Page 256 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

REVOLUTE–PRISMATIC (RP) ARM WITH PARALLEL LINKS 231

Note also that these same obstacles A and B create another two shadows,
which are not of importance in our current task but might show up with some
other M-line: the arm endpoint will not be able to reach any point within the
figures described by points in the left part of Figure 5.24a, (4, 5, 5 , 1 , 1, 2, 3 , 4)

and (0, 10 , 6 , 0). An obstacle that extends into the arm’s dead zone forms only
one shadow, which extends from the obstacle all the way to the W-space outer
periphery. An obstacle X that is in the shadow of another obstacle Y will never
interact with the arm. If obstacle X partially intersects the shadow of obstacle Y,
it will be treated by the arm as a part of Y.
Therefore, with the exception of obstacles that extend into the arm’s dead zone,
any virtual obstacle in the workspace of this RP arm includes the actual obstacle
itself plus two shadow components: one front shadow and one rear shadow. The
front shadow extends from the obstacle to the outer boundary of W-space [see the
curve (6, 7, 8, 9, 10, 11, 12, 6) in Figure 5.24a]; the rear shadow extends from the
arm origin O into the periphery, but in general not to the boundary, of W-space
[see the curve (1, 0, 5, 4, 3, 2, 1) in Figure 5.24a].
Unlike the more complicated situation with the RR arm (Section 5.2), in the
case of our RP arm the virtual lines of obstacles are always simple curves. In
a special case when the M-line crosses the dead zone, the latter can be treated
simply as an obstacle interfering with the rear end of link l 2 .
Two independent variables, one angular and the other linear displacement, can
be represented by the surface of a cylinder. The C-space of our arm is therefore
a cylinder whose flat sides, called base circles, correspond to the first joint value,
θ 1 , and whose height corresponds to the second joint value, l 2 .The C-space of
the example in Figure 5.24a is shown in Figure 5.24b. Shown in the figure are
the images of an M-line and of the shadow components of virtual obstacles A and
B that interfere with the arm motion. In order to not overcrowd the picture, for
the complementary two shadow components only their projections on the lower
base circle, (1 , 5 ) and (6 , 10 ), are shown.

Unlike the RR arm, each point in W-space of our RP arm has only one
arm solution. That is, there is one-to-one mapping between W-space and the
corresponding C-space, as compared to the one-to-two mapping in the case of
the RR arm. Because of this, and also because virtual lines in W-space are always
simple curves, the virtual boundaries in C-space are also simple curves. Recall
that this constitutes a necessary condition for the basic path planning procedures
(Section 3.3). In general, each virtual boundary in C-space of the RP arm consists
of a combination of three distinct segments:

1. A curve formed when the front or the rear end of link l 2 follows the
boundary of the actual obstacle; for obstacle B in Figure 5.24b this seg-
ment passes through points 7-8-9.
2. A vertical straight-line segment formed when points of the arm body
other than the arm endpoint touch the obstacle while passing around it;
in Figure 5.24b this consists of lines 7-6-12 and 9-10-11).