Resistors, Markov chains and dynamic path planning

Dynamic planning involves continuously updating a map by sensing changes in the environment and planning appropriate actions, with all tasks sharing common computational resources. We use harmonic functions for dynamic planning. Analogous representations of harmonic functions as Markov chains and resistor networks are used to develop the notion of escape probability and energy dissipation. These measures are used to indicate convergence (event that permits resources to be devote to non-planning tasks) more robustly than monitoring maximum or average field changes between iterations. The convergence of the harmonic function is related quadratically to the number of grid elements. An example of an irregular sampling strategy - quad tree - is developed for harmonic functions, which is complete yet imprecise. Quad trees are not a sufficient sampling strategy for addressing the exponential growth of multiple dimensions and therefore current investigations include other sampling strategies or dimensional parallelization.