  We present a semi-real-time algorithm for minimal-time optimal path planning
based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ)
equations. Partial differential equation (PDE) based optimal path planning
methods are well-established in the literature, and provide an interpretable
alternative to black-box machine learning algorithms. However, due to the
computational burden of grid-based PDE solvers, many previous methods do not
scale well to high dimensional problems and are not applicable in real-time
scenarios even for low dimensional problems. We present a semi-real-time
algorithm for optimal path planning in the HJ formulation, using grid-free
numerical methods based on Hopf-Lax formulas. In doing so, we retain the
intepretablity of PDE based path planning, but because the numerical method is
grid-free, it is efficient and does not suffer from the curse of
dimensionality, and thus can be applied in semi-real-time and account for
realistic concerns like obstacle discovery. This represents a significant step
in averting the tradeoff between interpretability and efficiency. We present
the algorithm with application to synthetic examples of isotropic motion
planning in two-dimensions, though with slight adjustments, it could be applied
to many other problems.
