the base of material against which the wheel pushes and otherwise provide added thrust [25].

Slip contains information about the micro and macro interaction between soil particles and soil–tire interaction under dynamic load. Parameter identification and estimation method proposed here presupposes that it is better to work with slip as a variable, instead of the customary shear displacement (derived from slip) in traction modeling.

Physical properties or conditions contributing to identification and estimation of model parameters include soil moisture content, porosity of the soil, tire–soil interface contact geometry, as influenced under load. With this conjecture, the subsequent discussion presents proposed method and rationale for identification of contributing factors to parameterize and estimate slip modulus, $κ$⁠.

Figure 1 illustrates an example of tire bulge under load and subsequent indentation formed in the soil conforming to the shape of tire distortion (bulge and foot print elongation).

$τ/σ$ is the ratio of corresponding principal stresses. $dx$ and $dz$ are the incremental horizontal and vertical displacements under stress, respectively. $φm$ and $dz/dx$ represent the sliding friction between grains and interlocking friction, respectively.

Under applied vertical load and resulting deflection of the tire, the relationship between the compressive and shear forces exerted on the soil depends on the shape and extent of the lateral deformation (bulge) and longitudinal deformation of the tire. The geometric shape of the deformed tire influences the resulting soil displacement and thus the shape of the contact surface. A Cartesian coordinate system where z is positive downward with the ground free surface at $z=0$ with x and y orthogonal to z is chosen.

$δ$ as expressed in Eq. (7) can be interpreted as a parameter that blends the compaction and shear resistance of deformable soil under load applied on a deformable tire (Fig. 1). The perimeter C inherently retains information about the shear load-bearing surface area at the tire–soil interface.

A load cell mounted on a hydraulic press was used to measure load applied to a single-wheel axle. A displacement transducer was used to measure the wheel-center position (z) as tire is pressed into the soil. Perimeter C at the widest points around rut formed (Fig. 2) was measured using string and tape measure.

Moisture content of soil is the single most important factor that influences the strength characteristics (adhesion, cohesion, or friction) of soil under load [28]. In general, increased soil moisture content reduces interlocking of soil particles under load (e.g., soil particles can easily be crushed or reoriented in response to the magnitude and direction of loading). The reduction in the frictional interlocking ability of particles is dependent on the degree of saturation of soil by water.

In civil engineering applications two indices, plastic limit (PL) and liquid limit (LL) are often used as indicators of soil strength [29–33]. The LL represents soil water content where the consistency transitions from plastic state to viscous state. In traction mechanics, the moisture range of interest is from dry to some limiting moisture level approaching the LL.

The use of a form of liquidity index that accommodates soil moisture range from residual soil water content (dry soil) to liquid limit is more suited for our application to make the tire–soil interaction models employable over working soil moisture range.

The in-situ water content, w is the variable in Eq. (12). $Γ$ is the soil strength index. Substituting Eq. (12) into Eq. (13) we compute $Γ$ to generate families of curves showing soil strength indices over a broad range of moisture content for several soil types based on their respective LL. The in-situ water content (⁠$w$⁠), was varied from 1% to LL for the purpose of generating Fig. 3.¹ The LL is fixed for a given soil type and spanning between 21% and 118% for the specific range of soil types presented in Fig. 3. Figure 3 is shown to illustrate how $Γ$ changes with increasing in-situ soil water content.

From the foregoing discussion, we can infer the following:

1. It is intuitive that in loose soil initial deformation occurs in the soil followed by simultaneous deformation of soil and tire until soil compaction under the tire is such that there is minimum deformation or realignment of soil particles.

2. At higher water content, sinkage due to soil “flow” and hence less flotation of the tire will occur.

3. The soil strength index, $Γ$ is inversely proportional to tire sinkage and slip.

4. When the stiffness of soil is higher than the tire stiffness in the direction of load application bulging of the tire sidewall occurs increasing floatation of the tire (i.e., increasing tire–soil contact surface area).

