This chapter describes and discusses the experimental results and its discussion. First, the results of all the movements are presented. Next, a simulation of higher velocities is made, using the method of sub-sampling. And at last the results are discussed.

The optics magnification was kept fixed through the experiments. At the distance of 200 $mm$ the field of view was 50.67 $mm$ and at the distance of 300 $mm$ the field of view was of 77.65 $mm$.

The movements were captured by the camera and the proprietary software saves it in .avi video format. The videos were then processed in matlab, the frames were extracted and merged into an only image with all the frames of the movement. An example of these images is shown in Figure \ref{f:movements}. Figure (a) shows a movement using cork at distance 200 $mm$ and velocity 500 $mm/s$; figure (b) shows a movement using concrete at distance 200 $mm$ and velocity 500 $mm/s$.

Wile performing the first experiments it was found that the software was not working properly and it was not possible to capture images in full resolution ($1\times4096$ pixels), so the image acquisition was made in half resolution ($1\times2048$ pixels). The reason for this event is unknown and it might have been a malfunctioning driver.

Overall, the best results occurred using cross-correlation. In comparison with the methods of similarity, using cross-correlation were obtained higher values of coefficient of correlation. Also, the cross-correlation algorithm using FFT was, in average, ten times faster than the algorithm with the similarity coefficient methods. For those reasons, the following results are the ones made with cross-correlation.

\begin{figure}[htbp]
\centering
\subfloat[Cork.]{\includegraphics[width=7cm]{./Imagens/cork200mm500mms0deg.png}}
\quad
\subfloat[Concrete.]{\includegraphics[width=7cm]{./Imagens/concrete200mm500mms0deg.png}}
\caption{Illustration of the final image of the whole movement. Each line scan captured is merged vertically into one only image with all the scans made along the movement. These images are of size $n\times2048$, where $n$ is the number of scans.}
\label{f:movements}
\end{figure}

\section{Experimental Results}
In this section there will be presented the results of the experiments. The images taken with the camera were processed with the algorithm shown on section \ref{sec:correlation} using cross-correlation.

This section show the results using the algorithm described in section \ref{sec:algorithm}. After these results it was introduced an improvement to the algorithm, that will be explained in section \ref{section:dynamic}; the analysis was then repeated using that algorithm. Table \ref{t:pages} shows the plan of the results with the numbering for each subsection.

\begin{table}[h]
\centering
\caption{Page numbering of the results with the simple algorithm and with the dynamic sub-sampling.}
\begin{tabular}{l|l|l}
            & Simple analysis        & Dynamic sub-sampling \\ \hline
Parallel    & \pageref{simple_para}  & \pageref{dyna_para}  \\
Diagonal    & \pageref{simple_diag}  & \pageref{dyna_diag}  \\
Circular    & \pageref{simple_circ}  & \pageref{dyna_circ}  \\
Oscillatory & \pageref{simple_osc}   & \pageref{dyna_osc}   \\
Discussion  & \pageref{simple_resul} & \pageref{dyna_resul}
\end{tabular}
\label{t:pages}
\end{table}

\subsection{Parallel Movement}
\label{simple_para}
For the parallel movement the velocity estimated was highly similar for all of the materials. The following figures, show the plots for the velocity calculated with the data collected from camera. Each figure shows, also, the velocity plot from the robot (ground truth). The measurements presented were made on concrete and it should be emphasized that the plots for the other materials are similar. \\

Figure \ref{f:para_200_100mms} shows a big offset between the velocity calculated by the camera sensor and the ground truth. This is due to the fact that the correlation algorithm only detects a pixel displacement that is a multiple of 2. The pixel displacement calculated here was 2 and it is equivalent to 123.71 $mm/s$.

Figure \ref{f:para_200_500mms}, \ref{f:para_200_1000mms} and \ref{f:para_300_1000mms} show consistent measurements from the image sensor.

Figure \ref{f:para_300_100mms} show high fluctuations of the velocity, between 0 and 189.5735 $mm/s$. Here the successive frames were so similar to each other that the computed displacement oscillated. At some cases the displacement calculated was 2 pixels and at others had the value of 4, that caused the velocity fluctuation.

Figure \ref{f:para_300_500mms} shows an offset on the velocity measured by the sensor and also some fluctuations. Again, the offset was caused because the pixel displacement is always an integer and converting it to velocity induces error. \\

Table \ref{t:para_200} show the velocity errors for all the movements with all the materials. For the velocity it was calculated the average error and the maximum error using the robot as reference. It was also calculated the error for the total distance traveled. The results show that the tendency of the error is the same to all the materials. 

As expected, the error for 100 $mm/s$ at the distance of 200 $mm$ is high due to the offset calculated. The same happened for 500 $mm/s$ at the distance of 300 $mm$. The higher error values were, as expected, for 100 $mm/s$ at 300 $mm$ since the velocity measurements had high fluctuation. At the distance of 300 and velocity of 1000 $mm/s$ there was a small offset and that caused an error around 5$\%$. For the other cases the results were satisfactory with an error around 1$\%$. 
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/200_100.eps}
\caption{Parallel movement - Velocity plot - 100 $mm/s$ at distance of 200 $mm$.}
\label{f:para_200_100mms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/200_500.eps}
\caption{Parallel movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$.}
\label{f:para_200_500mms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/200_1000.eps}
\caption{Parallel movement - Velocity plot - 1000 $mm/s$ at distance of 200 $mm$.}
\label{f:para_200_1000mms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/300_100.eps}
\caption{Parallel movement - Velocity plot - 100 $mm/s$ at distance of 300 $mm$.}
\label{f:para_300_100mms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/300_500.eps}
\caption{Parallel movement - Velocity plot - 500 $mm/s$ at distance of 300 $mm$.}
\label{f:para_300_500mms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/parallel/300_1000.eps}
\caption{Parallel movement - Velocity plot - 1000 $mm/s$ at distance of 300 $mm$.}
\label{f:para_300_1000mms}
\end{figure}

\FloatBarrier

\begin{table}[h]
\centering
\caption{Parallel movement - Summary of error results.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                             &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ (mm)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{200}}                                    & 100                                                       & 23.71                                                                          & 24.20                                                & 23.71                                                                          & 24.00                                                & 23.40                                                                          & 22.92                                                & 23.71                                                                           & 24.00                                                 \\
\multicolumn{1}{l|}{}                                                        & 500                                                       & 1.03                                                                           & 0.50                                                 & 1.03                                                                           & 0.47                                                 & 1.03                                                                           & 0.65                                                 & 1.03                                                                           & 0.50                                                 \\
\multicolumn{1}{l|}{}                                                        & 1000                                                      & 1.03                                                                           & 0.76                                                 & 1.03                                                                           & 0.75                                                 & 1.03                                                                           & 0.77                                                 & 1.03                                                                           & 0.75                                                \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{300}}                                    & 100                                                       & 62.74                                                                          & 49.26                                                & 13.53                                                                          & 5.83                                                 & 22.37                                                                           & 12.21                                                 & 53.12                                                                           & 49.20                                                 \\
\multicolumn{1}{l|}{}                                                        & 500                                                       & 13.57                                                                          & 11.65                                                & 13.16                                                                          & 11.17                                                & 12.08                                                                          & 10.09                                                & 13.20                                                                           & 11.32                                                 \\
\multicolumn{1}{l|}{}                                                        & 1000                                                      & 5.21                                                                           & 4.68                                                 & 5.21                                                                           & 4.76                                                 & 5.21                                                                           & 4.67                                                 & 5.21                                                                           & 4.51                                                 \\ \hline
\end{tabular}
\label{t:para_200}
\end{table}


\subsection{Diagonal Movement}
\label{simple_diag}
For the diagonal movement the velocity plots were highly similar to the parallel one. In nature the two movements are very similar, only the diagonal has an angle applied to it. \\

Table \ref{t:diagonal_200} shows the average velocity and total distance relative error for all the movements with all the materials at the distance of 200 $mm$. Table \ref{t:diagonal_300} shows the results for a distance of 300 $mm$. It was calculated the average error and the maximum error using the robot as reference.

