A Geometric Optimization Approach to Detecting and Intercepting Dynamic Targets

Abstract

A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensing-pursuit problem considered in this paper is analogous to the Marco Polo game, in which the pursuer must capture multiple mobile targets that are sensed intermittently, and with very limited information. In this paper, the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections. Since the sensors are installed on robotic platforms and have limited range, the geometry of the platforms and of the sensors field-of- view play a key role in obstacle avoidance and target detection. Thus, a new cell decomposition approach is presented to formulate the probability of detection and the cost of operating the robots based on the geometric properties of the network. Numerical simulations verify the validity and flexibility of our methodology.

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A Geometric Optimization Approach to Detecting and Intercepting
Dynamic Targets
Silvia Ferrari, Chenghui Cai, Rafael Fierro, and Brent Perteet
Abstract—A methodology is developed to deploy a mobile
sensor network for the purpose of detecting and capturing
mobile targets in the plane. The sensing-pursuit problem
considered in this paper is analogous to the Marco Polo game,
in which the pursuer must capture multiple mobile targets that
are sensed intermittently, and with very limited information. In
this paper, the mobile sensor network consists of a set of robotic
sensors that must track and capture mobile targets based on
the information obtained through cooperative detections. Since
the sensors are installed on robotic platforms and have limited
range, the geometry of the platforms and of the sensors field-of-
view play a key role in obstacle avoidance and target detection.
Thus, a new cell decomposition approach is presented to
formulate the probability of detection and the cost of operating
the robots based on the geometric properties of the network.
Numerical simulations verify the validity and flexibility of our
methodology.
I. INTRODUCTION
The proliferation of reliable, low-cost sensor networks
and developments in autonomous vehicle technologies are
producing advanced surveillance systems, where both the
sensors and their platforms are characterized by a high degree
of functionality and reconfigurability. These networks are
expected to operate reliably in dynamic environments with
little human intervention. However, coordinating such large
heterogeneous sensor networks is extremely difficult and
requires the development of novel methods of communica-
tion, motion control, computation, proactive estimation and
sensing, and power management.
Cooperative pursuit strategies to detect, intercept and
capture intelligent evaders in cluttered environments are
described in [14]. However, the difficulty of solving pursuit-
evasion games and obtaining closed-form solutions of the
underlying optimization problem has motivated an intensive
research aimed at developing algorithmic approaches.
A variety of optimization techniques [4] have been ap-
plied to the coordination of robotic networks engaged in
distributed sensing tasks [7]. Optimal motion planning for
multiple robots is considered in [3]. Distributed motion
planning approaches are discussed in [10].
The problem of finding the configuration of a network
with multiple sensors that optimizes the number of tracks
intercepted is considered in [9]. A Bayesian network (BN)
S. Ferrari and C. Cai are with the Laboratory for Intelligent Systems
and Controls, Department of Mechanical Engineering & Materials Sci-
ence, Duke University, Durham, NC 27708-0300, USA {sferrari,
cc88}@duke.edu
R. Fierro and B. Perteet are with the MARHES Lab, School of Electrical
& Computer Engineering, Oklahoma State University, Stillwater, OK 74078-
5032, USA {rfierro, brent.perteet}@okstate.edu
1-4244-0989-6/07/$25.00 ©2007 IEEE.
approach is used to develop an automated computer player
for CLUE in [5]. In other related work [6] the authors
develop a decentralized motion coordination algorithm for
tracking tasks of dynamic targets. Strategies to search for
moving targets in a two-dimensional plane are considered in
[12]. Motion coordination strategies of multiple vehicles vis-
iting targets generated by a stochastic process are proposed
in [1].
In this paper, we develop a methodology that employs cell
decomposition algorithms to obtain a graph representation of
the robot configuration space that is void of obstacle and en-
ables target detections and integrates some of the techniques
reviewed. Specifically, the track-coverage definition in [9]
and robot motion planning are combined such that multiple
mission objectives are considered by the same formalism
and optimized by the same policy. The basic problem of
moving targets detection and interception is inspired by the
game Marco Polo described in [13] and is motivated by
many applications in surveillance. The goal of Marco Polo
is to capture a group of intelligent evaders as quickly as
possible using a team of autonomous pursuers that have an
intermittent knowledge of the evaders’ locations.
