Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required to pass through narrow passages in the free space. In [16] we proposed MAPRM, a method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space without having to first compute the medial axis; this was shown to increase sampling in narrow passages. In this paper we give details of the MAPRM algorithm for the case of a rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the C-obstacles or the medial axis. We give theoretical and experimental results to show this improves performance on problems involving narrow corridors and compare the performance to uniform random sampling from the free space.