When bodies are attached in a kinematic chain, degrees of freedom are
removed. Figure 3.9 shows two different ways in which
a pair of 2D links can be attached. The place at which the links are
attached is called a *joint*. For a *revolute joint*, one
link is capable only of rotation with respect to the other. For a
*prismatic joint* is shown, one link slides along the other. Each
type of joint removes two degrees of freedom from the pair of bodies.
For example, consider a revolute joint that connects
to
.
Assume that the point in the body frame of
is
permanently fixed to a point in the body frame of
.
This implies that the translation of
is completely determined
once and are given. Note that and depend on
, , and . This implies that
and
have a total of four degrees of freedom when attached. The
independent parameters are , , , and .
The task in the remainder of this section is to determine exactly how
the models of
,
, ,
are transformed when
they are attached in a chain, and to give the expressions in terms of
the independent parameters.

Consider the case of a kinematic chain in which each pair of links is attached by a revolute joint. The first task is to specify the geometric model for each link, . Recall that for a single rigid body, the origin of the body frame determines the axis of rotation. When defining the model for a link in a kinematic chain, excessive complications can be avoided by carefully placing the body frame. Since rotation occurs about a revolute joint, a natural choice for the origin is the joint between and for each . For convenience that will soon become evident, the -axis for the body frame of is defined as the line through the two joints that lie in , as shown in Figure 3.10. For the last link, , the -axis can be placed arbitrarily, assuming that the origin is placed at the joint that connects to . The body frame for the first link, , can be placed using the same considerations as for a single rigid body.

Steven M LaValle 2012-04-20