  Bi-directional search is a widely used strategy to increase the success and
convergence rates of sampling-based motion planning algorithms. Yet, few
results are available that merge both bi-directional search and asymptotic
optimality into existing optimal planners, such as PRM*, RRT*, and FMT*. The
objective of this paper is to fill this gap. Specifically, this paper presents
a bi-directional, sampling-based, asymptotically-optimal algorithm named
Bi-directional FMT* (BFMT*) that extends the Fast Marching Tree (FMT*)
algorithm to bi-directional search while preserving its key properties, chiefly
lazy search and asymptotic optimality through convergence in probability. BFMT*
performs a two-source, lazy dynamic programming recursion over a set of
randomly-drawn samples, correspondingly generating two search trees: one in
cost-to-come space from the initial configuration and another in cost-to-go
space from the goal configuration. Numerical experiments illustrate the
advantages of BFMT* over its unidirectional counterpart, as well as a number of
other state-of-the-art planners.
