Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization

arXiv:2606.29868 · cs.RO · Submitted 2026-06-29 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa.

Dev: And I'm Dev, here with Taro.

Rosa: We're looking at the paper: "Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization".

Dev: Here's the gist of the paper. Accurate, robust, and adaptive localization is essential for various robotic operations.

Rosa: This paper proposes a new message passing (MP) algorithm for realizing collaborative localization in a distributed manner.

Dev: Unified Message Passing Framework The algorithm unifies Gaussian belief propagation (GBP) and mean-field (MF) approximation, where GBP preserves dependencies among robot states, and MF enables estimation of noise statistics.

Rosa: To effectively handle non-conjugate terms from nonlinear measurement models, the algorithm adopts a parametric formulation in which these terms are treated by gradient estimators.

Title and authors: Rosa: So we're looking at this paper titled "Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization". I’m curious what that means when you take it apart.

Dev: It sounds like they're proposing a way to do localization where different robots can talk to each other, but they've built a system that unifies two different ways of thinking about how the robot states and the noise statistics should be handled.

Rosa: Exactly, and what I’m getting from the title is this message passing thing isn't just one method; it’s combining Gaussian belief propagation with mean-field approximation.

Taro: So, like they're trying to keep the robot states linked together using one part, but also estimate how noisy things are using another part simultaneously.

Dev: That’s the core idea—preserving dependencies among robot states through Gaussian belief propagation while mean-field helps them estimate noise statistics, which is usually done separately.

Rosa: And what they do next is address this issue of non-conjugate terms that pop up when you use nonlinear measurement models. They solve this by using a parametric formulation where these tricky terms get treated with gradient estimators.

Taro: Gradient estimators sound like they are trying to find the slope of the error function to figure out how to adjust the system correctly, which is usually a pain in real-time systems.

Dev: Right, and they go beyond just linearizing things or doing simple sampling; they actually design a normalizing flow based gradient estimator that lets them do learnable sampling.

Rosa: So, the key thing here is that this isn't just another localization trick; it’s a unified message passing framework that handles both the state structure and the noise modeling in a distributed way.

Taro: It suggests they're aiming for something where the system can handle those messy real-world measurements without breaking down when things get complicated.

The paper's summary: Rosa: Let’s look at what the paper actually says in its summary about this "Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization." It lays out how they tackle those complex measurement models.

Dev: They mention that they use a unified message passing framework, which combines Gaussian belief propagation and mean-field approximation, where GBP preserves dependencies among robot states, and MF enables estimation of noise statistics.

Rosa: That’s the foundation—GBP keeps the state connections intact while MF helps them estimate how noisy the measurements are. They handle non-conjugate terms from nonlinear measurement models by using a parametric formulation where these terms are treated by gradient estimators.

Taro: So, when you have a nonlinear model, instead of trying to solve it directly in every step, they set up an optimization problem where these terms get handled via specific estimators like linearization or sampling or their normalizing flow approach.

Dev: They introduce three specific strategies for those non-conjugate terms: first linearization using a first-order Taylor expansion; second sampling combining Lemma one with Bonnet’s theorem and Price’s theorem for Monte Carlo integration; and third they have these NF-based approaches utilizing importance sampling formulated via Real NVP to make the sampling more efficient <ref:2606.29868#pg1>.

Rosa: And what makes this approach special is that they optimize directly over the natural parameters of candidate distributions limited to exponential family, which lets them separate factors with closed-form updates from those needing dedicated treatment.

Taro: So, they’re trying to keep the math manageable by separating the easy parts from the hard parts of their estimation problem.

Dev: They construct a Lagrangian function for this optimization over expectation parameter space to facilitate that process, which ties everything together into one coherent update rule set.

The paper's improvements: Rosa: Now let’s talk about the actual improvements the authors claim in "Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization." They focus on how this unified method actually performs compared to existing localization algorithms.

Dev: They extend the method to support states on Lie groups, which is important because robot states often involve rotations, and this allows them to handle those three dee rotation manifolds accurately <ref:2606.29868#pg3>.

