Unscented Kalman Filter (UKF)

The classical nonlinear estimator that the UKF was designed to improve upon. The EKF linearizes the nonlinear process and measurement models by computing a first-order Taylor series expansion around the current state estimate, effectively approximating the system with a locally linear model.

• Key limitation: Requires the computation of Jacobian matrices, which can be analytically complex or undefined for discontinuous functions
• Linearization error: Introduces bias and can cause filter divergence when the true dynamics are highly nonlinear
• Computational cost: Similar to UKF for low-dimensional systems, but the UKF avoids the derivation burden
• Use case: Still preferred when analytical Jacobians are readily available and the system exhibits mild nonlinearity

A sequential Monte Carlo method that represents the posterior distribution using a large set of weighted random samples called particles. Unlike the UKF's deterministic sigma points, particle filters use stochastic sampling to approximate arbitrary distributions.

• Key advantage: Can represent highly non-Gaussian, multimodal distributions that the UKF cannot capture
• Computational cost: Requires hundreds to thousands of particles, making it significantly more expensive than the UKF's 2n+1 sigma points
• Degeneracy problem: Suffers from particle impoverishment, requiring resampling strategies that add algorithmic complexity
• Use case: Preferred for global localization, kidnapped robot problems, and tracking with ambiguous data associations

The core mathematical innovation that powers the UKF. The unscented transform selects a minimal set of deterministically chosen sample points—called sigma points—that exactly capture the mean and covariance of a probability distribution. These points are propagated through the nonlinear function, and the transformed points are recombined to compute the posterior statistics.

• Accuracy: Captures the posterior mean and covariance accurately to the 3rd order (Taylor series) for Gaussian inputs, versus the EKF's 1st-order approximation
• Sigma point selection: Typically uses 2n+1 points for an n-dimensional state, with scaling parameters controlling spread
• No Jacobians required: Works with black-box nonlinear functions, making it ideal for complex simulation-based models
• Variants: Includes the Scaled UT and Spherical Simplex UT for improved numerical stability and reduced point sets

A closely related derivative-free filter that uses a spherical-radial cubature rule to compute Gaussian-weighted integrals encountered in nonlinear Bayesian filtering. The CKF selects 2n equally weighted cubature points, avoiding the UKF's potential for negative weights.

• Numerical stability: Guarantees positive weights, preventing covariance matrix non-positive-definiteness issues that can occur with UKF scaling parameters
• Accuracy: Matches the UKF's 3rd-order accuracy for Gaussian integrals but with a simpler, parameter-free point selection
• Point count: Uses exactly 2n points (versus UKF's 2n+1), offering marginally lower computational cost
• Use case: Preferred in high-dimensional systems where UKF parameter tuning becomes challenging and numerical robustness is critical

A Monte Carlo approximation of the Kalman filter designed for very high-dimensional systems, such as weather forecasting and reservoir modeling. The EnKF propagates a large ensemble of state vectors and uses the sample covariance for updates.

• Scalability: Handles state spaces with millions of dimensions where the UKF's sigma point approach becomes computationally prohibitive
• Ensemble size: Typically uses 20-100 ensemble members, far fewer than the theoretical state dimension
• Key difference: Uses a stochastic ensemble rather than deterministic sigma points, introducing sampling error but enabling massive parallelism
• Use case: Dominant in geophysical data assimilation where state dimension vastly exceeds available computational resources for full covariance propagation

An enhanced variant that re-linearizes the measurement update by iteratively re-computing sigma points around the updated posterior estimate. This reduces linearization error when the measurement function is highly nonlinear.

• Mechanism: After the standard UKF update, the posterior mean is used as the new linearization center, and the update is repeated until convergence
• Convergence: Typically converges in 2-5 iterations, adding modest computational overhead
• Benefit: Approaches the performance of a full maximum a posteriori estimate without requiring a particle filter's computational burden
• Use case: Critical in applications with severe measurement nonlinearities, such as bearing-only tracking and range-only localization