## 2 Principle and Modeling
### Equivalent Inertia Modeling of Tracked Vehicles  
Formula (6) shows that in order to obtain \(\omega_{em}(s)\), the inertia of the whole vehicle must be equivalent to the driving wheel. Equivalent analysis of the inertia of the tracked vehicle shows that  
\[\frac{1}{2}(m_{v}v_{v}^{2}+m_{t}v_{t}^{2}+n_{l}J_{l}\omega_{l}^{2}+J_{d}\omega_{ d}^{2}+J_{i}\omega_{i}^{2}+n_{b}J_{b}\omega_{b}^{2})=\frac{1}{2}J_{e}\omega_{d}^{2} \tag{7}\]  
where, \(m_{v}\) means the vehicle mass (excluding track), \(v_{v}\) means the vehicle speed, \(m_{t}\) means the track mass, \(v_{t}\) means the track speed, \(n_{l}\) means the number of load wheels, \(J_{l}\) means the load wheel rotation inertia around the axle center, \(\omega_{l}\) means the load wheel angular speed, \(J_{d}\) means the drive wheel rotation inertia around the axle center, \(\omega_{d}\) means the drive wheel angular speed, \(J_{i}\) means the induced wheel rotation around the axle center Inertia, \(\omega_{i}\) means the angular velocity of the inducer, \(n_{b}\) means the number of supporting roller, \(J_{b}\) means the inertia of the supporting roller rotating around the axle center, and \(\omega_{b}\) means the angular velocity of the supporting roller. Equation (7) shows that the translational inertia and rotational inertia of all parts of the vehicle can be expressed by the inertia revolving around the driving wheel. In order to calculate the equivalent inertia accurately, it should be divided into three parts, as shown in Formula (8). \[J_{e}=J_{v}+J_{t}+J_{o} \tag{8}\]  
Here, \(J_{v}\) means the vehicle body equivalent inertia (excluding the track and wheel rotational inertia); \(J_{t}\) means the equivalent inertia of the track; and \(J_{o}\) means other equivalent inertia, including that of the driving wheel, induced wheel, and supporting roller. These three kinds of inertia are analyzed separately below. Figure 5 shows the relationship between body speed and driving wheel speed. Figure 4: Chart of car body inclination angle and resistance moment analysis. According to Figure 5, we can get  
\[v_{v}=\omega_{d}R(1-\delta) \tag{9}\]  
where \(\delta\) means the slip rate between the track and ground. So for the body, there is  
\[\frac{1}{2}m_{v}v_{v}^{2}=\frac{1}{2}J_{v}\omega_{d}^{2} \tag{10}\]  
By combining the two forms, we can get  
\[J_{v}=m_{v}(R-R\delta)^{2} \tag{11}\]  
When \(\delta=0\), the original format becomes  
\[J_{v}=m_{v}R \tag{12}\]  
The biggest difference between a tracked vehicle and traditional vehicle is that a tracked vehicle is equipped with high-quality tracks [32]. Because the track is irregular and the absolute speed of each part when driving is different, it should be divided into four parts, including the upper, lower, front, and rear parts, when calculating. Assuming that the tracked vehicle is moving at a uniform speed on the ground with no relative slip, that is, the slip rate \(\delta=0\), the inertia of the grounding part of the track can be ignored. However, in a normal driving process, due to bad ground conditions (such as marsh, snow, etc. ), the slip rate is usually not stable, so it is necessary  
By understanding three velocities as three vectors, we can get  
\[\vec{v_{tnp}}=\vec{v_{tnp}}+\vec{v_{tnp}} \tag{13}\]  
According to the Vector Superposition Rule,  
\[v_{tnp}{}^{2}=v_{tnp}{}^{2}+v_{tnp}{}^{2}-2v_{tnp}v_{tnp}cos\pi \tag{14}\]  
where \(\pi\) is the vector angle between \(\vec{v_{tnp}}\) and \(\vec{v_{tnp}}\). For \(v_{tnp}\), there is  
\[v_{tnp}=v_{v}=\omega_{d}R(1-\delta) \tag{15}\]  
Furthermore, for \(v_{tnp}\), there is  
\[J_{e}=J_{v}+J_{t}+J_{o} \tag{8}\]0  
In addition, according to the law of the conservation of energy,  
\[J_{e}=J_{v}+J_{t}+J_{o} \tag{8}\]1  
By simultaneously considering formulae (13) to (17), we can get  
\[J_{e}=J_{v}+J_{t}+J_{o} \tag{8}\]2  
For the front part of the track, the velocity relationship is shown in \[J_{e}=J_{v}+J_{t}+J_{o} \tag{8}\]3 In the same way, we can derive the equivalent inertia.