Residual-based RAIM in Navigation

• Residual-based RAIM is an integrity monitoring method that uses residuals from over-determined measurement models to detect and exclude faults in navigation systems.
• It employs techniques like weighted least squares, solution separation, and innovation checks across various applications including GNSS, visual localization, and multi-sensor fusion.
• The approach enhances system reliability by computing protection levels and adapting to challenges such as redundancy, non-Gaussian noise, and time-correlated errors.

Residual-based Receiver Autonomous Integrity Monitoring (RAIM) is an integrity-monitoring architecture in which redundant measurements are tested against an estimated navigation or localization state through residuals, innovations, or solution-separation quantities. In the cited literature, it appears in its classical GNSS form as weighted least-squares or solution-separation consistency checking, and in adapted forms for vector tracking, PPP, vision-based landing, visual localization, cellular positioning, and multi-sensor factor-graph optimization. Its core functions are fault detection, fault exclusion, and, in the more formal variants, protection-level computation or equivalent position-error bounding (Li et al., 2019, Dunik et al., 2024).

1. Core estimation structure and redundancy requirements

Residual-based RAIM is built on an overdetermined measurement model. In linearized form, the cited works write this either as $dy = H\,dx + e$, as in optimization-based visual localization, or as $z_k = H_k x_k + \nu_k$, as in Kalman-filter-based tracking. A state estimate is obtained by weighted least squares or recursive filtering, and the residual is the component of the measurement vector that the estimated state cannot explain. In visual localization this residual is

$$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$

whereas in EKF/UKF-based vector tracking the analogous consistency quantity is the innovation

$r_k = z_k - \hat{z}_k.$

Both forms implement the same principle: integrity is inferred from the mismatch between predicted and observed measurements (Li et al., 2019, Wu et al., 2019).

A second structural element is redundancy. In the GPS vector delay-locked loop (VDLL) and the HAPS-aided GPS study, the receiver can detect an abnormal observation only when the number of observations is at least $5$, and can exclude one only when the number of observations is at least $6$. The VDLL paper further emphasizes that acceptable geometry is required, because the RAIM layer is intended to check consistency among satellite solutions rather than merely flag low signal strength (Wu et al., 2019, Zheng et al., 2023).

Advanced RAIM generalizes the same idea from a single residual vector to solution separation. In the ARAIM formulation summarized in the cited work, the receiver computes a full solution using all pseudorange measurements and multiple sub-solutions that omit one measurement or one fault hypothesis. The integrity statistic is then the difference between the full solution and a sub-solution, for example

$d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$

with thresholds driven directly by the allocated false-alert probability. This places solution separation within the same residual-based lineage, but with explicit multi-hypothesis structure (Dunik et al., 2024).

2. Residual statistics and decision rules

Residual-based RAIM does not use a single universal statistic; the exact quantity depends on the estimator, the measurement modality, and the integrity objective. The concrete instantiations below appear in the cited GNSS, PPP, vision, and robust-estimation systems (Li et al., 2019, Wu et al., 2019, Li et al., 2020, Valentin et al., 13 Aug 2025, Zheng et al., 2023).

Setting	Residual or statistic	Decision/use
Classical WLS / visual localization	$\lambda = \epsilon^T W \epsilon$, with $\lambda \sim \chi^2_{n-m}$ under $H_0$	Global consistency test
VDLL vector tracking	$z_k = H_k x_k + \nu_k$0	Detect degraded satellite/channel; support FDE
PPP/LiDAR urban canyon	$z_k = H_k x_k + \nu_k$1, $z_k = H_k x_k + \nu_k$2	Inlier/outlier screening; reject associated carrier phase
Vision-based landing	$z_k = H_k x_k + \nu_k$3, with dof $z_k = H_k x_k + \nu_k$4	ACCEPT/REJECT model output
HAPS-aided GPS robust RAIM	$z_k = H_k x_k + \nu_k$5	Inflate variance of abnormal observations

In the classical chi-square form, the weighted sum of squared residuals is compared with a threshold derived from the $z_k = H_k x_k + \nu_k$6-quantile of the central chi-square distribution. In the UAV visual-localization formulation, the null hypothesis is “all measurements are consistent / no outliers,” and the alternative is “one or more outliers exist.” Under $z_k = H_k x_k + \nu_k$7, $z_k = H_k x_k + \nu_k$8 follows a central chi-square law with degrees of freedom $z_k = H_k x_k + \nu_k$9; under outliers, it follows a non-central chi-square law (Li et al., 2019).

