Bicriteria trajectory planning in demining operations

Bicriteria trajectory planning in demining

operations

Michael Morin1 Armand Fruitet2 Irène Abi-Zeid3

Claude-Guy Quimper1

1Dep. of Computer Science and Software Engineering, Université Laval, Québec, Canada2Dép. SES, Université Toulouse le Mirail, Toulouse, France

3Dep. of Operations and Decision Systems, Université Laval, Québec, Canada

B [email protected]

This work has been possible due to the collaboration with Yvan R. Petillot from the OceanSystems Lab, Edinburgh, Scotland, UK

May 28, 2014

M. Morin et al. (U.Laval, U.T.leMirail) Hybrid Algorithm for CPPIED May 28, 2014 1 / 44

IntroductionCoverage?

Coverage path planning (CPP) problem [Choset, 2001]

Plan an agent's path to guarantee complete coverage of theenvironment [Mannadiar and Rekleitis, 2010]

Objectives

Minimize the traveled distanceMinimize the number of turns

O�-line and sensor-based approaches

Usual order of priority of the objectives: distance then turns


IntroductionContext

Context [Reed et al., 2003]

Underwater minesweeping operations using robots

A naval mine [Lippsett, 2004] A robot underwater [Linder, 2006]


IntroductionContext

The robots: REMUS1 [Nicholson and Healey, 2008, Fang and Anstee, 2010]

Swim long distancesConstant speed and altitudeInfrequent turns

A REMUS 100 swimming in Western Greenland [Kukulya, 2012]

1Remote Environmental Monitoring UnitS


IntroductionContext

SensorsSidescan sonar�Gap �ller� forward looking sonar

A side-scan sonar [Wikipedia, 2013]


Coverage path planning with imperfect extended detectionSpeci�c formalism

Coverage path planning problem with imperfect extended detection(CPPIED) [Drabovich, 2008]

Imperfect sensors: (conditional) detection probability [Gage, 1995]

Minimal required coverage instead of complete coverageExtended detection range

Objectives

Minimize the traveled distanceMinimize the number of turns

O�-line path planning

Grids of square cells


Coverage path planning with imperfect extended detectionSeabed types

ff

f

r

c

c

c

f

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r

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f complex

sand ripples

�at

The di�erent seabed type of a cell in�uences the sensor's performance.


Coverage path planning with imperfect extended detectionSensors scans

1 scan

2 scans

3 scans

4 scans

Scans with a range

of r = 3 cells; darker

cells have a higher

scan frequency.


Coverage path planning with imperfect extended detectionMinimal required coverage

De�nition

A feasible path achieves the minimal required coverage in each cell of

matrix D.

ff

f

r

c

c

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Minimal required coverage on a seabed grid


Coverage path planning with imperfect extended detectionConditional detection probability (imperfect sensor)

De�nition

The conditional detection probability of a scan in a given cell is a function

of

the seabed type, and

the distance (and the maximal lateral range r).


Coverage path planning with imperfect extended detectionConditional detection probability

.9

.9

.8

.8

.9

.9

.9

.9

.8

.8

.9

.9

.6

.4

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.1

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complex

sand ripples

�at

Scans with a range of r = 3

cells on a heterogeneous

seabed

The distance between the robot's position and the scanned cell in�uences

the sensor's performance.

pscan =

pscan(1, c) pscan(2, c) pscan(3, c)pscan(1, r) pscan(2, r) pscan(3, r)pscan(1, f) pscan(2, f) pscan(3, f)

=

0.7 0.3 0.10.8 0.4 0.20.9 0.6 0.3


Coverage path planning with imperfect extended detectionCoverage

De�nition

Let C be the current coverage matrix.a

aThe initial coverage is null.


