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Landmarks and Triangulation in Navigation 157

a feasible way to initialize the position of a mobile robot automatically, which
can be found in Reference 3.

4.3.3 KF-Based Navigation Algorithm
The triangulation algorithm is difficult to implement when the robot moves
around since it is necessary to compensate the time frame as each of three
landmarks is detected at different robot positions. Skewis and Lumelsky [12]
proposed a triangulation algorithm to attack this problem. However, there was
no satisfactory result being obtained after the algorithm was tested. This is
mainly due to the following reasons:

• Each of the landmarks is in a single strip and not encoded, that is,
indistinguishable from one another.
• Noisy readings come from the laser scanner as some angle measure-
ments are caused by random objects.
• In general, the robot environment is nonconvex. Therefore, not all
landmarks can be seen by the laser scanner. Moreover some land-
marks may be obscured by dynamic objects such as humans and
other robots.

The KF algorithm is a natural choice for robot localization since it provides
a convenient way to fuse the data from multiple sensors, for example, the laser
scanner and odometry. However, it normally requires a linear dynamic model
and a linear output model. However, in this research, both models are nonlinear
as follows:

x(k + 1) = f(x(k), u(k)) + w(k) (4.3)

z(k + 1) = h(B i , x(k)) + v(k) (4.4)

where f(x(k), u(k)) is the nonlinear state transition function of the robot. w(k) ∼
N(0, Q(k)) indicates a Gaussian noise with zero mean and covariance Q(k).
h(B i , x(k)) is the nonlinear observation model and v(k) is Gaussian noise with
zero mean and covariance R(k).
The control vector is calculated by two optical encoders at each cycle time k:

T
u(k) =[ d, θ] (4.5)
and the state transition function of the robot is

 
x(k) + d cos θ
f(x(k), u(k)) =  y(k) + d sin θ  (4.6)
θ(k) + θ

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c004” — 2006/3/31 — 16:42 — page 157 — #9