## 2 Materials and Methods
### Dynamics Analysis of Slope Steering  
The steering of tracked vehicles on slopes can be divided into four typical conditions, as shown in Figure 3a. The steering process of the tracked vehicle is influenced by the slope of the ground, the driving force, the drag force, and the centrifugal force. A schematic diagram of the spatial steering of tracked vehicles is shown in Figure 3b, and a sketch of the steering dynamics analysis is presented in Figure 3c.  
Figure 3: (**a**) Typical steering conditions of tracked vehicles on a slope; (**b**) gravity and component directions of tracked vehicles; (**c**) ground steering force diagram of tracked vehicles.  
In Figure 3b, \(O-XYZ\) is the attitude coordinate system based on the tracked vehicle body; let the angle of the tracked vehicle \(OY\) axis relative to the horizontal plane be \(\gamma\), the angle of the ox axis relative to the horizontal plane be \(\alpha\); \(G\) is the weight of the whole vehicle, and the component forces of \(G\) on the \(OX\), \(OY\) axis are Gcos\(\alpha\) and Gcos\(\gamma\). In Figure 3c, \(F_{1}\) and \(F_{2}\) are the driving forces of the tracks on both sides; \(F_{R1}\) and \(F_{R2}\) are the rolling resistance of the tracks; \(M_{\mu}\) is the steering resistance moment; \(R_{e}\) is the distance from the steering center to the vehicle centerline (\(X\)-axis in the figure). \(R^{\prime}\) is the distance from the center of the vehicle plane (point C) to the steering center (O); \(S_{O}\) is the longitudinal offset of the steering center; \(\beta\) is the angle between the direction of the vehicle centrifugal force and the \(Y\)-axis; \(h\) is the height of the mass center. The expressions for the driving force and rolling resistance on both sides are:  
\[F_{1}=\begin{cases}0.5\varphi G\cdot\cos\gamma\frac{T_{1}\mu}{r}>0.5\varphi G \cdot\cos\gamma\\ \frac{T_{1}\mu\gamma}{r}\frac{T_{1}\mu\gamma}{r}\leqslant 0.5\varphi G\cdot\cos \gamma\end{cases} \tag{8}\]  
\[F_{2}=\begin{cases}0.5\varphi G\cdot\cos\gamma\frac{T_{2}\mu\gamma}{r}>0.5 \varphi G\cdot\cos\gamma\\ \frac{T_{2}\mu\gamma}{r}\frac{T_{2}\mu\gamma}{r}\leqslant 0.5\varphi G\cdot\cos \gamma\end{cases} \tag{9}\]  
\[F_{R1}=F_{R2}=\frac{fG\cos\alpha}{2} \tag{10}\]  
In Equations (9)-(11), \(T_{1}\) and \(T_{2}\) are motor torque on both sides, respectively; \(\varphi\) is ground adhesion coefficient; \(i\) is the side transmission ratio; \(r\) is the radius of the drive wheel; \(\eta\) is the transmission efficiency from motor to track, where \(\eta=0.941(0.97-0.03v)\)[16]; \(f\) is the rolling resistance coefficient.  
The longitudinal and transverse components of the centrifugal force are:  
\[\begin{cases}F_{cx}=mR^{\prime}\omega^{2}\sin\beta\\ F_{cy}=mR^{\prime}\omega^{2}\cos\beta\end{cases} \tag{11}\]  
The longitudinal offset of the steering center is:  
\[S_{o}=\frac{LF_{cy}}{2\psi G\cdot\cos\alpha} \tag{12}\]  
In Equation (13), \(L\) is the grounding length of the tracks; \(\psi\) is the lateral attachment coefficient; \(G\) is the fully loaded weight of the vehicle.  
According to the literature [17], the steering resistance moment can be expressed as:  
\[M_{\mu}=\frac{1}{4}\mu G\cdot\cos\alpha\cdot L\bigg{[}1-\bigg{(}\frac{2S_{o} }{L}\bigg{)}^{2}\bigg{]} \tag{13}\]  
The steering resistance coefficient can be calculated from the Nifty empirical formula [15]:  
\[\mu=\frac{\mu_{\max}}{0.925+0.15\rho}(\rho\geqslant 0.5) \tag{14}\]  
In Equation (15), \(\rho\) is the relative steering radius, \(\rho=\frac{R}{B}\).  
From the dynamic equilibrium equation of the tracked vehicle, the driving force of the both sides tracks can be expressed as:  
\[\begin{cases}F_{2}+F_{1}-(F_{R1}+F_{R2}+F_{w}\pm G\cdot\sin\alpha)-F_{cx}=m \dot{v}_{c}\\ (F_{2}-F_{1})\frac{B}{2}+(F_{R1}-F_{R2})\frac{B}{2}-M_{\mu}=J\omega\end{cases} \tag{15}\]