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\begin{document}
\LARGE
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\twocolumn[% 
\vspace{5mm} 

\begin{center} 

{\bf Real-Time Tracking of an Unpredictable Target\\ 
Amidst Unknown Obstacles} 
\end{center} \large

\begin{center} 
{\bf Cheng-Yu Lee$^{(1)}$ \ \ \ Hector Gonz\'alez-Ba\~{n}os$^{(2)}$ \ \ \ 
Jean-Claude Latombe$^{(1)}$}
\end{center} \normalsize 

\begin{center} 
(1) Robotics Laboratory, Computer Science Department, Stanford 
University, CA, USA\\ 
(2) Honda's Fundamental Research Labs., Mountain View, CA, USA\\ 
Email:
chengyu@robotics.stanford.edu, hhg@hra.com, latombe@cs.stanford.edu
\end{center} ] 


\small 
\begin{center} {\bf Abstract} \end{center}

This paper describes a new approach to keep a moving target into the
field of view of a robot, despite obstacles.  Unlike previous
work, this approach does not assume that a complete or partial
model of the target's behavior is available, nor that a map of the
obstacles is given. At approximately 10 Hz, the robot constructs a
local map of the environment from range data, localizes the target in
this map, and computes its new velocity command. This command is
selected to minimize a function that measures the risk that the target
may escape the robot's field of view by crossing an occlusion ray
created by an obstacle. The risk function defined in this paper
combines a reactive term, whose minimization handles immediate
threats, and a look-ahead term, whose minimization improves the
robot's future ability to react to evading motions of the target.  The
experimental robotic system has been remarkably successful at tracking
a target performing brisk maneuvers and swift turns behind obstacles.

\section{Introduction} 
 
This paper describes a robot, called the {\em observer}, whose task is
to keep a moving target into its field of view. While previous
tracking techniques assume that both a model of the target's bahavior
and a map of the obstacles are given, our approach makes no such
assumptions. At approximately 10 Hz, the implemented observer
constructs a local map of the environment from sensed range data,
localizes the target in this map, and computes a new velocity command
that is sent to its servo-controller.  This command is selected to
minimize a measure of the risk that the target gets hidden by
obstacles.

\begin{figure}[thp!]
\centerline{
\psfig{figure=observer.PS,height=1.2in} 
}
\label{f:system}
\caption{The observer (right) and the target}
\end{figure}

Our observer is a mobile robot (SuperScout platform of Nomadic
Technology) equipped with a polar laser range sensor (see Fig. 1). The
target is another mobile robot. The observer and the target operate in
an in-door environment with solid and mostly static obstacles. The
observer's range finder provides a 180-dg horizontal scan of the
environment. Both the target's position and the local obstacle map are
extracted from this scan. Despite the paucity of its sensing
equipment, our observer has been remarkably successful at tracking a
target performing brisk maneuvers and swift turns behind obstacles in
non-engineered environments. The observer also tolerates transcient
obstacles, though it does not explicitly distinguish them from
stationary obstacles.  We attribute much of its success to both our
new risk-based tracking approach and the specific risk function
embedded in this approach, which combines a reactive and a look-ahead
term. These concepts are presented in this paper.

\section{Previous Work} 

{\bf Automatic target tracking and interception.}
Tracking has been widely studied in the field of automatic control, for such 
applications as missile control~\cite{missile}. The tracker and the target moves
in 3-D space. Unlike in our work, obstacles are few (if any) and occlusion of the 
target is not a concern. Instead, the emphasis is on dealing with tight 
dynamic restrictions on the tracker's motion.

\vspace*{0.05in}

\noindent {\bf Visual tracking.}  Here the goal is to track an object
in a sequence of images~\cite{huttenlocher}, despite partial
occlusions of the target by other objects, and aspect changes
and self-occlusions of the target as it moves. There is no attempt to
prevent occlusions from happening by moving the camera
appropriately. Visual tracking and sensor placement are two
complementary subproblems in target tracking among obstacles. Our work
addresses the second subproblem.

\vspace*{0.05in}

\noindent
{\bf Visual servo-control.}
Here visual tracking is used to control the motion of a 
robot~\cite{hutchinson}. The robot has a certain goal, 
like docking against another machine. Visual feedback is used to track an object 
-- e.g., the machine against which the robot intends to dock -- in order to adjust 
the robot's trajectory. Again, there is no attempt to move the 
camera to avoid undesirable occlusions.

