Foundations — Unscented Kalman Filter (UKF) — sigma points, better for nonlinear

Before you can read a single line of the parent note, you need a vocabulary. This page builds every symbol it uses, starting from a smart-but-blank slate. Nothing here assumes you've seen probability, matrices, or filters before.

1. What is a "state"? The symbol $x$

Picture it: a single point floating in a room. If the room has $n$ directions you can move, the point needs $n$ numbers to pin it down.

Why the topic needs it: the entire filter's job is to keep a good estimate of $x$ over time. Everything else — noise, covariance, sigma points — exists only to answer "where is $x$, really?"

• $n$ ::: the number of entries in $x$, called the dimension. A 2D position → $n=2$.
• $\bar{x}$ (read "x-bar") ::: our single best guess for $x$ — the average, the center of the cloud.
• $\hat{x}$ (read "x-hat") ::: the filter's estimate of $x$. The little hat means "estimated, not the true value."

2. Uncertainty is a cloud, not a point

We never know $x$ exactly. Our knowledge is really a fuzzy cloud of possibilities centered on $\bar{x}$.

The bell-shaped fuzziness above is the Gaussian (or "normal") distribution. It is the single most common way to describe "roughly here, give or take."

• Gaussian $\mathcal{N}(\bar{x},P)$ ::: shorthand for "a bell-shaped cloud centered at $\bar{x}$ with spread $P$." Build on Nonlinear Systems later; here just picture the haze.

3. Measuring the spread: variance ${\sigma }^2$ and covariance $P$

How do we put a number on "how wide is the haze"?

Picture it: $\sigma$ is the radius of the haze in 1D — one step out from the middle catches most (about $68\%$) of the cloud.

But a state has many directions, and uncertainty can be wide in one direction, narrow in another, and tilted. One number won't do.

Why the topic needs it: the UKF's slogan is "push the distribution, not the function." The distribution is $(\bar{x},P)$ — a center and an ellipse. Everything downstream is about transforming this ellipse.

• $P$ ::: covariance = the uncertainty ellipse of the state.
• $P_y$ ::: the covariance of the output $y$ after we transform.
• $Q$ ::: extra uncertainty added each step because the physics itself is imperfect (process noise).
• $R$ ::: uncertainty in the sensor readings (measurement noise).
• $A^{\top }$ (read "A-transpose") ::: flip a matrix across its diagonal; turns a column into a row. Needed because $(\mathcal{X}-\bar{x})(\mathcal{X}-\bar{x}{)}^{\top }$ is how you build a covariance from a single offset — an outer product that produces a full grid, not one number.

4. The nonlinear map: $f$, $h$, and why "nonlinear" hurts

Why the topic needs it: the UKF exists only because $f$ and $h$ curve. If they were straight, the plain Kalman Filter (linear) would already be perfect.

5. The tool the EKF leans on: the derivative and Taylor series

To straighten a curve, the EKF uses calculus.

Why the topic needs it (as a foil): you must understand the Jacobian to appreciate the punchline — the UKF never computes one. Where $f$ has no clean derivative (e.g. $h=\Vert x\Vert$ at the origin), the EKF stalls and the UKF sails on.

6. The matrix square root: $\sqrt{P}$ and Cholesky

The UKF places its sample points along the shape of the ellipse $P$. To step "one ellipse-radius" in each direction, it needs a matrix square root.

Picture it: for a 1D variance ${\sigma }^2$, the square root is just $\sigma$ — one standard-deviation step. In $n$ dimensions, $\sqrt{P}$ generalizes "$\sigma$" into the tilted, stretched directions of the ellipse.

Why the topic needs it: the sigma-point formula ${\mathcal{X}}_i=\bar{x}\pm (\sqrt{(n+\lambda )P}{)}_i$ literally cannot be written without a matrix square root. Its columns are the arrows pointing along each principal axis of the uncertainty ellipse.

7. The stars of the show: sigma points, weights, Greek tuning knobs

• ${\mathcal{X}}_i$ (calligraphic X, read "curly-X sub i") ::: the $i$-th sigma point (an input sample).
• ${\mathcal{Y}}_i$ ::: that point after being pushed through $f$: ${\mathcal{Y}}_i=f({\mathcal{X}}_i)$.
• ${\mathcal{Z}}_i$ ::: a sigma point pushed through the measurement model $h$.
• $W_i^{(m)}$ ::: the mean weight of point $i$ — how much it counts when averaging to find $\bar{y}$.
• $W_i^{(c)}$ ::: the covariance weight of point $i$ — how much it counts when building the spread $P_y$. Differs from $W^{(m)}$ only at the center point.

The Greek knobs that decide how far out the points sit and how the weights are set:

• $\lambda$ (lambda) ::: a scaling number, $\lambda ={\alpha }^2(n+\kappa )-n$, setting the reach of the points.
• $\alpha$ (alpha) ::: small (e.g. $10^{-3}$); controls the spread — small $\alpha$ keeps points hugging the mean.
• $\kappa$ (kappa) ::: a secondary scaling number, usually $0$ or $3-n$.
• $\beta$ (beta) ::: injects prior knowledge of the cloud's shape; $\beta =2$ is best for a Gaussian.

Why the topic needs them: these seven symbols are the Unscented Transform. Master this list and the parent's boxed formulas read like plain sentences.

8. How it all connects

Read it top to bottom: a state carries an uncertainty ellipse ($P$); its square root places sigma points, weighted by the Greek knobs; pushing them through the nonlinear maps is the Unscented Transform, which the filter runs each cycle — sidestepping the Jacobian the EKF needs.

Equipment checklist

Self-test: can you answer each before revealing?