This section illustrates the effect of nature sensing actions, but only for the nondeterministic case. General methods for computing probabilistic I-states are covered in Section 11.6.

Let . Motions are generated by integrating the velocity , which is expressed as and . For simplicity, assume is applied for all time, which is a command to move right. The nature action interferes with the outcome. The robot tries to make progress by moving in the positive direction; however, the interference of nature makes it difficult to predict the direction. Without nature, there should be no change in the coordinate; however, with nature, the error in the direction could be as much as , after seconds have passed. Figure 11.24 illustrates the possible resulting motions.

Sensor observations will be made that alleviate the growing cone of
uncertainty; use the sensing model from Figure 11.11, and
suppose that the measurement error is . Suppose there is a
disc in
of radius larger than , as shown in Figure
11.23a. Since the true state is never further than
from the measured state, it is always possible to determine
whether the state passed above or below the disc. Multiple possible
observation histories are shown in Figure 11.23a.
The observation history need not even be continuous, but it is drawn
that way for convenience. For a disc with radius less than , there
may exist some observation histories for which it is impossible to
determine whether the true state traveled above or below the disc; see
Figure 11.23b. For other observation histories, it
may still be possible to make the determination; for example, from the
uppermost trajectory shown in Figure 11.23b it is
known for certain that the true state traveled above the disc.

To control the robot, a motion command is given in the form of an action . Nature interferes with the motions in two ways: 1) The robot tries to travel some distance , but there is some error , for which the true distance traveled, , is known satisfy ; and 2) the robot tries to move in a direction , but there is some error, , for which the true direction is known to satisfy . These two independent errors can be modeled by defining a 2D nature action set, . The transition equation is then defined so that the forward projection is as shown in Figure 11.25.

Some nondeterministic I-states will now be constructed. Suppose that the initial state is known, and history I-states take the form

(11.73) |

The first sensor observation, , is useless because the initial state is known. Equation (11.29) is applied to yield . Suppose that the action is applied, indicating that the robot should move horizontally to the right. Equation (11.30) is applied to yield , which looks identical to the shown in Figure 11.25. Suppose that an observation is received as shown in Figure 11.26a. Using this, is computed by taking the intersection of and , as shown in Figure 11.26b.

The next step is considerably more complicated. Suppose that and that (11.30) is applied to compute
from
. The shape shown in Figure
11.26c is obtained by taking the union of
for
all possible
. The resulting shape is composed
of circular arcs and straight line segments (see Exercise
13). Once is obtained, an intersection is taken
once again to yield
, as
shown in Figure 11.27. The process repeats in the same way
for the desired number of stages. The complexity of the region in
Figure 11.26c provides motivation for the approximation
methods of Section 11.4.3. For example, the
nondeterministic I-states could be nicely approximated by ellipsoidal
regions.

Steven M LaValle 2020-08-14