\vspace*{0.05in}

\noindent {\bf Guarding an art gallery.} The basic art-gallery problem
is to compute the positions of fixed observers (guards), such that
they collectively see a given space~\cite{orourke}. Finding the
minimum number of guards in a polygonal space is NP-hard, but
approximation algorithms exist~\cite{gonzalez}.  Such an algorithm
could be used to compute the placements of sensors such that the
target would always be visible from one of them. Instead, our work
assumes that there is only one sensor, but this sensor moves.

\vspace*{0.05in}

\noindent {\bf Off-line planning.}  A similar problem as ours is
considered in~\cite{becker}, assumming that both a global map of the
obstacles and the target's trajectory are given in advance. An
off-line planner backprojects visibility regions to compute a tracking
trajectory. This trajectory is then executed in open loop by the
observer. The assumptions are too constraining to lead a practical
robotic system, but this method has been applied to computer graphic
animation~\cite{li}.

\vspace*{0.05in}

\noindent {\bf Game-theoretic approach.}  The tracking algorithm
in~\cite{lavalle} is based on a game-theoretic approach and minimizes
a cost function using dynamic programming.  When the target's
trajectory is known in advance, along with the global obstacle map, an
optimal trajectory is computed off-line.  When the target's trajectory
is not known, a probabilistic model of the target's behavior is used
to compute and update in real-time a strategy maximizing the
probability that the observer keeps the target in view within some
time horizon $h$. But the computational cost grows quickly with $h$,
hence $h$ must be small. Systems built using this
method~\cite{spie,lavalle,rafael} give reasonably good results when
the target does not move too swiftly.  The risk-based method presented
below does not require a model of the target's behavior nor does it
use costly dynamic programming. Experimental results show that it is
faster and handles swift targets more reliably.

\vspace*{0.05in}

\noindent {\bf Observer self-localization.}  When the observer uses a
global map, it must localize itself in this map.  The above
game-theoretic approach is extended in~\cite{fabiani} to take into
account the impact of self-localization uncertainties on tracking. A
cost function allows the observer to decide at each stage of the
tracking process whether it is preferable to perform the best tracking
motion, or to deviate from this motion in order to see a landmark and
achieve better self-localization. Our method does not make use of a
prior global map. It eliminates the need for the observer to localize
itself, hence saving precious computational time and avoiding sporadic
major self-localization errors.


\section{Tracking Procedure}
\label{s:procedure}

Our tracking approach is based on minimizing a function $\varphi$ of
the observer's position that estimates the {\em risk} that the target
may escape the observer's field of view by crossing an occlusion ray
created by an obstacle. The commanded velocity is set to $-\nabla
\varphi$. In our implementation it is updated at 10~Hz.

\begin{figure}[thp!]
\centerline{
\psfig{figure=scan2.PS,height=1.3in} 
}
\caption{Range scan and extracted information}
\label{f:scan}
\end{figure}

Each sensor reading provides a 180-dg horizontal scan of the
environment made of 360 points at regular angular spacing. This scan
is first segmented into maximal sequences of adjacent points. The
target is then recognized (as a circular arc of known radius) and
localized in the observer's coordinate system. Polygonal lines
(polylines) are fit through the remaining sequences of points. Each
endpoint of a polyline is an {\em occluding vertex} through which an
{\em occluding ray} is erected.  Fig.~\ref{f:scan} shows the outcome
of the treatment of a scan.  The localized target is shown in plain
(blue) line. The occlusion rays are the straight line segments
converging toward the observer (the black point at the base of the
figure). The outer circular arcs bounding the shaded area corresponds
to the sensor's maximum range.

\begin{figure}[thp!]
\centerline{
\psfig{figure=ept.PS,height=1.2in} 
}
\caption{Escape-path tree}
\label{f:ept}
\end{figure}

The tracking algorithm then computes the shortest escape path of the
target to every occlusion ray. The obtained paths form a tree that we
call the {\em escape-path tree} ({\sc ept} for short).  An {\sc ept}
is shown in Fig.~\ref{f:ept} (here, the paths have been computed as if
the target was a point).  The paths in the {\sc ept} are used to
assign an individual escape risk $\varphi_e$ to each occlusion ray
$e$.  These individual risks are aggregated into a global risk
function $\varphi$.  As indicated below, this function is
differentiable, hence the algorithm easily calculates $-\nabla
\varphi$.

