Page 185 - Autonomous Mobile Robots

Landmarks and Triangulation in Navigation 169

and dual landmark case. The method proposed here is the extension of Boley’s
LSE [24].

4.4.3.1 Single-landmark LSE (SLSE)
The single-landmark LSE algorithm is based on the coordinates shown in
Figure 4.9a. The origin point is placed at the landmark, and the x-axis is set
parallel to the robot moving trajectories. From each sampling point on a robot
trajectory, the bearings to the landmark are measured as α i (i = 0, 1, ... , n),
by using the methods in Section 4.2. The position of each sample is noted in
T
the vector z i = (x i , y i ) , the distances between each position, and z 0 , which
can be obtained from the readings of odometry, are noted as d i = z i − z 0 =
T
(x i − x 0 , y 0 ) . It is easy to observe that:
x 0 + d i
tan(α i ) = (4.29)
y 0
Rewriting Equation (4.29), we have:

x 0 cos(α i ) − y 0 sin(α i ) =−d i cos(α i ) (4.30)

Row-by-row collecting all the equations for i = 1, ... , n, wehaveover
determined equations which can be expressed as:

Az 0 = b (4.31)

where
   
cos(α 1 ) − sin(α 1 ) −d 1 cos(α 1 )
cos(α 2 ) − sin(α 2 ) −d 2 cos(α 2 )
   
. .  , z 0 = , .
  x 0  
. . y 0 .
A =  b =  
 . .   . 
cos(α n ) − sin(α n ) −d n cos(α n )
Using the Least Square method, we can estimate z as follows:
T −1 T
z 0 = (A A) A b (4.32)
In this method, we adopt samples at the positions z i (i = 1, ... , n)asthe
reference sample (RS) to estimate the robot position z 0 .

4.4.3.2 Dual-landmark LSE (DLSE)
In this case, the coordinates are built on two landmarks. The original point is set
to one landmark and the x-axis point to the other one. The coordinate values of

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c004” — 2006/3/31 — 16:42 — page 169 — #21