Page 340 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THREE-LINK XXP ARM MANIPULATORS 315

any set E p ={(j 1 ,j 2 )}⊂ J p , the inverse minimal projection is P m −1 (E p ) =
(j 1 ,j 2 ,l 3 ) | (j 1 ,j 2 ) ∈ E p and l 3 ∈ I}.

The minimal projection of a single point is empty. Hence P m ({j s }) =
P m ({j t }) =∅.If E ⊂ J is homeomorphic to a sphere and stretches over the
whole range of l 3 ∈ I,then P m (E) is the intersection between J p and the max-
imum cylinder that can be inscribed into O J and whose axis is parallel to l 3 .
If a set E ⊂ J presents a cylinder whose axis is parallel to the axis l 3 ,then
P c (E) = P m (E).
In general, O J is not homeomorphic to a sphere and may be composed of
many components. We extend the notion of a cylinder as follows:

Definition 6.3.9. A subset E ⊂ J presents a generalized cylinder if and only if
P c (E) = P m (E).

Type 1 and Type 2 obstacles present such generalized cylinders. It is easy to
see that for any E p ⊂ J p , P −1 (E p ) is a generalized cylinder, and P c (P −1 (E p ))
m m
= E p .
We will now consider the relationship between a path p J in J and its projection

= P c (p J ) in J p .
p J p
Lemma 6.3.4. For any set E ⊂ J and any set E p ⊂ J p ,if E p ∩ P c (E) =∅,
then P m −1 (E p ) ∩ E =∅.

−1
If P m (E p ) ∩ E =∅,thenwehave P c (P m −1 (E p ) ∩ E) = P c (P m −1 (E p )) ∩
P c (E) = E p ∩ P c (E) =∅. Thus a contradiction.
The next statement provides a sufficient condition for the existence of a path
in joint space:

Lemma 6.3.5. For a given joint space obstacle O J in J and the corresponding
projection P c (O J ), if there exists a path between P c (j s ) and P c (j t ) in J p that
avoids obstacle P c (O J ), then there must exist a path between j s and j t in J that
avoids obstacle O J .

(t) ={(j 1 (t), j 2 (t))} be a path between P c (j s ) and P c (j t ) in J p avoid-
Let p J p
−1
ing obstacle P c (O J ). From Lemma 6.3.4, P m (p J p (t)) ∩ O J =∅ in J. Hence,
(t), (1 − t)l 3s + t · l 3t )}∈ P −1 (t)})
for example, the path p J (t) ={(p J p m ({p J p
connects positions j s and j t in J and avoids obstacle O J .
To find the necessary condition, we use the notion of a minimal projection.
The next statement asserts that a zero overlap between two sets in J implies a
zero overlap between their minimal projections in J p :

Lemma 6.3.6. For any sets E 1 ,E 2 ⊂ J,if E 1 ∩ E 2 =∅,then P m (E 1 ) ∩
P m (E 2 ) =∅.