Page 298 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

INTRODUCTION 273

develop efficient heuristics. This is all true, and indeed more true for arms with
revolute joints—but these difficulties have been formulated for the motion plan-
ning problem with complete in formation. Notice that these difficulties above did
not prevent us from designing rather elegant sensor-based planning algorithms
for 2D arms with revolute joints, even in the workspace with arbitrarily complex
obstacles. The question now is how far we can go with the 3D case.
It was said before that this is a difficult area of motion control and algorithm
design. As we will see in Chapter 7, human intuition is of little help in designing
reasonable heuristics and even in assessing proposed algorithms. Doing research
requires expertise from different areas, from topology to sensing technology.
There are still many unclear issues. Much of the exciting research is still waiting
to be done. Jumping to the end of this chapter, today there are still no provable
algorithms for the 3D kinematics with solely revolute joints. While this type of
kinematics is just one mechanism among others in today’s robotics, it certainly
rules the nature.
As outlined in Section 5.1.1, we use the notion of a separable arm [103],
which is an arm naturally divided into the major linkage responsible for the
arm’s position planning (or gross motion), and the minor linkage (the hand)
responsible for the orientation planning of the arm’s end effector. As a rule,
existing arm manipulators are separable. Owing to the fact that three degrees of
freedom (DOF) is the minimum necessary for reaching an arbitrary point in 3D
space, and another three DOF are needed to provide an arbitrary orientation for
the tool—six DOF in total as a minimum—many 3D arm manipulators’ major
linkages include three links and three joints, and so do typical robot hands. Our
motion planning algorithms operate on the major linkage—that is, on handling
gross motion and making sure that the hand is brought into the vicinity of the
target position. The remaining “fine tuning” for orientation is usually a simpler
task and is assumed to be done outside of the planning algorithm. For all but
very few unusual applications, this is a plausible assumption.
While studying topological characteristics of the robot configuration space
for a few 3D kinematics types, we will show that obstacle images in these
configuration spaces exhibit a distinct property that we call space monotonicity:
For any point on the surface of the obstacle image, there exists a direction
along which all the remaining points of the configuration space belong to the
obstacle. Furthermore, the sequential connection of the arm links results in the
property called space anisotropy of the configuration space, whereby the obstacle
monotonicity presents itself differently along different space axes.
The space monotonicity property provides a basis for selecting directions of
arm motion that are more promising than others for reaching the target posi-
tion. By exploiting the properties of space monotonicity and anisotropy, we will
produce motion planning algorithms with proven convergence. No explicit or
implicit beforehand calculations of the workspace or configuration space will be
ever needed. All the necessary calculations will be carried out in real time in the
arm workspace, based on its sensing information. No exhaustive search will ever
take place.