Page 334 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THREE-LINK XXP ARM MANIPULATORS 309

The purpose of distinguishing between the front and rear parts of a pris-
matic (sliding) link as follows: When the front (respectively, rear) part of link
L 3 approaches an obstacle, the only reasonable local direction for maneuvering
around the obstacle is by decreasing (respectively, increasing) the joint variable
l 3 . This makes it easy to decide on the direction of motion based on the local con-
tact only. Since the point set is L 3 ((j 1 ,j 2 , 0)) ∩ L 3 ((j 1 ,j 2 , 1) ⊂ L 3 ((j 1 ,j 2 ,l 3 ))
for any l 3 ∈ I, and it is independent of the value of joint variable l 3 ,the setis
considered a part of L 2 . These definitions are easy to follow on the example
of the PPP (Cartesian) arm that we considered in great detail in Section 6.2:
See the arm’s effective workspace in Figure 6.3a and its complete workspace in
Figure 6.3b.
The robot workspace may contain obstacles. We define an obstacle as a rigid
body of an arbitrary shape. Each obstacle is of finite volume (in 2D of finite area),
and its surface is of finite area. Since the arm workspace is of finite volume (area),
these assumptions imply that the number of obstacles present in the workspace
must be finite. Being rigid bodies, obstacles cannot intersect. Formally, we have
the following definition:

Definition 6.3.3. In the 2D (3D) case, an obstacle O k , k = 1, 2,... , is the inte-
2
3
rior of a connected and compact subset of ( ) satisfying
=∅, (6.2)
O k 1 ∩ O k 2 k 1 = k 2

M
!
We use notation O = O i to represent a general obstacle, where M is the
k=1
number of obstacles in W.
Definition 6.3.4. ThefreeW-space is

W f = W − O.

Lemma 6.3.1 follows from Definition 6.3.1.

Lemma 6.3.1. W f is a closed set.

The robot arm can simultaneously touch more than one obstacle in the work-
space. In this case the obstacles being touched effectively present one obstacle
for the arm. They will present a single obstacle in the joint space.

Definition 6.3.5. An obstacle in J-space (J-obstacle) O J ⊂ J is defined as

O J ={j ∈ J : L(j) ∩ O =∅}.
Theorem 6.3.1. O J is an open set in J.