Coordinated Target Assignment and Intercept for

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Coordinated Target Assignment and Intercept for
Proceedings of the 2002 IEEE
International Conference on Robotics & Automation
Washington, DC • May 2002
Coordinated Target Assignment and Intercept
for Unmanned Air Vehicles
Randal W. Beard∗
Timothy W. McLain†
Abstract
is feasible for each UAV on the team. Third, for each
UAV with velocity constraints v ∈ [Vmin , Vmax ], determine a path (specified via waypoints), such that if
the UAV were to fly along straight-line paths, it could
complete the path in the specified TOT, while satisfying the given velocity constraints. Fourth, transform
each waypoint path into a feasible trajectory for the
UAV. By feasible we mean that the turning rate constraints, and the velocity constraints are not violated
along the trajectory. Also, the trajectory should have
the same TOT as the path specified by waypoints.
In this paper we offer a complete solution to this
problem. Our solution is derived via a systematic hierarchical approach to multiple vehicle systems. Many
of the subproblems outlined above have previously
been addressed in the literature. The assignment
problem is a well known optimization problem and is
similar to problems addressed in [1]. Our approach is
based on satisficing decision theory [2]. The simultaneous rendezvous problem has been addressed in [3, 4].
A similar approach for simultaneous target intercept
will be used in this paper. The problem of planning
waypoint paths through a cluttered environment has
been addressed in a number of works including [5, 6, 7].
The trajectory generation problem has been addressed
in [8, 9, 10] to mention just a few. In this paper we will
derive a novel trajectory generator that allows realtime generation of the trajectories such that the pathlength of the trajectory is equal to the corresponding
Voronoi path.
This paper presents an end-to-end solution to the
battlefield scenario where M unmanned air vehicles
are assigned to strike N known targets, in the presence of dynamic threats. The problem is decomposed
into the subproblems of (1) cooperative target assignment, (2) coordinated UAV intercept, (3) path planning, and (4) feasible trajectory generation. The design technique is based on a hierarchical approach to
coordinated control. Detailed simulation results are
presented.
1
Introduction
Consider a battlefield scenario where a group of M
unmanned air vehicles (UAVs) are required to drop
munitions on N known target locations. In addition,
on the battlefield there are a number of known antiaircraft facilities. Suppose that each of the targets is
equipped with anti-aircraft facilities with radars that
detect the location of attacking aircraft, but that these
facilities are only capable of firing on one aircraft at
a time. Therefore to maximize the probability of success, it is desirable to have multiple UAVs arrive on
the boundary of each target’s radar detection region
simultaneously. The rationale is that simultaneous arrival of multiple UAVs on the detection boundary will
confuse the software or operators of the anti-aircraft
facility, enabling a non-empty subset of the attacking
UAVs to drop their munitions.
The problem can therefore be decomposed into several sub-problems. First, given M UAVs with N targets, assign each vehicle to a target, such that each
target has, if possible, multiple UAVs assigned to it,
and such that preference is given to high priority targets. Second, for each team of UAVs assigned to a
single target determine an estimated time-over-target
(TOT) that ensures simultaneous intercept and that
2
System Architecture
A detailed schematic of the system architecture for
a single UAV is shown in Figure 1. At the lowest
level of the architecture is the controlled UAV which
we assume to be equipped with trajectory tracking
controllers.
In Figure 1, the target manager, path planner, and
intercept manager work together to generate waypoint
paths for the UAV. The path planner generates k best
paths from the specified UAV to a specific target. In
addition, the path planner returns information about
each path, namely the estimated fuel expenditure and
the estimated threat exposure. Both the target manager and the intercept manager make calls to the path
∗ Assistant Professor, Electrical and Computer Engineering,
BYU, Provo, Utah 84602, [email protected], corresponding author.
† Associate Professor, Mechanical Engineering, BYU, Provo,
Utah 84602, [email protected]
‡ Assistant Professor, Computer Science, BYU, Provo, Utah
84602, [email protected]
0-7803-7272-7/02/$17.00 © 2002 IEEE
Michael Goodrich‡
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1
Communication
Manager
Target
Manager
Target
Target
&
Possible
paths
Team
Intercept
Manager
from the initial vehicle location to the target location
are derived from a k-best paths graph search [7] of a
Voronoi diagram that is constructed from the known
threat, vehicle, and target locations [12]. Figure 2
shows a Voronoi diagram created for a set of specified
threats, UAV location, and target location.
