The lattices developed in this section were introduced in
[290] for analyzing the kinodynamic planning problem
in the rigorous approximation algorithm framework for NPhard
problems [765]. Suppose that there are two or three
independent double integrators. The analysis shows that the computed
solutions are approximately optimal in the following sense. Let
and be two positive constants that define a function

(14.25) 
Let denote the time at which the termination action is applied.
A state trajectory is called
safe if for all
, the ball of radius
that is
centered at does not cause collisions with obstacles in .
Note that the ball radius grows linearly as the speed increases. The
robot can be imagined as a disk with a radius determined by speed.
Let , , , and be given (only a point goal
region is allowed). Suppose that for a given problem, there exists a
safe state trajectory (resulting from integrating
any
) that terminates in after time .
It was shown that by choosing the appropriate (given by a
formula in [290]), applying breadthfirst search to
the reachability lattice will find a
safe trajectory that takes time at most
, and approximately connects to
(which means that the closeness in depends on
). Furthermore, the running time of the algorithm is
polynomial in
and the number of primitives used to define
polygonal obstacles.^{14.5} One of the key steps in the analysis is to show that any state
trajectory can be closely tracked using only actions from and
keeping them constant over . One important aspect is that
it does not necessarily imply that the computed solution is close to
the true optimum, as it travels through (only the execution times
are close). Thus, the algorithm may give a solution from a different
homotopy class from the one that contains the true optimal trajectory.
The analysis was extended to the general case of openchain robot arms
in [288,441].
Steven M LaValle
20200814