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

An Efficient Semi-Real-Time Algorithm for Path Planning in the Hamilton-Jacobi Formulation