Here, it is possible to see that the tendency of the error is the same as for the parallel movement. In fact, the results were similar in the two cases, being even equal in most of the movements. This is an evidence that this measuring method is robust even when the image sensor has a rotation relative to the line of movement.

\FloatBarrier
% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Diagonal movement - errors for a separation of 200 $mm$.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                                  &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l|}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Angle\\ ($^{\circ}$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{2.5}}                                         & 100                                                       & 23.71                                                                          & 24.37                                                & 23.71                                                                          & 24.27                                                & 23.32                                                                          & 23.10                                                & 23.71                                                                           & 24.37                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.03                                                                           & 0.21                                                 & 1.03                                                                           & 0.32                                                 & 1.03                                                                           & 0.08                                                 & 1.03                                                                           & 0.23                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 1.03                                                                           & 0.07                                                 & 1.03                                                                           & 0.03                                                 & 1.03                                                                           & 0.10                                                 & 1.03                                                                           & 0.10                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{5}}                                           & 100                                                       & 23.71                                                                          & 25.35                                                & 23.71                                                                          & 24.66                                                & 22.97                                                                          & 22.42                                                & 23.71                                                                           & 25.20                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.03                                                                           & 0.30                                                 & 1.03                                                                           & 0.27                                                 & 1.03                                                                           & 0.01                                                 & 1.03                                                                           & 0.31                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 1.03                                                                           & 0.53                                                 & 1.03                                                                           & 0.02                                                 & 1.03                                                                           & 0.15                                                 & 1.03                                                                           & 0.43                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{10}}                                          & 100                                                       & 23.71                                                                          & 19.84                                                & 23.71                                                                          & 26.13                                                & 23.32                                                                          & 24.18                                                & 23.71                                                                           & 18.20                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.03                                                                           & 0.86                                                 & 1.03                                                                           & 0.98                                                 & 1.03                                                                           & 0.90                                                 & 1.03                                                                           & 0.76                                                 \\
\multicolumn{1}{l|}{}                                                             & 100                                                       & 1.06                                                                           & 0.34                                                 & 1.02                                                                           & 0.67                                                 & 1.03                                                                           & 0.61                                                 & 1.03                                                                           & 0.40                                                
\end{tabular}
\label{t:diagonal_200}
\end{table}

\FloatBarrier
% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Diagonal movement - errors for a separation distance of 300 $mm$.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                                  &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Angle\\ ($^{\circ}$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{2.5}}                                         & 100                                                       & 67.04                                                                          & 52.56                                                & 16.19                                                                          & 7.48                                                 & 23.45                                                                           & 28.36                                                 & 56.12                                                                           & 40.21                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 13.48                                                                          & 12.38                                                & 11.59                                                                          & 10.33                                                & 9.51                                                                           & 8.56                                                 & 13.20                                                                           & 12.46                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 5.21                                                                           & 3.94                                                 & 5.21                                                                           & 3.97                                                 & 5.21                                                                           & 4.02                                                 & 5.21                                                                           & 3.90                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{5}}                                           & 100                                                       & 51.21                                                                          & 38.33                                                & 13.60                                                                          & 6.36                                                 & 0.81                                                                           & 1.74                                                 & 49.30                                                                           & 47.65                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 3.49                                                                           & 4.30                                                 & 10.27                                                                          & 9.32                                                 & 10.93                                                                          & 9.95                                                 & 3.20                                                                           & 4.12                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 5.21                                                                           & 3.36                                                 & 5.21                                                                           & 4.00                                                 & 5.21                                                                           & 3.98                                                 & 5.21                                                                           & 3.42                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{10}}                                          & 100                                                       & 43.78                                                                          & 30.20                                                & 9.10                                                                           & 4.90                                                 & 0.53                                                                           & 0.98                                                 & 39.25                                                                           & 37.24                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 2.79                                                                           & 1.32                                                 & 6.05                                                                           & 4.12                                                 & 3.69                                                                           & 4.39                                                 & 2.54                                                                           & 1.23                                                 \\
\multicolumn{1}{l|}{}                                                             & 100                                                       & 5.21                                                                           & 3.47                                                 & 5.21                                                                           & 3.27                                                 & 5.21                                                                           & 3.32                                                 & 5.21                                                                           & 3.12                                                
\end{tabular}
\label{t:diagonal_300}
\end{table}

\FloatBarrier
\subsection{Circular Movement}
\label{simple_circ}
Figures \ref{f:200_100_2}, \ref{f:200_500_5}, \ref{f:200_1000_10}, \ref{f:300_100_5}, \ref{f:300_500_10}, \ref{f:300_1000_2} show the generic velocity plots for the circular movement at the various velocities and distances. It shows also the trajectory made by the robot versus the one estimated by the camera and gyroscopes.

Tables \ref{t:concrete_200}, \ref{t:concrete_300}, \ref{t:cork_200}, \ref{t:cork_300}, \ref{t:stone_200}, \ref{t:stone_300}, \ref{t:asphalt_200}, \ref{t:asphalt_300} show the error and final position deviation for all the experiments. The tables show the average velocity error and total distance error, they also show the $X$ and $Y$ coordinates deviation and the euclidean distance between the end path points estimated and the ground truth. The final position of the robot was $(x, y) = (500, 0)$ and the $X$, $Y$ deviations are calculated as the difference between these and the ground truth.

In general, the estimated trajectories made with the camera and gyroscopes showed small deviation from the reference. Except for the experiments at 100 $mm/s$ wish had high errors in the longitudinal velocity estimation.  

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_100_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_100_2.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s - diameter of 2 m - at distance 200 mm.}
\label{f:200_100_2}
\end{figure}


\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_500_5.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s - diameter of 5 m - at distance 200 mm.}
\label{f:200_500_5}
\end{figure}


\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/200_1000_10.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s - diameter of 10 m - at distance 200 mm.}
\label{f:200_1000_10}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_100_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_100_5.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s - diameter of 5 m - at distance 300 mm.}
\label{f:300_100_5}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_500_10.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s - diameter of 10 m - at distance 300 mm.}
\label{f:300_500_10}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular/300_1000_2.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s - diameter of 2 m - at distance 300 mm.}
\label{f:300_1000_2}
\end{figure}


\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for concrete at a distance separation of 200 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Concrete}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 23.71                                                     & 25.34                                                & 120.38   & 13.24    & 121.10                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.07                                                 & 12.34    & 0.55     & 12.35                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 11.15                                                     & 8.97                                                 & 60.69    & 1.90     & 60.72                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 23.71                                                     & 26.65                                                & 132.97   & 0.72     & 132.97                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.13                                                 & 7.36     & 1.19     & 7.45                                                                \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 0.49                                                 & 12.84    & 0.74     & 12.86                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 23.71                                                     & 24.12                                                & 120.40   & 16.39    & 121.51                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.69                                                 & 8.21     & 19.11    & 20.80                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 1.03                                                 & 12.83    & 19.00    & 22.93                                                              
\end{tabular}
\label{t:concrete_200}
\end{table}             


