Unscented Kalman Filter

Main Ideas / Concepts (UKF vs EKF)

• Unscented Kalman Filter (UKF) is presented as an alternative to the Extended Kalman Filter (EKF) when:
  • The state transition model and/or the observation model are nonlinear.
  • Nonlinearities are strong enough that EKF’s linearization approach performs poorly.

Why EKF Can Fail in Highly Nonlinear Systems

• EKF works by locally linearizing nonlinear functions using Jacobians (a first-order Taylor expansion).
• This makes the filter only near-optimal when the system is nearly linear.
• For systems that behave “chaotically” or have complex nonlinear dynamics, EKF’s linear approximation error can accumulate over time, degrading performance.

Core UKF Improvement

• UKF uses an Unscented Transformation (UT):
  • It avoids propagating uncertainty via Jacobian-based linearization.
  • Instead, it uses sampling of the prior distribution via sigma points to capture how nonlinear functions transform mean/covariance more accurately.

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Detailed Methodology (UKF Workflow)

A) Unscented Transformation (UT): How UKF Propagates Uncertainty

Inputs (Prior Beliefs)

• Prior state belief is typically Gaussian with:
  • mean μ
  • covariance Σ

Step 1: Create Sigma Points

• Choose 2L + 1 sigma points for an L-dimensional state.
• Sigma points are placed around the prior mean x̄ (μ) using a square-root of the covariance (e.g., via Cholesky decomposition), such that their weighted sample moments match the desired mean and covariance.

Step 2: Propagate Each Sigma Point Through Nonlinear Functions

• Pass sigma points through nonlinear models:
  • state transition function f
  • observation function h (sometimes denoted g in parts of the narration)
• This yields a transformed set of sigma points.

Step 3: Recompute Mean and Covariance

• Compute:
  • transformed mean (predicted/posterior mean)
  • transformed covariance
• UKF does not assume the transformed distribution remains Gaussian using only mean/covariance similarity transforms.
• The covariance update comes from the sampled transformed sigma points, not from Jacobian-based linear similarity transforms.

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B) Prediction Step (Time Update)

• Given sigma points built from the current belief (mean/covariance):

Step 1: Pass Sigma Points Through the State Transition Model

• For each sigma point (X_j):
  • (X_j’ = f(X_j))

Step 2: Compute Predicted State Mean

• Weighted average of transformed sigma points using first-order weights (w_j^a).

Step 3: Compute Predicted State Covariance

• Use second-order weights (w_j^c) and add process noise covariance Q.

Key point: UKF computes predicted mean/covariance from propagated sigma points rather than Jacobians.

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C) Correction Step (Measurement Update)

Step 1: Pass Sigma Points Through the Observation Model

• For each predicted sigma point:
  • (Z_j’ = h(X_j’))

Step 2: Compute Predicted Measurement Statistics

• Use weights (w_j^a) and (w_j^c).
• Add measurement noise covariance R to the predicted measurement covariance.

Step 3: Compute Cross-Covariance

• Compute the cross-covariance between transformed state sigma points and transformed measurement sigma points:
  • (C_{xz})

Step 4: Compute Kalman Gain

• (K = C_{xz} \cdot S^{-1}), where (S) is the predicted measurement covariance.

Step 5: Update State Mean and Covariance

• Use the innovation term:
  • (Z - \bar{Z}) (actual measurement minus predicted measurement)
• Updated mean/covariance incorporate the innovation via the Kalman gain (K).

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Sigma Points: Number and Placement

• For an L-dimensional state:
  • Total sigma points: 2L + 1

Dimensionality Example

• If n = 2, then L = 2 ⇒ 2L + 1 = 5 sigma points.
• The narration also mentions a simplified illustrative case (e.g., “mean 0 and variance 1”) that resembles a small geometric teaching example (possibly focusing conceptually on a 3-point structure).

General Sigma Point Construction

• Uses Cholesky decomposition:
  • (P = A A^T)
• Sigma points are:
  • centered at mean μ
  • offset by scaled columns of A in both positive and negative directions
  • plus the central point.

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Role of UT vs EKF (Why UKF Often Works Better)

• EKF:

  • Uses Jacobian-based linearization.
  • Assumes transformed mean/covariance can be approximated by Jacobian similarity transforms.

• UKF:

  • Uses sigma point sampling to more faithfully represent the effects of nonlinear transformations on uncertainty.
  • Emphasizes that matching only the first two moments (mean, covariance) may not be sufficient to guarantee correct behavior of higher moments in highly nonlinear mappings; the UT improves the approximation of how distributions evolve.

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Additional Notes: Parameters and Computational Tradeoffs

UKF Naming Note

• “Unscented” is described as arbitrary naming choice (related to the creator’s initials and to avoid confusing similar names).

UKF Parameterization (α, β, κ)

Many UKF implementations include hyperparameters that affect sigma point placement:

• α (alpha): controls the spread of sigma points around the mean (often small)
• κ (kappa): affects scaling/spread (often small or zero)
• β (beta): incorporates prior knowledge of the distribution shape (commonly β = 2 for Gaussian assumptions)

These parameters change sigma point locations and thus can affect results.

Computational Cost

• UKF avoids Jacobians, but requires:
  • propagating multiple sigma points
• Therefore, it typically costs more computation than EKF.

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Demonstrated Example / Benchmark (Matlab/Simulink)

• The video implements UKF vs EKF using the Mackey–Glass delay differential equation, a common benchmark for nonlinear/chaotic prediction.
• Why Mackey–Glass: it is highly nonlinear and includes delay, producing complex/chaotic time series where linearization (EKF) struggles.

Pipeline in Simulink (High-Level)

• Generate noisy measurements:
  • add process noise Q and measurement noise R
  • use zero-order hold to align with sampling assumptions
• Use built-in Simulink blocks for:
  • UKF (and separately EKF)
• Provide two model functions:
  • state transition f (Mackey–Glass right-hand side)
  • observation h (identity in this setup: measurement is the state itself)

Result Claim

• UKF tracks the chaotic series better than EKF (lower error / lower mean squared error).
• EKF has higher error.
• Occasional spikes/drops are attributed to time delay effects.

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Speakers / Sources Featured

• Jeffrey (Evan) referenced in the narration as the person behind the “unscented” naming choice (spelling appears inconsistent in auto-subtitles).
• Evan van der Merwe (explicitly referenced via an original paper)

Original Research / References

• “The Unscented Kalman Filter for Nonlinear Estimation” by Evan van der Merwe and others (mentioned as the source of the equations)
• Wikipedia is mentioned as an earlier source for equation structure, with narration stating equations also came from Wikipedia and the cited paper.