Jethro's Braindump

Robotics Probabilistic Generative Laws


world state at time \(t\)
measurement data at time \(t\) (e.g. camera images)
control data (change of state in the environment) at time \(t\)

State Evolution

\begin{equation} p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t}) \end{equation}

State Transition Probability

\begin{equation} p(x_t | x_{0:t-1} z_{1:t-1}, u_{1:t}) = p ( x_t | x_{t-1}, u_t) \end{equation}

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

Measurement Probability

\begin{equation} p(z_t | x_{0:t}, z_{1:t-1}, u_{1:t}) = p(z_t | x_t) \end{equation}

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