\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for concrete at a distance separation of 300 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Concrete}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                      & 100                                                       & 4.74                                                      & 3.09                                                 & 22.92    & 11.12    & 25.48                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 24.17                                                     & 20.55                                                & 115.17   & 1.10     & 115.17                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 3.89                                                 & 35.58    & 1.50     & 35.61                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 40.58                                                     & 27.55                                                & 138.34   & 7.98     & 138.57                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 6.02                                                      & 5.42                                                 & 19.34    & 0.42     & 19.35                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.11                                                 & 31.21    & 0.63     & 31.22                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 64.52                                                     & 51.30                                                & 257.22   & 13.86    & 257.59                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 13.24                                                     & 11.38                                                & 51.15    & 18.99    & 54.56                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.67                                                 & 31.05    & 18.97    & 36.39                                                              
\end{tabular}
\label{t:concrete_300}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for cork at a distance separation of 200 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Cork}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                       & 100                                                       & 23.71                                                     & 25.18                                                & 119.52   & 13.20    & 120.25                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.02                                                 & 12.88    & 0.02     & 12.88                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 9.82                                                      & 7.82                                                 & 55.89    & 2.13     & 55.93                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 23.71                                                     & 24.32                                                & 120.82   & 3.90     & 120.89                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.04                                                 & 7.75     & 1.25     & 7.85                                                                \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 0.37                                                 & 13.01    & 0.81     & 13.04                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 23.71                                                     & 24.01                                                & 119.93   & 16.38    & 121.04                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.69                                                 & 9.07     & 19.19    & 21.23                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 1.02                                                 & 13.58    & 19.04    & 23.39                                                              
\end{tabular}
\label{t:cork_200}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for cork at a distance separation of 300 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Cork}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 5.21                                                      & 3.04                                                 & 19.92    & 4.00     & 20.32                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 24.17                                                     & 20.53                                                & 115.68   & 1.01     & 115.68                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 3.98                                                 & 36.13    & 1.58     & 36.17                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 3.22                                                      & 0.41                                                 & 0.63     & 3.79     & 3.84                                                                \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 8.39                                                      & 7.31                                                 & 44.78    & 0.06     & 44.78                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.17                                                 & 31.75    & 0.60     & 31.76                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 11.07                                                     & 4.20                                                 & 21.11    & 16.61    & 26.87                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 10.43                                                     & 8.27                                                 & 34.93    & 19.35    & 39.93                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.74                                                 & 31.98    & 18.99    & 37.19                                                              
\end{tabular}
\label{t:cork_300}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for stone at a distance separation of 200 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Stone}                                                                                                                                                                                   \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                       & 100                                                       & 18.58                                                     & 15.88                                                & 69.39    & 19.68    & 72.12                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.04                                                      & 0.09                                                 & 11.27    & 0.14     & 11.27                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 4.38                                                      & 3.14                                                 & 32.13    & 1.80     & 32.18                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 22.60                                                     & 21.77                                                & 104.14   & 4.94     & 104.26                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.19                                                 & 7.31     & 1.27     & 7.42                                                                \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 0.34                                                 & 12.13    & 0.76     & 12.15                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 23.04                                                     & 22.26                                                & 108.09   & 16.04    & 109.27                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.81                                                 & 8.52     & 19.23    & 21.03                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 1.04                                                 & 13.32    & 19.06    & 23.26                                                              
\end{tabular}
\label{t:stone_200}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for stone at a distance separation of 300 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Stone}                                                                                                                                                                                   \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                       & 100                                                       & 5.20                                                      & 3.53                                                 & 22.59    & 10.35    & 24.85                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 24.17                                                     & 20.70                                                & 115.50   & 1.02     & 115.50                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.06                                                 & 36.28    & 1.41     & 36.31                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 4.52                                                      & 3.82                                                 & 19.76    & 3.18     & 20.01                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 8.40                                                      & 7.44                                                 & 44.79    & 0.22     & 44.79                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.17                                                 & 31.30    & 0.55     & 31.30                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 3.88                                                      & 3.80                                                 & 19.99    & 17.07    & 26.28                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 5.90                                                      & 4.25                                                 & 15.72    & 19.22    & 24.83                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.86                                                 & 32.27    & 19.01    & 37.45                                                              
\end{tabular}
\label{t:stone_300}
\end{table}


\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for asphalt at a distance separation of 200 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Asphalt}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 23.71                                                     & 25.34                                                & 120.38   & 13.24    & 121.10                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.07                                                 & 12.34    & 0.55     & 12.35                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 11.15                                                     & 8.97                                                 & 60.69    & 1.90     & 60.72                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 23.71                                                     & 26.65                                                & 132.97   & 0.72     & 132.97                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.13                                                 & 7.36     & 1.19     & 7.45                                                                \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 0.49                                                 & 12.84    & 0.74     & 12.86                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 23.71                                                     & 24.12                                                & 120.40   & 16.39    & 121.51                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                      & 0.69                                                 & 8.21     & 19.11    & 20.80                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                      & 1.03                                                 & 12.83    & 19.00    & 22.93                                                              
\end{tabular}
\label{t:asphalt_200}
\end{table}             

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - errors for asphalt at a distance separation of 300 $mm$.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Asphalt}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Distance\\ Separation\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 4.74                                                      & 3.09                                                 & 22.92    & 11.12    & 25.48                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 24.17                                                     & 20.55                                                & 115.17   & 1.10     & 115.17                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 3.89                                                 & 35.58    & 1.50     & 35.61                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 40.58                                                     & 27.55                                                & 138.34   & 7.98     & 138.57                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 6.02                                                      & 5.42                                                 & 19.34    & 0.42     & 19.35                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.11                                                 & 31.21    & 0.63     & 31.22                                                               \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 64.52                                                     & 51.30                                                & 257.22   & 13.86    & 257.59                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 13.24                                                     & 11.38                                                & 51.15    & 18.99    & 54.56                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                      & 4.67                                                 & 31.05    & 18.97    & 36.39                                                              
\end{tabular}
\label{t:asphalt_300}
\end{table}


\FloatBarrier
\subsection{Oscillatory Movement}
\label{simple_osc}
For the oscillatory movement the velocity measurements were made at the initial distance of 200 $mm$, increasing to 225 $mm$, back to 200 $mm$ and then decreasing to 175 $mm$. At the middle the robot did a small pause. 

The following figures show the velocity plots for each material. Figure \ref{f:osc_concrete} shows the velocity plot for concrete. An error measuring the velocity can be seen, caused by the changes in magnification. Figure \ref{f:osc_cork} shows the result for cork and it is similar to the previous one. Figure \ref{f:osc_stone} shows the result for stone, here the error was higher due to the high fluctuation of the values calculated. Stone is a material almost texture-less and the changes in magnification cause blur. This blur induces errors in the velocity estimation. The other materials are less sensitive to the blur as they have bigger grains on its texture.

Table \ref{t:osc} shows the errors for these experiments. The average and maximum error were calculated for velocity and the error for total distance traveled was also calculated.


\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/oscil/concrete.eps}
\caption{Oscillatory movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$ - concrete}
\label{f:osc_concrete}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/oscil/cork.eps}
\caption{Oscillatory movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$ - Cork.}
\label{f:osc_cork}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/oscil/stone.eps}
\caption{Oscillatory movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$ - stone}
\label{f:osc_stone}
\end{figure}


% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Error Results for oscillatory movement.}
\begin{tabular}{l|l|l|l}
         & \begin{tabular}[c]{@{}l@{}}Velocity\\ ($mm/s$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. Velocity\\ Error ($\%$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Total Distance\\ Error ($\%$)\end{tabular} \\ \hline
Concrete & \multirow{4}{*}{500}                                        & 2.42                                                                 & 0.32                                                                  \\
Cork     &                                                             & 2.16                                                                 & 0.61                                                                  \\
Stone    &                                                             & 12.84                                                                & 8.78                                                                  \\
Asphalt  &                                                             & 2.53                                                                 & 0.33                                                                  \\ \hline
Concrete & \multirow{4}{*}{1000}                                       & 2.43                                                                 & 0.33                                                                  \\
Cork     &                                                             & 2.16                                                                 & 0.61                                                                  \\
Stone    &                                                             & 12.22                                                                & 8.85                                                                  \\
Asphalt  &                                                             & 2.61                                                                 & 0.40                                                                 
\end{tabular}
\label{t:osc}
\end{table}

\FloatBarrier
\subsection{Results Discussion}
\label{simple_resul}
While carrying ou the experiments the sensor showed an overall good result, with measurements similar to the ones made on the robot. Although, there are several conditions in which the precision is affected resulting in increased error. \\

In all the experiments made at 100 $mm/s$ the error was very high. This is due to the fact that the implemented algorithm only computes pixel displacements that are integers. In this case the real displacement was 0.04 $mm$. At the distance of 200 $mm$, the displacement calculated was 2 pixels and that is equivalent to 0.04948 $mm$. 

