Unscented Kalman Filter (UKF) — sigma points, better for nonlinear: Visual walkthrough

Step 1 — What is "uncertainty" as a picture?

WHAT. Before we transform anything, we must agree on what we are transforming. In filtering, we never know a quantity exactly — we know a best guess and a spread of doubt around it.

The best guess is the mean, written $\overset{x}{ˉ}$ (read "x-bar"). It is the point you'd bet on.
The doubt is the covariance, written $P$ . In one dimension $P = σ^{2}$ , where $σ$ ("sigma") is how far a typical sample strays from $\overset{x}{ˉ}$ . In two dimensions $P$ is a little table of numbers that describes an ellipse of doubt — long in directions we're unsure about, short where we're confident.

WHY. A Kalman filter only ever tracks these two things: where (mean) and how sure (covariance). So the entire game is: given a mean and a spread coming in, what mean and spread come out after a nonlinear map?

PICTURE. The blue dot is $\overset{x}{ˉ}$ ; the shaded ellipse is $P$ — one standard deviation of doubt in every direction.

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

Step 2 — Why not just push the whole cloud through $f$ ?

WHAT. We have a nonlinear function $y = f (x)$ . "Nonlinear" means: its graph is a curve, not a straight line — equal steps in $x$ do not give equal steps in $y$ .

WHY. The honest answer to "what happens to the cloud?" would be: take infinitely many points from the cloud, send each through $f$ , and look at the resulting cloud. That is a Particle Filter — correct but expensive. The Extended Kalman Filter (EKF) goes to the opposite extreme: it replaces the curve $f$ with its tangent line at $\overset{x}{ˉ}$ (a Taylor Series Expansion cut off after the linear term) and pushes the ellipse through the line. Fast, but the curvature is thrown away.

PICTURE. Watch the curved orange function bend a symmetric input cloud into a lop-sided output cloud. The dashed grey tangent line — the EKF's world — keeps it symmetric and puts the mean in the wrong place.

Step 3 — The clever points: choosing sigma points

WHAT. We replace the whole cloud with a tiny, hand-picked set of points called sigma points, written $X_{i}$ (script-X, index $i$ ). In $n$ dimensions we use $2 n + 1$ of them:

one at the center, $X_{0} = \overset{x}{ˉ}$ ,
and a matched pair along each principal axis of the ellipse — one on the $+$ side, one on the $-$ side.

X_{0} = center point \overset{x}{ˉ}, X_{i} = \overset{x}{ˉ} + step out along axis i ((n + λ) P)_{i}, X_{i + n} = \overset{x}{ˉ} - step the other way ((n + λ) P)_{i}

Term by term:

$P$ is the matrix square root of the covariance (usually the Cholesky Decomposition factor $S$ with $S S^{⊤} = P$ ). It is the object that turns "size of doubt" into "how far to step." In 1D it is just $σ$ .
$(\dots)_{i}$ means "the $i$ -th column" — one direction of the ellipse.
$(n + λ)$ is a scale dial ( $λ$ = "lambda"): bigger dial ⇒ points sit further out. We meet $λ$ properly in Step 6.

WHY. Symmetric pairs guarantee the point cloud's mean stays exactly at $\overset{x}{ˉ}$ (a $+$ step is cancelled by its $-$ twin). Stepping along the axes of $P$ makes the cloud's spread match $P$ . Two moments matched, by construction.

PICTURE. The five red dots for a 2D ellipse: center plus two symmetric pairs, each pair straddling the ellipse along one of its axes.

Recall Why

$2 n + 1$ and not more? Why is a deterministic minimal set enough? ::: Because a Kalman filter only cares about two moments (mean + covariance); $2 n + 1$ points is exactly enough to pin both down. More points would only add sampling noise.

Step 4 — Meet the tuning knobs $λ, α, κ, β$

WHAT. Before we can weight the dots, we must name the three little numbers that shape the scale dial $λ$ and one more that tunes the spread. They are:

λ = α^{2} (n + κ) - n

$α$ ("alpha", a small positive number, e.g. $1 0^{- 3}$ ): controls the spread — how far the outer dots sit from the mean. Small $α$ ⇒ dots hug the center.
$κ$ ("kappa", a secondary knob, usually $0$ or $3 - n$ ): a second lever on the scale, kept for legacy reasons.
$β$ ("beta"): folds in prior knowledge of the shape of the distribution. For a Gaussian input the optimal value is $β = 2$ .

