Search Problem
2008-10-20
FL seminar
Search Problem
Cooperative Optimal Search Control
Centralized Control and Distributed Control
FL08-15-2
To locate a target to be found,
deploying agents with the available resources.
„ search and rescue operations
„ detecting lost objects
Objective
„ formulate the Optimal Search Control Problem
Mamoru Saito
‰ maximize the probability of finding the target
‰ reduce the control energy consumption
„ analyze the agent’s behavior
Outline
Types of Search Problems
OneOne-sided Search
the searcher chooses his strategy
the target neither chooses a
strategy nor reacts to the search
„ review (single agent)
the searcher and the target
can choose their strategies
ex.
ex.
„ Cooperative Search
TwoTwo-sided Search
Stationary ‰ search for lost car keys
‰ hide-and-seek
Target ‰ search for natural resources
‰ Centralized Control
Cooperative Search
ex.
‰ Distributed Control
Moving
Target
ex.
‰ search for
a life raft in the ocean
‰ the target moves randomly
the target attempts to make
itself as detectable as possible
NonNon-Cooperative Search
the target tries
to remain undetected
ex. Pursuit-Evasion Game
Outline
Problem Setting
Search Area :
„ review (single agent)
the position
of the agent
„ Cooperative Search
‰ Centralized Control
‰ Distributed Control
observation
point
target
1
Problem Setting
continuous-time linear system
agent：continuous-
Main Result
observation time (obs. time) :
„ formulate Optimal Search Control Problem
‰ approximate solution
„ the analysis of the agent’s behavior
obs. point
position：
‰ the expectation value of how many times the targets appears
without being found
‰ a necessary and sufficient condition for the agent’s state and
input to converge respectively to a periodic trajectory
waypoint
the obs. points set
the obs. points row
position
obs. point
1
random walk based
proposed method
4
A target appears randomly and stays for
[s]
The target appears at time
1
0
−1
−2 −2
−1
0
1
2
search level ( small → good )
3
search level
good bad
1
0.8
0.6
0.4
0.2
0
2
2
2
sensing accuracy
z −displacement[m]
0.8
1
0.2
0
0
1
initial position
Outline
„ review (single agent)
0.6
0.465
0.4
2
3
z1−displacement[m]
4
5
0
0
5
10
15
j
Cooperative Search
Cooperative Search is useful for more efficient search.
„ Centralized Control
„ Distributed Control
„ Cooperative Search
: the number of the agent
‰ Centralized Control
‰ Distributed Control
remark :
The superscript
Centralized Cooperative Search
Centralized Cooperative Optimal Search Control Problem
means the state of
th agent.
Simulation
system
s.t.
solution:
reduce a combinatorial optimization
sort
to minimize the control energy consumption
2
Simulation (g=1)
Simulation (g=1)
1
3
2
1
0
3
0.6
0.4
0.357
0.2
2
2
3
z −displacement[m]
4
2
4
6
1000
0
0
8
1
2
3
z −displacement[m]
4
2
4
6
8
10
control energy consumption
5
1
random walk based method
Simulation (g=6)
1
8000
random walk based
proposed method
search level
0.8
4
z2−displacement[m]
4
3
2
1
0
3
0.6
0.4
0.2
2
0
0
5000
4000
3000
2000
2
4
6
8
10
12
0
0
14
5
j
search level
0
4
6000
1000
0.161
1
random walk based
proposed method
7000
control energy consumption
20 steps
z2−displacement[m]
1500
step k
Simulation (g=6)
2
3
z −displacement[m]
2000
search level
0
1
1
2500
j
5
proposed method
0
random walk based
proposed method
3000
500
0
0
1
0
1
control energy consumption
4
z2−displacement[m]
z2−displacement[m]
search level
0.8
4
0
3500
random walk based
proposed method
10 steps
5
0
1
1
2
3
z −displacement[m]
4
10
step k
15
20
control energy consumption
5
1
proposed method
random walk based method
Simulation (g=k+f)
Simulation (g=k+f)
( for stationary target )
1
3
2
1
3
2
3
z1−displacement[m]
4
proposed method
5
0.6
0.4
0.2
2
random walk based
proposed method
3000
2500
2000
1500
1000
500
0
0
1
2
4
step k
search level
0
0
1
control energy consumption
4
z2−displacement[m]
z2−displacement[m]
search level
0.8
4
0
3500
random walk based
proposed method
8 steps
0
1
2
3
z1−displacement[m]
4
6
8
0
0
2
4
step k
6
8
control energy consumption
5
random walk based method
3
Outline
Distributed Cooperative Search
proposed method
1. divide the search area at time
shared search area
„ review (single agent)
2. Each agent computes the optimal control input.
„ Cooperative Search
Objective
‰ Centralized Control
1. share the search area optimally
‰ Distributed Control
2. maximize the probability of finding the target
3. minimize the control energy consumption
Multi-Objective Optimization
Divide the Search Area
Discrete-Time Lloyd Algorithm
Voronoi Partition
Lemma
(19)
⇒
(locally) minimizes
ref.
proof: omitted.
Distributed Control Algorithm (Outline)
J. Cortes, S. Martinez, T. Karatas, and F. Bullo.
Coverage Control for Mobile Sensing Networks.
IEEE Transactions on Robotics and Automation, 20(2), pp.243-255, 2004.
Distributed Cooperative Search
Distributed Cooperative Optimal Search Control Problem (DCOSCP)
at time
and renew the Voronoi Partition
.
at the other time
optimal search control in
remark :
Voronoi Partition may not be optimal division
in this approach to cooperative search.
s.t.
s.t.
if
This is a synchronous method.
4
Distributed Control Algorithm
Simulation (g=1)
20 steps
4
z2−displacement[m]
z2−displacement[m]
4
3
2
1
0
1
2
3
z1−displacement[m]
4
1
0
5
Simulation (g=1)
1
4
5
random walk based method
Simulation (g=4)
6000
0.6
0.4
0.2
0
0
5
10
j
random walk based
proposed method
5000
4
3000
2000
1000
0
0
15
search level
40 steps
4000
5
10
step k
15
20
control energy consumption
4
z2−displacement[m]
0.8
z2−displacement[m]
control energy consumption
random walk based
proposed method
3
2
1
0
3
2
1
0
0
1
2
3
z −displacement[m]
4
5
0
1
1
proposed method
Simulation (g=4)
1
control energy consumption
0.8
0.6
0.4
0.2
12000
5
10
15
20
j
search level
25
30
35
4
5
1
random walk based method
Conclusion
random walk based
proposed method
Centralized Cooperative Optimal Search Control
Distributed Cooperative Optimal Search Control
10000
8000
6000
Future Works
4000
2000
0
0
2
3
z −displacement[m]
Conclusion and Future Works
14000
random walk based
proposed method
search level
2
3
z −displacement[m]
1
proposed method
search level
2
0
0
1
3
0
0
Another Distributed Search method
10
20
step k
30
40
control energy consumption
‰ how to divide the search area
‰ asynchronous
Collision Avoidance
5