Page 276 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 251

Definition 5.8.2. An obstacle, O k , k = 1, 2,...,M, is the interior of a connected
3
and compact subset of satisfying
=∅, (5.6)
O k 1 ∩ O k 2 k 1 = k 2
M
!
When the index k is not important, we use notation O = O i to represent
k=1
the union of all obstacles in WS.
Definition 5.8.3. The free workspace is

W f = W − O
Lemma 5.8.2 follows from Definition 5.8.1.

Lemma 5.8.2. W f is a closed set.
In WS the robot may simultaneously touch more than one obstacle. In such cases
the obstacles involved effectively present one obstacle for the robot; in CS they
present a single body.

Definition 5.8.4. Configuration space obstacle (CSO) O C ⊂ C is defined as

O C ={j ∈ C : L(j) ∩ O =∅}.
The free configuration space (FCS) is

C f = C − O C

CSO may consist of many separate components. For convenience, we use the
term “configuration space obstacle” to also refer to a component of O C when the
exact meaning is obvious from the context. A workspace obstacle can map into
any large but finite number of disconnected CSO components (Theorem 5.8.2).

Theorem 5.8.1. O C is an open set in C.
∗
Proof: Let j ∈ O C . By Definition 5.8.4, there exists a point x ∈ L such that
y = x(j ) ∈ O.Since O is an open set (Definition 5.8.2), there must exist an
∗
> 0 such that the neighborhood U(y, ) ⊂ O. On the other hand, since x(j)
7
is a continuous function from C to W, there exists δ> 0 such that for all
j ∈ U(j ,δ), x(j) ∈ U(y, ) ⊂ O; thus, U(j ,δ) ⊂ O C ,and O C is an open set.
∗
∗
Q.E.D.
The theorem gives rise to this statement:
Corollary 5.8.1. FCS is a closed set.

Being a closed set, C f = C f . Thus, points on C f boundary can be considered
reachable by the robot.

7 If x ∈ L is a reference point on the robot, then x(j) is the forward kinematics with respect to x
and is thus continuous [8].