WHY name them now. The covariance weight below uses $α$ and $β$ directly, and the worked example uses $κ$ — so we must have them defined before we write those formulas. No symbol before its picture.

PICTURE. The knobs on a "spread panel": $α$ slides dots in/out, $β$ re-shapes the doubt, $κ$ trims the scale — all feeding into the single dial $λ$ .

Step 5 — The weights: making the dots reproduce mean and covariance

WHAT. Each sigma point carries a weight, written $W_{i}$ ("W" for weight). There are two weight lists: $W_{i}^{(m)}$ for reconstructing the mean, and $W_{i}^{(c)}$ for reconstructing the covariance.

W_{0}^{(m)} = \frac{λ}{n + λ}, W_{i}^{(m)} = W_{i}^{(c)} = \frac{1}{2 ( n + λ )} (i \geq 1)

$W_{0}^{(m)}$ is the fat weight on the center point.
Every outer point shares the small weight $\frac{1}{2 ( n + λ )}$ — the same on both sides of a pair (symmetry again).

WHY two lists? The covariance list differs from the mean list only at the center:

W_{0}^{(c)} = \frac{λ}{n + λ} + Gaussian tuning (β = 2 ideal) (1 - α^{2} + β)

This lets us inject a distribution-tuning term — built from the $α$ and $β$ we just defined in Step 4 — into the spread without moving the mean. Pure bookkeeping.

Sanity check the mean weights sum to 1 (an unbiased average must). The mean list has one center weight $W_{0}^{(m)}$ plus $2 n$ equal outer weights:

W_{0}^{(m)} + i = 1 \sum 2 n W_{i}^{(m)} = \frac{λ}{n + λ} + 2 n \cdot \frac{1}{2 ( n + λ )} = \frac{λ}{n + λ} + \frac{n}{n + λ} = \frac{λ + n}{n + λ} = 1 ✓

PICTURE. The same five dots, now drawn as circles whose area is the weight — a big central disk, four small satellites.

Step 6 — Push through, then read off the answer

WHAT. Now the payoff. Send each sigma point through the true function:

Y_{i} = f (X_{i})

Then rebuild the output statistics as weighted sums:

\overset{y}{ˉ} = i = 0 \sum 2 n W_{i}^{(m)} Y_{i}, P_{y} = i = 0 \sum 2 n W_{i}^{(c)} (Y_{i} - \overset{y}{ˉ}) (Y_{i} - \overset{y}{ˉ})^{⊤}

$\overset{y}{ˉ}$ = weighted average of the transformed dots = the new mean.
$(Y_{i} - \overset{y}{ˉ})$ = how far a transformed dot lands from the new mean.
$(\dots) (\dots)^{⊤}$ = the "outer product," which turns a displacement into a little ellipse-piece; summing them rebuilds the output ellipse $P_{y}$ .

WHY. Because we used the real $f$ on each dot, the transformed dots land on the true curve — so their weighted spread captures the curvature the EKF's tangent line missed. This is accurate to 3rd order for Gaussian inputs, versus 1st order for the EKF.

PICTURE. Left: input dots on the ellipse. Middle: the curve carries each dot up to the output axis. Right: the recovered output mean (star) and spread — notice the mean has shifted away from where the tangent line would put it.

Step 7 — Edge case: what the dial $λ$ does, and its extremes

WHAT. Recall the step size is $(n + λ) P$ , so $λ$ controls how far the outer dots sit from the center — and $λ$ is set entirely by the Step-4 knobs $α, κ$ .

WHY / the cases.

Large $α$ (points far out): the dots sample the function far from $\overset{x}{ˉ}$ , feeling far-field curvature — good if $f$ is gently curved, risky if it bends wildly out there.
Tiny $α = 1 0^{- 3}$ (the "scaled UT"): $n + λ = α^{2} (n + κ) \approx 3 \times 1 0^{- 6}$ , so dots sit at $\pm 3 \times 1 0^{- 6} σ \approx \pm 1.7 \times 1 0^{- 3} σ$ — hugging the mean, sampling locally. The $(1 - α^{2} + β)$ term in $W_{0}^{(c)}$ re-inflates the covariance so the spread stays correct even though the dots moved in.
Degenerate $P = 0$ (zero uncertainty): the square root is $0$ , all dots pile onto $\overset{x}{ˉ}$ , and the UT reduces to plain $\overset{y}{ˉ} = f (\overset{x}{ˉ})$ — exactly right, since a point with no doubt just gets mapped through.
Guarding the root: if you ever pick $λ < - n$ the step becomes imaginary — never do this; keep $κ \geq - n$ so $n + λ = α^{2} (n + κ) \geq 0$ .

