A new contact and road model for multi-body dynamic simulation of wheeled vehicles on soft-soil terrain

Abstract

In this paper, a new high-performance and memory-efficient contact and road model is developed. Specifically, the road is modeled as a rectangular structured grid of deformable springs in the vertical direction, thus enabling fast execution. The new road model stands out due to its ability to handle large road scenarios by allocating computer memory dynamically for each spring, resulting in efficient memory utilization. Furthermore, each spring represents a small road patch that entails various information, such as the soil elevation, the soil properties, and the soil compaction, allowing for complicated simulations incorporating spatially varying soil properties and phenomena related to the multi-pass effect. In addition, using the new contact model, complex terrain geometries are handled in a computationally efficient way by approximating locally the irregular road profile with a suitable equivalent plane. For this, two different strategies are proposed, namely the radial basis function (RBF) interpolation method and the 3D envelo** contact model. Finally, the proposed techniques are implemented in Altair MotionSolve, a comprehensive multi-body simulation software for complex mechanical systems. In particular, a single-wheel test bed is initially examined followed by a four-wheeled rover model and the next-generation NATO reference mobility model (NG-NRMM). In all cases, the proposed model is validated by using available experimental data. Lastly, a case involving both wheeled and tracked vehicles is also examined by using a shared road model.

1 Introduction

The tire/soft-soil interaction and the corresponding modeling is of utmost importance to properly predict the mobility of vehicles in a wide range of engineering applications, such as agricultural machines [1], planetary explorations rovers [2, 3] as well as construction, mining, and military vehicles [4]. Therefore, a significant amount of research effort has been concentrated during the last years on develo** appropriate contact and road models for the corresponding interaction. The modeling techniques of this interaction can be split into two broad categories: a) high-fidelity physics models; b) and semi-empirical models [5].

In the first category, a high degree of accuracy can, generally, be achieved with the drawback of significantly increased computational resources necessary for the simulation. Typical applications include the use of discrete element method (DEM) [6, 7] as well as the finite element method (FEM) [8, 28, 29], is used. The core idea of this method is founded on using a series of ellipses to scan the road profile and, thus, produce an effective road plane. Therefore, the same technique is employed here for deriving an equivalent plane that locally approximates the irregular profile of a deformable terrain since, to the best of the authors’ knowledge, this methodology has not been examined from this point of view.

In addition, a variety of vehicle models are used for testing the new techniques proposed in this work. For this, the proposed methods are implemented in Altair MotionSolve, a comprehensive multi-body simulation software for complex mechanical systems. Subsequently, a single-wheel test bed is initially used to confirm the validity of the single tire’s longitudinal and lateral forces under combined slip conditions. Following that, the developed methods are tested in various advanced simulations to illustrate their effectiveness and applicability in real-life engineering applications. Specifically, a four-wheeled planetary exploration rover and the FED-Alpha vehicle of the next generation NATO reference mobility model [30, 31] are tested in various complex maneuvers. Lastly, a case involving a wheeled utility terrain vehicle (UTV) and a tracked personnel carrier vehicle is handled by introducing the concept of a shared road model.

The organization of this paper is as follows. First, the new road model is presented along with a description of obstacles’ definition. Following that, the core idea of the new contact model is introduced, while a thorough description of the two different strategies is carried out next. In particular, the equations for the radial basis function interpolation method and the 3D envelo** contact model are formulated. Subsequently, the theoretical approach of this paper is completed by displaying the essential parts of the tire/soft-soil interaction. Then, the examined models are presented along with the extracted numerical results that demonstrate the validity and effectiveness of the proposed methods. Finally, the most important conclusions are summarized in the last section.

2 Road modeling

Within this work, the road surface is represented by a height-field (HF), thus enabling the use of rough terrain geometries. Specifically, a rectangular structured grid of deformable springs in the vertical (z-axis) direction is used, as shown in Fig. 1. Based on this approach, each spring represents a small road patch for which the necessary information is stored. The properties of each spring include the soil elevation ($z$ coordinate), the soil properties, and the information regarding the compaction of the soil. Therefore, the HF data structure is augmented with the soil properties and the soil compaction parameters, as depicted in Fig. 1. This allows the execution of advanced simulations incorporating spatially varying soil properties and phenomena related to the multi-pass effect by using the new road model.

Fig. 1

Road modeled as an HF data structure, augmented with the soft-soil properties and multi-pass parameters

In addition, a constant grid spacing is used for the description of the road surface, which is set automatically based on the minimum tire width present in the model. This approach has been preferred over the case of nonstructured grids since the rectangular grid structure used here results in fast execution of the simulation code. This stems from the fact that, given specific x and y coordinates, the current spring can be identified in an extremely easy and computationally efficient way since a constant grid spacing is used for the whole soil area.

In general, this approach comes with the drawback of significantly increased memory allocation since the road discretization must be the same even for soil areas that the vehicle will not interact with. Therefore, a proper treatment for this issue is also proposed in this work. Specifically, in a wide range of applications, it is necessary to predict the vehicle mobility over a large soil area. In these cases, it would be extremely inefficient to allocate the necessary memory for the whole grid of springs from the beginning of the simulation. A distinct feature of the new road model lies in its ability to handle such cases by allocating the memory needed for the springs when it is first needed. Specifically, each spring is created, and the corresponding properties are stored when it gets pressed for the first time, thus leading to greatly improved memory allocation of the simulation code.

