Weighted Line Fitting Algorithms forMobile Robot Map Building and

Efficient Data RepresentationSam Pfister, Stergios Roumeliotis, Joel Burdick

Mechanical Engineering, California Institute of Technology

Overview:

• Motivation

• Problem Formulation

• Experimental Results

• Conclusion, Future Work

Motivation

Problem Formulation

- Improved data compression- Effective data correspondence- Increased robustness to outliers and noise

Raw PointData

Fit Line

Geometric Representation

Weighted Line Fitting

Correspondence and Merging

Goal : Efficient data representation

m1

b

(x1, y1)

(x2, y2)(x,y,)

Candidate Geometric Representations End points : [x1 y1 x2 y2] - 4D representation

Point + Orientation : [x y ] - 3D representation

Slope Intercept : [m b] y=mx+b - 2D representation

R

Polar Form : [R ] - 2D representation

Selected Geometric Representation

Polar line form • Minimal representation : L = [R,]

• Endpoints maintained as scalar value pairs : S1, S2

• Uncertainty maintained as 2x2 covariance matrix : PL =

R

R

R

R

S1

S2

RR bounds bounds Combined RR, bounds

Fit LineNoisy PointsTrue Line

Line Fit Simulation Single Run

Weighted Line Fitting : Motivation

Least Squares Fit vs. Weighted Fit

Fit LinesNoisy PointsTrue Line

Monte Carlo Simulation 100 Runs

Fit LinesNoisy PointsTrue Line

Monte Carlo Simulation100 Runs

Fit LineNoisy PointsTrue Line

Line Fit Simulation Single Run

Weighted Line Fitting : Formulation

Initial Point Grouping- Hough Transform (Duda & Hart [72])

Point Uncertainty Modelling- Zero mean gaussian assumption

- Laser rangefinder uncertainty parameters determined experimentally

dk

d

Robot Pose k

Qk

Laser RangeScan Data

Weighted Line Fitting : Formulation

Point Error Formulation : k

Point Error Formulation : Pk

Likelihood of obtaining errors {k} given line

Maximum Likelihood Estimation

•Position displacement estimate obtained in closed form

•Orientation estimate found using series expansion approximation

Non-linear Optimization Problem

Line Correspondence and Merging

Line Correspondence : 2 Test

Line Merge

Can merge non-overlapping line segments

Hallway Data

Results :

Kalman Filter Lab Run 1

KF Lab Run 2

Results :

Conclusions and Future Work

Developed general approach for working with line segments in a probabilistic framework

Showed that accurate error modelling can significantly improve line segment extraction accuracy and can enable robust line segment correlation.

Future Work:

Method can likely be extended for use in image processing applications as well as applications using other other range sensors (radar, ultrasound, etc.)

• requires specific sensor error models