Rosa: Beyond that extension, they demonstrate that this MP-NF algorithm achieves the highest estimation accuracy in their simulations and experiments. Specifically, they show it significantly outperforms algorithms of the same class, even surpassing centralized algorithms like GTSAM.

Taro: That’s a big deal for reliability because it means the system is very accurate when things are running on an autonomous surface vehicle system with odometry, GNSS, and UWB ranging.

Dev: They show that introducing the normalizing flow is actually worthwhile because they achieve sufficiently high accuracy within only five iterations in their simulations.

Rosa: And for real-world deployment, they also find that the performance of MP-NF and GTSAM is comparable in field experiments. They show translational and rotational root mean square errors for all algorithms in Figure seven with MP-NF achieving the highest accuracy for translational estimation.

Taro: What’s interesting is how robust it seems; they show strong generalization capabilities, meaning even when the graph structures change, like on different environments or graphs, the performance stays consistently reliable.

Dev: They show this by looking at their evaluation datasets in Figure thirteen where the first ten histograms are from real evaluation data and only the last two are from training data.

Conclusion: Rosa: So to wrap up on "Normalizing Flow-Enhanced Message Passing for Multirobot Collaborative Localization," this paper introduces a unified message passing algorithm that integrates Gaussian belief propagation and mean-field approximation.

Dev: This combination maintains the state dependencies using GBP while the mean-field part estimates noise statistics, which boosts robustness and adaptability when dealing with real robot systems.

Rosa: They handle nonlinear measurement models through gradient estimators, specifically using linearization, sampling, or their NF-based techniques for learnable sampling of those complex terms.

Taro: The extension to Lie groups is key because it lets the system estimate practical states that involve rotations and allows it to operate in three dee rotation manifolds <ref:2606.29868#pg3>.

Dev: They’ve validated this on multirobot localization involving odometry, GNSS, and UWB, with simulations and autonomous surface vehicle experiments demonstrating the accuracy, robustness, and adaptability improvements.

Rosa: Overall, the paper shows how combining these techniques can lead to a system that is accurate in simulation and comparable in the field when deployed.

Taro: What’s really telling me is that this method has strong generalization capabilities, meaning it should be deployable on physical systems even if it was only trained in simulation.

Southeast University · Southern University of Science and Technology · University of Groningen

cs.RO

Submitted: 2026-06-29

Updated: 2026-10-29

Importance score: 92/100

The gist: Accurate, robust, and adaptive localization is essential for various robotic operations.

Key concepts

Unified Message Passing Framework: This framework combines Gaussian belief propagation (GBP) to track robot state dependencies with mean-field approximation to estimate noise statistics. This allows the system to simultaneously manage the uncertainty of where robots are and how noisy their sensors are.
Normalizing Flow (NF)-based Gradient Estimator: Since nonlinear measurement models often have complex terms, this estimator uses a normalizing flow formulation, trained end-to-end. This enables learnable sampling techniques that efficiently calculate gradients for optimization, improving the algorithm's ability to handle non-standard noise.
Parametric Inference: The method optimizes over the natural parameters of candidate distributions belonging to an exponential family. This structure allows the algorithm to separate simple updates from complex ones, facilitating efficient computation when dealing with various measurement models.