Best
path
Path
Planner
30
target
20
Target
3
3
10
Way−points
Flag
Trajectory
Generator
Desired Trajectory
Trajectory
Controlled
UAV
X (North)
1,2,4,5
0
1,2,4,5
1,2,4
2,3
−10
5
1,5
4
−20
start
Figure 1: System Architecture for a Single UAV.
−30
−10
10
20
Y (East)
30
40
50
Figure 2: Threat-based Voronoi diagram. The threats
are represented by dots.
planner. The role of the target manager is to assign
each UAV a target. The role of the intercept manager
is to ensure that when multiple UAVs are assigned
to the same target by the target manager, that they
arrive on the radar detection boundary of the target
simultaneously. The design of the path planner, target manager, and intercept manager, are discussed in
Section 2.1, Section 2.2, and Section 2.3 respectively.
The trajectory generator in Figure 1 receives a set
of waypoints, which specify the desired path of the
UAV. The waypoints received by the trajectory generator are not parameterized in time, they are simply a
set of X, Y , and h coordinates. The role of the trajectory generator is to generate a time trajectory for the
UAV that is feasible within its dynamic constraints.
The design of the trajectory generator is discussed in
Section 2.4.
The communication manager shown in Figure 1
facilitates communication between different UAVs.
Each UAV implements a separate target manager, intercept manager, and path planner. Therefore the
decisions reached by these functional blocks must be
synchronized among the different UAVs. The primary
role of the communication manager is to ensure this
synchronization. The design of communication managers for multiple cooperating autonomous vehicles
has been discussed in [11].
2.1
0
Each edge of the Voronoi diagram is assigned two
costs: a threat cost, specified as
Ã
!
N
X
1
1
1
+ 4
+ 4
Jthreat,i = Li
,
d41/6,i,j
d1/2,i,j
d5/6,i,j
j=1
(1)
where N is the total number of threats and d1/2,i,j is
the distance from the 1/2 point on the ith edge to the
jth threat, and a length cost, specified as Jlength,i =
Li . The total cost for traveling along an edge comes
from a weighted sum of the threat and length costs:
Ji = kJlength,i + (1 − k)Jthreat,i ,
(2)
where k ∈ [0, 1].
With the cost determined for each of the Voronoi
edges, the Voronoi diagram is searched to find the
set of lowest-cost candidate paths using a variation
of Eppstein’s k-best paths algorithm [7]. In Figure 2,
the five best paths to the target are shown.
A great advantage of the Voronoi diagram approach
is that it reduces the path-planning problem from
an infinite-dimensional search, to a finite-dimensional
graph search. This important abstraction makes the
path-planning problem feasible in near-real-time.
Path Planner
2.2
The Path Planner shown in Figure 1 is used by
both the Target Manager and the Intercept Manager.
The output of the Path Planner is a set of waypoints
and a commanded velocity for each vehicle. Paths
Target Manager
Assume that there are M targets, N UAVs, and P
threats. The task is to assign each vehicle to a target
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such that the overall group cost is mitigated, while
maximizing the number of targets that are destroyed.
In assigning targets to UAVs, there are four objectives. First, minimize the group path length to the targets, second, minimize group threat exposure, third,
maximize the number of vehicles attacking each target
(to maximize survivability), and fourth, maximize the
number of targets attacked. We refer to these objectives as the ShortPath, AvoidThreats, MaxForce, and
MaxSpread objectives respectively.
The ShortPath and AvoidThreats objectives are myopic objectives, whereas the MaxForce and MaxSpread
objectives are team objectives. The proposed solution
methodology is based on the satisficing paradigm [2]
which is used to select a set of potential targets that
are acceptable for each UAV. The set of targets that
are selected are then used by each agent to negotiate
an acceptable group assignment.
The first step is to use the path planner described
in the previous section to generate k best paths to
each target. Associated with these k best paths is
a median length cost, denoted as J¯length (Vi , Tj ) and a
median threat exposure cost denoted as J¯threat (Vi , Tj ),
where Vi is the ith UAV and Tj is the jth target. The
median operator throws away targets that have only
one good path but keeps targets that have at least k/2
reasonable paths.
For an individual UAV, close targets are deemed acceptable, while targets with large threat exposure are
deemed rejectable. Normalized measures of acceptability and rejectability can be defined as
given by
SVi = {Tj : µAi (Tj ) ≥ bi µRi (Tj )},
where bi is a selectivity index and can be used to ensure that the cardinality of SVi is greater than zero,
but small enough to enable an efficient search.
