11.1.1 Sensors

As the name suggests, *sensors* are designed to sense the state.
Throughout all of this section it is assumed that the state space,
, is finite or countably infinite, as in Formulations 2.1
and 2.3. A *sensor* is defined in terms of two
components: 1) an *observation space*, which is the set of
possible readings for the sensor, and 2) a *sensor mapping*, which
characterizes the readings that can be expected if the current state
or other information is given. Be aware that in the planning model,
the state is not really given; it is only assumed to be given when
modeling a sensor. The sensing model given here generalizes the one
given in Section 9.2.3. In that case, the sensor provided
information regarding instead of because state spaces
were not needed in Chapter 9.

Let denote an *observation space*, which is a finite or
countably infinite set. Let denote the *sensor mapping*.
Three different kinds of sensor mappings will be considered, each of
which is more complicated and general than the previous one:

**State sensor mapping:**In this case, , which means that given the state, the observation is completely determined.**State-nature sensor mapping:**In this case, a finite set, , of*nature sensing actions*is defined for each . Each nature sensing action, , interferes with the sensor observation. Therefore, the state-nature mapping, , produces an observation, , for every and . The particular chosen by nature is assumed to be unknown during planning and execution. However, it is specified as part of the sensing model.**History-based sensor mapping:**In this case, the observation could be based on the current state or any previous states. Furthermore, a nature sensing action could be applied. Suppose that the current stage is . The set of nature sensing actions is denoted by , and the particular nature sensing action is . This yields a very general sensor mapping,(11.1)

Many examples of sensors will now be given. These are provided to illustrate the definitions and to provide building blocks that will be used in later examples of I-spaces. Examples 11.1 to 11.6 all involve state sensor mappings.

(11.2) |

The limitation of this sensor is that it only tells whether is odd or even. When combined with other information, this might be enough to infer the state, but in general it provides incomplete information.

(11.3) |

(11.4) |

This sensor provides very limited information because it only indicates on which side of the boundary the state may lie. It can, however, precisely determine whether .

(11.5) |

An obvious generalization can be made for any state space that is formed from Cartesian products. The sensor may reveal the values of one or more components, and the rest remain hidden.

A special case of the *bijective sensor* is the *identity
sensor*, for which is the identity function. This was essentially
assumed to exist for all planning problems covered before this chapter
because it immediately yields the state. However, any bijective
sensor could serve the same purpose.

From the examples so far, it is tempting to think about partitioning based on sensor observations. Suppose that in general a state mapping, , is not bijective, and let denote the following subset of :

which is the

Next consider some examples that involve a state-action sensor mapping. There are two different possibilities regarding the model for the nature sensing action:

**Nondeterministic:**In this case, there is no additional information regarding which will be chosen.**Probabilistic:**A probability distribution is known. In this case, the probability, , that will be chosen is known for each .

It is sometimes useful to consider the state-action sensor model as a probability distribution over for a given state. Recall the conversion from to in (9.28). By replacing by , the same idea can be applied here. Assume that if the domain of is restricted to some , it forms an injective (one-to-one) mapping from to . In this case,

(11.7) |

If the injective assumption is lifted, then is replaced by a sum over all for which .

(11.8) |

It is always known that . Therefore, if is received as a sensor reading, one of the following must be true: , , or .

(11.9) |

In this case, if , it is no longer known for certain whether . It is possible that or . If , then it is possible for the sensor to read , 0, or .

(11.10) |

Under the nondeterministic model for the nature sensing action, the sensor provides no useful information regarding the state. Regardless of the observation, it is never known whether is even or odd. Under a probabilistic model, however, this sensor may provide some useful information.

It is once again informative to consider preimages. For a state-action sensor mapping, the preimage is

In comparison to state sensor mappings, the preimage sets are larger for state-action sensor mappings. Also, they do not generally form a partition of . For example, the preimages of Example 11.8 are , , and . This is not a partition because every preimage contains 0. If desired, can be directly defined for each , instead of explicitly defining nature sensing actions.

Finally, one example of a history-based sensor mapping is given.

Steven M LaValle 2020-08-14