The overall tracking algorithm is the following:

\vspace*{0.05in}

\noindent
{
{\bf Procedure} {\sc track}\\
\hspace*{0.1in} Repeat:\\
\hspace*{0.2in} {\bf 1.} Acquire a scan.\\
\hspace*{0.2in} {\bf 2.} Localize the target.\\
\hspace*{0.2in} {\bf 3.} If step 2 failed, then exit with failure.\\
\hspace*{0.2in} {\bf 4.} Extract occluding vertices and occlusion rays.\\
\hspace*{0.2in} {\bf 5.} Compute the {\sc ept}.\\
\hspace*{0.2in} {\bf 6.} Set the velocity command to $- \nabla \varphi$.
}

\vspace*{0.05in}

An efficient algorithm given in~\cite{icra02,cyl-thesis} computes the
{\sc ept} by exploiting geometric properties of shortest escape paths
in star-shaped polygons. Below, we focus on the description of the
risk function $\varphi$.

\begin{figure}[thp!]
\begin{center}
\begin{tabular}{c}
\psfig{figure=approx1.PS,height=2.0in}\\ 
(a)\\
\psfig{figure=approx2.PS,height=2.0in}\\
(b) 
\end{tabular}
\end{center}
\caption{Approximate strategies}
\label{fig:approx}
\end{figure}

\section{Risk Function}

In the absence of any model of the target's behavior, a tracking
approach suggested in~\cite{lavalle} is to compute the velocity
command that maximizes the escape time for the worst-case trajectory
of the target. However, when a global map of the obstacles is given,
this computation is extremely costly.  When no map is given, it is
simply impossible. This led us to investigate approximations of this
strategy.

\subsection{Escape Risk for Occlusion Ray}

For a moment, let us focus our attention to a single occlusion ray $e$
extracted from the current scan, and let $\tau_e$ denote the shortest
escape path of the target to $e$.  If the target's motion is not
restricted by any nonholonomic or dynamic constraint, the ratio of the
length of $\tau_e$ by the target's maximal velocity is exactly the
minimum time $\varepsilon$ that the target needs to escape the {\em
current} field of view of the observer through $e$. If the target's
motion is restricted, this ratio remains in general a good lower-bound
approximation of this time. Hence, a possible approximate strategy for
the observer is to move in the direction that yields the greatest
differential increase of the length of $\tau_e$. To illustrate
consider the example of Fig.~\ref{fig:approx}(a), where the path
$\tau_e$ consists of a single line segment between the target's
position and an escape point $p$ in an occlusion ray $e$ erected from
the occluding vertex $v$. The observer's motion that yields the
greatest differential increase of the length of $\tau_e$ is
perpendicular to the line supporting $e$.

We can push this reasoning further. During the time $\varepsilon$ it
would take for the target to reach $p$ by moving at maximal velocity
along $\tau_e$, the observer can reach any point within a circle of
radius $\delta$ centered at its current position, where $\delta$ is
the product of the observer's maximal velocity by $\varepsilon$
(assuming that no nonholonomic or dynamic constraint limits the
observer's motion). Consider the tangent line to this circle drawn
from $v$.  A better direction of motion for the observer than the
vector shown in Fig.~\ref{fig:approx}(a) is the vector shown in
Fig.~\ref{fig:approx}(b), which points toward the tangency point.  It
combines a component perpendicular to the line supporting $e$ and a
component in this line pointing toward $v$.  The key observation that
must be made here is that the time needed by the target to escape the
observer's current field of view provides some ``slack'' that the
observer can exploit to reach a position where it will have greater
ability to react to future evading moves of the target. More
specifically, by getting closer to the occluding vertex $v$, while it
still has time to do so, the observer will later be able to perform
swinging motions leading the occluding edge $e$ to rotate at a faster
rate around $v$.

This discussion yields the risk function $\varphi_e$
for one occlusion edge $e$.  We define $\varphi_e$ to be of
the form:
\begin{equation}
\varphi_e = c \, r^2 \, f(1/h),
\label{eq:varphi}
\end{equation}
where $c$ is a scaling factor, $r$ is the distance between the
observer's position and the occluding vertex $v$, 
$h$ is the length of the shortest path $\tau_e$ ending in $e$,
and $f$ is an increasing function. 
The experimental results of Section~\ref{s:experiments}
where obtained for $f(x) = \ln (x^2 + 1)$. Hence,
$\varphi_e$ goes to $+\infty$ when $h$ goes to 0, 
and to 0 when the observer gets arbitrarily close to
$v$. 

