Foundations — Kalman filter derivation — predict step, update step

This page assumes you know nothing. Before you touch the predict/update equations in the parent topic, you must be fluent in a small pile of symbols. We build them one at a time, each from a picture, each earning the next.

0. The single picture to hold in your head

Everything is about a bump. A "guess with doubt" is drawn as a bell curve: the peak sits at your best guess, and the width says how unsure you are. Narrow bump = confident. Wide bump = clueless.

1. Scalars, vectors, and the state $x$

The hidden thing we care about is the state $x$. In GNC it usually bundles several physical quantities:

$x=\left[\begin{array}{c}p\\ v\end{array}\right]⟸\text{position }p\text{ and velocity }v,\text{ stacked}$

$x_k$ means "the state at time step $k$". The little $k$ is a clock tick: $k-1$ is the previous instant, $k$ is now, $k+1$ is next. We build State-space representation out of exactly this indexing.

2. The hat $\hat{x}$ — guess versus truth

This distinction is the most-missed idea in the whole subject. Every time you see a hat, remind yourself: this is a belief, not a fact.

3. Variance ${\sigma }^2$ and the width of the bump

4. The covariance matrix $P$

One number (${\sigma }^2$) captures doubt about one quantity. But $x$ is a vector, so we need doubt about each entry and how their doubts are linked. That bookkeeping is the matrix $P$.

The symbol $\mathbb{E}[\cdot ]$ means expectation — the long-run average if you repeated the experiment forever. Picture: run the noisy world a million times, average the result. That is all $\mathbb{E}$ ever means.

The superscript $\top$ (as in $A^{\top }$) is transpose: flip a matrix over its diagonal, turning rows into columns. Picture: a grid reflected across its top-left-to-bottom-right line. We need it because a column times a row builds a whole grid — exactly the outer product inside $P$.

5. The Gaussian $\mathcal{N}(0,Q)$ and noise

Two noises appear on the parent:

• $w_k\sim \mathcal{N}(0,Q)$ — process noise. Picture: little random shoves the physics model fails to predict (wind gusts, unmodelled forces). $Q$ = how big those shoves are.
• $v_k\sim \mathcal{N}(0,R)$ — measurement noise. Picture: the jitter on a sensor reading. $R$ = how shaky the sensor is.

6. The transforming matrices $F$, $B$, $H$

These three matrices are the "machines" that connect states, controls, and measurements.

7. The measurement $z$, the innovation $y$, and the gain $K$

The superscript minus, as in ${\hat{x}}_k^{-}$ and $P_k^{-}$, marks a prior — before the measurement is folded in ("predicted, not yet corrected"). No minus = posterior, after correction. Picture: minus = "coasting", no-minus = "just snapped to the sensor".

8. Two calculus tools the derivation borrows

The parent finds the best gain by minimising doubt. Two tools do that:

How it all feeds the topic

Once these foundations are solid, the two big steps (predict, update) are just this machinery applied twice: push the bump forward with $F$ (widen by $Q$), then squeeze it toward the sensor with $K$. From here you can also climb to the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) for nonlinear worlds, all resting on Bayesian inference.

Equipment checklist

Test yourself — cover the right side and answer: