%
%
\begin{figure*}[t]
  \centering
  \subfigure[Accurate estimation by current state-of-the-art and our framework]{\includegraphics[width=\columnwidth]{graphics/analysis/chair07-accuracy-jun10.pdf}\label{fig:chair-accurate}}
  \subfigure[Failed estimation by current state-of-the-art
  ]{\includegraphics[width=\columnwidth]{graphics/analysis/chair06-accuracy-fail.pdf}\label{fig:chair-fail}}

  \caption{Comparison of $SE(3)$ pose for a chair estimated via
    fiducial markers (Tag), current state-of-the-art (Katz) and our framework
    (Ours). (a) The figures show the strong performance of our framework, as
    compared to marker-based solutions and current
    state-of-the-art algorithms, to robustly track and
    estimate the $SE(3)$ pose of a chair being manipulated on multiple
    occasions. (b) Current state-of-the-art, however, fails to robustly
    estimate the $SE(3)$ pose on certain trials. }
  \label{fig:chair-motion-accuracy-comparison}
\end{figure*}
%
%

\subsubsection{Pose Estimation Accuracy}
%
For each visual demonstration, we compared the segmentation and
$SE(3)$ pose of each object part estimated by our method with those
produced by Katz.  We also obtained pose estimates for each object
part by tracking attached fiducial markers.  Synchronization across
pose observations was ensured by evaluating only poses in the set
intersection of the timestamps of each pose sequence. For each
overlapping time step, we compared the relative pose of the estimated
object segment obtained from both algorithms with that obtained via
fiducial markers (Figure~\ref{fig:pose-estimation-eval}). For
consistency in evaluation, the $SE(3)$ poses of individual object
parts were initialized identically for both algorithms.

Figure~\ref{fig:chair-motion-accuracy-comparison} compares the
absolute $SE(3)$ poses estimated by the three methods described above,
given observations a chair being moved on the ground plane.
Figure~\ref{fig:chair-accurate} illustrates a scenario in which both
algorithms, ours and Katz's, perform reliably.  Katz's method is
within $2.1~$cm and $2.64\,^{\circ}$ of the ground truth pose produced
with fiducial markers.  Our method achieves comparable accuracy of
$1.7~$cm and $2.11\,^{\circ}$.  Using data from another demonstration,
Katz's method failed to track the object motion robustly, resulting in
drift and incorrect motion estimates
(Figure~\ref{fig:chair-fail}). Such failures can be attributed
to: \textit{(i)} the KLT tracker that is known to cause drift 
during feature tracking; \textit{(ii)} SVD least squares minimization
in the relative pose estimation stage, without appropriate outlier
rejection.

For a variety of articulated objects
(Table~\ref{tab:pose-estimation-comparison}), our method achieves
average accuracies of $2.5~$cm and $4.5\,^{\circ}$ with respect to
ground truth estimated from noisy Kinect RGB-D data. In comparison,
Katz's method~\cite{katz2013interactive} achieved average accuracies
of $3.7~$cm and $9.9\,^{\circ}$ for the same objects.  Our method
achieved an error of less than $10~$cm and $25\,^{\circ}$
in 35 of 41 demonstrations, vs.~22 of 41 for Katz.

% Pose estimation eval fig
%
\begin{figure}[!h]
  \centering
    \includegraphics[width=\columnwidth]{graphics/pipeline-figure/pose-estimation-eval.png}\\
  \caption{Pose estimation accuracy of our method, compared 
    to that achieved using fiducial markers.}
  \label{fig:pose-estimation-eval}
\end{figure}

\begin{table*}[t]
  \centering
  \scriptsize
  \begin{minipage}[t]{0.9\columnwidth}
    \centering
    {\setlength{\tabcolsep}{0.4em}
        \begin{tabular}{|l|c|c|c|c|c|c|}
          \hline
          \multirow{3}{*}{\textbf{Dataset}} & \multicolumn{ 3}{c|}{\textbf{Katz
              et al.}} & \multicolumn{ 3}{c|}{\textbf{Ours}} \\ \cline{ 2- 7}
          \multicolumn{ 1}{|c|}{} & \multicolumn{ 2}{c|}{\textbf{Average Error}}
          & 
          \multirow{2}{*}{\parbox{0.75cm}{\centering\textbf{Success Rate}}} & 
          \multicolumn{ 2}{c|}{\textbf{Average Error}} & 
          \multirow{2}{*}{\parbox{0.75cm}{\centering\textbf{Success Rate}}} \\ \cline{ 2- 3}\cline{ 5- 6}
          
          \multicolumn{ 1}{|c|}{} & \textbf{Pos.} & \textbf{Orient.} &
          \multicolumn{ 1}{c|}{} & \textbf{Pos.} & \textbf{Orient.} & \multicolumn{ 1}{c|}{} \\ \hline
          \multicolumn{1}{|l|}{Chair} & 3.6~cm & 13.2$\,^{\circ}$ & 4/6 & \textbf{2.3}~cm & \textbf{4.5}$\,^{\circ}$ & \textbf{5/6} \\ %\hline
          \multicolumn{1}{|l|}{Door} & 6.0~cm & 6.8$\,^{\circ}$ & 6/7 & \textbf{5.0}~cm & \textbf{5.5}$\,^{\circ}$ &  \textbf{7/7} \\ %\hline
          \multicolumn{1}{|l|}{Drawer} & 6.1~cm & 18.0$\,^{\circ}$ & 3/7 & \textbf{3.7}~cm & \textbf{3.0}$\,^{\circ}$ & \textbf{6/7} \\ %\hline
          \multicolumn{1}{|l|}{Fridge} & 2.2~cm & 8.1$\,^{\circ}$ & 4/8 & \textbf{1.0}~cm & \textbf{2.9}$\,^{\circ}$ &  \textbf{6/8}\\ %\hline
          \multicolumn{1}{|l|}{Laptop} & 0.4~cm & 2.3$\,^{\circ}$ & 2/5 & \textbf{0.3}~cm & \textbf{6.4}$\,^{\circ}$ &  \textbf{4/5}\\ %\hline
          \multicolumn{1}{|l|}{Microwave} & 4.3~cm & 14.2$\,^{\circ}$ & 2/4 & \textbf{1.9}~cm & \textbf{6.9}$\,^{\circ}$ & \textbf{4/4} \\ %\hline
          \multicolumn{1}{|l|}{Printer} & - & - & 0/2 & \textbf{0.5}~cm & \textbf{2.3}$\,^{\circ}$ &  \textbf{2/2}\\ %\hline
          \multicolumn{1}{|l|}{Screen} & \textbf{2.6}~cm & 24.9$\,^{\circ}$ & 1/2 & 3.4~cm &
          \textbf{3.5}$\,^{\circ}$ &  \textbf{1/2}\\ \hline
          \multicolumn{1}{|l|}{\textbf{Overall}} & 3.7~cm & 9.9$\,^{\circ}$ & 24/41 & \textbf{2.5}~cm &
          \textbf{4.5}$\,^{\circ}$ &  \textbf{35/41}\\ \hline
        \end{tabular}
      }
\captionof{table}{Comparison of $SE(3)$ pose estimates produced by
  our method and Katz's, with marker-based pose estimates used as
  ground truth. 
}
\label{tab:pose-estimation-comparison}
\end{minipage}\hfill
\begin{minipage}[t]{1.1\columnwidth}
  \centering
  {\setlength{\tabcolsep}{0.4em}
    \centering
    \begin{tabular}{|l|c|c|c|c|c|c|c|c|}
      \hline

          \multirow{3}{*}{\textbf{Dataset}} & \multicolumn{ 4}{c|}{\textbf{Katz
              et al.}} & \multicolumn{ 4}{c|}{\textbf{Ours}} \\ \cline{ 2- 9}
          \multicolumn{1}{|c|}{} & \multicolumn{ 2}{c|}{\textbf{Model Fit Error}} &
          \multirow{2}{*}{\parbox{1.2cm}{
              \centering{\renewcommand{\arraystretch}{1}\textbf{Param. Est. Error}}}} &
          \multirow{2}{*}{\parbox{0.75cm}{\centering\textbf{Success Rate}}} &
          \multicolumn{2}{c|}{\textbf{Model Fit Error}} &
          \multirow{2}{*}{\parbox{1.2cm}{\centering\textbf{Param. Est. Error}}} & 
          \multirow{2}{*}{\parbox{0.75cm}{\centering\textbf{Success Rate}}} \\ \cline{ 2- 3}\cline{ 6- 7}
          \multicolumn{1}{|c|}{} & \textbf{Pos.} & \textbf{Orient.} &
          \multicolumn{ 1}{c|}{} & \multicolumn{ 1}{c|}{} &
          \textbf{Pos.}& 
          \textbf{Orient.}& 
          \multicolumn{1}{c|}{} & \multicolumn{ 1}{c|}{} \\ \hline

