Page 337 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

312 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

j t drawn from S s and S t , respectively, is not practical because S s and S t are
likely to contain an infinite number of points. To simplify the problem further,
in Section 6.3.5 we will introduce the notion of configuration space.

6.3.2 Monotonicity of Joint Space
3
Let L i (j) ⊂ be the set of points that link L i occupies when the manipula-
tor joint value vector is j ∈ J, i = 1, 2, 3 (Definition 6.3.1). Define joint space
obstacles resulting from the interaction of L i with obstacle O as Type i obsta-
3
(j) ⊂ be, respectively, the set of points
cles. For link L 3 ,let L 3 + (j) and L 3 −
that the front part and rear part of link L 3 occupy when the joint value vector
is j ∈ J (Definition 6.3.2). Define Type 3 + and Type 3 − J-obstacles respectively
with an obstacle O. More precisely:
resulting from the interaction of L 3 + and L 3 −
Definition 6.3.6. The Type iJ-obstacle, i = 1, 2, 3, is defined as

O Ji ={j ∈ J|L i (j) ∩ O =∅} (6.3)
Similarly, the Type 3 + and Type 3 − J-obstacles are defined as

(j) ∩ O =∅}
O J3 + ={j ∈ J|L 3 + (j) ∩ O =∅} and O J3 − ={j ∈ J|L 3 −
(6.4)
and O J = O J1 ∪ O J2 ∪ O J3 . We will also need
Note that O J3 = O J3 + ∪ O J3 −

∩
notation for the intersection of Type 3 + and Type 3 − obstacles: O J3∩ = O J3 +
.
O J3 +
We now show that the underlying kinematics of the XXP robot arm results in
a special topological properties of J-space, which is best described by the notion
of J-space monotonicity:
J-Space Monotonicity. In all cases of arm interference with obstacles, there

is at least one of the two directions along the l 3 axis, such that if a value l of link
3
L 3 cannot be reached because of the interference with an obstacle, then no value

l >l (in case of contact with the front part of link L 3 ) or, inversely, l <l (in

3 3 3 3
1

case of contact with the rear part of link L 3 )or l ∈ I (in case of contact with
3
link L 1 or L 2 ) can be reached either.
J-space monotonicity results from the fact that link L 3 of the arm manipulator
presents a generalized cylinder. Because links are chained together successively,
the number of degrees of freedom that a link has differs from one link to another.
As a result, a specific link, or even a specific part—front or rear—of the same
link can produce J-space monotonicity in one or more directions. A more detailed
analysis appears further.

Lemma 6.3.2. If j = (j 1 ,j 2 ,l 3 ) ∈ O J1 ∪ O J2 ,then j = (j 1 ,j 2 ,l ) ∈ O J1 ∪

3
1
O J2 for all l 3 ∈ I .