Sensor-Based Mapping andSensor Fusion

By Rashmi Patel

Overview

Bayes Rule Occupancy Grids [Alberto Elfes, 1987] Vector Field Histograms-[J. Borenstein,

1991] Sensor Fusion [David Conner, 2000]

Posterior (Conditional) probability – probability assigned to a event given some evidence

Conditional Prob. Example- Flipping coins: P(H) = 0.5 P(H | H) = 1 P(HH) = 0.25 P(HH | first flip H) =0.5

Bayes Rule

Bayes Rule- continued

Bayes Rule:P(A|B) = P(A)P(B|A) P(B)

The useful thing about Bayes Rule is that it allows you to “turn” conditional probabilities around

Example: P(Cancer) = 0.1, P(Smoker) = 0.5, P(S|C) = 0.8

P(C|S) = ?

P(C|S) = P(S|C)*P(C) / P(S) = 0.16

Occupancy Grids [Elfes]

In the mid 80’s Elfes starting implementing cheap ultrasonic transducers on an autonomous robot

Because of intrinsic limitations in any sonar, it is important to compose a coherent world-model using information gained from multiple reading

Occupancy Grids Defined

The grid stores the probability that Ci = cell(x,y) is occupied O(Ci) = P[s(Ci) = OCC](Ci)

Phases of Creating a Grid: Collect reading generating O(Ci) Update Occ. Grid creating a

map Match and Combine maps from

multiple locations

x

y

Ci

Occupancy Grids Sonar Pattern

24 transducers, in a ring, spaced 15 degrees a part

Sonars can detect from 0.9-35 ft

Accuracy is 0.1 ft

Main sensitivity in a 30 cone

-3db sensitivity from middle 15

(1/2 response)Beam Pattern

22.5 deg

Occupancy Grids Sonar Model

Probability Profile- Gaussian p.d.f. is used but that is variable

p(r | z,)=1/(2r)*exp[ (-(r-z)2/2r2) - ((2)/

2)]

Where r is the sensor reading and z is actual distance

Range Measurement

Probably Empty

Rmin

Ranging error

Somewhere Occupied

Distance R

angle

Occupancy Grids Discrete sonar model

.45 .48 .49 .5 0.8

.45 .4 .38 .35 .32 .31 .32 .38 .41 .45 0.9

NA 0 .01 .02 .05 .09 .14 .22 .25 .3 .35 .4 .44 1

.45 .4 .38 .35 .32 .31 .32 .38 .41 .45 0.9

.49 0.5 0.8

EX.

Occupancy Grids Notation

Definitions: Ci is a cell in the

occupancy grid

s(Ci) is the state of cell Ci

(i.e. value of that cell)

OCC means OCCUPIED and whose value is 1

x

y

Ci

Occupancy Grids Bayes Rule

Applying Baye’s Rule to a single cell s(Ci) with sensor reading r:

P[s(Ci) = OCC | r] = P[r | s(Ci) = OCC] * P[s(Ci) = OCC] --------------------------------------------------------------------------------

p[r | s(Ci)] * P[s(Ci)]

Where p(r) = p[r | s(Ci)] * P[s(Ci)] summed over the cells that intercept the sensor model

Then apply this to all the cells creating a local map for each sensor

Occupancy Grids Bayes Rule Implemented

P[s(Ci) = OCC | r] = P[r | s(Ci) = OCC] * P[s(Ci) = OCC] -----------------------------------------------------------------------------------------

p[r | s(Ci) = OCC] * P[s(Ci) = OCC]

P[s(Ci) = OCC | r] is the probability that a cell is Occupied given a sensor reading r

P[r | s(Ci) = OCC] is the probability that sensor reading is ‘r’ given the state of cell Ci (this value is found by using the sensor model)

P[s(Ci) = OCC] is the probability that the value of cell Ci is 1 or that s(Ci) = OCC (this value is taken from the occupancy grid)

