  State estimation for legged robots is challenging due to their highly dynamic
motion and limitations imposed by sensor accuracy. By integrating Kalman
filtering, optimization, and learning-based modalities, we propose a hybrid
solution that combines proprioception and exteroceptive information for
estimating the state of the robot's trunk. Leveraging joint encoder and IMU
measurements, our Kalman filter is enhanced through a single-rigid body model
that incorporates ground reaction force control outputs from convex Model
Predictive Control optimization. The estimation is further refined through
Gated Recurrent Units, which also considers semantic insights and robot height
from a Vision Transformer autoencoder applied on depth images. This framework
not only furnishes accurate robot state estimates, including uncertainty
evaluations, but can minimize the nonlinear errors that arise from sensor
measurements and model simplifications through learning. The proposed
methodology is evaluated in hardware using a quadruped robot on various
terrains, yielding a 65% improvement on the Root Mean Squared Error compared to
our VIO SLAM baseline. Code example: https://github.com/AlexS28/OptiState
