A steering model for articulated tracked vehicle considering soil deformation on track–soil interaction

Abstract

With the advantage of steering performance, articulated tracked vehicles have excellent mobility in off-road application. However, in current models for steering performance, soil deformation on the interaction between track and soil cannot always be taken into account. Therefore, steering performance cannot always be calculated accurately. In order to solve the problem, it is essential to propose a steering model which can take the effect of soil deformation on track–soil interaction into consideration. In this article, a steering model of articulated tracked vehicle is proposed on track–soil interaction. Moreover, in order to improve steering performance, a track–soil sub-model is developed that can consider soil deformation on track–soil interaction. Using this steering model based on track–soil sub-model, steering performance can be calculated more accurately. Simulation studies and experimental results are in strong agreement with the theoretical results in this article. The results show that equipped with the track–soil sub-model, the proposed steering model can be used to accurately predict steering performance. The steering model of articulated tracked vehicle proposed in this article can provide a basis for other similar vehicles.

Introduction

In recent years, the researches on steering performance have made considerable achievements. At first, Steeds proposed skid-steer concept ¹ and then a lot of work had been investigated from all aspects. ² However, unlike a skid-steer vehicle, an articulated vehicle uses an articulated mechanism to achieve a steering path. That means for a skid-steer vehicle, the efficiency of steering performance will be degraded due to the slip and skid phenomenon. However, for an articulated vehicle, the efficiency will not be degraded because of articulated mechanism. Moreover, tracked vehicles have excellent mobility in off-road application.^3,4 And with the combination of steering performance and mobility, the applications of articulated tracked vehicles have been spread widely.

Most of works of articulated vehicles are focused on stability and dynamics such as heavy trucks, tractor-implement, and wheel loader. In Iida and colleagues,^5,6 a steering model of articulated wheel loader on level ground is developed based on the sideslip angle; in Azad et al., ⁷ a linear model of articulated steering vehicle is developed based on the disturbed motion; in Polotski and Hemami, ⁸ a theoretical model is developed for the motion of wheel load truck on a horizontal plane; in Li et al., ⁹ a mathematical model of articulated steering wheel loader is developed to evaluate the steering performance. However, little work has been implemented focused on the machine–soil interaction. In fact, machine–soil interaction plays a significant role in deterring steering performance. In order to predict steering performance more accurately, it is necessary to focus on machine–soil interaction for articulated vehicle.

For tracked vehicles, some researches have been focused on track–soil interaction. At first, Bekker ¹⁰ proposed empirical relations of pressure-sinkage and shear stress-slippage on track–soil interaction. Then Wong ¹¹ extended Bekker’s principles and proposed a model of tracked vehicle to predict the normal pressure distribution on track–soil interaction. Joseph and Paramsothy ¹² used multi-body dynamics model to evaluate the influence on track–soil interaction. However, such researches were obtained based on the various constraint situation. In fact, soil deformation generated by side bulldozing plays an important role in determining the steering performance. If the force caused by side bulldozing is always not considered on track–soil interaction, ¹³ the steering resistance torque cannot be calculated accurately. Therefore, in order to calculate steering performance more accurately, it is necessary to develop a steering model that can take the effect of soil deformation into consideration on track–soil interaction.

The contributions of this article are constructing a detailed steering model of articulated tracked vehicle and developing a track–soil sub-model on track–soil interaction. The track–soil sub-model can consider soil deformation generated by side bulldozing on track–soil interaction. Therefore, equipped with the track–soil sub-model, the steering model can take the effect of soil deformation into consideration on track–soil interaction, which will serve as a basis for other similar vehicles. This article is organized as follows: In section “A steering model of articulated tracked vehicle,” a steering model of articulated tracked vehicle is constructing. In section “A track–soil sub-model of articulated tracked vehicle,” a detailed track–soil sub-model of articulated tracked vehicle is developed. The simulation results are discussed in section “Numerical simulation,” and in section “Experiments,” the experimental results are discussed. Finally, conclusions are made in section “Conclusion.”

A steering model of articulated tracked vehicle

In order to improve steering performance, a steering model of articulated tracked vehicle need to be proposed. Unlike a skid-steer vehicle, an articulated tracked vehicle uses an articulated mechanism to achieve a steering path. ¹⁴ Figure 1 gives the virtual prototypes of articulated tracked vehicle and articulated mechanism. In Figure 1, the articulated mechanism consists of a pair of steering cylinders, a pair of pitching cylinders, an articulated frame, and a hinge support. When an articulated tracked vehicle is steering, articulated mechanism uses steering cylinders to steer a path. That means steering resistance torque can be overcome using articulated mechanism, so that the efficiency of steering performance cannot be degraded.

