Page 216 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

PLANAR REVOLUTE–REVOLUTE (RR) ARM 191

b 1
b 2
b 7
b 8
b 10
b 3 b 9
A b 11
b 6
b 4 b 12
b 5
b 16
b 17
a 1 a 7 b 13 B
a 2
a 6 a 11
a 3 a 5
a 10
a
a 4 a 12
a 9
a 14 a 8
a 13
b 14
a 16 b 15
’
’’ θ 1
θ 1
a 17
q = 0
C O
Figure 5.4 An example of interaction between obstacles. The shadow (shaded area)
behind obstacle A is the result of interaction between obstacles A and B. If the arm moves
through the positions (a 1 ,b 1 ), (a 2 ,b 2 ), ...,(a 17 ,b 17 ), at any moment it is in contact with
either obstacle A or B. This means that the arm will perceive these two obstacles as one
obstacle. Because of obstacle C, link l 1 cannot realize any angle values θ 1 in the range

θ <θ 1 <θ .
1 1
of them. The shape of a shadow depends on the shape, size, and position in
W-space of the corresponding actual obstacle that creates the shadow, as well
as on the arm links’ shapes and dimensions. An obstacle can form disconnected
shadows, as in the case of obstacle B (Figure 5.3). Or, obstacles can interact in
forming shadows; this happens, for example, when two or more points of the
arm body touch two or more actual obstacles simultaneously, as at position (a 8 ,
b 8 ) in Figure 5.4.
Definition 5.2.1. A virtual obstacle X is an area (or areas) in W-space, no points
of which can be reached by the arm endpoint because of the arm’s possible inter-
ference with the actual obstacle X.

Thus a virtual obstacle consists of the corresponding actual obstacles and their
shadows. In W-space a virtual obstacle forms one or more compact areas (see
Figure 5.4). Whereas topologically this combination presents little of interest in
W-space, we will see below that it possesses interesting properties in the arm’s
C-space.