Page 148 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

FROM A POINT ROBOT TO A PHYSICAL ROBOT 123

3.7 FROM A POINT ROBOT TO A PHYSICAL ROBOT

So far our mobile robot has been a point. What changes if we try to extend the
motion planning algorithms considered in this chapter to real mobile robots with
5
flesh and finite dimensions? A theoretically correct way to address this question
is to replace the original problem of guiding the robot in the two-dimensional
workspace (W-space) with its reflection in the corresponding configuration space
(C-space). (This will be done systematically in Chapter 5 when considering the
motion planning problem for robot arm manipulators.) C-space is the space of the
robot’s control variables, its degrees of freedom.In C-space the robot becomes
a point. Since our robot has two degrees of freedom, which correspond to its
6
coordinates X and Y in the planar W-space, its C-space is also a plane. Obstacles
will change accordingly.
If the robot is of a simple convex shape—for example, circular or rectan-
gular, as most mobile robots are, or can be satisfactorily approximated by such
shapes—the corresponding C-space can be obtained simply by “growing” the
obstacles to compensate for the robot’s “shrinking” to a point. This well-known
approach has been used already in the earlier works on motion planning (see
Section 2.7). For simple robot shapes the C-space is “almost the same” as the
W-space, and motion planning can be done in W-space, keeping in mind this
transformation. One can see, for example, that asking whether the circular robot
R of diameter D shown in Figure 3.22 can pass between two obstacles, O 1 and
O 2 , is equivalent to asking if the minimum distance between the grown obstacles,
each grown by D/2, is more than zero.
Recall that explicitly building the C-space is possible only in the paradigm
of motion planning with complete information (the Piano Mover’s model). Since

R
0
A

B
O 1 O 2

T
Figure 3.22 Effect of robot shape and geometry on motion planning.

5 A related question is, What kind of sensing does such a robot need in order to protect its whole
body from potential collisions? This will be considered in more detail in Chapters 5 and 8.
6 Including other control variables—for example, the robot orientation—would make C-space three-
or even higher-dimensional and will complicate the problem accordingly. In practice, the effect of
orientation can be often considered independent from the translation controls in X and Y directions.
Then the said complication can be avoided. These more special questions are not pursued in this
text. Some of these are discussed in Ref. 64.