Figure 1.

Virtual prototypes of articulated tracked vehicle and articulated mechanism.

As mentioned above, steering performance of articulated tracked vehicle is different from the performance of skid-steer vehicle. For an articulated tracked vehicle, an articulated mechanism is used to accomplish relative steering between front and rear units. Figure 2 gives the steering motion of articulated tracked vehicle. In Figure 2, a navigational reference frame $N(X,Y,Z)$ is defined.

Figure 2.

The steering motion of articulated tracked vehicle.

For the front unit of articulated tracked vehicle, the advancing velocity is denoted by V, the accelerations are denoted by $a_x$ and $a_y$ in x-axis and y-axis, and the position is denoted by $(X,Y)$ . Therefore, the kinematic equation of front unit can be described as follows

$$V=\sqrt{V_x^2+V_y^2}$$ (1)

$$\{\begin{array}{c}{a}_x={\stackrel{·}{V}}_x+V_y\stackrel{·}{\theta }\hfill \\ \multicolumn{1}{c}{a_y={\stackrel{·}{V}}_y-V_x\stackrel{·}{\theta }}\end{array}$$ (2)

$$\{\begin{array}{c}X=X_0-\underset{0}{\overset{t}{\int }}V\cos \varphi \mathrm{dt}\hfill \\ \multicolumn{1}{c}{Y=Y_0-\underset{0}{\overset{t}{\int }}V\sin \varphi \mathrm{dt}}\end{array}$$ (3)

where $V_x$ and $V_{\mathrm{y}}$ denote the components of advancing velocity V in x-axis and y-axis, respectively. Variable $\theta$ denotes yaw angle of front unit, and $\varphi$ denotes directional angle.

Figure 3 shows the forces generated by soil deformation on track–soil interaction. According to balance principle, the balance equations from the dynamic balance between all forces and moments about the z-axis acting on the front unit of articulated tracked vehicle can be calculated by

$$\{\begin{array}{c}m({\stackrel{·}{V}}_x+V_y\stackrel{·}{\theta })=(F_{x1}+F_{x2})-P_x\hfill \\ \multicolumn{1}{c}{m({\stackrel{·}{V}}_y-V_x\stackrel{·}{\theta })=(F_{y1}+F_{y2})-P_y}\\ \multicolumn{1}{c}{I_z\stackrel{··}{\theta }=\frac{B}{2}(F_{x2}-F_{x1})-(F_{y1}+F_{y2})\frac{L}{2}-P_{y}l+M}\end{array}$$ (4)

where m denotes the mass of front unit, L denotes the length of track–soil contact, and B denotes the width. The variable $F_{x1}$ and $F_{x2}$ denote the longitudinal forces of two tracks in x-axis; $F_{y1}$ and $F_{y2}$ denote the lateral forces of two tracks in y-axis. The variable $P_x$ and $P_{\mathrm{y}}$ denote the forces acting on articulation point for front unit in x-axis and y-axis, respectively. The variable l denotes the distance from articulation point to center of gravity for front unit, and M denotes the steering torque acting on articulation point for front unit.

Figure 3.

The forces generated by soil deformation on track–soil interaction.

And for the rear unit of articulated tracked vehicle, the advancing velocity is denoted by $V\prime$ . The kinematic equation of rear unit can be written as

$$\{\begin{array}{c}\theta \prime =\theta -\zeta \hfill \\ \multicolumn{1}{c}{{V\prime }_x=V_x\cos \zeta +(V_x+l\stackrel{·}{\theta })\sin \zeta }\\ \multicolumn{1}{c}{{V\prime }_y=l\prime \theta \prime -V_x\sin \zeta +(V_y+l\stackrel{·}{\theta })\cos \zeta }\end{array}$$ (5)

where $V{\prime }_x$ and $V{\prime }_y$ denote the components of advancing velocity $V\prime$ in x-axis and y-axis, respectively, and $\theta \prime$ denotes the yaw angle of rear unit. Variable $\zeta$ denotes the articulate angle of articulated tracked vehicle and $l\prime$ denotes the distance from articulation point to center of gravity for rear unit.

