# Kinematic Analysis of an Omnidirectional Tracked Vehicle
## 1 Introduction  
Omnidirectional robots and vehicles represent a class of machines specifically designed to enable movement in any direction without changing the angular orientation of their body [1]. This capability is achieved through the use of specialized wheels or tracks [2], or alternatively, by employing conventional wheels in conjunction with additional systems. The majority of research in the field of omnidirectional robots predominantly concentrates on different types of wheeled vehicles [3, 4]. The earliest known wheel specifically designed to enable omnidirectional motion is the omni wheel, patented in 1919 [5]. This design incorporates a series of free rollers mounted around the wheel's circumference. These non-driven free rollers are barrel shaped to reduce vibration during operation and oriented perpendicularly to the wheel axis [6]. Another well-known solution enabling omnidirectional movement is the Mecanum wheel, also referred to as the \"Swedish wheel\" or \"Ilon wheel\". Like the omni wheel, it includes rollers on its perimeter; however, in this case, the rollers are commonly mounted at a 45\({}^{\circ}\) angle to both the wheel axis and the lateral plane. When integrated with appropriate control algorithms and mechanical configurations, Mecanum wheels allow a vehicle to move in any direction, at any angle [7]. Another type of omnidirectional wheel is the Lidiadir wheel patented in 2016 by [PERSON] [8]. This design consists of torus tires mounted on rims, on which they rest against rows of transversely mounted rollers--both free and driven. Movement along the wheel axis is achieved through the rotation of the entire wheel, while lateral movement results from moving the torus tire freely over the rollers [7, 9]. Omnidirectional motion can also be realized using spherical elements. In such solutions, the intermediary element between the robot's corpus and the ground is a sphere [10]. Depending on the particular solution, robots equipped with this type ofomidirectional drive have from one to several driven spheres. The sphere, due to its shape, can roll in a free direction without resistance, which is used to achieve omnidirectional motion. In addition to the developments characterized above, typical cylindrical wheels can also be adapted for omnidirectional movement. Each wheel then has two drives--a rolling drive and a rotational drive on an axis perpendicular to the ground plane [11]. However, due to the large number of drives required to control a chassis equipped with such wheels, in many cases, the choice of this solution proves uneconomical [12]. For this reason, this solution is typically reserved for applications where certain aspects unique to these wheels are required [13], such as providing the ability to move in any direction without changing the orientation of the body under heavy load. Omnidirectional mobility is not limited to wheeled solutions. It can also be achieved using other elements mediating between the body of the mobile robot and the ground. One such approach involves the use of specially engineered omnidirectional crawler systems, which can be implemented in tracked vehicles. Tracked vehicles with omnidirectional capabilities offer potential advantages in specialized operational scenarios, where they may outperform both conventional tracked systems and wheeled omnidirectional platforms. These scenarios include rescue and inspection robots operating in urban environments where accidents or disasters have occurred. Tight spaces, typically in technical corridors or ventilation shafts, can present challenges for tracked vehicles. Areas directly affected by disasters, such as passageways partially covered by debris or technical platforms twisted by falling elements, can present an obstacle that is impossible to overcome for wheeled omnidirectional vehicles. Vehicles on omnidirectional chassis are able to maneuver and operate in such conditions. Omnidirectional crawler systems can generally be classified into two categories: active and passive.