Page 271 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

246 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

We observed that by studying the configuration space of each particular arm
and making appropriate modifications in the basic sensor-based path planning
procedure (which take into account topological properties of the arm workspace
and configuration space), the basic algorithms developed in Chapter 3 for point
mobile robots can be applied to planning the arm motion in space with arbitrary
and previously unknown obstacles. Realization of these algorithms requires avail-
ability of sensing hardware that provides information about potential collision at
every point of the arm body. We further learned in Section 5.2.5 that the devel-
oped algorithms can be extended to take advantage of more complex nontactile
sensing, using the ideas of “algorithms with vision” developed in Section 3.6.
One may find it odd that each kinematic arm configuration in Figure 5.1
requires its own motion planning algorithm. While in general this is rather nat-
ural (after all, animals with different kinematics of their body have different
gaits—compare a cat and a kangaroo), it would be indeed interesting to approach
all these arms in a unified manner and attempt a unified motion planning strategy
that would serve them all. In this section we will attempt a unified theory, and a
unified motion planning algorithm, of planar arm manipulators.
Two comments on what follows:

• Looking ahead, we will see, in particular, that the topology of configura-
tion space and planning algorithms for all arm configurations depicted in
Figure 5.1 are special cases of the RR (revolute–revolute) arm. The conse-
quence of this is that the sensor-based motion planning algorithm for RR
Arm is the universal 2-link-Arm Algorithm. While, on the positive side, this
means that one algorithm can serve all cases of 2D arm manipulators, it also
means, on the negative side, that in cases of simpler arm configurations one
would use a strategy that is more complex than the case in hand requires.
• The theory developed in this section is somewhat more complex than that
presented elsewhere in this text (and it will not be used in the following
chapters). It requires a prior exposure to topology. If this presents a problem
to the reader, and/or if the comment above convinces the reader that a unified
motion planning algorithm may be of a limited value, the reader can skip
the rest of this section.

Let us recall some basics. The sought commonality of the five two-link arms
of Figure 5.1 lies in the connectivity properties of their free space. Clearly, a path
between two points in space exists if and only if both points lie in a connected
area [or volume, in the three-dimensional (3D) case] of the free space. According
to our model (Section 5.1.1), we deal with the case of highly incomplete infor-
mation, with a situation when input information appears in real time and is of
strictly local nature, as when coming from robot sensors. Since potential obsta-
cles in the robot’s environment are not known beforehand, the hope is that the
robot can (a) infer some essential topological properties of the scene from a few
incomplete encounters with obstacles and (b) use this knowledge to guarantee
the solution to the motion planning task.