Page 351 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

326 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

In this section we intend to show how a 2D motion planning algorithm—we
will call it A p —can be carried out in B f . We assume that A p operates according
to our usual model—that is, using local information from the robot tactile sensors,
the paths it produces are confined to generic paths, obstacle boundaries, and
8
space boundaries, if any—and that it guarantees convergence. As before, it is
also assumed that all decisions as to what directions the robot will follow along
the generic path or along an obstacle boundary are made at the corresponding
intersection points.
Without loss of generality, side walls of the C-space, if any, are simply
treated below as Type I and Type II obstacles, the C-space ceiling is treated
as a Type III + obstacle, and the C-space floor is treated as a Type III − obstacle.
Since the robot start and target positions are not necessarily in B f , our first
step is to bring the robot to B f . This is easily achieved by moving from j s
downward until a Type III − obstacle is encountered; that is, the link L 3 of the
robot either reaches its joint limit or touches an obstacle with its rear end. Then,
the robot switches to A p , which searches for point j directly above or below j t ,

t
with the following identification of path elements:
.
• Generic path—the intersection curve of V and ∂O 3 −
).
and ∂(O 1 ∪ O 2 ∪ O 3 +
• Obstacle boundary—the intersection curve of ∂O 3 −
If A p terminates without reaching j , then the target j t is not reachable. On

t
the other hand, if j is reached, then the robot moves directly toward j t . Along

t
this path segment the robot will either reach j t or encounter an obstacle, in which
case the target is not reachable. This shows how a motion planning algorithm
for a compact 2D surface can be “lifted” into 3D to solve the motion planning
problem of an XXP arm.
6.3.8 Step Planning
Similar to 2D algorithms in Chapter 5, realization of the above 3D motion plan-
ning algorithms requires a complementary lower-level control piece for step
calculation. The required procedure for step calculation is similar to the one
sketched in Section 5.2.3 for a 2D arm, except here the tangent to the C-obstacle
at the local point of contact is three-dimensional. Since, according to the motion
planning algorithm, the direction of motion is always unique—the curve along
which the arm moves is either an M-line, or an intersection curve between an
obstacle and one of the planes M-plane or V-plane, or an intersection curve
between obstacles—the tangent to the C-obstacle at the contact point is unique
as well. More detail on the step calculation procedure can be found in Refs. 106
and 115.
8 The question of taking advantage of a sensing medium that is richer than tactile sensing (vision,
etc.) has been covered in great detail in Section 3.6 and also in Section 5.2.5; hence we do not dwell
on it in this chapter.