In all the experiments the pixel displacement calculated was always 2 or a multiple of 2, this can be seen in the example in Figure \ref{f:pixel} were there is a plot of the pixel displacement correspondent to a movement at 1000 $mm/s$. One possible explanation for this event might be explained with the taping method of the camera. The camera used in the experiments features a quad tap method to increase the acquisition speed. This method, as described in section \ref{section:image_sensors}, divides the linear sensor in front/even, front/odd, rear/even and rear/odd. The camera software recorded the images in half resolution and it is possible that the software divided the full image and stored the odd section as the first line and the even section as the second line, and so on. However, this could not be proved, because when the problem arose, the camera was no longer available to make a more detailed investigation.

\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{./Imagens/pixel.eps}
\caption{Parallel movement - Velocity plot - 100 $mm/s$ at distance of 200 $mm$ with dynamic sub-sampling.}
\label{f:pixel}
\end{figure}

It is also possible to see that successive frames captured can be so similar that the algorithm fails to compute the correct displacement. This fact suggests that the measuring system should have a variable acquisition rate or the algorithm should adapt to the current velocity. Overall, the results were better at the distance of 200 $mm$. At 300 $mm$ the errors were higher and the only acceptable results were at the velocity of 1000 $mm/s$. This leads to the conclusion that increasing the field of view reduces the ability to measure small velocities.

The results for the diagonal movements show that having the camera sensor with an angle of up to 10$^{\circ}$ does not have an effect on the velocity measured. At the distance of 200 $mm$ the errors were the same as the parallel movement. At 300 $mm$ the errors were higher and the only, as previously, and the only matching with the parallel movement was at the velocity of 1000 $mm/s$. This leads to the same conclusion that increasing the field of view reduces the ability to measure small velocities. These results are important as they prove that the vehicle can have sideways movements and the sensor still being able to measure the velocity accurately. Also, these results show that the camera sensor will operate properly even if it is poorly installed.

The circular movement showed similar results to the previous ones. It is clear that a line scan sensor can measure longitudinal velocity even when the movement is not linear. A automobile has a typical turning diameter of about 10 $m$ and here was shown that the sensor can perform well to diameters as low as 2 $m$. The trajectories estimated were close to the ground truth, except when the longitudinal velocity is poorly estimated. The error in the trajectory is mostly due to the longitudinal velocity deviations and not due to gyroscopes deviations.

For the oscillatory movement it was shown that measuring the velocity with a small change on the optics magnification is possible, although with higher errors than the previous situations. The measurements showed some fluctuations but on average the error was small and the total distance computed had a very small error, except for stone were the average error was 12.84 $\%$. The surface texture is an important variable in this type of movement, as a texture-less surface like stone can induce higher errors. It might be possible to solve this problem with telecentric lenses, which have a magnification invariable with the target distance.

Looking at the tables of results it is possible to see that the value of the average error value was the same in many different situations. A value that appeared often was 1.03$\%$ for the target velocities of 500 and 1000 $mm/s$. This is due to the low resolution. At 500 $mm/s$ the value of displacement calculated was 8. At 1000 $mm/s$ the displacement calculated was exactly the double the previous, 16 pixels, and velocity is the double so the error is exactly the same.

\FloatBarrier
\section{Results with Dynamic Sub-Sampling}
\label{section:dynamic}
After the previous results it is clear that this method has high error when dealing with low velocities; to improve the measurements, the resolution of the sensor would need to be higher. Another alternative to improve the precision is to reject the lines that are close to each other. In this section is proposed an improvement to the algorithm to increase the accuracy at low velocities.

This algorithm consists in using one reference line, then, each line captured is correlated to the reference until a minimum displacement is found. After the minimum displacement is found the current line becomes the new reference. The process is illustrated in Figure \ref{f:dyna_algo}. The distance can be calculated with the algorithm for the similarity coefficients or cross-correlation.
 
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/algorithm.pdf}
\caption{Illustration of the algorithm for dynamic sub-sampling. Where $n$ represents the line number and $d\_min$ represents the minimum displacement.}
\label{f:dyna_algo}
\end{figure}

This will reduce the sampling rate because the intermediate displacements are discarded but will improve the ability to measure low velocities. This was considered to be Dynamic Sub-Sampling.

This algorithm was applied to all the experiments, as in the previous section, and the results are shown in the following subsections. The minimum displacement required was 16 pixels.

\subsection{Parallel Movement}
\label{dyna_para}
the velocity estimated was highly similar for all of the materials. Figures \ref{f:para_200_100mms_d}, \ref{f:para_200_500mms_d}, \ref{f:para_200_1000mms_d}, \ref{f:para_300_100mms_d}, \ref{f:para_300_500mms_d}, \ref{f:para_300_1000mms_d} show the plots for the velocity calculated with the data collected from camera. Each figure shows, also, the velocity plot from the robot (ground truth). The measurements presented were made on concrete and it should be emphasized that the plots for the other materials are similar. \\

Table \ref{t:para_200_d} show the velocity errors for all the movements with all the materials. For the velocity it was calculated the average error using the robot as reference. It was also calculated the error for the total distance traveled. The results show that the tendency of the error is the same to all the materials.\\

With sub-sampling the big offsets are no longer present, as expected, and the velocity plots are now more similar to the ground truth plots.
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/200_100.eps}
\caption{Parallel movement - Velocity plot - 100 $mm/s$ at distance of 200 $mm$ with dynamic sub-sampling.}
\label{f:para_200_100mms_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/200_500.eps}
\caption{Parallel movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$ with dynamic sub-sampling.}
\label{f:para_200_500mms_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/200_1000.eps}
\caption{Parallel movement - Velocity plot - 1000 $mm/s$ at distance of 200 $mm$ with dynamic sub-sampling.}
\label{f:para_200_1000mms_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/300_100.eps}
\caption{Parallel movement - Velocity plot - 100 $mm/s$ at distance of 300 $mm$ with dynamic sub-sampling.}
\label{f:para_300_100mms_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/300_500.eps}
\caption{Parallel movement - Velocity plot - 500 $mm/s$ at distance of 300 $mm$ with dynamic sub-sampling.}
\label{f:para_300_500mms_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/parallel_dyna/300_1000.eps}
\caption{Parallel movement - Velocity plot - 1000 $mm/s$ at distance of 300 $mm$ with dynamic sub-sampling.}
\label{f:para_300_1000mms_d}
\end{figure}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Parallel movement - Summary of error results with dynamic sub-sampling.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                             &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ (mm)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{200}}                                    & 100                                                       & 1.03                                                                           & 0.95                                                 & 1.03                                                                           & 0.95                                                 & 0.96                                                                           & 0.95                                                 & 1.03                                                                           & 0.95                                                 \\
\multicolumn{1}{l|}{}                                                        & 500                                                       & 1.03                                                                           & 0.99                                                 & 1.03                                                                           & 0.96                                                 & 1.03                                                                           & 0.98                                                 & 1.03                                                                           & 0.99                                                 \\
\multicolumn{1}{l|}{}                                                        & 1000                                                      & 1.03                                                                           & 2.41                                                 & 1.03                                                                           & 2.30                                                 & 1.03                                                                           & 2.35                                                 & 1.03                                                                           & 2.23                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{300}}                                    & 100                                                       & 1.09                                                                           & 1.07                                                 & 1.03                                                                           & 0.94                                                 & 0.99                                                                           & 0.96                                                 & 1.07                                                                           & 1.03                                                 \\
\multicolumn{1}{l|}{}                                                        & 500                                                       & 1.10                                                                           & 0.69                                                 & 1.11                                                                           & 0.56                                                 & 1.11                                                                           & 0.70                                                 & 1.10                                                                           & 0.70                                                 \\
\multicolumn{1}{l|}{}                                                        & 1000                                                      & 5.21                                                                           & 4.93                                                 & 5.21                                                                           & 5.03                                                 & 5.21                                                                           & 4.94                                                 & 5.21                                                                           & 4.92                                                 \\ \hline
\end{tabular}
\label{t:para_200_d}
\end{table}

