A Hybrid Algorithm for Coverage Path Planning With ... · where cell (x 0;y 0) is the current robot...

A Hybrid Algorithm for Coverage Path Planning

With Imperfect Sensors

Michael Morin1 Irène Abi-Zeid2 Yvan Petillot3

Claude-Guy Quimper1

1Department of Computer Science and Software Engineering2Department of Operations and Decision Systems, Université Laval, Québec, Qc, Canada

3Ocean Systems Laboratory, Heriot-Watt University, Edinburgh, Scotland, UK

[email protected]

November 6, 2013

M. Morin et al. (U.Laval, Oc.Syst.Lab.) Hybrid Algorithm for CPPIED November 6, 2013 1 / 28

IntroductionContext

Context [Reed et al., 2003]

Underwater minesweeping operationsTo plan the path of an AUV1

REMUSplatform [Nicholson and Healey, 2008, Fang and Anstee, 2010]

Swim long distancesConstant speed and altitudeInfrequent turnsVery little capacity to sense and avoid obstacles

Sensors

Sidescan sonar�Gap �ller� forward looking sonar

1Autonomous Underwater VehicleM. Morin et al. (U.Laval, Oc.Syst.Lab.) Hybrid Algorithm for CPPIED November 6, 2013 2 / 28

IntroductionGeneral formalism

The coverage path planning (CPP) problem [Choset, 2001]2

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

Objectives (in order of priority)

Minimize the traveled distance, and the number of turns

O�-line and sensor-based approaches

We assume an o�-line path planning problem on grids of square cells.

2More applications may be found in the work of [Williams, 2010,Stack and Smith, 2003, Fang and Anstee, 2010, Oksanen and Visala, 2009,Galceran and Carreras, 2012, De Carvalho et al., 1997].


IntroductionSpeci�c formalism

The CPP with imperfect extended detection (CPPIED)problem [Drabovich, 2008]

Imperfect sensors: (conditional) detection probability [Gage, 1995]Minimal required coverage instead of complete coverageExtended detection range

Objectives (in order of priority)

Minimize the traveled distance, and number of turns


The CPPIED ProblemSeabed types

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sand ripples

�at

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


The CPPIED ProblemSensors scans

1 scan

2 scans

3 scans

4 scans

Scans with a rangeof r = 3 cells; darkercells have a higherscan frequency.


The CPPIED ProblemMinimal required coverage

De�nition

A feasible path achieves the minimal required coverage in each cell ofmatrix D.

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Minimal required coverage on a seabed grid


The CPPIED ProblemConditional detection probability (imperfect sensor)

De�nition

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

the seabed type, and

the distance (and the maximal lateral range r).


The CPPIED ProblemConditional detection probability

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complex

sand ripples

�at

Scans with a range of r = 1cell on an heterogeneousseabed with

The seabed type in�uences the sensor's performance.

pscan =

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

=

0.70.80.99


The CPPIED ProblemConditional detection probability

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Scans with a range of r = 3cells on a homogeneous of�at seabed

The distance between the robot's position and the scanned cell in�uencesthe sensor's performance.

pscan =[pscan(1, f) pscan(2, f) pscan(3, f)

]=[0.9 0.6 0.3

]M. Morin et al. (U.Laval, Oc.Syst.Lab.) Hybrid Algorithm for CPPIED November 6, 2013 10 / 28

The CPPIED ProblemCoverage

De�nition

Let C be the current coverage matrix.a

aThe initial coverage is null.


The CPPIED ProblemUpdated 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). (1)

pscan =


=

0.70.80.99

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The CPPIED ProblemUpdated coverage

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.


The CPPIED ProblemProblem 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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The CPPIED ProblemOptimal coverage path

De�nition

A lexicographically optimal path achieves the required coverage andminimizes

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

the number of turns.

aLexicographic order suggested in [Drabovich, 2008].

pscan =


=

0.70.80.99

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The CPPIED ProblemA complex coverage path

More than 21,000 cells of a complex seabed topology

Complex seabeds and/or distant visibility often lead to a complexcoverage path...

... clever algorithms are required.


The DpSweeper algorithmConcepts

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 DpSweeper algorithmConcepts

The good:

Computing the optimal segment length for independent segments isdone in polynomial-time using dynamic programming.3

The bad:

Independent segments do not allow for minimal required coverage inmany instances.

The pretty:

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

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

3Using Kadane's algorithm [Bentley, 1984] for computing a segment's length.M. Morin et al. (U.Laval, Oc.Syst.Lab.) Hybrid Algorithm for CPPIED November 6, 2013 18 / 28

The DpSweeper algorithmA CPPIED problem decomposition in two steps

DpSweeper does the following:1 It greedily constructs a partial path

made of disconnected segments inset S to guarantee that the requiredcoverage is achieved (dynamicprogramming).

