Risk-Sensitive Crowd Navigation with Adaptive Ellipsoidal Conformal Prediction

Ruihan A. Li; Ziyao Guo; Yingying Li

cs.ROOct 5, 2026

Risk-Sensitive Crowd Navigation with Adaptive Ellipsoidal Conformal Prediction

Authors: Ruihan A. Li, Ziyao Guo, Yingying Li

Organizations: Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL

Abstract

Safe crowd navigation under distribution shift requires uncertainty representations that capture structured human-motion prediction errors and safety objectives that account for rare but consequential failures. Existing uncertainty-aware methods typically represent prediction errors using isotropic regions, which can be either overly conservative or poorly aligned with directional motion uncertainty. We introduce a risk-aware navigation framework that uses anisotropic conformal ellipsoids to translate structured prediction uncertainty into an episode-level conditional value-at-risk (CVaR) signal that regulates the Lagrangian safety penalty. In particular, adaptive ellipsoidal conformal prediction (AECP) captures directional prediction errors and adaptively calibrates uncertainty regions under distribution shift, while the resulting CVaR-regulated navigation policy is optimized using Lagrangian proximal policy optimization. We evaluate our proposed method under both in-distribution settings and out-of-distribution (OOD) settings involving shifts in pedestrian motion patterns. Compared with state-of-the-art baselines, our method maintains competitive in-distribution performance while improving success rates by 5.68-7.44 percentage points and reducing collision rates by 5.44-6.64 percentage points across OOD settings. We further deploy the trained policy without fine-tuning on a physical robot with onboard perception and CPU-only inference, showing that the full pipeline is feasible in physical crowd navigation.

Figures & tables

Fig. 1 : Prediction errors from Gumbel Social Transformer two-step trajectory predictions for pedestrian 1 in the OOD crossing-flow scenario (see Section IV ). Grey points denote samples of prediction errors/residuals. The blue dashed contour illustrates a circular uncertainty region, while the red contour illustrates an ellipsoidal uncertainty region. The residuals exhibit clear directional anisotropy: a circular region must expand uniformly to cover errors along the dominant direction, whereas an ellipsoidal region can adapt both orientation and relative axis lengths.

Scenario	Method	SR $\uparrow$	CR $\downarrow$	TR $\downarrow$	NT $\downarrow$	PL $\downarrow$	ITR $\downarrow$	SD $\uparrow$
In-distribution	SF	$11.68{\scriptstyle\pm 3.22}\%$	$21.28{\scriptstyle\pm 0.33}\%$	$67.04{\scriptstyle\pm 3.51}\%$	$30.72{\scriptstyle\pm 2.58}$	$27.17{\scriptstyle\pm 0.16}$	$3.82{\scriptstyle\pm 0.08}\%$	$0.432{\scriptstyle\pm 0.002}$
	ORCA	$64.48{\scriptstyle\pm 3.14}\%$	$31.28{\scriptstyle\pm 1.11}\%$	$4.24{\scriptstyle\pm 2.20}\%$	$24.07{\scriptstyle\pm 1.15}$	$13.14{\scriptstyle\pm 0.60}$	$1.49{\scriptstyle\pm 0.14}\%$	$0.490{\scriptstyle\pm 0.005}$
	CrowdNav++	$87.60{\scriptstyle\pm 3.32}\%$	$12.40{\scriptstyle\pm 3.32}\%$	$0.00{\scriptstyle\pm 0.00}\%$	$14.03{\scriptstyle\pm 0.33}$	$13.12{\scriptstyle\pm 0.32}$	$7.65{\scriptstyle\pm 0.12}\%$	$0.417{\scriptstyle\pm 0.009}$
	GenSafeNav	$97.28{\scriptstyle\pm 0.44}\%$	$2.72{\scriptstyle\pm 0.44}\%$	$0.00{\scriptstyle\pm 0.00}\%$	$17.51{\scriptstyle\pm 0.51}$	$17.11{\scriptstyle\pm 0.52}$	$3.07{\scriptstyle\pm 0.10}\%$	$0.4467{\scriptstyle\pm 0.0096}$
	Ours (AECP)	$97.76{\scriptstyle\pm 0.36}\%$	$2.24{\scriptstyle\pm 0.36}\%$	$0.00{\scriptstyle\pm 0.00}\%$	$16.98{\scriptstyle\pm 0.41}$	$16.66{\scriptstyle\pm 0.42}$	$4.29{\scriptstyle\pm 0.24}\%$	$0.419{\scriptstyle\pm 0.004}$
OOD-Crossing Flow	SF	$1.84{\scriptstyle\pm 0.54}\%$	$16.80{\scriptstyle\pm 0.40}\%$	$81.36{\scriptstyle\pm 0.36}\%$	$32.98{\scriptstyle\pm 1.43}$	$19.86{\scriptstyle\pm 0.49}$	$4.25{\scriptstyle\pm 0.71}\%$	$0.354{\scriptstyle\pm 0.003}$

TABLE I : Main test results. Results are reported as mean

\pm

standard deviation. Lower is better for CR, TR, NT, PL, and ITR; higher is better for SR and SD. ITR and SD reflect proximity-comfort tradeoffs.

