Page 265 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

240 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

(Figure 5.28b) indicates that the arm is dealing with a Type II obstacle (see
Section 5.2.1 for the definition of Type I and Type II obstacles).
As we know from the RR arm study, having explored one curve of a Type
II obstacle virtual boundary brings in additional global knowledge: It tells the
arm that somewhere there is another, second curve of the virtual boundary. It
also tells the arm in which direction it can find this unknown curve. Now the
arm knows it has to find and start exploring the second curve in order to draw
conclusions about the target reachability. Because of the topology of a common
cylinder, if one simple curve of an obstacle boundary connects to both base
circles of C-space, in order for the virtual obstacle to be separated from the rest
of the C-space cylinder, the two endpoints of the second curve of the boundary
must lie at the two base circles as well. This means that the only way to reach
the second curve is to go in the direction opposite to the current M-line. For
that, a complementary M-line will be used. If, after reaching and following the
second curve of the virtual boundary neither the M-line nor the target is met, this
means the target T cannot be reached.
If an obstacle happen to interfere with the first link, as in the example in
Figure 5.29a, the resulting virtual boundary forms a band around the C-space
cylinder (Figure 5.29b). If the arm attempts to pass around the obstacle, starting,
say, in the position 1, it will follow the whole virtual boundary, passing through
points (1, 2, 3, 4, 5, 6, 1) and thus making a full circle. For the PR arm, this is
the first example so far where the arm makes a full circle in θ 2 . It suggests that,
similar to the RR arm before, we may need a counter to indicate the fact of making
afullcirclein θ 2 . Denote the counter C 2 , to emphasize that it corresponds to our
second joint variable, θ 2 . (There will be no counter C 1 .) The fact of completing
a full circle will be detected by the counter C 2 as follows: Starting at the hit
point, the counter integrates the angle θ 2 , taking into account the sign. If a full
circle is made without ever meeting the M-line (that is, the value of C 2 is 2π),
then the target cannot be reached.
Continuing our observations of various special cases, if two (or more) obstacles
happen to interfere with the first link the way it appears in Figure 5.30, two bands
are formed on the surface of the C-space cylinder. In this example the distance
between the obstacles along the line OO 1 is longer than the length of link l 2 ;
this makes the two bands connect with each other in two places, forming two
disconnected free areas in C-space. As a result, in Figure 5.30a position T cannot
be reached from the arm position S.
Here is what will happen in this example under the algorithm: Starting at S,the
arm follows the M-line, hits obstacle A at point 1 (Figure 5.30), goes through
points (2, 3, 9, 8, 7) while trying to pass around the obstacle, and eventually
returns to the hit point 1 without ever encountering the M-line. This trajectory of
the arm endpoint in shown in Figure 5.30a. However, unlike in the example in
Figure 5.29, the counter C 2 will now contain zero. Obviously, the target cannot
be reached. [One may note that this outcome is quite different from the outcome
in a similar situation with an RR arm, (Section 5.2).]