According to balance principle, the balance equations from the dynamic balance between all forces and moments about the z′-axis acting on the rear unit of articulated tracked vehicle can be calculated by

$$\{\begin{array}{c}m\prime (\stackrel{·}{V}{\prime }_x+V{\prime }_y\stackrel{·}{\theta }\prime )=(F{\prime }_{x1}+F{\prime }_{x2})+P{\prime }_x\hfill \\ \multicolumn{1}{c}{m\prime (\stackrel{·}{V}{\prime }_y-V{\prime }_x\stackrel{·}{\theta }\prime )=(F{\prime }_{y1}+F{\prime }_{y2})-P{\prime }_y}\\ \multicolumn{1}{c}{I{\prime }_z\stackrel{··}{\theta }\prime =\frac{B\prime }{2}(F{\prime }_{x2}-F{\prime }_{x1})-(F{\prime }_{y1}+F{\prime }_{y2})\frac{L\prime }{2}+P{\prime }_{y}l\prime -M\prime }\end{array}$$ (6)

where $m\prime$ denotes the mass of rear unit, $L\prime$ denotes the length of track–soil contact, and $B\prime$ denotes the width. The variable $F{\prime }_{x1}$ and $F{\prime }_{x2}$ denote the longitudinal forces of two tracks in x-axis; $F{\prime }_{y1}$ and $F{\prime }_{y2}$ denote the lateral forces of two tracks in y-axis. The variable $P{\prime }_x$ and $P{\prime }_y$ denote the forces acting on articulation point for rear unit in x-axis and y-axis, respectively. The variable $M\prime$ denotes the steering torque acting on articulation point for rear unit.

Figure 4 shows the motion relationship of force and steering torque on articulated point. The force and steering torque acting on the articulated point can be written as

$$\{\begin{array}{c}{P}_x=P{\prime }_x\cos \zeta +P{\prime }_y\sin \zeta \hfill \\ \multicolumn{1}{c}{P_y=P{\prime }_x\sin \zeta -P{\prime }_y\cos \zeta }\\ \multicolumn{1}{c}{M=-M\prime }\end{array}$$ (7)

Figure 4.

The force and steering torque on articulated point.

In order to obtain the steering model of articulated tracked vehicle, substituting equations (1)–(3), (5), and (7) into equations (4) and (6), then the steering model can be presented directly by

$$\begin{array}{c}\left[\begin{array}{ccc}m+m\prime \hfill & 0\hfill & -m\prime l\sin \zeta \hfill \\ \multicolumn{1}{c}{0}& m+m\prime \hfill & m(l+l\prime \cos \zeta )\hfill \\ \multicolumn{1}{c}{-m\prime l\prime \sin \zeta }& m\prime (l\prime +l\cos \zeta )\hfill & \begin{array}{c}{I}_z+m\prime l(l+l\prime \cos \zeta )\hfill \\ \multicolumn{1}{c}{+I{\prime }_z+m\ell (l\prime +l\cos \zeta )}\end{array}\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}{\stackrel{·}{V}}_x\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{V}}_y}\\ \multicolumn{1}{c}{\stackrel{··}{\theta }}\end{array}\right]=\left[\begin{array}{c}F(1)\hfill \\ \multicolumn{1}{c}{F(2)}\\ \multicolumn{1}{c}{F(3)}\end{array}\right]\hfill \end{array}$$ (8)

where $F(1)-F(3)$ can be written as

$$\begin{array}{c}F(1)=f(1)-f(3)\cos \zeta -f(4)\sin \zeta -m\prime l\prime \sin \zeta ·\stackrel{··}{\zeta }\hfill \\ F(2)=f(2)-f(3)\sin \zeta +f(4)\cos \zeta +m\prime l\prime \cos \zeta ·\stackrel{··}{\zeta }\hfill \\ F(3)=f(5)-\mathrm{lf}(3)\sin \zeta +\mathrm{lf}(4)\cos \zeta +f(6)\hfill \\ +l\prime f(4)+(m\prime \mathrm{ll}\prime \cos \zeta +I{\prime }_z+m\prime l{\prime }^2)\stackrel{··}{\zeta }\hfill \end{array}$$

