# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 2 Materials and Methods
### Kinematic Model
\tag{6}\]  
Based on the determined values of the velocity vectors \(\overset{\rightarrow}{b}\) and \(\overset{\rightarrow}{f}\), it is possible to determine the velocity \(\overset{\rightarrow}{v_{\text{L}}}\) of the point L of the module in question based on the following equation:  
\[\overset{\rightarrow}{v_{\text{L}}}=\overset{\rightarrow}{\dot{b}}+\dot{q }_{2}=\overset{\rightarrow}{f}+\overset{\rightarrow}{\dot{q}_{1}} \tag{7}\]\[\stackrel{{\rightarrow}}{{v_{\rm L}}}= \begin{bmatrix}v_{\rm L}^{x}\\ v_{\rm L}^{y}\end{bmatrix} = \begin{bmatrix}\dot{b}\cos\,\delta_{2}\\ \dot{b}\sin\,\delta_{2}+\dot{q}_{2}\end{bmatrix} = \begin{bmatrix}\dot{f}\,\cos \,\delta_{1}\\ \dot{f}\,\sin\,\delta_{1}+\dot{q}_{1}\end{bmatrix}  \tag{8}\]  
The orientation of the vector \(\stackrel{{\rightarrow}}{{v_{\rm L}}}\) is denoted as \(\gamma\). The angle can be calculated using the atan2 function based on the following formula:  
\[\gamma=atan2 \begin{pmatrix}v_{\rm L}^{y},v_{\rm L}^{x}\end{pmatrix}  \tag{9}\]  
The atan2 function returns the angle of the straight line drawn through the centre of the coordinate system and the given point and the \(x\)-axis. The proposed kinematic model of a single omni-track module is controlled by two drives. It allows the body to move in any direction without having to change the angular orientation of the body. It therefore differs from the existing models of typical tracked vehicles, in which the angular orientation of the body must be changed to achieve movement in a direction other than the main axis. In further steps, it is possible to develop the proposed model by analyzing any number of modules (pairs of track running systems) to obtain more complex vehicles. The article considers an omni-track vehicle consisting of two track modules connected by a body \(k\). The geometric centre of the body \(k\) contains point R, which will be the subject of further considerations. The modules are arranged in the same direction, in a completely overlapping manner. During the simulation, only the translational motion of the vehicle will be studied, which means that both modules must be controlled in the same way. This means that if the kinematic excitations \(q_{1}=q_{3}\) and \(q_{2}=q_{4}\) and the velocities \(\dot{q}_{1}=\dot{q}_{3}\) and \(\dot{q}_{2}=\dot{q}_{4}\) are maintained, the equality (10) will hold. The vector \(\stackrel{{\rightarrow}}{{S_{\rm L}}}\) and \(\stackrel{{\rightarrow}}{{v_{\rm L}}}\) displacement determines the velocity of point L, which is equal to the displacement and velocity of points R and P on the body \(k\):  
\[\stackrel{{\rightarrow}}{{S_{\rm L}}}=\stackrel{{ \rightarrow}}{{S_{\rm R}}}=\stackrel{{\rightarrow}}{{S_{\rm P }}} \tag{10}\] \[\stackrel{{\rightarrow}}{{v_{\rm L}}}=\stackrel{{ \rightarrow}}{{v_{\rm R}}}=\stackrel{{\rightarrow}}{{v_{\rm P }}}\]  
Based on Equation (4), assuming the spacing between the theoretical sliders is equal to the distance between the tracks d, we can determine the formula for the new position \(\stackrel{{\rightarrow}}{{S_{R}}}\) of point R at the given excitations \(q_{1}=q_{3}\) and \(q_{2}=q_{4}\):  
\[\stackrel{{\rightarrow}}{{S_{R}}}= \begin{bmatrix}S_{R}^{x}\\ S_{R}^{y}\end{bmatrix} = \begin{bmatrix}b\cos\,\delta_{2}\\ b\sin\,\delta_{2}+q_{2}\end{bmatrix} = \begin{bmatrix}f\,\cos\,\delta_{1}\\ f\sin\,\delta_{1}+q_{1}\end{bmatrix}  \tag{11}\]  
whereas the velocity \(\stackrel{{\rightarrow}}{{v_{R}}}\) and its orientation \(\gamma\) are analogous to the case based on Equations (8) and (9):  
\[\stackrel{{\rightarrow}}{{v_{R}}}= \begin{bmatrix}v_{R}^{x}\\ v_{R}^{y}\end{bmatrix} = \begin{bmatrix}\dot{b}\cos\,\delta_{2}\\ \dot{b}\sin\,\delta_{2}+\dot{q}_{2}\end{bmatrix} = \begin{bmatrix}\dot{f}\,\cos \,\delta_{1}\\ \dot{f}\,\sin\,\delta_{1}+\dot{q}_{1}\end{bmatrix}  \tag{12}\]  
\[\gamma=atan2 \begin{pmatrix}v_{R}^{y},v_{R}^{x}\end{pmatrix}