There are two competing objectives that affect how
the team behavior is evaluated: the MaxForce objective prefers assigning every vehicle to the same target, and the MaxSpread objective prefers attacking as
many targets as possible. These may be competing
objectives. We will address each in turn.
From the MaxForce perspective, the more vehicles
assigned to a target the better. We can represent this
objective by assigning a value to the team size for each
assigned target. In other words, we can identify a
number that encodes how much better a larger team
is than a smaller team. We use a monotonically increasing function that increases dramatically between
one and three team members, indicating that a minimally acceptable team consists of two members. The
monotonic function we selected is the sigmoid
UMaxForce (Gj ) =
µRi (Tj ) =
1
,
1 + e−3(m(Gj )−2)
where
m(Gj ) = |{Vi : Vi → Gj }|
´
´ ³
³
maxTk J¯length (Vi , Tk ) − J¯length (Vi , Tj )
´,
³
´
³
µAi (Tj ) =
maxTk J¯length (Vi , Tk ) − minTk J¯length (Vi , Tk )
³
(5)
(3)
´
J¯threat (Vi , Tk ) − minTk J¯threat (Vi , Tj )
³
´.
³
´
maxTk J¯threat (Vi , Tk ) − minTk J¯threat (Vi , Tk )
(4)
The acceptability function µA assigns the closest target the highest measure of acceptability (equal to one),
the most distant target the lowest measure of acceptability (equal to zero), and all other targets some assignment in the range [0, 1]. The rejectability function
µR assigns the target with the most threat exposure
the highest measure of rejectability (equal to one),
the target with the least threat exposure the lowest
measure of rejectability (equal to zero), and all other
targets some assignment in the range [0, 1]. Notice
that both of these functions can be computed in order kN M time.
For each vehicle, we can identify the set of assignments that are individually satisficing. This set is
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is the number of vehicles assigned to target Gj .
We can use the value of the team size, encoded
in UMaxForce (Gj ), to estimate the value of an assignment. We encode the value of the assignment as
Y
JMaxForce =
UMaxForce (Gj ).
(6)
Gj ∈{Gj :m(Gj )>0}
This value takes the product of each team-size value
over the set of assigned targets (the set of targets that
have at least one vehicle attacking it). Using the product has a double effect. First, it lets targets with small
team sizes dramatically reduce the value of the overall assignment. Second, it allows the preference for
large team sizes to be tempered by the preference for
having multiple targets attacked; i.e., it accounts for
the MaxSpread objective. Since it is unlikely that all
agents can feasibly attack the same target (it is unlikely that such an assignment is individually satisficing), the product avoids unbalanced team sizes (with,
for example, five agents assigned to one target and
only one to another target) from receiving high preference. Thus, multiple nearly equal-sized teams are
preferred to multiple, differently-sized teams.
For a scenario with five UAV’s and six targets, Figure 3 shows the targets assigned to each UAV and its
associated path as a solid line. In addition, the satisficing paths for the UAV in the bottom left hand corner
are plotted as dotted lines. Note the preference for
nearby targets and targets that do not require passing
too close to threats.
30
Resource Allocation Manager
team and target
assignments
mission
status
20
Cooperative Path Manager
2
subject to TOT∗ ∈ ∩ S
STOT,1
X (North) km
10
n
Choose TOT∗ to minimize Σ Ji
^
TOT∗
i=1
TOT,i
^
J1(TOT)
UAV n
0
UAV 2
UAV 1
1
−10
3
Determine feasible TOT range: STOT,1
^
Calculate coord. function: J1(TOT)
J1 = f(ξ1,V1)
−20
−30
−10
^
J1 = h(TOT)
TOT = g(ξ1,V1)
0
10
20
Y (East) km
30
40
Plan path (determine ξ1,V1)
to match TOT∗ that
minimizes J1(ξ1,V1)
path description
(to trajectory generator)
ξ1,V1
50
Figure 4: Cooperative Path Planning Algorithm
Figure 3: Target assignment for case study scenario.
Solid lines connect each vehicle to its assigned target. The dotted lines represent satisficing paths for
the UAV in the bottom left corner.
2.3
ordination function, is utilized. In this problem, the
coordination function Jˆi models the cost to UAV i of
achieving a particular TOT: Jˆi = h(T OT ). The coordination function is determined from the relations
for T OTi and Ji . Based on the candidate paths determined from the Voronoi diagram and the feasible
range of UAV air speeds, the feasible T OT range for
each UAV on the team can also be determined. The
set of feasible T OT ranges for UAV i is denoted by
ST OT,i . Coordination function information for two
UAVs composing a team is shown in Figure 5. Note
that each candidate path produces a line segment in
the plot.
