A novel robust and slip-aware speed estimation framework is developed and experimentally verified for mobile robot navigation by designing proprioceptive robust observers at each wheel. The observer for each corner is proved to be consistent, in the sense that it can provide an upper bound of the mean square estimation error (MSE) timely. Under proper conditions, the MSE is proved to be uniformly bounded. A covariance intersection fusion method is used to fuse the wheel-level estimates, such that the updated estimate remains consistent. The estimated slips at each wheel are then used for a robust consensus to improve the reliability of speed estimation in harsh and combined-slip scenarios. As confirmed by indoor and outdoor experiments under different surface conditions, the developed framework addresses state estimation challenges for mobile robots that experience uneven torque distribution or large slip. The novel proprioceptive observer can also be integrated with existing tightly-coupled visual-inertial navigation systems.