2.1 Obstacle definition

Using this road model, it is possible to define generic-shaped obstacles, while the option to assign different soft-soil properties (compared to the whole soil area) for the obstacle’s region is also enabled. In particular, a variety of predefined obstacle templates can be readily used, which are superimposed on the existing soft-soil road profile. For this, a few basic parameters, including their dimensions and their position and orientation, are the only essential input data for their definition, as shown in Fig. 2.

Fig. 2

Obstacle’s geometry, superimposed on the existing soft-soil road profile

Within this soft-soil tire model, the following obstacle types can be created:

• Rectangular obstacle

• Circular obstacle

• Bump obstacle

• Ramp obstacle

• Roof obstacle

• Sine obstacle

• Sine-sweep obstacle (linear/logarithmic)

• Plank obstacle (beveled edges/round edges)

• Arbitrary obstacle

More specifically, using the last option, a generic obstacle can be created, as depicted in Fig. 3, where the obstacle’s region can have an arbitrary shape. A rectangular grid structure is still used for the road model, while an irregular terrain surface can be defined inside the obstacle’s region by assigning proper soil elevation (z) values to the enclosed springs. Furthermore, in all cases, it is possible to assign different soft-soil properties for the obstacle’s enclosing area by creating a new material that includes all the necessary parameters. Lastly, each obstacle can also be identified as rigid, thus leading to a tire-rigid road interaction. In such cases, a well-established tire model for rigid road, namely the Fiala tire model, is used to model the corresponding interaction.

Fig. 3

Arbitrary obstacle, superimposed on the existing soft-soil road profile

2.2 Shared road model

At this point, it is worth noting that the new road model can also be used for predicting the mobility of tracked vehicles on soft-soil terrain. This allows the analysis of cases where both wheeled and tracked vehicles co-exist in a model and, thus, a shared road model must be used since the mobility of each vehicle is strongly influenced by the soil compaction created by the other vehicles. For this, the same set of soft-soil parameters is used for both cases, while the soil compaction parameters are updated after each pass of the wheels or the track links.

3 Contact modeling

Concerning the tire–soil contact model, a computationally efficient approach is developed here, which enables the use of irregular terrain profiles. Specifically, instead of performing the necessary calculations for the tire–soil interaction on a per-node basis, the rough terrain surface is locally approximated with a proper equivalent plane. Therefore, a much faster execution of the simulation code is achieved since the complexity of the examined contact problem is significantly reduced.

In particular, two contact methods are developed and presented in this work, namely the radial basis function (RBF) interpolation method and the 3D envelo** contact model. However, the core idea in both models remains the same. That is, based on the above-described methodology, both contact models use the springs enclosed in the tire contact patch area (see Fig. 4) to calculate the equivalent soil elevation and local inclinations of the road surface.

Fig. 4

Tire–soil contact patch and springs enclosed in the tire contact patch area

3.1 Radial basis function (RBF) interpolation

In the first case, a radial basis function interpolation process [26] is employed to calculate representative values for the soil elevation and the partial derivatives of the road surface. For this, assuming that the tire–soil contact patch area consists of $n$ spring regions with coordinates $\left ( \underline{x}_{i} = \left ( x_{i}, y_{i} \right ), z_{i} \right )$, $i=1,\dots ,n$, the weights $w_{i}$, $i=1,\dots ,n$, are initially determined using the equations

$$ z\left ( \underline{x}_{j} \right ) = \sum _{i = 1}^{n} w_{i}\varphi \left ( r_{ij} \right ),\quad j = 1, \ldots ,n, $$

(1)

where the distance $r_{ij}$ is given by

$$ r_{ij} = \left \| \underline{x}_{j} - \underline{x}_{ i} \right \| $$

(2)

and $\varphi$ is a radial basis function. The system of equations (Eq. (1)) can be written in the equivalent matrix form:

$$ \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \varphi \left ( \left \| \underline{x}_{ 0} - \underline{x}_{ 0} \right \| \right ) & \varphi \left ( \left \| \underline{x}_{ 0} - \underline{x}_{ 1} \right \| \right ) & \cdots & \varphi \left ( \left \| \underline{x}_{ 0} - \underline{x}_{ n} \right \| \right ) \\ \varphi \left ( \left \| \underline{x}_{ 1} - \underline{x}_{ 0} \right \| \right ) & \varphi \left ( \left \| \underline{x}_{ 1} - \underline{x}_{ 1} \right \| \right ) & \cdots & \varphi \left ( \left \| \underline{x}_{ 1} - \underline{x}_{ n} \right \| \right ) \\ \vdots & \vdots & \ddots & \vdots \\ \varphi \left ( \left \| \underline{x}_{ n} - \underline{x}_{ 0} \right \| \right ) & \varphi \left ( \left \| \underline{x}_{ n} - \underline{x}_{ 1} \right \| \right ) & \cdots & \varphi \left ( \left \| \underline{x}_{ n} - \underline{x}_{ n} \right \| \right ) \end{array}\displaystyle \right ]\left ( \textstyle\begin{array}{c} w_{0} \\ w_{1} \\ \vdots \\ w_{n} \end{array}\displaystyle \right ) = \left ( \textstyle\begin{array}{c} z\left ( \underline{x}_{ 0} \right ) \\ z\left ( \underline{x}_{ 1} \right ) \\ \vdots \\ z\left ( \underline{x}_{ n} \right ) \end{array}\displaystyle \right ). $$

