Do not linearize the function — sample it. Pick 2n+1 deterministic sigma points that match the state's mean and covariance, push each through the true nonlinearity, and rebuild the mean and covariance from what comes out. No Jacobian. Accurate to second order on curved dynamics.
source Julier & Uhlmann, A New Extension of the Kalman Filter to Nonlinear Systems (1997), SPIE Vol. 3068, pp. 182–193 · DOI 10.1117/12.280797. Rendered, not quoted.
State x ~ N(m, P). The unscented transform draws a fixed set of points that exactly carry the first two moments:
χ₀ = m, W₀ = κ/(n+κ)
χᵢ = m ± √(n+κ)·Sᵢ, Wᵢ = 1/2(n+κ)
S Sᵀ = P (Cholesky factor of P)
Push every χ through the real f. The output mean is Σ Wᵢ f(χᵢ); the output covariance is Σ Wᵢ (f(χᵢ)−ŷ)(·)ᵀ. This is exact for any quadratic — a full order beyond the EKF's tangent line.
Parent: the-extended-kalman-filter. The EKF freezes f at a single point — its Jacobian tangent line — and pays a bias on curvature. Julier & Uhlmann (1997) keep the same predict/update skeleton but propagate points, not a derivative: no Jacobian to derive, and the mean stays correct through the bend.
Live re-check of the engine's output against the closed-form true moments. Green while honest; flips red the instant window 6 tampers with the sigma set.
m, P loading…
A 2-D Gaussian and a genuinely nonlinear map f(x) = [x₀², x₁²].
| point | x₀ | x₁ | W |
|---|
computing…
wallThe UT is exact to second order, not all orders. Feed it a strongly cubic-or-higher f, or a bimodal / heavy-tailed prior, and the moment match degrades — the transformed set no longer summarizes the true density. On a non-positive-definite P the Cholesky factor does not exist and the sigma set is undefined. Scaling κ<−n or a poorly chosen (α,β,κ) can push weights negative and yield an indefinite output covariance.
"The UKF is just the EKF with more points." No — the EKF's mean is f(m), biased by ½·tr(Hess·P); the UKF mean is exact for quadratics, off by zero on this f.
"More sigma points = more accuracy." No — 2n+1 is fixed by the moment-matching, not a resolution knob; extra points buy nothing and can only mis-weight.
"It's a particle filter." No — the points are deterministic and moment-matched, not random samples; there is no importance weighting and no resampling.
The disclosed planted void: drop the 2n spread points and keep only χ₀. The transform collapses to a naive f(m) — it loses the covariance and the second-order mean correction. The witness in 7 catches it.