Spatial Datastructures for Robotics

Parth Sarthi Pandey

University of Hamburg

Faculty of Mathematics, Informatics and Natural Sciences

Department of Informatics

05. December 20161

Outline

1. Motivation

2. Spatial Datastructures

KD-Trees

Range Trees

Quad Trees

Octrees

3. Applications

Collision Avoidance

N-objects Octrees – as solution

Other technologies for Collision Avoidance

4. Conclusion

5. References

2

Motivation

Efficient data representation - Necessity for

algorithmic development

Each data structure serves a specific purpose.

Crucial to choose the most appropriate.

3

KD Trees

Space partitioning data structure for organizing points in a k-dimensional

space.

A Binary Tree.

Every node is a k-dimensional point.

4

KD Trees

Building a KD-tree

{(1,9), (2,3), (4,1),

(3.7), (5,4), (6,8),

(7,2), (8,8), (7,9),

(9,6)}

Pick random

dimension, find

median, split data,

repeat.

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[2] [3]

KD Trees

Nearest-Neighbor-

Queries

NN for point (7,4)

Build time: O(nlogn)

Query time: O(√n+k)

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[2] [3]

KD Trees

Nearest-Neighbor-Queries

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[10]

Range Trees

Query time:

Space:

Recursively defined

multi-level binary

search tree.

Range Queries

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[4]

Quad-Trees

Tree Data Structure.

Partitions 2D space.

Recursively subdivides

into four quadrants or

regions.

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[11]

Octree

3D equivalent of Quad-

tree.

Each node subdivides

the space into

eight octants.

Decomposition Rule.

Region Octree.

N-object Octree.

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[1]

Collision Avoidance

The Problem ?

Robot Arm Collision.

N-object Octree as a solution.

The Experiment: [1]

Maintain a model of the robots’ workspace.

Update and evaluate the model to detect imminent

collisions.

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Collision Avoidance using N-

object Octree

World modelled by solid

primitives.

Representing complex

objects as the union of

simpler primitives.

2 types:

Cylspheres

Rectangular solids

12

[1]

Collision Avoidance using N-

object Octree

OCTREE GENERATION

Decomposition for sample

environment using a five-

object quadtrce.

Starts with a cube containing

all the objects.

Splits when the no. of objects

in a node is greater than N.

(e.g N = 5)

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[1]

Collision Avoidance using N-

object Octree

OCTREE UPDATION

3 approaches:

Completely rebuild the octree for each cycle.

Delete and reinsert moving primitives.

Changes the structure of the octree when :

Primitive exits a node.

Primitive moves into a new node.

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Collision Avoidance using N-

object Octree

Raise an Alarm

Not for - Compatible primitives

Safety buffer

Extended Primitive

Thumb Rule:

Two extended primitives may intersect only if they share a

node in the N-objects octree

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Experimental Results

Naïve Algorithm:

Checks for possible

intersections with all

other primitives in the

world.

Octree Algorithm:

Checks for possible

intersections with

primitives sharing one

node.

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[1]

Other Technologies for

Collision Avoidance

Sensors

[5] – Capacitive Sensor as Whiskers

[7] – Capacitive Sensor as Sensitive Skin

[9] – A Ring of Proximity Sensors

Laser Based Range Scanner [8]

Dijkstra’s & A* Algorithms [6]

Path finding & Traversals

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Conclusion

Spatial data structures is a huge field.

Its applications include collision avoidance, path

planning, trajectory optimization, mapping etc.

KD Trees – nearest neighbor search.

Range Trees – shortest path searches.

Quad & Octrees – collision avoidance.

N-object Octree used for robot arm collision

avoidance.

We Cannot Understand the Present Until we Know the

Past. [A Wise Man]

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References

[1] Clifford A. Shaffer, Member, IEEE, and Gregory M. Herb “A Real-Time Robot Arm Collision Avoidance System ” IEEE TRANSATIONS ON ROBOTICS AND AUTOMATION. VOL X. NO 2 APRIL 1992

[2] Andrew W. Moore “Efficient Memory-based Learning for Robot Control“ PhD. Thesis; Technical Report No. 209, Computer Laboratory, University of Cambridge. 1991

[3] Victor Lavrenko -https://www.youtube.com/watch?v=TLxWtXEbtFE

[4] M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications: 5. Orthogonal Range Searching, Springer, 3rd edition, 2008

[5] Thomas Schlegl , Torsten Kroger , Andre Gaschler , Oussama Khatib and Hubert Zangl “Virtual Whiskers —Highly Responsive Robot Collision Avoidance” 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) November 3-7, 2013. Tokyo, Japan

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References (contd..)

[6] Jur van den Berg, Dave Ferguson and James Kuffner “Anytime Path Planning and Replanning in Dynamic Environments” Proceedings of the 2006 IEEE International Conference on Robotics and Automation Orlando, Florida - May 2006

[7] Tin Lun Lam, Hoi Wut Yip, Huihuan Qian and Yangsheng Xu“Collision Avoidance of Industrial Robot Arms using an Invisible Sensitive Skin” 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-12, 2012. Vilamoura, Algarve, Portugal

[8] Andrea Cherubini and François Chaumette ‘Visual navigation of a mobile robot with laser-based collision avoidance’ The International Journal of Robotics Research 32(2) 189–205 © The Author(s) 2013

[9] Angel Soriano, Enrique J. Bernabeu, Angel Valera, and Marina Vallés “Collision Avoidance of Mobile Robots Using Multi-Agent Systems ” Instituto U. de Automática e Informática Industrial, Universitat Politècnica de Valencia, Camino de Vera s/n 46022 Valencia, Spain

[10] https://en.wikipedia.org/wiki/K-d_tree

[11] https://en.wikipedia.org/wiki/Quadtree

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