(3)

Following that, the interpolated value for the soil elevation is derived by using the equation

$$ z\left ( \underline{x} \right ) = \sum _{i = 1}^{n} w_{i}\varphi \left ( \left \| \underline{x} - \underline{x}_{ i} \right \| \right ). $$

(4)

It is worth noting that a significant advantage of the RBF method lies in its ability to work both with structured and nonstructured grids. In addition, this contact model can be used as a general framework for the calculation of the equivalent plane, depending on the radial basis function used. More specifically, typical radial basis functions that can be used include the Gaussian, the multiquadric, and the inverse multiquadric function, which are provided by the following expressions:

$$\begin{aligned} &\text{Gaussian:} \quad \varphi \left ( r \right ) = e^{ - \left ( \varepsilon r \right )^{2}}, \end{aligned}$$

(5)

$$\begin{aligned} &\text{Multiquadric:}\quad \varphi \left ( r \right ) = \sqrt{1 + \left ( \varepsilon r \right )^{2}}, \end{aligned}$$

(6)

$$\begin{aligned} &\text{Inverse multiquadric:}\quad \varphi \left ( r \right ) = \frac{1}{\sqrt{1 + \left ( \varepsilon r \right )^{2}}} . \end{aligned}$$

(7)

In the above equations, the position vector $\underline{x} = \left ( x,y \right )$ corresponds to the point $C_{CP}$ of the tire contact patch area, as shown in Fig. 5. This point is derived by projecting the wheel center $C_{W}$ on the road surface by taking into account the wheel orientation. In addition, $\varepsilon$ represents the shape parameter that controls the flatness of the different radial basis functions. Herein, the shape parameter is calculated based on the average distance between the springs enclosed in the contact patch area.

Fig. 5

Three-dimensional contact patch area and projection of wheel center on the road surface

Apparently, the Gaussian and inverse multiquadric radial basis functions exhibit a local response since their value decreases with increasing distance $r$, as illustrated in Fig. 6. Conversely, the value of the multiquadric function increases with increasing distance, thus a global response is achieved. Therefore, a prominent feature of this contact model stems from its ability to easily adapt on the application’s needs, based on the selection of the radial basis function. In addition, using this contact model, the local inclinations of the road surface are calculated in an analytical and efficient way [27].

Fig. 6

Gaussian, multiquadric and inverse multiquadric radial basis functions’ values for $\varepsilon =0.5$ and $\varepsilon =3$

3.2 3D envelo** contact model

Concerning the second contact model, the 3D envelo** method is a well-established model for irregular nondeformable (rigid) terrain [28, 29]. In this paper, the same technique is applied to construct an effective plane, which, locally, approximates the irregular profile of a deformable terrain. The core idea is founded on using a series of ellipses to scan the road profile and, thus, to produce an effective road plane that is defined by three quantities. Namely, the modified effective height $w '$, the effective forward slope $\beta _{y}$, and the effective road camber angle $\beta _{x}$. The simplest form of this contact model is depicted in Fig. 7.

Fig. 7

3D envelo** contact model (Reproduced from [28])

Specifically, the shape of the ellipses is defined by the parameters $a_{e}$, $b_{e}$, and $c_{e}$ according to the expression

$$ \left ( \frac{x}{a_{e}} \right )^{c_{e}} + \left ( \frac{z}{b_{e}} \right )^{c_{e}} = 1. $$

(8)

These parameters are calculated by introducing the user-defined dimensionless parameters [28]

$$ p_{ae} = \frac{a_{e}}{R}, $$

(9)

$$ p_{be} = \frac{b_{e}}{R}, $$

(10)

$$ p_{ce} = c_{e}. $$

(11)

In addition, the tandem base length $l_{s}$ is defined by introducing the user-defined dimensionless parameter

$$ p_{ls} = \frac{l_{s}}{2a}, $$

(12)

where $2a$ denotes the length of the tire contact patch, while the parameter $2 b_{EC}$ corresponds to the tire’s width. Once the necessary parameters are defined, the height of the center of each ellipsis Z is calculated at every time point by scanning the road profile underneath the ellipsis and determining the corresponding height based on the ellipsis–road intersection. A thorough presentation of the underlying theory in addition to guidelines for the selection of the associated parameters can be found in [28].