The remainder of the paper is organized as follows.
We formulate the dynamic tracking problem in Section II.
Section III describes the methodology that allows a group
of pursuers to detect and intercept dynamic targets. The
control strategies for capturing evaders are detailed in Section
IV. Section V contains simulation results. Finally, we draw
conclusions in Section VI.
II. PROBLEM STATEMENT
We consider a game that takes place in a simply connected,
convex polygon S ⊂ R2with boundary ∂S and populated
by convex obstacles Oj⊂ S.
The pursuers or mobile sensor agents are nonholonomic
vehicles that can be modeled using the unicycle model
˙ xi
˙ yi
˙θi
p
=
=
=
vi
vi
ωi
pcosθi
psinθi
p,
p
p,
(1)
pp,
where qi= (xi
R2. The input to pursuer i is ui
R2.
The model of the target agents or evaders is given by
p, yi
p, θi
p) ∈ SE(2) and pi= [xi
p= [vi
p
p
yi
p]T∈
ωi
p]Tand up∈ U ⊂
˙ xj
˙ yj
τ
=
=
cj
cj
xτ,
yτ,
τ
(2)
Proceedings of the 2007 American Control Conference
Marriott Marquis Hotel at Times Square
New York City, USA, July 11-13, 2007
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where τj= [xj
and cj
along straight lines
τ
yτare constants. In other words evaders move
yj
τ]T∈ R2denotes the position of target j,
xτand cj
yj
τ(t) =cj
yτ
cj
xτ
xj
τ(t) + yj
τ(0) −cj
yτ
cj
xτ
xj
τ(0),
where (xj
θj, thus θj := arctan(cj
distance from the jthtarget position, τj, to the closest
pursuer, i.e., eji= mind(τj,pi). Then the pursuer i is said
to capture the target j when eji< ε. The threshold value ε
is called the capture threshold for an interval ∆ccalled the
capture timeframe.
In the actual Marco Polo game, when requested by Marco
(i.e., pursuer) all Polos (i.e., evaders) must communicate
their positions to Marco. Let σj be a random variable
representing the time period between communications for
the jthtarget. At the instant of communication from target
j, its exact position within S is known by the pursuer.
Following this, the target may continue to move but the
pursuer receives no updated information until the next com-
munication from target j. We consider here a modified
version of the Marco Polo problem. First, no communication
between targets and pursuers takes place. Second, pursuers
have a limited detection disc Diso that targets are detected
if they enter Di. At any time t, the set of detections is
denoted by Zt
target positions τjobtained since the initial time t = 0, e.g.,
Zt
previous discussion, the problem can be stated as follows
Problem 2.1: Given a set P of N pursuers and a set T of
M target agents within a specified game area S, find a set
of policies ui
maximizes the probability of intercepting partially-observed
and unobserved tracks and minimizes the time tcrequired to
capture fully-observed targets in T .
To solve the above problem, we assume that the au-
tonomous pursuers each perform their own detection process-
ing to determine the time and location of a detection event. A
track-before-detect approach [15] is considered, in which the
individual sensor detections are used to form a hypothetical
track before declaring a positive detection. This approach
is suited to systems comprised of sensors that are relatively
simple and produce few observations for each moving target,
have no prior knowledge of its track, and are subject to
frequent false alarms. Once a track has been formed from at
least k individual detections that have occurred at different
moments in time, an upper-level controller determines the
purssuer in P that is in the best position to pursue it.
In this approach, it is convenient to classify tracks based on
the following definitions. An unobserved track refers to the
state (e.g., position, velocity, and acceleration) of an evader
(2) for which there have been no detections up to the present
time, t. A partially-observed track refers to the track of an
evader that is formed from 0 < l < k individual sensor
detections obtained up to time t. An evader track is said to
be fully-observed when it is formed from l > k individual
τ(0), yj
τ(0)) ∈ ∂S. The heading of a target is
yτ,cj
xτ). Let eji be the Euclidean
j, and symbolizes all measurements of the
j= {z1(t1),z2(t1),...,zi(t?),...,zj(t)}. Based on the
p= ci(pi,Zt
τ) ∈ U for all pursuers in P which
sensor detections obtained up to time t. Only after a track
is fully-observed, the target is considered to be positively
detected. Then, the formed track is used to compute a pursuit
strategy on line, minimizing the individual capture time tcj.