\subsection{Diagonal Movement}
\label{dyna_diag}
For the diagonal movement the velocity plots were highly similar to the parallel one. Again, with sub-sampling there no more big offsets on the measurements.

Table \ref{t:diagonal_200_d} shows the average velocity and total distance relative error for all the movements with all the materials at the distance of 200 $mm$. Table \ref{t:diagonal_300} shows the results for a distance of 300 $mm$. It was calculated the average error and the distance using the robot as reference.

Here it is possible to see that the tendency of the error is the same as for the parallel movement. In fact, the results were similar in the two cases, having even equal results in most of the movements. As seen in the previous section,the method continues robust even when the image sensor has a rotation relative to the line of movement.


% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Diagonal movement - errors for a distance separation of 200 $mm$ with dynamic sub-sampling.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                                  &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Angle\\ ($^{\circ}$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{2.5}}                                         & 100                                                       & 1.02                                                                           & 0.79                                                 & 1.02                                                                           & 0.79                                                 & 0.96                                                                           & 0.71                                                 & 1.02                                                                           & 0.79                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.03                                                                           & 0.93                                                 & 1.03                                                                           & 0.21                                                 & 1.03                                                                           & 0.30                                                 & 1.03                                                                           & 0.62                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 1.03                                                                           & 1.70                                                 & 1.03                                                                           & 1.59                                                 & 1.03                                                                           & 1.86                                                 & 1.03                                                                           & 1.42                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{5}}                                           & 100                                                       & 1.02                                                                           & 0.08                                                 & 1.03                                                                           & 0.56                                                 & 0.92                                                                           & 0.40                                                 & 1.02                                                                           & 0.13                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.03                                                                           & 0.29                                                 & 1.03                                                                           & 0.18                                                 & 1.03                                                                           & 0.28                                                 & 1.03                                                                           & 0.33                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 1.03                                                                           & 1.12                                                 & 1.03                                                                           & 1.63                                                 & 1.03                                                                           & 1.74                                                 & 1.03                                                                           & 1.24                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{10}}                                          & 100                                                       & 0.97                                                                           & 3.83                                                 & 0.37                                                                           & 1.19                                                 & 0.81                                                                           & 0.32                                                 & 0.86                                                                           & 1.10                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 0.99                                                                           & 0.46                                                 & 1.00                                                                           & 0.44                                                 & 1.01                                                                           & 0.63                                                 & 1.00                                                                           & 0.75                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 1.07                                                                           & 2.01                                                 & 1.07                                                                           & 1.34                                                 & 1.03                                                                           & 1.07                                                 & 1.07                                                                           & 1.95                                                
\end{tabular}
\label{t:diagonal_200_d}
\end{table}

% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Diagonal movement - errors for a distance separation of 300 $mm$ with dynamic sub-sampling.}
\begin{tabular}{ll|ll|ll|ll|ll}
                                                                                  &                                                           & \multicolumn{2}{l|}{Concrete}                                                                                                         & \multicolumn{2}{l}{Cork}                                                                                                              & \multicolumn{2}{l|}{Stone}                                                                                                            & \multicolumn{2}{l}{Asphalt}                                                                                                           \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Angle\\ ($^{\circ}$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{2.5}}                                         & 100                                                       & 1.09                                                                           & 1.19                                                 & 1.01                                                                           & 1.07                                                 & 1.09                                                                           & 1.02                                                & 1.09                                                                           & 1.12                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 1.10                                                                           & 1.40                                                 & 1.11                                                                           & 1.14                                                 & 1.11                                                                           & 1.31                                                 & 1.10                                                                           & 1.39                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 5.21                                                                           & 4.20                                                 & 5.21                                                                           & 4.29                                                 & 5.21                                                                           & 4.21                                                 & 5.21                                                                           & 4.23                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{5}}                                           & 100                                                       & 0.64                                                                           & 1.43                                                 & 0.97                                                                           & 1.31                                                 & 1.10                                                                           & 1.55                                                 & 0.86                                                                           & 1.21                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 0.16                                                                           & 0.69                                                 & 0.93                                                                           & 1.02                                                 & 1.10                                                                           & 1.41                                                 & 0.65                                                                           & 0.70                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 5.21                                                                           & 3.64                                                 & 5.21                                                                           & 4.23                                                 & 5.21                                                                           & 4.20                                                 & 5.21                                                                           & 3.72                                                 \\ \hline
\multicolumn{1}{l|}{\multirow{3}{*}{10}}                                          & 100                                                       & 0.28                                                                           & 1.19                                                 & 0.40                                                                           & 1.07                                                 & 0.94                                                                           & 1.91                                                 & 0.31                                                                           & 1.02                                                 \\
\multicolumn{1}{l|}{}                                                             & 500                                                       & 0.97                                                                           & 0.10                                                 & 1.23                                                                           & 0.44                                                 & 0.33                                                                           & 1.23                                                 & 0.99                                                                           & 0.24                                                 \\
\multicolumn{1}{l|}{}                                                             & 1000                                                      & 5.26                                                                           & 3.76                                                 & 5.21                                                                           & 3.55                                                 & 5.21                                                                           & 3.58                                                 & 5.21                                                                           & 3.40                                                
\end{tabular}
\label{t:diagonal_300_d}
\end{table}


\FloatBarrier
\subsection{Circular Movement}
\label{dyna_circ}
Figures \ref{f:200_100_2_d}, \ref{f:200_500_5_d}, \ref{f:200_1000_10_d}, \ref{f:300_100_5_d}, \ref{f:300_500_10_d}, \ref{f:300_1000_2_d} show the generic velocity plots for the circular movement at the various velocities and distances. It shows also the trajectory made by the robot versus the one estimated by the camera and gyroscopes.

Tables \ref{t:concrete_200_d}, \ref{t:concrete_300_d}, \ref{t:cork_200_d}, \ref{t:cork_300_d}, \ref{t:stone_200_d}, \ref{t:stone_300_d}, \ref{t:asphalt_200_d}, \ref{t:asphalt_300_d} show the error and final position deviation for all the experiments. The tables show the average velocity error and total distance error, they also show the coordinates deviation and the euclidean distance between the final position estimated and the ground truth.

An improvement using dynamic sub-sampling is clear, mostly for the plots at 100 $mm/s$ because of the attenuation of the offset seen in the previous section.