In this simple form of the 3D envelo** contact model, the necessary quantities are calculated by using the equations

$$ w' = \frac{Z_{f,left} + Z_{r,left} + Z_{f,right} + Z_{r,right}}{4} - b_{e}, $$

(13)

$$ \tan \beta _{y} = \frac{Z_{r,left} - Z_{f,left} + Z_{r,right} - Z_{f,right}}{2l_{s}}, $$

(14)

$$ \tan \beta _{x} = \frac{Z_{f,left} - Z_{f,right} + Z_{r,left} - Z_{r,right}}{2\left ( 2b_{EC} \right )}. $$

(15)

In general, the simple form with two parallel tandems, shown in Fig. 7, is not sufficient to obtain accurate results for sharp irregularities. Therefore, more parallel tandems and intermediate cams are added, as illustrated by Fig. 8. Specifically, parallel tandems are added along the tire’s width, while the intermediate cams are positioned between the front and rear edge of the tire contact patch. In addition, the intermediate cams between the left and right sides are not necessary since their contribution cancel out when calculating the effective road camber angle [29]. The same equations, as presented earlier for the simple 3D envelo** contact model, apply again, but now extended for the case of multiple parallel tandems and intermediate cams. In particular, the three necessary quantities are provided by the expressions

$$ w' = \frac{1}{n}\sum _{j = 1}^{n} \left ( \frac{Z_{fj} + Z_{rj}}{2} \right ) - b_{e}, $$

(16)

$$ \tan \beta _{y} = \frac{1}{n}\sum _{j = 1}^{n} \left ( \frac{Z_{rj} - Z_{fj}}{l_{s}} \right ), $$

(17)

$$ \tan \beta _{x} = \frac{1}{m}\sum _{i = 1}^{m} \left ( \frac{Z_{i,n} - Z_{i,1}}{2b_{EC}} \right ), $$

(18)

where $m$ is the number of longitudinal cams and $n$ is the number of parallel tandems.

Fig. 8

3D envelo** contact model with n = 6 parallel tandems and m = 5 longitudinal cams (reproduced from [28])

The advantage of this contact model stems from its ability to provide improved accuracy in cases of short wavelengths (sharp steps) in the road height. However, it constitutes a more computationally demanding methodology in comparison to the above-presented RBF interpolation method. It should be emphasized that in this work it is possible to define which of the tires of a model (if any) will use the 3D envelo** contact method. Therefore, both tires that use the RBF interpolation method and the 3D envelo** contact model can co-exist in a model, leading to an optimum combination of accuracy and computational efficiency.

4 Tire–road interaction

In this section, the theoretical approach of this paper is completed by displaying the essential parts of the tire–soil interaction. Specifically, the soil modeling is initially presented, followed by the description of the normal and shear stress distribution over the tire–soil contact patch area. Subsequently, the equations for the modeling of the bulldozing resistance are presented, while the methodology used to take into account the tire deformability is explained next. Lastly, the core idea and the equations used for the multi-pass effect are illustrated at the final part of this section.

4.1 Soil modeling

The soil model is based on the classical and well-established expression introduced by Bekker for the pressure–sinkage relationship. More specifically, the pressure is derived by employing the Bekker formula [12, 13]

$$ \sigma \left ( h \right ) = \left ( \frac{k_{c}}{b} + k_{\phi} \right )h^{n}, $$

(19)

where $b$ is the length of the shorter side of the rectangular contact patch, $h$ is the sinkage, and $k_{c}$, $k_{\varphi}$, and $n$ represent empirical coefficients. Moreover, regarding the soil failure, a variety of criteria exist in the literature. Here, the widely used Mohr–Coulomb failure criterion is employed, which states that the maximum soil shear strength is provided by the expression

$$ \tau _{\max ,s}\left ( \sigma \right ) = c + \sigma \tan \phi , $$

(20)

where $c$ is the soil apparent cohesion and $\varphi$ is the angle of internal shearing resistance of the soil material.

4.2 Contact patch area

First, the tire–soil contact patch area, which is defined by the entry angle $\theta _{f}$ and the exit angle $\theta _{r}$, is determined as a function of the sinkage $h$, the tire’s radius $R$, and the elastic sinkage $h_{e}$ by employing the equations

$$ \theta _{f} = \cos ^{ - 1}(1 - h / R), $$

(21)

$$ \theta _{r} = \cos ^{ - 1}(1 - h_{e} / R). $$

(22)

Specifically, the tire–soil contact patch constitutes the region defined by the wheel contact angles $\theta _{f}$ and $\theta _{r}$ in addition to the tire width, as shown in Fig. 9. Then, the springs enclosed in the contact patch area are, initially, identified. Subsequently, an effective road plane is constructed by using the soil elevation values of these springs, as explained in Sect. 3, through the determination of the equivalent soil elevation and local inclinations of the road surface.