Intercept Manager
Once teams have been formed and targets assigned
to each team, it is necessary to plan the trajectories
that each vehicle will take to its respective target. In
military applications, such as the one presented here,
the ability to prescribe and coordinate the timing of
target interception is critically important. In the architecture developed here, the vehicles composing a
team are required to strike their target simultaneously to maximize the element of surprise and enhance
mission effectiveness. The challenge becomes one of
determining the best time over target (TOT) specification for the team in light of the threat scenario
and dynamic capabilities of the vehicles. The output
of the intercept manager is a set of waypoints and a
commanded velocity for each vehicle. The waypoints
specified avoid the threats and allow for simultaneous
arrival of the vehicles at the assigned target.
The strategy for cooperatively planning UAV paths
employed here is presented in detail in [4]. Figure 4
depicts graphically the steps of the cooperative path
planning algorithm.
Candidate paths to the target for UAV i are parameterized by the waypoints ξi and the UAV forward
speed Vi . Time over target for each UAV is a function of the speed of the UAV along the selected path:
T OTi = f (ξi , Vi ). Similarly, the cost (threat and fuel)
to travel along any path to the target is a function
of the path and the speed: Ji = g(ξi , Vi ). To choose
the best TOT for the team in a cooperative manner,
some information regarding the candidate paths for
the UAVs must be exchanged. Rather than passing
ξi and Vi around among all of the UAVs, a more efficient parameterization of information, called the co-
0.6
0.5
path cost
0.4
0.3
0.2
UAV 3
0.1
UAV 4
0
100
150
200
250
300
350
400
450
500
TOT (sec)
Figure 5: Coordination Functions for Two UAVs
Coordination function and feasible T OT range information is exchanged among vehicles to enable cooperative path planning. At the team level, the cooperative planning problem is simplified to finding T OT ∗
that minimizes the collective threat exposure of the
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4
team
T OT ∗ = arg min
n
X
Jˆi
local reachability region
i=1
subject to T OT ∗ ∈ ∩ST OT,i . From Figure 5 it can be
seen that this optimization results in a team-optimal
value of T OT ∗ = 160.5 sec. Time over target T OT
is called the coordination variable. By requiring individual UAVs to match the team-optimal value of
the coordination variable T OT ∗ , cooperation is ensured. We note that this optimization problem has
been greatly simplified by searching over the coordination variable instead of the original path parameters (ξi , Vi ) of the system. For example, in the coordinated strike problem, the coordination variable is
one-dimensional (T OT ), whereas the original path parameterization is of much higher dimension.
Once T OT ∗ has been determined, each UAV must
determine which of the candidate paths ξi to take and
the appropriate flight speed Vi so that its own threat
exposure Ji = f (ξi , Vi ) is minimized and the T OT ∗
value is matched. In implementation, the best path
for a specified T OT was determined when the coordination function was computed. At this point, it is
simply a matter of looking up the trajectory waypoints
in memory and computing the appropriate velocity.
Waypoint and velocity information is then supplied to
the trajectory generator on the UAV.
2.4
u_1 = − c
u_1 = + c
x
y
Figure 6: Local reachability region of the trajectory
generator.
Ri = Vid /c. Note that as the desired velocity increases
the minimum turning radius increases.
Consider the problem of switching from one
straight-line segment to another in minimum time at a
constant velocity. If the trajectory is not constrained
to pass through the waypoint connecting the two lines,
then a time-optimal trajectory is shown as a dashed
line in Figure 7.
Trajectory Generator
Given a set of waypoints and the desired velocity Vid , the task of the Trajectory Generator (TG) is
to generate time-parameterized trajectories that are
feasible within the dynamic constraints of the vehicle. The essential idea is to use a nonlinear filter that
has a mathematical structure similar to the kinematics of the vehicle to generate trajectories that smooth
through the waypoints in real-time, while the vehicle traverses the trajectory. The UAV dynamics given
in [13] suggest the following structure for the Trajectory Generator for constant altitude maneuvers:
waypoint
u_1 = −c
u_1 = + c
u_1 = −c
u_1 = + c
Ẋid = Vid cos ψid
Ẏid = Vid sin ψid
ψ̇id = u1
ḣdi = 0,
(7)
Figure 7: Time-optimal transitions between path segments. The dashed line is not constrained to go
through the waypoint. The solid line is constrained
to pass through the waypoint.