Assumptions on the agents dynamics are that the maximum
velocity Vp,τmaxof all agents (pursuers and targets) is known,
the translational speed of target j is uniformly distributed in
[0,Vτmax], and Vpmax> Vτmax.
Pursuers always operate in one of two modes: detection
or pursuit. The objectives of an agent in detection mode
can be summarized as follows: (i) Avoid n fixed obstacles
{O1,...,On}; (ii) maximize the probability of intercepting
m partially-observed tracks {R1
the probability of intercepting unobserved tracks. On the
other hand, the objective of an agent in pursuit mode is to
minimize the time tcjrequired to capture the target τj, based
on its track Rj
information sensed during the pursuit, which may be used
to update the track (e.g., as shown in [2]).
The following sections describe a methodology for plan-
ning the motions of the pursuers, in order to meet all of the
above objectives.
θ,...,Rm
θ}; and maximize
θ, the pursuer’s initial position pi(t0), and any
III. METHODOLOGY
The geometry of the pursuers is assumed to be a convex
polygon denoted by Ai, with a configuration qiwith respect
to a fixed Cartesian frame FS such that Ai(qi) ∈ S. The
robot workspace S is populated with n fixed obstacles Oj,
j = 1,2,...,n, to be avoided, and by m targets or partially-
observed tracks Rj
a disk Direpresent the field of view of an omnidirectional
sensor installed on Ai, with radius ri, and such that Di(qi) ∈
S. Henceforth, the sensor installed on Aican detect the target
on the track Rj
applications where the sensors field of view is much larger
than the robot geometry, and hence a robot can sense a target
without necessarily being close enough to capture it.
θ, i = 1,2,...,m, to be intercepted. Let
θwhen Di(qi)∩Rj
θ?= ∅. We are interested in
A. Robot Motion Planning in the Presence of Multiple
Targets and Obstacles
The partially-observed tracks are viewed as targets from
which it is desirable to collect additional measurements,
before investing in the costly resources needed to capture
them, such as deploying one or more robots to chase one
of many possible targets. Then, the subsets of S where the
sensor can collect measurements from the partially-observed
tracks can be defined as
Definition 3.1 (C-Target): The target track Rj
in the ithpursuer configuration space C to the C-target region
CRj= {qi∈ C | Di(qi) ∩ Rj?= ∅}.
The boundary of a C-target is the curve followed by the
origin of FAiwhen Dislides in contact with the boundary
of Rj
targets boundaries are obtained by growing Rj
by the radius riwithin S. C-obstacles are defined as in [11]
and used together with the C-targets to obtain a roadmap
θin S maps
θ. With the assumed robot and sensor geometries, the C-
θisotropically
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representation of the robot configuration space, as explained
in the following paragraphs.
Well known path planning techniques, such as cell decom-
position [11] or roadmap methods, are modified to account
for the presence of the targets, which play a role opposite to
obstacles. In classical cell decomposition, the obstacle-free
configuration space, defined as,
Ci
free= C \
n?
k=1
COk= {qi∈ C | Ai(qi)∩(
n?
k=1
Ok) = ∅}
(3)
is decomposed into a finite set of cells, {κ1,...,κf}, within
which a path free of obstacles can be easily generated. To
account for the presence of targets, let an observation cell,
¯ κl, be defined as a subset in configuration space with the
property that every robot configuration qiin it enables sensor
measurements from at least one track, i.e., qi∈ ¯ κl⊂ CRj
implies both qi∈ Ci
and qi∈ CRj.
The following definition, taken from [11], is useful when
the C-targets are grown isotropically by a disk, Di:
Definition 3.2 (Generalized Polygon): A
polygon is a subset of R2that is homeomorphic to the
closed unit disc and whose boundary is a closed-loop
sequence of straight line segments and circular arcs.