5. It is also intuitive that the soil strength in the immediate vicinity of the rotating tire can reduced due to vibration induced by rotational kinetic energy. The degree of change in strength is dependent on initial strength state of the ground and influences slip and tractive force generated at the interface.

With these observations in mind, the next section describes the method of estimation of parameters and aggregates them to define the parameter $κ$ (i.e., $κ=f(δ,Γ,s)$⁠).

In tire sinkage/rut formation under static or quasi-static normal load (Fig. 1) shear displacement predominantly occurred in the vertical plane or near vertical plane. The contact geometry and sinkage of the tire is influenced by the soil strength index, and tire compliance. When tire is under applied torque slip-sinkage occurs on a plane tangential to the tire–soil contact surface or in a plane within the soil when adhesion force is high, due to translational force transferred from the axle to the tire ground interface and angular velocity of the rotating tire–wheel assembly. Therefore, additional parameter to define the aggregate parameter, $λ=1κ$ in a way that encapsulates the tire–soil interaction under applied torque is needed. I put forward the following to do so.

Slip vector under applied normal load and no torque is predominantly in or near the vertical plane depending on the amount of tire sidewall bulge. Upon application of torque, the slip vector becomes a composite of slip due to sinkage and tangential slip as the tire spins. The side slip vector is considered negligible for tire driven in a straight path. Change in deformation of tire affects vertical displacement, z. Change in z affects the dynamic rolling radius of the tire which in turn affects slip. The rate of rise of the tractive force at lower slip (i.e., slip until peak traction is reached) and subsequent changes in the magnitude and slope of the traction curve as slip further increases is encapsulated in slip modulus.

Asserting that slip modulus, $κ$ is a function of applied load (normal load and torque), slip, and the resultant contact surface geometry; a term to incorporate slip in defining the scale and shape functions is developed using Fig. 4 and geometric interpretation of hyperbolic trigonometric identities.

The use of a secant line as an average rate of change of slope of a traction curve (shown in Fig. 4) by sweeping through the $μ−s$ space is proposed to retain the variable rate of change of $μ$ with respect to slip. Figure 4 is a unit hyperbola and $μ−s$ curve plotted on the same Cartesian coordinates. We will make use of hyperbolic trigonometric function identities to extract a term contributing to $κ$⁠.

In Fig. 4, as point A approaches point C along $y=x2−1$ and the slope of $OA¯$ approaches zero. Notice also that as point A approaches point C, point M $[(1,tanh(a))=(1,tanθ)]$ moves along the vertical line $CH¯$⁠. Owing to $ddatanh(a)=1−tanh2(a)$ and hyperbolic trig identity $tanh2(a)+sech2(a)=1$ we can express $ddatanh(a)=sech2(a)$ or $tanh2(a)=1−sech2(a)$⁠. Observing that the domains for $sech(a)$⁠, (⁠$0<sech(a)≤1$⁠), and slip, s$(0≤s≤1)$ overlap and both abscissa in Fig. 4, we can rewrite $1−sech2(a)$ as $1−s2$⁠.

In Fig. 4, when point A coincides with point C, point G sliding along the x-axis reaches the origin (point O) (i.e., $tanh2(a)=1−sech2(a)=0$⁠). This corresponds to when slip, $s=0.$ Similarly, $lima→∞sech(a)=0$ corresponds to $s=1$⁠. It is noticeable from Fig. 4 that $tanθ=tanh(a)$ is the secant slope of $OM¯$ as point A moves along the unit hyperbola.

We exploit the trigonometric relationships described here to formulate Eq. (15) where $1−s2$ is posited to contribute to coefficient λ in Eq. (4).

Substituting, $λ$ in Eq. (4) with the expanded term in Eq. (16), we obtain Eq. (17).