          \multicolumn{1}{|l|}{Door} & 1.9~cm & 6.7$\,^{\circ}$ & 1.9$\,^{\circ}$ & 4/7 & \textbf{0.4}~cm & \textbf{4.7}$\,^{\circ}$ & \textbf{1.8}$\,^{\circ}$ & \textbf{5/7} \\ %\hline
          \multicolumn{1}{|l|}{Drawer} & 2.0~cm & 7.3$\,^{\circ}$ & 2.5$\,^{\circ}$ & 2/7 & \textbf{1.7}~cm &
          \textbf{3.1}$\,^{\circ}$ & \textbf{2.0}$\,^{\circ}$ & \textbf{6/7} \\ %\hline
          \multicolumn{1}{|l|}{Fridge} & 0.5~cm & 6.5$\,^{\circ}$ & 5.6$\,^{\circ}$ & 4/8 & \textbf{0.4}~cm &
          \textbf{5.8}$\,^{\circ}$ & \textbf{3.5}$\,^{\circ}$ & \textbf{5/8}\\ %\hline
          \multicolumn{1}{|l|}{Laptop} & - & - & - & 0/5 & \textbf{0.2}~cm &
          \textbf{6.4}$\,^{\circ}$ & \textbf{6.1}$\,^{\circ}$ & \textbf{4/5}\\ %\hline
          \multicolumn{1}{|l|}{Microwave} & 7.0~cm & \textbf{1.2}$\,^{\circ}$ & \textbf{0.2}$\,^{\circ}$ & 2/4 & \textbf{6.5}~cm &
          4.1$\,^{\circ}$ & 0.3$\,^{\circ}$ & \textbf{3/4} \\ %\hline
          \multicolumn{1}{|l|}{Printer} & \textbf{0.9}~cm & 0.8$\,^{\circ}$ & 1.5$\,^{\circ}$ & 1/2 & 2.1~cm &
          \textbf{0.2}$\,^{\circ}$ & \textbf{1.4}$\,^{\circ}$ & \textbf{1/2}\\ 
          \multicolumn{1}{|l|}{Screen} & - & - & - & 0/2 & \textbf{0.9}~cm &
          \textbf{0.7}$\,^{\circ}$ & \textbf{3.2}$\,^{\circ}$ & \textbf{1/2}\\ \hline
          \multicolumn{1}{|l|}{\textbf{Overall}} & 2.1~cm & 5.4$\,^{\circ}$ & 2.8$\,^{\circ}$ & 13/35 & \textbf{1.6}~cm &
          \textbf{4.6}$\,^{\circ}$ & \textbf{2.9}$\,^{\circ}$ & \textbf{25/35}\\ \hline
        \end{tabular}
      }
      \captionof{table}{Comparison of kinematic model and parameter
        estimation performance by our method and Katz's, with
        marker-based model estimation used as ground truth. 
}
      \label{tab:model-estimation-comparison}
    \end{minipage}
\end{table*} 

\subsubsection{Model Estimation Accuracy} Once the $SE(3)$ poses of the object
parts are estimated, we compare the kinematic structure and model
parameters of the articulated object estimated by our method with
those produced by Katz. As in our other experiments, we use the
kinematic structure and model parameters identified from fiducial
marker-based solutions as a
baseline. Table~\ref{tab:model-estimation-comparison} summarizes the
model estimation and parameter estimation performance achieved with
our method and Katz's. The model fit error is defined as the average
spatial and orientation error between the $SE(3)$ observations and the
estimated articulation manifold (i.e.~prismatic or rotational
manifold). For the dataset of articulated objects evaluated
(Table~\ref{tab:model-estimation-comparison}), our method achieved an
average model fit error of $1.6~$cm spatially, and $4.6\,^{\circ}$ in
orientation, an improvement over Katz's method (average model fit
errors of $2.1~$cm and $5.4\,^{\circ}$ respectively).  Of 35
demonstrations evaluated, our method determined the correct kinematic
structure and accurate parameters in 25 cases, whereas Katz did so
in only 13 cases.
% XXX Katz's method incorrectly recovered a prismatic
% link for the laptop (perhaps due to its small range of motion), and
% performed poorly for the projector screen (perhaps due to significant
% sensor noise).

We also compared the model parameters estimated by our method and
Katz's method with ground truth from markers, by transforming poses
estimated by both methods into the fiducial marker's reference frame
based on the initial configuration of the articulated object. This
allows us to directly compare model parameters estimated through our
proposed framework, the current state-of-the-art and marker-based
solutions. In each demonstration, the model parameters estimated via
our method are closer to the marker-based solution than those obtained
by Katz.

% In general,
% lower model fit errors imply a more accurate kinematic model. 

% For most of the dataset evaluated, our framework is capable the average model fit
% errors are reported 

%   kinematic
% model estimation using fiducial marker-based solutions. On each of the evaluated
% demonstrations, our proposed framework correctly identifies the kinematic model
% of the observed object, and estimates its model parameters with accuracies
% similar to that of marker-based solutions, and better that current state-of-the-art. The average position and orientation
% error is computed by determining the average relative error between the
% projected pose described by the kinematic model, and the estimated poses
% provided by the pose estimation step. As indicated in
% Table~\ref{tab:model-estimation-comparison}, we are able to obtain model
% estimates that exhibit only up to $1.6$~cm average positional error, and up to
% $4.6\,^{\circ}$ average orientation error. COMPARED TO Katz.

% - COMPARE angle between the Rotational Axis/Prismatic Axis as computed via our
% algorithm and the current state-of-the-art with that obtained via ground truth. 
