This paper describes the foundations and algorithms of a new probabilistic roadmap (PRM) planner that is: (1) singlequery –i.e., it does not pre-compute a roadmap, but uses the two input query configurations to explore as little space as possible; (2) bi-directional i.e., it searches the robot’s free space by concurrently building a roadmap made of two trees rooted at the query configurations – and (3) applies a systematic lazy collision-checking strategy –i.e., it postpones collision tests along connections in the roadmap until they are absolutely needed. Several observations motivated this collision-checking strategy: (1) PRM planners spend more than 90% of their time checking collision; (2) most connections in a PRM are not on the final path; (3) the collision test for a connection is the most expensive when there is no collision; and (4) the probability that a short connection is collision-free is large. The strengths of single-query and bidirectional sampling technique...