Forward projections under a fixed plan

Forward projections can now be defined under the constraint that a particular plan is executed. The specific expression of actions is replaced by $ \pi $. Each time an action is needed from a state $ x \in X$, it is obtained as $ \pi (x)$. In this formulation, a different $ U(x)$ may be used for each $ x \in X$, assuming that $ \pi $ is correctly defined to use whatever actions are actually available in $ U(x)$ for each $ x \in X$.

First we will consider the nondeterministic case. Suppose that the initial state $ x_1$ and a plan $ \pi $ are known. This means that $ u_1 = \pi (x_1)$, which can be substituted into (10.10) to compute the one-stage forward projection. To compute the two-stage forward projection, $ u_2$ is determined from $ \pi (x_2)$ for use in (10.11). A recursive formulation of the nondeterministic forward projection under a fixed plan is

\begin{displaymath}\begin{split}X_{k+1}(x_1,\pi ) = \{ x_{k+1} \in X \;\vert\; &...
...mbox{ and } x_{k+1} = f(x_k,\pi (x_k),\theta_k)\} . \end{split}\end{displaymath} (10.28)

The probabilistic forward projection in (10.10) can be adapted to use $ \pi $, which results in

$\displaystyle P(x_{k+2}\vert x_k,\pi ) = \sum_{x_{k+1} \in X} P(x_{k+2}\vert x_{k+1},\pi (x_{k+1})) P(x_{k+1}\vert x_k,\pi (x_k)) .$ (10.29)

The basic idea can be applied $ k-1$ times to compute $ P(x_k \vert x_1,
\pi )$.

A state transition matrix can be used once again to express the probabilistic forward projection. In (10.15), all columns correspond to the application of the action $ u$. Let $ M_\pi $, be the forward projection due to a fixed plan $ \pi $. Each column of $ M_\pi $ may represent a different action because each column represents a different state $ x_k$. Each entry of $ M_\pi $ is

$\displaystyle m_{i,j} = P(x_{k+1} = i \; \vert \; x_k = j, \;\pi (x_k)) .$ (10.30)

The resulting $ M_\pi $ defines a Markov process that is induced under the application of the plan $ \pi $.

Steven M LaValle 2012-04-20