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Goal inference with Rao-Blackwellized Particle Filters
Yixuan Wang, Dan P. Guralnik, Warren E. Dixon
TL;DR
Inferring an agent’s goal from noisy trajectory observations is difficult because intent is latent. The paper uses an RBPF to infer intent from closed-loop motion and analytically marginalizes the linear–Gaussian state. Simulations show fast, accurate recovery despite noise, with the reduced estimator maintaining lower final error states than the complete estimator.
Problem
Method
An RBPF samples goal location, radius, and arrival time while analytically propagating the conditionally linear–Gaussian state, and forms complete and reduced weighted-mixture intent estimates.
Source: §1 · p. 1Results
Simulations show fast, accurate intent recovery despite noise, and the Reduced Estimator consistently maintains lower final error states than the complete estimator.
Source: §6 · p. 5, §5.2 · p. 5Takeaways & Limitations
The results motivate future work on designing intent-obfuscating controllers.
Source: §6 · p. 5A large number of particles is needed to obtain a representative sample of the intent domain.
Source: §3.1 · p. 2
Abstract
from arXiv · showhide
Inferring the eventual goal of a mobile agent from noisy observations of its trajectory is a fundamental estimation problem. We initiate the study of such intent inference using a variant of a Rao-Blackwellized Particle Filter (RBPF), subject to the assumption that the agent's intent manifests through closed-loop behavior with a state-of-the-art provable practical stability property. Leveraging the assumed closed-form agent dynamics, the RBPF analytically marginalizes the linear-Gaussian substructure and updates particle weights only, improving sample efficiency over a standard particle filter. Two difference estimators are introduced: a Gaussian mixture model using the RBPF weights and a reduced version confining the mixture to the effective sample. We quantify how well the adversary can recover the agent's intent using information-theoretic leakage metrics and provide computable lower bounds on the Kullback-Leibler (KL) divergence between the true intent distribution and RBPF estimates via Gaussian-mixture KL bounds. We also provide a bound on the difference in performance between the two estimators, highlighting the fact that the reduced estimator performs almost as well as the complete one. Experiments illustrate fast and accurate intent recovery for compliant agents, motivating future work on designing intent-obfuscating controllers.
1. INTRODUCTION
Intent inference from noisy trajectories matters for early decisions in safe autonomy, but overlapping trajectories make goals uncertain. This work applies an RBPF to continuous-space intent inference and develops weighted-mixture estimators.
1. INTRODUCTION
- Intent inference supports early decisions by letting robots reason over plausible futures before conflicts arise, despite overlapping trajectories from different goals and arrival times.The motivating settings include navigation, assistance, and surveillance.Source: §1 · p. 1
- Existing intent-inference and obfuscation approaches largely use finite-state abstractions, offering limited guidance for continuous state spaces with nonlinear dynamics.Source: §1 · p. 1
- The proposed RBPF samples intent parameters while analytically propagating the conditionally linear–Gaussian state, updating particle weights from trajectory observations via a Kalman update.Intent is represented by goal location, radius, and arrival time.Source: §1 · p. 1
- The paper develops complete and reduced weighted-mixture intent estimators, bounds their information-leakage difference, and reports fast, accurate recovery that degrades gracefully under measurement noise.Source: §1 · p. 1
2. PROBLEM FORMULATION
The problem is to infer an agent’s goal location, radius, and desired arrival time from its trajectory. The adversary uses an assumed stabilizing closed-loop model with bounded disturbance to estimate that intent.
2. PROBLEM FORMULATION
- The intent is θ∗=(x∗, r∗, t∗): a goal center, positive radius, and desired arrival time; the adversary passively infers it from the agent’s trajectory.The agent and adversary move in R^n for n=2 or 3.Source: §2 · p. 2
- The intent parameters are restricted to a predefined domain with bounded goal locations, radii, and arrival times.Source: §2 · p. 2
- The adversary may know the agent dynamics but typically cannot observe its control inputs in real time, and assumes closed-loop behavior stabilizes the goal within the allotted time.Source: §2 · p. 2
- The disturbance bound ¯d is assumed known to the adversary as a representation of the agent’s system capabilities in the environment.Source: §2 · p. 2
- Under the stated closed-loop condition, the trajectory enters the goal-radius ball by the desired arrival time and remains there afterward.Source: §2 · p. 2
- Using the assumed behavior model and stability result, the adversary seeks to infer the true intent from real-time state observations.Source: §2 · p. 2
3. INTENTION INFERENCE
The section adapts an RBPF to infer intent from noisy trajectories by sampling intent hypotheses and filtering their state trajectories. It then represents estimates as distributions, including a reduced form over effective particles.
3.1 RBPF Model Development
- For each intent particle, a Kalman filter updates the state estimate, while the intent parameter remains fixed during that particle’s lifetime.The intent parameters are sampled from a uniform prior over Θ.Source: §3 · p. 2, §3.1 · pp. 2–3
- The highest-weight particle can be a false positive, while the weighted-average particle location may lie outside every predicted goal region.These cases motivate estimating intent from particle distributions rather than relying only on point estimates.Source: §3.1 · p. 3
- Intent particles are scored against noisy trajectory observations, and resampling discards particles with negligible weights.Each particle represents a possible trajectory under its intent hypothesis.Source: §3.1 · pp. 2–3
3.2 Estimation and Inference
- Each intent parameter is represented by a continuous probability distribution over an extended space, with localization and timing precision set by system parameters.The distributions are constructed as products over goal location, radius, and arrival time.Source: Representation of Intent by a… · p. 3
- The complete estimator averages representative particle distributions, while the reduced estimator restricts that mixture to effective particles.The distribution retains information that its first moment may not; the top-Neff representation contains the maximum information among the three estimators described.Source: Density-Based Estimators of Intent · p. 4, The Reduced Estimator · p. 4
4. INTENT ESTIMATORS: REDUCED VS. COMPLETE
The section establishes lower bounds for information-leakage estimates and uses their discrepancy to conservatively compare the complete and reduced estimators.
