Unscented Kalman Filter
Also known as: UKF, Sigma-Point Kalman Filter
The Unscented Kalman Filter (UKF) is a nonlinear state estimation algorithm that approximates nonlinear systems without requiring explicit Jacobian computation. Introduced by Julier and Uhlmann in 1997, the UKF uses the unscented transform—a deterministic method to capture mean and covariance statistics through a carefully chosen set of sample points (sigma points)—making it more accurate than the Extended Kalman Filter for highly nonlinear systems while avoiding the computational burden of derivative calculations.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use the UKF when you have a strongly nonlinear system and want accuracy better than the EKF without computing Jacobians. It is ideal for robotics (especially 3D attitude estimation), navigation with nonlinear sensor models (radar, bearing angles), and systems where linearization is a poor approximation. The UKF is more computationally expensive than EKF (by a small factor), so avoid it only if computation is severely constrained or system nonlinearity is mild (EKF sufficient).
Strengths & limitations
- More accurate than EKF for highly nonlinear systems; captures third-order moments without explicit Jacobians.
- No need to compute or code Jacobian matrices; reduces implementation errors and improves robustness.
- Minimal tuning required; sigma point placement is determined by well-established formulas.
- Provably convergent for a wider class of nonlinear systems compared to EKF.
- Works well with non-Gaussian noise if the noise statistics can be approximated by a Gaussian mixture.
- Slightly more computationally expensive than EKF (2n+1 function evaluations per time step instead of Jacobian).
- Still assumes Gaussian distributions; non-Gaussian noise can cause filter divergence.
- Sigma point propagation requires storing and manipulating more matrices than EKF.
- No universally optimal choice of alpha and beta parameters; system-dependent tuning may be needed.
- Can still diverge if initial estimate is far from truth or model mismatch is severe.
Frequently asked
What are sigma points and how are they chosen?
Sigma points are a deterministically chosen set of sample points that capture the mean and covariance of a Gaussian distribution. For n-dimensional state, 2n+1 sigma points are placed at: x and x ± sqrt((λ+n) P), where λ = α^2(n+κ)−n. The parameters α and κ control spread: α (typically 0.001-1) controls distance from mean, κ is a secondary scaling parameter (often 0).
How does UKF compare to EKF in terms of accuracy and computation?
UKF is typically more accurate for nonlinear systems (third-order vs. first-order moment matching). Computationally, UKF requires 2n+1 evaluations of the nonlinear function per step, while EKF requires one function evaluation plus one Jacobian evaluation. For low-dimensional systems (n<20), the overhead is small; for high-dimensional systems, EKF can be faster despite lower accuracy.
Can I use UKF with multimodal (non-Gaussian) distributions?
Standard UKF assumes Gaussian distributions and will perform poorly with multimodal distributions. For multimodal posteriors, use a Particle Filter, which represents the distribution as a weighted set of particles. Alternatively, use the Gaussian Mixture UKF, which maintains multiple Gaussian components.
How do I tune the UKF parameters α, β, and κ?
Standard choice: α = 0.001 (places sigma points very close to mean), κ = 0, β = 2 (optimal for Gaussian). For non-Gaussian noise, try β = 2 (Gaussian optimal), larger β emphasizes outer sigma points. For systems with strong nonlinearity, increase α (e.g., 0.1) to spread sigma points wider. Empirically test on simulated data.
Sources
- Julier, S. J., & Uhlmann, J. K. (1997). A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control, 45(3), 477-482. link ↗
- Wan, E. A., & Van Der Merwe, R. (2000). The unscented Kalman filter for nonlinear estimation. Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, 153-158. link ↗
- Sarkka, S. (2013). Bayesian Filtering and Smoothing. Cambridge University Press. DOI: 10.1017/CBO9781139344203 ↗
How to cite this page
ScholarGate. (2026, June 3). Unscented Kalman Filter. ScholarGate. https://scholargate.app/en/control-theory/unscented-kalman-filter
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Extended Kalman FilterControl Theory↔ compare
- Linear Quadratic GaussianControl Theory↔ compare