Fig. 9

Wheel contact angles and contact patch area

4.3 Normal and shear stress distribution

Then, the normal and the shear stress distribution at the contact patch area is determined as a function of the wheel angle $\theta$. More specifically, using the pressure–sinkage relationship proposed by Bekker (see Eq. (19)), the normal stress distribution is provided by

$$ \sigma \left ( \theta \right ) = \left \{ \textstyle\begin{array}{c} R^{n}\left ( \frac{k_{c}}{b} + k_{\phi} \right )\left ( \cos \theta - \cos \theta _{f} \right )^{n},\quad \theta _{m} \le \theta \le \theta _{f} \\ R^{n}\left ( \frac{k_{c}}{b} + k_{\phi} \right )\left ( \cos \left ( \theta _{f} - \frac{\theta - \theta _{r}}{\theta _{m} - \theta _{r}}\left ( \theta _{f} - \theta _{m} \right ) \right ) - \cos \theta _{f} \right )^{n},\quad \theta _{r} \le \theta \le \theta _{m} \end{array}\displaystyle \right ., $$

(23)

where n is the sinkage exponent defined in Eq. (19). In the above equation, the parameter $\theta _{m}$ represents the wheel angle at which the normal stress is maximized, given by the formula [14]

$$ \theta _{m} = \left ( c_{1} + c_{2}\kappa \right )\theta _{f}, $$

(24)

where $c_{1}$ and $c_{2}$ denote parameters that depend on the wheel–soil interaction and $\kappa$ represents the longitudinal slip. Furthermore, the normal stress distribution, given by Eq. (23), is modified in this work to account for the soil dam** effect [8]. Specifically, the normal stress distribution is provided by

$$ \sigma \left ( \theta \right ) = \left \{ \textstyle\begin{array}{c} R^{n}\left ( \frac{k_{c}}{b} + k_{\phi} \right )\left ( \cos \theta - \cos \theta _{f} \right )^{n} + \frac{c_{s}v_{c}}{A_{c}},\quad \theta _{m} \le \theta \le \theta _{f} \\ R^{n}\left ( \frac{k_{c}}{b} + k_{\phi} \right )\left ( \cos \left ( \theta _{f} - \frac{\theta - \theta _{r}}{\theta _{m} - \theta _{r}}\left ( \theta _{f} - \theta _{m} \right ) \right ) - \cos \theta _{f} \right )^{n} + \frac{c_{s}v_{c}}{A_{c}},\quad \theta _{r} \le \theta \le \theta _{m} \end{array}\displaystyle \right . $$

(25)

where $c_{s}$ denotes the soil dam**, $v_{c}$ represents the soil’s compression rate/velocity, and $A_{c}$ is the contact patch area.

Following that, the shear stress distributions $\tau _{x} (\theta )$ and $\tau _{y} (\theta )$, in the longitudinal (x) and in the lateral (y) direction respectively, are calculated using the similar expressions [15, 16]

$$ \tau _{x}\left ( \theta \right ) = \tau _{\max} \left ( 1 - \exp \left ( - \frac{j_{x}\left ( \theta \right )}{k_{x}} \right ) \right ), $$

(26)

$$ \tau _{y}\left ( \theta \right ) = \tau _{\max} \left ( 1 - \exp \left ( - \frac{j_{y}\left ( \theta \right )}{k_{y}} \right ) \right ). $$

(27)

In the above equations, the parameters $k_{x}$ and $k_{y}$ represent the shear modules, which are provided by the equations [17]

$$ k_{x} = k_{x0}\alpha + k_{x1}, $$

(28)

$$ k_{y} = k_{y0}\alpha + k_{y1}, $$

(29)

where $\alpha$ denotes the slip angle. Moreover, the soil deformations $j_{x}$ and $j_{y}$ are derived as functions of the wheel angle $\theta$ by employing the expressions [14, 16]

$$ j_{x}\left ( \theta \right ) = R\left [ \theta _{f} - \theta - \left ( 1 - \kappa \right )\left ( \sin \theta _{f} - \sin \theta \right ) \right ], $$

(30)

$$ j_{y}\left ( \theta \right ) = R\left ( 1 - \kappa \right )\left ( \theta _{f} - \theta \right )\tan \alpha . $$

(31)

It should be emphasized that, for the maximum shear stress $\tau _{max}$ (see Eqs. (26)–(27)), a modified approach is followed here to take into account the coefficient of friction at the tire–soil interface [19]. More specifically, the maximum shear stress at the tire–soil interface $\tau _{max,\ ts}$ is initially approximated as a function of the pressure $\sigma$ and the friction coefficient $\mu _{s}$ by employing the equation

$$ \tau _{\max ,ts}\left ( \sigma \right ) = \mu _{s}\sigma . $$

(32)

Subsequently, the minimum of the two constituent parts, namely the maximum soil shear strength (see Eq. (20)) and the maximum shear stress at the tire–soil interface is utilized here for the shear stress calculation [19]. That is,

$$ \tau _{\max} \left ( \sigma \right ) = \min \left ( \mu _{s}\sigma , c + \sigma \tan \phi \right ). $$

(33)

4.4 Bulldozing resistance

Moreover, in this work, the lateral force comprises two different parts. Specifically, apart from the shear force exerted on the contact patch due to the tangential stresses $\tau _{y}$, the bulldozing force, which acts on the side face of the wheel, due to the tire’s sinkage, is also incorporated [16, 17]. Herein, the Hegedus resistance estimation method is used to calculate the bulldozing force [16, 17]. Using this approach, a bulldozing resistance $R_{b}$ is developed per unit width of a blade, which is given by the equation

$$ R_{b} = \frac{\cot X_{c} + \tan \left ( X_{c} + \phi \right )}{1 - \tan \alpha '\tan \left ( X_{c} + \phi \right )}\left \{ hc + \frac{1}{2}\rho _{s}h^{2}\left [ \left ( \cot X_{c} - \tan \alpha ' \right ) + \frac{\left ( \cot X_{c} - \tan \alpha ' \right )^{2}}{\tan \alpha ' + \cot \phi} \right ] \right \}, $$