One can regard $\varphi_e$ as being made of two terms:
\begin{enumerate}
\item The {\em reactive} term, $f(1/h)$: Its minimization 
increases the length of the escape path to $e$.
\item The {\em look-ahead} term, $r^2$: Its minimization
improves the future ability of the observer to 
react to target's evading moves.
\end{enumerate}

The component of the commanded velocity associated 
with $\varphi_e$ is:
\[ - \nabla \varphi_e \; = \; - 2cr f(1/h) \nabla r \,
+ \, c \frac{r^2}{h^2} f'(1/h) \nabla h. \]
It can easily be computed in closed form by treating
$r$ and $h$ as functions of the observer's 
position~\cite{icra02,cyl-thesis}. When $h$ is very small,
there is an immediate threat; then, the dominant
term is the reactive term and the commanded velocity 
is almost perpendicular to the supporting line of $e$
(direction of $\nabla h$). Instead, if $h$ is
large, the dominant term is the look-ahead component
and the commanded velocity has a substantial component
pointing toward $v$ (direction of $-\nabla r$).

\subsection{Global Risk Function}

Let us now consider all occlusion rays and all escape paths
in the {\sc ept}.
We use Eq.~(\ref{eq:varphi}) to compute the risk associated 
with every occlusion edge. Note that the shortest escape path 
(over all possible occlusion edges) may not result in the 
greatest risk. An occlusion ray with a longer escape path to it
may represent a greater risk if the observer is located further away
from this ray's occluding vertex. So, the issue is:
How can we combine the individual risks associated with the occlusion edges
to define the global risk $\varphi$?

One possibility is to identify the occlusion ray $e$ that yields the
greatest risk $\varphi_e$ and set $\varphi = \varphi_e$.  But, as the
observer and the target move, the greatest-risk occlusion ray, and so
the commanded velocity, may change abruptly and frequently. In an
early implementation, this resulted into a shattering control causing
erratic moves of the observer.  To avoid this problem, we define the
global risk function $\varphi$ by averaging the escape risks
associated with several occlusion rays.

In the ``standard'' average, the risks associated with all occlusion rays are counted 
equally. But many escape paths often share the same initial
structure. Such paths are then over-represented in $\varphi$ at the expense of the
solitary escape paths. Two averages that work better in practice are the following: 
\begin{itemize}
\item {\em Recursive average}: The risks of the {\sc ept} nodes sharing the same
parent are first averaged and the result
is propagated upward to the parent. 
\item {\em Tournament average}: The global risk is the
average of the risks associated with the $k$ greatest-risk occlusion rays
(with $k = 3$ or 4).
\end{itemize}
Since the gradient is a linear operation, we apply directly these averaging
techniques to the gradients of the individual risks. No risk $\varphi_e$,
nor $\varphi$, is ever computed by the tracking algorithm. The experimental results
presented below were obtained with the recursive average.

\section{Experimental Results}
\label{s:experiments}

We have implemented the above tracking approach a
SuperScout mobile robot of Nomadic Technologies equipped with an
LMS200 laser range finder of Sick OpticElectronic. The tracking algorithm
runs on an off-board 410MHz Pentium II processor connected to the robot by
wireless ethernet. The target, a joysticked Nomad-200 robot, is modeled as 
a disc of non-zero radius and the escape paths are computed accordingly. 
The SuperScout robot is an almost cylindrical nonholonomic vehicle with 
zero-turning radius. A simple trajectory follower is used to 
track the flow of the successive velocity commands 
$- \nabla \varphi$~\cite{icra02,cyl-thesis}. In addition, we use a second regulator
to deal with the fact that the range sensor has limited range (8 meters) 
and a 180-dg cone. This regulator is a simple proportional feedback of 
the target's deviation from an ideal position in the
sensor's field of view. The maximal linear velocities of
the observer and the target are approximately the same,
with the observer slightly faster (0.5~m/sec).

\begin{figure}[thp!]
\centerline{
\psfig{figure=corner.ps,height=2.2in} 
}
\caption{Tracking a target around a corner}
\label{f:ex1}
\end{figure}

We have conducted various experiments with this system. In some
experiments, we have compared the risk-based algorithm with a pure
visual-tracking algorithm. This second algorithm only tries to keep
the target close to an ideal position in the sensor's field of view.
It ignores obstacles, hence does not take potential occlusion into
account.  Unlike the risk-based algorithm, the pure visual-tracking
algorithm is easily tangled by a target moving toward an obstacle
corner and swiftly turning around this corner. Fig.~\ref{f:ex1} shows
a record of the target's and observer's trajectories in one experiment
with the risk-based algorithm. The successive positions of the target
and the observer are shown with triangles and circles,
respectively. Segments connects the pairs of simultaneous
positions. Note how well the observer anticipates the risk that the
target may turn around the corner and swings around this corner.