Likelihood Prior

Normalize

Occupancy Grids Implementation Ex

Let the red oval be the somewhere Occupied region

The Yellow blocks are in the sonar sector

The black lines are the boundaries of that sonar sector

P(r) = Sum over all of those yellow block using the sonar model to figure out the Probability P(r)

x

y

Occupied Range

Occupancy Grids Multiple Sonars

Combining Readings from Multiple Sonars:

The Grid is updated sequentially for t sensors {r}t = {r1,…,rt}

To update for new sensor reading rt+1:

P[s(Ci) = OCC | rt+1] =

P[rt+1 | s(Ci) = OCC] * P[s(Ci) = OCC|{r}t] --------------------------------------------------------------------------------

p[rt+1 | s(Ci)] * P[s(Ci) |{r}t]

Occupancy Grids Equations

P[s(Ci) = OCC | rt+1] =

P[rt+1 | s(Ci) = OCC] * P[s(Ci) = OCC|{r}t] -------------------- -----------------------------------------------------------------------------------------

P[rt+1 | s(Ci) = OCC] * P[s(Ci) = OCC|{r}t]

P[s(Ci) = OCC | rt+1] is the probability that a cell is Occupied given a sensor reading r

P[rt+1 | s(Ci) = OCC] is the probability that sensor reading is ‘r’ given the state of cell Ci (this value is found by using the sensor model)

P[s(Ci) = OCC|{r}t] is the probability that the value of cell Ci is 1 or that s(Ci) = OCC (this value is taken from the occupancy grid)

Occupancy Grids Multiple Maps

Matching Multiple Maps Each new map must be

integrated with existing maps from past sensor readings

The maps are integrated by finding the best rotation and translation transform which results in the maps having best correlation in overlapping areas

Occupancy Grid

Occupancy Grids Matching Maps Ex

Example 1: A simple translation of maps

Center of Robot at (2,2)

0 0 0 0 0

0 .9 .9 .9 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

Map 1: Map 2: After translating 0 .8 .8 .8 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0

0 .98 .98 .98

0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0

New Map: Combined map 1&2

P(cell3) = P(cell1)+P(cell2)- P(cell1)*P(cell2)

Occupancy Grids Vs Certainty Grids Occupancy Grids and Certainty Grids basically

the same in the method that is used to Collect readings to generate Probability Occupied

and for Certainty grids Probability Empty Create grid from different sonars Match Maps to register from other locations

Difference arise from the fact that Occ. Grids use conditional prob. to determine Probability Occupied while Certainty Grids use simpler math models

Occupancy Grids Vs Certainty Grids

Both have a P.d.f for the sonar model

However the major difference is in finding the probability that a cell is occupied First Pempty is computed for a cell Then Poccupied is computed using Pocc = 1-

Pemp Then Pocc is normalized over the sonar beam and

combined with the value of that cell from other sonars and Pocc(sonar reading r)

Vector Field Histograms [Borenstien]

The VFH allows fast continuous control over a mobile vehicle

Tested on CARMEL using 24 ultrasonic sensors placed in a ring around the robot

The scan times range from 100 to 500 ms depending on the level of safety wanted

Vector Field Histograms Notation

The VFH uses a two dimensional Cartesian Histogram grid similar to certainty grids [Elfes]

Definition: CVmax = 15 Cvmin = 0 d is the distance returned by the

sonar Increment value is 3 Decrement value is –1 VCP is Vehicle center point Obstacle vector-vector point from cell

to VCP

Vector Field Histograms Histogram Grid

The histogram grid is incremented differently from the certainty grid

The only cell incremented in the grid is the cell which is distance d away and lying on the acoustic axis of the sonar

Similarly only the cells on the acoustic axis and are less than distance d are decremented

Vector Field HistogramsPolar Histogram

Next the 2-D histogram grid is converted into a 1-D grid called the Polar histogram