where $f(1)-f(6)$ can be written as

$$\begin{array}{c}f(1)=(F_{x1}+F_{x2})-m{V}_y\stackrel{·}{\theta }\hfill \\ f(2)=(F_{y1}+F_{y2})+m{V}_x\stackrel{·}{\theta }\hfill \\ f(3)=m\prime \stackrel{·}{\zeta }[(V_y+l\stackrel{·}{\theta })\cos \zeta -V_x\sin \zeta ]\hfill \\ +(F{\prime }_{y1}+F{\prime }_{y2})-m\prime V_y\stackrel{·}{\theta }\prime \hfill \\ f(4)=m\prime \stackrel{·}{\zeta }[(V_y+l\stackrel{·}{\theta })\sin \zeta +V_x\cos \zeta ]\hfill \\ +(F{\prime }_{y1}+F{\prime }_{y2})+m\prime V{\prime }_x\stackrel{·}{\theta }\prime \hfill \\ f(5)=\frac{B}{2}(F_{x2}-F_{x1})-(F_{y1}+F_{y2})\frac{L}{2}\hfill \\ f(6)=\frac{B\prime }{2}(F{\prime }_{x2}-F{\prime }_{x1})-(F{\prime }_{y1}+F{\prime }_{y2})\frac{L\prime }{2}\hfill \end{array}$$

The steering model of articulated tracked vehicle can be obtained by equation (8). Obviously, the forces $F_{x1}$ $F_{x2}$ $F_{y1}$ $F_{y2}$ and $F{\prime }_{x1}$ $F{\prime }_{x2}$ $F{\prime }_{y1}$ $F{\prime }_{y2}$ generated by soil deformation on track–soil interaction play an important role in determining the steering performance for an articulated tracked vehicle. If the forces $F_{x1}$ $F_{x2}$ $F_{y1}$ $F_{y2}$ and $F{\prime }_{x1}$ $F{\prime }_{x2}$ $F{\prime }_{y1}$ $F{\prime }_{y2}$ can be calculated accurately, the steering model can predict steering performance more accurately. Therefore, in the next section, a track–soil sub-model need to be developed for calculating the forces $F_{x1}$ $F_{x2}$ $F_{y1}$ $F_{y2}$ and $F{\prime }_{x1}$ $F{\prime }_{x2}$ $F{\prime }_{y1}$ $F{\prime }_{y2}$ on soft soil.

A track–soil sub-model of articulated tracked vehicle

As mentioned in section “A steering model of articulated tracked vehicle,” if the forces $F_{x1}$ $F_{x2}$ $F_{y1}$ $F_{y2}$ and $F{\prime }_{x1}$ $F{\prime }_{x2}$ $F{\prime }_{y1}$ $F{\prime }_{y2}$ generated by soil deformation on track–soil interaction can be calculated accurately, a steering model of articulated tracked vehicle can be improved. Therefore, in this section, the detailed track–soil sub-model is developed. And in the track–soil sub-model, soil deformation need to be considered. Therefore, using this track–soil sub-model, the steering model can predict steering performance more accurately.

Kinematic analysis of a track

When an articulated tracked vehicle is steering, every track has its ground instantaneous center, which is denoted by $O_{\mathrm{si}}$ or $O{\prime }_{\mathrm{si}}$ ( $i=1,2$ ). On the other hand, each track has its geometric center, which is denoted by $O_i$ or $O{\prime }_i$ ( $i=1,2$ ). In order to develop track–soil sub-model of articulated tracked vehicle, the following assumption has been made: ¹⁵ the center of mass is coincided with the geometric center.

Figure 5 presents the instantaneous center and geometric center of a track. In Figure 5, the longitudinal offset $D_i$ is produced by lateral external force, and $D_i$ can be calculated by ¹⁶

$$D_i=\frac{V_y}{\stackrel{·}{\theta }}$$ (9)

Figure 5.

The instantaneous center and geometric center of a track.

