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170 Autonomous Mobile Robots

two landmarks are known as (0, 0) and (D,0), where D is the distance between
two landmarks.
T
For each sampling point z i , both landmarks are measured as (α 1i , α 2i ) , and
two equations will be generated:

x 0 + d xi

 tan(α 1i ) =

y 0 + d yi

(4.33)
x 0 + d xi − D


 tan(α 2i ) =

y 0 + d yi
where d xi and d yi are the displacement of the sampling points in two directions,
which are obtained from the readings of the odometer.
Rewriting Equation (4.33), and collecting the equations for each RS,
we have:
Az 0 = b (4.34)
where
 
cos(α 11 ) − sin(α 11 )
. .
. .
 
 . . 
 

cos(α 1n ) 
− sin(α 1n ) x 0
A =   , z 0 =
 cos(α 21 ) − sin(α 21 )  y 0


. .
 
 . . 
 . . 
cos(α 2n ) − sin(α 2n )
 
sin α 11 d y1 − d x1 cos α 11
.
.
 
 . 
 
 sin α 1n d yn − d xn cos α 1n 
 
b =  
sin α 21 d y2 − (d x2 − D) cos α 21
.
 
.
 
 . 
sin α 2n d2 − (d x2 − D) cos α 2n
The position z 0 can also be deduced from Equation (4.32).
4.4.4 Implementation and Results
The experiments are carried out using a “Logitech QuickCam” web camera
(1/4 CCD sensor, 4.9 mm lens). In our implementation, the mobile robot is

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c004” — 2006/3/31 — 16:42 — page 170 — #22