Page 217 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

192 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

Definition 5.2.2. Passing around an obstacle presents a continuous motion of
the arm, during which the arm is constantly in contact with the corresponding
physical obstacle(s).
It is clear from Figure 5.4 that two or more actual obstacles may be interpreted
by the arm as a single virtual obstacle. In Figure 5.4, at any position from the
set (a 1 ,b 1 ), (a 2 ,b 2 ), ...,(a 17 ,b 17 ) the arm is in contact with at least one of the
actual obstacles A and B. Hence the two obstacles will be interpreted as one.
Definition 5.2.3. A virtual line is a curve in W-space that the arm endpoint fol-
lows when passing around an obstacle. The virtual line forms the boundary of a
virtual obstacle in W-space.

A virtual line is not necessarily a smooth curve. For example, if the arm
endpoint follows a sharp corner on an obstacle, or if the arm contacts some
obstacle while passing around another obstacle [as in the link position (a 8 ,b 8 ),
Figure 5.4], the virtual line may form sharp turns. Nor is a virtual line necessarily
a non-self-intersecting curve (see virtual boundary of obstacle B, Figure 5.3),
differing in this respect from the boundaries of physical two-dimensional objects.
We will discuss this issue later, when analyzing the arm C-space properties.
Points of contact on the arm may undergo a discontinuous jump when passing
around obstacles. This can happen because of the shapes of obstacles and arm
links involved, or because of the arm–obstacle interaction. In Figure 5.4, for
example, during link l 2 motion through positions (a 1 ,b 1 ), (a 2 ,b 2 ), and so on, an
instant before position (a 8 ,b 8 ) link l 2 is in contact with obstacle A; an instant
after position (a 8 ,b 8 ) the link is in contact with obstacle B. Accordingly, in this
short period the contact point on the arm jumps from a point of contact on one
side of link l 2 to a completely different point on the link’s other side.
Note, however, that even in such cases there will be no discontinuity in the
2
virtual curve. For example, in the area of point b 8 , which corresponds to the
jump of the contact point mentioned above (Figure 5.4), the virtual line remains
continuous. There will be more on the virtual line continuity in our analysis of
the arm C-space.
Observe also that some distinct pieces of the virtual line may be associated
with the same physical curve. Such is, for example, a part of the virtual line
(b 14 ,b 8 ) (Figure 5.3), which is a part of obstacle A boundary. When trying to
do a complete “rotation” by the arm around A, the arm endpoint will follow the
curve segment (b 14 ,b 8 ) twice, once in each of the two directions.
The requirement of continuous contact while passing around the obstacle is
equivalent to adding a constraint on the arm motion. In general, the arm’s position
relative to obstacles is described by one of these three situations:
1. No contact with obstacles takes place; the motion is unconstrained, and all
points in the vicinity of the arm endpoint are available for its next position.

2 Given the physics of the underlying phenomenon, this is not surprising: Physical motion is contin-
uous, so the arm endpoint must be moving through a continuous curve.