Alternatively, the pill-shape C-targets can be approximated
by a convex polygon. Then, given the initial and final
configurations, q0 and qf, a connectivity graph including
observation cells can be obtained by the following algorithm:
(I) Decompose the configuration space that is void of any
C-obstacles or C-targets, and is defined as:
free(or, equivalently, qi?∈ COj, for ∀j)
generalized
Ci
void
=
C \ {
n?
k=1
COk ∪
r?
n?
j=1
CRj}
=
{qi∈ C | Ai(qi) ∩ (
k=1
Ok) = ∅,
Di(qi) ∩ (
r?
j=1
Rj) = ∅}
(4)
(II) Decompose each obstacle-free C-target,
CRj \
n?
k=1
COk,
(5)
thereby obtaining the set of observation cells.
(III) Construct a connectivity graph G using both void (I)
and observation cells (II).
A sweeping-line algorithm can be used to decompose a
non-convex generalized polygon with ν vertices into O(ν)
convex generalized polygons in O(νlogν) time (see Section
5.1 in [11]). An illustrative example of workspace and
corresponding cell decomposition is shown in Fig. 1.
B. Search Area Coverage
The observation cells in the connectivity graph represent
robot configurations that enable additional individual sensor
Fig. 1. Example of cell decomposition (dashed lines) for a workspace with
four C-obstacles (black regions) and one C-target CR (patterned region)
corresponding to 2 < k prior detections. The sensor with range r is installed
on a robot with a square platform geometry A.
detections of targets that may be moving along partially-
observed tracks. Additionally, the team of pursuers must also
detect new targets that may have just recently entered the
search area S or that may have been missed up to the present
time, t. Since the targets are always in motion, maximizing
the area coverage of the sensors may not be the best strategy
for obtaining at least k detections from each target. In fact
these detections may take place at different moments in time,
anywhere along a target track, and still lead to a positive
detection provided they form a feasible track. The problem
of intercepting a target moving along a straight line across
S by means of k omnidirectional sensors in a network of
size n is referred to as track-coverage [9]. It can be viewed
within the context of geometric transversal theory, where:
Definition 3.3: A family of k convex sets in Rnis said
to have a d-transversal if it is intersected by a common d-
dimensional flat (or translate of a linear subspace).
When, d = 1 and the transversal is also known as a line-
stabber of the family of convex sets. In the track coverage
problem, d = 1, and a track intercepted by k sensors is
said to be a stabber of the corresponding fields of view, e.g.,
{D1,...,Dk}, in R2.
Then, it can be shown [9] that the family of stabbers of
one disk Di, with radius riand centered at pi= [xi, yi]T
in R2, can be described by a two-dimensional cone defined
by the unit vectors,
?
and,
?
for the y-intercept by. Where, αidenotes half of the opening
angle of the coverage cone and its trigonometric functions
ˆhi=
cosαi
sinαi
−sinαi
cosαi
?
vi
?vi?= Q+
iˆ vi,
(6)
ˆli=
cosαi
−sinαi
sinαi
cosαi
?
vi
?vi?= Q−
iˆ vi,
(7)
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are easily computed from the position pi,
sinαi=
ri
?vi?=
ri
?
x2
1,i+ (x2,1− b)2
??vi? − (ri)2
by]T. Also, let all unit vectors of the
(8)
and,
cosαi=
?vi?
(9)
with vi ≡ pi− [0
same class be ordered according to their direction sine in
R2. Then, for a fixed y-intercept by, the family of stabbers
of a family of k disks with different radii and coordinates
can be described by a cone, referred to as k-coverage cone,
that is defined by two unit vectors:
ˆl∗≡ sup{ˆl1,...,ˆlk},
The opening angle of a k-coverage cone defined with respect
to the y-axes,
ψ = sin−1||ˆl∗׈h∗||
is a measure over the set of line stabbers for the family of k
disks, and is used to derive the probability of detection for
unobserved target tracks.