\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_100_vel.eps}}
\quad
\subfloat[]{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_100_2.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s - diameter of 2 m - at distance 200 mm with dynamic sub-sampling.}
\label{f:200_100_2_d}
\end{figure}


\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_500_5.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s - diameter of 5 m - at distance 200 mm with dynamic sub-sampling.}
\label{f:200_500_5_d}
\end{figure}


\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/200_1000_10.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s - diameter of 10 m - at distance 200 mm with dynamic sub-sampling.}
\label{f:200_1000_10_d}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_100_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_100_5.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s - diameter of 5 m - at distance 300 mm with dynamic sub-sampling.}
\label{f:300_100_5_d}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_500_10.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s - diameter of 10 m - at distance 300 mm with dynamic sub-sampling.}
\label{f:300_500_10_d}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=6cm]{./Imagens/circular_dyna/300_1000_2.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s - diameter of 2 m - at distance 300 mm with dynamic sub-sampling.}
\label{f:300_1000_2_d}
\end{figure}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for concrete at separation distance of 200 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Concrete}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 6.13                                                    & 5.31                                               & 35.56  & 7.97   & 36.45                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 13.71                                                   & 12.58                                              & 66.02  & 1.90   & 66.05                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 27.63                                                   & 29.53                                               & 156.35 & 0.53  & 156.35                                                            \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 1.03                                                    & 2.57                                               & 15.10  & 1.13   & 15.14                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 0.71                                              & 7.20   & 1.50   & 7.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.28                                               & 17.86  & 1.22   & 17.90                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.03                                                    & 1.03                                               & 5.69   & 2.18   & 6.09                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 1.25                                               & 8.34   & 0.55  & 8.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.68                                               & 17.70  & 0.61  & 17.71                                                             
\end{tabular}
\label{t:concrete_200_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for concrete at separation distance of 300 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Concrete}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 5.21                                                    & 4.27                                                & 27.13  & 7.19   & 28.07                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 5.43                                                    & 4.41                                               & 30.81   & 0.68  & 30.81                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.38                                                    & 4.53                                               & 33.87  & 0.23  & 33.87                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 0.55                                                    & 0.58                                              & 0.62  & 2.28   & 2.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 0.41                                                   & 0.31                                              & 1.61    & 1.24   & 2.03                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.52                                               & 28.22  & 1.02   & 28.24                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.11                                                    & 1.07                                               & 4.37   & 2.31   & 4.95                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.07                                                    & 0.61                                              & 1.03   & 0.73  & 1.26                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.93                                               & 28.32  & 0.72  & 28.33                                                            
\end{tabular}
\label{t:concrete_300_d}
\end{table}

\FloatBarrier

\begin{table}[h]
\centering
\caption{Circular movement - Errors for cork at separation distance of 200 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Cork}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 4.94                                                    & 4.12                                               & 30.11  & 8.76   & 31.36                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 11.65                                                   & 11.33                                              & 57.03   & 1.89   & 57.06                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 23.50                                                   & 27.19                                              & 146.83 & 2.67   & 146.85                                                            \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 1.03                                                    & 0.71                                              & 5.01   & 1.96   & 5.37                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 0.61                                              & 7.29   & 1.70    & 7.48                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.03                                               & 17.33  & 1.44   & 17.39                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.03                                                    & 1.03                                               & 5.79   & 2.17   & 6.19                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 1.20                                               & 9.07   & 0.50  & 9.09                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.64                                               & 17.77  & 0.52  & 17.77                                                            
\end{tabular}
\label{t:cork_200_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for cork at separation distance of 300 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Cork}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 5.32                                                    & 4.27                                                & 27.09  & 5.99   & 27.74                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 8.63                                                    & 7.58                                               & 46.48  & 0.27  & 46.48                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 8.71                                                    & 7.87                                               & 50.38  & 0.27  & 50.38                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 0.70                                                   & 0.51                                              & 5.33   & 2.29   & 5.80                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.64                                                     & 2.02                                               & 13.40  & 1.22   & 13.45                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.51                                               & 28.92   & 1.16   & 28.95                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 0.62                                                   & 0.45                                              & 0.92  & 2.29   & 2.46                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 0.91                                                   & 0.03                                             & 1.57   & 0.51  & 1.65                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 5.18                                               & 30.41  & 0.60    & 30.42                                                            
\end{tabular}
\label{t:cork_300_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for stone at separation distance of 200 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Stone}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 2.60                                                    & 1.66                                               & 17.34  & 8.34   & 19.24                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 5.46                                                     & 5.47                                               & 24.70  & 2.79   & 24.86                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 10.86                                                   & 15.76                                              & 94.32  & 5.01    & 94.45                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 1.03                                                    & 0.71                                              & 4.96   & 1.96   & 5.33                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 0.52                                              & 6.65   & 1.65   & 6.85                                                               \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 1.92                                               & 16.48  & 1.43   & 16.54                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.03                                                    & 1.030                                              & 5.70   & 2.18   & 6.11                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 1.13                                               & 7.81   & 0.41  & 7.82                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.70                                               & 17.56  & 0.47  & 17.57                                                            
\end{tabular}
\label{t:stone_200_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for stone at separation distance of 300 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Stone}                                                                                                                                                                                    \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 5.26                                                    & 4.27                                                & 27.19  & 7.12   & 28.11                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 8.20                                                   & 7.12                                               & 44.37  & 0.45  & 44.37                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 7.87                                                    & 7.13                                               & 46.80  & 0.16  & 46.80                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 0.77                                                   & 0.63                                              & 5.91   & 2.34   & 6.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 2.30                                                    & 2.33                                              & 14.94   & 1.20   & 14.99                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.54                                               & 28.98  & 1.10   & 29.00                                                              \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 0.65                                                   & 0.45                                              & 0.85  & 2.29   & 2.44                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 0.89                                                   & 0.01                                             & 1.32  & 0.51  & 1.42                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 5.18                                               & 30.15  & 0.60  & 30.16                                                            
\end{tabular}
\label{t:stone_300_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for asphalt at separation distance of 200 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Asphalt}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{200}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 6.13                                                    & 5.30                                               & 35.56  & 7.97   & 36.44                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 13.70                                                   & 12.58                                              & 66.01  & 1.89   & 66.04                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 27.62                                                   & 29.53                                               & 156.34 & 0.52  & 156.34                                                            \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 1.03                                                    & 2.57                                               & 15.10  & 1.13   & 15.14                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 0.71                                              & 7.20   & 1.49   & 7.35                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.27                                               & 17.85  & 1.22   & 17.89                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.030                                                   & 1.03                                               & 5.68   & 2.18  & 6.09                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.03                                                    & 1.24                                               & 8.34   & 0.55  & 8.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 1.03                                                    & 2.68                                               & 17.69  & 0.60  & 17.70                                                             
\end{tabular}
\label{t:asphalt_200_d}
\end{table}

\FloatBarrier
\begin{table}[h]
\centering
\caption{Circular movement - Errors for asphalt at separation distance of 300 $mm$ using dynamic sub-sampling.}
\begin{tabular}{lll|ll|lll|}
                                                                               &                                                                                &                                                           & \multicolumn{5}{l|}{Asphalt}                                                                                                                                                                                \\ \cline{4-8} 
                                                                               &                                                                                &                                                           & \multicolumn{2}{l|}{Error}                                                                                       & \multicolumn{3}{l|}{Deviation}                                                            \\ \hline
\multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Separation\\ Distance\\ ($mm$)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Diameter\\ ($mm$)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Target\\ Velocity\\ (mm/s)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. \\ Vel.\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Dist.\\ Travel.\\ (\%)\end{tabular} & X ($mm$) & Y ($mm$) & \begin{tabular}[c]{@{}l@{}}Euclidean\\ Distance\\ ($mm$)\end{tabular} \\ \hline
\multicolumn{1}{l|}{\multirow{9}{*}{300}}                                      & \multicolumn{1}{l|}{\multirow{3}{*}{2}}                                        & 100                                                       & 5.21                                                    & 4.27                                                & 27.13  & 7.19   & 28.06                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 5.42                                                    & 4.40                                               & 30.80   & 0.67  & 30.81                                                             \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.38                                                    & 4.53                                               & 33.86  & 0.23  & 33.86                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{5}}                                        & 100                                                       & 0.55                                                    & 0.58                                              & 0.61  & 2.28   & 2.36                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 0.40                                                   & 0.30                                              & 1.61    & 1.23   & 2.03                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.51                                               & 28.22  & 1.02   & 28.24                                                             \\ \cline{2-8} 
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{\multirow{3}{*}{10}}                                       & 100                                                       & 1.10                                                    & 1.06                                               & 4.37   & 2.30   & 4.94                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 500                                                       & 1.06                                                    & 0.61                                              & 1.02   & 0.73  & 1.26                                                              \\
\multicolumn{1}{l|}{}                                                          & \multicolumn{1}{l|}{}                                                          & 1000                                                      & 5.21                                                    & 4.92                                               & 28.32  & 0.72  & 28.33                                                            
\end{tabular}
\label{t:asphalt_300_d}
\end{table}



