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.

翻译：在杂乱环境中实现可靠导航，不仅要求感知输出具有准确性，还需配备适用于安全控制的不确定性集合。逆向感知契约通过将感知估计映射至以高置信度包含真实值的集合，提供了这种连接。然而，现有的IPC方法将不确定性实例化为单一的椭球集合，并依赖确定性信任分数来引导机器人运动。这种表示方法无法捕捉细粒度感知误差的多模态与非规则结构，通常导致集合过于保守并降低导航性能。本研究提出了基于高斯混合的逆向感知契约，该方法将IPC扩展为使用源自高斯混合模型的椭球置信集合的并集来表示不确定性。这一设计超越了确定性的单集合抽象，能够以形式化保证捕获细粒度、多模态及非凸的误差结构。本文提出了一种学习框架，用于训练GM-IPC以考虑概率包含性、分布匹配和空域惩罚，从而确保预测集合的有效性和紧致性。我们进一步证明，所得的不确定性表征可被整合至下游规划框架中，实现实时安全导航，在保持概率安全性的同时，实现更少保守且更具适应性的机器人运动。