Page 257 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

232 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

3. A segment that is a part of one of the base circles (e.g., line 11-12,
Figure 5.24b); the inside points of this segment cannot be reached by the
arm.

In order to apply the basic path planning procedure to this arm, the algorithm
has to reflect specifics of moving along the C-space cylinder. Similar to the RR
arm, one concern in our case is whether obstacle boundaries may be formed
by more than one simple curve. Recall that if a virtual boundary is formed
by one simple (closed) curve, it is called a Type I obstacle, and if the virtual
boundary has more than one simple (closed) curve, it is a Type II obstacle (see
Section 5.2.2). Starting with one specific case, observe that if a ring-like actual
obstacle appears in W-space, positioned so that it separates the arm from the
W-space outer boundary, the result will be a band-like virtual obstacle in C-
space—formally, a Type II obstacle. One simple closed curve of the band can
be reached by the arm, whereas the other, formed by one of the base circles, is
inaccessible to the arm. Because of this, and in spite of the fact that the virtual
boundary has two closed curves, from the standpoint of path planning we will
treat it as a Type I obstacle.
As another case, observe the arm shown in Figure 5.1c, where l 1 = 0. If
an obstacle extends from W-space into the dead zone, it is easy to see that
in C-space a swath-like virtual obstacle will appear, whose virtual boundary
in C-space includes two separate “simple curves,” plus two vertical lines each
connecting the opposite base circles of the C-space cylinder. This is a real Type
II obstacle. Similar to the RR arm, if during the arm motion one such curve of a
Type II obstacle has been completely explored by the arm without ever meeting
the M-line, it is clear that the second curve has to be explored as well. To do
that, the complementary M 2 -line will be used.
As with the Cartesian arm studied above, the choice of the local direction
for following the virtual obstacle by our RP arm happens to be unique. Once
the arm encounters an obstacle, one of two possible cases arises. If the con-
tact is a front contact—that it, it corresponds to the front part of the arm
contacting the obstacle—then only such a local direction is meaningful that
corresponds to decreasing values of l 2 . As one can see in Figure 5.24a, the oppo-
site local direction would never bring the arm any closer to the target. If, on the
other hand, the contact is a rear contact, then only such local direction should
be chosen that corresponds to increasing values of l 2 . The reachability test is
built in a manner similar to this test for the RR arm (Section 5.2.2), taking
into account the simpler structure of the RP arm’s C-space (see the algorithm
below).
How will the arm tell a front contact from a rear contact? By our model, the
arm’s sensing lets it know which point of its body contacts the obstacle. The arm
also knows at all times which point of link l 2 is at the joint point of the link.
This information allows the arm to always distinguish a front contact from a rear
contact.