Page 352 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

OTHER XXX ARMS 327

6.3.9 Discussion
As demonstrated in this section, the kinematic constraints of any XXP arm major
linkage result in a certain property—called monotonicity—of the arm joint space
and configuration space (C-space or C f ). The essence of the monotonicity prop-
erty is that for any point on the surface of a C-space obstacle, there exists at
least one direction in C-space that corresponds to one of the joint axes, such
that no other points in space along this direction can be reached by the arm. The
monotonicity property allows the arm to infer some global information about
obstacles based on local sensory data. It thus becomes an important component
in sensor-based motion planning algorithms. We concluded that motion planning
for a three-dimensional XXP arm can be done on a 2D compact surface, B f ,
which presents a deformation retract of the free configuration space C f .
We have further shown that any convergent 2D motion planning algorithm for
moving a point on a compact surface (torus, in particular) can be “lifted” into
3D for motion planning for three-joint XXP robot arms. The strategy is based on
the monotonicity properties of C-space.
Given the arm’s start and target points j s ,j t ∈ C f and the notions “above”
and “below” as defined in this section, the general motion planning strategy for
an XXP arm can be summarized as consisting of these three steps:

1. Move from j s to j ,where j ∈ B f is directly above or below j s ;

s
s

2. find a path between j and j within B f ,where j ∈ B f is directly above

s t t
or below j t ;and

3. move from j to j t .
t
Because of the monotonicity property, motion in Steps 1 and 3 can be achieved
via straight line segments. In reality, Step 2 does not have to be limited to the
plane: It can be “lifted” into 3D by modifying the 2D algorithm respectively,
thus resulting in local optimization and shorter paths. With the presented theory,
and with various specific algorithms presented in this and previous chapters, one
should have no difficulty constructing one’s own sensor-based motion planning
algorithms for specific XXP arm manipulators.
6.4 OTHER XXX ARMS

One question about motion planning for 3D arm manipulators that still remains
unanswered in this chapter is, How can one carry out sensor-based motion plan-
ning for XXR arm manipulators—that is, arms whose third joint is of revolute
type? At this time, no algorithms with a solid theoretical foundation and with
guaranteed convergence can be offered for this group. This exciting area of
research, of much theoretical as well as practical importance, still awaits for its
courageous explorers.
In engineering terms, one kinematic linkage from the XXR group, namely
RRR, is of much importance among industrial robot manipulators. On a better