Using Incomplete Online Metric Maps for Topological Exploration with the Gap Navigation Tree Liz Murphy and Paul Newman Mobile Robotics Research Group,

Using Incomplete Online Metric Maps for Topological Exploration with the Gap

Navigation Tree

Liz Murphy and Paul Newman

Mobile Robotics Research Group, Oxford University, UK

Oxford Mobile Robotics Group

The Problem

• Path Planning: How to get from A->B?• Navigation: Knowing where you are when going from A

to B• Exploration: Deciding what B is.

• Possible Approach: Gap Navigation Tree– B. Tovar, R. Murrieta and S. M. LaValle, Distance-Optimal

Navigation in an Unknown Environment without Sensing Distances, IEEE Transactions on Robotics (2006)

– Very lightweight representation of space• Uses a minimalistic sensor (gap sensor) based on a simple range

scanner to detect gaps and build a tree structure

– Returns a complete topological map of a simply-connected 2D region

– Idea of using such a lightweight process to ‘drive’ an underlying SLAM system is appealing


Summary of Gap Navigation Tree

• Gap = discontinuity in depth information in the robot’s field of view

• GNT requires a Gap Sensor to identify and recognize gaps as robot moves– Ideal, infinite range sensor– Returns cyclical order of gaps as seen from

the robot, no metric representation required• Gaps correspond to regions of free

space– Lie at the boundary between visible region

and the shadow region not visible to robot• Tree structure encodes the relationship

between these regions in a topological map– Each gap is a node in the tree– Reflects combinatorial changes in the

visibility region of the robot– Reflects how the environment appears

relative to the position of the robot


Overview

• Summary of Gap Navigation Tree operation• Highlight the difficulties with practical

implementation of the gap sensor• Present a probabilistic Gap Sensor to detect and

track gaps• Present an architecture to enable online

exploration and map building using an existing SLAM system, the probabilistic gap sensor and the Gap Navigation Tree algorithm


Critical Gap Events

• Gaps appear or disappear as robot crosses an inflection ray

• Gaps split or merge as the robot crosses a bitangent ray

• Update tree in accordance– Appear/disappear add or remove

node from root of tree– Merge add or remove

• Children of root node are those gaps currently seen by Gap Sensor

• All other gaps in the tree were once seen by sensor but have merged

• Differentiate between gaps which have been completely explored and those which represent unseen areas

• Can be used to drive exploration by chasing down branches of the tree until all regions have been seen

Appear

a b c a bDisappear

a b c d c

a b

Split

Merge

Above figure: http://planning.cs.uiuc.edu

Above figure: http://planning.cs.uiuc.edu


GNT Operation


But in reality … Issues with adopting the GNT

• Predicated on existence of perfect gap sensor – Capable of tracking gaps perfectly even as the robot moves

across non-smooth points in the boundary and the gaps jump discontinuously

• Predicated on ability to observe world with infinite resolution– In reality sensor range limitations produce gaps (may or may

not be real gaps) at the end of corridor walls

• Not immediately applicable to integration with discrete laser based SLAM system:– Map is discrete– Sensor is discrete


Why complicate things with a map?

• Laser generates samples from continuous surfaces. – Naïve policy and common pathologic sensing

geometry can produce faux gaps.

• Motivates use of accumulation of data to mitigate this– Precisely what SLAM does for us

• Idea: Use SLAM map to generate a superior Gap Sensor, not “ideal”.– Take a probabilistic approach so that we can

still track gaps even when the range sensor fails to find correct gaps

• Encapsulate the uncertainty in sensing by representing gap location at time=k by a probability distribution gk

Faux Gap


Gap Detection

• Decide on angular resolution

• Allocate each laser data point to an angular bin

• Take the closest point from each bin to compute the visibility map– Gives an approximation to

the visible region• Differentiate to find gap

location gk– Test for gaps against

threshold– Resolve to an [x,y] location

(with associated covariance)– Location is a Tangent Point

as line between robot and TP is tangential to the corner


Gap Tracking in a Sampled Representation

• Using pose-based SLAM• Map is dense aggregate of individual points• Local shape of map can be used to generate a

probabilistic model of gap motion across time steps– Helps us to cope with the addition of a chunk of point

cloud data to the SLAM map

t=k t=k+1t=k-1


Formulation of Gap model from SLAM Map

• Gap Motion Model– Given the location of the gap at t=k-1

and the current state of the map, model the distribution of its current location as a 1st order Markov process

• Prediction– Integrate out the last estimate of the

gap’s location to come up with the predicted location at t=k

• Prediction gives us p(gkpred)

– Used to reconcile against p(gkdetect)

),|( 1 kkk mggp

111 )|(),|()|( kkkkkkkk dgmgpmggpmgp

km 1

1kg

2

1kg3

1kg

4

1kg


Gap Motion Model

• Is a gaussian:

• Γ is a function

– D is a diagonal scaling matrix – Used to exaggerate the covariance of the nearest

neighbours to encompass potential gap movement

• where

Tkk VVDmg ),(

)),(,(),|( 1111 kkkkkkk mgPgNmggp

)],(cov[ kkT gmKVV


Data Association

• Reconcile detected and predicted gk

• Mimic the operation of the ideal angular gap sensor by converting our gk=[xk,yk] locations to bearing measurement– Do this for both detected and

predicted gk • Χ2 test used to determine

associations• Multiple associations show that

splitting or merging is occurring– Set relatively wide threshold for

the Χ2 test to allow these to be captured

1

1gk (detected)

gk (predicted)

Two new associationsto the same old gap indicates split

tk-1 tk


Integration

• SLAM mapping generates dense aggregate map

• Map allows implementation of virtual gap sensor

• Tracking of gaps over time substantially increases performance and stability of GNT implementation

SLAM System

Gap Navigation Tree

Map informsGap Tracking

GNT drives Navigation process


Conclusions

• GNT can be driven by data from a far from GNT-ideal sensor

• Good use can be made of the metric structure of the local map to aid in the understanding of the perception of apparent gap behaviour.

• SLAM map need not be global – so we can still hold onto a light weight exploration

formulation

• Future work– Further improve virtual gap sensor by using Gaussian

Processes to model discrete sensor data or add policy to actively increase local metric map sample density


Questions

Questions?


Results

Documents

Using Incomplete Online Metric Maps for Topological Exploration with the Gap Navigation Tree Liz Murphy and Paul Newman Mobile Robotics Research Group,