By summing the opening angles of the k-coverage cones
corresponding to all possible intercepts, obtained by dis-
cretizing the border of S, the following function is derived
for the probability of detection of unobserved tracks:
ˆh∗≡ inf{ˆl1,...,ˆlk}
(10)
Pk
S(X)=
δ2
2π(L2+ δ2)
L2
?
by=0
[H(ψ)ψ + H(ϕ)ϕ] (11)
+
δ1
2π(L1+ δ1)
L1
?
bx=0
[H(ζ)ζ + H(ρ)ρ]
Where, the coverage function of a configuration of sensors
X = {p1,...,pn}, given by,
L2
?
L2
?
is derived in [9], and ζ, ϕ, and ρ denote the opening angles
defined with respect to the axis x, y?, and x?, respectively.
The Heaviside function H(·) guarantees that ifˆl∗?ˆh∗the
opening angle of the coverage cone is equal to zero. As an
example, these angles are illustrated in Fig. 2, for a network
with n = 2 sensors and k = 2 required detections. Also,
it can be shown that total track-coverage is given by the
constant,
?δ2+ L2
for any k and n, where δ1 and δ2 are the discretization
intervals for (x,x?) and (y,y?), respectively. Therefore, (11)
can be used to measure the probability of detection of
a sensor configuration X, with n sensors, and k-required
detections per track, given uniform probability distributions
for their points of entry along ∂S.
2 · Tk
S(X)=
by=0
H(ψ)ψ +
L1
?
bx=0
H(ζ)ζ
(12)
+
by?=0
H(ϕ)ϕ +
L1
?
bx?=0
H(ρ)ρ
Tmax
S
= π
δ2
+δ1+ L1
δ1
?
(13)
x
k=2-Coverage cones
y
?
y
b
p2
x
b
y’
x’
?
?
?
' y
b
' x
b
L1? L2 Game Area, S
p1
FS
?
FBS
B
Fig. 2.
of view centered at p1 and p2, and a rectangular area of interest S, with
perimeter ∂S shown in bold.
Coverage cone definition illustrated for two sensors with fields
C. Information-driven Motion Planning
Information-driven motion planning is a new problem that
arises when the robots navigate a workspace for the purpose
of obtaining sensor information from multiple targets. In
classical robot motion planning, the goal is to reach a
final configuration qf by traveling an obstacle-free path of
minimum cost. By utilizing the free-space decomposition ap-
proach presented in Section III-A, the presence of the targets
is accounted for, and the robot configurations that enable
sensor measurements (i.e., lead to non-empty intersections
of targets and sensor field of view) are represented by obser-
vation cells in the connectivity graph. While not sufficient
to guarantee observation, visiting these cells increases the
probability of observing the corresponding target.
The sensor planning objectives are expressed in terms
of a reward function that is assigned to every arc in the
connectivity graph
R(κl,κı) = B(κl,κı) − J(κl,κı)
representing the expected profit of the information that will
be obtained by moving from a configuration qi ∈ κl to a
configuration qi∈ κı, that is, the associated benefit minus
its cost. Then, the optimal pursuer’s path is the sequence of
adjacent cells or channel that maximizes the total reward
(14)
µ∗≡ {κ0,...,κf}∗= argmax
µ
?
(κl,κı)∈µ
R(κl,κı),
(15)
and connects κ0to κfin the connectivity graph G. Suppose
κlis an observation cell that is obtained from the decompo-
sition of the jthC-target: ¯ κl⊂ CRj. Then, one benefit of
visiting the lthcell in G is the probability of detecting the
target τj
PR(¯ κl⊂ CRi) = Pr{Dji= 1 | eji≤ ri},
where Dji represents the event that the ithsensor reports
a detection when the jthtarget comes within its detection
range. In this paper, it is assumed to be equal to one for
(16)
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simplicity, and when a cell is void PR= 0. In general, it can
be estimated from knowledge of the measurement process,
and can be made dependent on location and distance from
the target.
Another benefit of moving to a configuration within a
cell κlis the gain in probability of detection of unobserved
tracks ∆Pk
cost of traveling to any cell in G is comprised of distance
and, possibly, of energy and computational power required
by sensor measurements, if the cell is an observation cell.