The lateral offset $A_i$ is produced by slip and skid, and $A_i$ can be calculated by ¹⁷

$$A_i=\frac{V_x}{\stackrel{·}{\theta }}$$ (10)

The slip angle $\beta$ , slip angular velocity $\stackrel{·}{\beta }$ , and direction angle $\varphi$ can be calculated by ¹⁸

$$\{\begin{array}{c}\beta ={\tan }^{-1}\left(\frac{V_y}{V_x}\right)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\beta }=\frac{(V_x{\stackrel{·}{V}}_y-V_y{\stackrel{·}{V}}_x)}{V^2}}\end{array}$$ (11)

Hence

$$\varphi =\theta -\beta$$ (12)

The force caused by soil shear on track–soil interaction

In order to improve steering performance, soil deformation should be taken into account on track–soil interaction. When an articulated tracked vehicle is steering on soft soil, the effect of soil shear will lead soil deformation on track–soil interaction. In order to present the soil shear deformation, according to Janosi-Hanamoto’s soil shear relationship, the shear stress can be obtained by ¹⁹

$$\tau =p(x,y)\mu [1-\exp (-j/K)]$$ (13)

where $\tau$ denotes soil shear stress, $\mu$ denotes coefficient of friction on track–soil interaction, K denotes soil shear modulus, and j denotes shear deformation that can be described by

$$j=\sqrt{j_x^2+j_y^2}$$ (14)

where $j_x$ and $j_y$ denote the components of shear deformation in x-axis and y-axis, respectively, and $j_x$ and $j_y$ can be calculated by

$$\{\begin{array}{c}{j}_x=(R\prime \pm \frac{B}{2}+x)[\cos \left(\frac{L/2-D_i-y}{r}\right)-1]-y\sin \left(\frac{L/2-D_i-y}{r}\right)\hfill \\ \multicolumn{1}{c}{j_y=(R\prime \pm \frac{B}{2}+x)\sin \left(\frac{L/2-D_i-y}{r}\right)-(\frac{L}{2}-D_i)+y\cos \left(\frac{L/2-D_i-y}{r}\right)}\end{array}$$ (15)

where $R\prime$ denotes the distance between turning center and center of mass; r denotes radius of driving wheel.

Figure 6 illustrates soil shear deformation of a track on track–soil interaction. The force $d{F}_i$ of each infinitesimal element acts on every micro-unit dxdy of a track, and its direction is opposite to absolute velocity direction. Then force $d{F}_i$ can be described by

$$d{F}_i=\tau \mathrm{dA}=\mu p[1-\exp (-j/K)]\mathrm{dxdy}$$ (16)

where p denote ground pressure, and it can be calculated by ²⁰

$$p=\frac{F_{\mathrm{zi}}}{\mathrm{bL}}$$ (17)

where $F_{\mathrm{zi}}$ denotes the loading acting on the ith track, and b denotes the width of a track. The components of $d{F}_i$ in x-axis and y-axis can be calculated by

$$\{\begin{array}{c}d{F}_{\mathrm{xi}}=d{F}_i\cos \varphi =d{F}_i\frac{y}{\sqrt{x^2+y^2}}\hfill \\ \multicolumn{1}{c}{d{F}_{\mathrm{yi}}=-d{F}_i\sin \varphi =-d{F}_i\frac{x}{\sqrt{x^2+y^2}}}\end{array}$$ (18)

where $\phi$ denotes soil internal friction angle. Then the longitudinal force $F_{\mathrm{xi}}^{(1)}$ and lateral force $F_{\mathrm{yi}}^{(1)}$ generated by soil shear deformation on track–soil interaction can be calculated by

$$\{\begin{array}{c}{F_{\mathrm{xi}}}^{(1)}=\int \int d{F}_{\mathrm{xi}}=\underset{-(0.5L+D_i)}{\overset{0.5L-D_i}{\int }}\underset{A_i-0.5b}{\overset{A_i+0.5b}{\int }}\frac{y}{\sqrt{x^2+y^2}}\frac{\mu F_{\mathrm{zi}}}{\mathrm{bL}}[1-\exp (-j/K)]\mathrm{dxdy}\hfill \\ \multicolumn{1}{c}{{F_{\mathrm{yi}}}^{(1)}=\int \int d{F}_{\mathrm{yi}}=\underset{-(0.5L+D_i)}{\overset{0.5L-D_i}{\int }}\underset{A_i-0.5b}{\overset{A_i+0.5b}{\int }}\frac{-x}{\sqrt{x^2+y^2}}\frac{\mu F_{\mathrm{zi}}}{\mathrm{bL}}[1-\exp (-j/K)]\mathrm{dxdy}}\end{array}$$ (19)

Figure 6.

Soil shear deformation of a track on track–soil interaction.

