Page 208 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

INTRODUCTION 183

suffices to say here that the workspace of what we call “planar arms” is not
necessarily planar, but it remains two-dimensional.

5.1.1 Model and Definitions
The Robot Arm. The arm body consists of two links, l 1 and l 2 , and two joints,
J 0 and J 1 . Joint J 0 is fixed and is the origin of the reference system. See different
arm configurations in Figures 5.1a to 5.1e. Solely for presentation purposes we
represent the links as straight-line segments, of lengths l 1 and l 2 , respectively.
The arm’s configuration is defined by its kinematics—that is, by the way its
joints connect the links together. In the arms in Figure 5.1 a link may be rotating
about its joint, in which case the joint is a revolute joint, or it may be sliding in
it, in which case the joint is a prismatic joint.
Accordingly, in the case of a revolute joint the corresponding link is of a
constant length, and in case of a prismatic joint the corresponding link is of a
variable length. An arm solution for a given point P in the arm workspace (W-
space)—equivalent terms that we may use are arm position, arm coordinates,and
link positions—is defined by a pair of variables called joint values. Joint values
are either angles (as in Figure 5.1a) or linear translations (as in Figure 5.1b), or a
combination of both (as in Figures 5.1c, 5.1d, 5.1e). An equivalent presentation
for the same solution P is given, for example, by Cartesian coordinates (x, y)of
the link endpoints, a p and b p ; b p also designates the arm endpoint position at
point P. Positive directions and zero positions for joint values of the five arms
are shown in Figures 5.1a to 5.1e.
As said above, for the considered class of path planning algorithms, the shape
of arm links and of obstacles—for example, the fact of their convexity or con-
cavity—is of no importance. Without loss of generality, and solely for better
visualization and material presentation, line segment links and circular obstacles
are used in most figures of this section.
The arm is capable of performing the following actions:

1. Moving the arm endpoint through a prescribed simple curve (called main
line or M-line) that connects the arm’s start (S)and target (T ) positions.
2. When the arm body hits an obstacle, identifying the point(s) of the arm
body that is in contact with the obstacle.
3. Following the obstacle boundary.
The first of these operations implies that the arm is capable of computing coordi-
nates of consecutive points along the M-line and, if necessary, transforming them
into the corresponding joint values (using, for example, appropriate procedures
of inverse kinematics [8]).
The sole purpose of the second operation is to provide the local information
needed to pass around the obstacle. At any moment, when at least one point of
the arm body is in contact with an obstacle, the arm identifies coordinates of
the points of contact on the arm body relative to the arm’s internal reference