Statistical environment representation to support navigation of mobile robots in

unstructured environments

Sumare workshop 13.12.00

Stefan Rolfes

Maria Joao Rendas

rolfes,rendas@i3s.unice.fr

Outline

• Short introduction to the problem

• Novel environment representation (RCS models)

• Navigation using RCS models as a map

• Simulation results

• Conclusion

Mobile robot navigation

• Global supervision (GPS, beacons, cameras)

• Feature based approach (mapping and recognition)

Common approaches :Common approaches :

Basic requirement: localisation capacities

True robot pose

Estimated robot pose

Map

Observations

• Recognition

• Estimation of deviation

Navigation in unstructured environments

ProblemsProblems

(1) in unstructured environments (unreliable feature description)

mismatch leads to erroneous pose estimation

(2) in underwater scenarios (no GPS available)

no external pose information

Solution under studySolution under study

Statistical environment description of natural scenes

Natural scenes

We consider that natural, unstructured scenes can be described as a collection of closed sets:

Observation : Objects that occur in natural scenes tend to form patches (alga, stone fields, …)

;1

ii

K

KiK (family of closed sets)

Statistical versus feature based description

Feature description :

Mapping individual features

(Shape description of salient features)

Statistical description :

Captures global characteristics

• Spatial distribution

• Morphological characteristics (size, boundary length,…..)

size

p(size)

Statistical environment description: Example

Image processing

Posidonie (Villefranche)

Distribution of the orientation

)(P Statistics

Random Closed Set

)(1

iii

Each model is defined by a parameter vector ),(

Family of models : ,,, 21 MMM ),( iM

},,{ 21 l

},,{ 21 K

Doubly stochastic process :

1) Random point process (germ process)

describes spatial distribution of objects

2) Shape process (grain process)

determines the geometry of the objects

Examples of Random Closed Sets

Uniform distribution

Cluster process Line process

Non isotropic distribution

The hitting capacity

Analytical forms of can be found for some model types)(KT

);()( KPKT

K

Theorem : Knowledge of the hitting capacities for all compact sets

is equivalent to knowledge of the model parameter

• The sequence of locations (germs) of the closed sets is a stationary Poisson process of intensity

},,{ 21 l

• The sequence (grains) are i.i.d. realisations of random closed sets with distribution

9Simple RCS model : Boolean Models

)))((exp(1)( 02 KKT

E

Already used in biological / physical contexts to model natural scenes

Analytical expression for the hitting capacity :

},,{ 21 K

Map of the environment

4A

1A2A

3A

Segmentation of the workspace : ,1

ii

A

)(ii MA

Non isotropic

),(xx

isotropic

x Map of the environment

Pose estimation : Bayesian approach

An optimal estimate of the robot’s state is obtained by (MMSE):

kk

kkk dXYXpXX )(ˆ

Past observations : ,,,1 kk YYY iii ZDY ,

Dynamic model: kkk wXfX )(1

kY memoryless observations: )()( 1 kkk YpYYp

Optimal filter

)( 11

kk YXp )( 1k

k YXp )( kk YXp )()( 1

kkk

k XYpYXp

Assuming and to be uncorrelatedkD kZ

))(ˆ()()( kkkkkk XTpXDpXYp

Need to be characterized

The a-posteriori density is obtained :

Prediction Filtering

Characterisation of ))(ˆ( kk XTp

Good approximation by Gaussian densities

Approximation of the optimal filter by an Extended Kalman Filter (easy computation)

Perceptual observations memoryless ?

kX

1kX

Observation window

Overlapping observation area

kX

3kX

Observations not memoryless :

Requires random sampling of the image

Observations memoryless :

Use of perceptual observations periodically

Simulated environment

Bolean model (discs of random radii)

),),(()( 21 rrxx Map (RCS model parameters):

Generation

Realisation

Simulation results (1)

Simulation results (2)

Simulation results (3)

Pose estimation

Use of perceptual observations

Only odometry

Conclusions

• We proposed a novel environment description (not relying

• and demonstrated the feasibility of mobile robot navigation

A lot of future work

• Characterisation of more complex RCS models suitable to

• Address the Model testing (using MDL or ML)

• Solve the problem of joint mapping and localisation

describe natural scenes

on individual feature description) by RCS models

based on these descriptions