2 It optimally links segments to createa path that is within the robot'sphysical constraints (TSPreduction).


DpSweeper achieves required coverage

Initialize the coverage matrix C to the null matrix.

While the required coverage is not attained do:Maximize coverage gains by using horizontal and vertical independentsegment sets:

Compute H∗, an optimal set of independent horizontal segments.Compute V∗, an optimal set of independent vertical segments.Choose the set, between H∗ and V∗, that has the highest coverage.

Compute the coverage matrix C.

Suppose that the lateral range is r = 1.

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Each cell contains the number of scans left whichdepends on:

the seabed types;

the required coverage matrix; and

the conditional probability of detection.








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the seabed types;










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the seabed types;











the seabed types;




The DpSweeper algorithmPart 2: Reducing to a TSP and Reconstructing the Path P

DpSweeper plans the optimal tour and reconstructs the path

Reducing to a TSP and obtaina an optimal tour:

minimize the length (our �rst criterion).

Reconstruct the path from the optimal tour.

aTo �nd the lowest cost cycle, and therefore the optimal solution path in thereduction, we chose the Concorde solver [Applegate et al., 2011]


Experimentation

No Ocean bed O Required coverage Dij (∀i, j) Detection prob. pscan

1 0.9

0.6 0.65 0.620.7 0.85 0.80.99 0.99 0.99

2 0.9

0.7 0.8 0.750.99 0.99 0.990.91 0.95 0.92

3 0.9

0.8 0.8 0.80.8 0.8 0.80.91 0.91 0.91

4 0.75

0.51 0.51 0.510.8 0.8 0.80.91 0.91 0.91

5 0.75

0.51 0.51 0.510.8 0.8 0.80.99 0.99 0.99

6 0.9

0.6 0.6 0.60.8 0.8 0.80.91 0.91 0.91

7 0.85

0.6 0.6 0.60.8 0.8 0.80.91 0.91 0.91


Results

Comparison of DpSweeper to HCPP [Drabovich, 2008]

HCPP† DpSweeper

Solution‡ Solution‡ TSP Time (s)

No Length Turns Length Turns nodes TSP Total

1 3777 40 3755 41 66 < 1 < 1

2 4599 60 4628 61 99 < 1 1

3 5556 82 5321 95 144 < 1 1

4 5494 77 5363 75 117 < 1 1

5 5119 114 5151 107 162 1 2

6 6689 136 5681 270 402 45 52

7 7141 137 5731 174 273 12 15

† The solving times of the HCPP method are not published.

‡ Shorter is better, ties are broken on the number of turns.


Conclusion

Contributions:

Coverage path planning with imperfect sensor de�nitionThe DpSweeper algorithm

DpSweeper advantages:

A high performance even on the hardest instancesA high quality path is obtained within a few secondsFlexible and without �ne-tuning


References I

[Applegate et al., 2011] Applegate, D., Bixby, R., Chvatal, V., and Cook, W. (2011).Concorde TSP solver.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.

[De Carvalho et al., 1997] De Carvalho, R. N., Vidal, H., Vieira, P., and Ribeiro, M. (1997).Complete coverage path planning and guidance for cleaning robots.In Proc. of IEEE International Symposium on Industrial Electronics (ISIE'97), volume 2,pages 677�682, Guimarães, Portugal.

[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.


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.

[Galceran and Carreras, 2012] Galceran, E. and Carreras, M. (2012).Coverage path planning for marine habitat mapping.In Proc. of OCEANS 2012 MTS/IEEE, pages 1�8, Hampton Roads, VA, USA.

[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.

[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.

[Oksanen and Visala, 2009] Oksanen, T. and Visala, A. (2009).Coverage path planning algorithms for agricultural �eld machines.Journal of Field Robotics, 26(8):651�668.


References III

[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.

[Stack and Smith, 2003] Stack, J. R. and Smith, C. M. (2003).Combining random and data-driven coverage planning for underwater mine detection.In Proc. of OCEANS 2003 MTS/IEEE, volume 5, pages 2463�2468, San Diego, CA.

[Williams, 2010] Williams, D. P. (2010).On optimal AUV track-spacing for underwater mine detection.In Proc. of IEEE International Conference on Robotics and Automation (ICRA'10), pages4755�4762, Anchorage, AK.


Acknowledgment

Thank you for yourattention.

We wish to thank P. Giguère for hisinsightful suggestions, and the OceanSystems Lab for providing us with theproblem instances.

... we can't simply ��sh� for mines.


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