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Efficient planning of informative paths for
multiple robots
Amarjeet Singh*, Andreas Krause+, Carlos Guestrin+, William J. Kaiser*, Maxim Batalin*
* Center for Embedded Networked Sensing, University of California, Los Angeles+ Machine Learning Department, Carnegie Mellon University
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Predicting spatial phenomena in large environments
Constraint: Limited fuel for making observations
Fundamental Problem: Where should we observe to maximize the collected information?
Biomass in lakes Salt concentration in rivers
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How to quantify collected information?
Mutual Information (MI): reduction in uncertainty (entropy) at unobserved locations
[Caselton & Zidek, 1984]
MI = 4Path length = 10
MI = 10Path length = 40
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Selecting the sensing locations
Lake Boundary
G1
G2
G3
G4
Greedy selection of sampling
locations is (1-1/e) ~ 63% optimal [Guestrin et. al, ICML’05]
Result due to Submodularity of MI: Diminishing returns
Greedy may lead to longer paths!
Greedily select the locations that provide the most amount of information
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Greedy - reward/cost maximization
Available Budget = B
s
Reward = B
Cost = B
rewardcost
= 2
rewardcost
= 1
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Greedy - reward/cost maximization
Available Budget = B-
s
B
B
BToo far!
Greedy Reward = 2
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Greedy - reward/cost maximization
Available Budget = 0
s
B
B
Greedy Reward = 2
Optimal Reward = B
Greedy can be arbitrarily poor!
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Informative path planning problem
maxp MI(P) MI – submodular function
Lake Boundary
Start- sFinish- t
P
C(P) · B Informative path planning – special
case of Submodular Orienteering Best known approximation algorithm –
Recursive path planning algorithm [Chekuri et. Al, FOCS’05]
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Recursive path planning algorithm
[Chekuri et.al, FOCS’05]
Start (s)Finish (t)
vm
Recursively search middle node vm
P1
P2
Solve for smaller subproblems P1 and P2
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vm2
Recursive path planning algorithm
[Chekuri et.al, FOCS’05]
Start (s)Finish (t)
P1vm1
vm3
Maximum reward
Recursively search vm C(P1) · B1
Lake boundary
vm
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Recursive path planning algorithm
[Chekuri et.al, FOCS’05]
Start (s)Finish (t)
P1
vm
Recursively search vm C(P1) · B1
Commit to the nodes visited in P1
Recursively optimize P2 C(P2) · B-B1
P2
Maximum reward
Committing to nodes in P1 before optimizing P2 makes the algorithm greedy!
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Quasi-polynomial running time O(B*M)log(B*M)
B: Budget
RewardChekuri ¸RewardOptimal
log(M) M: Total number of nodes in the graph
60 80 100 120 140 160
Cost of output path (meters)
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Exe
cutio
n T
ime
(S
eco
nd
s)OOPS!
Small problem with 23 sensinglocations
Recursive path planning algorithm[Chekuri et.al, FOCS’05]
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60 80 100 120 140 16010
0
105
102
103
104
101
Exe
cutio
n T
ime
(se
con
ds)
Cost of output path (meters)
Almost a day!!
Recursive path planning algorithm[Chekuri et.al, FOCS’05]
Quasi-polynomial running time O(B*M)log(B* M)
B: Budget
RewardChekuri ¸RewardOptimal
log(M) M: Total number of nodes in the graph
Small problem with 23 sensinglocations
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Our contributions
Algorithm with significantly improved running time exploiting recursive path planning
Spatial decomposition of sensing region Branch and bound - Calculating bounds using
submodularity and other heuristics to prune search space
Extended single robot path planning to multiple robots with strong approximation guarantee
Extensive empirical evaluation on several real world sensing datasets
Including data collected using robotic boat at Lake Fulmor, California
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Lake Boundary
Spatial decomposition into cells
Ending node t
Starting node s
Ending Cell Ct
Starting Cell Cs
Search for middle Cell Cm
Perform recursive path planning on cells
P1P2
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Ending Cell Ct
Starting Cell Cs
Middle Cell Cm
P1P2
Greedily select locations without path cost constraint: 1-1/e optimal
Node selection inside the cell
Incoming path Exiting path
G1
G2 G4
G3
Tradeoff: Larger cell size ) Faster Execution, Increased additional traveling
cost Smaller cell size ) Slower Execution, Reduced additional traveling
cost
Small cells: Traveling cost
inside cell can be ignored
Additional cost for traveling to the
sensing locations
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Recursive Path Planning
Approximation guarantees
Running time:
O((B*N)log B*N)
Required budget:O(B)
Collected Reward ¸(1-1/e) RewardOptimal
log(N) N: Total number of cells in the graph
80 100 120 140 160Cost of output path (meters)
6010
0
105
102
103
104
101
Exe
cutio
n T
ime
(se
con
ds)
Efficient Path Planning
Approx. a day
Approx. 2 min.
