# Omni Wheel Arrangement Evaluation Method Using Velocity Moments
## 2 Related Work  
Many methods have addressed the relationship between the velocity of a robot's Omni wheels and its center of mass's linear and rotational velocity, employing inverse kinematic models. [PERSON] et al., [PERSON] et al., and [PERSON] et al. derived inverse kinematic models of a four-wheeled Omni wheel robot with axles facing the center of the robot [12; 13; 14]. [PERSON] et al. proposed an inverse kinematic model for robots with any arbitrary number of wheels in a point-symmetric arrangement with axles oriented toward the robot center [7]. Figure 2: The example of omnidirectional wheels. (1 [[https://www.nexusrobot.com/product/4-inch100](https://www.nexusrobot.com/product/4-inch100) mm-double-aluminum-omni-wheel-bearing-rollers-14054.html]([https://www.nexusrobot.com/product/4-inch100](https://www.nexusrobot.com/product/4-inch100) mm-double-aluminum-omni-wheel-bearing-rollers-14054.html), accessed on 18 January 2023. 2 [[https://www.nexusrobot.com/product/4-inch-100](https://www.nexusrobot.com/product/4-inch-100) mm-mecanum-wheel-rightbearing-rollers-14]([https://www.nexusrobot.com/product/4-inch-100](https://www.nexusrobot.com/product/4-inch-100) mm-mecanum-wheel-rightbearing-rollers-14) 094r.html, accessed on 18 January 2023). [PERSON] et al. generalized this model to be independent of Omni wheel orientation; however, their validation only addresses models with axles oriented toward the center [15]. This generalized inverse kinematics has also been used by [PERSON] et al., and [PERSON] et al. in their work on robots with axles oriented toward the center [16; 17]. Our previous work shows that robots can move in all directions, even in a point-symmetric arrangement with axles not oriented toward the robot center [18]. Thus, imposing axle orientation and symmetry is not a requirement; but rather a self-imposed rule for robot design. However, this limitation can become an issue for particular applications; for example, [PERSON] et al. mention the need for asymmetrically shaped machines in the agricultural fields [10]. As a method for determining whether a mecanum wheel robot can move in all directions, [PERSON] et al. proposed using the intersections of bottom-rollers axles of any three mecanum wheels on the robot [11]. If the number of intersections is 2 or 3, the robot can achieve omnidirectional motion. However, not all wheel arrangements that meet the requirements will have adequate operability. This unsuitable arrangement is also similar to Omni wheels. Since this loss of operability is due to the robot's automatic turning, we focused on the relationship between the speed of each wheel and the robot's turning. Using this relationship, We achieve omnidirectional movement for good operability even in an asymmetric arrangement. Regarding fault tolerance for omnidirectional mobile robots, most literature has focused on mecanum wheels, with only a few studies on Omni wheels. [PERSON] et al. have achieved fault tolerance of a four-wheel mecanum wheel robot by fuzzy control [19]. This study shows that the failed wheel can reach its destination with or without obstacles, assuming it is free to rotate. On the other hand, it has yet to be shown that the control is effective in different wheel arrangements. [PERSON] et al. show that fault tolerance can be achieved using a weighted pseudo-inverse of the Jacobian matrix derived for kinematics in a symmetric four-wheel mecanum wheel robot [20]. This study shows that it can cope with actuator output anomalies. [PERSON] et al. validated fault-tolerance control for one- and two-wheel failures of a four-wheel mecanum wheel robot [21]. This study assumes that the failed wheel is free to rotate, as in [PERSON] et al. As in [PERSON] et al., a weighted pseudo-inverse matrix is used for fault tolerance control. The results show that fault tolerance was influential in the case of a single-wheel failure but could have worked better in the case of a two-wheel failure, depending on the wheel configuration.