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Efficient planning of informative paths for multiple robots

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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

Salt concentration in rivers

Biomass in lakes

Constraint:Limited fuel for making observations

Fundamental Problem:Where should we observe to maximize the collected information?

MI = 4

Path length = 10

MI = 10

Path length = 40

- Mutual Information (MI): reduction in uncertainty (entropy) at unobserved locations
[Caselton & Zidek, 1984]

G1

G4

G2

G3

Lake Boundary

Greedy selection of sampling locations is (1-1/e) ~ 63% optimal

[Guestrin et. al, ICML’05]

- Result due to Submodularity of MI:
- Diminishing returns

Greedily select the locations that provide the most amount of information

Greedy may lead to longer paths!

reward

reward

= 2

= 1

cost

cost

Available Budget = B

Reward = B

2

Cost = B

s

B

B

2

B

Too far!

Available Budget = B-

Greedy Reward = 2

s

B

2

B

Greedy can be arbitrarily poor!

Available Budget = 0

Greedy Reward = 2

Optimal Reward = B

s

Lake Boundary

maxp MI(P)

- MI – submodular function

C(P)·B

- Informative path planning – special case of Submodular Orienteering
- Best known approximation algorithm – Recursive path planning algorithm
[Chekuri et. Al, FOCS’05]

P

s

Start-

t

Finish-

- Recursively search middle node vm

- Solve for smaller subproblems P1 and P2

Start (s)

P2

Finish (t)

P1

vm

vm

vm3

vm1

vm2

Lake boundary

- Recursively search vm
- C(P1) · B1

P1

Start (s)

Finish (t)

Maximum reward

Committing to nodes in P1 before optimizing P2 makes the algorithm greedy!

- Recursively search vm
- C(P1) · B1

- Commit to the nodes visited in P1

- Recursively optimize P2
- C(P2) · B-B1

Maximum reward

P1

Start (s)

P2

Finish (t)

vm

RewardOptimal

log(M)

5000

4500

4000

3500

3000

Execution Time (Seconds)

2500

2000

1500

1000

500

0

60

80

100

120

140

160

Cost of output path (meters)

- RewardChekuri

¸

- M: Total number of nodes in the graph

- Quasi-polynomial running timeO(B*M)log(B*M)
- B: Budget

OOPS!

Small problem with 23 sensing

locations

5

10

RewardOptimal

4

10

log(M)

3

10

Execution Time (seconds)

2

10

Almost a day!!

1

10

0

10

60

80

100

120

140

160

Cost of output path (meters)

- RewardChekuri

¸

- M: Total number of nodes in the graph

- Quasi-polynomial running timeO(B*M)log(B* M)
- B: Budget

Small problem with 23 sensing

locations

- Algorithm with significantlyimproved 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

Ending Cell Ct

Ending node t

P1

P2

Starting node s

Lake Boundary

Search for middle Cell Cm

Starting Cell Cs

Perform recursive path planning on cells

Incoming path

Exiting path

Ending Cell Ct

G1

G3

P1

G2

G4

P2

Middle Cell Cm

Starting Cell Cs

Small cells:Traveling cost inside cell can be ignored

Additional cost for traveling to the sensing locations

- Greedily select locations without path cost constraint: 1-1/e optimal

- Tradeoff:
- Larger cell size)Faster Execution,Increased additional traveling cost
- Smaller cell size)Slower Execution,Reduced additional traveling cost

(1-1/e) RewardOptimal

log(N)

5

10

Recursive Path Planning

Approx. a day

4

10

Approx. 2 min.

3

10

Execution Time (seconds)

2

10

Efficient Path Planning

1

10

0

10

60

80

100

120

140

160

Cost of output path (meters)

Too slow for larger problems!

- Collected Reward

¸

- N: Total number of cells in the graph

- Running time:
O((B*N)log B*N)

Required budget:O(B)

Small problem – 23 sensing locations

Upto 400 meters calculated within approx. 15 min.

5

10

4

10

Execution Time (seconds)

3

10

2

10

200

250

300

350

400

450

Cost of output path (meters)

Larger problem – 109 sensing locations

Search space represented as SUM-MAX tree (similar to AND-OR tree)

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

maxP1,P2,P3 MI(P1UP2UP3)

- MI – submodular function

C(P1)·B; C(P2)·B; C(P3)·B

P3

P1

s

t

P2

- 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

P3

P1

s

t

P2

RewardOptimal

Sequential allocation for multiple robots –Greedy over paths

Greedy selection of nodes with no path cost constraint

Recursive Greedy path planning

1 +

RewardOpt

RewardRG

¸

(=log(M))

Arbitrarily Poor

??

RewardMR

¸

- Works for any single robot path planning algorithm
- Independent of number of robots used

Obtain cell path exploiting submodularity, branch and bound

Spatial Decomposition

A

D

Sequential Allocation for multi-robot path planning

Greedy node selection within visited cells to get node path

B

C

Precipitation data collected from 167 regions in Pacific NW, during the years 1949-1994

52 sensor motes used to monitor temperature at Intel Research laboratory, Berkeley

Robotic boat measuring surface temperature and chlorophyll at Lake Fulmor, California

5

10

8

7

4

10

6

Higher information quality

Execution Time (seconds)

5

3

4

10

3

2

2

10

10

15

20

25

30

35

10

15

20

25

30

35

Cost of output path (meters)

Cost of output path (meters)

No. of cells = 36

No. of cells = 16

No. of cells = 25

Lower is better

Precipitation Dataset

10

9

Chekuri

Algorithm

8

Higher information quality

7

6

Proposed Efficient

Algorithm

5

4

60

80

100

120

140

160

Cost of output path (meters)

- Reduced execution time by several factors
- Similar collected reward

Intel Laboratory Temperature Dataset

14

12

10

Higher information quality

8

6

4

200

250

300

350

400

450

Cost of output path (meters)

Efficient informative path planning algorithm

Known heuristic

[Chao et. al’ 96]

Lake Temperature Dataset

1 Robot

16

15

14

13

2 Robots

Total RMS Error

12

3 Robots

11

10

9

8

200

250

300

350

400

450

Cost of output path per robot (meters)

Robot-3

Robot-1

Robot-2

Lower is better

- 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