Since these quantities may vary slightly within each cell,
they are computed in reference to the geometric centroid, ¯ qi,
such that the distance-cost associated with moving between
two nodes κl→ κıin G is the Euclidean distance,
d(κl,κı) ≡ max||A(¯ qı) − A(¯ ql)||,
and the gain in probability of detection is
Sthat when negative translates into a cost. The
(17)
∆Pk
S(κl,κı) ≡ Pk
S(Xı) − Pk
S(Xl),
(18)
where Xlrepresents the set of pursuers’ locations when the
ithpursuer is positioned such that the center of Dicoincides
with the centroid of κl, i.e., pi ⊂ ¯ ql. So, in summary, the
reward function can be written as
R(κl,κı) = w1PR(κl) + w2∆Pk
S(κl,κı) − w3d(κl,κı).
(19)
In this paper, sensor power and energy considerations are
neglected for simplicity. And, the weights w1, w2, and w3
are chosen by the designer based on the desired tradeoff
between these competing objectives.
IV. CONTROL STRATEGY
Once the connectivity graph has been obtained, and the
rewards computed for each arc in G, the optimal channel
µ∗is obtained using the graph searching algorithm A∗[11].
Next, the output of the motion planner is mapped into a set
of waypoints which in turn are used by a trajectory generator
and trajectory tracking controller. In this section, we focus
on a simple yet effective strategy to capture a full-observed
target. Several control algorithms can be implemented such
as a proportional controller and a more sophisticated leader-
follower approach. In the latter case the target to capture is
seen as the leader to follow.
We describe an effective approach that allows a pursuer to
capture a target along a trajectory of minimal length [8]. We
assume that the target is moving in a straight line with con-
stant velocity and intercepting along that line. This strategy
depicted in Fig. 3 is based upon the geometry of the problem
and taking into account the kinematic constraints of the
pursuer. The strategy attempts to intercept the target at a point
δ. The interception point is calculated by determining the
time for both the pursuer and target to reach that point. The
initial states of the pursuer and target are p0= (xp,yp,θp)
and τ0= (xτ,yτ,θτ), respectively. The interception point is
defined as
δ =
yt+ tcvtsinθt
?
xt+ tcvtcosθt
?
Fig. 3.Control strategy to capture a target.
and the time to interception becomes
tc=rψ + ?c − δ?cosα
vp
,
(20)
where the distance traveled by the pursuer is the distance
along the arc p0p1plus the straight line distance between p1
and δ. The arc radius is the turn radius of the pursuer and
is defined as r =
interception point δ in (20) which is passed to the controller
may be solved numerically using an iterative algorithm or a
Newton method.
vp
ωp. Using the geometric definition, the
V. SIMULATION RESULTS
This simulation scenario considers a rectangular environ-
ment with multiple targets and multiple pursuit sensors using
reward function weights w1 = 1, w2 = 1, and w3 = 1.
Before the simulation scenario begins, five sensors – one with
sensing radius 1.5 m, one with sensing radius 1.25 m, and
three with sensing radii of 1 m – with platforms measuring
0.25 m square are placed in the 10 m by 10 m environment
to maximize the probability of detecting tracks with k = 2.
Obstacle and coverage maps are generated for each sensor
corresponding to placement in each cell. Initially, all sensors
are in detection mode and each is a candidate to switch to
the pursuit mode when target tracks become fully observed.
In this scenario, two targets enter the environment at
different locations and headings and with different velocities.
As they move along their trajectories, they are detected by
the sensors . The sensors remain motionless since each target
has been detected only once. After the second detection of
a target, the network hypothesizes the target track based on
previous detections and deploys the sensor which receives
the highest reward (or lowest cost) as obtained by the A*
graph searching algorithm to move to obtain an additional
detection of the target (Fig. 4). When the second target
becomes partially detected, the same track hypothesis and
sensor deployment occurs. At the point that the first target’s
track becomes fully observed, the network again evaluates
the reward (distance) and deploys the best sensor, denoted
by its green color in Fig. 5, to pursue the target. The same
pursuit is performed when the second target is fully observed.
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