\FloatBarrier
\subsection{Oscillatory Movement}
\label{dyna_osc}
The following figures show the velocity plots for each material. Figure \ref{f:osc_concrete_d} shows the velocity plot for concrete, Figure \ref{f:osc_cork_d} shows the result for cork and Figure \ref{f:osc_stone_d} shows the result for stone.

Table \ref{t:osc_d} shows the errors for these experiments. The average and maximum error were calculated for velocity and the error for total distance traveled was also calculated.

% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Error Results for oscillatory movement using dynamic sub-sampling.}
\begin{tabular}{l|l|l|l}
         & \begin{tabular}[c]{@{}l@{}}Velocity\\ ($mm/s$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. Velocity\\ Error ($\%$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Total Distance\\ Error ($\%$)\end{tabular} \\ \hline
Concrete & \multirow{4}{*}{500}                                        & 2.65                                                                 & 1.26                                                                  \\
Cork     &                                                             & 4.96                                                                 & 2.77                                                                  \\
Stone    &                                                             & 5.69                                                                 & 3.70                                                                  \\
Asphalt  &                                                             & 2.55                                                                 & 1.16                                                                  \\ \hline
Concrete & \multirow{4}{*}{1000}                                       & 2.45                                                                 & 1.36                                                                  \\
Cork     &                                                             & 4.91                                                                 & 2.87                                                                  \\
Stone    &                                                             & 5.53                                                                 & 3.10                                                                  \\
Asphalt  &                                                             & 2.65                                                                 & 1.26                                                                 
\end{tabular}
\label{t:osc_d}
\end{table}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/oscil_dyna/concrete.eps}
\caption{Oscillatory movement - Velocity plot - 500 $mm/s$ at distance of 200 $mm$ - concrete with dynamic sub-sampling.}
\label{f:osc_concrete_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/oscil_dyna/cork.eps}
\caption{Oscillatory movement - Velocity plot - 100 $mm/s$ at distance of 200 $mm$ - Cork with dynamic sub-sampling.}
\label{f:osc_cork_d}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/oscil_dyna/stone.eps}
\caption{Oscillatory movement - Velocity plot - 100 $mm/s$ at distance of 200 $mm$ - stone with dynamic sub-sampling.}
\label{f:osc_stone_d}
\end{figure}

\FloatBarrier
\subsection{Results Discussion}
\label{dyna_resul}
On the parallel movement the use of dynamic sub-sampling improved the results mostly on the experiments at 100 $mm/s$. On the other situations the average error was the same or had small variations. The velocity plots now have an acceleration phase that fits better with the ones from the ground truth opposed to the degree-like acceleration seen on the previous section.

The results were consistent on the diagonal movements and the same tendency seen previously occurred here too. The error results are very similar to the ones on the parallel movement showing that the angle of the image sensor had approximately none impact.

For circular movements the improvements made by this algorithm were similar to the previous situations. For 100 $mm/s$ the error was significantly reduced, in some cases from around 23$\%$ to around 5$\%$. Although with a diameter of 2 $m$ the estimates for 500 and 1000 $mm/s$ had an increased error. This is due to the fact that 2 meters of diameter is a very tight curvature and that produces high rotation between the successive line scans and thus a weaker similarity; then this algorithm reduces the sampling rate by discarding some line scans and the correlation is made with more distant lines wish have even weaker similarity. The similarity between lines is, of course, an essential parameter and as the similarity decreases the error increases. Overall, for the other cases, the errors maintained very similar to results with the previous algorithm. For tighter curvatures the correlation only works with good results if sub-sampling is not used. For this reason the algorithm may dynamically changes its minimum distance of pixel displacement based on the data from the gyroscopes, to not compromise the sampling rate. Basically as the curvature increases this minimum needs to be decreased. This solution would change its level of sub-sampling to better suit the needs.

For the oscillatory movement the results were similar to the ones using the previous algorithm. Only there was an improvement for the experiments with stone. In this case, with the previous algorithm and as stated in Figure \ref{f:osc_stone}, the velocity measured was in some cases zero; with the sub-sampling it was imposed a minimum displacement to be read and that eliminated those bad estimates.

Looking at the tables of results it is possible to see that the value of the average error value was the same in many different situations. A value that appeared often was 1.03$\%$ for the target velocities of 500 and 1000 $mm/s$. This is due to the low resolution. At 500 $mm/s$ the value of displacement calculated was 8. At 1000 $mm/s$ the displacement calculated was exactly the double the previous, 16 pixels, and velocity is the double so the error is exactly the same.

This algorithm using sub-sampling reveals clear improvements when dealing with low velocities without compromise for higher velocities. Although for it to operate with guarantees it should adjust its "level" of sub-sampling based on how much the vehicle is turning.

\FloatBarrier
\section{Parallel Movement With Gyroscopes}
It is important to study the influence of the gyroscopes readings and its deviation to the movement estimate when moving straight forward. In this case, due to the gyroscope's drift, an error of orientation will appear. To study this effect it was used the previous velocity plots from the parallel movement together with the gyroscopes readings. The results are shown in Figures \ref{f:gyros_200_100}, \ref{f:gyros_200_500}, \ref{f:gyros_200_1000}, \ref{f:gyros_300_100}, \ref{f:gyros_300_500}, \ref{f:gyros_300_1000}. Table \ref{t:gyros_results} shows the final positions of each simulation using the longitudinal velocity estimated by the camera and the gyroscope's readings. The reference was calculated using the same longitudinal velocity estimations but with a reference angular velocity of 0 degrees per second.

On table \ref{t:gyros_results} it is possible to see that the deviation is small and the higher the longitudinal velocity error the higher the final position deviation. Including the gyroscopes data in this simulation had very little impact on the movement estimation. This proves that even when the movement is a straight line, the gyroscope's drift does not degrade the position estimates. However, these deviations can be significant as the distance increases, then higher precision, low drift gyroscopes may be used.

In Figure \ref{f:gyros_300_100} it is possible to see that the velocity plot has high oscillations. In this case the displacements calculated were 0 or 2 pixels, equivalent to a velocity of 0 or 189.5735 $mm/s$. This happened because the successive lines were so similar and the resolution was low causing the correlation to fail in some cases.

