Page 251 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

226 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

5.4 PRISMATIC–PRISMATIC (PP, OR CARTESIAN) ARM

This arm is the second one, arm (b), among the five arms shown in Figure 5.1.
The reason it is called Cartesian is that the displacement in each of its joints
translates directly in exactly the same motion of the arm endpoint in Cartesian
plane (see Figure 5.23). The arm has two prismatic (sliding) joints, with joint
values l 1 and l 2 —hence its other name, a PP Arm. The boundaries of the arm’s
W-space are limited by the rectangle whose sides are equal to the maximum
lengths of links l 1 and l 2 . We assume that no obstacles outside W-space can
interfere with the arm motion; hence the path planning problem is limited to
the arm’s W-space rectangle. The M-line is defined as a straight-line segment
connecting the arm’s starting and target points, S and T .
From Section 5.2.1, a shadow of an obstacle X is the area of workspace no
point of which can be reached by the arm endpoint due to interference of X
with the arm motion. Functionally, then, for the arm an obstacle shadow is the
same as the physical obstacle that causes it. Observe in Figure 5.23 that for any
obstacle in W-space of the PP arm, no points to the right of an obstacle can
ever be reached by the arm endpoint. We will see that this property of this arm
kinematics makes motion planning a rather simple task.
The boundary of an obstacle plus the boundary of the shadow that it forms
produces the virtual line of the obstacle. The said property of the PP arm—that
no points to the right of an obstacle can be reached by the arm endpoint—means

l
l 1 2max

S

l 2 b
2 1 l
a 3 1max
4 A
A
l 1 5 6 7

T
0 l 2

Figure 5.23 Cartesian (PP) arm. The virtual obstacle includes the actual obstacle and
its shadow (shaded).