# Omni Wheel Arrangement Evaluation Method Using Velocity Moments
## 7 Conclusions  
In production, there is a need for wheeled omnidirectional mobile robots that can perform precise movements in a confined work environment with high efficiency. In addition, mobile robots may need to be asymmetric depending on the application. This manuscript presents a design methodology for omnidirectional mobile robots with asymmetric wheel arrangements. In particular, Omni wheels, which have less vibration than mecanum wheels, are used for wheeled omnidirectional mobile robots. It is known that the operability of Omni wheel robots is reduced by the wheel arrangement. We hypothesize that an imbalance in propulsive forces causes this reduced operability. In addition, we assumed that the propulsive force generated by the wheels depends on the ground speed of the wheels. We then proposed a wheel placement method that takes into account the moment balance of the wheel ground speed on the robot's turning center. Omni wheel robots lose operability when control of some of their wheels is lost. We hypothesized that even with this loss of operability, the robot could move straight ahead in at least one direction. In this manuscript, we propose a method for estimating the direction in which the robot will move straight ahead using the balance of velocity moments. To evaluate the effectiveness of the proposed method, we compare the movements of 4- and 6-wheel Omni wheel robots in 8 directions in arrangements that satisfy and do not satisfy the velocity moment balance. It was confirmed that when the equilibrium of the velocity moments is satisfied, the 4- and 6-wheel robots go straight according to inverse kinematics. When the equilibrium is not satisfied, the robots spontaneously turn. These experimental results show that the equilibrium of the velocity moments can be used to obtain the \"easy-to-operate Omni wheel arrangement\" that follows inverse kinematics. It was confirmed that the angular velocity is proportional to the commanded velocity when the equilibrium of the velocity moments is not satisfied. It was also suggested that this angular velocity is proportional to the sum of the velocity moments. Since the velocity moment is proportional to the robot's velocity, the relationship between the sum of the velocity moments and the angular velocity of the robot was obtained. The results showed that the sum of the velocity moments and the robot's angular velocity are proportional for the 4- and 6-wheel robots. The proportionality constant of the angular velocity to the sum of the velocity moments is the sensitivity to the sum. Experimental results showed that this sensitivity is affected by the wheel arrangement and the number of wheels on the robot. In particular, the results verified for the wheel arrangement suggest that this sensitivity is related to the root mean square of the distance between the velocity vector generated by the wheels and the robot's center of mass. Conceptualization, M.H. and K.O. ; software, M.H. ; validation, R.M. ; data curation, M.H. ; writing--original draft preparation, M.H. ; writing--review and editing, R.M. and K.O. ; supervision, R.M. ; project administration, R.M. and K.O. ; All authors have read and agreed to the published version of the manuscript. This research received no external funding. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Arrangement**} & \multirow{2}{*}{**Average**} & \(I\) **[m]** & & \(I^{\prime}\) **[m]** & \\  & & **RMS** & **Average** & **RMS** \\ \hline
4-wheeled Not & & & & \\ E-to-O 1 & 0.254 & 0.261 & 0.177 & 0.197 \\
4-wheeled Not & & & & \\ E-to-O 2 & 0.231 & 0.233 & 0.173 & 0.201 \\
4-wheeled Not & & & & \\ E-to-O 3 & 0.223 & 0.225 & 0.151 & 0.179 \\
6-wheeled Not & & & & \\ E-to-O & 0.297 & 0.309 & 0.212 & 0.278 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average and root-mean-square of the distance \(l\) of the wheel from the origin and the distance \(l^{\prime}\) between the wheel’s resulting velocity vector and the origin in each arrangement. **Institutional Review Board Statement:** Not applicable.