If the robot consists of multiple bodies, the situation is more complicated. The definition in (4.34) only implies that the robot does not collide with the obstacles; however, if the robot consists of multiple bodies, then it might also be appropriate to avoid collisions between different links of the robot. Let the robot be modeled as a collection, , of links, which may or may not be attached together by joints. A single configuration vector is given for the entire collection of links. We will write for each link, , even though some of the parameters of may be irrelevant for moving link . For example, in a kinematic chain, the configuration of the second body does not depend on the angle between the ninth and tenth bodies.

Let denote the set of *collision pairs*, in which each
collision pair,
, represents a pair of link indices
, such that
. If appears in
, it means that and are not allowed to be in a
configuration, , for which
.
Usually, does not represent all pairs because consecutive links
are in contact all of the time due to the joint that connects them.
One common definition for is that each link must avoid collisions
with any links to which it is not attached by a joint. For
bodies, is generally of size ; however, in practice it
is often possible to eliminate many pairs by some geometric analysis
of the linkage. Collisions between some pairs of links may be
impossible over all of , in which case they do not need to appear
in .

Using , the consideration of robot self-collisions is added to the definition of to obtain

Thus, a configuration is in if at least one link collides with or a pair of links indicated by collide with each other.

Steven M LaValle 2020-08-14