Page 91 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

66 A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS

Theorem 2.9.4. For any finite maze, Fraenkel’s algorithm generates a path of
length P such that

P ≤ 2D + 2 p i (2.24)
i
where D is the length of M-line, and p i are perimeters of obstacles in the maze.

In other words, the worst-case estimates of the length of generated paths for
Trumaux’s, Tarry’s, and Fraenkel’s algorithms are identical. The performance of
Fraenkel’s algorithm can be better, and never worse, than that of the two other
algorithms. As an example, if the graph presents a Euler graph, Fraenkel’s robot
will traverse each edge only once.

2.9.2 Maze-to-Graph Transition
It is interesting to note that until the advent of robotics, all work on labyrinth
search methods was limited to graphs. Each of the strategies above is based solely
on graph-theoretical considerations, irrespective of the geometry and topology of
mazes that produce those connectivity graphs. That is why constructs like the
M-line are foreign to those methods. (M-line was not of course a part of the
works above; it was introduced here to make this material consistent with the
algorithmic work that will follow.) One can only speculate with regard to the
reasons: Perhaps it might be the power of Euler’s ideas and the appeal of models
of graph theory.
Whatever the reason, the universal substitution of mazes by graphs made the
researchers overlook some additional information and some rich problems and
formulations that are relevant to physical mazes but are easily lost in the transition
to general graphs. These are, for example: (a) the fact that any physical obstacle
boundary must present a closed curve, and this fact can be used for motion
planning; (b) the fact that the continuous space between obstacles present an
infinite number of options for moving in free space between obstacles; and (c)
the fact that in space there is a sense of direction (one can use, for example, a
compass) which disappears in a graph. (See more on this later in this and next
chapter.)
Strategies that take into account such considerations stay somewhat separate
from the algorithms cited above that deal directly with graph processing. As input
information is assumed in these algorithms to come from on-line sensing, we will
call them sensor-based algorithms and consider them in the next section, before
embarking on development and analysis of such algorithms in the following
chapters.

2.9.3 Sensor-Based Motion Planning

The problem of robot path planning in an uncertain environment has been first
considered in the context of heuristic approaches and as applied to autonomous