# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 2 Materials and Methods
### Kinematic Model
This means that the connection between the ground and the free roller can be treated as a class I translational pair. The orientation of the roller relative to the track does not change during movement. Therefore, both of these elements can be replaced by one body consisting of two bodies fixed relative to each other at an angle corresponding to the orientation angle of the roller. The movement of the track relative to the vehicle body is always rectilinear. Therefore, the connection of the track with the body can be presented as a class I progressive pair. In order to enable the movement of the tracks relative to the body, this track must be equipped with one drive. The proposed model of a single omnidirectional vehicle module consists of a base 0, two guides 1 and 2, each of which consists of two bodies oriented and fixed relative to each other, and a body k, with two parallel mounted sliders. The diagram of the proposed simplification of the omni-track vehicle mechanism is shown in Figure 7, while the kinematic diagram is shown in Figure 8. For the purposes of the presented example, it was assumed that this is a model of the L\({}_{12}\) module, with the geometric centre at point L, which consists of running systems 1 and 2, modelled by links 1 and 2. The mobility of the mechanism in question was calculated using the formula for the mobility of the wedge mechanism:  
\[W=2(k-1)-p_{1} \tag{3}\]  
\(W\)--mobility;  
\(k\)--number of bodies;  
\(p_{1}\)--number of pairs of class I. For \(k=4\) and \(p_{1}=4\), mobility \(W\) is \(W=2\). The discussed mechanism (Figure 8) has two degrees of freedom. This means that two kinematic excitations (drives) are necessary for control. The first one, called \(q_{1}\), describes the displacement of link 1 relative to body k. The second drive, called \(q_{2}\), forces the displacement between link 2 and body k. In this model, link k represents the body of the omni-tracked vehicle. Links 1 and 2 are the left and right tracks (\(g_{1}\) and \(g_{2}\)), respectively. This means that the velocities of the drives \(\hat{q}_{1}\) and \(\hat{q}_{2}\) are equal to the linear movement velocities of the tracks relative to body k of the vehicle. The analysis of the discussed mechanism was performed using vector equations. In the proposed mechanism, the position of the links can be described by a one vector constraint equation (Figure 9). Figure 8: Kinematic diagram of a single module of an omni-track vehicle. 0—base, 1, 2—guides, k—vehicle body, \(\overset{\rightarrow}{q}_{1}\), \(\overset{\rightarrow}{q}_{2}\)—kinematic excitations. Figure 7: Omni-track vehicle: (**a**) view of the omni-track vehicle chassis, (**b**) schematic view of the omni-track vehicle chassis, (**c**) view of a single omni-track vehicle module, marked with a red circle in (**a**), (**d**) schematic view of a single omni-track vehicle module marked with a red circle in (**b**), and (**e**) kinematic model of a single omni-track vehicle module. 0—base, 1, 2—guides, k—vehicle body. The equations of this mechanism can be written as  
Figure 6.0  
The lengths of vectors \(\stackrel{{\rightarrow}}{{a}}\), \(\stackrel{{\rightarrow}}{{q}}_{1}\), \(\stackrel{{\rightarrow}}{{d}}\) and\(\stackrel{{\rightarrow}}{{q}}_{2}\) and angles \(\delta_{1}\) and \(\delta_{1}\) are known (Figure 9). Vectors \(\stackrel{{\rightarrow}}{{q}}_{1}\) and \(\stackrel{{\rightarrow}}{{q}}_{2}\) describe the given kinematic excitations, i.e., the displacement of the tracks relative to the body k. Vectors \(\stackrel{{\rightarrow}}{{a}}\) and \(\stackrel{{\rightarrow}}{{d}}\) define the geometry of the system. The lengths of vectors \(\stackrel{{\rightarrow}}{{b}}\) and \(\stackrel{{\rightarrow}}{{f}}\) are sought, based on which the displacement of the vehicle body k relative to the base 0 can be determined. Vector \(\stackrel{{\rightarrow}}{{d}}\) is parallel to the \(x\)-axis of the global coordinate system and is located on the base. Its length is equal to the distance between the conventional places of attachment of the sliders.