The *random loop generator* greatly improves the chance of
obtaining closure by iteratively restricting the range on each of the
active variables. The method requires that the active variables
appear sequentially along the chain (i.e., there is no interleaving of
active and passive variables). The coordinates of are
obtained sequentially as follows. First, compute an interval, ,
of allowable values for . The interval serves as a loose bound
in the sense that, for any value
, it is known for
certain that closure cannot be obtained. This is ensured by
performing a careful geometric analysis of the linkage, which will be
explained shortly. The next step is to generate a sample in
, which is accomplished in [244] by picking a random
point in . Using the value , a bounding interval is
computed for allowable values of . The value is
obtained by sampling in . This process continues iteratively
until and are obtained, unless it terminates early
because some
for . If successful termination
occurs, then the active variables are used to find values
for the passive variables. This step still might fail, but the
probability of success is now much higher. The method can also be
applied to linkages in which there are multiple, common loops, as in
the Stewart-Gough platform, by breaking the linkage into a tree and
closing loops one at a time using the loop generator. The performance
depends on how the linkage is decomposed [244].

Steven M LaValle 2012-04-20