Developed mainly in control theory literature, linear sensing models are some of the most common and important. For all of the sensors in this family, assume that (nonsingular linear transformations allow the sensor space to effectively have lower dimension, if desired). The simplest case in this family is the identity sensor, in which . In this case, the state is immediately known. If this sensor is available at every stage, then the I-space collapses to by the I-map .
Now nature sensing actions can be used to corrupt this perfect state observation to obtain . Suppose that is an estimate of , the current state, with error bounded by a constant . This can be modeled by assigning for every , as a closed ball of radius , centered at the origin:
A more typical probabilistic sensing model can be made by letting and defining a probability density function over all of . (Note that the nondeterministic version of this sensor is completely useless.) One of the easiest choices to work with is the multivariate Gaussian probability density function,
The sensing models presented so far can be generalized by applying linear transformations. For example, let denote a nonsingular matrix with real-valued entries. If the sensor mapping is , then the state can still be determined immediately because the mapping is bijective; each contains a unique point of . A linear transformation can also be formed on the nature sensing action. Let denote an matrix. The sensor mapping is
In general, and may even be singular, and a linear sensing model is still obtained. Suppose that . If is singular, however, it is impossible to infer the state directly from a single sensor observation. This generally corresponds to a projection from an -dimensional state space to a subset of whose dimension is the rank of . For example, if
Steven M LaValle 2020-08-14