## 4 Motion Verification
### Results  
The traversed paths for each configuration and target trajectories are shown in Figures 5 and 6. From the experiment results, we can observe that the robots tend to prefer moving in the same direction as their wheels, making the shape of the trajectory different for both configurations. As it can be seen in Figure 5, for the conventional robot, the path looks like a rhomboid, while for our proposed design, it looks like a square. These results can be explained by the DC motors not being driven correctly at a slow speed. This is confirmed when looking at Figure 7, which shows that the actual velocities of the robot do not follow target velocities accurately. This is a problem that can be mostly attributed to the controller; therefore, it can be stated that the LQR controller used does not suffice. As we are not interested in robot control but in understanding how the different configurations affect performance, we did not improve on this controller. Instead, we focus on how wheel placement can affect trajectory control. While not due to an inherent mechanical design issue, these experiments prove that trajectory errors can result from poor control and mechanical design. Finally, we evaluate the trajectory error by measuring the travel direction error between measured locations and their corresponding target locations. The median of direction error for each robot configuration/test is shown in Tables 2-5. The median is used because it correctly evaluates the direction the robot moves by excluding the effects of outliers caused by lagging control. To assess if the median errors for the conventional and proposed arrangements are statistically significant, we performed a two-tailed \(t\)-test, with a rejection probability of 5%. For the circular pattern (Tables 1 and 2), we have that the difference in median distance errors for both configurations was not significantly different (t = 1.7; \(p<0.106\)), while the \begin{table}
\begin{tabular}{l l} \hline \hline
**Material/Condition** & **Detail** \\ \hline Based Robot & NEXUS ROBOT’s 4 WD 58 mm Omni Wheel Arduino Robot Kit 1  \\ Wheel Material & Nylon + Polyethylene \\ Flooring & Lonseal’s Lonleam Plain 2  \\ Standard Applied to Flooring & JIS A 5705 \\ Flooring Material & vinyl resin \\ Flooring Size & 2 mm (thickness) \(\times\) 1.82 m \(\times\) 3 m \\ Flooring laying method & direct placement (no adhesion) \\ Camera & Logitech’s C930e 3  \\ Camera Mounting Height & 2 m \\ Room Temperature & 23 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The materials and conditions used in the experiments. difference in median angular errors was significant (t = 3.8; \(p\) < 0.0013). This shows that our proposed arrangement can provide the movement with less postural blur than the conventional type. Figure 5: The robot trajectory during each run using circle pattern. (**a**) Conventional robot; (**b**) proposed robot. For the square pattern (Tables 3 and 4), we found significant differences in the distance errors (t =13.3; \(p\) < 9.47 \(\times\) 10\({}^{-11}\)) and angular errors (t = 6.6; \(p\) < 3.37 \(\times\) 10\({}^{-6}\)). Given these values, we can ascertain that for the square pattern, our proposed arrangement improved performance. The time required to complete the trajectory for each robot configuration/test is shown in Table 6. A two-tailed t-test with a rejection probability of 5% was applied to the time required for the two robots. We found significant differences in the time required for circle  
Figure 6: The robot trajectory during each run using square pattern. (**a**) Conventional robot; (**b**) proposed robot. pattern (t = 6.5; \(p\) < 4.11 \(\times\) 10\({}^{-6}\)) and square pattern (t = 13.1; \(p\) < 1.2 \(\times\) 10\({}^{-10}\)). As a result of the above, the proposed arrangement has the potential to operate as accurately or more accurately as a conventional type and still operate at high speed.