Page 213 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

188 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

that for all practical purposes an obstacle can be of arbitrary shape. There can
be only a finite number of obstacles present in W-space. Any disk or a straight
line passing through W-space intersects a finite set of obstacles. Being rigid
bodies, obstacles cannot intersect. Two or more obstacles may touch each other,
in which case the arm will treat them as one obstacle. Only such configurations
of sets of obstacles are considered for which, at any position of the arm, at
least some arm motion is possible. Only continuous motion of robot links is
allowed.
At any given moment, the arm knows its current coordinates θ 1 and θ 2 ,as
well as coordinates of the target position T . The starting position S is known to
be reachable; that is, when the arm is in the position S, no arm links intersect any
obstacles. It is not known whether position T is reachable and, if so, whether T
can be reached from S. The arm is said to be moving in free space when it has
no contacts with obstacles. Repeating the description given in the section above,
the arm is assumed to be capable of the following actions:

1. Moving the arm endpoint through a prescribed simple curve (called main
line or M-line) that connects points S and T .
2. Identifying the point(s) of contact on the arm body when the arm hits an
obstacle.
3. Following the obstacle boundary.

The first operation implies that the arm is capable of computing coordinates of
consecutive points along the M-line and transforming them into the corresponding
pairs (θ 1 , θ 2 ).
The purpose of the second operation is to provide information needed to pass
around an obstacle. This is done with the help of the arm’s tactile sensing.
When at least one point of an arm link is in contact with an obstacle, relative
coordinates of the point(s) of contact can be identified in the link reference
system. Note that the identification is a local operation that does not require
any additional information about the environment. Assume, for example, that
the arm is covered with a “skin” with densely spaced tactile sensors, so that
when a sensor contacts an obstacle, the point of contact on the arm body is
known.
For the third operation, imagine that, while in contact with an obstacle, the
arm follows the obstacle boundary as if some weak force pushes it against the
obstacle. Therefore, at any moment during such motion, there is a variable point
or points of contact between the obstacle boundary and the arm body. 1
If the arm endpoint follows the obstacle up to the W-space boundary—for
example, points on the outer circle in Figure 5.2—it is not clear whether the
arm is still in contact with the obstacle on the boundary. To avoid an ambiguity,
assume that no point of the W-space boundary can be a point of contact between
an obstacle and the arm.

1 A similar ability is considered in works on compliance control of robot wrists (see, e.g., Ref. 104).