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_100_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_100.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s at distance 200 mm.}
\label{f:gyros_200_100}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_500.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s at distance 200 mm.}
\label{f:gyros_200_500}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/200_1000.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s at distance 200 mm.}
\label{f:gyros_200_1000}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_100_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_100.eps}}
\caption{Velocity and position plot for velocity of 100 mm/s at distance 300 mm.}
\label{f:gyros_300_100}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_500_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_500.eps}}
\caption{Velocity and position plot for velocity of 500 mm/s at distance 300 mm.}
\label{f:gyros_300_500}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_1000_vel.eps}}
\quad
\subfloat{\includegraphics[width=7cm]{./Imagens/with_gyros/300_1000.eps}}
\caption{Velocity and position plot for velocity of 1000 mm/s at distance 300 mm.}
\label{f:gyros_300_1000}
\end{figure}

% Please add the following required packages to your document preamble:
% \usepackage{multirow}
\begin{table}[h]
\centering
\caption{Final positions for parallel movement with gyroscope's data.}

\begin{tabular}{l|l|l|l|l|l}
Distance             & \begin{tabular}[c]{@{}l@{}}Velocity\\ ($mm/s$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}X reference\\ ($mm$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}X estimated\\ ($mm$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Y reference\\  ($mm$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Y estimated\\ ($mm$)\end{tabular} \\ \hline
\multirow{3}{*}{200} & \multirow{3}{*}{500}                                        & 621.00                                                       & 615.78                                                       & 0.00                                                          & 1.22                                                         \\
                     &                                                             & 497.50                                                       & 498.06                                                       & 0.00                                                          & 0.58                                                         \\
                     &                                                             & 496.20                                                       & 498.66                                                       & 0.00                                                          & 0.53                                                         \\ \hline
\multirow{3}{*}{300} & \multirow{3}{*}{1000}                                       & 704.30                                                       & 659.15                                                       & 0.00                                                          & 1.27                                                         \\
                     &                                                             & 558.25                                                       & 555.07                                                       & 0.00                                                          & 0.79                                                         \\
                     &                                                             & 497.66                                                       & 499.56                                                       & 0.00                                                          & 0.47                                                        
\end{tabular}
\label{t:gyros_results}
\end{table}



\FloatBarrier
\section{Correlation with Interpolation}

One limitation of the algorithms used is that they only estimate a pixel displacement that is an integer. For low velocities this resolution might be small and that can induce high errors. A solution that can reduce the error is by doing the correlation with interpolation to estimate displacements that are floats.

The interpolation is a quadratic interpolation of 3 uniformly spaced samples. After applying one of the algorithms there will be a correlation coefficient for each equally spaced shift. This method performs a quadratic interpolation between the maximum correlation coefficient and its neighbors.

In this section there are presented four examples with the application of this improvement. Figure \ref{f:inter_300_500_10}, \ref{f:inter_200_500_0}, \ref{f:inter_300_500_5}, \ref{f:inter_200_100_0} show the velocity and pixel displacement plots of the original algorithm (a) versus the algorithm with interpolation (b).

\begin{table}[h]
\centering
\caption{Error results for Normal Correlation Vs Correlation with Interpolation.}
\begin{tabular}{l|l|l|l|l|}
          & \multicolumn{2}{l|}{Without Interpolation}                                                                                             & \multicolumn{2}{l|}{With Interpolation}                                                                                                \\ \cline{2-5} 
          & \begin{tabular}[c]{@{}l@{}}Avg. Velocity\\ Error ($\%$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Distance\\ Error ($\%$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Avg. Velocity\\ Error ($\%$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Distance\\ Error ($\%$)\end{tabular} \\ \hline
Example 1 & 5.12                                                                 & 5.12                                                            & 4.55                                                                & 4.63                                                           \\
Example 2 & 1.03                                                                 & 0.92                                                            & 0.89                                                                 & 0.88                                                            \\
Example 3 & 6.17                                                                 & 4.48                                                            & 3.20                                                                 & 2.42                                                            \\
Example 4 & 1.14                                                                & 0.95                                                           & 1.13                                                                & 0.94                                                          
\end{tabular}
\label{t:inter_results}
\end{table}


Analyzing table \ref{t:inter_results} it is possible to see that using interpolation approximates the velocity estimates to the ground truth and the average error decreases. Interpolation can then be successfully used to increase the accuracy of the estimates with no compromise in computation speed, since interpolation is a simple operation.

\begin{figure}[htbp]
\centering
\subfloat[Whitout interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/300_1000_0_normal.eps}}
\quad
\subfloat[With interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/300_1000_0_inter.eps}}
\caption{Example 1 - 1000 $mm/s$ at distance of 300 mm.}
\label{f:inter_300_500_10}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat[Without interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/200_500_0_normal.eps}}
\quad
\subfloat[With interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/200_500_0_inter.eps}}
\caption{Example 2 - 500 $mm/s$ at distance of 200 mm.}
\label{f:inter_200_500_0}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat[Without interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/300_500_5_inter.eps}}
\quad
\subfloat[With interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/300_500_5_normal.eps}}
\caption{Example 3 - 500 $mm/s$ at distance of 300 mm.}
\label{f:inter_300_500_5}
\end{figure}

\begin{figure}[htbp]
\centering
\subfloat[Without interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/200_1000_0_normal.eps}}
\quad
\subfloat[With interpolation]{\includegraphics[width=7cm]{./Imagens/interpolations/200_1000_0_inter.eps}}
\caption{Example 4 - 1000 $mm/s$ at distance of 200 mm.}
\label{f:inter_200_100_0}
\end{figure}


\FloatBarrier
\section{Sub-sampling}

Sub-sampling is the process of reducing the sampling rate. When performing a sub-sample of the video data, one can simulate a higher displacement and thus a higher velocity movement. For example, using the 100 $mm/s$ video and doing the computing on intervals of 5 frames, the displacement will be 5 times higher and thus the velocity, also 5 times higher. With sub-sampling it is possible to make a extrapolation of the data to higher velocities.

To prove that using sub-sampling gives the same results as using the full sample some simulations were made. A sample from 100 $mm/s$ was used to simulate a movement with a velocity of 500 and 1000 $mm/s$ the errors were exactly the same as the ones calculated using the full sample. The same was done with a sample form 500 $mm/s$ simulating a movement of 1000 $mm/s$ and the error was also equal. This leads to the conclusion that it is safe to do sub-sampling and extrapolate to higher velocities.

Next, simulations of 5, 10 and 20 $m/s$ are made with the sample taken at 100 $mm/s$ on concrete. The results are shown in Figure \ref{f:5ms}, \ref{f:10ms} and \ref{f:20ms} and the error results are presented on Table \ref{t:subsamp}. These results show that at a sampling rate of 2500 Hz the sensor can measure velocity up to 10 $m/s$ with an average error of 0.2 $\%$ and a maximum error of 5.15 $\%$. At 20 $m/s$ the results begin to show relevant fluctuations and a maximum error of 35.67 $\%$ is reached.

\begin{table}[h]
\centering
\caption{Error results for sub-sampling simulations.}
\begin{tabular}{l|l|l}
Velocity ($m/s$) & Avg. Error (\%) & Max. Error (\%) \\ \hline
5                & 0.2             & 3.918           \\
10               & 0.18            & 5.15            \\
20               & 1.05            & 35.67          
\end{tabular}
\label{t:subsamp}
\end{table}

The results of the sub-sampling show that increasing the displacement increases the global and the average error. At an acquisition frequency of 2500 Hz the measurements are acceptable up to 10 $m/s$, if dealing with higher velocities one should increase the sampling rate. This leads to the conclusion that for car-like velocities it is necessary to increase the sampling rate. For example if one wants to measure a velocity up to 120 $km/h$ or 33.333 $m/s$, it would be necessary a sampling rate of 3.3333 $\times$ 2500 = 8333.33 $Hz$.

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/subsample/5ms.eps}
\caption{Velocity plot at 5 $m/s$ using sub-sampling from sample of 100 $mm/s$.}
\label{f:5ms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/subsample/10ms.eps}
\caption{Velocity plot at 10 $m/s$ using sub-sampling from sample of 100 $mm/s$.}
\label{f:10ms}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{./Imagens/subsample/20ms.eps}
\caption{Velocity plot at 20 $m/s$ using sub-sampling from sample of 100 $mm/s$.}
\label{f:20ms}
\end{figure}


