Jethro's Braindump

Robotics Probabilistic Generative Laws

Notation

$x_{t}$: world state at time $t$
$z_{t}$: measurement data at time $t$ (e.g. camera images)
$u_{t}$: control data (change of state in the environment) at time $t$

State Evolution

$p (x_{t} | x_{0 : t - 1} z_{1 : t - 1}, u_{1 : t})$

State Transition Probability

$p (x_{t} | x_{0 : t - 1} z_{1 : t - 1}, u_{1 : t}) = p (x_{t} | x_{t - 1}, u_{t})$

The world state at the previous time-step is a sufficient summary of all that happened in previous time-steps.

Measurement Probability

$p (z_{t} | x_{0 : t}, z_{1 : t - 1}, u_{1 : t}) = p (z_{t} | x_{t})$

The measurement at time-step $t$ is often just a noisy projection of the world state at time-step $t$ .

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