Page 278 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 253

the obstacle. Mathematically, the removal of such “false contacts” translates into
the following condition, which guarantees that each CSO component has at least
one interior point:
∗
Condition 5.8.1. Let j ∈ C satisfy (5.7); that is, there exists u ∈ L such that
∗
w = u(j ) ∈ O. For given δ> 0 and > 0, define O = O ∩ U(w, δ), L =
L ∩ U(u, δ), and O C ={j ∈ U(j , ) : L(j) ∩ O =∅}. For any given γ>
∗
0, there must exist ∈ (0,γ ) and δ ∈ (0,γ ) such that O C =∅.
Theorem 5.8.2. An obstacle in WS can map into any large but finite number of
CSO components in CS.

Proof: We first design a simplified example showing that a simple obstacle in
WS can mapintotwo CSO components in CS. In Figure 5.35, the WS obstacle
O produces two separate CSO components, each resulting from the interaction
between O and each of the two vertical walls on the robot. Clearly, one can
add additional vertical walls to the robot (and reduce the size of the obstacle if
necessary) so that the number of CSO components will increase as well. This
way one can create as many CSO components as one wishes.
On the other hand, by Condition 5.8.1, a CSO component must have an interior
point. Also, by Theorem 5.8.1, CSO is an open set, and so its any interior point
must have a neighborhood of positive radius r that is entirely enclosed in a
CSO component. Thus the CSO component must occupy in CS a finite volume
(area). By Lemma 5.8.1, C has a finite volume or area; hence the number of CSO
components in CS must be finite. Q.E.D.

Figure 5.36a demonstrates another case of a “false contact,” more compli-
cated than the previous one. The corresponding CSO indeed has interior points,
Figure 5.36b. By our definition of contact, at the configuration shown the robot
is not in contact with the obstacle because it cannot exert any force upon the

O
O c O c
0 l
Robot
l

(a) (b)
Figure 5.35 Illustration for Theorem 5.8.2. A single physical obstacle, O, can produce
more than one CSO component. (a) WS: A simple robot with one translational joint. (b)
CS: The corresponding two separate CSO components.