# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 1 Introduction
The first category encompasses vehicles utilizing crawler-based chassis in which, in addition to the primary driven tracks, supplementary active components are integrated into the individual links of the track. Transverse motion is facilitated by actuating these auxiliary elements, which typically take the form of rollers or mini-tracks mounted perpendicularly to the robot's longitudinal axis. Omnidirectional mobility is thus achieved through a coordinated combination of movement along the main axis--resulting from the rolling motion of the track--and lateral displacement--enabled by the activation of these transverse drive elements. A conceptual design of such a drive system is presented in [14], where the authors address the challenge of navigating a tracked vehicle across loose terrain featuring numerous obstacles. The concept of a crawler drive with active elements can also be found in article [15], which discusses issues related to the experimental testing of a mobile robot equipped with two parallel plate crawlers with active transverse drive. The second category of omnidirectional tracked vehicles includes platforms equipped with a minimum of three independently driven tracks (Figure 1), with additional elements on the links, usually in the form of free rollers. An example of a single crawler segment with a non-driven roller is shown in Figure 2.  
There are solutions that include chassis equipped with both three and four independently driven tracks, located at either 120\({}^{\circ}\) or 90\({}^{\circ}\) angles to each other [16; 17; 18], as seen in Figure 3. These vehicles operate in a similar manner to platforms based on Kiwi Drive chassis [19]. Among omni-track vehicles with passive rollers, the most frequently studied configurations are platforms featuring four equally spaced, independently driven tracks. The classification of such systems is typically based on the relative arrangement of the track pairs and the orientation of the passive rollers mounted on individual links. A distinction is made here between groups of vehicles with free rollers arranged symmetrically or asymmetrically. In terms of track configuration, the systems are further classified as non-overlapping, partially overlapping, or fully overlapping (Figure 4). One of the earliest studies on such chassis was published in 2014 [20], analyzing the kinematics of a robot equipped with symmetrically arranged, fully overlapping tracks. The paper also presented results from a simplified simulation of a robot motion using symmetrical, partially overlapping tracks, which revealed a curvature in the actual motion trajectory compared to the planned motion during both longitudinal and lateral movements. A publication from 2015 [21] covered the issues raised in [20]. It discusses a chassis consisting of three tracks located at an angle of 120 deg, four tracks located at an angle of 90 deg, and a symmetrical, completely non-overlapping chassis in two variants in the location  
Figure 1: Omnidirectional tracked robot. Figure 3: Schematics of omni-tracked vehicles with free rollers, with the non-parallel arrangement of tracks. From (**left**) tracks at 120° and (**right**) tracks at 90°. Figure 2: Solid model and view of a free roller on a single-track link. of the free rollers. The study examined the influence of roller angles on the achieved speed and presented field test results of a prototype robot with a chassis in a symmetrical, completely non-overlapping arrangement. The research included the measurement of current consumption during the execution of specific types of movement, as well as the effect of the robot's angular orientation on the efficiency of the movement of climbing hills. In 2018, another study [22] described the construction of two vehicle demonstrators with a symmetrical, completely non-westerly track system. The first vehicle, measuring 0.8 \(\times\) 1.2 m (width and length), was equipped with additional drives to lift the tracks.