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

TL;DR

Gaussian Mixture-based Inverse Perception Contract is introduced, which extends IPC to represent uncertainty with unions of ellipsoidal confidence sets derived from Gaussian mixture models, enabling fine-grained, multi-modal, and non-convex error structures to be captured with formal guarantees.

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 (5)

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

Theorems & Definitions (8)

• Definition 1: Ellipsoidal Confidence Region
• Definition 2: Gaussian Mixture–based IPC
• Remark 1
• Lemma 1: PAC generalization bound over trials
• Remark 2
• Corollary 1: Consistency of ERM shalev2014understanding for GM-IPC
• proof
• Remark 3