Coverage path planning with imperfect extended detectionUpdated coverage

De�nition

C′ is the updated coverage matrix after one or more scans, i.e., C′ij is the

(bayesian) updated coverage in cell (i , j) after a scan in (i , j):

C′ij := Cij + (1− Cij)× pscan(d(x ′, y ′, i , j),Oij).

pscan =

pscan(1, c)pscan(1, r)pscan(1, f)

=

0.70.80.99

ff

f

r

c

c

c

f

f

r

r

r

r

f

f

r

r

f

f

f

f

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f

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f

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r

r

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f

.99

.99

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.7

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≈1

≈1

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96

≈1



The following cases may occur in our bayesian update:

a mine may have already been detected in (i , j) with a probability Cij ;orit has not been detected before with a probability 1− Cij , and

it will be detected now with a conditional probabilitypscan(d(x , y , i , j),Oij ).

The updated coverage of a cell (i , j), C′ij , is de�ned as:

Previously detected︷︸︸︷Cij +︸︷︷︸

or

No mine detected yet︷︸︸︷(1− Cij) ×︸︷︷︸

and

A mine is detected now︷︸︸︷pscan(d(x ′, y ′, i , j),Oij),

where cell (x ′, y ′) is the current robot position.








or


and










or


and










or


and










or


and




Coverage path planning with imperfect extended detectionProblem instance

De�nition

A CPPIED problem instance consists of:

a seabed type set T ;a seabed matrix O;

a required coverage matrix D;

a conditional detection function pscan; and

an initial position (i init, j init) (optional).

pscan =


=

0.70.80.99

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f

r

c

c

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r

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f

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.5

.5


Coverage path planning with imperfect extended detectionE�cient coverage path

De�nition

An e�cient path achieves the required coverage and minimizes

the length of the path (distance, robot's moves, etc.); and

the number of turns.

pscan =


=

0.70.80.99

ff

f

r

c

c

c

f

f

r

r

r

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f

f

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f

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≈1

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96

≈1


An Algorithmic StoryA complex coverage path

More than 21,000 cells of a complex seabed topology

Path planning on a cell by cell basis is not realistic due to the

intractability of the problem...


An Algorithmic StoryA complex coverage path

More than 21,000 cells of a complex seabed topology

... clever algorithms are required.


The HCPP algorithm [Drabovich, 2008]

The problem was �rst a lexicographic optimization problem:

Heterogeneous Coverage Path Planning algorithm [Drabovich, 2008]



Parameterized Heuristic

HCPP is a parameterized heuristic algorithm

Construct the path segment by segment.

The segment length is entirely determined by the algorithms

parameters.




The HCPP algorithm does the following:1 Con�gure the parameters on a given instance or set of instances.

The parameters that lead to the shortest path with the lowest numberof turns are retained.

2 Using the parameters from the con�guration phase:

Generate one path segment.Link the segment to the current path.If the required coverage is not achieved, iterate.



We started to explore the problem in its bicriteria form



6500 7000 7500 8000 8500140

160

180

200

220

240

260

280

300

320

340

Path length

Numnber

ofturns

HCPP solutions



6500 7000 7500 8000 8500140

160

180

200

220

240

260

280

300

320

340

Path length

Numnber

ofturns

HCPP solutionsHCPP Pareto




The HCPP algorithm does the following:1 First con�gure the parameters on a given instance or set of instances.

The parameters that lead to the shortest path with the lowest numberof turns are retained.







The parameters that lead to the shortest path with the lowest numberof turns are retained.We may as well retain the parameters which minimize the number of

turns and then the length.







The parameters that lead to the shortest path with the lowest numberof turns are retained.We may as well retain the parameters which minimize the number of

turns and then the length.



Alternatively we may retain all the solutions and approximate the Pareto

front.


The DpSweeper algorithm [Morin et al., 2013]

Dynamic Programming Sweeper algorithm [Morin et al., 2013]



Hybrid?

Dynamic programming

Traveling salesman problem reduction

External solver: Concorde solver [Applegate et al., 2011]

... The Dynamic Programming Sweeper (DpSweeper) algorithm...



