Support-Safe Variational Hybrid Filtering for Contact-Mode and Sparse-Law Recovery

Contact-rich robot dynamics are hybrid: a single observation can match several latent states and contact regimes (free, impact, stick--slip). A standard amortized filter that places no probability on a feasible contact transition will permanently lose the branch the robot actually follows. We introduce VHYDRO, a variational hybrid dynamics learner that prevents this branch loss. At each step, VHYDRO mixes the learned proposal with a feasible transition law before sampling and importance weighting. This keeps every feasible contact alternative covered. VHYDRO jointly infers a continuous latent state and a discrete contact mode, and fits a sparse port-Hamiltonian law to each recovered regime. On top of this, three guarantees connect: support coverage stabilizes filtering, the stabilized filter concentrates the discrete contact posterior on coherent regimes, and mode-pure segments admit sparse port-Hamiltonian recovery. The recovery error separates cleanly into filtering, derivative, mode-impurity, and physics-residual parts. Three empirical findings track the same mechanism. Under heavy occlusion the support-safe filter stays usable while a non-defensive proposal collapses. On ManiSkill demonstrations and on four canonical Sawyer/BridgeData task families the discrete state forms temporally coherent contact-regime segments that post-hoc segmenters and mode-free baselines miss. On hybrid systems with known equations the mode-conditioned sparse fit recovers the active physical terms; purely predictive baselines do not.