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Landmarks and Triangulation in Navigation 159

where ∇B is the Jacobean matrix of the measurement function:

∂h ∂h
∇h = , , −1 (4.15)
∂x ∂y
For each measurement, a validation gate is used to decide whether it
is a match or not:
T
v(k + 1)S(k + 1)v (k + 1) ≤ G (4.16)

If it is true, the current measurement is accepted. Otherwise, it is
disregarded.
Step 4: Updating — It corrects the prediction error from odometry
readings.
The filter gain is updated by:

T −1
W(k + 1) = P(k + 1/k)∇h S (k + 1) (4.17)
The robot state is then calculated by:

x(k + 1/k + 1) = ˆ x(k + 1/k) + W(k + 1)v(k + 1) (4.18)

The covariance is updated by:

T
P(k + 1/k + 1) = P(k + 1/k) − W(k + 1)S(k + 1)W (k + 1)
(4.19)

Step 5: Return to Step 1 to recursively implement the four steps earlier.

The algorithm is essentially very simple although it contains some very use-
ful features. It produces the estimate of the current robot position at each cycle
by integrating odometry data with only one angle measurement from the laser
scanner. Recursively, it combines every new measurement with measurements
made in the past to estimate the robot position, or “make a compromise.” This
can be seen as a pseudo “triangulation” technique in a dynamic sense.

4.3.4 Implementation and Results
The proposed navigation algorithm based on angle-only observations was
implemented in our robotics research laboratory. The mobile robot equipped
with a rotating laser scanner and single-stripe landmarks were fixed on the walls
within the laboratory. The mobile robot was commanded to follow a close-loop
route at a speed of 0.3 m/sec. The route is near circular with a diameter of 4 m.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c004” — 2006/3/31 — 16:42 — page 159 — #11