Page 274 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

TOPOLOGY OF ARM’S FREE CONFIGURATION SPACE 249

Under what conditions will CSO boundaries present manifolds? We will show
below that if a certain unrestrictive spatial relationship between the robot and
obstacle is satisfied—for example, under no condition is the robot immobilized,
nor does it need to squeeze between two obstacles as in Figure 5.33—then FCS
is uniformly locally connected (ULC). Although the ULC property is not sufficient
to ensure manifold boundaries for the general case, we will show that for the
two-dimensional (2D) case the ULC property guarantees that FCS is bounded by
manifolds—in this case, simple closed curves.
We proceed as follows. First, a general robot with d translational and/or rev-
olute degrees of freedom will be defined. CS is defined as a Euclidean space
formed by the robot parameter variables. A physical obstacle is the interior of a
connected compact point set in WS. We will show that CSO is a closed subset
of CS (Corollary 5.8.1).
Then we will study the interaction between the robot and obstacles, and will
define a set of conditions that correspond to certain undesirable degenerate situ-
ations (Conditions 5.8.1 to 5.8.4), such as when a part of or the whole robot is
immobilized or when the robot can move between two obstacles only by simul-
taneously touching them both (Figure 5.33). We will show in Section 5.8.3 that
after these situations are removed, CSO presents a uniformly locally connected
subset of CS.
We will then show in Section 5.8.4 that ULC is a necessary condition for
an open subset of a compact space to have manifold boundaries. This is also a
sufficient condition for a 2D open subset of a compact space to have manifold
boundaries—that is, simple closed curves. We will thus conclude that FCS of a
2-DOF robot is bounded by simple closed curves. More details pertinent to the
material in this section can be found in [107].

5.8.1 Workspace; Configuration Space
As said above, kinematically a robot arm manipulator is an assembly of rigid links
connected to each other by joints that permit the links’ motion relative to each
other [8]. Joints and links form kinematic pairs. As in prior sections, without
loss of generality we limit the types of kinematic pairs to either translational
(prismatic) or rotational (revolute). The degrees of freedom (DOF) of a robot are
often referred to as its mobility, which is the number of independent variables
that must be specified in order to locate all the links relative to each other.
The 9-DOF robot shown in Figure 5.32 has nine joints and nine links. The robot
base is a link in which two translational joints l 1 and l 2 are implemented. The
number of degrees of freedom of a robot is not necessarily equal to the number
of links or the number of joints. Closed kinematic chains often have fewer DOF
than the number of their links (joints). For example, a triangle-shaped planar closed
kinematic chain with three links and three revolute joints has mobility zero.
We choose arbitrarily d independent joints, J 1 ,...,J d ,toform a d-DOF robot,
and we parameterize the robot configuration using the corresponding joint vari-
ables. With a reference system defined at its connecting joint, each kinematic
pair can be specified by four scalar parameters. So, for the joint i, a i is the link