The.Algorithm.Design.Manual.Springer-Verlag.1998

Motion Planning
There is a wide range in the complexity of motion planning problems, with many factors to consider:
● Is your robot a point? - When the robot is a point, the problem becomes finding the shortest path
from to t around the obstacles, also known as geometric shortest path. The most readily
implementable approach constructs the visibility graph of the polygonal obstacles, plus the points
and t. This visibility graph has a vertex for each obstacle vertex and an edge between two obstacle
vertices if they ``see'' each other without being blocked by some obstacle edge.
A brute-force algorithm to construct the visibility graph tests each candidate edge against the
obstacle edges for a total time of time. By weighting each edge of the visibility graph with
its length and using Dijkstra's shortest-path algorithm (see Section ) on this graph, we can find
the shortest path from to t in time bounded by the time to construct the graph.
● How is your robot free to move? - Motion planning becomes considerably more difficult when the
robot becomes a polygon instead of a point. Now we must make sure that all of the corridors we
use are wide enough to permit the robot to pass through. The complexity depends upon the
number of degrees of freedom the robot has to move. Is it free to rotate as well as to translate?
Does the robot have links that are free to bend or to rotate independently, as in an arm with a
hand? Each degree of freedom corresponds to a dimension in the search space. Therefore, the
more freedom, the harder it is to compute a path between two locations, although it becomes more
likely that such a path exists.
● Can you simplify the shape of your robot? - Motion planning algorithms tend to be complex and
time-consuming. Anything you can do to simplify your environment would be a win. In
particular, consider replacing your robot by an enclosing disk. If there is a path for the disk, there
will be a path for whatever is inside of it. Further, since any orientation of a disk is equivalent to
any other orientation, rotation provides no help in finding a path Therefore, all movements are
limited to translation only.
● Are motions limited to translation only? - When rotation is not allowed, the expanded obstacles
approach can be used to reduce the problem to that of motion planning for a point robot, which is
simply the shortest path in a graph. Pick a reference point on the robot, and replace each obstacle
by the Minkowski sum of the object and the robot (see Section ). This creates a larger obstacle,
defined as the robot walks a loop around the obstacle while maintaining contact with it. Finding a
path from the initial position of the reference point to the goal point amidst these fattened
obstacles defines a legal path for the polygonal robot.
● Are the obstacles known in advance? - Thus far we have assumed that the robot starts out with a
map of its environment. This is not always true, or even possible, in applications where the
obstacles move. There are two approaches to solving motion planning problems without a map.
The first approach explores the environment, building a map of what has been seen, and then uses
this map to plan a path to the goal. A simpler strategy, which will fail in environments of
sufficient complexity, proceeds like a sightless man with a compass. Walk in the direction
towards the goal until progress is blocked by an obstacle, and then trace a path along the obstacle
until the robot is again free to proceed directly towards the goal.
file:///E|/BOOK/BOOK5/NODE197.HTM (2 of 4) [19/1/2003 1:31:58]