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THREE-LINK XXP ARM MANIPULATORS 313

Consider Figure 6.13. If j ∈ O J1 ,then L 1 (j) ∩ O =∅.Since L 1 (j) is inde-

pendent of l 3 ,then L 1 (j ) ∩ O =∅ for all j = (j 1 ,j 2 ,l ). Similarly, if j ∈ O J2 ,
3
then L 2 (j) ∩ O =∅.Since L 2 (j) is independent of l 3 ,then L 2 (j ) ∩ O =∅ for

j = (j 1 ,j 2 ,l ) with any l ∈ I.
3 3
for all
3
Lemma 6.3.3. If j = (j 1 ,j 2 ,l 3 ) ∈ O J3 + ,then j = (j 1 ,j 2 ,l ) ∈ O J3 +
for all l <l 3 .

3
3
l >l 3 .If j = (j 1 ,j 2 ,l 3 ) ∈ O J3 − ,then j = (j 1 ,j 2 ,l ) ∈ O J3 − 3
Using again an example in Figure 6.13, if j ∈ O J3 ,then L 3 (j) ∩ O =∅.

Because of the linearity and the (generalized cylinder) shape of link L 3 , L 3 (j ) ∩

O =∅ for all j = (j 1 ,j 2 ,l ) and l >l 3 . A similar argument can be made for
3 3
the second half of the lemma.
Let us call the planes {l 3 = 0} and {l 3 = 1} the floor and ceiling of the
=∅, then its intersection
joint space. A corollary of Lemma 6.3.3 is that if O 3 +
=∅, then its intersection with
with the ceiling is not empty. Similarly, if O 3 −
the floor is nonempty. We are now ready to state the following theorem, whose
proof follows from Lemmas 6.3.2 and 6.3.3.
Theorem 6.3.2. J-obstacles exhibit the monotonicity property along the l 3 axis.
This statement applies to all configurations of XXP arms. Depending on the spe-
cific configuration, though, J-space monotonicity may or may not be limited to
the l 3 direction. In fact, for the Cartesian arm of Figure 6.15 the monotonic-
ity property appears along all three axes: Namely, the three physical obstacles
O 1 ,O 2 ,and O 3 shown in Figure 6.15a produce the robot workspace shown in
Figure 6.15b and produce the configuration space shown in Figure 6.15c. Notice
that the Type 3 obstacles O 1 and O 2 exhibit the monotonicity property only along
the axis l 3 , whereas the Type 2 obstacle O 3 exhibits the monotonicity property
along two axes, l 1 and l 2 . A Type 1 obstacle (not shown in the figure) exhibits the
monotonicity property along all three axes (see Figure 6.6 and the related text).
6.3.3 Connectivity of J f
We will now show that for XXP arms the connectivity of J f can be deduced from
the connectivity of some planar projections of J f . From the robotics standpoint,
this is a powerful result: It means that the problem of path planning for a three-
joint XXP arm can be reduced to the path planning for a point robot in the plane,
and hence the planar strategies such as those described in Chapters 3 and 5 can
be utilized, with proper modifications, for 3D planning.
∼
Let J p be the floor {l 3 = 0}. Clearly, J f = J 1 × J 2 . Since the third coordinate
of a point in J f is constant zero, we omit it for convenience.

Definition 6.3.7. Given a set E ={j 1 ,j 2 ,l 3 }⊂ J, define the conventional pro-
jection P c (E) of E onto space J p as P c (E) ={(j 1 ,j 2 ) |∃l ,(j 1 ,j 2 ,l ) ∈ E}.
∗
∗
3 3