Page 303 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

278 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

For the sensing mechanism, we assume that the robot arm is equipped with
a kind of “sensitive skin” that covers the surfaces of arm links and allows any
point of the arm surface to detect a contact with an approaching obstacle. Other
sensing mechanisms are equally acceptable as long as they provide information
about potential obstacles at every point of the robot body. Depending on the
nature of the sensor system, the contact can be either physical—as is the case
with tactile sensors—or proximal. As said above, solely for presentation purposes
we assume that the arm sensory system is based on tactile sensing. 3

The Task. Given the start and target positions, S and T , with coordinates
S = (l 1 S ,l 2 S ,l 3 S ) and T = (l 1 T ,l 2 T ,l 3 T ), respectively, the robot is required
to generate a continuous collision-free path from S to T if one exists. This may
require the arm to maneuver around obstacles. The act of maneuvering around
an obstacle refers to a motion during which the arm is in constant contact with
the obstacle. Position T may or may not be reachable from S; in the latter case
the arm is expected to make this conclusion in finite time. We assume that the
arm knows its own position in space and those of positions S and T at all times.

Environment and Obstacles. The 3D volume in which the arm operates is the
robot environment. The environment may include a finite number of obstacles.
Obstacle positions are fixed. Each obstacle is a 3D rigid body whose volume and
outer surface are finite, such that any straight line may have only a finite number
of intersections with obstacles in the workspace. Otherwise obstacles can be of
arbitrary shape. At any position of the arm, at least some motion is possible. To
avoid degeneracies, the special case where a link can barely squeeze between
two obstacles is treated as follows: We assume that the clearance between the
obstacles is either too small for the link to squeeze in between, or wide enough
so that the link can cling to one obstacle, thus forming a clearance with the
other obstacle. The number, locations, and geometry of obstacles in the robot
environment are not known.

W-Space and W-Obstacles. The robot workspace (W-space or W) presents
a subset of Cartesian space in which the robot arm operates. It includes the
effective workspace, any point of which can be reached by the arm end effector
(Figure 6.3a), and the outside volumes in which the rear ends of the links may
also encounter obstacles and hence also need to be protected by the planning
algorithm (Figure 6.3b). Therefore, W is the volume occupied by the robot arm
when its joints take all possible values l = (l 1 ,l 2 ,l 3 ), l i = [0,l i max ], i = 1, 2, 3.
Denote the following:

• v i is the set of points reachable by point J i , i = 1, 2, 3;
• V i is the set of points (the volume) reachable by any point of link l i . Hence,

3 On adaptation of “tactile” motion planning algorithms to more complex sensing, see Sections 3.6
and 5.2.5.