We have also successfully experimented with our observer in environments with transient 
obstacles, obtained by suddenly adding an obstacle in the field of view of the observer and 
either leaving it there, or removing it a short amount of time later. The
observer has proven to be tolerant to such obstacles~\cite{cyl-thesis}. 

Fig.~\ref{f:ex2} shows snapshots of another experiment where the observer
successfully tracks the target along a long tour throughout our laboratory
environment. The tour starts in a corridor, then traverses a large area 
with many different objects (chairs, boxes), and finally ends in another corridor.
The scans and {\sc ept}'s corresponding to these snapshots are displayed
in Fig.~\ref{f:ex2-bis}.

In a number of experiments, the observer also failed to track the target
correctly. Most failures occurred because the observer either failed to 
detect the target though it was visible, or mis-located it. Such
failures occurred mainly in the presence of cylindrical objects
(e.g., waste baskets) similar to the target. But imagine yourself
having to track a target in a dark environment by using a flashlight
projecting only a plane of light! Given the paucity of its sensor,
our observer has been remarkably successful.

\section{Conclusion} 

This paper describes a new approach to target tracking based on risk
minimization. It proposes a risk function that combines a reactive
term, to deal with immediate threats, and a look-ahead term, to better
use available time. Experiments have shown that the observer is able
to keep the target in sight in both densely and sparsely cluttered
environments, even when the target performs sudden and swift moves.
An obvious improvement is to equip the observer with an additional
sensor, for instance a stereo-vision system, to better detect and
localize the target. An interesting extension will be to perform
tracking and environment mapping simultaneously. A more difficult
extension would be to coordinate a team of independent observers with
limited communication to track multiple targets~\cite{rafael}.

\vspace*{0.05in}
\noindent {\small
{\bf Acknowledgements:} We thank Vicky Chao for her help in running the 
experiments.}

{\small
\begin{thebibliography}{10}

\bibitem{spie}
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M.~Stein (ed.), vol. 3840, pp.~210-221, 1999.

\bibitem{becker}
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Robotics (ISER)}, pp.~153-160, 1995.

\bibitem{fabiani} P.~Fabiani, H. Gonz\'alez-Ba\~nos, J.C.~Latombe,
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\bibitem{gonzalez}
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pp. 232-240, 2001.

\bibitem{icra02}
H.~Gonz\'alez-Ba\~nos, C.Y.~Lee, J.C.~Latombe, ``Real-Time Combinatorial Tracking of a
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Robotics \& Automation}, 2002.

\bibitem{hutchinson}
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\bibitem{huttenlocher}
D.P.~Huttenlocher, J.J.~Noh, W.J.~Rucklidge, ``Tracking Non-Rigid Objects
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\bibitem{lavalle}
S.~LaValle, H. Gonz\'alez-Ba\~nos, C.~Becker, J.C.~Latombe, 
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pp.~731--736, 1997.

\bibitem{cyl-thesis}
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\bibitem{rafael}
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\end{thebibliography}}


%
\begin{figure}[thp!]
\begin{center}
\begin{tabular}{c}
\psfig{figure=tour1.ps,height=1.4in}\\ 
\psfig{figure=tour3.ps,height=1.4in}\\ 
\psfig{figure=tour5.ps,height=1.4in}\\ 
\psfig{figure=tour6.ps,height=1.4in}\\ 
\psfig{figure=tour8.ps,height=1.4in}\\ 
\psfig{figure=tour10.ps,height=1.4in}
\end{tabular}
\end{center} 
\caption{Tracking a target performing a long tour}
\label{f:ex2}
\end{figure}
%
%
\begin{figure}[thp!]
\begin{center}
\begin{tabular}{c}
\psfig{figure=ept1.ps,height=1.4in}\\ 
\psfig{figure=ept3.ps,height=1.4in}\\ 
\psfig{figure=ept5.ps,height=1.4in}\\ 
\psfig{figure=ept6.ps,height=1.4in}\\ 
\psfig{figure=ept8.ps,height=1.4in}\\ 
\psfig{figure=ept10.ps,height=1.4in}
\end{tabular}
\end{center} 
\caption{Scans and {\sc ept}'s}
\label{f:ex2-bis}
\end{figure}
%


%\bibliography{mthesis}
\end{document}