## 1 Introduction  
In contrast to non-holonomic vehicles, such as regular automobiles, a holonomic vehicle can move in any direction without rotating its body. This extra mobility gives the holonomic vehicle an advantage over non-holonomic vehicles. A holonomic vehicle is ideal for narrow or small working environments, where turning the vehicle is hard or impossible. Holonomic vehicles are also useful when performing difficult driving maneuvers, for example, parallel parking is much easier to perform in a holonomic vehicle than in a standard non-holonomic vehicle. Because of the usefulness of holonomic and Omni-directional vehicles, engineers over the years have developed unique ways to make vehicles holonomic. Because of the advantages of the holonomic vehicle, it has been applied under various conditions, as follows. One type of holonomic vehicle is called the power caster vehicle (PCV). The PCV's design basically looks like a powered office chair. The PCV is a stable holonomic vehicle that can smoothly navigate all of the usual indoor obstacles [1]. It was successfully integrated with the Programmable Universal Machine for Assembly, or Programmable Universal Manipulation Arm (PUMA) robotic arm in Nomadic Technologies XR4000 robot [2]. To help the physically challenged, a holonomic wheelchair with a ball and wheel mechanism was created [3,4]. The OmniMate is a holonomic vehicle that was created for hazardous environments. It requires few external corrections because it uses an internal position control correction device that allows it to correct for bumps and other floor irregularities [5; 6]. There have been several types of wheels developed for the application in holonomic vehicles, as follows. One vehicle design that allows for holonomic movement incorporates Mecanum wheels. Mecanum wheels consist of a ring of angled periphery rollers. With varying wheel directions and speeds, the combination of forces produced allow the vehicle to move in any direction [7; 8]. Another wheel design manufactured for holonomic vehicles is made up of several smaller wheels attached perpendicularly along the periphery of the main wheel [9]. A frequently used design for mobile robots is a three-wheel design, with wheels arranged at the vertices of an equilateral triangle [10]. An interesting approach to create a holonomic vehicle is in the shape of a sphere. By controlling pendulums that are mounted in the spherical body, the spherical vehicle is able to move in any direction [11]. A different way to use a sphere in a holonomic vehicle is to use it as a tire. A single rider holonomic vehicle was created, using a basketball as a tire, and four force-transfer-mechanisms for movement [12]. Various models for the control of the holonomic movement of land-based vehicles, underwater vehicles, indoor airship or aerial robots, have been explored in previous studies [9; 13; 14; 15]. For holonomic control, each wheel of the holonomic vehicle drives the robot forward and rotates the frame of the robot; this is because the wheels are located on the frame periphery. The number of traction forces which can be obtained is equal to the number of activated motors, and the driving direction and the sum of the resultant forces are calculated from these forces generated by the motors [9]. Therefore, conventional methods used redundant thrusters to realize motion in each direction, which resulted in high weight and costs. In a study by [PERSON] et al. on underwater vehicles (SMART-E), all six degrees of freedom (forward/back, left/right, up/down, yaw, roll and pitch) were controlled by using three thrusters [16]. The three thrusters were rotated by a servo motor about 180\({}^{\circ}\) around a rotating axis to achieve holonomic 3D movements, and full holonomic propulsion of underwater vehicles was realized based on the mathematical force model used in the work by [PERSON] [13; 16].