Planning for optimal motion is a central problem in
robotics. Historically, a computational geometry perspective has
been a dominant approach towards solving this problem, in which a
feasible path is considered to be a solution. In this talk, we
present a drastically different solution of the planning problem,
which relies on computing the feedback control policy as a
solution of Hamilton-Jacobi-Bellman partial differential
equation (HJB-PDE). By introducing the Delaunay mesh refinement
algorithm, we propose an incremental HJB-PDE solver that shares
many benefits of geometric path planning algorithms, such as
probabilistic completeness and asymptotic convergence. In
addition to theoretical benefits, planning directly for feedback
control eliminates unnecessary path following middleware during
the plan execution stage in practice. Numerical experiments,
reveal that the proposed feedback planning algorithm outperforms
many previous path planners when applied to low-dimensional
problems. When the dimension number is higher than five, however,
both geometric path planning algorithms and incremental PDE
solvers fail to converge in reasonable time.