Page 272 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 247

The sensor-based approach is thus a topological approach. The question being
posed is as follows: Is there a solution to the robot motion planning problem based
on topological (rather than geometrical and algebraic) characteristics of the space
at hand—that is, a solution that does not require knowing shapes and dimensions
of the objects involved, be it the robot itself or obstacles in its environment? Such
procedures would allow robots to operate in a previously unknown environment
filled with arbitrarily shaped obstacles. As we saw in the prior sections, a positive
answer to this question carries significant advantages: It allows, first, to drop the
computationally expensive requirement of algebraic representation and, second,
drop the equally expensive requirement of complete information. The algorithms
developed in earlier sections suggest that at least in some cases of arm kinematics
the answer to the said question is “yes.”
For the topological approach to work correctly, in the prior sections of this
chapter it was vital that obstacle boundaries presented appropriate manifolds in
the corresponding configuration space. In this section we will study the spa-
tial relationships between the robot and obstacles and develop a set of condi-
tions under which the obstacle boundaries present manifolds in the configuration
space [107]. The analysis makes use of topology of the arm workspace and does
not require algebraic representations.
Recall that kinematically a robot arm manipulator consists of connected rigid
bodies, links and joints, which together possess some—say, d —degrees of freedom
(d-DOF), d = 1, 2,... . The spatial arrangement (position and orientation) of the
links and joints in the robot arm’s workspace makes its kinematic configuration.
In all practical cases a robot configuration can be uniquely described by a finite
number of parameters. Assuming that each DOF of the robot is implemented via
the (most popular in practice) translational (prismatic and sliding are other terms
used) or rotational (revolute) joint, each joint value represents such a parameter.
For example, the three-dimensional robot in Figure 5.32 has nine degrees of
freedom (9-DOF), and so its configuration can be described by the nine-tuple
(l 1 ,l 2 ,θ 3 ,θ 4 ,θ 5 ,θ 6 ,θ 7 ,l 8 ,l 9 ). Here translational variables l 1 ,l 2 ∈ describe the
Cartesian coordinates of the robot base unit; revolute variables θ 3 . ..θ 7 ∈ [0, 2π),
respectively, parameterize the “waist,” left and right “shoulders,” and “elbows”
of the robot arms; and translational variables l 8 and l 9 relate to the left- and
right-hand effectors, respectively.
Two different sets of the nine-tuple parameters above would describe two
distinct robot positions (configurations would be another term) in space. The col-
lection of all possible robot configurations define the robot configuration space
(C-space). To emphasize the theoretical nature of this section, we will drop the
terms W-space and C-space that we used above for the workspace and configu-
ration space and we will use instead abbreviations WS and CS, accordingly.
Due to the presence of obstacles in WS, some regions in CS are not reachable;
these regions collectively form the configuration space obstacle, denoted CSO or
O C . A reachable configuration is called a free configuration (FC); the subspace
that contains all free configurations is called the free configuration space, FCS.
Points in CS represent robot configurations. A path in CS represents continuous