One problem that typically arises in mobile robotics is that optimal
motion plans bring robots too close to obstacles. Recall from Section
6.2.4 that the shortest Euclidean paths for motion
planning in a polygonal environment must be allowed to touch obstacle
vertices. This motivated the maximum clearance roadmap, which was
covered in Section 6.2.3. A grid-based approximate
version of the maximum clearance roadmap can be made. Furthermore, a
navigation function can be defined that guides the robot onto the
roadmap, then travels along the roadmap, and finally deposits the
robot at a specified goal. In [588], the resulting navigation
function is called *NF2*.

Assume that there is a single goal state, . The computation of a maximum clearance navigation function proceeds as follows:

- Instead of , assign to be the set of all states from which motion in at least one direction is blocked. These are the states on the boundary of the discretized collision-free space.
- Perform wavefront iterations that propagate costs in waves outward from the obstacle boundaries.
- As the wavefronts propagate, they will meet approximately at the
location of the maximum clearance roadmap for the original, continuous
problem. Mark any state at which two wavefront points arrive from
opposing directions as a
*skeleton state*. It may be the case that the wavefronts simply touch each other, rather than arriving at a common state; in this case, one of the two touching states is chosen as the skeleton state. Let denote the set of all skeleton states. - After the wavefront propagation ends, connect to the skeleton by inserting and all states along the path to the skeleton into . This path can be found using any search algorithm.
- Compute a navigation function over by treating all other states as if they were obstacles and using the wavefront propagation algorithm. This navigation function guides any point in to the goal.
- Treat as a goal region and compute a navigation function using the wavefront propagation algorithm. This navigation function guides the state to the nearest point on the skeleton.
- Combine and as follows to obtain . For every , let . For every remaining state, the value is assigned, in which is the nearest state to such that . The state can easily be recorded while is computed.

Steven M LaValle 2020-08-14