In tracking architectures, the residual is often embedded in a navigation filter rather than derived explicitly in parity-space form. The VDLL receiver over a land mobile satellite channel does not present a parity-space derivation, but its RAIM logic is naturally tied to the innovation $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$0. This is operationally important because, in a vector loop, one corrupted channel can spread to the whole coupled system and cause loss of lock on all satellites (Wu et al., 2019).

Other systems use thresholded residual filtering rather than full classical RAIM. In P$$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$1-LOAM, the residual vector $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$2 is computed from the linearized SF-PPP model, the per-epoch statistics $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$3 are formed, and satellite $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$4 is declared an inlier if $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$5; when a pseudorange observation is rejected, the associated carrier-phase observation is also treated as an outlier. In the runway-landing adaptation, the residual is a corrected reprojection mismatch whitened by the predicted covariance, and the acceptance test uses a chi-squared density or threshold after accounting for pose sensitivity (Li et al., 2020, Valentin et al., 13 Aug 2025).

3. Fault detection, exclusion, and protection levels

Residual-based RAIM typically separates monitoring from response. In the VDLL formulation, RAIM is the monitoring logic and fault detection and exclusion (FDE) is the response logic: RAIM detects inconsistency, FDE isolates or removes the offending satellite or channel, and the vector loop continues operating on the remaining healthy channels. This is specifically intended to prevent failure propagation from a single degraded tracking channel to the entire vector-tracking solution (Wu et al., 2019).

In feature-based visual localization, exclusion is iterative rather than one-shot. The UAV visual-localization paper introduces Iterative Parity Space Outlier Rejection (IPSOR): compute residuals and $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$6, remove the feature contributing most to the test statistic when $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$7, recompute pose and residuals, and repeat until the residual test passes or too few inliers remain. The line-feature monocular localization paper adopts the same general FDE pattern with a greedy Chi-squared-based exclusion strategy that repeatedly removes the observation with the largest residual until the weighted sum of squared residuals becomes statistically consistent (Li et al., 2019, Zheng et al., 2022).

Protection levels are the formal counterpart of detection and exclusion. In the visual-localization RAIM adaptation, after outlier rejection the protection level in direction $$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$8 is

$$\epsilon = dy - H\,d\hat{x} = \big(I - H(H^TWH)^{-1}H^TW\big)dy,$$9

with $r_k = z_k - \hat{z}_k.$0 in the experiments. The first term is the worst undetected-fault contribution compatible with the residual threshold, and the second is the Gaussian noise contribution from the estimated covariance (Li et al., 2019).

The line-feature monocular method extends the same logic to multiple outliers and to both translation and rotation. Its direction-wise protection level is

$r_k = z_k - \hat{z}_k.$1

and the paper applies this to six axes: $r_k = z_k - \hat{z}_k.$2 and $r_k = z_k - \hat{z}_k.$3. The factor-graph GNSS/INS/Vision work preserves the same architecture at a higher level of estimator complexity: its position-error bound consists of a nominal covariance term $r_k = z_k - \hat{z}_k.$4 plus sensor-specific fault terms derived from linearized GNSS pseudorange, IMU pre-integration, and visual residuals (Zheng et al., 2022, Tian et al., 2024).

4. Statistical assumptions and advanced formulations

Residual-based RAIM is usually built on explicit statistical assumptions about nominal errors, false alerts, and fault observability. The standard chi-square forms assume Gaussian nominal errors and thresholding from an allocated false-alert budget. The ARAIM false-alert-allocation study shows that this allocation is itself delicate when pseudorange noise is time-correlated: the per-sample false-alert probability must be treated as conditional and time-varying, not as a white-noise constant. In the illustrative example with $r_k = z_k - \hat{z}_k.$5 s and $r_k = z_k - \hat{z}_k.$6, the conditional probability at $r_k = z_k - \hat{z}_k.$7 is about $r_k = z_k - \hat{z}_k.$8, the moving-average estimate is about $r_k = z_k - \hat{z}_k.$9, the common approach yields a maximum VPL about $5$0 cm too low, and the apparent $5$1 availability becomes about $5$2 when evaluated with the corrected conditional probability (Dunik et al., 2024).

