Gaussian Mixture-Based Inverse Perception Contract for Uncertainty-Aware Robot Navigation

Bingyao Du; Joonkyung Kim; Yiwei Lyu

cs.ROMar 4, 2026

Gaussian Mixture-Based Inverse Perception Contract for Uncertainty-Aware Robot Navigation

Authors: Bingyao Du, Joonkyung Kim, Yiwei Lyu

Organizations: The author is with the Department of Computer Science, Columbia University, New York, NY 10027, USA. · The authors are with the Department of Computer Science and Engineering, Texas A&M University, College Station, TX 77843, USA.

Abstract

Reliable navigation in cluttered environments requires perception outputs that are not only accurate but also equipped with uncertainty sets suitable for safe control. An inverse perception contract (IPC) provides such a connection by mapping perceptual estimates to sets that contain the ground truth with high confidence. Existing IPC formulations, however, instantiate uncertainty as a single ellipsoidal set and rely on deterministic trust scores to guide robot motion. Such a representation cannot capture the multi-modal and irregular structure of fine-grained perception errors, often resulting in over-conservative sets and degraded navigation performance. In this work, we introduce Gaussian Mixture-based Inverse Perception Contract (GM-IPC), which extends IPC to represent uncertainty with unions of ellipsoidal confidence sets derived from Gaussian mixture models. This design moves beyond deterministic single-set abstractions, enabling fine-grained, multi-modal, and non-convex error structures to be captured with formal guarantees. A learning framework is presented that trains GM-IPC to account for probabilistic inclusion, distribution matching, and empty-space penalties, ensuring both validity and compactness of the predicted sets. We further show that the resulting uncertainty characterizations can be leveraged in downstream planning frameworks for real-time safe navigation, enabling less conservative and more adaptive robot motion while preserving safety in a probabilistic manner.

Figures & tables

Panel of Figure 1 — Fig. 1 : Overview of GM-IPC. (a) Perception module: the robot camera captures the point cloud of environment, and the perception module predicts obstacle bounding boxes. Points inside the boxes are mapped to a 2D binary occupancy map. (b) IPC network: The map is processed by a convolutional encoder, with the robot state processed by an MLP. Their representations are then fused through one fusion layer, followed by 3 linear heads to produce the Gaussian parameters (c) Uncertainty sets: given a confidence value $p$ , the boundaries of all Gaussians are extracted to cover the obstacle region.

Fig. 2 : Overview of navigation pipeline. At each time step, raw sensor readings $\textnormal{P}_{t}$ are processed by the perception module to produce perceptual estimates $\hat{\textnormal{y}}_{t}$ . The IPC network then generates a Gaussian Mixture uncertainty set, which is passed to the MPC-CBF planner to compute a sequence of safe future states ${(\textnormal{p}_{h},\textnormal{v}_{h})}_{h=1}^{H}$ . The first control output is applied to the robot, and the process repeats at the next time step. White blocks denote intermediate results, while gray blocks represent modules.

Scenario	Model	Mean Inclusion Rate		% Steps $\geq 95\%$		Compactness
Scenario	Model	Final	All	Final	All	Final ( $\mu\pm\sigma$ )
Chair	GMM	1.000	0.999	100.0	99.8	$9.52\pm 0.46$
	Ellip	0.999	0.989	100.0	96.6	11.74 $\pm$ 0.35
Sofa	GMM	0.997	0.993	100.0	97.8	10.50 $\pm$ 1.00
	Ellip	0.994	0.988	96.7	93.7	$9.48\pm 0.39$
Multi Sofa	GMM	0.999	0.998	100.0	99.0	5.58 $\pm$ 0.30

TABLE I : Validity of IPC models.

Loss Function	Mean Inclusion Rate		% Steps $\geq 95\%$		Compactness
Loss Function	Final	All	Final	All	Final ( $\mu\pm\sigma$ )
Full	0.997	0.993	100.0	97.8	10.50 $\pm$ 1.00
No Empty Penalty	1.0	1.0	100.0	100.0	1.00 $\pm$ 0.00
No NLL	0.997	0.993	100.0	97.0	9.43 $\pm$ 0.64

TABLE II : Ablation study on loss functions.

Panel of Figure 5 — Fig. 3 : Ablation study on the loss function in the single-sofa case. (a) full loss function, (b) without the empty penalty term, and (c) without the NLL term.

Scenario	Ellip-IPC	GM-IPC
Mixed	0.817	0.982
Two sofas	0.773	0.935
Single sofa	0.953	0.990
Single chair	0.894	0.929

TABLE III : Average solved confidence level under equal union area of ellipsoids (30 trials per scenario; aggregated across steps).

Panel of Figure 7 — Fig. 4 : Coverage under equal union area of ellipsoids: (a) GMM single-sofa, (b) Ellipsoid single-sofa, (c) GMM multi-sofa, (d) Ellipsoid multi-sofa.

Scenario	Model	Steps	Len	Eff.	Ctrl	Succ.
Mixed	Ellip-1	434.2	12.69	0.90	0.18	0.76
Mixed	Ellip-2	408.3	11.95	0.95	0.12	0.70
	GMM	403.1	11.62	0.97	0.12	0.88
Multi Sofa	Ellip-1	440.0	12.79	0.89	0.22	0.74
Multi Sofa	Ellip-2	412.4	12.08	0.94	0.13	0.86
	GMM	397.2	11.42	0.99	0.13	0.90

TABLE IV : Navigation evaluation results across scenarios (50 trials each).

Panel of Figure 9 — Fig. 5 : Navigation trajectories in multi-object scenarios using IPC. Yellow regions indicate the ground-truth obstacles, and blue points are the observations from the perception module. (a, e) Two-sofa case with GM-IPC, (b) Two-sofa case with Ellip-IPC, (c, f) Mixed case with GM-IPC, (d) Mixed case with Ellip-IPC.

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Abstract

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