## 5 Simulation Results
### Results  
First of all, let us compare the modified Pure Pursuit path-tracking algorithm with the original, conventional version under the same conditions and for SR 0.1 s.  
Figure 13 clearly reveals the benefits of using the modified proposal (MPP) versus the original, conventional one (CPP)--that is, MPP is able to track the path more accurately than CPP. This fact is confirmed by the cost indexes presented in Table 1, where \(J_{1}\) and \(J_{2}\) are noticeably worsened for the original algorithm. However, \(J_{3}\) points out a slight reduction of time that CPP needs to take in order to follow the trajectory. Note that the indexes are divided by the best case--that is, the SR 0.1 s MPP. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(J_{1}\) & \(J_{2}\) & \(J_{3}\) \\ \hline SR 0.1 s—Modified Pure Pursuit (MPP) & 1.0 & 1.0 & 1.0 \\ SR 0.1 s—Conventional Pure Pursuit (CPP) & 14.3 & 6.96 & 0.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cost index results for Pure Pursuit comparison. Figure 16: Path-tracking trajectory error over time. Figure 15: Comparison results for a Lissajous reference. The main conclusions of the comparison are as follows:  
* Cost index \(J_{3}\) presents values very similar to the nominal one for every case, except for SR 0.3 --that is, the case where direct measurements are sensed at sampling period \(T=0.3\) s, which has a value 2.7 times higher. * SR 0.3 s shows the worst behavior (noticeable oscillations), which is confirmed by the clear rise of every cost index. On average, \(J_{1}\) and \(J_{2}\) increase their values by 42.8 and 17.5 times, respectively. * In general, the NUDREKF cost indices are lower than the DREKF cost indices when appearing in nonuniform sampling patterns, beyond managing noisy and scarce data, and existing process nonlinearities. As can be seen in Figure 14, the higher \(P\) is, the worse the cost value will be. This rule is more evident for the DREKF. Note that the worst case for NUDREKF--that is, when \(P=0.5\)--presents lower \(J_{1}\), \(J_{2}\), and \(J_{4}\) than the best case for DREKF--that is, when \(P=0.1\). * NUDREKF \(P=0.1\) shows accurate path tracking, despite having scarce position measurements (10 less times), and assuming dropouts, noise, and nonlinearities. The cost indexes confirm the achievement of satisfactory control properties, since \(J_{1}\) and \(J_{2}\) are slightly worsened with respect to the nominal case (4.8 and 2.4 times higher, respectively). NUDREKF \(P=0.3\) shows slightly worse path tracking with respect to NUDREKF \(P=0.1\), increasing 0.5, 1, and 0.3 points in \(J_{1}\), \(J_{2}\), and \(J_{4}\), respectively. NUDREKF \(P=0.5\) worsens the path-tracking behavior a bit more regarding NUDREKF \(P=0.1\), and increases 0.7, 1.2, and 0.5 points in \(J_{1}\), \(J_{2}\), and \(J_{4}\), respectively. In summary, the NUDREKF strategy preserves a satisfactory trajectory tracking performance under conditions of loss of measurements. * In contrast, DREKF \(P=0.1\) shows slightly worse path tracking with respect to NUDREKF \(P=0.5\), and quite worse behavior regarding NUDREKF \(P=0.1\). As can be seen in Figure 15, DREKF \(P=0.3\) and DREKF \(P=0.5\) seem to achieve good path tracking; however, the correction of the position is taken only a few times and it is generally based on odometry measurements, obtaining poor performance on the cost index \(J_{1}\), \(J_{2}\) and \(J_{4}\). If the experiment were extended over time, the tracking would suffer from drift in the estimation of the robot. * Figure 16 depicts the path-tracking trajectory error for the different cases evaluated. As expected (see Figure 16a), SR 0.1 s presents the lowest error, being practically null except for the four slight spikes corresponding to the four curves of the trajectory, and SR 0.3 s shows the highest error, which reaches up to 0.6 m. Between both single-rate cases, the error value for the dual-rate approaches can be found.