# Comparison of Multi-Tracked Running Gears in Terms of Obstacle Negotiation Capabilities
## 3 Simulation Model  
### Model Construction  
The model (Figure 4) of the track systems was made in the multi-body dynamics simulation program--MSC.ADAMS. The structures of the two-track module were reduced to a one-track system, containing a drive wheel (2), idler (3), and four road wheels (4), with the body mass concentrated in the geometric center of the track system (5). By changing the length and location of the constraints of the connecting element (7) of the modules (1), various (A, B, C, D, E) multi-tracked system solutions were achieved.  
A rubber track belt (6) was modeled using 38 segments connected to each other by force constraints (Figure 5). In this way, the belt structure's susceptibility was taken into account. The forces \(F_{x}\), \(F_{y}\) applied to the segment along the X and Y axes and moment \(M_{z}\) occurring in the Z axis are described by the equations:  
\[M_{z}(t)=M_{zk}(t)-M_{zc}(t)=k_{z}\cdot\zeta_{Z1}(t)-c_{z}\cdot\zeta_{Z1}(t) \tag{2}\]  
\[F_{x}(t)=k_{x}\cdot(x_{2}(t)-x_{1}(t))-c_{x}\cdot(x_{2}(t)-x_{1}(t)) \tag{3}\]  
\[F_{y}(t)=k_{y}\cdot(y_{2}(t)-y_{1}(t))-c_{y}\cdot(y_{2}(t)-y_{1}(t)), \tag{4}\]  
where \(M_{zk}(t),M_{zc}(t)\) are the components of the bending moment caused by the angular stiffness and angular damping of the track segments connections, respectively; \(k_{x},k_{y},k_{z}\), \(c_{x},c_{y},c_{z}\) are the stiffness and damping coefficients in three directions, determined based on the identification tests available in the previous publications [50]. The  
Figure 4: Multi-tracked running gear simulation model visualization.  
values of these parameters were identified as \(\mathrm{k_{x}},\mathrm{k_{y}}=100000\frac{\mathrm{N}}{\mathrm{mm}}\), \(c_{x},c_{y}=1000\frac{\mathrm{N}\cdot\mathrm{g}}{\mathrm{mm}}\), \(k_{z}=150\frac{\mathrm{N}\cdot\mathrm{mm}}{\mathrm{c_{z}}}=8\frac{\mathrm{N} \cdot\mathrm{mm}}{\mathrm{s}\cdot\mathrm{s}}\) and \(c_{z}=8\frac{\mathrm{N}\cdot\mathrm{mm}}{\mathrm{s}\cdot\mathrm{s}}\).  
The key part of the model was the description of the track-ground contact, the diagram of which is shown in Figure 6. This type of model has already been used to study obstacle overcoming by UGVs [51]. The track belt was discretized, so that the ground normal reaction forces \(N_{ym}\) and horizontal reaction forces \(F_{xn}\) only affected the loaded track segments in contact with the ground. In this way, the dynamically changing and irregular contact surface was taken into account. The ground normal reactions \(N_{yn}\) are provided by the CONTACT function available in the program:  
\[N_{yn}=\left\{\begin{matrix}0\;\;if\;\;z<0\\ k\cdot(y_{1}-y)^{e}-c\dot{y}\cdot STEP(y,\;y_{1}-d,1,y_{1},0)\;\;if\;\;z\geq 0 \end{matrix}\right. \tag{5}\]  
where \(y_{1}\) is the distance of the segment center from its horizontal edge, \(y\) is the distance of the segment center and the ground, \(k\), \(c\), and \(e\) are the contact stiffness and damping parameters and the contact force exponent, and \(d\) is the penetration depth.  
Figure 5: Constraint model scheme for track belt segments.  
In the case of ground contact, the segment was also loaded with horizontal shear reaction force \(F_{xn}\), described by the Wong equation for a compact loam [49], calculated as a function of the ground deformation j:  
\[F_{xn}=A_{n}\cdot\tau_{max}\cdot K_{r}\left(1+\left(\frac{1}{K_{r}\left(1-\frac{ 1}{\epsilon}\right)}-1\right)\cdot e^{(1-\frac{j}{K_{\omega}})}\right)\cdot(1- e^{(-\frac{-j}{K_{\omega}})}), \tag{6}\]  
where \(A_{n}\) is the segment surface area, \(K_{r}\)\(K_{\omega}\) represent the soil parameters of the ratio of the residual stress and shear displacement where the maximum shear stress occurs, respectively. Motion resistance was taken into account by means of the appropriate torque on the drive wheel \(Mr\) calculated as  
\[M_{r}=N\cdot f\cdot\tau, \tag{7}\]  
where \(N\) is the platform weight, \(f\) is the motion resistance coefficient, and \(r\) is the drive wheel radius.