Page 258 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

REVOLUTE–PRISMATIC (RP) ARM WITH PARALLEL LINKS 233

Because of the unique choice of the local direction, there is no need to investi-
gate the whole curve of the virtual boundary. If, while passing around the obstacle
in the chosen local direction, the arm reaches one of the limits of l 2 , it can safely
conclude that it is dealing with a Type II obstacle, so the arm should start looking
for the second curve of the virtual boundary using the complementary M-line.
The procedure is further simplified through the use of the following statement
similar to the one in Section 5.2.2:

Lemma 5.5.1. For the two-link revolute–prismatic (RP) arm, if position T is
reachable from the starting position S, then there exists a path from S to T such
that it corresponds to a monotonic change of the joint value θ 1 .

In the motion planning procedure, a flag is used to indicate processing of
each of the two curves of a Type II virtual boundary. When the complementary
M-line is introduced, the numbering of hit and leave points starts over; L o = S.
The distance used is a Euclidean distance in W -space. Assume the M 1 -line is
the shorter of the two complementary M-lines. The procedure RP-Arm Algorithm
includes the following steps.

1. Establish an M 1 -line as the M-line. Set the flag down. Set j = 1. Go to
Step 2.
2. From point L j−1 , the arm moves along the M-line until one of the following
occurs:
(a) Target T is reached. The procedure stops.
(b) An obstacle is encountered and a hit point, H j , is defined. In case
of a front contact, choose the local direction such that it corresponds
to decreasing values of l 2 . In the case of a rear contact, choose the
local direction such that it corresponds to increasing values of l 2 .Go
to Step 3.
3. The arm follows the virtual boundary until one of the following occurs:
(a) The target is reached. The procedure stops.
T
S
(b) Current joint value θ 1 is outside the interval (θ ,θ ). The target cannot
1 1
be reached. The procedure stops.
(c) The M-line is met at a distance d from T such that d< d(H j ,T ).
Point L j is defined. Increment j.Goto Step2.
(d) The value l 2 approaches one of its limits, and the flag is down (i.e.,
the first curve of the virtual boundary of a Type II obstacle has been
processed). Set the flag up. Set j = 1. Establish an M 2 -line as the
M-line. Move the arm back to S.GotoStep2.
(e) The value l 2 approaches one of its limits, and the flag is up (i.e., the
second curve of the virtual boundary of a Type II obstacle has been
processed). The target cannot be reached. The procedure stops.