A convenient, alternative formulation can be given by allowing nature to act twice:

- First, a nature action, , is chosen but is unknown to the robot.
- Following this, a
*nature observation action*is chosen to interfere with the robot's ability to sense .

- A nonempty, finite set called the
*action space*. - A nonempty, finite set called the
*nature action space*. - A nonempty, finite set called the
*observation space*. - For each
, a nonempty set
called the
*nature observation action space*. - A sensor mapping .
- A function
called the
*cost function*.

such that | (9.27) |

For a probabilistic model, a distribution is specified (often, this may reduce to ). Suppose that when the domain of is restricted to some , then it forms an injective mapping from to . In other words, every nature observation action leads to a unique observation, assuming is fixed. Using and , is derived as

If the injective assumption is lifted, then is replaced by a sum over all for which . In Formulation 9.6, the only difference between the nondeterministic and probabilistic models is the characterization of , which represents a kind of measurement interference. A strategy still takes the form . A hybrid model is even possible in which one nature action is modeled nondeterministically and the other probabilistically.

Steven M LaValle 2012-04-20