A second stress point is non-Gaussian nominal error. The jackknife ARAIM work argues that conventional Gaussian-bounded nominal models can be overly conservative when real-world satellite clock and orbit errors are heavy-tailed. It therefore replaces position-domain solution separation with a range-domain jackknife detector, extends the detector to multiple simultaneous faults, and derives an integrity-risk bound that propagates hypothetical fault vectors through the position solution. In worldwide simulations, the proposed method keeps the dual-constellation $5$3 percentile VPL below $5$4 m in most locations, whereas baseline ARAIM suffers VPL inflation over $5$5 m due to heavy-tailed Galileo signal-in-space range errors and unavailability events account for about $5$6 at a $5$7 m vertical alert limit (Yan et al., 6 Jul 2025).

These limitations have motivated alternative integrity architectures that are explicitly positioned against residual-based RAIM. The Bayesian 3D cellular RAIM paper treats classical RAIM as frequentist and conservative because it relies on residual or solution-separation tests and then bounds the error after discarding measurements or selecting a fault mode. Its Bayesian alternative computes the exact posterior of the position and clock-bias vector as a Gaussian mixture via factor-graph message passing and reports over $5$8 protection-level reduction at a comparable computational cost. The simplified 1D Bayesian case study makes the same argument in a tractable setting: residual-based advanced RAIM uses solution-separation thresholds and conservative mode-wise overbounds, whereas Bayesian RAIM computes exact posterior fault probabilities and derives the protection level directly from the posterior tail probability (Ding et al., 2024, Ding et al., 2022).

5. Domain-specific realizations

In GNSS tracking, residual-based RAIM is often deployed as a runtime safeguard against coupled-estimator failure. In the LMS multipath-fading VDLL receiver, RAIM/FDE reduces deep RMS error at interruption points, prevents signal unlock, and improves tracking continuity; the paper reports that EKF outperformed UKF in Rayleigh fading, while UKF outperformed EKF in Rician fading (Wu et al., 2019). In urban-canyon PPP/LiDAR fusion, the "P3-LOAM: PPP based LiDAR Odometry and Mapping" system (Li et al., 2020) uses LiDAR-SLAM-assisted pseudorange residual filtering with $5$9; PPP after RAIM has MAE $6$0 m, RMSE $6$1 m, max error $6$2 m, and availability $6$3, while the fused P$6$4-LOAM system reaches MAE $6$5 m, RMSE $6$6 m, max error $6$7 m, with $6$8 availability (Li et al., 2020).

In visual perception, RAIM is adapted by replacing pseudoranges with reprojection measurements constrained by known geometry. The runway-landing system uses predicted keypoint means and diagonal covariances, solves a weighted nonlinear least-squares pose problem, and then performs a whitened residual consistency test. The resulting pipeline runs at about $6$9–$d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$0 Hz and the spatial Soft Argmax regressor achieves sub-pixel keypoint error, as low as $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$1 pixels; under a simulated fault that shifts the far runway point by about $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$2 meters, the residual statistics are clearly separated from nominal statistics and the monitor reliably detects the inconsistency (Valentin et al., 13 Aug 2025). In UAV visual localization on EuRoC, the RAIM-inspired protection level bounds the true translational error much more reliably than the commonly used $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$3 method: with $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$4-pixel covariance, $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$5 often bounds only about $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$6–$d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$7 of frames in $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$8, whereas the protection level bounds about $d_n = \hat{x}_0 - \hat{x}_n = (S_0 - S_n)w,$9–$\lambda = \epsilon^T W \epsilon$0 or more depending on the sequence (Li et al., 2019).

