Page 301 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

276 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

We will first analyze the PPP arm (Section 6.2), an arm with three slid-
ing (prismatic) joints (it is often called the Cartesian arm), and will develop a
sensor-based motion planning strategy for it. Similar to the 2D Cartesian arm,
the SIM algorithm for a 3D Cartesian arm turns out to be the easiest to visu-
alize and to design. After mastering in this case the issues of 3D algorithmic
machinery, in Section 6.3 we will turn our attention to the general case of an
XXP linkage. Similar to the material in Section 5.8, some theory developed in
Section 6.2.4 and Sections 6.3 to 6.3.6 is somewhat more complex than most of
other sections.
As before, we assume that the arm manipulator has enough sensing to sense
nearby obstacles at any point of its body. A good model of such sensing mech-
anism is a sensitive skin that covers the whole arm body, similar to the skin on
the human body. Any other sensing mechanism will do as long as it guarantees
not missing potential obstacles. Similar to the algorithm development for the 2D
case, we will assume tactile sensing: As was shown in prior chapters, the algo-
rithmic clarity that this assumption brings is helpful in the algorithm design. We
have seen in Sections 3.6 and 5.2.5 that extending motion planning algorithms
to more information-rich sensing is usually relatively straightforward. Regarding
the issues of practical realization of such sensing, see Chapter 8.

6.2 THE CASE OF THE PPP (CARTESIAN) ARM

The model, definitions, and terminology that we will need are introduced in
Section 6.2.1. The general idea of the motion planning approach is tackled in
Section 6.2.2. Relevant analysis appears in Sections 6.2.3 and 6.2.4. We formu-
late, in particular, an important necessary and sufficient condition that ties the
question of existence of paths in the 3D space of this arm to existence of paths
in the projection 2D space (Theorem 6.2.1). This condition helps to lay a foun-
dation for “growing” 3D path planning algorithms from their 2D counterparts.
The corresponding existential connection between 3D and 2D algorithms is for-
mulated in Theorem 6.2.2. The resulting path planning algorithm is formulated
in Section 6.2.5, and examples of its performance appear in Section 6.2.6.

6.2.1 Model, Definitions, and Terminology
For the sake of completeness, some of the material in this section may repeat the
material from other chapters.

Robot Arm. The robot arm is an open kinematic chain consisting of three links,
l 1 , l 2 ,and l 3 ,and three joints, J 1 , J 2 ,and J 3 ,of prismatic (sliding) type [8]. Joint
axes are mutually perpendicular (Figure 6.2). For convenience, the arm endpoint
P coincides with the upper end of link l 3 . Point J i , i = 1, 2, 3, also denotes the
center point of joint J i , defined as the intersection point between the axes of link
l i and its predecessor. Joint J 1 is attached to the robot base O and is the origin