Estimating traversability in unstructured environments requires conditioning on robot embodiment, as the same terrain can be traversable for one platform and unsafe for another. Existing methods often transfer predictions across morphologies through late-stage trajectory filtering rather than encoding platform constraints in the learned representation. We propose Capability-Aware Traversability (CAT), a framework that embeds physical limits directly into the spatial feature space. CAT grounds dense supervision masks in physical trajectories through an interactive annotation pipeline and modulates semantic terrain maps with robot-specific traversability vectors through Spatially-Adaptive Denormalization (SPADE) blocks. Across human-annotated and trajectory-aligned datasets, CAT leads all ranking-based metrics, improving AUROC by 11.0% on physically executed trajectories and AUPRC by 15.8% on human traces over the strongest baseline. Ablations show that spatial conditioning and per-robot prototypes produce capability sensitivity beyond generic path prediction. Deployments on a legged quadruped and a wheeled skid-steer demonstrate embodiment-aware obstacle avoidance on embedded hardware at 4.8 Hz.

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