The same residual logic now appears in heterogeneous positioning systems. The smartphone-oriented "Guardian Positioning System (GPS) for Location Based Services" (Liu et al., 14 May 2025) extends RAIM to GNSS, Wi-Fi, cellular, Bluetooth, GeoIP/RTT, and onboard sensors, reports a detection accuracy improvement of up to $\lambda = \epsilon^T W \epsilon$1–$\lambda = \epsilon^T W \epsilon$2 compared to state-of-the-art algorithms and location providers, and detects attacks within $\lambda = \epsilon^T W \epsilon$3 seconds, with a low false positive rate (Liu et al., 14 May 2025). The Gaussian-mixture extension with opportunistic information reports more than $\lambda = \epsilon^T W \epsilon$4 better detection performance than a Kalman-filter baseline and $\lambda = \epsilon^T W \epsilon$5–$\lambda = \epsilon^T W \epsilon$6 improvement over a location-fusion baseline, with an additional $\lambda = \epsilon^T W \epsilon$7–$\lambda = \epsilon^T W \epsilon$8 gain from outlier exclusion (Liu et al., 2024). In Wi-Fi rogue-AP detection, subset-based Gaussian-mixture RAIM achieves about $\lambda = \epsilon^T W \epsilon$9–$\lambda \sim \chi^2_{n-m}$0 true positive under one attack type, and detection accuracy rises from $\lambda \sim \chi^2_{n-m}$1 to $\lambda \sim \chi^2_{n-m}$2 at $\lambda \sim \chi^2_{n-m}$3 as the sampling ratio increases (Liu et al., 2024). An anchor-free multi-robot formulation replaces satellite pseudoranges with inter-robot ranges and couples residual screening to sequential convex programming and ADMM for distributed detection, identification of affected robots, and localization-error reconstruction (Vijay et al., 2024).

6. Limitations, misconceptions, and current research directions

A recurrent misconception is that every RAIM-labeled method computes a formal integrity risk and a classical protection level. Several practical systems in the cited literature do not. The VDLL LMS receiver uses residual-based RAIM conceptually, but does not explicitly derive a parity-space RAIM test, a chi-square threshold, a formal residual norm statistic, explicit fault probability or missed-detection probability formulas, or detailed satellite-by-satellite exclusion equations (Wu et al., 2019). P$\lambda \sim \chi^2_{n-m}$4-LOAM explicitly states that there is no sequential subset test, parity-space test, solution separation, or explicit protection level computation; its monitor is a thresholded residual filtering scheme (Li et al., 2020). The HAPS-aided GPS study uses a modified Danish estimation method with C/N$\lambda \sim \chi^2_{n-m}$5-based variance modeling and does not derive protection levels or formal integrity risk in the aviation sense (Zheng et al., 2023).

Another limitation is dependence on residual calibration and model validity. The runway-landing adaptation states that if predicted uncertainties are too small, false alarms increase; if they are too large, the test becomes less sensitive and faults may pass. The same paper notes that if pixel errors become strongly correlated across keypoints, RAIM’s effectiveness degrades, and in the extreme case of perfectly correlated errors a consistent but wrong reprojection can make fault detection impossible (Valentin et al., 13 Aug 2025). The multimodal LBS frameworks likewise rely on the assumption that some benign subset remains because it is difficult to compromise all benign anchors all the time; coordinated spoofing therefore weakens subset-consistency logic even when it does not eliminate it (Liu et al., 14 May 2025, Liu et al., 31 Oct 2025).

Recent work also questions the strict separation between estimation and monitoring. The GNSS-camera particle-filter study argues that post-estimation residual checking is biased relative to joint state estimation and integrity monitoring, and instead uses a particle distribution, camera-derived position probabilities, KL-divergence-based fusion, and a PAC-Bayesian upper bound on hazardously misleading information (Mohanty et al., 2021). The Bayesian 3D cellular RAIM paper makes a related argument in the snapshot setting: excluding a measurement after using it to decide exclusion can violate the integrity accounting, whereas posterior-based monitoring keeps all measurements in the probabilistic model and derives bounds from the full posterior mixture (Ding et al., 2024).

This suggests that residual-based RAIM is no longer only a GNSS pseudorange detector. In current research it functions as a general measurement-consistency paradigm, ranging from classical chi-square residual tests and solution separation to robust reweighting, subset-consistency monitoring, range-domain jackknife detection, and residuals embedded in factor graphs, while its principal challenges remain redundancy, geometry, covariance credibility, multi-fault behavior, time correlation, and non-Gaussian nominal error.