Path Planning with Objectives Minimum Length and Maximum Clearance - - PowerPoint PPT Presentation

Path Planning with Objectives

Mansoor Davoodi, Arman Rouhani, Maryam Sanisales Summer 2020

Minimum Length and Maximum Clearance

All the pictures have been created by the presenter himself

Institute for Advanced Studies in Basic Sciences Zanjan, Iran

Outline

• Introduction
• Literature review
• Problem definition
• Inputs and outputs
• Optimal paths
• Contribution
• Conclusion

1/19

If you optimize everything, you will always be unhappy.

Donald Knuth

• Donald Knuth

2/19

Path Planning Problem (PP)

• A challenging problem in computer science and robotics
• A practical problem in industries, game design and daily routine work
• The classic PP problem (single objective)
• Workspace containing: obstacles, start point s and destination point t
• Obstacle-free path for a robot starting from s and ending at t : s-t-path

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• Multiobjective PP problem

Path Planning Problem Variations

• Single-Objective
• Bi-Objective

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O2 O1 O3 O1 O2 O3

Literature Review

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2020

Welzl Hershberger Elshamli Castillo Ahmed and Deb Davoodi Geraerts . . . Gosh Mitchell Clarkson

Bi-Objective

Clearance as the Second Objective

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s

𝑄

s

𝑄′

Problem Definition

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• Bi-objective PP problem
• Minimizing Length as the first objective
• Maximizing Clearance as the second objective
• Input: Simple non-intersecting polygons with total n vertices
• Output: Set of Pareto optimal solutions
• Length : Manhattan , Clearance : Euclidean

𝑢 𝑡 𝑢

𝜌1

𝑡 𝑡

Pareto Optimal Solutions

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Extreme Pareto Optimal Paths

𝑢

𝑄

1

𝑄

3

𝑄

2

𝜌1

𝑢

𝑄

1

𝑄

3

𝑄

2

𝜌2

𝑄

1

𝑄

3

𝑄

2

𝜌3

𝑄

1

𝑄

3

𝑄

2

𝜌4

𝑡

𝜌2 𝜌3 𝜌4

𝑀(𝜌) C(𝜌)

(𝑏) (𝑒) (𝑐) (𝑑) (𝑓)

Literature Review Cont.

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2020

Welzl Hershberger Elshamli Castillo Ahmed and Deb Davoodi Wein Geraerts Davoodi

• Deterministic algorithm in grid workspace
• O(n3) time algorithm in continuous workspace

. . . Gosh Mitchell Clarkson

Contribution

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Algorithm

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• SP algorithm

Tangent points

• Rectilinear path
• Computing tangent points
• Performing Dijkstra
• Constructing tree T

Algorithm Cont.

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• Tree T
• Segment dragging

• Type3 event: Two obstacles intersect at some clearance λ

Algorithm Cont.

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Algorithm Cont.

• Intervals

SP

O(n) intervals Intersection Event

Tree MWVD

Type1 & Type2

DS SP Tree MWVD

Type1 & Type2

DS SP Tree MWVD

Type1 & Type2

DS

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• Type1 event: A tangent point reaches an MWVD edge

Algorithm Cont.

Type1 critical clearances in each interval are stored in a HEAP structure

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• Type2 event: Two tangent points reach the same x or y coordinate.

Algorithm Cont.

Type2 critical clearances in total are stored in a SORTEDLIST

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Complexity

SP

Intersection Event

Tree MWVD

Type1 & Type2

DS

O(n) O(n log n) O(n log n) O(1) Heap: O(log n) List: O(1) Type2 : O(n2 * n log n) = O(n3 log n) Type1 : O(n * n log n) = O(n3 log n) O(n)

In total

x O(n)

O(n3 log n)

Conclusion

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• Bi-objective PP problem
• Minimizing Length and Maximizing Clearance
• O(n3 log n) time algorithm
• Extreme Pareto optimal solutions
• Length : Manhattan , Clearance : Euclidean
• ( 𝟑,1)-approximation

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• 1. Clarkson, K., Kapoor, S., Vaidya, P.: Rectilinear shortest paths through polygonal obstacles in o (n (log n) 2) time. In:

Proceedings of the third annual symposium on Computational geometry. pp. 251{257 (1987)

• 2. Davoodi, M.: Bi-objective path planning using deterministic algorithms. Robotics and Autonomous Systems 93, 105-

115 (2017)

• 3. De Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational geometry. In: Computational geometry,
• pp. 1-17. Springer (1997)
• 4. Geraerts, R.: Planning short paths with clearance using explicit corridors. In: 2010 IEEE International Conference on

Robotics and Automation. pp. 1997-2004. IEEE (2010)

• 5. Ghosh, S.K., Mount, D.M.: An output-sensitive algorithm for computing visibility graphs. SIAM Journal on Computing

20(5), 888-910 (1991)

• 6. Hershberger, J., Suri, S.: An optimal algorithm for euclidean shortest paths in the plane. SIAM Journal on Computing

28(6), 2215{2256 (1999)

• 7. Hwang, F.K.: An o (n log n) algorithm for rectilinear minimal spanning trees. Journal of the ACM (JACM) 26(2), 177-

182 (1979)

• 8. Inkulu, R., Kapoor, S.: Planar rectilinear shortest path computation using corridors. Computational Geometry 42(9),

873-884 (2009)

• 9. Mitchell, J.S.: L 1 shortest paths among polygonal obstacles in the plane. Algorithmica 8(1-6), 55-88 (1992)
• 10. Wein, R., Van den Berg, J.P., Halperin, D.: The visibility-voronoi complex and its applications. Computational

Geometry 36(1), 66-87 (2007)

References

Thank You