Page 277 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

252 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

5.8.2 Interaction Between the Robot and Obstacles
Below we will need the property of space uniform local connectedness (ULC).
To derive it, we need to properly define the notion of a contact between the robot
and an obstacle. To this end, four conditions will be stated (Conditions 5.8.1 to
5.8.4) that together define a contact. Mathematically, at position (joint vector)
∗
j , robot L is in contact with obstacle O if
∗
∗
L(j ) ∩ O =∅ and L(j ) ∩ O =∅ (5.7)
The first relation in (5.7) states that j ∈O C (Definition 5.8.4), while the second
∗
∗
relation states that j ∈ ∂O C ,where ∂O C refers to the boundary of O C . However,
there may be situations where both relations of (5.7) hold but no obstacle exists
in CS. Consider a robot manipulator with a fixed base, one link, and one revolute
joint, along with a circular obstacle centered at the robot base O, as shown in
Figure 5.34. Here relation (5.7) is satisfied at every robot configuration. Note,
though, that the link can rotate freely in WS; this means that there are no obstacles,
and hence no obstacle boundaries, in CS.
Therefore, robot configurations that satisfy Eq. (5.7) do not necessarily corre-
spond to points on CSO boundaries. We modify the notion of contact by imposing
additional conditions on the admissible robot and obstacle spatial relationships.
As with any physical system, the term “contact” implies an existence of a force
at the point of contact between the robot and the obstacle. In other words, for an
object to present an obstacle for the robot, it must be possible for the robot to
move in the direction of the force if the object were removed. With this definition
of a contact, the robot shown in Figure 5.34 is not in contact with the obstacle
at any position θ because at a point of “contact” it cannot exert a force upon

Θ
O

Figure 5.34 Shown is a single-link “robot” with a revolute joint at point O, along with
a circular obstacle (shaded) also centered at O. With no obstacles in CS, the link can
freely rotate about point O.