# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 2 Materials and Methods
### Kinematic Model
In kinematic analysis, the rectilinear guide can perform only a sliding motion in the slider in the direction parallel to the slider. This means that for this analysis, it does not matter at what distance from each other they are; the points at which the sliders will be located will be assumed. The angular orientation \(\delta_{1}\), \(\delta_{2}\) of these sliders is identical to the angle \(\alpha\) of the orientation of the free rollers. Vector \(\stackrel{{\rightarrow}}{{b}}\) is the length of the extension of guide 2 in relation to the previously assumed point at which the slider was located on the base. It should be emphasized that this slider is only a theoretical construct--its presence is necessary to introduce interaction between the track and the ground into the model. The orientation of vector \(\stackrel{{\rightarrow}}{{b}}\) described as \(\delta_{2}\) is forced by the direction of guide 2, while its length is variable. Angle \(\delta_{2}\) depends on the design parameters of the omni-tracks. Its value is 90\({}^{\circ}\) greater than angle \(\alpha_{2}\), defining the angular orientation of the free axis of the roller \(o_{2}\) in relation to the transverse axis of the track (Figure 10). As a result, the vector \(\stackrel{{\rightarrow}}{{b}}\) and the free roller axis \(o_{2}\) are perpendicular to each other. The vector \(\stackrel{{\rightarrow}}{{q}}_{2}\) is parallel to the \(y\)-axis of the global coordinate system. Its length is the same as the linear displacement of the track g\({}_{2}\) relative to the body. This means that it is a known value resulting from the set control. The vector \(\stackrel{{\rightarrow}}{{d}}\) is parallel to the \(x\)-axis of the global coordinate system. It is the distance between the model elements representing the tracks. This vector represents the track spacing in a single drive module of an omni-track vehicle. The vector \(\overrightarrow{q_{1}}\) is the same as the linear displacement of the tracks g\({}_{2}\) relative to the body. It is parallel to the \(y\)-axis, and its length results from the adopted control. The vector \(\overrightarrow{f}\), like the vector \(\overrightarrow{b}\), represents the length of slide extension 1 relative to the theoretically adopted slider located on the base. It is oriented at an angle \(\delta_{1}\), which depends on the angle of orientation of the free roller on the track under consideration. The value \(\delta_{1}\) is 90\({}^{\circ}\) smaller than the angle \(\alpha_{2}\). In addition, the vector \(\overrightarrow{f}\) closes the chain. By projecting the vectors onto the individual axes, the following system of equations can be obtained, describing the position of the various components of the mechanism:  
\[\left\{ \begin{array}{rl}&0=\text{a}+b\ cos\ \delta_{2}-d-f\ cos\ \delta_{1}\\ &0=b\ sin\ \delta_{2}-q_{1}+q_{2}-f\ sin\ \delta_{1}\end{array} \right. \tag{4}\]  
The values of \(q_{1}\) and \(q_{2}\) are given. The values sought are \(b\) and \(f.\) Solving the above system of equations will determine the values of b and f. Thus, given the known values of \(q_{1}\) and \(q_{2}\), the displacement \(\Delta\ r_{L}\) of point L with respect to the ground can be determined from the following relationship:  
\[\Delta\ r_{L}=f+q_{1}=b+q_{2} \tag{5}\]  
After differentiating the equations describing the position with respect to time t once, a system of equations describing the relationship between velocities in the mechanism in question will be formed:  
\[\left\{ \begin{array}{rl}&0=\dot{b}\ cos\ \delta_{2}-\dot{f}\ cos\ \delta_{1}\\ &0=\dot{b}\ sin\ \delta_{2}-\dot{q}_{1}+\dot{q}_{2}-\dot{f}\ sin\ \delta_{1}\end{array} \right.