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for all valid disparities. Because of this separability, the C-space expansion can
be implemented very efficiently. In a first step, an intermediate disparity image is
generated by horizontally expanding all disparity values from the stereo disparity
map using the
u
1
/
u
2
look-up table. In a second step, each pixel in the intermediate
disparity image is expanded vertically using the
v
1
/
v
2
look-up table and the expansion
column is stored with a new disparity value from the
d
new
look-up table in the final
C-space disparity map.
4.3.4 Collision Checking in Disparity Space
The queries from the motion planner come as a sequence of short linear 3D trajectory
segments. The collision checker module takes each segment that is defined through
its start and end point, projects it into the current C-space disparity map
D
cm
, and
checks all pixels that are located on the straight line between the projected start
and end point for collision. Collision checking itself depends on the reconstructed
disparity value
d
of a point
p
s
on the segment and the actual disparity of the
underlying pixel
p
cm
in the C-space map. If the disparity of
p
s
is larger than the
disparity of
p
cm
, the pixel is classified as
safe
. If the disparity of
p
s
is smaller than
p
cm
the point on the trajectory is located behind an obstacle, and it is classified
depending on the disparity difference
(
p
s
)
d
(
p
s
)>
d
(
p
cm
)
:
SAFE
d
(
p
s
)<
d
(
p
cm
)
∧
d
(
p
s
)
−
d
(
p
cm
)<
k
:
COLLISION
d
(
p
s
)<
d
(
p
cm
)
∧
d
(
p
s
)
−
d
(
p
cm
)
≥
k
:
OCCLUDED
(4.16)
p
s
∈
D
cm
:
OUTSIDE
d
(
p
cm
)
=
invalid
:
NO_DATA
If the difference is smaller than a threshold, a collision occurred. If it is larger,
the trajectory point is labeled as
occluded
as shown in Fig.
4.10
. If the C-space
map contains no valid disparity data at a checked pixel location
p
cm
, the trajectory
segment is labeled as
no_data
—which can be caused, e.g., by a nontextured surface
when applying a real-time stereo approach.
How the planner uses these different trajectory classifications is explained in the
following section.
4.3.5 Motion Planning over Closed-Loop Dynamics
Motion planning of aerial vehicles has several challenges. First, the state space is
high dimensional—6DOF position and orientation, and their time derivatives (veloc-
ity/angular velocity, etc.), resulting in at least 12 states. Second, the system is very
agile and consequently has poor stability, and naively propagating the open-loop vehi-
