Erdmann proposed a practical way to compute effective motion commands by separating the reachability and recognizability issues [311,312]. Reachability refers to characterizing the set of points that are guaranteed to be reachable. Recognizability refers to knowing that the subgoal has been reached based on the history I-state. Another way to interpret the separation is that the effects of nature on the configuration transitions is separated from the effects of nature on sensing.

For reachability analysis, the sensing uncertainty is neglected. The
notions of forward projections and backprojections from Section
10.1.2 can then be used. The only difference here is that
they are applied to continuous spaces and motion commands (instead of
). Let denote a subset of
. Both weak
backprojections,
, and strong backprojections,
, can be defined. Furthermore, *nondirectional
backprojections* [283],
and
, can be defined,
which are analogous to (10.25) and
(10.26), respectively.

Figure 12.46 shows a simple problem in which the task is
to reach a goal edge with a motion command that points downward. This
is inspired by the peg-in-hole problem. Figure
12.47 illustrates several backprojections from the goal
region for the problem in Figure 12.46. The action is
; however, the actual motion lies within the shown cone
due to nature. First suppose that contact with the obstacle is not
allowed, except at the goal region. The strong backprojection
is given in Figure 12.47a. Starting from any point in
the triangular region, the goal is guaranteed to be reached in spite
of nature. The weak backprojection is the unbounded region
shown in Figure 12.47b. This indicates configurations
from which it is *possible* to reach the goal. The weak
backprojection will not be considered further because it is important
here to *guarantee* that the goal is reached. This is
accomplished by the strong backprojection. From here onward, it will
be assumed that *backprojection* by default means a strong
backprojection. Using weak backprojections, it is possible to develop an alternative framework of
*error detection and recovery* (EDR), which was introduced by Donald in [281].

Now assume that compliant motions are possible along the obstacle boundary. This has the effect of enlarging the backprojections. Suppose for simplicity that there is no friction ( in Figure 12.44a). The backprojection is shown in Figure 12.47c. As the robot comes into contact with the side walls, it slides down until the goal is reached. It is not important to keep track of the exact configuration while this occurs. This illustrates the power of compliant motions in reducing uncertainty. This point will be pursued further in Section 12.5.2. Figure 12.47d shows the backprojection for a different motion command.

Now consider computing backprojections in a more general setting. The
backprojection can be defined from any subset of
and
may allow a friction cone with parameter . To be included in
a backprojection, points from which sticking is possible must be
avoided. Note that sticking is possible even if
. For
example, in Figure 12.46, nature may allow the motion to
be exactly perpendicular to the obstacle boundary. In this case,
sticking occurs on horizontal edges because there is no tangential
motion. In general, it must be determined whether sticking is *possible* at each edge and vertex of
. Possible sticking from
an edge depends on , , and the maximum directional error
contributed by nature. The robot can become stuck at a vertex if it
is possible to become stuck at either incident edge.

Steven M LaValle 2012-04-20