# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 2 Materials and Methods  
### Kinematic Model  
The model proposed in this article enables the representation of a wide range of omni-track vehicles. It can be used to describe a mechanism (a pair of tracks, referredto as a module) with any configuration of free rollers, including asymmetric ones. Both tracks must be the same length, i.e., the same number of rollers in contact with the ground. Unbalanced forces may appear in the case of disproportion in length, which may affect the vehicle's trajectory. The model can also be used to describe a tracked vehicle composed of any number of modules and their mutual configuration. An omni-track vehicle with fully overlapping tracks has four tread systems bn designated as \(n=1\), \(\ldots\), 4 and a body k. Adjacent tread systems can be divided into pairs, hereinafter referred to as modules. The first module, referred to hereafter as L\({}_{12}\), consists of \(b_{1}\) and \(b_{2}\). The second module, referred to as P\({}_{34}\), consists of \(b_{3}\) and \(b_{4}\). A typical omni-track vehicle consists of two identical, parallel modules. At the geometric centre of the body, k, which is also the centre of the robot, a point R is determined, which is the same as the centre of the robot. In the geometric centre of module L\({}_{12}\), point L is determined, while in the geometric centre of module P\({}_{34}\), point P is determined. A schematic of the vehicle, along with a solid model, is shown in Figure 5.  
A single running system \(b_{n}\) consists of a track \(g_{n}\), a rotary drive mn, a gearbox, and a drive wheel. The wheel connected to the drive, whose speed is denoted as \(\omega_{n}^{M}\), is the active wheel of the gearbox. The gearbox sets the passive wheel in motion, whose speed is denoted as \(\omega_{n}^{B}\). The passive wheel is connected to the drive wheel of the track. The drive wheel forces the track \(g_{n}\) to move with a speed of \(v_{n}\), as shown in Figure 6.  
Figure 5: Omni-tracked vehicle with fully overlapping tracks: (**a**) view of the solid model and (**b**) schematic of the vehicle. Figure 6: Construction of the omni-track running system \(b_{n}\). The gear ratio between the drive and the passive gear \(i_{MBn}\) can be described by the formula:  
\[i_{MBn}=\frac{\omega_{n}^{M}}{\omega_{n}^{B}}=\frac{r_{n}^{B}}{r_{n}^{M}} \tag{1}\]  
\(n\)--tracked running gear number \(n=1,\ldots,4\);  
\(r_{n}^{B}\)--radius of the passive wheel of running gear \(n\);  
\(r_{n}^{M}\)--radius of the active wheel of running gear \(n\);  
\(\omega_{n}^{B}\)--angular velocity of the passive wheel of running gear \(n\);  
\(\omega_{n}^{M}\)--angular velocity of the active wheel of running gear \(n\). The linear velocity \(v_{n}\) of the gn track can be described by the formula:  
\[v_{n}=\omega_{n}^{M}\;i_{MBn}\;r_{n}^{KN} \tag{2}\]  
\(r_{n}^{KN}\)--radius of the driving wheel of the running gear \(n\). Equation (2) describes the velocity \(v_{n}\) of the movement of the track section gn in contact with the ground relative to the vehicle body k. This velocity will be referred to in the latter part of this article as the track velocity. Based on the calculated velocities of the individual tracks, it is possible to determine the direction and velocity of the movement of point R, located on the robot body k. The body moves in a translational motion, so the velocity vectors of all points belonging to a given body will be equal. It can, therefore, be assumed that in order to determine the direction and value of the velocity vector of point R, it is sufficient to determine the orientation and value of the velocity vector of a single module, i.e., point L in the case of module L\({}_{12}\) or point P in the case of module P\({}_{34}\). A kinematic model of a single module consisting of two running systems was proposed. All resistance to the motion was omitted. It was assumed that there would be no slippage during the interaction of the free roller with the flat ground.