##

8.2.3 Grid-Based Navigation Functions for Motion Planning

To consider feedback plans for continuous spaces, vector fields and
other basic definitions from differential geometry will be needed.
These will be covered in Section 8.3; however, before
handling such complications, we first will describe how to use the
ideas presented so far in Section 8.2 as a discrete
approximation to feedback motion planning.

Examples 8.1 and 8.2 have already defined
feedback plans and navigation functions for 2D grids that contain
obstacles. Imagine that this model is used to approximate a motion
planning problem for which
. Section
5.4.2 showed how to make a topological graph that
approximates the motion planning problem with a grid of samples. The
motions used in Example 8.1 correspond to the
1-neighborhood definition, (5.37). This idea was
further refined in Section 7.7.1 to model approximate
optimal motion planning by moving on a grid; see Formulation
7.4. By choosing the Manhattan motion model, as defined in
Example 7.4, a grid with the same motions considered
in Example 8.1 is produced.

To construct a navigation function that may be useful in mobile
robotics, a high-resolution (e.g., to points per axis) grid
is usually required. In Section 5.4.2, only a few points
per axis were needed because feedback was not assumed. It was
possible in some instances to find a collision-free path by
investigating only a few points per axis. During the execution of a
feedback plan, it is assumed that the future states of the robot are
not necessarily predictable. Wherever the robot may end up, the
navigation function in combination with the local operator must
produce the appropriate action. If the current state (or
configuration) is approximated by a grid, then it is important to
reduce the approximation error as much as possible. This is
accomplished by setting the grid resolution high. In the feedback
case, the grid can be viewed as ``covering'' the whole configuration
space, whereas in Section 5.4.2 the grid only represented a
topological graph of paths that cut across the
space.^{8.3}

**Subsections**
Steven M LaValle
2020-08-14