# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 2 Materials and Methods
### Simulation Model  
The considerations in Section 2.1 apply to the kinematic model. Any aspects due to slippage between rollers and the ground or uneven drives are ignored here. However, these effects will affect the actual platform. For this reason, the simulation studies of driving dynamics have been carried out, which will demonstrate the validity of using the developed kinematic model to control the platform under development. The numerical simulation will verify that the proposed kinematic model captures the behaviour of the dynamic model with sufficient accuracy.  
Numerical studies were carried out in the HEXAGON ADAMS 2022 system for the dynamic analysis of multibody systems. It developed a solid model of an omni-track vehicle consisting of four rigid bodies numbered 1,... 4, with base 0 and body k. Links 1,... 4 and modelling tracks \(g_{1},\ldots g_{4}\) are connected to body \(k\) by translational joint pairs. Translational motion kinematic forcing \(q_{1},\ldots q_{4}\) is placed between the individual beams and the body. The change in the value of kinematic excitations over time was described by a linear function. The free rollers are designated as \(rl_{1},\ldots r_{8}\). Each beam is connected to two rollers by revolute joint pairs. The rollers \(rl_{1},\ldots r_{8}\) are fixed with respect to the main axis of the vehicle at an angle of \(\alpha\). Rollers connected to beams 1 and 3 were oriented at angles \(\alpha_{1}\) and \(\alpha_{3}\), while rollers connected to beams 2 and 4 were oriented at angles \(\alpha_{2}\) and \(\alpha_{4}\). Contact forces were defined between the substrate and the rollers using contact forces. The Coulomb friction model was used. The parameters of these forces, selected on the basis of the literature [33; 34; 35], are shown in Table 1. The friction coefficient between the rollers and the substrate corresponds to the friction between rubber and concrete. The model has a mobility of 18, of which 4 displacements \(q_{1},\ldots q_{4}\) are controlled. The view of the model with markings is shown in Figure 12. Figure 13 shows the geometric dimensions of the model. The total weight of the model was 100 kg. The vehicle mass was concentrated primarily in the section representing the body \(k\). Its mass was set at 86 kg.  
\begin{table}
\begin{tabular}{c c c c} \hline Stiffness & 20 N/mm & Static Coefficient & 1 \\ \hline Force Exponent & 2.2 & Dynamic Coefficient & 0.85 \\ \hline Damping & 1 Ns/mm & Stiction Transition Velocity & 10 mm/s \\ \hline Penetration Depth & 1 mm & Friction Transition Velocity & 2000 mm/s \\ \hline \end{tabular}
\end{table}
Table 1: Dynamic simulation parameters.  
Figure 11: Graphical representation of the driving parameters of the omni-track vehicle.  
A dynamic simulation was performed. The simulation step was 5 ms. During the first second of the simulation, the translation pairs did not move. Only the gravitational force acted on the model, balanced by the force of the substrate on the rollers. This gave the model time to stabilize. Then, one second of motion was simulated. The GSTIFF solver was used for the simulation. The error parameter was set to 0.001, formulation I3.  
A visualization of an example simulation is shown in Figure 14, while Figure 15 shows the trajectory obtained during this simulation.  
Figure 12: View of the solid model of the omni-tracked vehicle with symmetrical tracks, completely overlapping. Red arrows for graphical representation of contact forces.  
Figure 13: Dimensions of the solid model of the omni-tracked vehicle with symmetrical tracks, completely overlapping.