Unscented Kalman Filter (UKF)

Problem Statement

For strongly nonlinear dynamics, first-order Jacobian linearization can degrade EKF accuracy. The UKF uses deterministic sigma-point propagation to capture higher-order effects without symbolic Jacobians.

Model and Formulation

Given state mean \mu and covariance P, construct 2n+1 sigma points:

$$ \chi_0 = \mu,\quad \chi_i = \mu \pm \left(\sqrt{(n+\lambda)P}\right)_i $$

Propagate each point through nonlinear models, then recover moments:

$$ \hat{\mu} = \sum_i W_i^{(m)}\chi_i,\quad \hat{P} = \sum_i W_i^{(c)}(\chi_i-\hat{\mu})(\chi_i-\hat{\mu})^\top + Q $$

Algorithm Procedure

1. Generate sigma points using (\alpha,\beta,\kappa) scaling.
2. Propagate points through process model for prediction.
3. Project predicted points into measurement space.
4. Compute gain from cross-covariance and update posterior state.

Tuning Guidance

• Use small \alpha (1e-3 to 1e-1) for local spread control.
• Set \beta=2 for approximately Gaussian priors.
• Increase process noise if sigma clouds collapse under model mismatch.

Failure Modes and Diagnostics

• Poor scaling parameters can produce non-positive definite covariance.
• Non-Gaussian heavy-tailed noise can still break Gaussian-moment assumptions.
• Monitor covariance eigenvalues to detect numerical instability.

Implementation and Execution

python -m uav_sim.simulations.estimation.ukf

Evidence

References

• Wan and Van der Merwe, The Unscented Kalman Filter for Nonlinear Estimation (2000)
• Julier and Uhlmann, Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE (2004)

Related Algorithms

• Extended Kalman Filter
• Particle Filter