Page 279 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

254 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

q
O 2p
O c

q O c
l
0 2p l
(a) (b)
Figure 5.36 A 2-DOF robot with one translational joint and one revolute joint. (a) WS:
The configuration shown is singular. (b) CS: The two resultant components of CSO almost
touch each other—almost because at the corresponding point the robot is not in contact
with obstacles, and hence the two CSO components are disjoint.

obstacle in the vertical direction. Thus the configuration corresponds not to a
point on the CSO boundaries, but to an interior point of FCO.
Note that the robot configuration shown in Figure 5.36 is a singular configu-
ration. That is, in this configuration a certain direction of motion in WS—here,
along the vertical line—is impossible (or, theoretically, requires infinite joint
velocities). In this configuration the robot cannot exert a force in the upward
vertical direction. On the other hand, a small change in any joint variable will
enable the robot to exert a force onto the obstacle. In CS this means that the two
CSO components are very close but not touching each other (Figure 5.36b). We
require that if the robot touches obstacles in two configurations that are arbitrarily
close to each other, then the robot shall be able to move from one configura-
tion to the other while maintaining the contact. This translates into the following
condition:

Condition 5.8.2. Let O, L, and O C be as defined in Condition 5.8.1.
2
2
1
1
∀c ,c ∈ O C , c = c , there exists a continuous path p C : I → O C such that
1
2
p C (0) = c , p C (1) = c , p C (t) ∈ O C for t ∈ I.
We further restrict that only such interactions between the robot and an obstacle
are allowed in which, for any robot configuration, at least some robot motion is
possible:
∗ ∗
Condition 5.8.3. For any j ∈ C f and any > 0, there exists c ∈ U(j , ) ⊂
C f , such that c = j and c and j are connected within C f .
∗
∗
Condition 5.8.3 rules out the possibility that C f contains isolated points. A degen-
erate case where the robot can barely squeeze between two or more obstacles
while being in contact with both of them is not allowed. Removing this case