4. INTENT ESTIMATORS: REDUCED VS. COMPLETE
- Theorem 4 gives lower bounds for the information-leakage estimates of the complete and reduced estimators.The bounds are stated with constants Cx, Cr, and Ct.Source: §4 · p. 4
- The bounds express complete-estimator leakage using data available to the adversary, even without precise agent-state information.Source: §4 · p. 4
- The discrepancy between the lower bounds serves as a conservative metric for the performance difference between the two estimators.Source: §4 · p. 4
5. NUMERICAL EXPERIMENTS
The experiments evaluate intent inference in planar goal-directed navigation. Across the reported trials, the reduced estimator has lower final error, while the RBPF identifies the goal well before halfway through the trajectory.
5.1 Simulation Setup
- The evaluation measures real-time recovery of goal center, radius, and arrival time in planar navigation, using KL divergence as its primary success metric.The RBPF uses 1,200 particles; inference time is when KL divergence falls below 50 and remains bounded.Source: §5 · p. 4, §5.1 · p. 5
5.2 Results and Analysis
- Over 100 Monte Carlo trials, the Reduced Estimator maintained lower final error states, with no information advantage for the Complete Estimator except possibly in rare cases.Table 1 summarizes terminal accuracy and inference speed.Source: Table 1 · p. 5, §5.2 · p. 5
- As the agent commits to a direction, likelihood updates collapse the initially diffuse posterior around the correct goal.At the beginning, particles cover both candidate goals and have high variance.Source: §5.2 · p. 5
- The RBPF identified the correct goal well before halfway through the trajectory, and the final estimate matched ground truth with negligible error.Source: §5.2 · p. 5
6. CONCLUSION
The paper implements goal-directed intent inference as an RBPF over goal location, radius, and arrival time, using analytic Kalman updates to marginalize linear–Gaussian dynamics. It reports fast, accurate recovery in simulations and identifies richer models, multi-agent settings, and obfuscation-controller design as future work.
6. CONCLUSION
- Analytic Kalman updates marginalize the linear–Gaussian dynamics in an RBPF over (x, r, t), improving sample efficiency relative to standard particle filters.The framework evaluates information leakage with computable KL lower bounds and produces posterior marginals reproducible across tasks.Source: §6 · p. 5
- Simulations demonstrate fast, accurate intent recovery despite noise.The conclusion describes future work on richer motion models, non-Gaussian observation channels, KL-based obfuscation controllers, and multi-agent extensions.Source: §6 · p. 5
- Future obfuscation-controller design is framed around minimizing KL-based leakage subject to task constraints while tracking the reference path.The paper also identifies extensions to multi-agent settings.Source: §6 · p. 5
Appendix A. PROOF OF LEMMA 2
The proof introduces the trajectory error state and a Lyapunov function candidate.
Appendix A. PROOF OF LEMMA 2
- The error state is defined as e = x − x(θ).The passage then introduces V(e) as a Lyapunov function candidate.Source: Appendix A · p. 5
2. Following (Aubin
The proof differentiates the Lyapunov function along a complete solution of the differential inclusion and uses a bound on λ for times after t(θ).
2. Following (Aubin
- For a complete solution of the differential inclusion, the proof applies the chain rule to V composed with the error state.It notes that V is smooth.Source: §2 · p. 6
- The derivative satisfies V̇ ∈ −2λV + 2d̄e⊤B for almost all t.The proof then proceeds to an inequality and integration after algebraic manipulation.Source: §2 · p. 6
- For all t ≥ t(θ), the proof states that it suffices for λ to satisfy λ ≥ 2d̄r(θ).This follows after establishing positivity of a factor in (A.6) and bounding the exponential terms.Source: §2 · p. 6
Appendix B. PROOF OF THEOREM 4
The proof derives KL-divergence lower bounds for the reduced estimator using Gaussian-mixture bounds, then combines bounds for the intent components.
Appendix B. PROOF OF THEOREM 4
- The proof applies lower bounds for KL divergences between a Gaussian and a Gaussian mixture to obtain a lower bound for the reduced estimator’s x term.The passage identifies the result as an example of the component-wise bounds.Source: Appendix B · p. 6
- The position dimension n is restricted to 2 or 3 in this paper.Source: Appendix B · p. 6
- The proof collects the three component terms into a lower bound for the reduced estimator’s information quantity.It notes that similar upper bounds exist and that an information convergence analysis is left for future work.Source: Appendix B · p. 6
- The proof invokes Proposition 5, attributed to Wang et al. (2026, Section 4.2), for the effective-weight bounds.It separately introduces a bound on k and bounds involving Bν and dν.Source: Appendix B · p. 6
- The proof states that the bounds are sharp.Source: Appendix B · p. 6