(34)

where $\rho _{s}$ is the soil density. Moreover, the destructive angle $X_{c}$, based on Bekker’s theory, is approximated by

$$ X_{c} = 45^{ \circ} - \frac{\phi}{2}. $$

(35)

4.5 Tire deformability

To account for the tire’s deformability, a larger substitute circle is used to describe the tire–soil contact patch area [13, 19, 32], as shown in Fig. 10. More specifically, to calculate the diameter of the substitute circle, an iterative procedure is followed until the soil vertical reaction force and the tire vertical force are balanced. The former is calculated through integration of the normal and shear stresses at the contact patch area, while for the latter, the tire’s vertical stiffness is used along with the corresponding deformation. More specifically, the soil vertical reaction force is given by

$$ F_{z,s} = bR^{*}\int _{\theta _{r}^{*}}^{\theta _{f}^{*}} \left [ \tau _{x}\left ( \theta ^{*} \right )\sin \theta ^{*} + \sigma \left ( \theta ^{*} \right )\cos \theta ^{*} \right ] d\theta ^{*}, $$

(36)

while the tire vertical force is provided by

$$ F_{z,t} = K_{t} f, $$

(37)

where $K_{t}$ denotes the tire’s vertical stiffness and $f$ stands for the resulting deformation of the tire. In addition, an extra equation is needed, which relates the tire’s deformation and the radius of the substitute circle $R^{*}$. For this, the following expression, originally proposed by Bekker, is used:

$$ \frac{R^{*}}{R} = \left ( \sqrt{1 + \frac{f}{h}} + \sqrt{\frac{f}{h}} \right )^{2}. $$

(38)

The last three equations are solved iteratively until

$$ \left | F_{z,s} - F_{z,t} \right | < \varepsilon _{F}, $$

(39)

where the value of the numerical threshold $\varepsilon _{F}$ constitutes a function of the tire’s maximum vertical load. Specifically, for a given value of the tire sinkage, which is easily determined based on the wheel’s position at each time point, the resulting tire deformation $f$ and the radius of the substitute circle $R^{*}$ must be calculated so that the forces given by Eqs. (36) and (37) will be in balance. In particular, the value of the tire deformation is, initially, modified. Then, the radius of the substitute circle $R^{*}$ is determined by using Eq. (38). Subsequently, the forces given by Eqs. (36) and (37) are calculated, and the condition given by Eq. (39) is evaluated. If this condition is not satisfied, then the value of the tire deformation is, again, modified and the above-described procedure is repeated.

Fig. 10

Substitute circle method to account for the tire’s deformability

It should be noted that, in cases of negligible deformation of the tire, an extra functionality can be enabled to model the tire as nondeformable (rigid) and, thus, simplify and accelerate the calculation of the vertical force $F_{z}$. Specifically, no iterations are required in this case since only Eq. (36) is used where the quantities $R^{*}$ and $\theta ^{*}$ are replaced by $R$ and $\theta$ respectively.

4.6 Multi-pass effect

Lastly, regarding the multi-pass effect, the response of the soil to repetitive normal load needs to be established. More specifically, the mathematical description of the normal stress distribution must be modified in cases of existing precompaction of the soil [18, 19]. In general, one part of the induced soil deformation is elastic (elastic sinkage), and the remaining part (plastic sinkage) is irreversible. Therefore, a proper way to distinguish between these two constituent parts of the total sinkage needs to be established.

Within this work, the following equation is used for the calculation of the elastic sinkage:

$$ h_{e} = \frac{\sigma \left ( \theta _{m} \right )}{K_{s}}, $$

(40)

where $K_{s}$ denotes the soil’s elastic stiffness.

4.7 Soil library

In all semi-empirical soft soil tire models, the proper selection of the associated parameters is, undoubtedly, a challenging task from the user’s perspective, while crucial for the development of reliable and accurate models at the same time. Specifically, some physical tests are, generally, required for the determination of these parameters. These experiments are very expensive, require special equipment, and take a significant amount of time to be executed.

Based on this, a ready-to-use soil library has already been created, which can be used as a very good starting point. The developed soil library, which is based on the existing literature [1, 14, 17, 18, 33], includes the associated properties for a variety of soil types. Then, the default values for the relevant properties can be modified if the user has some additional knowledge on the specific soil type used. Lastly, it should be noted that the developed soil library is already integrated into Altair’s multi-body simulation software, MotionSolve.

5 Numerical results and discussion

In the current section, a full set of numerical results is presented to demonstrate the validity and effectiveness of the new methods. In particular, a single wheel test bed is initially examined, followed by a planetary exploration rover. Then, the FED-Alpha vehicle of the next generation NATO reference mobility model is tested in various complex maneuvers. Lastly, an advanced simulation involving both wheeled and tracked vehicles, namely a utility terrain vehicle and a personnel carrier vehicle, is also carried out by employing a shared road model.

