Page 280 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 255

simplifies the theory and is not restrictive for practice. Stated more precisely,
this condition is as follows:

∗
Condition 5.8.4. Let j ∈ C satisfy (5.7), and L(j ) ∩ O contain at least two
∗
i
i
i
2
1
∗
points. Let u ,u ∈ L be such that w = u (j ) ∈ O, i = 1, 2. For given δ and ,
i
i
i
i
i
i
i
define O = O ∩ U(w ,δ ), L = L ∩ U(u ,δ ), and O ={c ∈ U(j , ) :
∗
C
i
i
1
2
1
2
L (c) ∩ O =∅}, i = 1, 2. For any , δ ,δ > 0, O ∩ O =∅.
C C
We can now define the term “contact” in mathematical terms:
Definition 5.8.5. The robot is in contact with an obstacle if and only if Eq. (5.7)
and Conditions 5.8.1 to 5.8.4 are all satisfied.
5.8.3 Uniform Local Connectedness
Together, Conditions 5.8.2 and 5.8.4 bring about an important topological prop-
erty of CSO,the uniform local connectedness (ULC). ULC guarantees that ∂O C
presents manifolds in the 2D case.
Definition 5.8.6. Let E be a subset of a space X, and let x be any point of X.The
set E is locally connected at x if, given any positive , there exists a positive δ such
that any two points of E ∩ U(x, δ) are joined by a connected set in E ∩ U(x, ).
Note that x in this definition is not necessarily a point in E.However,if x ∈E,
then E is certainly locally connected at x, since for sufficiently small δ,there
are no points in E ∩ U(x, δ).
Definition 5.8.7. A space or a set of points is uniformly locally connected if,
given a positive , there exists a positive δ such that all pairs of points, x and y,
of distance x − y <δ are joined by a connected subset of the space, of diameter
less than .
Theorem 5.8.3. The open set CSO is uniformly locally connected.
Proof: Since CS is compact and CSO is open and locally connected (Theo-
rem 5.8.1), according to Ref. 110, VI.13.1, we only need to prove that CSO is
locally connected at CSO boundary points.
∗
Let j ∈ C satisfy Eq. (5.7). If L(j ) ∩ O contains only one single point,
∗
then Condition 5.8.2 guarantees that O C is locally connected at j . Now assume
∗
i
i
i
∗
L(j ) ∩ O contains at least two points. Let u ∈ L satisfy w = u (j ) ∈ O;
∗
i
i
i
let and δ be such as to satisfy Condition 5.8.2 with respect to u , i = 1, 2.
i
1
2
Let = min( , ),and let O be as defined in Condition 5.8.4. According
C
2
1
+
to Condition 5.8.4, there exists a point c ∈ O ∩ O . According to Con-
C
C
i i i
∗
+
dition 5.8.2, every point c, c ∈ O ⊂ U(j ,δ ) is connected to c ∈ O ⊂
C
C
2
2
1
i
i
1
U(j ,δ ) within O . Thus, any two points c ,c ∈ O ∪ O are connected
∗
C C C
1
2
in U(j , max( , )),and so O C is locally connected at j . Q.E.D.
∗
∗