First, we consider a simple example that uses the sign sensor of Example 11.3.
The general expression for a history I-state at stage is
The solution can even be implemented with sensor feedback because the action depends only on the current sensor value. Let be defined as
The next example provides a simple illustration of solving a problem without ever knowing the current state. This leads to the goal recognizability problem  (see Section 12.5.1).
The problem shown in Figure 11.7 serves two purposes. First, it is an example of sensorless planning [321,394], which means that there are no observations (see Sections 11.5.4 and 12.5.2). This is an interesting class of problems because it appears that no information can be gained regarding the state. Contrary to intuition, it turns out for this example and many others that plans can be designed that estimate the state. The second purpose is to illustrate how the I-space can be dramatically collapsed using the I-map concepts of Section 11.2.1. The standard nondeterministic I-space for this example contains I-states, but it can be mapped to a much smaller derived I-space that contains only a few elements.
There are no sensor observations for this problem. However, nature interferes with the state transitions, which leads to a form of nondeterministic uncertainty. If an action is applied that tries to take one step, nature may cause two or three steps to be taken. This can be modeled as follows. Let
Since there are no sensor observations, the history I-state at stage is
With perfect information, this would be trivial; however, without sensors the uncertainty may grow very quickly. For example, after applying the action from the initial state, the nondeterministic I-state becomes . After it becomes . A nice feature of this problem, however, is that uncertainty can be reduced without sensing. Suppose that for stages, we repeatedly apply . What is the resulting I-state? As the corner state is approached, the uncertainty is reduced because the state cannot be further changed by nature. It is known that each action, , decreases the coordinate by at least one each time. Therefore, after nine or more stages, it is known that . Once this is known, then the action can be applied. This will again increase uncertainty as the state moves through the set of left states. If is applied nine or more times, then it is known for certain that , which is the required goal state.
A successful plan has now been obtained: 1) Apply for nine stages, 2) then apply for nine stages. This plan could be defined over ; however, it is simpler to use the I-map from Example 11.12 to define a plan as . For such that , . For such that , . For , . Note that the plan works even if the initial condition is any subset of . From this point onward, assume that any subset may be given as the initial condition.
Some alternative plans will now be considered by making other derived I-spaces from . Let be an I-map from to a set of three derived I-states. Let , in which denotes ``goal,'' denotes ``left,'' and denotes ``any.'' The I-map, , is
To address this problem, consider a new I-map,
, which is sufficient. There are derived
I-states, which include as defined previously, for
, and for
. The I-map is defined as
for the smallest value of such that
is a subset of
. If there is no
such value for , then
, for the smallest
value of such that is a subset of
. Now the plan is defined
. Although the plan is larger, the robot does not need to
represent the full nondeterministic I-state during execution. The
correct transitions occur. For example, if
at , then is obtained. If
is applied at
, then is obtained. From there, is applied to
yield . These actions can be repeated until eventually and
are reached. The resulting plan, however, is not an improvement
over the original open-loop one.
Steven M LaValle 2012-04-20