5.1 Single-wheel test bed

To start with, the single tire’s longitudinal and lateral forces are validated under combined slip conditions. Specifically, in Fig. 11 (a), the simulation results (MotionSolve) for the longitudinal force as a function of the longitudinal slip are compared with the corresponding experimental data [17] for the case of slip angle equal to 5 degrees. Similarly, in Fig. 11 (b), the lateral force is examined for the case of slip angle equal to 10 degrees.

Fig. 11

Single-wheel test bed results under combined slip conditions: ($a$) Longitudinal force – Longitudinal slip for $\alpha =5\ [deg]$ and ($b$) Lateral force – Longitudinal slip for $\alpha =10\ [deg]$ (source of experimental data: [17])

5.2 Planetary exploration rover

Next, a four-wheeled planetary exploration rover [17] is examined in various maneuvers. This vehicle, shown in Fig. 12, consists of eleven (11) bodies and ten (10) joints in total, while the wheels are connected to the main body through rocker suspension. In addition, all four wheels of the rover possess a steering degree of freedom.

Fig. 12

MotionView model of four-wheeled planetary exploration rover

First, a steering maneuver is examined where the front wheels are steered by 30 degrees. Specifically, in Fig. 13 (a), the main body’s center of mass x and y position is shown, while in Fig. 13 (b), the vehicle’s yaw angle as a function of the simulation time is presented. As can be seen from both graphs, a very good correlation is achieved between the simulation results and the corresponding experimental data [17].

Fig. 13

Steering maneuver results: ($a$) Main body’s center of mass position and ($b$) Vehicle’s yaw angle – Simulation time (source of experimental data: [17])

Moreover, the path of the vehicle and the resulting soil compaction are illustrated in Fig. 14 both for the simulated and the experimental case. It should also be emphasized that the total simulation time of this maneuver is 37 [s], while the necessary CPU time is 35.33 [s]. The simulation was performed by using a single core of an Intel^® Core™ i7-10875H Processor @ 2.30 GHz. Therefore, it is of great interest that the CPU time is comparable with the simulation time, even by using a single core for running the simulation.

Fig. 14

Path of the vehicle and resulting soil compaction for the steering maneuver

Until this point, a flat road surface was used in all examined cases. Using this as a solid starting point, the developed soft-soil tire model is also validated and tested in much more complex terrain geometries. To start with, a slope traversing maneuver with a slope angle of 10 degrees is examined in Fig. 15. In particular, the simulation results for the main body’s center of mass position are compared with the corresponding experimental data for three different cases [34]. Specifically, for the case marked with the blue curve, none of the four wheels of the rover model is steered, whereas only the front wheels are steered by 15 degrees for the case denoted with the red curve. Similarly, all four wheels of the rover model are steered by 15 degrees for the case marked with the black curve. Moreover, the necessary CPU time for this maneuver is 37.59 [s], whereas the total simulation time is 12.5 [s]. Again, it should be highlighted that the CPU time is comparable with the simulation time, even by using a single core, thus illustrating the computational efficiency of the proposed methods.

Fig. 15

Slope traversing maneuver ($\psi =10\ [deg]$): Comparison of simulation results and experimental data (source of experimental data: [34])

Regarding the numerical values of the soil properties for both the single-wheel test bed and the exploration rover simulation cases, the vast majority is provided by the corresponding publications, which include the experimental data [17, 34]. In addition, the soil’s elastic stiffness $K_{s}$ is determined by taking into account that the wheel sinkage ratio is approximately equal to 1 according to [17]. The coefficient of friction is selected equal to 0.8 based on the acceptable ranges derived from previous experimental observations [35], while the soil dam** value is equal to 0.5 [kNs/m]. Lastly, the RBF interpolation method was used in these maneuvers for modeling the tire–soil contact.

5.3 Next generation NATO reference mobility model (NG-NRMM)

Following that, the capabilities of the proposed soft-soil tire model are also demonstrated by using available experimental data from the next generation NATO reference mobility model [31]. For this, the FED-Alpha vehicle is initially modelled in MotionView, as shown in Fig. 16. This vehicle model consists of 101 bodies and 97 joints in total, while double wishbone suspensions with air-springs, coil springs, and selective damper are employed. Then, a series of advanced soft terrain tests is used to demonstrate the effectiveness of the proposed methods in complex, real-life applications.

Fig. 16

Model of FED-Alpha vehicle in MotionView

Specifically, the model representation is grouped in different subsystems corresponding to the same assemblies in the actual FED-Alpha vehicle. The relatively large number of bodies and joints is justified by the complexity of the examined vehicle and the necessity of providing an accurate description of all subsystems to capture the dynamic behavior of the vehicle in the best possible way. In particular, the majority of bodies and joints correspond to the suspension and driveline subsystems. For instance, the suspension stabilizer bar consists of a series of bodies that are properly connected through joints, thus leading to a significant number of bodies and joints for the suspension subsystem. Similarly, the need for an accurate representation of the center, front, and rear differentials (all-wheel drive configuration) leads to a significant number of bodies and joints for the driveline subsystem.

