## Distance function concept

A grid-search algorithm aims at building a distance function u : Q2 ^ H , which is solution of the functional minimization problem defined as follows:

where {Cx } is the set of all the possible curves between the source xstart and the current configuration x within Q. For the sake of notational simplicity, and assuming that the source of exploration xstart is fixed, we note u(xstart, x) = u(x).

The distance function u may be related to the value function concept in reinforcement learning. The difference lies only in the fact that value functions are refined in an iterative process (called learning), whereas the distance function is built from scratch. In the path planning literature one can find other names for the distance function, such as navigation function (LaValle, 2006), convex-map (Melchior et al., 2003) or multi-valued distance map (Kimmel et al., 1998).

Once the distance function has been found through the goal configuration, the optimal path is the one which follows the gradient descent over the distance function from the goal to the start configuration. This backtracking technique is reliable as no local minima have been exhibited during the exploration process.

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