Page 269 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

244 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

Hence the arm will pass point 1 and continue following the obstacle boundary.
It will pass point 2 and then point 3. At point 8 the arm will encounter again
obstacle A (albeit at this new point), switch to following it, encounter at point
9 obstacle B, follow it to point 10 where it meets obstacle D, and follow that
obstacle to point 11, where it—finally!—meets the M-line again. Since point 11
is closer to T than the last hit point (point 4), point 11 is defined as the second
leave point.
Notice that by the time the arm reaches point 1 for the second time, on its
way to point 11 it will have encountered all four obstacles A, B, C, and D.
Since during this motion the arm is passing from one obstacle to the other in
an uninterrupted obstacle following, with no path segments in free space, the
arm will perceive all four as one obstacle. [This is very clear from the C-space
picture (Figure 5.31b).] Thereafter, starting at point 11, the arm will switch to
following the M-line and eventually will arrive happily at the target point T .
The torturous exercise that we have just gone through is given only for those
courageous souls who want to understand how a local cycle is formed and why
it causes no difficulty to the algorithm. The arm will have no idea that it went
through a local cycle. The theory developed here and in Section 5.2.1 guarantees
that the cycle will not become an infinite loop. The different cases that we
have considered thus far are necessary only to establish such a guarantee in the
algorithm.
We emphasize again that the described path has been produced by the arm
equipped with simple tactile sensing. As described in Section 5.2.5, a more
advanced sensing will in general improve the performance—that is, shorten the
path—quite markedly.

The Algorithm. We are now ready to formulate the sensor-based motion-plan-
ning algorithm for the prismatic-revolute (PR) arm. The following notation and
parameters will be used in the procedure:

• Parameter F is used to handle an open curve of a Type II obstacle. It is set
according to this rule:

F =+1 when the arm, while following a virtual boundary, reaches the
upper limit of joint value l 1 .
F =−1 when, while following a virtual boundary, the arm reaches the
lower limit of joint value l 1 .
F = 0 at the first hit point of the virtual boundary.

• A flag is used to distinguish between the first and second open curves of
the virtual boundary of a Type II obstacle.
• Counter C 2 is used to handle closed curves of virtual boundaries.
• Complementary M-lines, M 1 and M 2 , are defined as before.
• Hit, H j ,and leave, L j , points are defined as before (Section 5.2.1); L o = S.