Probe Sets for Nonlinear Kalman Filtering: Multi-Center Gains and Covariance Recalibration

The gist The introduction presents probe sets to improve gain selection and covariance recalibration in nonlinear Kalman filters, showing that probing preserves typical accuracy while substantially reducing covariance inconsistency and rare large-error trajectories

Probe Set Framework

The paper introduces probe sets as small covariance-scaled collections of states at which the filter repeats its local approximation The filter evaluates its usual moment rule at each member, and then selects the gain that minimizes the average reported covariance across these local models This construction applies to different KF variants without requiring a specific approximation rule The proposed probe geometry is related to reduced sigma-point constructions, but the purposes are different, as each probe member serves as the center of a complete local moment approximation

Optimality and Bounds

Proposition 2 shows that the minimizing gain is Kpm:= C S−1, with C and S being averages over the probe members Proposition 3 establishes covariance guarantees without requiring the local moment approximation to be exact, showing that for quadratic measurements under an exact Gaussian prediction, probing provides a conservative covariance report and cannot increase the true error covariance under stated assumptions Specifically, the probe gain satisfies Prc((C, S); Kpm) = P − C tS−1 C⊤

Covariance Recalibration and Back-out

Covariance recalibration repeats the moment calculation at the updated state using the predicted covariance, and for the probe framework, this results in a recalibration probe that retains the spread P A back-out rule is triggered when JM(Prep) > JM(P), which compares sums of variances before and after the update The output state estimate and covariance are then determined based on this comparison, returning to the predicted values if the uncertainty increases

Experimental Validation

Simulations across two systems, four KF variants, and 300 randomized setups per system show that probing preserves typical accuracy while improving covariance consistency and reducing large-error tails The results show that probing preserves typical accuracy while substantially reducing covariance inconsistency and the rare large-error trajectories that dominate the root mean square of per-run errors F4 has the lowest geometric-mean RMS of per-run RMSE for every KF variant in both applications, while its median is within 5% of the best method in all eight combinations

Conclusion

Conventional nonlinear KFs approximate measurement moments at a single state, which can lead to inaccurate corrections and overconfident covariance estimates when the local model varies substantially across the uncertainty region Covariance-scaled probe sets account for this variation in gain selection and covariance recalibration The analysis shows why probing can be advantageous when prediction uncertainty is understated Future work will explore the use of probe sets in other state estimation methods, including moving horizon estimation

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Kalman filters (KFs) select the gain as a function of two moments: the state–measurement cross-covariance and the innovation covariance For linear systems with correctly specified moments, both can be computed exactly, and the KF minimizes mean-square error among all linear estimators [1]

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The central covariance of the candidate estimation error, averaged over the current state and measurement conditional on past measurements, is P t+(K) = P t − KCt⊤ − C tK⊤ + KStK⊤

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A nonlinear KF cannot generally recover (C t, S t) from the predicted mean and covariance alone Instead, it applies a local moment map

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The conventional center pair and gain are θ0:= ΘP (ˆx−) = (C0, S0), K0:= C0S−1 0 The next identity isolates the covariance penalty caused by an inaccurate gain and expresses it in terms of moment error [19]

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The conventional covariance report is Pconv = Prc(θ0; K0) = P − C0S−1 C⊤ 0 Thus one local moment pair selects the correction and also determines the reported covariance at its estimated minimum

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The update probe represents a collection of possible states around the predicted mean We select a gain that minimizes the average reported error covariance Prc(K):= 1/(n + 1) X i Prc((C i, S i); K) Proposition 2 shows that the minimizing gain is Kpm:= C S−1, with C and S being averages over the probe members

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Proposition 3 bounds the true innovation covariance and the true error covariance of a fixed-gain correction by their probe-based reports for every moment map of the stated form At Kpm = C tS−1, the generalized Joseph report reduces to P − C tS−1 C⊤, which proves (31)

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Proposition 4 compares each local moment map with its own probed version If υ ≥ 1 + δ, their true error covariances satisfy trP−1P t+(Kpm) ≤ trP−1P t+(K0)

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Figure 4 and Table 2 show that probing substantially improves covariance consistency F4’s ANEES ranges from 0.98 to 1.

1. Bold gain selection for nonlinear Kalman filters: The system can select a gain that minimizes the average reported covariance across these local models by evaluating probe sets, small covariance-scaled collections of states at which the filter repeats its local approximation.

2. Covariance consistency improvement: The system will exhibit substantially reducing covariance inconsistency by implementing the probe framework, as it minimizes the average reported covariance and accounts for variations among local moment maps through the term slope-disagreement covariance (25).

3. Improved performance under understated uncertainty: When prediction uncertainty is understated, the system can achieve a lower normalized true error variance than single-center updates by using the probe gain, as shown in Proposition 4, which states that probing improves the gain in normalized true error variance when prediction uncertainty is sufficiently understated.

4. Enhanced large-error trajectory handling: The filter will reduce rare large-error trajectories that dominate the root mean square of per-run errors by using the probe framework, as simulations show F4 has a lower RMS of per-run RMSE compared to baselines across 300 randomized setups.

5. Adaptive uncertainty management: The system implements a back-out rule triggered when "JM(Prep) > JM(P), allowing it to withdraw the update and return to the predicted mean and covariance" if the revised local approximation assigns greater uncertainty than the previous estimate, thereby preventing an increase in the true error covariance.