Page 207 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

182 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

• Arm with two prismatic joints, typically referred to as a Cartesian arm or
a PP arm (Figure 5.1b).
• RP (revolute–prismatic) arm, which has a revolute joint followed by a par-
allel prismatic link (Figure 5.1c).
• Another RP arm—a revolute link followed by a perpendicular prismatic
link (Figure 5.1d).
• PR (prismatic–revolute) arm—prismatic joint followed by a revolute joint
(Figure 5.1e).

In the next section, a general model of the arm manipulator and of the envi-
ronment in which it operates will be outlined, along with necessary definitions.
Any modifications that the model may require for a specific arm configuration
will appear in the corresponding sections. Next we will consider in detail the first
of the five two-link arms, the RR (revolute–revolute) arm (Figure 5.1a). We will
study interactions between the arm and the obstacles in the arm’s workspace,
eventually deriving a path planning algorithm with guaranteed convergence.
In the sections that follow the RR arm study, we will study in a similar
fashion, except in a more brisk pace, each of the remaining four arms depicted
in Figure 5.1. When developing the corresponding motion planning procedures,
we will observe that the algorithmic issues for these arms turn out to be simpler
compared to the RR arm.
For a reader familiar with the Piano Mover’s techniques, it will come perhaps
as a surprise that in principle each of the two-link arms shown in Figure 5.1 will
require its own version of the sensor-based motion planning algorithm. While a
detailed reasoning as to why that is so will come later, here the reader is invited
to accept this fact as a law of nature. Indeed, nature operates in a similar fashion:
Constraints imposed on animals’ motion by the kinematics of their limbs and
bodies make each species move differently from others. Each species’ “algorithm”
for obstacle avoidance differs from other species’ algorithms. Humans are no
exception: A person who lost his leg in an accident will have to re-learn the use
of legs. He will learn to walk around or step over objects in ways dramatically
different from how he handled this task before the accident.
On the other hand, there will be much in common between motion plan-
ning algorithms for different arm manipulators. Based on this observation, in
Section 5.8.4 we will attempt to build a unified theory of planar manipulators.
This will allow us to derive planar arm algorithms as a special case of one gen-
eral strategy. We will see, in particular, that the arm configurations b, c, d, and e
in Figure 5.1 are in some sense special cases of the RR (revolute–revolute) arm.
As a concluding remark in this introductory section, one should keep in mind
that the planar two-link arms shown in Figure 5.1 can be arranged in more
kinematic arrangements, and hence more geometries, beyond those shown in the
figure. Note, for example, that in the RR arm planar arm shown in Figure 5.1a
the axes of both joints are parallel. A number of other configurations of RR arms
can be obtained by manipulating the mutual arrangement of those axes. We will
consider this variety and its effect on motion planning later, in Section 5.3. It