PICTURE. The same input ellipse with three settings of $α$ : wide dots, tight dots, and the collapsed $P = 0$ case — with the covariance-correction weight shown re-inflating the tight case.

Step 8 — Edge case: the non-differentiable map (why no Jacobian)

WHAT. So far the output of $f$ was called $y$ . In a real filter the same machinery is used on the measurement function $h$ : it takes the state $x$ and predicts what a sensor should read. To keep the standard filter notation, when the function is $h$ we rename its output $z$ (the predicted measurement), its mean $\overset{z}{ˉ}$ , and its covariance $P_{z z}$ — these are exactly the $y, \overset{y}{ˉ}, P_{y}$ of Step 6 with $f$ replaced by $h$ . Nothing new, only new letters:

$z = h (x)$ ::: the predicted measurement (plays the role of $y = f (x)$ ).
$\overset{z}{ˉ} = \sum_{i} W_{i}^{(m)} h (X_{i})$ ::: its mean (the role of $\overset{y}{ˉ}$ ).
$P_{z z} = \sum_{i} W_{i}^{(c)} (h (X_{i}) - \overset{z}{ˉ}) (h (X_{i}) - \overset{z}{ˉ})^{⊤}$ ::: its spread (the role of $P_{y}$ ).
$P_{x z} = \sum_{i} W_{i}^{(c)} (X_{i} - \overset{x}{ˉ}) (h (X_{i}) - \overset{z}{ˉ})^{⊤}$ ::: the cross-covariance — brand new — measuring how the input $x$ and output $z$ move together.

Now take a 2D state and $h (x) = ∥ x ∥$ (distance from origin). Its derivative is $\partial h / \partial x = x^{⊤} /∥ x ∥$ , which blows up at the origin — division by zero.

WHY. The EKF needs that derivative to propagate uncertainty, so it fails right at $x = 0$ . The UT never differentiates. Every term of $P_{x z}$ above is a plain evaluation of $h$ at a dot — no slope, no singularity. This $P_{x z}$ is exactly the ingredient the Kalman gain $K = P_{x z} P_{z z}^{- 1}$ needs, replacing the Jacobian entirely.

PICTURE. Four sigma points around $\overset{x}{ˉ}$ near the origin; the cone $∥ x ∥$ has a sharp tip (no tangent plane) at the center, yet each dot still gets a clean output value.

The one-picture summary

Everything at once: an input Gaussian → sigma points chosen along $P$ → each dot pushed through the true curve → weighted recovery of the shifted output mean and skewed output covariance, with the failed EKF tangent line shown ghosting behind for contrast.

Recall Feynman retelling — say it like a story

We start with a guess and a fog of doubt around it — a dot and an ellipse. We don't trust ourselves to bend the whole ellipse through a curvy function, so we cheat cleverly: we drop a handful of stones — one in the middle, and matched pairs stepping out along each direction the fog reaches. How far they step is set by a dial $λ$ , built from small knobs $α, κ$ (and we always keep $n + λ \geq 0$ so the step is a real distance, not an imaginary one). We weight the stones so that, together, they are the original fog (same center, same spread), and one extra knob $β$ tunes the spread weight for Gaussian fog ( $β = 2$ is ideal). Then we throw every stone through the real curve — not a fake straight-line stand-in — and watch where they land. The weighted average of where they land is our new best guess; the weighted spread is our new fog. When the function is a sensor we just rename the output $z$ and also track how input and output move together, $P_{x z}$ — and even when the sensor has a sharp tip where slopes explode, our stones don't care, they only ever evaluate, never differentiate.

Recall The whole cycle in one flow

flowchart TD
  A["mean x-bar and covariance P"] --> B["build 2n+1 sigma points"]
  B --> C["push each through true f"]
  C --> D["weighted mean gives y-bar"]
  C --> E["weighted spread gives P_y"]
  D --> F["output belief"]
  E --> F