Expected Shortest Paths for Landmark-Based Robot Navigation Amy Briggs Carrick Detweiler Daniel Scharstein Alexander Vandenberg-Rodes Department of Computer.

Expected Shortest Paths for Landmark-Based Robot Navigation

Amy BriggsCarrick DetweilerDaniel ScharsteinAlexander Vandenberg-Rodes

Department of Computer ScienceMiddlebury College

Student collaborators

Carrick Detweiler ‘04

David Ehringer ‘03

Deniz Sarioz ‘01

Alexander Vandenberg-Rodes

Lily Fu ‘03

Fafa Paku ‘02

Student collaborators

Darius Braziunas ‘00

Victor Dan ‘03

Huan Ding ‘03

Dan Knights ‘01

Jeff Lanza ‘01

Pete Wall ‘01

Related Work Fennema et al. 1990

Kavraki & Latombe 1994 Lazanas & Latombe 1995 Simmons & Koenig 1995 Taylor & Kriegman 1995 Nickerson et al. 1998 Owen & Nehmzow 1998 Blei & Kaelbling 1999 Mani et al. 1999 Thrun et al. 2000

Navigation using visual landmarks

Explore mode: build graph of landmark locations and visibility information Navigate mode: use graph to plan paths between landmarks

B

C

AD

Exploration Algorithm

Find an initial landmark and servo to it While untraversed edges exist in the graph, do: While current landmark has an untraversed

outgoing edge, do: Perform an observation Travel the shortest untraversed edge to another

landmark Update graph

Use ESP algorithm to go to the closest landmark with an untraversed outgoing edge

Exploration example

Navigation Algorithm

While not at destination landmark, do: Run ESP algorithm to find expected

shortest path Try to go to first landmark in path If that fails, do until successful:

Try next most optimal path If unable to leave current landmark:

Perform an observation, update graph, and rerun ESP algorithm

Navigation

Localization Algorithm

Perform a “partial explore” to build a graph based on the robot’s internal coordinatesCompare new graph with previously built graph of environmentCalculate x, y, and theta offsets between the two graphs

The ESP Problem

N landmarks with E edges

Yields N equations with E terms

Relation to MDPs

Problem is a special instance of aMarkov decision process (MDP)

Each visibility scenario is one state(2d states per node)

Our algorithms correspond to1. Value iteration2. Policy iteration

Expected shortest paths example

G

A

B

5

4

3

2

2

3

1.0

0.5

1.0

1.0

0.5

0.8

G

A

B

5

2

3

1.0

0.5 0.5

AGE Expected length of shortest path from A to G

)5,3,2min(25.0

)3,2min(25.0

)5,2min(25.0

)2(25.0

BGAG

BGAG

AG

AGAG

EE

EE

E

EE

Two Competing Algorithms

Value Iteration Evaluates system of equations Many iterations

Policy Iteration Solve linear system of equations Matlab sparse matrix solver

Algorithmic Evaluation

Generated realistic graphs (50,000+) Corridor graphs Multi-room graphs

Varied parameters Number of nodes and edges

Up to 25,000 nodes Range of edges: sparse to dense

Probability of edges Very low (.0001-.001) to full (.0001-1)

Iterations vs. Nodes

VI

PI

Sparse Dense

Running Time vs. Nodes

VI red

PI green

Sparse

Dense

Results

Value Iteration Many iterations to converge Very slow on sparse graphs

Policy Iteration Runtime only slightly affected by density Real-time performance on most graphs Higher memory requirements O(N2)

Demo

Acknowledgments

Thanks to Steve Abbott and Deniz Sarioz for insightful contributions, and to Darius Braziunas, Victor Dan, Cristian Dima, Huan Ding, David Ehringer, Lily Fu, Dan Knights, Jeff Lanza, Fafa Paku, and Peter Wall for their work on the implementation of the navigation framework.

Supported in part by NSF grants IIS-0118892, CCR-9902032, CAREER grant 9984485, POWRE grant EIA-9806108, by Middlebury College, by the Howard Hughes Medical Institute, and by the Council on Undergraduate Research.