where u1 is constrained by the dynamic capability of
the UAVs, namely the heading rate constraint −c ≤
u1 ≤ c. Note that if u1 = c, then the trajectory
generator given in Equation (7) traces out a righthanded circle as shown in Figure 6. Similarly if u1 =
−c then the TG traces out a left-handed circle. As
shown in Figure 6, the local reachability region of the
TG is bounded by these two circles. The radius of the
circles defining the local reachability region is given by
If (Xid , Yid , ψid , Vid ) is the configuration of the TG,
then the center of the two circles that bound the reach-
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5
ability region are given by
zright =
µ
Xid
Yid
¶
zleft =
µ
Xid
Yid
¶
+
Ã
+
Ã
Vid
d
c sin(ψi )
Vid
d
c cos(ψi )
−
made, the UAVs composing each team must coordinate to find a feasible team time-over-target (T OT ).
The target assignment problem was solved using
the satisficing and social welfare paradigms. The T OT
coordination problem was solved by encapsulating the
essential myopic information in coordination functions
that were then used to optimize the team T OT . The
path planning problem was solved via a Voronoi diagram and Eppstein’s k-best paths algorithm. The
trajectory generation problem was solved via a novel
real-time nonlinear filter that explicitly accounts for
the dynamic constraints of the vehicle.
!
Vid
d
c sin(ψi )
d
V
− ci cos(ψid )
!
Since the boundaries of the reachability region is given
by circles with known center and radius, finding the
intersection of the reachability region with lines and
circles can be done in a computationally efficient way.
A simple, real-time algorithm for trajectory generation that either guarantees minimum-time transition
between paths, or minimum-time transitions that are
constrained to go through the waypoint, is therefore
suggested by Figure 7, and is analyzed in detail in [14].
One of the disadvantages of both minimum time
transitions and transitions constrained to go through
the waypoint, is that the trajectories will have a different path length than the original Voronoi path.
Since the Voronoi path is used for determining intercept times, we would like to transition between path
segments, such that the path length of the feasible
trajectory is equal to the path length of the original
Voronoi path. From Figure 7 it is obvious that the
path length of a minimum time trajectory will be less
than the path length of the Voronoi path, and that
the path length of the constrained trajectory will be
greater than the path length of the Voronoi path. Let
κ ∈ [0, 1] parameterize the location of the center of
the turning radius circle along the bisector of the angle shown in Figure 7. The algorithm suggested by
Figure 7 can be modified with a preliminary step to
find κ such that the path lengths are equal [14].
The above algorithms are easily modified to account
for short path segments that are smaller than the radius Ri . To account for those cases, the trajectory
generator must look ahead several path segments.
There are several advantages to our approach.
First, it couples nicely with the Voronoi search algorithms described in the previous section. Second, the
approach has low computational overhead. In fact,
trajectories are generated in real-time, as the vehicles
move. Finally, it can be shown using optimal control
techniques that the approach minimizes the time that
the vehicle deviates from the Voronoi path.
Acknowledgments
This work was partially funded by AFOSR award
no. F49620-01-1-0091. The technical guidance of
Dr. Phil Chandler at AFRL is gratefully acknowledged. The authors would like to thank Mr. Erik Anderson for assistance with developing the trajectory
generator and Mr. Kevin Judd for assistance with developing the path planner.
References
[1] R. E. Burkard, “Selected topics in assignment problems,” tech. rep., Institute of Mathematics B, Technical University Graz, Movember 1999. To
appear in Discrete Applied Mathematics.
[2] M. A. Goodrich, “A satisficing approach to assigning vehicles to targets,”
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control of UAV rendezvous,” in Proc. of the ACC, June 2001.
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3
[12] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications. Springer-Verlag, 2000.
Conclusions and Discussion
[13] A. W. Proud, M. Pachter, and J. J. D’Azzo, “Close formation flight control,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference
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In this paper we have developed a complete system
architecture for the target assignment and coordinated
intercept problem. Coordination between UAVs take
place at two levels. At the highest level, the UAVs
must negotiate a target assignment vector that assigns
UAVs to targets. Once a target assignment has been
[14] R. W. B. Erik P. Anderson, “Constrained extremal trajectories and uav
path planning,” in Proceedings of the IEEE International Conference on Robotics
and Automation, (Washington D.C.), May 2002. (submitted).
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