Terminology

Localization — 53× in this paper · explained in 2 episodes
Accuracy — 47× in this paper · explained in 1 episode
Robustness — 25× in this paper · explained in 6 episodes
Normalizing Flow — 15× in this paper · explained in 4 episodes
Verification — 11× in this paper · explained in 2 episodes
Generalization — 9× in this paper · explained in 14 episodes
Core — 5× in this paper · explained in 2 episodes
Fixed Point — 4× in this paper · explained in 2 episodes
Parameterization — 4× in this paper · explained in 2 episodes
Generalization Capabilities — 3× in this paper · explained in 2 episodes
Clarity — 2× in this paper · explained in 3 episodes
Latent Variables — 2× in this paper · explained in 2 episodes
Monte Carlo Sampling — 2× in this paper · explained in 2 episodes
Normalizing Flows — 2× in this paper · explained in 4 episodes
Score Function — 2× in this paper · explained in 4 episodes
Adam — 1× in this paper · explained in 3 episodes
Calibration — 1× in this paper · explained in 9 episodes
Communication Overhead — 1× in this paper · explained in 2 episodes
Computational Efficiency — 1× in this paper · explained in 2 episodes
Computational overhead — 1× in this paper · explained in 2 episodes
Generative Model — 1× in this paper · explained in 2 episodes
Generative Models — 1× in this paper · explained in 4 episodes
Generator — 1× in this paper · explained in 2 episodes
Kullback-Leibler (KL) Divergence — 1× in this paper · explained in 6 episodes
Latent Space — 1× in this paper · explained in 7 episodes
Loss Function — 1× in this paper · explained in 2 episodes
Point Clouds — 1× in this paper · explained in 2 episodes
Pose Graph Optimization — 1× in this paper · explained in 2 episodes
Sensor Fusion — 1× in this paper · explained in 2 episodes
System Identification — 1× in this paper · explained in 3 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

Accurate, robust, and adaptive localization is essential for various robotic operations. This paper proposes a new message passing (MP) algorithm for realizing collaborative localization in a distributed manner.

Unified Message Passing Framework

The algorithm unifies Gaussian belief propagation (GBP) and mean-field (MF) approximation, where GBP preserves dependencies among robot states, and MF enables estimation of noise statistics. To effectively handle non-conjugate terms from nonlinear measurement models, the algorithm adopts a parametric formulation in which these terms are treated by gradient estimators. Beyond linearization and sampling, we further design a normalizing flow (NF)-based gradient estimator, enabling learnable sampling.

Gradient Estimation Strategies

The paper introduces three strategies for handling non-conjugate terms arising from nonlinear models:

Linearization, which adopts first-order Taylor expansion of the residual model.
Sampling, which combines Lemma 1 with Bonnet’s theorem and Price’s theorem to obtain gradients by Monte Carlo integration.
NF-based approaches, which utilize importance sampling formulated via Real NVP to improve sampling efficiency.

Parametric Inference and Optimization

The proposed MP algorithm directly optimizes over natural parameters of candidate distributions limited to exponential family. This parametric approach allows for a separation between factors that admit closed-form updates and the rest require dedicated treatment, i.e., conjugate and non-conjugate terms. The Lagrangian function of (19) is constructed to facilitate optimization over the expectation parameter space.

Extension to Lie Groups and Application

The method is extended to support states on Lie groups, which is common for robot state estimation. The algorithm is applied to multirobot localization task fusing odometry, global navigation satellite system (GNSS) measurements, and inter-robot ultra wideband (UWB) ranging. Simulations and experiments on autonomous surface vehicles (ASVs) demonstrate its improved accuracy, robustness, and adaptability.

Performance Validation

Simulations and experiments demonstrate that MP-NF achieves the highest estimation accuracy, not only significantly outperforming algorithms of the same class, but even surpassing centralized algorithm GTSAM. The results in Fig. 4 indicate that introducing NF is worthwhile, as they achieve sufficiently high accuracy within only 5 iterations. In field experiments, the performance of MP-NF and GTSAM is comparable. The rotational and translational RMSEs of all algorithms are shown in Fig. 7, with MP-NF achieving the highest accuracy for translational estimation. The results in Table III report the ARMSEs, SDS, and average time per iteration using new nonlinear ranging model.

Generalization and Robustness

The methods possess strong generalization capabilities, allowing rapid deployment on physical systems even when trained solely in simulation. The consistently reliable performance of MP-NF under these varying graph structures demonstrates its generalization capability. The results in Fig. 13 show that the first ten histograms, highlighted in green, are obtained from the evaluation datasets, whereas the final two, shown in blue, are from the training datasets. The consistently reliable performance of MP-NF under these varying graph structures demonstrates its generalization capability.