The Polar Histogram, H, has n angular sections with width

3 2 4 14 1 1 3 2 1

11 24 3 13 4 12 2 2 3 1

3 1 4 2 13 3 4 5 1 14 5 3

Robot

Active Grid, C*

ktarg

Vector Field Histograms H mapping

In order to generate H, we must map an every cell in the histogram grid into H

jijiji

ji

bdacm

xi

x

yj

y

,

2*,,

,

0

0arctan

VCP the to y ,x from direction the

position scell' active they ,x

position srobot' the y ,x

Where

,

th

00

jiji

ji i,j

x0, y0

xi, yj

mi,,j

i,,j

jiji

jiji

jiji

m

d

c

a,b

y ,x atvector obstacle the of magnitude the

VCP the and y ,x between distance the

y ,x of value certainty the

constants positive

.

,

*,

Vector Field Histograms Discrete H grid

Now that the Object vectors for every cell have been computed, we have to find the magnitude of each sector in H

jijik

ji

mh

k

,,

,int

density obstaclepolar the

ofsector the

of resolutionangular the

Where

k

ji

h

,yxk

H

Vector Field Histograms Threshold

Once the Polar Object Densities have been computed, H can be threshold to determine where the objects are located so that they can be avoided.

The choose of this threshold is important. Choosing to high a threshold and you may come too close to a object and too low may cause you to lose some valid paths

Sensor Fusion [D. Conner]

David C Conner PhD Student

Presentation on his thesis and the following paper:“Multiple camera, laser rangefinder, and encoder data fusion for navigation of a differentially steered 3-wheeled autonomous vehicle”

Sensor Fusion Navigator

2 front wheels are driven and third rear wheel is a caster

2 separate computer systems, PC handles sensor fusion and PLC handles motor control

180 degree laser range finder with 0.5 resolution

2 color CCD cameras

Navigator- 3wheeled differentially driven vehicle

Cameras and Frame Grabbers

Because the camera is not parallel to the ground the image must be transformed to correctly represent the ground

The correction is done using the Intel Image Processing Library (IPL)

Since there are two cameras the two images must be combined

The images are transformed into the vehicle coordinates and combined using the IPL functions

Cameras and Frame Grabbers

Image ConversionOnce a picture is captured It is converted to gray scale

It is blurred using a gaussian Convolution maskExample shown is below

Image Conversion continued

Then the threshold of image is taken to limit the amount of data in the image

The threshold value is chosen to be above the norm of the intensities from the gray scale histogram

Then resulting image is pixilated to store in a grid

Laser Range Finder SICK LMS-200 laser rangefinder return 361 data

points for a 180 degree arc with 0.5 degree resolution

Values above a certain range or ignored

Vector Field Histograms VFH are nice because it allows us to

easily combine our camera data and our laser rangefinder data to determine most accessible regions

Several types of polar obstacle density (POD) functions can be used (linear, quadratic, exponential)

POD = KC(a-b*d)

POD values-Laser Rangefinder

The POD values for the laser are determined by: Using the linear function shown above to

transform the laser data into POD values

Then for every two degrees a max of the POD values in that arc is chosen as the final value

POD values-Images

POD values for the image are pre-calculated and stored in a grid at startup

The pre-calculated values are multiplied by the pixilated image also stored in a grid. (The overlapping cells would be multiplied)

For every 2degree arc the cell with the highest POD values is chosen to the value of that arc

Combining VFH

The two VFH’s are then combined by taking the max POD for each sector

The max POD is chosen because that represents the closest object

Bibliography Elfes, A. “Occupancy Grids: A Stochastic Spatial

Representation for Active Robot Perception.” July 1990

Elfes, A. “Sonar-Based Real-World Mapping and Navigation”, June 1987

Borenstein, J. “The Vector Field Histogram- Fast Obstacle Avoidance for Mobile Robots”, June 1991

Conner, D. “Multiple camera, laser rangefinder, and encoder data fusion for navigation of a differentially steered 3-wheeled autonomous vehicle”,