Then, the steering resistance torque ${T_{\mathrm{oi}}}^{(1)}$ generated by soil shear deformation on track–soil interaction can be calculated by

$$\begin{array}{c}{T_{\mathrm{oi}}}^{(1)}=\int \int (y-A_i)d{F}_{\mathrm{xi}}-\int \int (x+D_i)d{F}_{\mathrm{yi}}\hfill \\ =\underset{-(0.5L+D_i)}{\overset{0.5L-D_i}{\int }}\underset{A_i-0.5b}{\overset{A_i+0.5b}{\int }}\frac{y(y-A_i)+x(x+D_i)}{\sqrt{x^2+y^2}}\frac{\mu F_{\mathrm{zi}}}{\mathrm{bL}}[1-\exp (-j/K)]\mathrm{dxdy}\hfill \end{array}$$ (20)

The force caused by side bulldozing on track–soil interaction

Figure 7 illustrates soil pressure deformation of a track on track–soil interaction, where R denotes bulldozing resistance of every micro-unit length, ${\phi }_{\omega }$ denotes side friction angle, C denotes cohesive force of every micro-unit area soil, and $\alpha$ denotes angle of failure surface. According to Bekker’s pressure subsidence relationship, the subsidence z of a track can be expressed by ²¹

$$z={\left[\frac{p}{(\frac{k_c}{b}+k_{\varphi })}\right]}^{\frac{1}{n}}={\left[\frac{F_{\mathrm{zi}}}{\mathrm{bL}(\frac{k_c}{b}+k_{\varphi })}\right]}^{\frac{1}{n}}$$ (21)

where $k_c$ denotes soil cohesion modulus, $k_{\phi }$ denotes soil friction deformation modulus, and n denotes soil deformation index.

Figure 7.

The soil pressure deformation of a track on track–soil interaction.

According to balance principle, the forces for every micro-unit length on track–soil interaction can be described by

$$\sum Y=R\cos {\phi }_{\omega }-\mathrm{zC}\cot \alpha -N\sin (\alpha -\phi )=0$$ (22)

$$\begin{array}{c}\multicolumn{1}{c}{\sum Z=0-R\sin {\phi }_{\omega }-{\gamma }_s{z}^2\frac{\cot \alpha }{2}}\\ -\mathrm{zC}+N\cos (\alpha +\phi )=0\hfill \end{array}$$ (23)

where ${\gamma }_s$ denotes soil bulk density.

Using equations (22) and (23), the bulldozing resistance R can be deduced by

$$R=\frac{{\gamma }_s{z}^2\frac{\cot \alpha }{2}+\mathrm{zC}[1+\cot \alpha ·\cot \alpha +\varphi ]}{\cos {\varphi }_{\omega }·\cot (\alpha +\varphi )-\sin {\varphi }_{\omega }}$$ (24)

In equation (24), the minimum value of R is corresponded to the maximum value of $\alpha$ . In order to obtain the maximum value of steering resistance torque which is caused by side bulldozing, equation (24) needs to be rewritten as

$$R=\frac{k_1{\cot }^2\alpha +k_2\cot \alpha +u_1}{k_3\cot \alpha +u_2}$$ (25)

where $k_1=1/2(r_s{z}^2)+\mathrm{zC}\cot \varphi ,k_2=1/2(r_s{z}^2)\cot \varphi ,k_3=\cos {\varphi }_{\omega }\cot \varphi -\sin {\varphi }_{\omega }$ , $u_1=\mathrm{zC}\cot \varphi$ , $u_2=-\cos {\varphi }_{\omega }-\sin {\varphi }_{\omega }\cot \varphi$ .

According to equation (25), in order to obtain the minimum value of R, the first time derivative of R is set equal to 0 that is calculated by

$$\stackrel{·}{R}=-\frac{k_1{k}_3{\cot }^2\alpha +2k_1{u}_2\cot \alpha +k_2{\mu }_2-k_3{u}_1}{{(k_3\cot \alpha +u_2)}^2{\sin }^2\alpha }=0$$ (26)

Using equation (26), the $\cot \alpha$ can be obtained by

$$\cot \alpha =\frac{-k_1{u}_2+\sqrt{k_1^2{u}_2^2-k_1{k}_2{k}_3{u}_2+k_1{k}_3^2{u}_1}}{k_1{k}_3}$$ (27)