Conclusion

This paper has presented a unified MP algorithm that integrates GBP and MF approximation, where the former maintains dependencies among robot states, and the latter estimates noise statistics to enhance robustness and adaptability. The MP algorithm optimizes over the natural parameter space of candidate distributions, while non-conjugate terms arising from nonlinear measurement models have been addressed through gradient estimators constructed by linearization, sampling, or NF-based techniques. End-to-end training for NF parameters has enhanced estimation accuracy. The MP algorithm has also been extended to Lie groups. The algorithm has been validated in multirobot localization involving odometry, GNSS, and UWB, with both simulations and ASV experiments demonstrating the improvements in accuracy, robustness, and adaptability.

APPENDIX A

EXAMPLES OF EXPONENTIAL FAMILY DISTRIBUTIONS

The base measures of the following distributions are all set as 1. The Gaussian distribution N(x; m, Σ) with mean vector m and covariance matrix Σ can be written in a minimal exponential family form by defining natural parameter λ and sufficient statistics T (x) in tuple notation [33], [34]. For notational clarity and to facilitate the derivations, we use vectorized representations of matrix natural parameters and sufficient statistics:

λ = Σ−1m−1/2Σ−1, T (x) = xvec xx⊤. The vectorized form of expectation parameters is µ = mvec Σ + mm⊤.

APPENDIX B

PROOF OF THEOREM 1

By setting ∇µa L = 0, ∀a ∈ ABP, ∇µiL = 0, ∀i ∈ I, and noticing that for a minimal exponential family defined as (8) with constant h(x), we have ∇µH(p(x;λ)) = −λ [48]. λa =∇µa Ua + X i∈NBP(a) L⊤ a,iλa,i, a ∈ ABP (57). λi = X a∈N(i) (λi − λa,i), i ∈ IBP (58).

APPENDIX C

PROOF OF LEMMA 1

Substituting factor (25) into definition (15), one has Ua = − 0.5 Z ba (xa) ba(ϕa) r⊤(xa)R−1 a(ϕa)r(xa)+ log det(Ra(ϕa))+ nra log 2π dxadϕa. Then, Ua = − 0.5 Z ba (xa) r⊤(xa)R−1 a dxa + c1=Z ba (xa) log N ra (xa); 0, R¯ a dxa + c, where c1 is a constant independent of xa and µa.

APPENDIX D

PROOF OF LEMMA 2

Let µa = col(µa,1, µa,2), where µa,1 has the same dimension as that of xˆa. We have xˆa = µa,1 and vec(Pa) = µa,2 − vec(µa,1µ⊤ a,1. According to chain rule, ∇µa,1Ua = ∂xˆtop∂µa,1∂Ua/∂xˆa+∂vectopPa∂µa,1∂Ua/∂vec(Pa)=∂Ua/∂xˆa− µ a,1⊗ I + I⊗ µ a,1⊤vec ∂Ua/∂Pa.

Improvements for AI systems

Unified Message Passing Algorithm for Collaborative Localization: The system can achieve accurate, robust, and adaptive localization by unifying Gaussian belief propagation (GBP) and mean-field (MF) approximation to preserve robot state dependencies while enabling noise statistics estimation.
Handling Non-Conjugate Terms via Learning: The AI system can effectively handle non-conjugate terms from nonlinear measurement models by adopting a normalizing flow based gradient estimator, which is trained end-to-end to tune NF parameters according to the behavior of the MP algorithm.
State Space Generalization: The method can support estimation of practical robotic states involving rotations and be extended to Lie group state spaces, allowing for accurate pose estimation in 3D rotation manifolds.
Adaptive Noise Modeling: The system can improve robustness by jointly inferring states and noise statistics, allowing the use of noise models beyond Gaussian assumptions, such as Gaussian-Student’s t mixture (GSTM), which is crucial for handling heavy-tailed UWB ranging noise.
Distributed Computation and Fusion: The AI system can realize distributed collaborative localization by partitioning variable nodes into a BP part (states) and an MF part (noise parameters), enabling local belief updates and decentralized fusion across multiple robots without relying on a central node.
Robustness to Outliers in UWB: By incorporating robustness through the chosen gradient estimator, the system can maintain higher estimation accuracy even when ranging outliers occur, as shown by MP-NF consistently reaching accuracy levels that other methods cannot attain even after many more iterations.