Page 205 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

180 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

Types of Sensing and Robot Geometry Versus Algorithms. Here and else-
where in this text, when we develop motion planning algorithms based on tactile
sensing and on a simplified shapes of robots (say, a point mobile robot or a stick
line arm manipulator links), this does not imply that tactile sensors or simplified
shapes are the only, or the recommended, modalities for an algorithm at hand:
(a) Any type of sensing (tactile, proximal, vision, etc.) can be used with such
algorithms, either directly or with small easily realizable modifications, provided
that sensing covers every point of the robot body. (b) The algorithm will work
with the robots or arm manipulator links of any shapes. See Section 1.2 and later
in Section 5.1.1.

Major and Minor Linkage. Following Ref. 103, we use the notion of a separable
arm, which is an arm manipulator that can be naturally divided into (a) the major
linkage responsible for the arm’s position planning (or gross motion) and (b) the
minor linkage responsible for the orientation of the arm’s end effector (its hand)
in the arm workspace. As a rule, existing arm manipulators are separable, and so
are the limbs of humans and animals (although theoretically this does not have
to be so).
The notions of major and minor linkages are tied to the notion of a minimal
configuration. For the major linkage a minimal configuration is the minimum
number of links (and joints) that the arm needs to be able to reach any point in its
workspace. For the two-dimensional (2D) case the minimal configuration includes
two links (and two joints). Looking ahead, in the three-dimensional (3D) case the
minimal configuration for the major linkage includes three links (and three joints).
In the algorithmic approach considered here, motion planning is limited to the
gross motion—that is, to the major linkage. The implicit assumption is that the
motion of the end effector (i.e., the minor linkage) can be planned separately, after
the arm’s major linkage arrives in the vicinity of the target position. For all but
very unusual applications, this is a plausible assumption. Although theoretically
this does not have to be so, providing orientation for the minor linkage is usually
significantly simpler than for the major linkage, simply because the hand is small.

Types of Two-Link Arm Manipulators. These kinematic pairs are called two-
dimensional (2D) arms, to reflect the fact that the end effector of any such
arm moves on a two-dimensional surface—or, in topological terms, in a surface
homeomorphic to a plane. With this understanding, we can call these arms planar
arms, as they are often called, although the said surface may or may not be a
plane. Only revolute and sliding joints will be considered, the two types that are
primary joints used in practical manipulators. Other types of joints appear very
rarely and are procedurally reducible to these two [103].
A revolute joint between two links is similar to the human elbow: One link
rotates about the other, and the angle between the links describes the joint value
at any given moment (see Figure 5.1a). In a sliding joint the link slides relative to
the other link; the linear displacement of sliding is the corresponding joint value
(see Figure 5.1b). Sliding joints are also known in the literature on kinematics
as prismatic joints.