Substituting equation (27) into equation (24), the $R_{min}$ can be obtained. Therefore, the steering resistance torque caused by side bulldozing on track–soil interaction can be calculated by

$$\begin{array}{c}\multicolumn{1}{c}{T_{\mathrm{oi}}^{(2)}=\underset{-0.5L}{\overset{0}{\int }}{R}_{min}·\cos {\varphi }_{\omega }·(-x)\mathrm{dx}}\\ +\underset{0}{\overset{0.5L}{\int }}{R}_{min}·\cos {\varphi }_{\omega }·\mathrm{xdx}-F_{\mathrm{yi}}^{(2)}{D}_i\hfill \end{array}$$ (28)

where $F_{\mathrm{yi}}^{(2)}$ denotes the force caused by side bulldozing on track–soil interaction, and $F_{\mathrm{yi}}^{(2)}$ can be calculated by

$$\begin{array}{c}\multicolumn{1}{c}{F_{\mathrm{yi}}^{(2)}=\underset{0}{\overset{(L/2+D_i)}{\int }}{R}_{min}·\cos {\varphi }_{\omega }\mathrm{dx}}\\ \multicolumn{1}{c}{+\underset{-(L/2+D_i)}{\overset{0}{\int }}{R}_{min}·\cos {\varphi }_{\omega }d}\\ \multicolumn{1}{c}{=(2D_i+L)R_{min}·\cos {\varphi }_{\omega }}\end{array}$$ (29)

and

$$T_{\mathrm{oi}}^{(2)}=R_{min}·\cos {\varphi }_{\omega }·(\frac{L^2}{4}-D_i^2)$$ (30)

Therefore, the force on track–soil interaction can be obtained by

$$\{\begin{array}{c}{F}_{\mathrm{xi}}={F_{\mathrm{xi}}}^{(1)}\hfill \\ \multicolumn{1}{c}{F_{\mathrm{yi}}={F_{\mathrm{yi}}}^{(1)}+{F_{\mathrm{yi}}}^{(2)}}\\ \multicolumn{1}{c}{T_{\mathrm{oi}}={T_{\mathrm{oi}}}^{(1)}+{T_{\mathrm{oi}}}^{(2)}}\end{array}$$ (31)

Substituting equations (9)–(30) into equation (31), the track–soil sub-model can be obtained. Using this detailed track–soil sub-model, the essential physics of track–soil interaction can be captured. Therefore by using the track–soil sub-model, steering performance can be calculated more accurately.

Substituting equation (31) into equation (8), a steering model of articulated tracked vehicle can be obtained. Equipped with the track–soil sub-model, the steering model can capture the essential physics of soil deformation on track–soil interaction. Therefore, based on this track–soil sub-model, the steering model of articulated tracked vehicle can be improved, and steering performance can be calculated more accurately.

Numerical simulation

In order to demonstrate the accuracy of steering model, numerical analyses have been performed using MATLAB and Recurdyn. The comparative studies are carried out between theoretical results and Recurdyn model. The parameters of articulated tracked vehicle are listed in Table 1.

Table 1.

Parameters of the articulated tracked vehicle.

Parameters	Variable	Value
Ground contact length	$L=L\prime$	4.3 m
Vehicle width	$B=B\prime$	2 m
Mass	$m=m\prime$	30,000 kg
Distances from articulation point to center of gravity	$l=l\prime$	2.78 m

The comparison in heavy clay soil

The heavy clay soil parameters are listed in Table 2. Table 3 illustrates the comparison with the steering resistance torque of every track when the articulated tracked vehicle is steering at $v=1\mathrm{m}/\mathrm{s}$ , $\zeta =5.5\circ$ . In Table 3, the simplified model is the model which does not consider soil deformation generated by side bulldozing on track–soil interaction in equations (29) and (30). Therefore, compared with the value of theoretical model proposed in this article, the value of simplified model has some error, and the value of theoretical model is close to the value of Recurdyn model.

Table 2.

Soil parameters of heavy clay.

Parameters	Variable	Value
Cohesive force of per unit area soil	C	4.83 kPa
Soil internal friction angle	$\phi$	13°
Side friction angle	${\phi }_{\omega }$	13°
Soil deformation index	n	0.5
Soil bulk density	${\gamma }_s$	16 kN m−3
Soil cohesion modulus	$k_c$	13.19 kN m−1.5
Soil friction deformation modulus	$k_{\phi }$	692.15 kN m−2.5

Table 3.

Steering resistance torques of every track.

