Wheel kinematics can support odometry for quadrupedal wheel-legged robots when LiDAR geometry provides weak motion constraints. However, slip and contact loss make rolling predictions unreliable, and contact state alone cannot determine their reliability across motion directions. We propose a tightly coupled LiDAR-inertial-kinematic odometry framework that retains analytical rolling predictions and learns per-wheel, direction-dependent process covariances from proprioceptive history. The estimator maintains moving wheel-center states, allowing the learned uncertainty to regulate the influence of rolling predictions through forward-kinematic corrections. Degeneracy-aware projection (DAP) complements this mechanism by suppressing LiDAR constraints along geometrically weak directions. The learned covariance network (CovNet) is trained through a differentiable filter using simulated velocity supervision and deployed on the real robot without fine-tuning. On the six room sequences with the highest degeneracy ratios in our indoor dataset, the method reduces mean per-sequence position root mean square error (RMSE) by 66.2% relative to FAST-LIO2 and by 31.0% relative to a fixed-covariance variant. In a severely degenerate corridor run, it achieves an endpoint position error of 0.198 m, compared with 7.531 m for FAST-LIO2. These results demonstrate robust odometry under challenging geometric and wheel-motion conditions. We plan to publicly release our code and multimodal dataset. Project website: https://weihairuo.github.io/wheel-motion-uncertainty/

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