A CPPIED problem decomposition in two steps

DpSweeper does the following:

1 It greedily constructs a partial path

made of disconnected segments in set

S to guarantee that the required

coverage is achieved (dynamic

programming).

2 It optimally links segments to create a

path that is within the robot's physical

constraints (TSP reduction).



Part 1: Greedily Choosing a Set of Segments S

De�nition

A segment is de�ned as a set of adjacent cells without any turn.

De�nition

Two segments are independent if their scans do not overlap.

Independent

segments (r = 3)

Dependent

segments (r = 3)

Overlapping

detections




The good:

Computing the optimal segment length for independent segments isdone in polynomial-time using dynamic programming.2

The bad:

Independent segments do not allow for minimal required coverage inmany instances.That is, overlapping segments are required to solve many instances.

The pretty:

When two detections (scans) overlap their order is not important:


2Using Kadane's algorithm [Bentley, 1984] for computing a segment's length.M. Morin et al. (U.Laval, U.T.leMirail) Hybrid Algorithm for CPPIED May 28, 2014 30 / 44



The good:


The bad:


The pretty:






The good:


The bad:


The pretty:






1

1

2

3

3

3

1

1

2

2

2

2

1

1

2

2

1

1

1

1

2

2

1

1

1

1

2

2

2

1

1

1

2

2

2

1

Suppose that the lateral range is r = 1.

Each cell contains the number of scans left

which depends on:

the seabed types;

the required coverage matrix; and

the conditional probability of detection.




1

2

2

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1



which depends on:

the seabed types;






1

1

1



which depends on:

the seabed types;








which depends on:

the seabed types;





Part 2: Reducing to a TSP and Reconstructing the Path P

Once S has been determined, we need to compute an optimal path

crossing each segment of S once.




First, we forget about the map...

... to think about an optimal TSP tour.




• • •

• • •

• • •

•••

•

•

•

•••

•••

Each segment is modeled as 3 vertices in the TSP.

The cost of the edges of a segment is null.




u v w

• • •

u′

v′

w′

• • •

We link the endpoints of each pair of segments.

The cost of the edges between the ends of two di�erent segments

is in number of moves (dotted edges).




• • •

• • •

• • •

•••

•

•

•

•••

•••

We obtain the optimal TSP tour using Concorde solver3.

3[Applegate et al., 2011]M. Morin et al. (U.Laval, U.T.leMirail) Hybrid Algorithm for CPPIED May 28, 2014 32 / 44



• • •

• • •

• • •

•••

•

•

•

•••

•••

We reconstruct the path.




DpSweeper plans the optimal tour and reconstructs the path

Reduce to a TSP and obtain an optimal tour:

minimize the length then the number of turns;or minimize the number of turns then the length.

Reconstruct the path from the optimal tour.



6500 7000 7500 8000 8500140

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180

200

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240

260

280

300

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Path length

Numnber

ofturns

HCPP solutionsHCPP Pareto



5500 6000 6500 7000 7500 8000 8500140

160

180

200

220

240

260

280

300

320

340

Path length

Numnber

ofturns

HCPP solutionsHCPP ParetoDpSweeper Pareto



Is the DpSweeper algorithm a �awless heuristic on all instances?



No, but it is especially e�ective in terms of path length.

5000 6000 7000 8000 9000 10000140

160

180

200

220

240

260

280

Path length

Numnb

erofturns

HCPP solutionsHCPP ParetoDpSweeper Pareto


Wrap Up

HCPP:

Lengthty con�guration (from hours to days)Per instance (or instance set) con�guration requiredFast for predetermined parameters ©

DpSweeper:

No parameter ©Fast even in presence of a TSP (running time under a minute)High performance in terms of path length ©


Conclusion

Contributions:

Coverage path planning with imperfect extended detection formalismThe extension of the HCPP algorithm to a bicriteria caseThe DpSweeper algorithm and its evaluation in a bicriteria context


Discussion

Thank you for yourattention.