The term $(1−s2)s$ in the exponent in Eq. (17) satisfies the constraint conditions: when $s=0$⁠, $μ(s)=0$ as no driving torque is applied and when $s=1$⁠, $μ(s)=0$ as at 100% slip there is no translational displacement of the driven tire (i.e., applied torque is overcome by motion resistance). Equation (17) without inclusion of the $(1−s2)$ term has the theoretical implication that $μ(s)=μmax$ at $s=∞$ where in fact $s∈[0,1]$⁠. Thus, the term $(1−s2)$ as introduced in Eq. (15) serves to satisfy boundary condition.

The influence of the coefficient $(1−s2)$ on $κ$ is negligible at low slip values and noticeable as slip increases. This is illustrated in Fig. 7 in the next section. The interpretation of this term in this case is that it accounts for reduction in soil strength at higher slip which is attributed to higher angular velocity of the tire–wheel assembly. As the angular velocity increases it can cause vibrations that loosen the soil and reduce its undisturbed/in-situ strength before being subjected to direct tire load. Implicit to this is the varying slip plane due to slip-sinkage and it manifests itself as added motion resistance in the term $(1−s2)$⁠. Also, embedded in the term $(1−s2)$ is the sliding friction component, $tanφm$ in Eq. (6).

Figure 5 summarizes traction model parameter estimation method described earlier.

Equation (4) is a traction model for tire shown in Fig. 6. Using values in Table 1, curve estimating $μ$ is created (Fig. 7).

The term $TrW$ represents the gross tractive coefficient and is modified by $1−s2$ to estimate the net peak traction coefficient, $μmax$⁠. $TrW$ multiplied by gradient $s2$ is considered an estimate of the motion resistance ratio as influenced by wheel angular velocity. In this way, the varying inertial effect of the rotating mass is intrinsically captured in the model along with the associated slip and soil strength changes that occur. An overlay of model using data in Table 1 and measured data are shown in Fig. 7 to illustrate the effect of the $1−s2$ term as slip increases. Figure 7 is intended for visualization and not to be construed as validation or verification of the model. The inclusion of the measured data is to acknowledge the variability that is encountered in-situ and also the downward trending of $μ$ after certain value of slip ratio is reached—consistent trend with curve representing $μ$ in Fig. 7. The measured data are from in-field measured condition where the soil shear strength is naturally variable due to heterogeneity in the soil particle size distribution and non-uniformity in the in-situ water content and porosity of the soil over the length of the test lane where measurement was taken. The measured data were obtained by measuring the dynamic vertical and longitudinal forces at the wheel end using a 44.5 kN load cell. Wheel shaft encoder was used to measure the angular velocity of the driven tire. Translational speed maintained was measured using GPS speedometer.

• A novel approach for model parameter identification and estimation based on observations and consideration of the underlying engineering mechanics principles is demonstrated.

• The method addresses the scalability and large displacement limitations of current traction models.

• In principle, the proposed method does not conflict with traditional soil strength measurement and estimation techniques such as cone index, direct shear, or pressure sinkage measurements. It parameterizes intergranular friction of soil particles, lubrication due to moisture, particle realignment, and adhesion of soil under load and induced slip for large deformation/distortion.

• Soil strength index as proposed is a good avenue to characterize shear strength of different soil types at soil moisture levels ranging from residual water content to liquid limit.

• The analogy in the formulation of Eqs. (4) and (2) or its variations is intentional. The method to calculate slip is the same in both cases. Where relevant historical tire and soil data are available, use of such data in the proposed model to extract information from past data for use as a priori knowledge may be feasible.

• With the evolution of sensor integration onto new vehicles, the model parameters identified are suitable for use with on-board control system as torque, the change in relative wheel position, dynamic loaded radius and slip can be quantified on the go.

No funds were received for this work.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent was not applicable. This article does not include any research in which animal participants were involved.

No data are available.

Any mistakes, inaccuracies or inconsistencies are the sole responsibility of the author.