Scenario	Method	SR $\uparrow$	CR $\downarrow$	TR $\downarrow$	NT $\downarrow$	PL $\downarrow$	ITR $\downarrow$	SD $\uparrow$
OOD-Crossing Flow	Ours (Circular)	$72.32{\scriptstyle\pm 1.53}\%$	$25.52{\scriptstyle\pm 1.75}\%$	$2.16{\scriptstyle\pm 0.22}\%$	$18.06{\scriptstyle\pm 0.86}$	$15.86{\scriptstyle\pm 0.25}$	$7.86{\scriptstyle\pm 0.19}\%$	$0.3669{\scriptstyle\pm 0.0029}$
OOD-Crossing Flow	Ours (AECP)	$77.68{\scriptstyle\pm 1.61}\%$	$21.36{\scriptstyle\pm 1.71}\%$	$0.96{\scriptstyle\pm 0.22}\%$	$17.38{\scriptstyle\pm 0.49}$	$15.31{\scriptstyle\pm 0.59}$	$7.87{\scriptstyle\pm 0.25}\%$	$0.368{\scriptstyle\pm 0.003}$
OOD-Merging Flow	Ours (Circular)	$80.64{\scriptstyle\pm 2.71}\%$	$15.20{\scriptstyle\pm 2.93}\%$	$4.16{\scriptstyle\pm 0.22}\%$	$28.34{\scriptstyle\pm 0.73}$	$26.44{\scriptstyle\pm 0.97}$	$5.04{\scriptstyle\pm 0.28}\%$	$0.3871{\scriptstyle\pm 0.0037}$
OOD-Merging Flow	Ours (AECP)	$84.80{\scriptstyle\pm 2.76}\%$	$13.28{\scriptstyle\pm 0.87}\%$	$1.92{\scriptstyle\pm 1.95}\%$	$27.15{\scriptstyle\pm 0.78}$	$25.08{\scriptstyle\pm 0.27}$	$6.14{\scriptstyle\pm 0.20}\%$	$0.388{\scriptstyle\pm 0.002}$
OOD-Dynamic Velocity	Ours (Circular)	$74.48{\scriptstyle\pm 3.06}\%$	$25.12{\scriptstyle\pm 3.06}\%$	$0.40{\scriptstyle\pm 0.00}\%$	$17.00{\scriptstyle\pm 0.31}$	$14.87{\scriptstyle\pm 0.07}$	$5.06{\scriptstyle\pm 0.37}\%$	$0.4063{\scriptstyle\pm 0.0050}$
	Ours (AECP)	$75.44{\scriptstyle\pm 2.63}\%$	$24.32{\scriptstyle\pm 2.42}\%$	$0.24{\scriptstyle\pm 0.22}\%$	$16.95{\scriptstyle\pm 0.33}$	$14.83{\scriptstyle\pm 0.21}$	$5.59{\scriptstyle\pm 0.43}\%$	$0.406{\scriptstyle\pm 0.002}$

TABLE II : Comparison between AECP and adaptive circular conformal prediction under OOD conditions. Both variants use DtACI-based online scale adaptation and the same CVaR-regulated PPO-Lagrangian training pipeline and differ only in uncertainty-set geometry. Results are reported as mean

\pm

standard deviation. Lower is better for CR, TR, NT, PL, and ITR; higher is better for SR and SD.

Scene	$\mathbf{k}$	$\mathbf{t}$	Circ. Cov.	Ellip. Cov.	Circ. Area	Ellip. Area
Crossing Flow	1	10	88.03%	88.40 %	0.102	0.088
Crossing Flow	1	40	52.28%	74.03 %	1.071	0.555
Crossing Flow	1	80	91.38 %	89.66%	2.064	0.453
Crossing Flow	2	10	89.51%	94.04 %	0.358	0.276
Crossing Flow	2	40	67.75%	78.28 %	0.985	0.569
Crossing Flow	2	80	89.66%	96.55 %	1.371	0.447

TABLE III : Circular and ellipsoidal coverage statistics at different prediction horizons

1\leq k\leq 2

and at evaluation times

t=10,40,80

for demonstration.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
Humans / arena size	$20$ / $12\times 12$ m
Simulation step / time limit	$0.25$ s / $50$ s
DtACI target miscoverage	$\alpha=0.10$
Prediction horizon / Considered Step	$5$ / $2$
Deployed DtACI radius	$\bar{\tau}\in[0,1]$
CVaR confidence level	$\beta=0.95$

Appendix

TABLE IV : Implementation and training parameters.

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