More information at

http://www.MichaelMorin.info


http://www.MichaelMorin.info

References I

[Applegate et al., 2011] Applegate, D., Bixby, R., Chvatal, V., and Cook, W. (2011).Concorde TSP solver.http://www.tsp.gatech.edu/concorde.Accessed: 2013/03/15.

[Bentley, 1984] Bentley, J. (1984).Programming pearls: algorithm design techniques.Communications of the ACM, 27(9):865�873.

[Choset, 2001] Choset, H. (2001).Coverage for robotics.Annals of Mathematics and Arti�cial Intelligence, 31(1-4):113�126.

[Drabovich, 2008] Drabovich, M. (2008).Automated mission trajectory planning for mine countermeasures operations.Master's thesis, Heriot Watt University, Edinburgh, Scotland, UK.

[Fang and Anstee, 2010] Fang, C. and Anstee, S. (2010).Coverage path planning for harbour seabed surveys using an autonomous underwater vehicle.In Proc. of OCEANS 2010 IEEE-Sydney, pages 1�8, Sydney, Australia.


http://www.tsp.gatech.edu/concorde

References II

[Gage, 1995] Gage, D. W. (1995).Many-robot MCM search systems.In Bottoms, A., Eagle, J., and Bayless, H., editors, Proc. of Autonomous Vehicles in MineCountermeasures Symposium, Monterey, CA.

[Kukulya, 2012] Kukulya, A. (2012).Remus 100 surveys glacier and remote fjord in western greenland.http://www.whoi.edu/cms/images/IMG_4705_244853.jpg.Woods Hole Oceanographic Institution, Accessed: 2014-01.

[Linder, 2006] Linder, C. (2006).GNU Octave.http:

//www.whoi.edu/cms/images/media-linder_belize1-belize1_cl_87419_99318.jpg.Woods Hole Oceanographic Institution, Accessed: 2014-01.

[Lippsett, 2004] Lippsett, L. (2004).Where are mines hiding on the sea�oor?Oceanus Magazine, 43(1).


http://www.whoi.edu/cms/images/IMG_4705_244853.jpg

http://www.whoi.edu/cms/images/media-linder_belize1-belize1_cl_87419_99318.jpg

http://www.whoi.edu/cms/images/media-linder_belize1-belize1_cl_87419_99318.jpg

References III

[Mannadiar and Rekleitis, 2010] Mannadiar, R. and Rekleitis, I. (2010).Optimal coverage of a known arbitrary environment.In Proc. of IEEE International Conference on Robotics and Automation (ICRA'10), pages5525�5530, Anchorage, AK.

[Morin et al., 2013] Morin, M., Abi-Zeid, I., Nguyen, T. T., Lamontagne, L., and Maupin, P.(2013).Search and surveillance in emergency situations - a gis based approach to construct optimalvisibility graphs.In Comes, T., Fiedrich, F., Fortier, S., Geldermann, J., and Müller, T., editors, Proceedingsof the 10th International Conference on Information Systems for Crisis Response andManagement, pages 452�456. Information Systems for Crisis Response and Management(ISCRAM).

[Nicholson and Healey, 2008] Nicholson, J. and Healey, A. (2008).The present state of autonomous underwater vehicle (AUV) applications and technologies.Marine Technology Society Journal, 42(1):44�51.

[Reed et al., 2003] Reed, S., Petillot, Y., and Bell, J. (2003).An automatic approach to the detection and extraction of mine features in sidescan sonar.28(1):90�105.


References IV

[Wikipedia, 2013] Wikipedia (2013).Side-scan sonar.http://en.wikipedia.org/wiki/Side-scan_sonar.Accessed: 2014-01.


http://en.wikipedia.org/wiki/Side-scan_sonar

Bicriteria trajectory planning in demining operations

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