Small problem – 23 sensing locations
Too slow for larger problems!
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Further improvement in running time
Search space represented as SUM-MAX tree (similar to AND-OR tree)
102
103
104
105
Exe
cutio
n T
ime
(se
con
ds)
200 250 300 350 400 450Cost of output path (meters)
Upto 400 meters calculated within approx. 15 min.
Pruned search space using branch and bound Upper bound exploiting
submodularity Lower bound exploiting
known heuristic [Chao et. al’
96] Several other tricks – see
paper
Larger problem – 109 sensing locations
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Multi robot informative path planning problem
maxP1,P2,P3 MI(P1 U P2 U P3)
MI – submodular function
s t
C(P1) · B; C(P2) · B; C(P3) · B
P2
P3
P1
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Multi robot path planning – Simple sequential allocation approach
s t
P2
P3
P1
Use algorithm for single robot instance to find path P1 for the first robot
Optimize for second robot (P2) committing to nodes in P1
Optimize for third robot (P3) committing to nodes in P1 and P2
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Performance evaluation
Works for any single robot path planning algorithm Independent of number of robots used
RewardMR¸
RewardOptimal
1 +
Greedy selection of
nodes with no path cost constraintArbitrarily Poor
Recursive Greedy path planning
RewardRG ¸RewardOpt
(=log(M))
Sequential allocation for multiple robots – Greedy over paths
??
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Efficient multi-robot information path planning
Spatial Decomposition Obtain cell path exploiting submodularity, branch and bound
A
B C
D
Greedy node selection within visited cells to get node path
Sequential Allocation for multi-robot path planning
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Robotic boat measuring surface temperature and chlorophyll at
Lake Fulmor, California
Empirical evaluation
SERVER
LAB
KITCHEN
COPYELEC
PHONEQUIET
STORAGE
CONFERENCE
OFFICEOFFICE50
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3739
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242526283032
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2729
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1
2
3540
52 sensor motes used to monitor temperature at Intel Research
laboratory, Berkeley
Precipitation data collected from 167 regions in Pacific NW, during
the years 1949-1994
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102
103
104
105
Exe
cutio
n T
ime
(sec
onds
)
10 15 20 25 30 35Cost of output path (meters)
Empirical evaluation – varying the cell size
Precipitation Dataset
No. of cells = 36 No. of cells = 25 No. of cells = 16
Low
er is be
tter
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Hig
her
info
rma
tion
qu
ality
10 15 20 25 30 35Cost of output path (meters)
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Hig
her
info
rma
tion
qu
ality
60 80 100 120 140 160Cost of output path (meters)
ChekuriAlgorithm
Proposed EfficientAlgorithm
Intel Laboratory Temperature Dataset
Empirical evaluation – reward comparison
• Reduced execution time by several factors• Similar collected reward
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Empirical evaluation – heuristic comparison
Lake Temperature Dataset
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6
8
10
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Hig
her
info
rma
tion
qu
ality
200 250 300 350 400 450Cost of output path (meters)
Efficient informative path planning algorithm
Known heuristic [Chao et. al’ 96]
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8
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To
tal R
MS
Err
or
200 250 300 350 400 450
Cost of output path per robot (meters)
Empirical evaluation – multi robot
Robot-1
Robot-3
Robot-2
1 Robot
2 Robots
3 Robots
Low
er is be
tter
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Conclusions
First efficient multi robot informative path planning algorithm with performance guarantee Exploits spatial decomposition Exploits submodularity and other heuristics for branch
and bound Near optimal extension of single robot path
planning algorithm to multiple robots Extensive empirical evaluation on several
real world sensor network datasets Including data collected using robotic boat in real lake Planning on a deployment at Lake Merced, California
with robotic boat in February