	Theoretical model ( $\mathrm{kN}\mathrm{m}$ )	Simplified model ( $\mathrm{kN}\mathrm{m}$ )	Recurdyn model ( $\mathrm{kN}\mathrm{m}$ )
Track 1	91.069	90.541	91.046
Track 2	76.209	75.682	76.189
Track 3	47.441	47.022	47.450
Track 4	52.621	52.203	52.619

Figure 8 shows the driving torque comparison theoretical model with Recurdyn model. In Figure 8, the articulated tracked vehicle is steering at $v=3\mathrm{m}/\mathrm{s}$ ; the values of articulated angles are $\zeta =5.5\circ$ , $11\circ$ , $16.5\circ$ , $22\circ$ , and $27.5\circ$ , respectively.

Figure 8.

The driving torque comparison theoretical model with Recurdyn model.

As shown in Figure 8, the value of driving torque at $\zeta =5.5\circ$ is 3684 N m for the theoretical model and the value is 3549 N m for the Recurdyn model. That means the value of theoretical model proposed in this article is similar to the value of Recurdyn model. On the other hand, it is clearly shown the relationship between the driving torque and articulated angle in Figure 8. The bigger the value of articulate angle is, the bigger the value of driving torque is.

Table 3 and Figure 8 demonstrate that the value of steering model is close to the value of Recurdyn model that means the proposed steering model can predict steering performance accurately. Therefore, the proposed steering model can be used to study steering problem of articulated tracked vehicle.

The comparison in different soft soil conditions

Using an articulated mechanism, an articulated tracked vehicle has excellent mobility in off-road application. Figure 9 presents the steering torque acting on articulation point of theoretical model proposed in different soft soil conditions.

Figure 9.

The steering torques of theoretical model in different soft soil conditions.

As shown in Figure 9, the average values of steering torque are 6794 N m on dry sand, 6032 N m on hard surface, and 10,873 N m on clayey soil, respectively. That means an articulated tracked vehicle needs more torque to steer on clayey soil because the adhesion of soil property leads soil deformation on track–soil interaction. Figure 9 demonstrates that the proposed steering model can capture the soil deformation and study steering problem of articulated tracked vehicle on soft soil.

Experiments

In order to verify the effectiveness of steering model proposed, the experimental platform is built for examining the steering performance of articulated tracked vehicle on soft soil.

The articulated tracked vehicle is experimented for research purposes. The system configuration is shown in Figure 10 where a denotes the computer, b denotes the wireless dynamic strain acquisition device, c denotes the resistance strain gauge, and d denotes the articulated tracked vehicle. The articulated tracked vehicle moves at $v=2\mathrm{m}/\mathrm{s}$ on soft soil.

Figure 10.

The system configuration of the articulated tracked vehicle for experiments.

Based on this experimental platform, the drive power is tested. At first, sets of data are tested on soft soil. Then the comparative studies between experiment and theoretical results are carried out. Figure 11 presents the results of articulated tracked vehicle on soft soil.

Figure 11.

The drive power of experiment and theoretical results.

As shown in Figure 11, the average values of drive power are 466.236 kW for experimental result, and 462.492 kW for theoretical result, respectively. That means the average value of theoretical model is similar to the average value of experimental result. Some difference probably lies in the difference of soil conditions between experimental results and theoretical model.

The comparisons indicate that the experimental results are reasonably in agreement with the theoretical results. The simulations and experimental results suggest that the proposed steering model of articulated tracked vehicle can consider soil deformation on track–soil interaction and calculate the steering performance more accurately. Therefore, the proposed model has the capable of study the steering problem for articulated tracked vehicle.

Conclusion

The steering model of articulated tracked vehicle considering soil deformation on track–soil interaction has been developed in this article. In particular, the track–soil sub-model is developed where the soil deformation generated by side bulldozing can be taken into consideration on track–soil interaction. Simulations and experimental results are in strong agreement with the theoretical results in this article. Therefore, equipped with the track–soil sub-model, this proposed steering model can be used to study steering problem of articulated tracked vehicle and accurately predict steering performance. The steering model of articulated tracked vehicle proposed in this article can provide a basis for other similar vehicles.

This steering model can capture the physics of track–soil interaction and calculate the steering performance accurately on a given soil condition. However, an articulated tracked vehicle will steer in unknown soft soil condition. In order to improve the steering model more accurately, the future work will focus on track–soil sub-model, which can capture the essential physics in the unknown soft soil conditions for real time.