The linear update misses what the polynomial catches.
The Unscented Kalman Filter handles nonlinearity by propagating sigma points through the true system dynamics instead of linearizing. But the measurement update — where observations correct the state estimate — remains linear: a gain matrix multiplied by the innovation. For strongly nonlinear measurements, this linear correction leaves residual error that accumulates.
Polynomial Unscented Kalman Filter (arXiv:2603.20259): enrich the measurement update with higher-order polynomial terms, computed via Conjugate Unscented Transformation. Instead of correcting the state estimate along a single linear direction, the correction follows a polynomial surface that matches the actual measurement geometry.
Tested on spacecraft navigation — Clohessy-Wiltshire relative motion and three-body orbital mechanics with non-Gaussian noise. The polynomial version achieves improved accuracy and covariance consistency compared to the linear update. The improvement is largest when the measurement function is strongly curved — precisely when the linear approximation fails most.
The structural insight: the UKF already handles nonlinearity in the prediction step (propagating sigma points). The limitation was in the update step (correcting with a linear gain). The fix is symmetry: make the update as nonlinear as the prediction. The system was half-modern and half-classical — nonlinear dynamics corrected by linear observations. Matching the sophistication of the two halves closes the gap.