Specifically, a variable slope climbing maneuver is initially performed on dry sand soil terrain. In this case, a rough road surface is encountered and the slope grade is gradually increased from 0% up to approximately 30%. In Fig. 17, a comparison of the simulation and the experimental results for the longitudinal slip of the front left tire is carried out. However, since a locked differential is used for performing this maneuver, this slip value also corresponds to the average wheel slip.

Fig. 17

Variable slope climbing maneuver: Comparison of simulation results and experimental data (source of experimental data: [31])

As can be seen, a very good agreement is observed between the simulation results and the respective experimental data. In addition, a rough terrain surface is encountered in this case, as shown in Fig. 18. Therefore, the 3D envelo** contact model is used for all four wheels of the vehicle for this simulation. Moreover, the resulting soil plastic sinkage can be clearly observed in the same figure.

Fig. 18

Variable slope climbing maneuver: Terrain surface and resulting soil plastic sinkage

In addition, a drawbar pull experiment is also performed on dry sand soil surface. In particular, the results of the drawbar pull as a function of the average longitudinal slip is presented in Fig. 19 for the simulation and the physical experiment. In this case, the RBF interpolation method is employed for modeling the resulting tire–soil contact since a flat terrain is used for the drawbar pull experiment. Based on Figs. 17 and 19, the effectiveness and applicability of the proposed methods is highlighted, even for cases of such complex vehicle models and terrain geometries.

Fig. 19

Drawbar pull experiment on dry sand soil surface: Comparison of simulation results and experimental data (source of experimental data: [31])

Concerning the numerical values of the soil properties, the vast majority is provided by the test data extracted during the NG-NRMM’s cooperative demonstration of technology event [31]. Moreover, the parameters $c_{1}$ and $c_{2}$ are chosen equal to 0.3 and 0.32, respectively, according to [14], while the coefficient of friction is set equal to 0.4. Lastly, the soil dam** value is chosen equal to 0.5 [kNs/m].

5.4 Wheeled and tracked vehicles – shared road model

Lastly, an advanced simulation that includes all the presented features is examined. Specifically, a complex road surface with generic-shaped obstacles is used, as shown in Fig. 20 (c). The terrain corresponds to dry sand soil type, and the numerical values of the necessary soil properties are selected according to [18]. In addition, both wheeled and tracked vehicles co-exist in this model. In particular, a utility terrain vehicle (UTV) and a tracked personnel carrier vehicle are employed, as illustrated in Fig. 20 (a), (b). Consequently, it should be noted that a generalization of the presented methods was performed for the case of tracked vehicles and for models including both wheeled and tracked vehicles through a shared road model.

Fig. 20

The constituent parts of the examined model: ($a$) Utility terrain vehicle, ($b$) Tracked personnel carrier vehicle, and ($c$) Road surface with generic-shaped obstacles

The examined model consists of 379 bodies and 389 joints, in total. Moreover, 854 contact pairs are defined, while the total number of bushings and forces used in this model is 398. Concerning the results of the analysis, the multi-pass effect is clearly demonstrated by observing the plastic sinkage resulting both from the UTV and from the tracked vehicle (see Fig. 21). In addition, both vehicles interact with the generic-shaped obstacles during their course.

Fig. 21

Multi-pass effect – Road surface with generic-shaped obstacles: Resulting soil plastic sinkage in shared road model

6 Conclusions

In the present study, a holistic approach for the simulation of wheeled vehicles on soft-soil terrain was proposed. For this, a new, fast, and memory-efficient road and contact model was developed. In addition, a generalization of the presented methods was also performed for tracked vehicles and for the cases where both wheeled and tracked vehicles co-exist in a model and, thus, a shared road model must be employed. Using the proposed methods, real-life applications involving complex phenomena can be properly handled.

Specifically, a distinct feature of the new road model stems from its ability to handle large road scenarios by allocating the memory for each spring when it is first needed, resulting in low memory allocation of the simulation code. Moreover, the height-field data structure is augmented with the soil properties and the soil compaction parameters in this work, while the rectangular grid structure used here leads to fast execution of the code. It is also worth noting that it is possible to define generic-shaped obstacles with different soft-soil properties in comparison to the whole soil area.

Subsequently, a computationally efficient tire–soil contact model, which enables the use of rough terrain geometries, was also presented here. More specifically, two different approaches were developed, namely the radial basis function interpolation method and the 3D envelo** contact model. Nonetheless, the core idea in both cases remains the same. That is, both contact models use the springs enclosed in the tire contact patch area to derive a proper equivalent plane and, thus, to provide the corresponding soil elevation and local inclinations of the road surface.

Finally, the proposed methods were implemented in Altair’s multi-body simulation software MotionSolve, and numerous engineering problems were examined to demonstrate their effectiveness and applicability in complex real-life applications. In particular, a detailed validation of the proposed soft-soil tire model was initially performed by using available experimental data. In all cases, a very good agreement was observed between the simulation results and the corresponding experimental data. Moreover, the developed model was tested in various advanced maneuvers, incorporating rough terrain geometries, passing over obstacles, and the phenomena related to the multi-pass effect. Lastly, a variety of wheeled and tracked vehicle models was used in these maneuvers, including a planetary exploration rover, a utility terrain vehicle, the FED-Alpha vehicle, and a tracked personnel carrier vehicle.