What is Cubature Kalman Filter?
A Cubature Kalman Filter is a nonlinear Gaussian state estimator that approximates transformed moments with equally weighted points from a spherical-radial cubature rule.
Quick Facts
| Specification | Official Specification |
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How It Works
Construct the spherical-radial cubature points
Factor covariance as P = S S^T, then form standardized points +sqrt(n)e_i and -sqrt(n)e_i. Mapping them through m + S xi_i gives 2n state points with weight 1/(2n). Their symmetry exactly reproduces the input mean and covariance and integrates a defined class of low-degree Gaussian-weighted polynomials.
Arasaratnam and Haykin's CKF paper derives the method from a third-degree spherical-radial cubature rule. The degree describes integration exactness for that rule, not a guarantee that an arbitrary nonlinear posterior is third-order accurate in every metric.
Propagate moments through both model stages
Prediction sends cubature points through the transition function, averages them, computes their centered covariance, and adds process noise. The measurement update regenerates points from the predicted Gaussian, propagates them through the sensor model, and computes measurement covariance plus state-measurement cross-covariance before applying a Kalman-style gain.
For non-additive noise, constrained states, angles, and manifolds, the point construction and residual operations must match the model geometry. Cholesky failure, negative variance from roundoff, or invalid point states require a square-root or geometry-aware design rather than silent clipping.
Compare CKF and UKF under the same contract
PyTCL's advanced-filter documentation shows the 2n equal-weight rule and contrasts it with configurable Unscented Transform points. CKF avoids alpha, beta, and kappa, but fewer tuning parameters do not establish universal superiority.
Compare state error, innovation calibration, factorization failures, runtime, and function evaluations under identical dynamics, observations, seeds, and noise assumptions. As dimension grows, cubature points move farther from the mean; strong curvature, multimodality, discontinuity, or hard constraints can still make the Gaussian moment approximation misleading.
Key Characteristics
- Uses a third-degree spherical-radial cubature rule in its standard form
- Evaluates two equally weighted deterministic points per state dimension
- Avoids analytic transition and measurement Jacobians
- Reconstructs Gaussian means, covariances, and cross-covariances
- Has no scaled-UKF alpha, beta, or kappa parameters
- Still depends on covariance factorization, model geometry, and Gaussian adequacy
Common Use Cases
- Nonlinear target tracking with differentiable or black-box models
- Navigation and sensor fusion without maintained Jacobian code
- State estimation with moderate dimension and smooth nonlinear dynamics
- Comparisons among EKF, UKF, CKF, and Particle Filter approximations
- Square-root nonlinear filtering for improved covariance reliability
Example
Loading code...Frequently Asked Questions
How many points does a standard CKF use?
The common third-degree spherical-radial CKF uses `2n` points for an `n`-dimensional state. They lie at positive and negative scaled coordinate directions and have equal weight. Other higher-degree cubature rules use different point sets.
How is a CKF different from a UKF?
Both propagate deterministic points through nonlinear functions. A standard CKF derives `2n` equal-weight points from a spherical-radial cubature rule, while a scaled UKF commonly uses `2n+1` points and configurable `alpha`, `beta`, and `kappa` weights.
Is a CKF always more accurate than an EKF or UKF?
No. Accuracy depends on nonlinearity, dimension, state geometry, initialization, noise assumptions, and numerical implementation. The CKF avoids Jacobians and certain UKF scaling choices, but benchmark results from one model cannot establish universal superiority.
Does a CKF handle non-Gaussian or multimodal posteriors?
Not in its standard form. It propagates points to approximate moments, then retains one Gaussian summary. A Particle Filter or mixture method may be more appropriate when separated modes, skew, truncation, or heavy tails materially affect decisions.
What should be monitored in a CKF implementation?
Monitor covariance factorization, minimum eigenvalues, innovation whiteness and normalized innovation squared, state error on truth data, point validity under constraints, function-evaluation cost, and sensitivity to initialization and noise covariances.