Components of a Kalman Filter

Term Rank Describes Description
\(A_t\) Matrix \((n × n)\) state change Describes how the state evolves from \(t - 1\) to \(t\) without controls or noise.
\(B_t\) Matrix \((n × l)\) control change Describes how the control \(u_t\) changes the state from \(t - 1\) to \(t\).
\(C_t\) Matrix \((k × n)\) maps state to observation Describes how to map the state \(x_t\), to an observation \(z_t\).

\(A_t\)

How does the state evolve if no command or control is executed.

\(B_t\)

term purpose
n The dimensionality of the state
l The dimensionality of the odometry 
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"prior knowledge of state" -> "P,x k-1|k-1"
"P,x k-1|k-1" -> "prediction step"
"next timestep" -> "P,x k-1|k-1"
"P,x k|k" -> "next timestep"j
"P,x k|k" -> "output estimate of state"
"update step" -> "P,x k|k"
measurements
                                 +-----------------+
                                 |  measurements   |
                                 +-----------------+
                                 +-----------------+
                                 |   update step   |
                                 +-----------------+
                                   |
                                   |
                                   v
+--------------------------+     +-----------------+
| output estimate of state | <-- |     P,x k|k     |
+--------------------------+     +-----------------+
                                   |
                                   |
                                   v
                                 +-----------------+
                                 |  next timestep  |
                                 +-----------------+
                                   |
                                   |
                                   v
+--------------------------+     +-----------------+
| prior knowledge of state | --> |   P,x k-1|k-1   |
+--------------------------+     +-----------------+
                                   |
                                   |
                                   v
                                 +-----------------+
                                 | prediction step |
                                 +-----------------+