Page 242 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

PLANAR REVOLUTE–REVOLUTE (RR) ARM 217

with its shape and dimensions shrinking and expanding from rectangle-like to
ellipse-like, and with shapes in between like in Figure 5.17. Animation of this
process makes for a wonderful movie: One sees a strange creature that is moving
while constantly changing its shape according to some mystifying law. The extent
of variability in the sensing range C-space image depends on the sensing range
r and the arm’s kinematics.
Calculation of the sensing range C-space image is an interesting though rather
involved task; there are many details and many special cases to attend to. With
good equations for the sensing range, one could improve motion planning algo-
rithms by providing a look-ahead optimization of the arm’s next few steps, or
attempt algorithms that take into account the arm dynamics, similar to the work
we did in Chapter 4. To my knowledge, today there are no published analyses
on this topic. As a first approximation, one can start with a simplified model of
the sensing range, presenting it as a circle whose radius changes as a function
of the arm position (θ 1 ,θ 2 ). A conservative approximation would be to model
the arm sensing by the maximum circle inscribed in the real sensing range. With
this model the robot would be safe, but much sensing would be wasted: In some
directions in the (θ 1 ,θ 2 ) plane the actual sensing will go much farther than the
circular model will indicate.
As the arm moves, its sensing range image in C-space “breathes,” shrinking
and expanding as it moves in the plane (θ 1 ,θ 2 ). The extent of such changes
depends on the motion. It is easy to see, for example, that if we fix angle θ 2 and
let angle θ 1 change, in C-space of Figure 5.17 the sensing range figure will move
horizontally, and its shape will remain the same. This is because the motion does
not involve any changes in the relative position of links l 1 and l 2 . Any motion
involving a change in angle θ 2 will cause changes in the shape of the sensing
range figure.
Except for the added calculation due to the variable sensing range in C-space,
incorporating proximity sensing in the arm motion planning algorithm is similar
to the analogous process for mobile robots (Section 3.6). One can combine, for
example, one of the VisBug algorithms for a mobile robot (Section 3.6) with the
RR-Arm Algorithm developed in this chapter. The fact that the latter is noticeably
more complex than Bug algorithms calls for a careful analysis. To date, there
are no published results in this area, in spite of its significant theoretical and
practical potential.
How proximity sensing can affect the RR-Arm Algorithm performance can
be seen in Figure 5.18. Here link l 2 happens to be attached to link l 1 not by its
endpoint, as in some of our prior figures, but by some other point on the link.
(This is a more realistic design; it often occurs in industrial arm manipulators.)
Note how elegant and economical the arm’s path becomes when the arm is pro-
vided with proximity sensing (Figure 5.18b), compared to its performance with
tactile sensing (Figure 5.18a). In fact, the robot path in Figure 5.18b is almost
the optimal path between the S and T locations; it could hardly be improved
even by a procedure operating with complete information. This of course will